research article robust fuzzy output feedback control for...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 197845 14 pageshttpdxdoiorg1011552013197845

Research ArticleRobust Fuzzy119867

infinOutput Feedback Control for a Class of

Nonlinear Uncertain Systems with Mixed Time Delays

Xiaona Song and Shanzhong Liu

Electronic and Information Engineering College Henan University of Science and Technology Luoyang 471023 China

Correspondence should be addressed to Xiaona Song xiaona 97163com

Received 25 June 2013 Accepted 18 August 2013

Academic Editor Baoyong Zhang

Copyright copy 2013 X Song and S Liu This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper studies the problem of delay-dependent robust119867infinoutput feedback control for a class of uncertain fuzzy neutral systems

with both discrete and distributed delays The system is described by a state-space Takagi-Sugeno fuzzy model with distributeddelays and norm-bounded parameter uncertainties The purpose is to design a fuzzy dynamic output feedback controller whichensures the robust asymptotic stability of the closed-loop fuzzy neutral system and satisfies an 119867

infinnorm bound constraint for all

admissible uncertainties In terms of linearmatrix inequalities sufficient conditions for the solvability of this problem are presentedFinally a numerical example is included to demonstrate the effectiveness of the proposed method

1 Introduction

Fuzzy control as a promising way to approach nonlinearcontrol problems has had an impact on the control com-munity [1ndash4] Furthermore the Takagi-Sugeno (T-S) fuzzydynamic model [5ndash9] is nonlinear system described by fuzzyIF-THEN rules which give local linear representations ofthe underlying systems [10 11] It has been shown that suchmodels can describe a wide class of nonlinear systems Henceit is important to investigate their stability analysis andcontroller design problems and in the past two decadesmanystability and control issues related to the T-S fuzzy systemshave been studied see for example [12ndash15] and the referencescited therein

On the other hand time delays exist commonly indynamic systems due to measurement transmission trans-port lags and so forth [16] which have been generallyregarded as a main source of instability and poor perfor-manceThus the analysis of time delay systems and controllerdesign for them is very important [17ndash20] Recently T-Sfuzzy systems with time delays have attracted a great dealof interests For example in [21] the stability analysis andstabilization problems for T-S fuzzy delay systems wereconsidered and state feedback fuzzy controllers and fuzzyobservers were designed The robust 119867

infincontrol problem

for T-S fuzzy systems with time delays was investigated in[22 23] and state feedback fuzzy controllers were designedthe corresponding results for the discrete case can be foundin [24 25] while in [26 27] the robust119867

infinoutput feedback

controllers were designed for the continuous and discretefuzzy time-delay systems respectively

Quite recently T-S fuzzy time-delay systems of neutraltypewere introduced in [28] where both the stabilization and119867infincontrol problemswere studied It should be point out that

distributed delays were not taken into accountWhile in [29]authors considered the problems of robust stabilization androbust 119867

infincontrol for uncertain T-S fuzzy neutral systems

with both discrete and distributed time delays However theresults in [28 29] were all delay-independent It is knownthat delay-dependent results are less conservative than delay-independent ones especially in the case when the size of thedelay is small On the other hand these obtained resultshowever are mainly dealt with through a state feedbackcontroller design method that requires all state variables tobe available In many cases this condition is too restrictiveSo it is meaningful to investigate the output feedback controlmethod To the best of our knowledge so far there areno results of delay-dependent robust 119867

infinoutput feedback

control for uncertain fuzzy neutral systemswith both discreteand distributed delays This motives the present studies

2 Mathematical Problems in Engineering

In this paper we consider the delay-dependent robust119867infin

output feedback control problem for fuzzy neutral systemswith both discrete and distributed delays The system to beconsidered is described by a state-spaceT-S fuzzymodel withmixed delays and norm-bounded parameter uncertaintiesThe distributed delays are assumed to appear in the stateequation and the uncertainties are allowed to be time varyingbut norm bounded The aim of this paper is to design a full-order fuzzy dynamic output feedback controller such that theresulting closed-loop system is robustly asymptotically stablewhile satisfying an 119867

infinnorm condition with a prescribed

level irrespective of the parameter uncertainties A sufficientcondition for the solvability of this problem is proposedin terms of linear matrix inequalities (LMIs) When theseLMIs are feasible an explicit expression of a desired outputfeedback controller is also given

Notation Throughout this paper for real symmetric matrices119883 and 119884 the notation 119883 ge 119884 (resp 119883 gt 119884) means that thematrix119883minus119884 is positive semidefinite (resp positive definite)119868 is an identity matrix with appropriate dimension N is theset of natural numbersL

2[0infin) refers to the space of square

summable infinite vector sequences sdot 2stands for the usual

L2[0infin) normThenotation119872119879 represents the transpose of

the matrix119872 Matrices if not explicitly stated are assumedto have compatible dimensions ldquolowastrdquo is used as an ellipsis forterms induced by symmetry

2 Problem Formulation

A continuous T-S fuzzy neutral model with distributeddelays and parameter uncertainties can be described by thefollowing

Plant Rule 119894 if 1199041(119905) is 120583

1198941and sdot sdot sdot and 119904

119901(119905) is 120583

119894119901 then

(119905) = [119860119894+ Δ119860119894(119905)] 119909 (119905) + [119860

1119894+ Δ1198601119894(119905)] 119909 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [119861119894+ Δ119861119894(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) = [119862119894+ Δ119862119894(119905)] 119909 (119905) + 119862

1119894119909 (119905 minus 120591

1) + 1198632119894119908 (119905)

119911 (119905) = 119864119894119909 (119905) + 119864

1119894119909 (119905 minus 120591

1) + 119866119894119906 (119905)

119909 (119905) = 120601 (119905) forall119905 isin [minus120591 0] 119894 = 1 2 119903

(1)

where 120583119894119895is the fuzzy set 119903 is the number of IF-THEN rules

and 1199041(119905) 119904

119901(119905) are the premise variablesThroughout this

paper it is assumed that the premise variables do not dependon control variables 119909(119905) isin R119899 is the state 119906(119905) isin R119898

is the control input 119910(119905) isin R119904 is the measured output119911(119905) isin R119902 is the controlled output 119908(119905) isin R119901 is the noisesignal 120591

119894gt 0 (119894 = 1 2 3) are integers representing the

time delay of the fuzzy systems 120591 = max1205911 1205912 1205913 119860119894 1198601119894

1198602119894 1198603119894 119861119894 119862119894 1198621119894 1198631119894 1198632119894 119864119894 1198642119894 and 119866

119894are known real

constant matrices Δ119860119894(119905) Δ119860

1119894(119905) Δ119860

2119894(119905) Δ119860

3119894(119905) Δ119861

119894(119905)

and Δ119862119894(119905) are real-valued unknown matrices representing

time-varying parameter uncertainties and are assumed to beof the form[Δ119860119894(119905) Δ119860

1119894(119905) Δ119860

2119894(119905) Δ119860

3119894(119905) Δ119861

119894(119905) Δ119862

119894(119905)]

= 119872119894119865119894(119905) [119873

011989411987311198941198732119894119873311989411987341198941198735119894] 119894 = 1 2 119903

(2)

where 119872119894 1198730119894 1198731119894 1198732119894 1198733119894 1198734119894 and 119873


constant matrices and 119865119894(sdot) N rarr R1198971times1198972 are unknown time-

varying matrix function satisfying

119865119894(119905)119879

119865119894(119905) le 119868 forall119905 (3)

The parameter uncertainties Δ119860119894(119905) Δ119860

1119894(119905) Δ119860

2119894(119905)

Δ1198603119894(119905) Δ119861

119894(119905) and Δ119862

119894(119905) are said to be admissible if both

(2) and (3) holdThen the final output of the fuzzy neutral system is

inferred as follows

(119905) =

119903

sum

119894=1

ℎ119894(119904 (119905)) [119860

119894+ Δ119860119894(119905)] 119909 (119905)

+ [1198601119894+ Δ1198601119894(119905)] 119909 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905) int

119905

119905minus1205913

119909 (119904) 119889119904]

+ [119861119894+ Δ119861119894(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) =

119903

sum

119894=1

ℎ119894(119904 (119905)) [119862

119894+ Δ119862119894(119905)] 119909 (119905)

+1198621119894119909 (119905 minus 120591

1) + 1198632119894119908 (119905)

119911 (119905) =

119903

sum

119894=1

ℎ119894(119904 (119905)) [119864

119894119909 (119905) + 119864

1119894119909 (119905 minus 120591

1) + 119866119894119906 (119905)]

(4)

where

ℎ119894(119904 (119905)) =

120603119894(119904 (119905))

sum119903

119894=1120603119894(119904 (119905))

120603119894(119904 (119905)) =

119901

prod

119895=1

120583119894119895(119904119895(119905))

119904 (119905) = [1199041(119905) 1199042(119905) sdot sdot sdot 119904

119901(119905)]

(5)

in which 120583119894119895(119904119895(119905)) is the grade of membership of 119904

119895(119905) in 120583

119894119895

Then it can be seen that

120603119894(119904 (119905)) ge 0 119894 = 1 119903

119903

sum

119894=1

120603119894(119904 (119905)) gt 0

(6)

Mathematical Problems in Engineering 3

for all 119905 Therefore for all 119905

ℎ119894(119904 (119905)) ge 0 119894 = 1 119903

119903

sum

119894=1

ℎ119894(119904 (119905)) = 1 forall119905

(7)

Now by the parallel distributed compensation (PDC) the fol-lowing full-order fuzzy dynamic output feedback controllerfor the fuzzy neutral system in (4) is considered

Control Rule 119894 if 1199041(119905) is 120583


119901(119905) is 120583

119894119901 then

(119905) = 119860

119896119894119909 (119905) + 119861

119896119894119910 (119905) (8)

119906 (119905) = 119862119896119894119909 (119905) 119894 = 1 2 119903 (9)

119909 (119905) = 0 119905 le 0 (10)

where 119909(119905) isin R119899 is the controller state and 119860119896119894 119861119896119894 and 119862

119896119894

are matrices to be determined later Then the overall fuzzyoutput feedback controller is given by

(119905) =

119903

sum

119894=1

ℎ119894(119904 (119905)) [119860

119896119894119909 (119905) + 119861

119896119894119910 (119905)] (11)

119906 (119905) =

119903

sum

119894=1

ℎ119894(119904 (119905)) 119862

119896119894119909 (119905) 119894 = 1 2 119903 (12)

From (4) and (11) one can obtain the closed-loop system as

119890 (119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times [119860119894119895+ Δ119860119894119895(119905)] 119890 (119905)

+ [1198601119894119895+ Δ1198601119894(119905)] 119890 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] 119890 (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)]

timesint

119905

119905minus1205913

119890 (119904) 119889119904 + 119863119894119895119908 (119905)

(13)

119911 (119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) [119864

119894119895119890 (119905) + 119864

1119894119890 (119905 minus 120591

1)]

(14)

where

119890 (119905) = [

(119905)

(119905)

]

119860119894119895= [

119860119894

119861119894119862119896119895

119861119896119895119862119894119860119896119895

]

Δ119860119894119895(119905) = [

Δ119860119894(119905) Δ119861

119894(119905) 119862119896119895

119861119896119895Δ119862119894(119905) 0

]

1198601119894119895= [

1198601119894

0

11986111989611989511986211198940] Δ119860

1119894(119905) = [

Δ1198601119894(119905) 0

0 0]

1198602119894= [

11986021198940

0 0] Δ119860

2119894(119905) = [

Δ1198602119894(119905) 0

0 0]

1198603119894= [

11986031198940

0 0] Δ119860

3119894(119905) = [

Δ1198603119894(119905) 0

0 0]

119863119894119895= [

1198631119894

1198611198961198951198632119894

] 119864119894119895= [119864119894119866119894119862119896119895]

1198641119894= [11986411198940]

(15)

Then the problem of robust fuzzy119867infincontrol problem to be

addressed in the previous is formulated as follows given anuncertain distributed delay fuzzy neutral system in (4) anda scalar 120574 gt 0 determine a dynamic output feedback fuzzycontroller in the form of (8) and (9) such that the closed-loopsystem in (13) and (14) is robustly asymptotically stable when119908(119905) = 0 and

1199112lt 120574119908

2 (16)

under zero-initial conditions for any nonzero 119908(119905) isin

L2[0infin) and all admissible uncertainties

3 Main Results

In this section an LMI approach will be developed tosolve the problem of robust output feedback 119867

infincontrol of

uncertain distributed delay fuzzy neutral systems formulatedin the previous section We first give the following resultswhich will be used in the proof of our main results

Lemma 1 (see [30]) LetADSW and119865 be realmatrices ofappropriate dimensions withW gt 0 and 119865 satisfying 119865119879119865 le 119868Then one has the following

(1) For any scalar 120598 gt 0 and vectors 119909 119910 isin R119899

2119909119879

DFS119910 le 120598minus1

119909119879

DD119879

119909 + 120598119910119879

S119879

S119910 (17)

(2) For any scalar 120598 gt 0 such thatW minus 120598DD119879 gt 0

(A +DFS)119879

Wminus1

(A +DFS)

le A119879

(W minus 120598DDT)

minus1

A + 120598minus1

S119879

S

(18)


Lemma 2 (see [31]) Given any matrices X Y and Z withappropriate dimensions such thatY gt 0 then one has

X119879

Z +Z119879

X le X119879

YX +Z119879

Yminus1

Z (19)

Theorem 3 The uncertain fuzzy neutral delay system in (13)and (14) is asymptotically stable and (16) is satisfied if thereexist matrices 119875 gt 0 119875

1gt 0 119875

2gt 0 119876

1gt 0 119876

2gt 0 119877

1 and

1198772and scalars 120598

119894119895gt 0 1 le 119894 le 119895 le 119903 such that the following

LMIs hold

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Π1119894119894

Π2119894119894

11987511986021198941198751198603119894119875119863119894119894

12059111198771

119864119879

119894119894Γ119894119894

119875119872119894119894

119873119879

119894119894

lowast 1198691

0 0 0 12059111198772119864119879

1119894Γ1119894119894

0 0

lowast lowast minus1198752

0 0 0 0 Γ2119894

0 0

lowast lowast lowast minus1198762

0 0 0 Γ3119894

0 0

lowast lowast lowast lowast minus1205742

119868 0 0 Γ4119894119894

0 0

lowast lowast lowast lowast lowast minus12059111198761

0 0 0 0

lowast lowast lowast lowast lowast lowast minus119868 0 0 0

lowast lowast lowast lowast lowast lowast lowast minus119869119879

2119869119879

2119872119894119894

0

lowast lowast lowast lowast lowast lowast lowast lowast minus120598119894119894119868 0

lowast lowast lowast lowast lowast lowast lowast lowast lowast minus120598minus1

119894119894119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 le 119903

[

[

Λ1119894119895

Λ2119894119895

Λ3119894119895

lowast Λ4119894119895

Λ5119894119895

lowast lowast Λ6119894119895

]

]

lt 0 1 le 119894 lt 119895 le 119903

(20)

whereΛ1119894119895

=

[

[

[

[

Π1119894119895+ Π1119895119894

Π2119894119895+ Π2119895119894

1198751198602119894+ 11987511986021198951198751198603119894+ 1198751198603119895

lowast 21198691

0 0


0


]

]

]

]

Λ2119894119895=

[

[

[

[

[

119875119863119894119895+ 119875119863119895119894212059111198771119864119879

119894119895+ 119864119879

119895119894Γ119894119895+ Γ119895119894

0 212059111198772119864119879

1119894+ 119864119879

1119895Γ1119894119895+ Γ1119895119894

0 0 0 Γ2119894+ Γ2119895

0 0 0 Γ3119894+ Γ3119895

]

]

]

]

]

Λ3119894119895=

[

[

[

[

[

[

[

119875119872119894119895119875119872119895119894119873119879

119894119895119873119879

119895119894

0 0 119879

1119894119879

1119895

0 0 119879

2119894119879

2119895

0 0 119879

3119894119879

3119895

]

]

]

]

]

]

]

Λ4119894119895=

[

[

[

[

minus21205742

119868 0 0 Γ4119894119895+ Γ4119895119894

lowast minus212059111198761

0 0

lowast lowast minus2119868 0


2

]

]

]

]

Λ5119894119895=

[

[

[

[

0 0 0 0

0 0 0 0

0 0 0 0

119869119879

2119872119894119895119869119879

21198721198951198940 0

]

]

]

]

Λ6119894119895=

[

[

[

[

[

minus120598119894119895119868 0 0 0

lowast minus120598119895119894119868 0 0

lowast lowast minus120598minus1

119894119895119868 0

lowast lowast lowast minus120598minus1

119895119894119868

]

]

]

]

]

Υ1= 119867119879

(1198751+ 1205912

31198762minus 1198771minus 119877119879

1)119867

Π1119894119895= Υ1+ 119875119860119894119895+ 119860119879

119894119895119875

Υ2= 119867119879

(1198771minus 119877119879

2) Π

2119894119895= Υ2+ 1198751198601119894119895

Γ119894119895= 119860119879

119894119895(1198752+ 12059111198761)

Γ1119894119895= 119860119879

1119894119895(1198752+ 12059111198761) Γ

2119894= 119860119879

2119894(1198752+ 12059111198761)

Γ3119894= 119860119879

3119894(1198752+ 12059111198761) Γ

4119894119895= 119863119879

119894119895(1198752+ 12059111198761)

1198691= minus1198751+ 1198772+ 119877119879

2 119869

2= 1198752+ 12059111198761

119872119894119895= [

119872119894

0

0 119861119896119895119872119894

] 119865119894(119905) = [

119865119894(119905) 0

0 119865119894(119905)]

119873119894119895= [

11987301198941198733119894119862119896119895

1198734119894

0

] 1119894= [

11987311198940

0 0]

2119894= [

11987321198940

0 0]

3119894= [

11987331198940

0 0]

(21)

Proof To establish the robust stability of the system in (13)we consider (13) with 119908(119905) equiv 0 that is

119890 (119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times [119860119894119895+ Δ119860119894119895(119905)] 119890 (119905)

+ [1198601119894119895+ Δ1198601119894(119905)] 119890 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] 119890 (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119890 (119904) 119889119904

(22)


For this system we define the following Lyapunov functioncandidate

119881 (119905) = 1198811(119905) + 119881

2(119905) + 119881

3(119905) + 119881

4(119905) + 119881

5(119905) + 119881

6(119905)

(23)

where

1198811(119905) = 119890(119905)

119879

119875119890 (119905)

1198812(119905) = int

119905

119905minus1205911

119890(119904)119879

1198751119890 (119904) 119889119904

1198813(119905) = int

119905

119905minus1205912

119890(119904)119879

1198752119890 (119904) 119889119904

1198814(119905) = int

0

minus1205911

int

119905

119905+120573

119890(120572)119879

1198761119890 (120572) 119889120572 119889120573

1198815(119905) = int

119905

119905minus1205913

[int

119905

119904

119890(120579)119879

119889120579]1198762[int

119905

119904

119890 (120579) 119889120579] 119889119904

1198816(119905) = int

1205913

0

119889119904int

119905

119905minus120573

(120579 minus 119905 + 120573) 119890(120572)119879

1198762119890 (120572) 119889120572 119889120573

(24)

The time derivative of 119881(119905) is given by

(119905) = 1(119905) +

2(119905) +

3(119905) +

4(119905) +

5(119905) +

6(119905)

(25)

where

1(119905) = 2119890(119905)

119879

119875 119890 (119905)

2(119905) = 119890(119905)

119879

1198751119890 (119905) minus 119890(119905 minus 120591

1)119879

1198751119890 (119905 minus 120591

1)

3(119905) = 119890(119905)

119879

1198752119890 (119905) minus 119890(119905 minus 120591

2)119879

1198752119890 (119905 minus 120591

2)

4(119905) = 120591

1119890(119905)119879

1198761119890 (119905) minus int

119905

119905minus1205911

119890(120572)119879

1198761119890 (120572) 119889120572

5(119905) = 2int

119905

119905minus1205913

(120579 minus 119905 + 1205913) 119890(119905)119879

1198762119890 (120579) 119889120579

minus [int

119905

119905minus1205913

119890(120579)119879

119889120579]1198762[int

119905

119905minus1205913

119890 (120579) 119889120579]

6(119905) =

1

2

1205912

3119890(119905)119879

1198762119890 (119905) minus int

119905

119905minus1205913

(120579 minus 119905 + 1205913) 119890(120579)119879

1198762119890 (120579) 119889120579

(26)

Now by Lemma 1 it can be shown that

2119890(119905)119879

1198762119890 (120579) le 119890(119905)

119879

1198762119890 (119905) + 119890(120579)

119879

1198762119890 (120579) (27)

Therefore

5(119905) le

1

2

1205912

3119890(119905)119879

1198762119890 (119905)

+ int

119905

119905minus1205913

(120579 minus 119905 + 1205913) 119890(120579)119879

1198762119890 (120579) 119889120579

minus [int

119905

119905minus1205913

119890(120579)119879

119889120579]1198762[int

119905

119905minus1205913

119890 (120579) 119889120579]

(28)

This together with (26) implies

5(119905) +

6(119905) le 120591

2

3119890(119905)119879

1198762119890 (119905) minus 120572(119905)

119879

1198762120572 (119905) (29)

where

120572 (119905) = int

119905

119905minus1205913

119890 (120579) 119889120579 (30)

Then there holds

(119905) le 2119890(119905)119879

119875 119890 (119905)

+ 119890(119905)119879

1198751119890 (119905) minus 119890(119905 minus 120591

1)119879

1198751119890 (119905 minus 120591

1)

+ 119890(119905)119879

1198752119890 (119905) minus 119890(119905 minus 120591

2)119879

1198752119890 (119905 minus 120591

2)

+ 1205911119890(119905)119879

1198761119890 (119905) minus int

119905

119905minus1205911

119890(120572)119879

1198761119890 (120572) 119889120572

+ 1205912

3119890(119905)119879

1198762119890 (119905) minus 120572(119905)

119879

1198762120572 (119905)

+ 2119890(119905)119879

1198771int

119905

119905minus1205911

119890 (120572) 119889120572

minus 2119890(119905)119879

1198771[119890 (119905) minus 119890 (119905 minus 120591

1)]

+ 2119890(119905 minus 1205911)119879

1198772int

119905

119905minus1205911

119890 (120572) 119889120572

minus 2119890(119905 minus 1205911)119879

1198772[119890 (119905) minus 119890 (119905 minus 120591

1)]

(31)

It follows from (22) and Lemma 2 that

2119890(119905)119879

119875 119890 (119905)

= 2119890(119905)119879

119875

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times [119860119894119895+ Δ119860119894119895(119905)] 119890 (119905)

+ [1198601119894119895+ Δ1198601119894(119905)] 119890 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] 119890 (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119890 (119904) 119889119904

le 2

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 119890(119905)

119879

119875119860119894119895120573 (119905)

+

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times [120576minus1

1119894119895119890(119905)119879

119875119872119894119895119872119879

119894119895119875119890 (119905)

+ 1205761119894119895120573(119905)119879

119879

119894119895119894119895120573 (119905)]

(32)

where

119860119894119895= [1198601198941198951198601119894119895

11986021198941198603119894]

120573 (119905) = [119890(119905)119879

119890(119905 minus 1205911)119879

119890(119905 minus 1205912)119879

120572(119905)119879

119890(120572)119879]

119894119895= [119873119894119895111989421198943119894]

(33)


It also can be verified that

119890(119905)119879

1198752119890 (119905)

=

1

2

119903

sum

119894=1

119903

sum

119895=1

119903

sum

119906=1

119903

sum

V=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) ℎ

119906(119904 (119905)) ℎV (119904 (119905)) 120573(119905)

119879

times [119860119894119895+119872119894119895119865119894(119905)119894119895]

119879

times 1198752[119860119906V +119872119906V119865119906 (119905) 119906V]

+ [119860119906V +119872119906V119865119906 (119905) 119906V]

119879

times 1198752[119860119894119895+119872119894119895119865119894(119905) 119894119895] 120573 (119905)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times [119860119894119895+119872119894119895119865119894(119905) 119894119895]

119879

1198752[119860119894119895+119872119894119895119865119894(119905) 119894119895]

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times [119860

119879

119894119895(119875minus1

2minus 120576minus1

2119894119895119872119894119895119872119879

119894119895)

minus1

119860119894119895+ 1205762119894119895119879

119894119895119894119895] 120573 (119905)

1205911119890(119905)119879

1198761119890 (119905)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times 119860

119879

119894119895[(12059111198761)minus1

minus 120576minus1

3119894119895119872119894119895119872119879

119894119895]

minus1

119860119894119895

+ 1205763119894119895119879

119894119895119894119895120573 (119905)

(34)

Hence with the support of the above conditions we have

(119905) le 2119890(119905)119879

119875 119890 (119905) + 119890(119905)119879

1198752119890 (119905)

+ 1205911119890(119905)119879

1198761119890 (119905) + 119890(119905)

119879

(1198751+ 1205912

31198762) 119890 (119905)

minus 119890(119905 minus 1205911)119879

1198751119890 (119905 minus 120591

1)

minus 119890(119905 minus 1205912)119879

1198752119890 (119905 minus 120591

2) minus int

119905

119905minus1205911

119890(120572)119879

1198761119890 (120572) 119889120572

minus 120572(119905)119879

1198762120572 (119905) + 2119890(119905)

119879

1198771int

119905

119905minus1205911

119890 (120572) 119889120572

minus 2119890(119905)119879

1198771[119890 (119905) minus 119890 (119905 minus 120591

1)]

+ 2119890(119905 minus 1205911)119879

1198772int

119905

119905minus1205911

119890 (120572) 119889120572 minus 2119890(119905 minus 1205911)119879

times 1198772[119890 (119905) minus 119890 (119905 minus 120591

1)]

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times [Ξ119894119895+ 119860119894119895(119875minus1

2minus 120576minus1

2119894119895119872119894119895119872119879

119894119895)

minus1

119860119894119895

+119860119894119895(120591minus1

1119876minus1

1minus 120576minus1

3119894119895119872119894119895119872119879

119894119895)

minus1

119860119894119895+ 120575119894119895119879

119894119895119894119895] 120573 (119905)

(35)

where

Ξ119894119895=

[

[

[

[

[

[

Σ1198941198951198751198601119894119895+ 1198771minus 119877119879

211987511986021198941198751198603119894

12059111198771

lowast minus1198751+ 1198772+ 119877119879

20 0 120591

11198772


0 0


0


]

]

]

]

]

]

Σ119894119895= 119875119860119894119895+ 119860119879

119894119895119875 + 1198751+ 1205912

31198762minus 1198771minus 119877119879

1+ 120576minus1

1119894119895119875119872119894119895119872119879

119894119895119875

(36)

for 1 le 119894 le 119895 le 119903 On the other hand by applying the Schurcomplement formula to (20) we have that for 1 le 119894 lt 119895 le 119903

Γ119894119895lt 0 (37)

and from this and (7) we have that (119905) lt 0 for all 120573(119905) = 0Therefore the system in (13) is robustly asymptotically stable

This completes the proof

Next we will establish the robust119867infinperformance of the

system in (13) and (14) under the zero initial condition Tothis end we introduce

119869 (119905) = int

119905

0

[119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904)] 119889119904 (38)

where 119905 gt 0 Noting the zero initial condition it can be shownthat for any nonzero 119908(119905) isinL

2[0infin) and 119905 gt 0

119869 (119905) = int

119905

0

[119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904) + (119904)] 119889119904 minus 119881 (119905)

le int

119905

0

[119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904) + (119904)] 119889119904

(39)

where 119881(119904) is defined in (23) and then we have

(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times 2119890(119905)119879

119875 [119860119894119895(119905) 119890 (119905) + 119860

1119894119895(119905) 119890 (119905 minus 120591

1)

+ 1198602119894(119905) 119890 (119905 minus 120591

2)

+1198603119894(119905) 120572 (119905) + 119863

119894119895119908 (119905)]


+ 119890(119905)119879

1198751119890 (119905) minus 119890(119905 minus 120591

1)119879

1198751119890 (119905 minus 120591

1)

+ 119890(119905)119879

1198752119890 (119905) minus 119890(119905 minus 120591

2)119879

1198752119890 (119905 minus 120591

2)

+ 1205911119890(119905)119879

1198761119890 (119905)

minus int

119905

119905minus1205911

119890(119905)119879

1198761119890 (120572) 119889120572 + 120591

2

3119890(119905)119879

1198762119890 (119905)

minus 120572(119905)119879

1198762120572 (119905)

+ 2119890(119905)119879

1198771int

119905

119905minus1205911

119890 (120572) 119889120572 minus 2119890(119905)119879


1)]

+ 2119890(119905 minus 1205911)119879

1198772int

119905

119905minus1205911

119890 (120572) 119889120572 minus 2119890(119905 minus 1205911)119879


1)]

(40)

Then by noting (14) and using Lemma 2 we have

119911(119904)119879

119911 (119904) =

119903

sum

119894=1

119903

sum

119895=1

119903

sum

119906=1

119903

sum

V=1

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) ℎ

119906(119904 (119904))

times ℎV (119904 (119904)) 120577(119904)119879

119864119879

119894119895119864119906V120577 (119904)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) 120577(119904)

119879

119864119879

119894119895119864119894119895120577 (119904)

(41)

It can be deduced that

119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904) + (119904)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) 120577(119904)

119879

Π1119894119895120577 (119904)

=

119903

sum

119894=1

ℎ119894(119904 (119904))

2

120577(119904)119879

Ξ1119894119894120577 (119904)

+ 2

119903

sum

119894119895=1119894lt119895

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) 120577(119904)

119879Ξ1119894119895+ Ξ1119895119894

2

120577 (119904)

(42)

where

Ξ1119894119895=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Σ1198941198951198751198601119894119895(119905) + 119877

1minus 119877119879

21198751198602119894(119905) 119875119860

3119894(119905) 119875119863

11989411989512059111198771

119864119879

119894119895119860119879

119894119895(119905) (1198752+ 12059111198761)

lowast 1198691

0 0 0 12059111198772119864119879

1119894119860119879

1119894119895(119905) (1198752+ 12059111198761)


0 0 0 0 119860119879

2119894(119905) (1198752+ 12059111198761)


0 0 0 119860119879

3119894(119905) (1198752+ 12059111198761)


119868 0 0 119863119879

119894119895(1198752+ 12059111198761)


0 0

lowast lowast lowast lowast lowast lowast minus119868 0

lowast lowast lowast lowast lowast lowast lowast minus(1198752+ 12059111198761)119879

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

120577 (119904) = [119890(119905)119879

119890(119905 minus 1205911)119879

119890(119905 minus 1205912)119879

120572(119905)119879

119908(119905)119879

119890(120572)119879]

119879

119864119894119895= [11986411989411989511986411198940 0 0 0]

(43)

for 1 le 119894 lt 119895 le 119903 Similar to the previous section applyingthe Schur complement formula to the LMI in (20) results inΞ1119894119895lt 0 1 le 119894 lt 119895 le 119903 which together with (39) gives 119869(119905) lt 0

for any nonzero 119908(119905) isin L2[0infin) and 119905 gt 0 Therefore we

have 1199112lt 120574119908

2

Now we are in a position to present a solution to therobust output feedback119867


Theorem4 Consider the uncertain fuzzy neutral delay systemin (1) and let 120574 gt 0 be a prescribed constant scalar The robust119867infin

problem is solvable if there exist matrices 119883 gt 0 119884 gt 01198761 1198762 Φ119894 Ψ119894 and Ω


119894119895 1 le 119894 le 119895 le 119903 such that

the following LMIs hold

[

[

[

[

[

[

Σ119894119894Σ1119894119894

Σ2119894119894

Σ3119894119894

1198775

lowast minus119877111198772119894119894

0 0

lowast lowast 1198773119894119894

0 1198776119894119894


0


]

]

]

]

]

]

lt 0 1 le 119894 le 119903 (44)


[

[

[

[

[

[

[

[

[

[

Σ119894119895+ Σ119895119894Σ1119894119895+ Σ1119895119894

Σ2119894119895

Σ4119894119895

Σ5119894119895

Σ6119894119895

0


2119894119895

0 0 0 0

lowast lowast 3119894119895

0 0 0 6119894119895

lowast lowast lowast minus119868 0 0 0


0 0

lowast lowast lowast lowast lowast minus119868 0

lowast lowast lowast lowast lowast lowast minus119868

]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 lt 119895 le 119903 (45)

[

minus119884 minus119868

minus119868 minus119883] lt 0 (46)

where

Θ119894119895

= [

119860119894119883 + 119883119860

119879

119894+ 119861119894Ψ119895+ Ψ119879

119895119861119879

119894119860119894+ Ω119879

119894

119860119879

119894+ Ω119894

119884119860119894+ 119860119879

119894119884 + Φ

119895119862119894+ 119862119879

119894Φ119879

119895

]

minus1198771minus 119877119879

1

Θ1119894119895= [

1198601119894119883 119860

1119894

0 1198841198601119894+ Φ1198951198621119894

] + 1198771minus 119877119879

2

Θ2119894= [

1198602119894

0

11988411986021198940] Θ

3119894= [

1198603119894

0

11988411986031198940]

Θ4119894119895= [

1198631119894

1198841198631119894+ Φ1198951198632119894

] Θ5119894119895= [

119883119864119879

119894+ Ψ119879

119895119866119879

119894

119864119879

119894

]

Θ6119894119895= [

119883119860119879

119894+ Ψ119879

119895119861119879

119894Ω119879

119895

119860119879

119894119860119879

119894119884 + 119862

119879

119894Φ119879

119895

]

Θ7119894119895= [

119872119894

0

119884119872119894Φ119895119872119894

]

Θ8119894119895= [

119883119873119879

0119894+ Ψ119879

119895119873119879

4119894119883119873119879

5119894

119873119879

0119894119873119879

5119894

] 1198693119894= [

119883119864119879

1119894

119864119879

1119894

]

1198694119894119895= [

1198831198601119894

0

1198601119894

1198601119894119884 + 119862

119879

1119894Φ119879

119895

]

1198695119894= [

119883119873119879

11198940

119873119879

11198940

] 1198696119894= [

119860119879

2119894119860119879

2119894119884

0 0

]

1198697119894= [

119860119879

3119894119860119879

3119894119884

0 0

]

1198698119894119895= [119863119879

1119894119863119879

1119894119884 + 119863

119879

2119894Φ119879

119895]

1198699= Π119879

2+ Π2minus 1198752minus 12059111198761

11986910119894119895

= [

119872119894

0

119884119872119894Φ119895119872119894

]

Σ119894119895=[

[

[

Θ119894119895

Θ1119894119895

Θ2119894

lowast minus119880 + 1198772+ 119877119879

20


]

]

]

Σ1119894119895=[

[

[

Θ3119894Θ4119894119895

12059111198771

0 0 12059111198772

0 0 0

]

]

]

Σ2119894119895=[

[

[

Θ5119894119895

Θ6119894119895

Θ7119894119895

Θ8119894119895

1198693119894

1198694119894119895

0 1198695119894

0 1198696119894

0 119879

2119894

]

]

]

Σ2119894119895=

[

[

[

[

Θ5119894119895+ Θ5119895119894

Θ6119894119895+ Θ6119895119894

Θ7119894119895

Θ7119895119894

Θ8119894119895

Θ8119895119894

1198693119894+ 1198693119895

1198694119894119895+ 1198694119895119894

0 0 1198695119894

1198695119895

0 1198696119894+ 1198696119895

0 0 119879

2119894119879

2119895

]

]

]

]

Σ3119894119895=[

[

[

Π119879

11205913Π119879

1119879119894119895

0 0 0

0 0 0

]

]

]

119879119894119895= [

0

1198841198601119894+ Φ1198951198621119894

]

1198775=[

[

0 0 0

119879111987910

0 0 0

]

]

1198791= [

119909

0

] Σ4119894119895=[

[

[

1198792119894119895

1198793119894119895

1198794119894119895

1198795119894119895

0 0 0 0

0 0 0 0

]

]

]

1198792119894119895= [

(119862119894119909 minus 119862

119895119909)

119879

0

]

1198793119894119895= [

0

Φ119895minus Φ119894

] 1198794119894119895= [

(Ψ119895minus Ψ119894)

119879

0

]

1198795119894119895= [

0

119884119861119894minus 119884119861119895

]

Σ5119894119895=[

[

Π119879

11205913Π119879

11198791198941198950

0 0 0 1198791

0 0 0 0

]

]


Σ6119894119895=[

[

119879119895119894

0 11987941198792

0 11987910 0

0 0 0 0

]

]

11987711= diag (119876

21205742

119868 119881) 1198772119894119895=[

[

0 11986971198940 119879

3119894

0 1198698119894119895

0 0

0 0 0 0

]

]

2119894119895=[

[

0 1198697119894+ 1198697119895

0 0 119879

3119894119879

3119895

0 1198698119894119895+ 1198698119895119894

0 0 0 0

0 0 0 0 0 0

]

]

1198773119894119895=

[

[

[

[

minus119868 0 0 0

lowast minus119869911986910119894119895

0



119894119895119868

]

]

]

]

3119894119895=

[

[

[

[

[

[

[

[

[

minus2119868 0 0 0 0 0

lowast minus2119869911986910119894119895

11986910119895119894

0 0

lowast lowast minus120598119894119895119868 0 0 0

lowast lowast lowast minus120598119895119894119868 0 0

lowast lowast lowast lowast minus120598minus1

119894119895119868 0

lowast lowast lowast lowast lowast minus120598minus1

119895119894119868

]

]

]

]

]

]

]

]

]

1198774= diag (119875minus1

1119876minus1

2119868)

1198776119894119895=

[

[

[

[

0 0 0

0 0 1198796119894119895

0 0 0

0 0 0

]

]

]

]

6119894119895=

[

[

[

[

[

[

[

[

0 0

1198795119894119895

1198793119894119895

0 0

0 0

0 0

0 0

]

]

]

]

]

]

]

]

1198796119894119895= [

0

119884119860119879

1119894+ Φ1198951198621119894

]

1198777= diag (119868 119868 119868) 119877

8= diag (1

2

119875minus1

1

1

2

119876minus1

2119868 119868)

Π1= [

119883 119868

119883 0] Π

2= [

119883 119868

119883 0]

(47)

Furthermore a desired robust dynamic output feedback con-troller is given in the form of (11) with parameters as follows

119860119870119894= (119883minus1

minus 119884)

minus1

(Ω119894minus 119884119860119894119883 minus 119884119861

119894Ψ119894minus Φ119894119862119894119883)119883minus1

119861119870119894= (119883minus1

minus 119884)

minus1

Φ119894 119862119870119894= Ψ119894119883minus1

1 le 119894 le 119903

(48)

Proof Applying the Schur complements formula to (44) and(45) and using Lemma 2 result in

[

[

Σ119894119894Σ1119894119894

Σ2119894119894

lowast minus119877111198772119894119894

lowast lowast 1198773119894119894

]

]

lt 0 1 le 119894 le 119903

[

[

[

Σ119894119895Σ1119894119895+ Σ1119895119894

Σ2119894119895


2119894119895


]

]

]

lt 0 1 le 119894 lt 119895 le 119903

(49)

where

Σ119894119895=

[

[

[

[

Θ119894119895+ Π119879

11198751Π1+ (1205913Π1)119879

1198762(1205913Π1) Θ1119894119895+ [

0 0

(1198841198601119894+ Φ1198951198621119894)119883 0

] Θ2119894

lowast minus119880 + 1198772+ 119877119879

20


]

]

]

]

Σ2119894119895=

[

[

[

[

Θ5119894119895

Θ6119894119895

Θ7119894119895

Θ8119894119895

1198693119894

1198694119894119895+ [

0 119883 (1198601119894119884 + 119862

119879

1119894Φ119879

119895)

0 0

] 0 1198695119894

0 1198696119894

0 119879

2119894

]

]

]

]

Σ119894119895= Σ119894119895+ Σ119895119894+

[

[

[

[

[

[

0 (119862119894119883 minus 119862

119895119883)

119879

(Φ119895minus Φ119894)

119879

+ (Ψ119895minus Ψ119894)

119879

(119884119861119894minus 119884119861119895)

119879

0 0 0

lowast 0 0 0 0

lowast lowast 0 0 0

lowast lowast lowast 0 0

lowast lowast lowast lowast 0

]

]

]

]

]

]

Σ2119894119895= Σ2119894119895+[

[

0 (Ψ119879

119895minus Ψ119879

119894) (119861119879

119894119884 minus 119861

119879

119895119884) + (119883119862

119879

119894minus 119883119862119879

119895) (Φ119879

119895minus Φ119879

119894) 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]

]

(50)


Note that

Π119879

2+ Π2minus 1198752minus 12059111198761le Π119879

2(1198752+ 12059111198761)minus1

Π2 (51)

Then by this inequality it follows from (49) that

[

[

Σ119894119894Σ1119894119894

Σ2119894119894

lowast minus119877111198772119894119894


]

]

lt 0 1 le 119894 le 119903

[

[

[

Σ119894119895Σ1119894119895+ Σ1119895119894

Σ2119894119895


2119894119895

lowast lowast 1198773119894119895

]

]

]

lt 0 1 le 119894 lt 119895 le 119903

(52)

where

3119894119895=

[

[

[

[

[

minus119868 0 0 0

lowast minusΠ119879

2(1198752+ 12059111198761)minus1

Π211986910119894119895

0



119894119895119868

]

]

]

]

]

1198773119894119895=

[

[

[

[

[

[

[

[

[

minus2119868 0 0 0 0 0

lowast minus2Π119879

2(1198752+ 12059111198761)minus1

Π211986910119894119895

11986910119895119894

0 0




119894119895119868 0


119895119894119868

]

]

]

]

]

]

]

]

]

(53)

Now set = Π

2Πminus1

1 (54)

Then by (46) it can be verified that gt 0 Noting theparameters in (48) and pre- and postmultiplying (3) by

diag (Πminus1198791 Πminus119879

1 119868 119868 119868 Π

minus119879

1 119868 Πminus119879

2 119868 119868) (55)

and its transpose respectively we have

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Π1119894119894

Π2119894119894

11987511986021198941198751198603119894119875119863119894119894

12059111

119864119879

119894119894Γ119894119894

119875119872119894119894

119873119879

119894119894

lowast 1198691

0 0 0 12059112119864119879

1119894Γ1119894119894

0 0


0 0 0 0 Γ2119894

0 0


0 0 0 Γ3119894

0 0


119868 0 0 Γ4119894119894

0 0


0 0 0 0



2119869119879

2119872119894119894

0



119894119894119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 le 119903

[

[

Λ1119894119895

Λ2119894119895

Λ3119894119895

lowast Λ4119894119895

Λ5119894119895


]

]

lt 0 1 le 119894 lt 119895 le 119903

(56)

where

Λ1= 119867119879

(1198751+ 1205912

31198762minus 1minus 119879

1)119867

Π1119894119895= Λ1+ 119875119860119894119895+ 119860119879

119894119895119875

Λ2= 119867119879

(1minus 119879

2)

Π2119894119895= Λ2+ 1198751198601119894119895

1= Π119879

11198771Π1

(57)

Finally byTheorem 3 the desired result follows immediately


Remark 5 Theorem 4 provides a sufficient condition for thesolvability of the robust119867

infinoutput feedback control problem

for uncertain fuzzy neutral systems with both discrete anddistributed delays It is worth pointing out that the result inTheorem 4 can be readily extended to the case with multipledelays

Remark 6 In Theorem 4 if we set 11986021= 0 119872

119894= 0 and

1198732119894= 0 (119894 = 1 2 119903) then the results in [32] are included

in our paper Also the controller design method in our papercan be the reference for designing the observer-based outputfeedback controllers and so forth

4 Simulation Example

In this section we provide one example to illustrate theoutput feedback 119867

infincontroller design approach developed

in this paperThe uncertain fuzzy system with distributed delays con-

sidered in this example is with two rulesPlant Rule 1 if 119909

1(119905) is 120583

1 then

(119905) = [1198601+ Δ1198601(119905)] 119909 (119905) + [119860

11+ Δ11986011(119905)] 119909 (119905 minus 120591

1)

+ [11986021+ Δ11986021(119905)] (119905 minus 120591

2)

+ [11986031+ Δ11986031(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198611+ Δ1198611(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) = [1198621+ Δ1198621(119905)] 119909 (119905) + 119862

11119909 (119905 minus 120591

1) + 11986321119908 (119905)

119911 (119905) = 1198641119909 (119905) + 119864

11119909 (119905 minus 120591

1) + 1198661119906 (119905)

(58)

Plant Rule 2 if 1199091(119905) is 120583

2 then

(119905) = [1198602+ Δ1198602(119905)] 119909 (119905) + [119860

12+ Δ11986012(119905)] 119909 (119905 minus 120591

1)

+ [11986022+ Δ11986022(119905)] (119905 minus 120591

2)

+ [11986032+ Δ11986032(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198612+ Δ1198612(119905)] 119906 (119905) + 119863

12119908 (119905)

119910 (119905) = [1198622+ Δ1198622(119905)] 119909 (119905) + 119862

12119909 (119905 minus 120591

1) + 11986322119908 (119905)

119911 (119905) = 1198642119909 (119905) + 119864

12119909 (119905 minus 120591

1) + 1198662119906 (119905)

(59)

where

1198601= [

005 0

minus02 minus006] 119860

11= [

005 002

minus005 001]

11986021= [

minus0002 0008

0006 0]

11986031= [

minus005 008

006 0] 119861

1= [

001 0

003 006]

1198621= [

minus003 minus01

02 002] 119862

11= [

minus003 minus001

002 002]

11986311= [

003

006] 119863

21= [

minus05

0]

1198641= [minus02 0] 119864

11= [minus002 0]

1198661= [0 minus003]

1198602= [

minus005 0


12= [

minus005 02

minus05 001]

11986022= [

0008 minus0002

minus0003 002]

11986032= [

005 minus02

minus03 02] 119861

2= [

001 minus008

002 005]

1198622= [

003 minus01

02 002]

11986212= [

minus003 minus001

002 02] 119863

12= [

minus001

002]

11986322= [

minus03

0]

1198642= [minus001 0] 119864

12= [minus001 0]

1198662= [0 002]

(60)

and Δ119860119894(119905) Δ119860

1119894(119905) Δ119860

2119894(119905) Δ119860

3119894(119905) Δ119861

119894(119905) Δ119862

119894(119905) (119894 =

1 2) can be represented in the form of (2) and (3) with

1198721= [

minus002

0] 119873

01= [minus02 01]

11987311= [minus02 02]

11987321= [minus002 001] 119873

31= [minus002 001]

11987341= [minus002 001]

11987351= [minus002 001] 119872

2= [

003

0]

11987302= 11987301 119873

12= 11987311

11987322= 11987321 119873

32= 11987331

11987342= 11987341 119873

52= 11987351

(61)


Then the final outputs of the fuzzy systems are inferred asfollows

(119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119860119894+ Δ119860119894(119905)] 119909 (119905)

+ [1198601119894+ Δ1198601119894(119905)] 119909 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [119861119894+ Δ119861119894(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119862119894+ Δ119862119894(119905)] 119909 (119905)

+1198621119894119909 (119905 minus 120591

1) + 1198632119894119908 (119905)

119911 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905)) 119864

119894119909 (119905) + 119864

1119894119909 (119905 minus 120591

1) + 119866119894119906 (119905)

(62)

where

ℎ1(1199091(119905)) =

1

3

for 1199091lt minus1

2

3

+

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

1 for 1199091gt 1

ℎ2(1199091(119905)) =

2

3


1

3

minus

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

0 for 1199091gt 1

(63)

In this example we choose the119867infin

performance level 120574 = 3In order to design a fuzzy119867

infinoutput feedback controller

for the T-S model we first choose the initial condition thatis 119909(0) = [03 minus01]

119879 and the disturbance input 119908(119905) isassumed to be

119908 (119905) =

1

119905 + 01

119905 ge 0 (64)

Then solving the LMIs in (44) (45) and (46) we obtain thesolution as follows

119883 = [

09868 06071

06071 13948]

119884 = [

129348 01814

01814 63384]

Ω1= [


minus00417 minus32117]

Ω2= [

82076 minus37031

minus58461 53292]

Φ1= [

31848 00758


Φ2= [

20547 01643


1205951= [


57867 minus778562]

1205952= [



(65)

Now by Theorem 4 a desired fuzzy output feedback con-troller can be constructed as in (11) and (12) with

1198601198961= [

873277 minus680397

1399266 minus1072394]

1198601198962= [

minus249856 158880

111983 minus122840]

1198611198961= [

109115 minus03725

165830 19936]

1198611198962= [

14642 minus00212

31940 05922]

1198621198961= [

minus160423 127577

202647 minus148918]

1198621198962= [

142596 minus104929

32304 minus11934]

(66)

With the output feedback fuzzy controller the simulationresults of the state response of the nonlinear system are givenin Figure 1 Figure 2 shows the control input while Figures3 and 4 present the corresponding measured output and thecontrolled output respectively

From these simulation results it can be seen that thedesigned fuzzy output feedback controller ensures the robustasymptotic stability of the uncertain fuzzy neutral system andguarantees a prescribed119867

infinperformance level

5 Conclusion and Future Work

The problem of delay-dependent robust output feedback119867infin

control for uncertain T-S fuzzy neutral systems with param-eter uncertainties and distributed delays has been studied In


x1(t)

x2(t)

0 10 20 30 40 50 60minus02

0

02

04

06

08

1

12

Time (s)

Stat

e res

pons

ex(t)

Figure 1 State response 1199091(119905) (continuous line) and 119909

2(119905) (dashed

line)

0 10 20 30 40 50 60minus100

minus50

0

50

100

150

200

Time (s)

Con

trol i

nput

u(t)

u1(t)

u2(t)

Figure 2 Control input 1199061(119905) (continuous line) and 119906

2(119905) (dashed

line)

terms of LMIs a sufficient condition for the existence of afull-order fuzzy dynamic output feedback controller whichrobustly stabilizes the uncertain fuzzy neutral delay systemsand guarantees a prescribed level on disturbance attenuationhas been obtained

Future work will mainly cover the problem of robustoutput feedback 119867

infincontrol for uncertain fractional-order

(FO) T-S fuzzy neutral systems with state and distributeddelays Firstly investigate the robust output feedback 119867

infin

control for FOT-S fuzzy systemswith state delays then studythe robust output feedback 119867

infincontrol for FO T-S fuzzy

systems with both state and distributed delays finally obtainthe results about the robust output feedback 119867

infincontrol

for uncertain FO T-S fuzzy neutral systems with state anddistributed delays

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China under Grant 61203047 the China

y1(t)

y2(t)

0 10 20 30 40 50 60minus1

0

1

2

3

4

5

Time (s)

Mea

sure

d ou

tput

y(t)

Figure 3 Measured output 1199101(119905) (continuous line) and 119910

2(119905)

(dashed line)

z1(t)

z2(t)

0 10 20 30 40 50 60Time (s)

minus40

minus20

0

20

40

60

80

100

120

Con

trolle

d ou

tput

z(t)

Figure 4 Controlled output 1199111(119905) (continuous line) and 119911

2(119905)

(dashed line)

Postdoctoral Science Foundation under Grant 2013T60670and the Projects of International Technology Cooperationunder Grant 2011DFA10440

References

[1] T Wang S Tong and Y Li ldquoRobust adaptive fuzzy controlfor nonlinear system with dynamic uncertainties based onbacksteppingrdquo International Journal of Innovative ComputingInformation and Control vol 5 no 9 pp 2675ndash2688 2009

[2] S Tong and Y Li ldquoObserver-based fuzzy adaptive control forstrict-feedback nonlinear systemsrdquo Fuzzy Sets and Systems vol160 no 12 pp 1749ndash1764 2009

[3] Y Li S Tong and T Li ldquoAdaptive fuzzy output feedback controlof uncertain nonlinear systems with unknown backlash-likehysteresisrdquo Information Sciences vol 198 pp 130ndash146 2012

[4] S Tong X He and H Zhang ldquoA combined backstepping andsmall-gain approach to robust adaptive fuzzy output feedbackcontrolrdquo IEEE Transactions on Fuzzy Systems vol 17 no 5 pp1059ndash1069 2009


[5] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[6] K Tanaka and M Sano ldquoOn the concepts of regulator andobserver of fuzzy control systemsrdquo inProceedings of the 3rd IEEEInternational Conference on Fuzzy Systems pp 767ndash772 1994

[7] J Yoneyama M Nishikawa H Katayama and A IchikawaldquoOutput stabilization of Takagi-Sugeno fuzzy systemsrdquo FuzzySets and Systems vol 111 no 2 pp 253ndash266 2000

[8] B Zhang J Lam S Xu and Z Shu ldquoRobust stabilizationof uncertain T-S fuzzy time-delay systems with exponentialestimatesrdquo Fuzzy Sets and Systems vol 160 no 12 pp 1720ndash17372009

[9] H Shen S Xu J Zhou and J Lu ldquoFuzzy 119867infin

filteringfor nonlinear Markovian jump neutral systemsrdquo InternationalJournal of Systems Science vol 42 no 5 pp 767ndash780 2011

[10] Y Niu J Lam XWang and DW C Ho ldquoAdaptive Hinfincontrol

using backstepping design and neural networksrdquo Journal ofDynamic Systems Measurement and Control vol 127 no 3 pp478ndash485 2004

[11] K Tanaka andM Sugeno ldquoStability analysis and design of fuzzycontrol systemsrdquo Fuzzy Sets and Systems vol 45 no 2 pp 135ndash156 1992

[12] X Ban X Gao X Huang and H Yin ldquoStability analysisof the simplest Takagi-Sugeno fuzzy control system usingpopov criterionrdquo International Journal of Innovative ComputingInformation and Control vol 3 no 5 pp 1087ndash1096 2007

[13] S Tong W Wang and L Qu ldquoDecentralized robust controlfor uncertain T-S fuzzy large-scale systems with time-delayrdquoInternational Journal of Innovative Computing Information andControl vol 3 no 3 pp 657ndash672 2007

[14] S Zhou G Feng J Lam and S Xu ldquoRobust 119867infin

controlfor discrete-time fuzzy systems via basis-dependent Lyapunovfunctionsrdquo Information Sciences vol 174 no 3-4 pp 197ndash2172005

[15] B Zhang S Xu G Zong and Y Zou ldquoDelay-dependentstabilization for stochastic fuzzy systems with time delaysrdquoFuzzy Sets and Systems vol 158 no 20 pp 2238ndash2250 2007

[16] H Gao J Lam C Wang and Y Wang ldquoDelay-dependentoutput-feedback stabilisation of discrete-time systems withtime-varying state delayrdquo IEE ProceedingsmdashControl Theory andApplications vol 151 pp 691ndash698 2004

[17] B Zhang W X Zheng and S Xu ldquoOn robust 119867infin

filtering ofuncertain Markovian jump time-delay systemsrdquo InternationalJournal of Adaptive Control and Signal Processing vol 26 no 2pp 138ndash157 2012

[18] B Zhang W X Zheng and S Xu ldquoFiltering of Markovianjump delay systems based on a new performance indexrdquo IEEETransactions on Circuits and Systems I vol 60 pp 1250ndash12632013

[19] H Shen S Xu J Lu and J Zhou ldquoPassivity-based control foruncertain stochastic jumping systems with mode-dependentround-trip time delaysrdquo Journal of the Franklin Institute vol349 no 5 pp 1665ndash1680 2012

[20] H Shen S Xu J Lu and J Zhou ldquoFuzzy dissipative controlfor nonlinear Markovian jump systems via retarded feedbackrdquoJournal of the Franklin Institute 2013

[21] Y-Y Cao and P Frank ldquoAnalysis and synthesis of nonlineartime-delay systems via fuzzy control approachrdquo IEEE Transac-tions on Fuzzy Systems vol 8 no 2 pp 200ndash211 2000

[22] K R Lee E T Jeung andH B Park ldquoRobust fuzzy119867infincontrol

for uncertain nonlinear systems via state feedback an LMIapproachrdquo Fuzzy Sets and Systems vol 120 no 1 pp 123ndash1342001

[23] B Zhang S Zhou and T Li ldquoA new approach to robustand non-fragile 119867

infincontrol for uncertain fuzzy systemsrdquo

Information Sciences vol 177 no 22 pp 5118ndash5133 2007[24] Y-Y Cao and PM Frank ldquoRobust119867

infindisturbance attenuation

for a class of uncertain discrete-time fuzzy systemsrdquo IEEETransactions on Fuzzy Systems vol 8 no 4 pp 406ndash415 2000

[25] B Zhang and S Xu ldquoDelay-dependent robust Hinfin

controlfor uncertain discrete-time fuzzy systems with time-varyingdelaysrdquo IEEE Transactions on Fuzzy Systems vol 17 no 4 pp809ndash823 2009

[26] K R Lee J H Kim E T Jeung and H B Park ldquoOutput feed-back robust H

infincontrol of uncertain fuzzy dynamic systems

with time-varying delayrdquo IEEE Transactions on Fuzzy Systemsvol 8 no 6 pp 657ndash664 2000

[27] S Xu and J Lam ldquoRobust Hinfin

control for uncertain discrete-time-delay fuzzy systems via output feedback controllersrdquo IEEETransactions on Fuzzy Systems vol 13 no 1 pp 82ndash93 2005

[28] S Xu J Lam and B Chen ldquoRobust 119867infin

control for uncertainfuzzy neutral delay systemsrdquo European Journal of Control vol10 no 4 pp 365ndash385 2004

[29] Y Li S Xu B Zhang and Y Chu ldquoRobust stabilization and119867infincontrol for uncertain fuzzy neutral systemswithmixed time

delaysrdquo Fuzzy Sets and Systems vol 159 no 20 pp 2730ndash27482008

[30] Y Wang L Xie and C E de Souza ldquoRobust control of a classof uncertain nonlinear systemsrdquo Systems amp Control Letters vol19 no 2 pp 139ndash149 1992

[31] L Xie and C E de Souza ldquoRobust 119867infin

control for linearsystems with norm-bounded time-varying uncertaintyrdquo IEEETransactions on Automatic Control vol 37 no 8 pp 1188ndash11911992

[32] X Song S Xu and H Shen ldquoRobust119867infincontrol for uncertain

fuzzy systems with distributed delays via output feedbackcontrollersrdquo Information Sciences vol 178 no 22 pp 4341ndash4356 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


In this paper we consider the delay-dependent robust119867infin

output feedback control problem for fuzzy neutral systemswith both discrete and distributed delays The system to beconsidered is described by a state-spaceT-S fuzzymodel withmixed delays and norm-bounded parameter uncertaintiesThe distributed delays are assumed to appear in the stateequation and the uncertainties are allowed to be time varyingbut norm bounded The aim of this paper is to design a full-order fuzzy dynamic output feedback controller such that theresulting closed-loop system is robustly asymptotically stablewhile satisfying an 119867

infinnorm condition with a prescribed

level irrespective of the parameter uncertainties A sufficientcondition for the solvability of this problem is proposedin terms of linear matrix inequalities (LMIs) When theseLMIs are feasible an explicit expression of a desired outputfeedback controller is also given

Notation Throughout this paper for real symmetric matrices119883 and 119884 the notation 119883 ge 119884 (resp 119883 gt 119884) means that thematrix119883minus119884 is positive semidefinite (resp positive definite)119868 is an identity matrix with appropriate dimension N is theset of natural numbersL

2[0infin) refers to the space of square

summable infinite vector sequences sdot 2stands for the usual

L2[0infin) normThenotation119872119879 represents the transpose of

the matrix119872 Matrices if not explicitly stated are assumedto have compatible dimensions ldquolowastrdquo is used as an ellipsis forterms induced by symmetry

2 Problem Formulation

A continuous T-S fuzzy neutral model with distributeddelays and parameter uncertainties can be described by thefollowing

Plant Rule 119894 if 1199041(119905) is 120583


119901(119905) is 120583

119894119901 then

(119905) = [119860119894+ Δ119860119894(119905)] 119909 (119905) + [119860

1119894+ Δ1198601119894(119905)] 119909 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [119861119894+ Δ119861119894(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) = [119862119894+ Δ119862119894(119905)] 119909 (119905) + 119862

1119894119909 (119905 minus 120591

1) + 1198632119894119908 (119905)

119911 (119905) = 119864119894119909 (119905) + 119864

1119894119909 (119905 minus 120591

1) + 119866119894119906 (119905)

119909 (119905) = 120601 (119905) forall119905 isin [minus120591 0] 119894 = 1 2 119903

(1)

where 120583119894119895is the fuzzy set 119903 is the number of IF-THEN rules

and 1199041(119905) 119904

119901(119905) are the premise variablesThroughout this

paper it is assumed that the premise variables do not dependon control variables 119909(119905) isin R119899 is the state 119906(119905) isin R119898

is the control input 119910(119905) isin R119904 is the measured output119911(119905) isin R119902 is the controlled output 119908(119905) isin R119901 is the noisesignal 120591

119894gt 0 (119894 = 1 2 3) are integers representing the

time delay of the fuzzy systems 120591 = max1205911 1205912 1205913 119860119894 1198601119894

1198602119894 1198603119894 119861119894 119862119894 1198621119894 1198631119894 1198632119894 119864119894 1198642119894 and 119866


constant matrices Δ119860119894(119905) Δ119860

1119894(119905) Δ119860

2119894(119905) Δ119860

3119894(119905) Δ119861

119894(119905)

and Δ119862119894(119905) are real-valued unknown matrices representing

time-varying parameter uncertainties and are assumed to beof the form[Δ119860119894(119905) Δ119860

1119894(119905) Δ119860

2119894(119905) Δ119860

3119894(119905) Δ119861

119894(119905) Δ119862

119894(119905)]

= 119872119894119865119894(119905) [119873

011989411987311198941198732119894119873311989411987341198941198735119894] 119894 = 1 2 119903

(2)

where 119872119894 1198730119894 1198731119894 1198732119894 1198733119894 1198734119894 and 119873


constant matrices and 119865119894(sdot) N rarr R1198971times1198972 are unknown time-

varying matrix function satisfying

119865119894(119905)119879

119865119894(119905) le 119868 forall119905 (3)

The parameter uncertainties Δ119860119894(119905) Δ119860

1119894(119905) Δ119860

2119894(119905)

Δ1198603119894(119905) Δ119861

119894(119905) and Δ119862

119894(119905) are said to be admissible if both

(2) and (3) holdThen the final output of the fuzzy neutral system is

inferred as follows

(119905) =

119903

sum

119894=1

ℎ119894(119904 (119905)) [119860

119894+ Δ119860119894(119905)] 119909 (119905)

+ [1198601119894+ Δ1198601119894(119905)] 119909 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905) int

119905

119905minus1205913

119909 (119904) 119889119904]

+ [119861119894+ Δ119861119894(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) =

119903

sum

119894=1

ℎ119894(119904 (119905)) [119862

119894+ Δ119862119894(119905)] 119909 (119905)

+1198621119894119909 (119905 minus 120591

1) + 1198632119894119908 (119905)

119911 (119905) =

119903

sum

119894=1

ℎ119894(119904 (119905)) [119864

119894119909 (119905) + 119864

1119894119909 (119905 minus 120591

1) + 119866119894119906 (119905)]

(4)

where

ℎ119894(119904 (119905)) =

120603119894(119904 (119905))

sum119903

119894=1120603119894(119904 (119905))

120603119894(119904 (119905)) =

119901

prod

119895=1

120583119894119895(119904119895(119905))

119904 (119905) = [1199041(119905) 1199042(119905) sdot sdot sdot 119904

119901(119905)]

(5)

in which 120583119894119895(119904119895(119905)) is the grade of membership of 119904

119895(119905) in 120583

119894119895

Then it can be seen that

120603119894(119904 (119905)) ge 0 119894 = 1 119903

119903

sum

119894=1

120603119894(119904 (119905)) gt 0

(6)



ℎ119894(119904 (119905)) ge 0 119894 = 1 119903

119903

sum

119894=1

ℎ119894(119904 (119905)) = 1 forall119905

(7)




119901(119905) is 120583

119894119901 then

(119905) = 119860

119896119894119909 (119905) + 119861

119896119894119910 (119905) (8)

119906 (119905) = 119862119896119894119909 (119905) 119894 = 1 2 119903 (9)

119909 (119905) = 0 119905 le 0 (10)


119896119894


(119905) =

119903

sum

119894=1

ℎ119894(119904 (119905)) [119860

119896119894119909 (119905) + 119861

119896119894119910 (119905)] (11)

119906 (119905) =

119903

sum

119894=1

ℎ119894(119904 (119905)) 119862

119896119894119909 (119905) 119894 = 1 2 119903 (12)


119890 (119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times [119860119894119895+ Δ119860119894119895(119905)] 119890 (119905)

+ [1198601119894119895+ Δ1198601119894(119905)] 119890 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] 119890 (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)]

timesint

119905

119905minus1205913

119890 (119904) 119889119904 + 119863119894119895119908 (119905)

(13)

119911 (119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) [119864

119894119895119890 (119905) + 119864

1119894119890 (119905 minus 120591

1)]

(14)

where

119890 (119905) = [

(119905)

(119905)

]

119860119894119895= [

119860119894

119861119894119862119896119895

119861119896119895119862119894119860119896119895

]

Δ119860119894119895(119905) = [

Δ119860119894(119905) Δ119861

119894(119905) 119862119896119895

119861119896119895Δ119862119894(119905) 0

]

1198601119894119895= [

1198601119894

0

11986111989611989511986211198940] Δ119860

1119894(119905) = [

Δ1198601119894(119905) 0

0 0]

1198602119894= [

11986021198940

0 0] Δ119860

2119894(119905) = [

Δ1198602119894(119905) 0

0 0]

1198603119894= [

11986031198940

0 0] Δ119860

3119894(119905) = [

Δ1198603119894(119905) 0

0 0]

119863119894119895= [

1198631119894

1198611198961198951198632119894

] 119864119894119895= [119864119894119866119894119862119896119895]

1198641119894= [11986411198940]

(15)



1199112lt 120574119908

2 (16)



3 Main Results


infincontrol of




2119909119879


119909119879

DD119879

119909 + 120598119910119879

S119879

S119910 (17)


(A +DFS)119879

Wminus1

(A +DFS)

le A119879

(W minus 120598DDT)

minus1

A + 120598minus1

S119879

S

(18)



X119879

Z +Z119879

X le X119879

YX +Z119879

Yminus1

Z (19)


1gt 0 119875

2gt 0 119876

1gt 0 119876

2gt 0 119877

1 and



LMIs hold

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Π1119894119894

Π2119894119894

11987511986021198941198751198603119894119875119863119894119894

12059111198771

119864119879

119894119894Γ119894119894

119875119872119894119894

119873119879

119894119894

lowast 1198691

0 0 0 12059111198772119864119879

1119894Γ1119894119894

0 0


0 0 0 0 Γ2119894

0 0


0 0 0 Γ3119894

0 0


119868 0 0 Γ4119894119894

0 0


0 0 0 0



2119869119879

2119872119894119894

0



119894119894119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 le 119903

[

[

Λ1119894119895

Λ2119894119895

Λ3119894119895

lowast Λ4119894119895

Λ5119894119895


]

]

lt 0 1 le 119894 lt 119895 le 119903

(20)

whereΛ1119894119895

=

[

[

[

[

Π1119894119895+ Π1119895119894

Π2119894119895+ Π2119895119894

1198751198602119894+ 11987511986021198951198751198603119894+ 1198751198603119895

lowast 21198691

0 0


0


]

]

]

]

Λ2119894119895=

[

[

[

[

[

119875119863119894119895+ 119875119863119895119894212059111198771119864119879

119894119895+ 119864119879

119895119894Γ119894119895+ Γ119895119894

0 212059111198772119864119879

1119894+ 119864119879

1119895Γ1119894119895+ Γ1119895119894

0 0 0 Γ2119894+ Γ2119895

0 0 0 Γ3119894+ Γ3119895

]

]

]

]

]

Λ3119894119895=

[

[

[

[

[

[

[

119875119872119894119895119875119872119895119894119873119879

119894119895119873119879

119895119894

0 0 119879

1119894119879

1119895

0 0 119879

2119894119879

2119895

0 0 119879

3119894119879

3119895

]

]

]

]

]

]

]

Λ4119894119895=

[

[

[

[

minus21205742

119868 0 0 Γ4119894119895+ Γ4119895119894


0 0



2

]

]

]

]

Λ5119894119895=

[

[

[

[

0 0 0 0

0 0 0 0

0 0 0 0

119869119879

2119872119894119895119869119879

21198721198951198940 0

]

]

]

]

Λ6119894119895=

[

[

[

[

[

minus120598119894119895119868 0 0 0

lowast minus120598119895119894119868 0 0


119894119895119868 0


119895119894119868

]

]

]

]

]

Υ1= 119867119879

(1198751+ 1205912

31198762minus 1198771minus 119877119879

1)119867

Π1119894119895= Υ1+ 119875119860119894119895+ 119860119879

119894119895119875

Υ2= 119867119879

(1198771minus 119877119879

2) Π

2119894119895= Υ2+ 1198751198601119894119895

Γ119894119895= 119860119879

119894119895(1198752+ 12059111198761)

Γ1119894119895= 119860119879

1119894119895(1198752+ 12059111198761) Γ

2119894= 119860119879

2119894(1198752+ 12059111198761)

Γ3119894= 119860119879

3119894(1198752+ 12059111198761) Γ

4119894119895= 119863119879

119894119895(1198752+ 12059111198761)

1198691= minus1198751+ 1198772+ 119877119879

2 119869

2= 1198752+ 12059111198761

119872119894119895= [

119872119894

0

0 119861119896119895119872119894

] 119865119894(119905) = [

119865119894(119905) 0

0 119865119894(119905)]

119873119894119895= [

11987301198941198733119894119862119896119895

1198734119894

0

] 1119894= [

11987311198940

0 0]

2119894= [

11987321198940

0 0]

3119894= [

11987331198940

0 0]

(21)


119890 (119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times [119860119894119895+ Δ119860119894119895(119905)] 119890 (119905)

+ [1198601119894119895+ Δ1198601119894(119905)] 119890 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] 119890 (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119890 (119904) 119889119904

(22)



119881 (119905) = 1198811(119905) + 119881

2(119905) + 119881

3(119905) + 119881

4(119905) + 119881

5(119905) + 119881

6(119905)

(23)

where

1198811(119905) = 119890(119905)

119879

119875119890 (119905)

1198812(119905) = int

119905

119905minus1205911

119890(119904)119879

1198751119890 (119904) 119889119904

1198813(119905) = int

119905

119905minus1205912

119890(119904)119879

1198752119890 (119904) 119889119904

1198814(119905) = int

0

minus1205911

int

119905

119905+120573

119890(120572)119879

1198761119890 (120572) 119889120572 119889120573

1198815(119905) = int

119905

119905minus1205913

[int

119905

119904

119890(120579)119879

119889120579]1198762[int

119905

119904

119890 (120579) 119889120579] 119889119904

1198816(119905) = int

1205913

0

119889119904int

119905

119905minus120573

(120579 minus 119905 + 120573) 119890(120572)119879

1198762119890 (120572) 119889120572 119889120573

(24)


(119905) = 1(119905) +

2(119905) +

3(119905) +

4(119905) +

5(119905) +

6(119905)

(25)

where

1(119905) = 2119890(119905)

119879

119875 119890 (119905)

2(119905) = 119890(119905)

119879

1198751119890 (119905) minus 119890(119905 minus 120591

1)119879

1198751119890 (119905 minus 120591

1)

3(119905) = 119890(119905)

119879

1198752119890 (119905) minus 119890(119905 minus 120591

2)119879

1198752119890 (119905 minus 120591

2)

4(119905) = 120591

1119890(119905)119879

1198761119890 (119905) minus int

119905

119905minus1205911

119890(120572)119879

1198761119890 (120572) 119889120572

5(119905) = 2int

119905

119905minus1205913

(120579 minus 119905 + 1205913) 119890(119905)119879

1198762119890 (120579) 119889120579

minus [int

119905

119905minus1205913

119890(120579)119879

119889120579]1198762[int

119905

119905minus1205913

119890 (120579) 119889120579]

6(119905) =

1

2

1205912

3119890(119905)119879

1198762119890 (119905) minus int

119905

119905minus1205913

(120579 minus 119905 + 1205913) 119890(120579)119879

1198762119890 (120579) 119889120579

(26)


2119890(119905)119879

1198762119890 (120579) le 119890(119905)

119879

1198762119890 (119905) + 119890(120579)

119879

1198762119890 (120579) (27)

Therefore

5(119905) le

1

2

1205912

3119890(119905)119879

1198762119890 (119905)

+ int

119905

119905minus1205913

(120579 minus 119905 + 1205913) 119890(120579)119879

1198762119890 (120579) 119889120579

minus [int

119905

119905minus1205913

119890(120579)119879

119889120579]1198762[int

119905

119905minus1205913

119890 (120579) 119889120579]

(28)


5(119905) +

6(119905) le 120591

2

3119890(119905)119879

1198762119890 (119905) minus 120572(119905)

119879

1198762120572 (119905) (29)

where

120572 (119905) = int

119905

119905minus1205913

119890 (120579) 119889120579 (30)

Then there holds

(119905) le 2119890(119905)119879

119875 119890 (119905)

+ 119890(119905)119879

1198751119890 (119905) minus 119890(119905 minus 120591

1)119879

1198751119890 (119905 minus 120591

1)

+ 119890(119905)119879

1198752119890 (119905) minus 119890(119905 minus 120591

2)119879

1198752119890 (119905 minus 120591

2)

+ 1205911119890(119905)119879

1198761119890 (119905) minus int

119905

119905minus1205911

119890(120572)119879

1198761119890 (120572) 119889120572

+ 1205912

3119890(119905)119879

1198762119890 (119905) minus 120572(119905)

119879

1198762120572 (119905)

+ 2119890(119905)119879

1198771int

119905

119905minus1205911

119890 (120572) 119889120572

minus 2119890(119905)119879

1198771[119890 (119905) minus 119890 (119905 minus 120591

1)]

+ 2119890(119905 minus 1205911)119879

1198772int

119905

119905minus1205911

119890 (120572) 119889120572

minus 2119890(119905 minus 1205911)119879

1198772[119890 (119905) minus 119890 (119905 minus 120591

1)]

(31)


2119890(119905)119879

119875 119890 (119905)

= 2119890(119905)119879

119875

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times [119860119894119895+ Δ119860119894119895(119905)] 119890 (119905)

+ [1198601119894119895+ Δ1198601119894(119905)] 119890 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] 119890 (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119890 (119904) 119889119904

le 2

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 119890(119905)

119879

119875119860119894119895120573 (119905)

+

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times [120576minus1

1119894119895119890(119905)119879

119875119872119894119895119872119879

119894119895119875119890 (119905)

+ 1205761119894119895120573(119905)119879

119879

119894119895119894119895120573 (119905)]

(32)

where

119860119894119895= [1198601198941198951198601119894119895

11986021198941198603119894]

120573 (119905) = [119890(119905)119879

119890(119905 minus 1205911)119879

119890(119905 minus 1205912)119879

120572(119905)119879

119890(120572)119879]

119894119895= [119873119894119895111989421198943119894]

(33)



119890(119905)119879

1198752119890 (119905)

=

1

2

119903

sum

119894=1

119903

sum

119895=1

119903

sum

119906=1

119903

sum

V=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) ℎ

119906(119904 (119905)) ℎV (119904 (119905)) 120573(119905)

119879

times [119860119894119895+119872119894119895119865119894(119905)119894119895]

119879

times 1198752[119860119906V +119872119906V119865119906 (119905) 119906V]

+ [119860119906V +119872119906V119865119906 (119905) 119906V]

119879

times 1198752[119860119894119895+119872119894119895119865119894(119905) 119894119895] 120573 (119905)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times [119860119894119895+119872119894119895119865119894(119905) 119894119895]

119879

1198752[119860119894119895+119872119894119895119865119894(119905) 119894119895]

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times [119860

119879

119894119895(119875minus1

2minus 120576minus1

2119894119895119872119894119895119872119879

119894119895)

minus1

119860119894119895+ 1205762119894119895119879

119894119895119894119895] 120573 (119905)

1205911119890(119905)119879

1198761119890 (119905)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times 119860

119879

119894119895[(12059111198761)minus1

minus 120576minus1

3119894119895119872119894119895119872119879

119894119895]

minus1

119860119894119895

+ 1205763119894119895119879

119894119895119894119895120573 (119905)

(34)


(119905) le 2119890(119905)119879

119875 119890 (119905) + 119890(119905)119879

1198752119890 (119905)

+ 1205911119890(119905)119879

1198761119890 (119905) + 119890(119905)

119879

(1198751+ 1205912

31198762) 119890 (119905)

minus 119890(119905 minus 1205911)119879

1198751119890 (119905 minus 120591

1)

minus 119890(119905 minus 1205912)119879

1198752119890 (119905 minus 120591

2) minus int

119905

119905minus1205911

119890(120572)119879

1198761119890 (120572) 119889120572

minus 120572(119905)119879

1198762120572 (119905) + 2119890(119905)

119879

1198771int

119905

119905minus1205911

119890 (120572) 119889120572

minus 2119890(119905)119879

1198771[119890 (119905) minus 119890 (119905 minus 120591

1)]

+ 2119890(119905 minus 1205911)119879

1198772int

119905

119905minus1205911

119890 (120572) 119889120572 minus 2119890(119905 minus 1205911)119879


1)]

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times [Ξ119894119895+ 119860119894119895(119875minus1

2minus 120576minus1

2119894119895119872119894119895119872119879

119894119895)

minus1

119860119894119895

+119860119894119895(120591minus1

1119876minus1

1minus 120576minus1

3119894119895119872119894119895119872119879

119894119895)

minus1

119860119894119895+ 120575119894119895119879

119894119895119894119895] 120573 (119905)

(35)

where

Ξ119894119895=

[

[

[

[

[

[

Σ1198941198951198751198601119894119895+ 1198771minus 119877119879

211987511986021198941198751198603119894

12059111198771

lowast minus1198751+ 1198772+ 119877119879

20 0 120591

11198772


0 0


0


]

]

]

]

]

]

Σ119894119895= 119875119860119894119895+ 119860119879

119894119895119875 + 1198751+ 1205912

31198762minus 1198771minus 119877119879

1+ 120576minus1

1119894119895119875119872119894119895119872119879

119894119895119875

(36)


Γ119894119895lt 0 (37)





119869 (119905) = int

119905

0

[119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904)] 119889119904 (38)



119869 (119905) = int

119905

0

[119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904) + (119904)] 119889119904 minus 119881 (119905)

le int

119905

0

[119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904) + (119904)] 119889119904

(39)


(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times 2119890(119905)119879

119875 [119860119894119895(119905) 119890 (119905) + 119860

1119894119895(119905) 119890 (119905 minus 120591

1)

+ 1198602119894(119905) 119890 (119905 minus 120591

2)

+1198603119894(119905) 120572 (119905) + 119863

119894119895119908 (119905)]


+ 119890(119905)119879

1198751119890 (119905) minus 119890(119905 minus 120591

1)119879

1198751119890 (119905 minus 120591

1)

+ 119890(119905)119879

1198752119890 (119905) minus 119890(119905 minus 120591

2)119879

1198752119890 (119905 minus 120591

2)

+ 1205911119890(119905)119879

1198761119890 (119905)

minus int

119905

119905minus1205911

119890(119905)119879

1198761119890 (120572) 119889120572 + 120591

2

3119890(119905)119879

1198762119890 (119905)

minus 120572(119905)119879

1198762120572 (119905)

+ 2119890(119905)119879

1198771int

119905

119905minus1205911

119890 (120572) 119889120572 minus 2119890(119905)119879


1)]

+ 2119890(119905 minus 1205911)119879

1198772int

119905

119905minus1205911

119890 (120572) 119889120572 minus 2119890(119905 minus 1205911)119879


1)]

(40)


119911(119904)119879

119911 (119904) =

119903

sum

119894=1

119903

sum

119895=1

119903

sum

119906=1

119903

sum

V=1

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) ℎ

119906(119904 (119904))

times ℎV (119904 (119904)) 120577(119904)119879

119864119879

119894119895119864119906V120577 (119904)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) 120577(119904)

119879

119864119879

119894119895119864119894119895120577 (119904)

(41)


119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904) + (119904)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) 120577(119904)

119879

Π1119894119895120577 (119904)

=

119903

sum

119894=1

ℎ119894(119904 (119904))

2

120577(119904)119879

Ξ1119894119894120577 (119904)

+ 2

119903

sum

119894119895=1119894lt119895

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) 120577(119904)

119879Ξ1119894119895+ Ξ1119895119894

2

120577 (119904)

(42)

where

Ξ1119894119895=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Σ1198941198951198751198601119894119895(119905) + 119877

1minus 119877119879

21198751198602119894(119905) 119875119860

3119894(119905) 119875119863

11989411989512059111198771

119864119879

119894119895119860119879

119894119895(119905) (1198752+ 12059111198761)

lowast 1198691

0 0 0 12059111198772119864119879

1119894119860119879

1119894119895(119905) (1198752+ 12059111198761)


0 0 0 0 119860119879

2119894(119905) (1198752+ 12059111198761)


0 0 0 119860119879

3119894(119905) (1198752+ 12059111198761)


119868 0 0 119863119879

119894119895(1198752+ 12059111198761)


0 0



]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

120577 (119904) = [119890(119905)119879

119890(119905 minus 1205911)119879

119890(119905 minus 1205912)119879

120572(119905)119879

119908(119905)119879

119890(120572)119879]

119879

119864119894119895= [11986411989411989511986411198940 0 0 0]

(43)



have 1199112lt 120574119908

2






119894119895 1 le 119894 le 119895 le 119903 such that


[

[

[

[

[

[

Σ119894119894Σ1119894119894

Σ2119894119894

Σ3119894119894

1198775

lowast minus119877111198772119894119894

0 0

lowast lowast 1198773119894119894

0 1198776119894119894


0


]

]

]

]

]

]

lt 0 1 le 119894 le 119903 (44)


[

[

[

[

[

[

[

[

[

[

Σ119894119895+ Σ119895119894Σ1119894119895+ Σ1119895119894

Σ2119894119895

Σ4119894119895

Σ5119894119895

Σ6119894119895

0


2119894119895

0 0 0 0


0 0 0 6119894119895



0 0



]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 lt 119895 le 119903 (45)

[


minus119868 minus119883] lt 0 (46)

where

Θ119894119895

= [

119860119894119883 + 119883119860

119879

119894+ 119861119894Ψ119895+ Ψ119879

119895119861119879

119894119860119894+ Ω119879

119894

119860119879

119894+ Ω119894

119884119860119894+ 119860119879

119894119884 + Φ

119895119862119894+ 119862119879

119894Φ119879

119895

]

minus1198771minus 119877119879

1

Θ1119894119895= [

1198601119894119883 119860

1119894

0 1198841198601119894+ Φ1198951198621119894

] + 1198771minus 119877119879

2

Θ2119894= [

1198602119894

0

11988411986021198940] Θ

3119894= [

1198603119894

0

11988411986031198940]

Θ4119894119895= [

1198631119894

1198841198631119894+ Φ1198951198632119894

] Θ5119894119895= [

119883119864119879

119894+ Ψ119879

119895119866119879

119894

119864119879

119894

]

Θ6119894119895= [

119883119860119879

119894+ Ψ119879

119895119861119879

119894Ω119879

119895

119860119879

119894119860119879

119894119884 + 119862

119879

119894Φ119879

119895

]

Θ7119894119895= [

119872119894

0

119884119872119894Φ119895119872119894

]

Θ8119894119895= [

119883119873119879

0119894+ Ψ119879

119895119873119879

4119894119883119873119879

5119894

119873119879

0119894119873119879

5119894

] 1198693119894= [

119883119864119879

1119894

119864119879

1119894

]

1198694119894119895= [

1198831198601119894

0

1198601119894

1198601119894119884 + 119862

119879

1119894Φ119879

119895

]

1198695119894= [

119883119873119879

11198940

119873119879

11198940

] 1198696119894= [

119860119879

2119894119860119879

2119894119884

0 0

]

1198697119894= [

119860119879

3119894119860119879

3119894119884

0 0

]

1198698119894119895= [119863119879

1119894119863119879

1119894119884 + 119863

119879

2119894Φ119879

119895]

1198699= Π119879

2+ Π2minus 1198752minus 12059111198761

11986910119894119895

= [

119872119894

0

119884119872119894Φ119895119872119894

]

Σ119894119895=[

[

[

Θ119894119895

Θ1119894119895

Θ2119894

lowast minus119880 + 1198772+ 119877119879

20


]

]

]

Σ1119894119895=[

[

[

Θ3119894Θ4119894119895

12059111198771

0 0 12059111198772

0 0 0

]

]

]

Σ2119894119895=[

[

[

Θ5119894119895

Θ6119894119895

Θ7119894119895

Θ8119894119895

1198693119894

1198694119894119895

0 1198695119894

0 1198696119894

0 119879

2119894

]

]

]

Σ2119894119895=

[

[

[

[

Θ5119894119895+ Θ5119895119894

Θ6119894119895+ Θ6119895119894

Θ7119894119895

Θ7119895119894

Θ8119894119895

Θ8119895119894

1198693119894+ 1198693119895

1198694119894119895+ 1198694119895119894

0 0 1198695119894

1198695119895

0 1198696119894+ 1198696119895

0 0 119879

2119894119879

2119895

]

]

]

]

Σ3119894119895=[

[

[

Π119879

11205913Π119879

1119879119894119895

0 0 0

0 0 0

]

]

]

119879119894119895= [

0

1198841198601119894+ Φ1198951198621119894

]

1198775=[

[

0 0 0

119879111987910

0 0 0

]

]

1198791= [

119909

0

] Σ4119894119895=[

[

[

1198792119894119895

1198793119894119895

1198794119894119895

1198795119894119895

0 0 0 0

0 0 0 0

]

]

]

1198792119894119895= [

(119862119894119909 minus 119862

119895119909)

119879

0

]

1198793119894119895= [

0

Φ119895minus Φ119894

] 1198794119894119895= [

(Ψ119895minus Ψ119894)

119879

0

]

1198795119894119895= [

0

119884119861119894minus 119884119861119895

]

Σ5119894119895=[

[

Π119879

11205913Π119879

11198791198941198950

0 0 0 1198791

0 0 0 0

]

]


Σ6119894119895=[

[

119879119895119894

0 11987941198792

0 11987910 0

0 0 0 0

]

]

11987711= diag (119876

21205742

119868 119881) 1198772119894119895=[

[

0 11986971198940 119879

3119894

0 1198698119894119895

0 0

0 0 0 0

]

]

2119894119895=[

[

0 1198697119894+ 1198697119895

0 0 119879

3119894119879

3119895

0 1198698119894119895+ 1198698119895119894

0 0 0 0

0 0 0 0 0 0

]

]

1198773119894119895=

[

[

[

[

minus119868 0 0 0

lowast minus119869911986910119894119895

0



119894119895119868

]

]

]

]

3119894119895=

[

[

[

[

[

[

[

[

[

minus2119868 0 0 0 0 0

lowast minus2119869911986910119894119895

11986910119895119894

0 0




119894119895119868 0


119895119894119868

]

]

]

]

]

]

]

]

]

1198774= diag (119875minus1

1119876minus1

2119868)

1198776119894119895=

[

[

[

[

0 0 0

0 0 1198796119894119895

0 0 0

0 0 0

]

]

]

]

6119894119895=

[

[

[

[

[

[

[

[

0 0

1198795119894119895

1198793119894119895

0 0

0 0

0 0

0 0

]

]

]

]

]

]

]

]

1198796119894119895= [

0

119884119860119879

1119894+ Φ1198951198621119894

]

1198777= diag (119868 119868 119868) 119877

8= diag (1

2

119875minus1

1

1

2

119876minus1

2119868 119868)

Π1= [

119883 119868

119883 0] Π

2= [

119883 119868

119883 0]

(47)


119860119870119894= (119883minus1

minus 119884)

minus1

(Ω119894minus 119884119860119894119883 minus 119884119861

119894Ψ119894minus Φ119894119862119894119883)119883minus1

119861119870119894= (119883minus1

minus 119884)

minus1

Φ119894 119862119870119894= Ψ119894119883minus1

1 le 119894 le 119903

(48)


[

[

Σ119894119894Σ1119894119894

Σ2119894119894

lowast minus119877111198772119894119894

lowast lowast 1198773119894119894

]

]

lt 0 1 le 119894 le 119903

[

[

[

Σ119894119895Σ1119894119895+ Σ1119895119894

Σ2119894119895


2119894119895


]

]

]

lt 0 1 le 119894 lt 119895 le 119903

(49)

where

Σ119894119895=

[

[

[

[

Θ119894119895+ Π119879

11198751Π1+ (1205913Π1)119879

1198762(1205913Π1) Θ1119894119895+ [

0 0

(1198841198601119894+ Φ1198951198621119894)119883 0

] Θ2119894

lowast minus119880 + 1198772+ 119877119879

20


]

]

]

]

Σ2119894119895=

[

[

[

[

Θ5119894119895

Θ6119894119895

Θ7119894119895

Θ8119894119895

1198693119894

1198694119894119895+ [

0 119883 (1198601119894119884 + 119862

119879

1119894Φ119879

119895)

0 0

] 0 1198695119894

0 1198696119894

0 119879

2119894

]

]

]

]

Σ119894119895= Σ119894119895+ Σ119895119894+

[

[

[

[

[

[

0 (119862119894119883 minus 119862

119895119883)

119879

(Φ119895minus Φ119894)

119879

+ (Ψ119895minus Ψ119894)

119879

(119884119861119894minus 119884119861119895)

119879

0 0 0

lowast 0 0 0 0

lowast lowast 0 0 0



]

]

]

]

]

]

Σ2119894119895= Σ2119894119895+[

[

0 (Ψ119879

119895minus Ψ119879

119894) (119861119879

119894119884 minus 119861

119879

119895119884) + (119883119862

119879

119894minus 119883119862119879

119895) (Φ119879

119895minus Φ119879

119894) 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]

]

(50)


Note that

Π119879

2+ Π2minus 1198752minus 12059111198761le Π119879

2(1198752+ 12059111198761)minus1

Π2 (51)


[

[

Σ119894119894Σ1119894119894

Σ2119894119894

lowast minus119877111198772119894119894


]

]

lt 0 1 le 119894 le 119903

[

[

[

Σ119894119895Σ1119894119895+ Σ1119895119894

Σ2119894119895


2119894119895

lowast lowast 1198773119894119895

]

]

]

lt 0 1 le 119894 lt 119895 le 119903

(52)

where

3119894119895=

[

[

[

[

[

minus119868 0 0 0


2(1198752+ 12059111198761)minus1

Π211986910119894119895

0



119894119895119868

]

]

]

]

]

1198773119894119895=

[

[

[

[

[

[

[

[

[

minus2119868 0 0 0 0 0


2(1198752+ 12059111198761)minus1

Π211986910119894119895

11986910119895119894

0 0




119894119895119868 0


119895119894119868

]

]

]

]

]

]

]

]

]

(53)

Now set = Π

2Πminus1

1 (54)



1 119868 119868 119868 Π

minus119879

1 119868 Πminus119879

2 119868 119868) (55)


[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Π1119894119894

Π2119894119894

11987511986021198941198751198603119894119875119863119894119894

12059111

119864119879

119894119894Γ119894119894

119875119872119894119894

119873119879

119894119894

lowast 1198691

0 0 0 12059112119864119879

1119894Γ1119894119894

0 0


0 0 0 0 Γ2119894

0 0


0 0 0 Γ3119894

0 0


119868 0 0 Γ4119894119894

0 0


0 0 0 0



2119869119879

2119872119894119894

0



119894119894119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 le 119903

[

[

Λ1119894119895

Λ2119894119895

Λ3119894119895

lowast Λ4119894119895

Λ5119894119895


]

]

lt 0 1 le 119894 lt 119895 le 119903

(56)

where

Λ1= 119867119879

(1198751+ 1205912

31198762minus 1minus 119879

1)119867

Π1119894119895= Λ1+ 119875119860119894119895+ 119860119879

119894119895119875

Λ2= 119867119879

(1minus 119879

2)

Π2119894119895= Λ2+ 1198751198601119894119895

1= Π119879

11198771Π1

(57)







119894= 0 and








1(119905) is 120583

1 then

(119905) = [1198601+ Δ1198601(119905)] 119909 (119905) + [119860

11+ Δ11986011(119905)] 119909 (119905 minus 120591

1)

+ [11986021+ Δ11986021(119905)] (119905 minus 120591

2)

+ [11986031+ Δ11986031(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198611+ Δ1198611(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) = [1198621+ Δ1198621(119905)] 119909 (119905) + 119862

11119909 (119905 minus 120591

1) + 11986321119908 (119905)

119911 (119905) = 1198641119909 (119905) + 119864

11119909 (119905 minus 120591

1) + 1198661119906 (119905)

(58)

Plant Rule 2 if 1199091(119905) is 120583

2 then

(119905) = [1198602+ Δ1198602(119905)] 119909 (119905) + [119860

12+ Δ11986012(119905)] 119909 (119905 minus 120591

1)

+ [11986022+ Δ11986022(119905)] (119905 minus 120591

2)

+ [11986032+ Δ11986032(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198612+ Δ1198612(119905)] 119906 (119905) + 119863

12119908 (119905)

119910 (119905) = [1198622+ Δ1198622(119905)] 119909 (119905) + 119862

12119909 (119905 minus 120591

1) + 11986322119908 (119905)

119911 (119905) = 1198642119909 (119905) + 119864

12119909 (119905 minus 120591

1) + 1198662119906 (119905)

(59)

where

1198601= [

005 0


11= [

005 002

minus005 001]

11986021= [

minus0002 0008

0006 0]

11986031= [

minus005 008

006 0] 119861

1= [

001 0

003 006]

1198621= [

minus003 minus01

02 002] 119862

11= [

minus003 minus001

002 002]

11986311= [

003

006] 119863

21= [

minus05

0]

1198641= [minus02 0] 119864

11= [minus002 0]

1198661= [0 minus003]

1198602= [

minus005 0


12= [

minus005 02

minus05 001]

11986022= [

0008 minus0002

minus0003 002]

11986032= [

005 minus02

minus03 02] 119861

2= [

001 minus008

002 005]

1198622= [

003 minus01

02 002]

11986212= [

minus003 minus001

002 02] 119863

12= [

minus001

002]

11986322= [

minus03

0]

1198642= [minus001 0] 119864

12= [minus001 0]

1198662= [0 002]

(60)

and Δ119860119894(119905) Δ119860

1119894(119905) Δ119860

2119894(119905) Δ119860

3119894(119905) Δ119861

119894(119905) Δ119862

119894(119905) (119894 =


1198721= [

minus002

0] 119873

01= [minus02 01]

11987311= [minus02 02]

11987321= [minus002 001] 119873

31= [minus002 001]

11987341= [minus002 001]

11987351= [minus002 001] 119872

2= [

003

0]

11987302= 11987301 119873

12= 11987311

11987322= 11987321 119873

32= 11987331

11987342= 11987341 119873

52= 11987351

(61)



(119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119860119894+ Δ119860119894(119905)] 119909 (119905)

+ [1198601119894+ Δ1198601119894(119905)] 119909 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [119861119894+ Δ119861119894(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119862119894+ Δ119862119894(119905)] 119909 (119905)

+1198621119894119909 (119905 minus 120591

1) + 1198632119894119908 (119905)

119911 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905)) 119864

119894119909 (119905) + 119864

1119894119909 (119905 minus 120591

1) + 119866119894119906 (119905)

(62)

where

ℎ1(1199091(119905)) =

1

3


2

3

+

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

1 for 1199091gt 1

ℎ2(1199091(119905)) =

2

3


1

3

minus

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

0 for 1199091gt 1

(63)






119908 (119905) =

1

119905 + 01

119905 ge 0 (64)


119883 = [

09868 06071

06071 13948]

119884 = [

129348 01814

01814 63384]

Ω1= [



Ω2= [

82076 minus37031

minus58461 53292]

Φ1= [

31848 00758


Φ2= [

20547 01643


1205951= [


57867 minus778562]

1205952= [



(65)


1198601198961= [

873277 minus680397

1399266 minus1072394]

1198601198962= [

minus249856 158880

111983 minus122840]

1198611198961= [

109115 minus03725

165830 19936]

1198611198962= [

14642 minus00212

31940 05922]

1198621198961= [

minus160423 127577

202647 minus148918]

1198621198962= [

142596 minus104929

32304 minus11934]

(66)








x1(t)

x2(t)

0 10 20 30 40 50 60minus02

0

02

04

06

08

1

12

Time (s)

Stat

e res

pons

ex(t)


2(119905) (dashed

line)

0 10 20 30 40 50 60minus100

minus50

0

50

100

150

200

Time (s)

Con

trol i

nput

u(t)

u1(t)

u2(t)


2(119905) (dashed

line)





infin




infincontrol


Acknowledgments


y1(t)

y2(t)

0 10 20 30 40 50 60minus1

0

1

2

3

4

5

Time (s)

Mea

sure

d ou

tput

y(t)


2(119905)

(dashed line)

z1(t)

z2(t)

0 10 20 30 40 50 60Time (s)

minus40

minus20

0

20

40

60

80

100

120

Con

trolle

d ou

tput

z(t)


2(119905)

(dashed line)


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











ℎ119894(119904 (119905)) ge 0 119894 = 1 119903

119903

sum

119894=1

ℎ119894(119904 (119905)) = 1 forall119905

(7)




119901(119905) is 120583

119894119901 then

(119905) = 119860

119896119894119909 (119905) + 119861

119896119894119910 (119905) (8)

119906 (119905) = 119862119896119894119909 (119905) 119894 = 1 2 119903 (9)

119909 (119905) = 0 119905 le 0 (10)


119896119894


(119905) =

119903

sum

119894=1

ℎ119894(119904 (119905)) [119860

119896119894119909 (119905) + 119861

119896119894119910 (119905)] (11)

119906 (119905) =

119903

sum

119894=1

ℎ119894(119904 (119905)) 119862

119896119894119909 (119905) 119894 = 1 2 119903 (12)


119890 (119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times [119860119894119895+ Δ119860119894119895(119905)] 119890 (119905)

+ [1198601119894119895+ Δ1198601119894(119905)] 119890 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] 119890 (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)]

timesint

119905

119905minus1205913

119890 (119904) 119889119904 + 119863119894119895119908 (119905)

(13)

119911 (119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) [119864

119894119895119890 (119905) + 119864

1119894119890 (119905 minus 120591

1)]

(14)

where

119890 (119905) = [

(119905)

(119905)

]

119860119894119895= [

119860119894

119861119894119862119896119895

119861119896119895119862119894119860119896119895

]

Δ119860119894119895(119905) = [

Δ119860119894(119905) Δ119861

119894(119905) 119862119896119895

119861119896119895Δ119862119894(119905) 0

]

1198601119894119895= [

1198601119894

0

11986111989611989511986211198940] Δ119860

1119894(119905) = [

Δ1198601119894(119905) 0

0 0]

1198602119894= [

11986021198940

0 0] Δ119860

2119894(119905) = [

Δ1198602119894(119905) 0

0 0]

1198603119894= [

11986031198940

0 0] Δ119860

3119894(119905) = [

Δ1198603119894(119905) 0

0 0]

119863119894119895= [

1198631119894

1198611198961198951198632119894

] 119864119894119895= [119864119894119866119894119862119896119895]

1198641119894= [11986411198940]

(15)



1199112lt 120574119908

2 (16)



3 Main Results


infincontrol of




2119909119879


119909119879

DD119879

119909 + 120598119910119879

S119879

S119910 (17)


(A +DFS)119879

Wminus1

(A +DFS)

le A119879

(W minus 120598DDT)

minus1

A + 120598minus1

S119879

S

(18)



X119879

Z +Z119879

X le X119879

YX +Z119879

Yminus1

Z (19)


1gt 0 119875

2gt 0 119876

1gt 0 119876

2gt 0 119877

1 and



LMIs hold

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Π1119894119894

Π2119894119894

11987511986021198941198751198603119894119875119863119894119894

12059111198771

119864119879

119894119894Γ119894119894

119875119872119894119894

119873119879

119894119894

lowast 1198691

0 0 0 12059111198772119864119879

1119894Γ1119894119894

0 0


0 0 0 0 Γ2119894

0 0


0 0 0 Γ3119894

0 0


119868 0 0 Γ4119894119894

0 0


0 0 0 0



2119869119879

2119872119894119894

0



119894119894119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 le 119903

[

[

Λ1119894119895

Λ2119894119895

Λ3119894119895

lowast Λ4119894119895

Λ5119894119895


]

]

lt 0 1 le 119894 lt 119895 le 119903

(20)

whereΛ1119894119895

=

[

[

[

[

Π1119894119895+ Π1119895119894

Π2119894119895+ Π2119895119894

1198751198602119894+ 11987511986021198951198751198603119894+ 1198751198603119895

lowast 21198691

0 0


0


]

]

]

]

Λ2119894119895=

[

[

[

[

[

119875119863119894119895+ 119875119863119895119894212059111198771119864119879

119894119895+ 119864119879

119895119894Γ119894119895+ Γ119895119894

0 212059111198772119864119879

1119894+ 119864119879

1119895Γ1119894119895+ Γ1119895119894

0 0 0 Γ2119894+ Γ2119895

0 0 0 Γ3119894+ Γ3119895

]

]

]

]

]

Λ3119894119895=

[

[

[

[

[

[

[

119875119872119894119895119875119872119895119894119873119879

119894119895119873119879

119895119894

0 0 119879

1119894119879

1119895

0 0 119879

2119894119879

2119895

0 0 119879

3119894119879

3119895

]

]

]

]

]

]

]

Λ4119894119895=

[

[

[

[

minus21205742

119868 0 0 Γ4119894119895+ Γ4119895119894


0 0



2

]

]

]

]

Λ5119894119895=

[

[

[

[

0 0 0 0

0 0 0 0

0 0 0 0

119869119879

2119872119894119895119869119879

21198721198951198940 0

]

]

]

]

Λ6119894119895=

[

[

[

[

[

minus120598119894119895119868 0 0 0

lowast minus120598119895119894119868 0 0


119894119895119868 0


119895119894119868

]

]

]

]

]

Υ1= 119867119879

(1198751+ 1205912

31198762minus 1198771minus 119877119879

1)119867

Π1119894119895= Υ1+ 119875119860119894119895+ 119860119879

119894119895119875

Υ2= 119867119879

(1198771minus 119877119879

2) Π

2119894119895= Υ2+ 1198751198601119894119895

Γ119894119895= 119860119879

119894119895(1198752+ 12059111198761)

Γ1119894119895= 119860119879

1119894119895(1198752+ 12059111198761) Γ

2119894= 119860119879

2119894(1198752+ 12059111198761)

Γ3119894= 119860119879

3119894(1198752+ 12059111198761) Γ

4119894119895= 119863119879

119894119895(1198752+ 12059111198761)

1198691= minus1198751+ 1198772+ 119877119879

2 119869

2= 1198752+ 12059111198761

119872119894119895= [

119872119894

0

0 119861119896119895119872119894

] 119865119894(119905) = [

119865119894(119905) 0

0 119865119894(119905)]

119873119894119895= [

11987301198941198733119894119862119896119895

1198734119894

0

] 1119894= [

11987311198940

0 0]

2119894= [

11987321198940

0 0]

3119894= [

11987331198940

0 0]

(21)


119890 (119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times [119860119894119895+ Δ119860119894119895(119905)] 119890 (119905)

+ [1198601119894119895+ Δ1198601119894(119905)] 119890 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] 119890 (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119890 (119904) 119889119904

(22)



119881 (119905) = 1198811(119905) + 119881

2(119905) + 119881

3(119905) + 119881

4(119905) + 119881

5(119905) + 119881

6(119905)

(23)

where

1198811(119905) = 119890(119905)

119879

119875119890 (119905)

1198812(119905) = int

119905

119905minus1205911

119890(119904)119879

1198751119890 (119904) 119889119904

1198813(119905) = int

119905

119905minus1205912

119890(119904)119879

1198752119890 (119904) 119889119904

1198814(119905) = int

0

minus1205911

int

119905

119905+120573

119890(120572)119879

1198761119890 (120572) 119889120572 119889120573

1198815(119905) = int

119905

119905minus1205913

[int

119905

119904

119890(120579)119879

119889120579]1198762[int

119905

119904

119890 (120579) 119889120579] 119889119904

1198816(119905) = int

1205913

0

119889119904int

119905

119905minus120573

(120579 minus 119905 + 120573) 119890(120572)119879

1198762119890 (120572) 119889120572 119889120573

(24)


(119905) = 1(119905) +

2(119905) +

3(119905) +

4(119905) +

5(119905) +

6(119905)

(25)

where

1(119905) = 2119890(119905)

119879

119875 119890 (119905)

2(119905) = 119890(119905)

119879

1198751119890 (119905) minus 119890(119905 minus 120591

1)119879

1198751119890 (119905 minus 120591

1)

3(119905) = 119890(119905)

119879

1198752119890 (119905) minus 119890(119905 minus 120591

2)119879

1198752119890 (119905 minus 120591

2)

4(119905) = 120591

1119890(119905)119879

1198761119890 (119905) minus int

119905

119905minus1205911

119890(120572)119879

1198761119890 (120572) 119889120572

5(119905) = 2int

119905

119905minus1205913

(120579 minus 119905 + 1205913) 119890(119905)119879

1198762119890 (120579) 119889120579

minus [int

119905

119905minus1205913

119890(120579)119879

119889120579]1198762[int

119905

119905minus1205913

119890 (120579) 119889120579]

6(119905) =

1

2

1205912

3119890(119905)119879

1198762119890 (119905) minus int

119905

119905minus1205913

(120579 minus 119905 + 1205913) 119890(120579)119879

1198762119890 (120579) 119889120579

(26)


2119890(119905)119879

1198762119890 (120579) le 119890(119905)

119879

1198762119890 (119905) + 119890(120579)

119879

1198762119890 (120579) (27)

Therefore

5(119905) le

1

2

1205912

3119890(119905)119879

1198762119890 (119905)

+ int

119905

119905minus1205913

(120579 minus 119905 + 1205913) 119890(120579)119879

1198762119890 (120579) 119889120579

minus [int

119905

119905minus1205913

119890(120579)119879

119889120579]1198762[int

119905

119905minus1205913

119890 (120579) 119889120579]

(28)


5(119905) +

6(119905) le 120591

2

3119890(119905)119879

1198762119890 (119905) minus 120572(119905)

119879

1198762120572 (119905) (29)

where

120572 (119905) = int

119905

119905minus1205913

119890 (120579) 119889120579 (30)

Then there holds

(119905) le 2119890(119905)119879

119875 119890 (119905)

+ 119890(119905)119879

1198751119890 (119905) minus 119890(119905 minus 120591

1)119879

1198751119890 (119905 minus 120591

1)

+ 119890(119905)119879

1198752119890 (119905) minus 119890(119905 minus 120591

2)119879

1198752119890 (119905 minus 120591

2)

+ 1205911119890(119905)119879

1198761119890 (119905) minus int

119905

119905minus1205911

119890(120572)119879

1198761119890 (120572) 119889120572

+ 1205912

3119890(119905)119879

1198762119890 (119905) minus 120572(119905)

119879

1198762120572 (119905)

+ 2119890(119905)119879

1198771int

119905

119905minus1205911

119890 (120572) 119889120572

minus 2119890(119905)119879

1198771[119890 (119905) minus 119890 (119905 minus 120591

1)]

+ 2119890(119905 minus 1205911)119879

1198772int

119905

119905minus1205911

119890 (120572) 119889120572

minus 2119890(119905 minus 1205911)119879

1198772[119890 (119905) minus 119890 (119905 minus 120591

1)]

(31)


2119890(119905)119879

119875 119890 (119905)

= 2119890(119905)119879

119875

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times [119860119894119895+ Δ119860119894119895(119905)] 119890 (119905)

+ [1198601119894119895+ Δ1198601119894(119905)] 119890 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] 119890 (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119890 (119904) 119889119904

le 2

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 119890(119905)

119879

119875119860119894119895120573 (119905)

+

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times [120576minus1

1119894119895119890(119905)119879

119875119872119894119895119872119879

119894119895119875119890 (119905)

+ 1205761119894119895120573(119905)119879

119879

119894119895119894119895120573 (119905)]

(32)

where

119860119894119895= [1198601198941198951198601119894119895

11986021198941198603119894]

120573 (119905) = [119890(119905)119879

119890(119905 minus 1205911)119879

119890(119905 minus 1205912)119879

120572(119905)119879

119890(120572)119879]

119894119895= [119873119894119895111989421198943119894]

(33)



119890(119905)119879

1198752119890 (119905)

=

1

2

119903

sum

119894=1

119903

sum

119895=1

119903

sum

119906=1

119903

sum

V=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) ℎ

119906(119904 (119905)) ℎV (119904 (119905)) 120573(119905)

119879

times [119860119894119895+119872119894119895119865119894(119905)119894119895]

119879

times 1198752[119860119906V +119872119906V119865119906 (119905) 119906V]

+ [119860119906V +119872119906V119865119906 (119905) 119906V]

119879

times 1198752[119860119894119895+119872119894119895119865119894(119905) 119894119895] 120573 (119905)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times [119860119894119895+119872119894119895119865119894(119905) 119894119895]

119879

1198752[119860119894119895+119872119894119895119865119894(119905) 119894119895]

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times [119860

119879

119894119895(119875minus1

2minus 120576minus1

2119894119895119872119894119895119872119879

119894119895)

minus1

119860119894119895+ 1205762119894119895119879

119894119895119894119895] 120573 (119905)

1205911119890(119905)119879

1198761119890 (119905)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times 119860

119879

119894119895[(12059111198761)minus1

minus 120576minus1

3119894119895119872119894119895119872119879

119894119895]

minus1

119860119894119895

+ 1205763119894119895119879

119894119895119894119895120573 (119905)

(34)


(119905) le 2119890(119905)119879

119875 119890 (119905) + 119890(119905)119879

1198752119890 (119905)

+ 1205911119890(119905)119879

1198761119890 (119905) + 119890(119905)

119879

(1198751+ 1205912

31198762) 119890 (119905)

minus 119890(119905 minus 1205911)119879

1198751119890 (119905 minus 120591

1)

minus 119890(119905 minus 1205912)119879

1198752119890 (119905 minus 120591

2) minus int

119905

119905minus1205911

119890(120572)119879

1198761119890 (120572) 119889120572

minus 120572(119905)119879

1198762120572 (119905) + 2119890(119905)

119879

1198771int

119905

119905minus1205911

119890 (120572) 119889120572

minus 2119890(119905)119879

1198771[119890 (119905) minus 119890 (119905 minus 120591

1)]

+ 2119890(119905 minus 1205911)119879

1198772int

119905

119905minus1205911

119890 (120572) 119889120572 minus 2119890(119905 minus 1205911)119879


1)]

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times [Ξ119894119895+ 119860119894119895(119875minus1

2minus 120576minus1

2119894119895119872119894119895119872119879

119894119895)

minus1

119860119894119895

+119860119894119895(120591minus1

1119876minus1

1minus 120576minus1

3119894119895119872119894119895119872119879

119894119895)

minus1

119860119894119895+ 120575119894119895119879

119894119895119894119895] 120573 (119905)

(35)

where

Ξ119894119895=

[

[

[

[

[

[

Σ1198941198951198751198601119894119895+ 1198771minus 119877119879

211987511986021198941198751198603119894

12059111198771

lowast minus1198751+ 1198772+ 119877119879

20 0 120591

11198772


0 0


0


]

]

]

]

]

]

Σ119894119895= 119875119860119894119895+ 119860119879

119894119895119875 + 1198751+ 1205912

31198762minus 1198771minus 119877119879

1+ 120576minus1

1119894119895119875119872119894119895119872119879

119894119895119875

(36)


Γ119894119895lt 0 (37)





119869 (119905) = int

119905

0

[119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904)] 119889119904 (38)



119869 (119905) = int

119905

0

[119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904) + (119904)] 119889119904 minus 119881 (119905)

le int

119905

0

[119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904) + (119904)] 119889119904

(39)


(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times 2119890(119905)119879

119875 [119860119894119895(119905) 119890 (119905) + 119860

1119894119895(119905) 119890 (119905 minus 120591

1)

+ 1198602119894(119905) 119890 (119905 minus 120591

2)

+1198603119894(119905) 120572 (119905) + 119863

119894119895119908 (119905)]


+ 119890(119905)119879

1198751119890 (119905) minus 119890(119905 minus 120591

1)119879

1198751119890 (119905 minus 120591

1)

+ 119890(119905)119879

1198752119890 (119905) minus 119890(119905 minus 120591

2)119879

1198752119890 (119905 minus 120591

2)

+ 1205911119890(119905)119879

1198761119890 (119905)

minus int

119905

119905minus1205911

119890(119905)119879

1198761119890 (120572) 119889120572 + 120591

2

3119890(119905)119879

1198762119890 (119905)

minus 120572(119905)119879

1198762120572 (119905)

+ 2119890(119905)119879

1198771int

119905

119905minus1205911

119890 (120572) 119889120572 minus 2119890(119905)119879


1)]

+ 2119890(119905 minus 1205911)119879

1198772int

119905

119905minus1205911

119890 (120572) 119889120572 minus 2119890(119905 minus 1205911)119879


1)]

(40)


119911(119904)119879

119911 (119904) =

119903

sum

119894=1

119903

sum

119895=1

119903

sum

119906=1

119903

sum

V=1

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) ℎ

119906(119904 (119904))

times ℎV (119904 (119904)) 120577(119904)119879

119864119879

119894119895119864119906V120577 (119904)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) 120577(119904)

119879

119864119879

119894119895119864119894119895120577 (119904)

(41)


119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904) + (119904)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) 120577(119904)

119879

Π1119894119895120577 (119904)

=

119903

sum

119894=1

ℎ119894(119904 (119904))

2

120577(119904)119879

Ξ1119894119894120577 (119904)

+ 2

119903

sum

119894119895=1119894lt119895

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) 120577(119904)

119879Ξ1119894119895+ Ξ1119895119894

2

120577 (119904)

(42)

where

Ξ1119894119895=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Σ1198941198951198751198601119894119895(119905) + 119877

1minus 119877119879

21198751198602119894(119905) 119875119860

3119894(119905) 119875119863

11989411989512059111198771

119864119879

119894119895119860119879

119894119895(119905) (1198752+ 12059111198761)

lowast 1198691

0 0 0 12059111198772119864119879

1119894119860119879

1119894119895(119905) (1198752+ 12059111198761)


0 0 0 0 119860119879

2119894(119905) (1198752+ 12059111198761)


0 0 0 119860119879

3119894(119905) (1198752+ 12059111198761)


119868 0 0 119863119879

119894119895(1198752+ 12059111198761)


0 0



]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

120577 (119904) = [119890(119905)119879

119890(119905 minus 1205911)119879

119890(119905 minus 1205912)119879

120572(119905)119879

119908(119905)119879

119890(120572)119879]

119879

119864119894119895= [11986411989411989511986411198940 0 0 0]

(43)



have 1199112lt 120574119908

2






119894119895 1 le 119894 le 119895 le 119903 such that


[

[

[

[

[

[

Σ119894119894Σ1119894119894

Σ2119894119894

Σ3119894119894

1198775

lowast minus119877111198772119894119894

0 0

lowast lowast 1198773119894119894

0 1198776119894119894


0


]

]

]

]

]

]

lt 0 1 le 119894 le 119903 (44)


[

[

[

[

[

[

[

[

[

[

Σ119894119895+ Σ119895119894Σ1119894119895+ Σ1119895119894

Σ2119894119895

Σ4119894119895

Σ5119894119895

Σ6119894119895

0


2119894119895

0 0 0 0


0 0 0 6119894119895



0 0



]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 lt 119895 le 119903 (45)

[


minus119868 minus119883] lt 0 (46)

where

Θ119894119895

= [

119860119894119883 + 119883119860

119879

119894+ 119861119894Ψ119895+ Ψ119879

119895119861119879

119894119860119894+ Ω119879

119894

119860119879

119894+ Ω119894

119884119860119894+ 119860119879

119894119884 + Φ

119895119862119894+ 119862119879

119894Φ119879

119895

]

minus1198771minus 119877119879

1

Θ1119894119895= [

1198601119894119883 119860

1119894

0 1198841198601119894+ Φ1198951198621119894

] + 1198771minus 119877119879

2

Θ2119894= [

1198602119894

0

11988411986021198940] Θ

3119894= [

1198603119894

0

11988411986031198940]

Θ4119894119895= [

1198631119894

1198841198631119894+ Φ1198951198632119894

] Θ5119894119895= [

119883119864119879

119894+ Ψ119879

119895119866119879

119894

119864119879

119894

]

Θ6119894119895= [

119883119860119879

119894+ Ψ119879

119895119861119879

119894Ω119879

119895

119860119879

119894119860119879

119894119884 + 119862

119879

119894Φ119879

119895

]

Θ7119894119895= [

119872119894

0

119884119872119894Φ119895119872119894

]

Θ8119894119895= [

119883119873119879

0119894+ Ψ119879

119895119873119879

4119894119883119873119879

5119894

119873119879

0119894119873119879

5119894

] 1198693119894= [

119883119864119879

1119894

119864119879

1119894

]

1198694119894119895= [

1198831198601119894

0

1198601119894

1198601119894119884 + 119862

119879

1119894Φ119879

119895

]

1198695119894= [

119883119873119879

11198940

119873119879

11198940

] 1198696119894= [

119860119879

2119894119860119879

2119894119884

0 0

]

1198697119894= [

119860119879

3119894119860119879

3119894119884

0 0

]

1198698119894119895= [119863119879

1119894119863119879

1119894119884 + 119863

119879

2119894Φ119879

119895]

1198699= Π119879

2+ Π2minus 1198752minus 12059111198761

11986910119894119895

= [

119872119894

0

119884119872119894Φ119895119872119894

]

Σ119894119895=[

[

[

Θ119894119895

Θ1119894119895

Θ2119894

lowast minus119880 + 1198772+ 119877119879

20


]

]

]

Σ1119894119895=[

[

[

Θ3119894Θ4119894119895

12059111198771

0 0 12059111198772

0 0 0

]

]

]

Σ2119894119895=[

[

[

Θ5119894119895

Θ6119894119895

Θ7119894119895

Θ8119894119895

1198693119894

1198694119894119895

0 1198695119894

0 1198696119894

0 119879

2119894

]

]

]

Σ2119894119895=

[

[

[

[

Θ5119894119895+ Θ5119895119894

Θ6119894119895+ Θ6119895119894

Θ7119894119895

Θ7119895119894

Θ8119894119895

Θ8119895119894

1198693119894+ 1198693119895

1198694119894119895+ 1198694119895119894

0 0 1198695119894

1198695119895

0 1198696119894+ 1198696119895

0 0 119879

2119894119879

2119895

]

]

]

]

Σ3119894119895=[

[

[

Π119879

11205913Π119879

1119879119894119895

0 0 0

0 0 0

]

]

]

119879119894119895= [

0

1198841198601119894+ Φ1198951198621119894

]

1198775=[

[

0 0 0

119879111987910

0 0 0

]

]

1198791= [

119909

0

] Σ4119894119895=[

[

[

1198792119894119895

1198793119894119895

1198794119894119895

1198795119894119895

0 0 0 0

0 0 0 0

]

]

]

1198792119894119895= [

(119862119894119909 minus 119862

119895119909)

119879

0

]

1198793119894119895= [

0

Φ119895minus Φ119894

] 1198794119894119895= [

(Ψ119895minus Ψ119894)

119879

0

]

1198795119894119895= [

0

119884119861119894minus 119884119861119895

]

Σ5119894119895=[

[

Π119879

11205913Π119879

11198791198941198950

0 0 0 1198791

0 0 0 0

]

]


Σ6119894119895=[

[

119879119895119894

0 11987941198792

0 11987910 0

0 0 0 0

]

]

11987711= diag (119876

21205742

119868 119881) 1198772119894119895=[

[

0 11986971198940 119879

3119894

0 1198698119894119895

0 0

0 0 0 0

]

]

2119894119895=[

[

0 1198697119894+ 1198697119895

0 0 119879

3119894119879

3119895

0 1198698119894119895+ 1198698119895119894

0 0 0 0

0 0 0 0 0 0

]

]

1198773119894119895=

[

[

[

[

minus119868 0 0 0

lowast minus119869911986910119894119895

0



119894119895119868

]

]

]

]

3119894119895=

[

[

[

[

[

[

[

[

[

minus2119868 0 0 0 0 0

lowast minus2119869911986910119894119895

11986910119895119894

0 0




119894119895119868 0


119895119894119868

]

]

]

]

]

]

]

]

]

1198774= diag (119875minus1

1119876minus1

2119868)

1198776119894119895=

[

[

[

[

0 0 0

0 0 1198796119894119895

0 0 0

0 0 0

]

]

]

]

6119894119895=

[

[

[

[

[

[

[

[

0 0

1198795119894119895

1198793119894119895

0 0

0 0

0 0

0 0

]

]

]

]

]

]

]

]

1198796119894119895= [

0

119884119860119879

1119894+ Φ1198951198621119894

]

1198777= diag (119868 119868 119868) 119877

8= diag (1

2

119875minus1

1

1

2

119876minus1

2119868 119868)

Π1= [

119883 119868

119883 0] Π

2= [

119883 119868

119883 0]

(47)


119860119870119894= (119883minus1

minus 119884)

minus1

(Ω119894minus 119884119860119894119883 minus 119884119861

119894Ψ119894minus Φ119894119862119894119883)119883minus1

119861119870119894= (119883minus1

minus 119884)

minus1

Φ119894 119862119870119894= Ψ119894119883minus1

1 le 119894 le 119903

(48)


[

[

Σ119894119894Σ1119894119894

Σ2119894119894

lowast minus119877111198772119894119894

lowast lowast 1198773119894119894

]

]

lt 0 1 le 119894 le 119903

[

[

[

Σ119894119895Σ1119894119895+ Σ1119895119894

Σ2119894119895


2119894119895


]

]

]

lt 0 1 le 119894 lt 119895 le 119903

(49)

where

Σ119894119895=

[

[

[

[

Θ119894119895+ Π119879

11198751Π1+ (1205913Π1)119879

1198762(1205913Π1) Θ1119894119895+ [

0 0

(1198841198601119894+ Φ1198951198621119894)119883 0

] Θ2119894

lowast minus119880 + 1198772+ 119877119879

20


]

]

]

]

Σ2119894119895=

[

[

[

[

Θ5119894119895

Θ6119894119895

Θ7119894119895

Θ8119894119895

1198693119894

1198694119894119895+ [

0 119883 (1198601119894119884 + 119862

119879

1119894Φ119879

119895)

0 0

] 0 1198695119894

0 1198696119894

0 119879

2119894

]

]

]

]

Σ119894119895= Σ119894119895+ Σ119895119894+

[

[

[

[

[

[

0 (119862119894119883 minus 119862

119895119883)

119879

(Φ119895minus Φ119894)

119879

+ (Ψ119895minus Ψ119894)

119879

(119884119861119894minus 119884119861119895)

119879

0 0 0

lowast 0 0 0 0

lowast lowast 0 0 0



]

]

]

]

]

]

Σ2119894119895= Σ2119894119895+[

[

0 (Ψ119879

119895minus Ψ119879

119894) (119861119879

119894119884 minus 119861

119879

119895119884) + (119883119862

119879

119894minus 119883119862119879

119895) (Φ119879

119895minus Φ119879

119894) 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]

]

(50)


Note that

Π119879

2+ Π2minus 1198752minus 12059111198761le Π119879

2(1198752+ 12059111198761)minus1

Π2 (51)


[

[

Σ119894119894Σ1119894119894

Σ2119894119894

lowast minus119877111198772119894119894


]

]

lt 0 1 le 119894 le 119903

[

[

[

Σ119894119895Σ1119894119895+ Σ1119895119894

Σ2119894119895


2119894119895

lowast lowast 1198773119894119895

]

]

]

lt 0 1 le 119894 lt 119895 le 119903

(52)

where

3119894119895=

[

[

[

[

[

minus119868 0 0 0


2(1198752+ 12059111198761)minus1

Π211986910119894119895

0



119894119895119868

]

]

]

]

]

1198773119894119895=

[

[

[

[

[

[

[

[

[

minus2119868 0 0 0 0 0


2(1198752+ 12059111198761)minus1

Π211986910119894119895

11986910119895119894

0 0




119894119895119868 0


119895119894119868

]

]

]

]

]

]

]

]

]

(53)

Now set = Π

2Πminus1

1 (54)



1 119868 119868 119868 Π

minus119879

1 119868 Πminus119879

2 119868 119868) (55)


[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Π1119894119894

Π2119894119894

11987511986021198941198751198603119894119875119863119894119894

12059111

119864119879

119894119894Γ119894119894

119875119872119894119894

119873119879

119894119894

lowast 1198691

0 0 0 12059112119864119879

1119894Γ1119894119894

0 0


0 0 0 0 Γ2119894

0 0


0 0 0 Γ3119894

0 0


119868 0 0 Γ4119894119894

0 0


0 0 0 0



2119869119879

2119872119894119894

0



119894119894119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 le 119903

[

[

Λ1119894119895

Λ2119894119895

Λ3119894119895

lowast Λ4119894119895

Λ5119894119895


]

]

lt 0 1 le 119894 lt 119895 le 119903

(56)

where

Λ1= 119867119879

(1198751+ 1205912

31198762minus 1minus 119879

1)119867

Π1119894119895= Λ1+ 119875119860119894119895+ 119860119879

119894119895119875

Λ2= 119867119879

(1minus 119879

2)

Π2119894119895= Λ2+ 1198751198601119894119895

1= Π119879

11198771Π1

(57)







119894= 0 and








1(119905) is 120583

1 then

(119905) = [1198601+ Δ1198601(119905)] 119909 (119905) + [119860

11+ Δ11986011(119905)] 119909 (119905 minus 120591

1)

+ [11986021+ Δ11986021(119905)] (119905 minus 120591

2)

+ [11986031+ Δ11986031(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198611+ Δ1198611(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) = [1198621+ Δ1198621(119905)] 119909 (119905) + 119862

11119909 (119905 minus 120591

1) + 11986321119908 (119905)

119911 (119905) = 1198641119909 (119905) + 119864

11119909 (119905 minus 120591

1) + 1198661119906 (119905)

(58)

Plant Rule 2 if 1199091(119905) is 120583

2 then

(119905) = [1198602+ Δ1198602(119905)] 119909 (119905) + [119860

12+ Δ11986012(119905)] 119909 (119905 minus 120591

1)

+ [11986022+ Δ11986022(119905)] (119905 minus 120591

2)

+ [11986032+ Δ11986032(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198612+ Δ1198612(119905)] 119906 (119905) + 119863

12119908 (119905)

119910 (119905) = [1198622+ Δ1198622(119905)] 119909 (119905) + 119862

12119909 (119905 minus 120591

1) + 11986322119908 (119905)

119911 (119905) = 1198642119909 (119905) + 119864

12119909 (119905 minus 120591

1) + 1198662119906 (119905)

(59)

where

1198601= [

005 0


11= [

005 002

minus005 001]

11986021= [

minus0002 0008

0006 0]

11986031= [

minus005 008

006 0] 119861

1= [

001 0

003 006]

1198621= [

minus003 minus01

02 002] 119862

11= [

minus003 minus001

002 002]

11986311= [

003

006] 119863

21= [

minus05

0]

1198641= [minus02 0] 119864

11= [minus002 0]

1198661= [0 minus003]

1198602= [

minus005 0


12= [

minus005 02

minus05 001]

11986022= [

0008 minus0002

minus0003 002]

11986032= [

005 minus02

minus03 02] 119861

2= [

001 minus008

002 005]

1198622= [

003 minus01

02 002]

11986212= [

minus003 minus001

002 02] 119863

12= [

minus001

002]

11986322= [

minus03

0]

1198642= [minus001 0] 119864

12= [minus001 0]

1198662= [0 002]

(60)

and Δ119860119894(119905) Δ119860

1119894(119905) Δ119860

2119894(119905) Δ119860

3119894(119905) Δ119861

119894(119905) Δ119862

119894(119905) (119894 =


1198721= [

minus002

0] 119873

01= [minus02 01]

11987311= [minus02 02]

11987321= [minus002 001] 119873

31= [minus002 001]

11987341= [minus002 001]

11987351= [minus002 001] 119872

2= [

003

0]

11987302= 11987301 119873

12= 11987311

11987322= 11987321 119873

32= 11987331

11987342= 11987341 119873

52= 11987351

(61)



(119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119860119894+ Δ119860119894(119905)] 119909 (119905)

+ [1198601119894+ Δ1198601119894(119905)] 119909 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [119861119894+ Δ119861119894(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119862119894+ Δ119862119894(119905)] 119909 (119905)

+1198621119894119909 (119905 minus 120591

1) + 1198632119894119908 (119905)

119911 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905)) 119864

119894119909 (119905) + 119864

1119894119909 (119905 minus 120591

1) + 119866119894119906 (119905)

(62)

where

ℎ1(1199091(119905)) =

1

3


2

3

+

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

1 for 1199091gt 1

ℎ2(1199091(119905)) =

2

3


1

3

minus

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

0 for 1199091gt 1

(63)






119908 (119905) =

1

119905 + 01

119905 ge 0 (64)


119883 = [

09868 06071

06071 13948]

119884 = [

129348 01814

01814 63384]

Ω1= [



Ω2= [

82076 minus37031

minus58461 53292]

Φ1= [

31848 00758


Φ2= [

20547 01643


1205951= [


57867 minus778562]

1205952= [



(65)


1198601198961= [

873277 minus680397

1399266 minus1072394]

1198601198962= [

minus249856 158880

111983 minus122840]

1198611198961= [

109115 minus03725

165830 19936]

1198611198962= [

14642 minus00212

31940 05922]

1198621198961= [

minus160423 127577

202647 minus148918]

1198621198962= [

142596 minus104929

32304 minus11934]

(66)








x1(t)

x2(t)

0 10 20 30 40 50 60minus02

0

02

04

06

08

1

12

Time (s)

Stat

e res

pons

ex(t)


2(119905) (dashed

line)

0 10 20 30 40 50 60minus100

minus50

0

50

100

150

200

Time (s)

Con

trol i

nput

u(t)

u1(t)

u2(t)


2(119905) (dashed

line)





infin




infincontrol


Acknowledgments


y1(t)

y2(t)

0 10 20 30 40 50 60minus1

0

1

2

3

4

5

Time (s)

Mea

sure

d ou

tput

y(t)


2(119905)

(dashed line)

z1(t)

z2(t)

0 10 20 30 40 50 60Time (s)

minus40

minus20

0

20

40

60

80

100

120

Con

trolle

d ou

tput

z(t)


2(119905)

(dashed line)


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











X119879

Z +Z119879

X le X119879

YX +Z119879

Yminus1

Z (19)


1gt 0 119875

2gt 0 119876

1gt 0 119876

2gt 0 119877

1 and



LMIs hold

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Π1119894119894

Π2119894119894

11987511986021198941198751198603119894119875119863119894119894

12059111198771

119864119879

119894119894Γ119894119894

119875119872119894119894

119873119879

119894119894

lowast 1198691

0 0 0 12059111198772119864119879

1119894Γ1119894119894

0 0


0 0 0 0 Γ2119894

0 0


0 0 0 Γ3119894

0 0


119868 0 0 Γ4119894119894

0 0


0 0 0 0



2119869119879

2119872119894119894

0



119894119894119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 le 119903

[

[

Λ1119894119895

Λ2119894119895

Λ3119894119895

lowast Λ4119894119895

Λ5119894119895


]

]

lt 0 1 le 119894 lt 119895 le 119903

(20)

whereΛ1119894119895

=

[

[

[

[

Π1119894119895+ Π1119895119894

Π2119894119895+ Π2119895119894

1198751198602119894+ 11987511986021198951198751198603119894+ 1198751198603119895

lowast 21198691

0 0


0


]

]

]

]

Λ2119894119895=

[

[

[

[

[

119875119863119894119895+ 119875119863119895119894212059111198771119864119879

119894119895+ 119864119879

119895119894Γ119894119895+ Γ119895119894

0 212059111198772119864119879

1119894+ 119864119879

1119895Γ1119894119895+ Γ1119895119894

0 0 0 Γ2119894+ Γ2119895

0 0 0 Γ3119894+ Γ3119895

]

]

]

]

]

Λ3119894119895=

[

[

[

[

[

[

[

119875119872119894119895119875119872119895119894119873119879

119894119895119873119879

119895119894

0 0 119879

1119894119879

1119895

0 0 119879

2119894119879

2119895

0 0 119879

3119894119879

3119895

]

]

]

]

]

]

]

Λ4119894119895=

[

[

[

[

minus21205742

119868 0 0 Γ4119894119895+ Γ4119895119894


0 0



2

]

]

]

]

Λ5119894119895=

[

[

[

[

0 0 0 0

0 0 0 0

0 0 0 0

119869119879

2119872119894119895119869119879

21198721198951198940 0

]

]

]

]

Λ6119894119895=

[

[

[

[

[

minus120598119894119895119868 0 0 0

lowast minus120598119895119894119868 0 0


119894119895119868 0


119895119894119868

]

]

]

]

]

Υ1= 119867119879

(1198751+ 1205912

31198762minus 1198771minus 119877119879

1)119867

Π1119894119895= Υ1+ 119875119860119894119895+ 119860119879

119894119895119875

Υ2= 119867119879

(1198771minus 119877119879

2) Π

2119894119895= Υ2+ 1198751198601119894119895

Γ119894119895= 119860119879

119894119895(1198752+ 12059111198761)

Γ1119894119895= 119860119879

1119894119895(1198752+ 12059111198761) Γ

2119894= 119860119879

2119894(1198752+ 12059111198761)

Γ3119894= 119860119879

3119894(1198752+ 12059111198761) Γ

4119894119895= 119863119879

119894119895(1198752+ 12059111198761)

1198691= minus1198751+ 1198772+ 119877119879

2 119869

2= 1198752+ 12059111198761

119872119894119895= [

119872119894

0

0 119861119896119895119872119894

] 119865119894(119905) = [

119865119894(119905) 0

0 119865119894(119905)]

119873119894119895= [

11987301198941198733119894119862119896119895

1198734119894

0

] 1119894= [

11987311198940

0 0]

2119894= [

11987321198940

0 0]

3119894= [

11987331198940

0 0]

(21)


119890 (119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times [119860119894119895+ Δ119860119894119895(119905)] 119890 (119905)

+ [1198601119894119895+ Δ1198601119894(119905)] 119890 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] 119890 (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119890 (119904) 119889119904

(22)



119881 (119905) = 1198811(119905) + 119881

2(119905) + 119881

3(119905) + 119881

4(119905) + 119881

5(119905) + 119881

6(119905)

(23)

where

1198811(119905) = 119890(119905)

119879

119875119890 (119905)

1198812(119905) = int

119905

119905minus1205911

119890(119904)119879

1198751119890 (119904) 119889119904

1198813(119905) = int

119905

119905minus1205912

119890(119904)119879

1198752119890 (119904) 119889119904

1198814(119905) = int

0

minus1205911

int

119905

119905+120573

119890(120572)119879

1198761119890 (120572) 119889120572 119889120573

1198815(119905) = int

119905

119905minus1205913

[int

119905

119904

119890(120579)119879

119889120579]1198762[int

119905

119904

119890 (120579) 119889120579] 119889119904

1198816(119905) = int

1205913

0

119889119904int

119905

119905minus120573

(120579 minus 119905 + 120573) 119890(120572)119879

1198762119890 (120572) 119889120572 119889120573

(24)


(119905) = 1(119905) +

2(119905) +

3(119905) +

4(119905) +

5(119905) +

6(119905)

(25)

where

1(119905) = 2119890(119905)

119879

119875 119890 (119905)

2(119905) = 119890(119905)

119879

1198751119890 (119905) minus 119890(119905 minus 120591

1)119879

1198751119890 (119905 minus 120591

1)

3(119905) = 119890(119905)

119879

1198752119890 (119905) minus 119890(119905 minus 120591

2)119879

1198752119890 (119905 minus 120591

2)

4(119905) = 120591

1119890(119905)119879

1198761119890 (119905) minus int

119905

119905minus1205911

119890(120572)119879

1198761119890 (120572) 119889120572

5(119905) = 2int

119905

119905minus1205913

(120579 minus 119905 + 1205913) 119890(119905)119879

1198762119890 (120579) 119889120579

minus [int

119905

119905minus1205913

119890(120579)119879

119889120579]1198762[int

119905

119905minus1205913

119890 (120579) 119889120579]

6(119905) =

1

2

1205912

3119890(119905)119879

1198762119890 (119905) minus int

119905

119905minus1205913

(120579 minus 119905 + 1205913) 119890(120579)119879

1198762119890 (120579) 119889120579

(26)


2119890(119905)119879

1198762119890 (120579) le 119890(119905)

119879

1198762119890 (119905) + 119890(120579)

119879

1198762119890 (120579) (27)

Therefore

5(119905) le

1

2

1205912

3119890(119905)119879

1198762119890 (119905)

+ int

119905

119905minus1205913

(120579 minus 119905 + 1205913) 119890(120579)119879

1198762119890 (120579) 119889120579

minus [int

119905

119905minus1205913

119890(120579)119879

119889120579]1198762[int

119905

119905minus1205913

119890 (120579) 119889120579]

(28)


5(119905) +

6(119905) le 120591

2

3119890(119905)119879

1198762119890 (119905) minus 120572(119905)

119879

1198762120572 (119905) (29)

where

120572 (119905) = int

119905

119905minus1205913

119890 (120579) 119889120579 (30)

Then there holds

(119905) le 2119890(119905)119879

119875 119890 (119905)

+ 119890(119905)119879

1198751119890 (119905) minus 119890(119905 minus 120591

1)119879

1198751119890 (119905 minus 120591

1)

+ 119890(119905)119879

1198752119890 (119905) minus 119890(119905 minus 120591

2)119879

1198752119890 (119905 minus 120591

2)

+ 1205911119890(119905)119879

1198761119890 (119905) minus int

119905

119905minus1205911

119890(120572)119879

1198761119890 (120572) 119889120572

+ 1205912

3119890(119905)119879

1198762119890 (119905) minus 120572(119905)

119879

1198762120572 (119905)

+ 2119890(119905)119879

1198771int

119905

119905minus1205911

119890 (120572) 119889120572

minus 2119890(119905)119879

1198771[119890 (119905) minus 119890 (119905 minus 120591

1)]

+ 2119890(119905 minus 1205911)119879

1198772int

119905

119905minus1205911

119890 (120572) 119889120572

minus 2119890(119905 minus 1205911)119879

1198772[119890 (119905) minus 119890 (119905 minus 120591

1)]

(31)


2119890(119905)119879

119875 119890 (119905)

= 2119890(119905)119879

119875

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times [119860119894119895+ Δ119860119894119895(119905)] 119890 (119905)

+ [1198601119894119895+ Δ1198601119894(119905)] 119890 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] 119890 (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119890 (119904) 119889119904

le 2

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 119890(119905)

119879

119875119860119894119895120573 (119905)

+

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times [120576minus1

1119894119895119890(119905)119879

119875119872119894119895119872119879

119894119895119875119890 (119905)

+ 1205761119894119895120573(119905)119879

119879

119894119895119894119895120573 (119905)]

(32)

where

119860119894119895= [1198601198941198951198601119894119895

11986021198941198603119894]

120573 (119905) = [119890(119905)119879

119890(119905 minus 1205911)119879

119890(119905 minus 1205912)119879

120572(119905)119879

119890(120572)119879]

119894119895= [119873119894119895111989421198943119894]

(33)



119890(119905)119879

1198752119890 (119905)

=

1

2

119903

sum

119894=1

119903

sum

119895=1

119903

sum

119906=1

119903

sum

V=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) ℎ

119906(119904 (119905)) ℎV (119904 (119905)) 120573(119905)

119879

times [119860119894119895+119872119894119895119865119894(119905)119894119895]

119879

times 1198752[119860119906V +119872119906V119865119906 (119905) 119906V]

+ [119860119906V +119872119906V119865119906 (119905) 119906V]

119879

times 1198752[119860119894119895+119872119894119895119865119894(119905) 119894119895] 120573 (119905)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times [119860119894119895+119872119894119895119865119894(119905) 119894119895]

119879

1198752[119860119894119895+119872119894119895119865119894(119905) 119894119895]

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times [119860

119879

119894119895(119875minus1

2minus 120576minus1

2119894119895119872119894119895119872119879

119894119895)

minus1

119860119894119895+ 1205762119894119895119879

119894119895119894119895] 120573 (119905)

1205911119890(119905)119879

1198761119890 (119905)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times 119860

119879

119894119895[(12059111198761)minus1

minus 120576minus1

3119894119895119872119894119895119872119879

119894119895]

minus1

119860119894119895

+ 1205763119894119895119879

119894119895119894119895120573 (119905)

(34)


(119905) le 2119890(119905)119879

119875 119890 (119905) + 119890(119905)119879

1198752119890 (119905)

+ 1205911119890(119905)119879

1198761119890 (119905) + 119890(119905)

119879

(1198751+ 1205912

31198762) 119890 (119905)

minus 119890(119905 minus 1205911)119879

1198751119890 (119905 minus 120591

1)

minus 119890(119905 minus 1205912)119879

1198752119890 (119905 minus 120591

2) minus int

119905

119905minus1205911

119890(120572)119879

1198761119890 (120572) 119889120572

minus 120572(119905)119879

1198762120572 (119905) + 2119890(119905)

119879

1198771int

119905

119905minus1205911

119890 (120572) 119889120572

minus 2119890(119905)119879

1198771[119890 (119905) minus 119890 (119905 minus 120591

1)]

+ 2119890(119905 minus 1205911)119879

1198772int

119905

119905minus1205911

119890 (120572) 119889120572 minus 2119890(119905 minus 1205911)119879


1)]

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times [Ξ119894119895+ 119860119894119895(119875minus1

2minus 120576minus1

2119894119895119872119894119895119872119879

119894119895)

minus1

119860119894119895

+119860119894119895(120591minus1

1119876minus1

1minus 120576minus1

3119894119895119872119894119895119872119879

119894119895)

minus1

119860119894119895+ 120575119894119895119879

119894119895119894119895] 120573 (119905)

(35)

where

Ξ119894119895=

[

[

[

[

[

[

Σ1198941198951198751198601119894119895+ 1198771minus 119877119879

211987511986021198941198751198603119894

12059111198771

lowast minus1198751+ 1198772+ 119877119879

20 0 120591

11198772


0 0


0


]

]

]

]

]

]

Σ119894119895= 119875119860119894119895+ 119860119879

119894119895119875 + 1198751+ 1205912

31198762minus 1198771minus 119877119879

1+ 120576minus1

1119894119895119875119872119894119895119872119879

119894119895119875

(36)


Γ119894119895lt 0 (37)





119869 (119905) = int

119905

0

[119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904)] 119889119904 (38)



119869 (119905) = int

119905

0

[119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904) + (119904)] 119889119904 minus 119881 (119905)

le int

119905

0

[119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904) + (119904)] 119889119904

(39)


(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times 2119890(119905)119879

119875 [119860119894119895(119905) 119890 (119905) + 119860

1119894119895(119905) 119890 (119905 minus 120591

1)

+ 1198602119894(119905) 119890 (119905 minus 120591

2)

+1198603119894(119905) 120572 (119905) + 119863

119894119895119908 (119905)]


+ 119890(119905)119879

1198751119890 (119905) minus 119890(119905 minus 120591

1)119879

1198751119890 (119905 minus 120591

1)

+ 119890(119905)119879

1198752119890 (119905) minus 119890(119905 minus 120591

2)119879

1198752119890 (119905 minus 120591

2)

+ 1205911119890(119905)119879

1198761119890 (119905)

minus int

119905

119905minus1205911

119890(119905)119879

1198761119890 (120572) 119889120572 + 120591

2

3119890(119905)119879

1198762119890 (119905)

minus 120572(119905)119879

1198762120572 (119905)

+ 2119890(119905)119879

1198771int

119905

119905minus1205911

119890 (120572) 119889120572 minus 2119890(119905)119879


1)]

+ 2119890(119905 minus 1205911)119879

1198772int

119905

119905minus1205911

119890 (120572) 119889120572 minus 2119890(119905 minus 1205911)119879


1)]

(40)


119911(119904)119879

119911 (119904) =

119903

sum

119894=1

119903

sum

119895=1

119903

sum

119906=1

119903

sum

V=1

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) ℎ

119906(119904 (119904))

times ℎV (119904 (119904)) 120577(119904)119879

119864119879

119894119895119864119906V120577 (119904)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) 120577(119904)

119879

119864119879

119894119895119864119894119895120577 (119904)

(41)


119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904) + (119904)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) 120577(119904)

119879

Π1119894119895120577 (119904)

=

119903

sum

119894=1

ℎ119894(119904 (119904))

2

120577(119904)119879

Ξ1119894119894120577 (119904)

+ 2

119903

sum

119894119895=1119894lt119895

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) 120577(119904)

119879Ξ1119894119895+ Ξ1119895119894

2

120577 (119904)

(42)

where

Ξ1119894119895=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Σ1198941198951198751198601119894119895(119905) + 119877

1minus 119877119879

21198751198602119894(119905) 119875119860

3119894(119905) 119875119863

11989411989512059111198771

119864119879

119894119895119860119879

119894119895(119905) (1198752+ 12059111198761)

lowast 1198691

0 0 0 12059111198772119864119879

1119894119860119879

1119894119895(119905) (1198752+ 12059111198761)


0 0 0 0 119860119879

2119894(119905) (1198752+ 12059111198761)


0 0 0 119860119879

3119894(119905) (1198752+ 12059111198761)


119868 0 0 119863119879

119894119895(1198752+ 12059111198761)


0 0



]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

120577 (119904) = [119890(119905)119879

119890(119905 minus 1205911)119879

119890(119905 minus 1205912)119879

120572(119905)119879

119908(119905)119879

119890(120572)119879]

119879

119864119894119895= [11986411989411989511986411198940 0 0 0]

(43)



have 1199112lt 120574119908

2






119894119895 1 le 119894 le 119895 le 119903 such that


[

[

[

[

[

[

Σ119894119894Σ1119894119894

Σ2119894119894

Σ3119894119894

1198775

lowast minus119877111198772119894119894

0 0

lowast lowast 1198773119894119894

0 1198776119894119894


0


]

]

]

]

]

]

lt 0 1 le 119894 le 119903 (44)


[

[

[

[

[

[

[

[

[

[

Σ119894119895+ Σ119895119894Σ1119894119895+ Σ1119895119894

Σ2119894119895

Σ4119894119895

Σ5119894119895

Σ6119894119895

0


2119894119895

0 0 0 0


0 0 0 6119894119895



0 0



]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 lt 119895 le 119903 (45)

[


minus119868 minus119883] lt 0 (46)

where

Θ119894119895

= [

119860119894119883 + 119883119860

119879

119894+ 119861119894Ψ119895+ Ψ119879

119895119861119879

119894119860119894+ Ω119879

119894

119860119879

119894+ Ω119894

119884119860119894+ 119860119879

119894119884 + Φ

119895119862119894+ 119862119879

119894Φ119879

119895

]

minus1198771minus 119877119879

1

Θ1119894119895= [

1198601119894119883 119860

1119894

0 1198841198601119894+ Φ1198951198621119894

] + 1198771minus 119877119879

2

Θ2119894= [

1198602119894

0

11988411986021198940] Θ

3119894= [

1198603119894

0

11988411986031198940]

Θ4119894119895= [

1198631119894

1198841198631119894+ Φ1198951198632119894

] Θ5119894119895= [

119883119864119879

119894+ Ψ119879

119895119866119879

119894

119864119879

119894

]

Θ6119894119895= [

119883119860119879

119894+ Ψ119879

119895119861119879

119894Ω119879

119895

119860119879

119894119860119879

119894119884 + 119862

119879

119894Φ119879

119895

]

Θ7119894119895= [

119872119894

0

119884119872119894Φ119895119872119894

]

Θ8119894119895= [

119883119873119879

0119894+ Ψ119879

119895119873119879

4119894119883119873119879

5119894

119873119879

0119894119873119879

5119894

] 1198693119894= [

119883119864119879

1119894

119864119879

1119894

]

1198694119894119895= [

1198831198601119894

0

1198601119894

1198601119894119884 + 119862

119879

1119894Φ119879

119895

]

1198695119894= [

119883119873119879

11198940

119873119879

11198940

] 1198696119894= [

119860119879

2119894119860119879

2119894119884

0 0

]

1198697119894= [

119860119879

3119894119860119879

3119894119884

0 0

]

1198698119894119895= [119863119879

1119894119863119879

1119894119884 + 119863

119879

2119894Φ119879

119895]

1198699= Π119879

2+ Π2minus 1198752minus 12059111198761

11986910119894119895

= [

119872119894

0

119884119872119894Φ119895119872119894

]

Σ119894119895=[

[

[

Θ119894119895

Θ1119894119895

Θ2119894

lowast minus119880 + 1198772+ 119877119879

20


]

]

]

Σ1119894119895=[

[

[

Θ3119894Θ4119894119895

12059111198771

0 0 12059111198772

0 0 0

]

]

]

Σ2119894119895=[

[

[

Θ5119894119895

Θ6119894119895

Θ7119894119895

Θ8119894119895

1198693119894

1198694119894119895

0 1198695119894

0 1198696119894

0 119879

2119894

]

]

]

Σ2119894119895=

[

[

[

[

Θ5119894119895+ Θ5119895119894

Θ6119894119895+ Θ6119895119894

Θ7119894119895

Θ7119895119894

Θ8119894119895

Θ8119895119894

1198693119894+ 1198693119895

1198694119894119895+ 1198694119895119894

0 0 1198695119894

1198695119895

0 1198696119894+ 1198696119895

0 0 119879

2119894119879

2119895

]

]

]

]

Σ3119894119895=[

[

[

Π119879

11205913Π119879

1119879119894119895

0 0 0

0 0 0

]

]

]

119879119894119895= [

0

1198841198601119894+ Φ1198951198621119894

]

1198775=[

[

0 0 0

119879111987910

0 0 0

]

]

1198791= [

119909

0

] Σ4119894119895=[

[

[

1198792119894119895

1198793119894119895

1198794119894119895

1198795119894119895

0 0 0 0

0 0 0 0

]

]

]

1198792119894119895= [

(119862119894119909 minus 119862

119895119909)

119879

0

]

1198793119894119895= [

0

Φ119895minus Φ119894

] 1198794119894119895= [

(Ψ119895minus Ψ119894)

119879

0

]

1198795119894119895= [

0

119884119861119894minus 119884119861119895

]

Σ5119894119895=[

[

Π119879

11205913Π119879

11198791198941198950

0 0 0 1198791

0 0 0 0

]

]


Σ6119894119895=[

[

119879119895119894

0 11987941198792

0 11987910 0

0 0 0 0

]

]

11987711= diag (119876

21205742

119868 119881) 1198772119894119895=[

[

0 11986971198940 119879

3119894

0 1198698119894119895

0 0

0 0 0 0

]

]

2119894119895=[

[

0 1198697119894+ 1198697119895

0 0 119879

3119894119879

3119895

0 1198698119894119895+ 1198698119895119894

0 0 0 0

0 0 0 0 0 0

]

]

1198773119894119895=

[

[

[

[

minus119868 0 0 0

lowast minus119869911986910119894119895

0



119894119895119868

]

]

]

]

3119894119895=

[

[

[

[

[

[

[

[

[

minus2119868 0 0 0 0 0

lowast minus2119869911986910119894119895

11986910119895119894

0 0




119894119895119868 0


119895119894119868

]

]

]

]

]

]

]

]

]

1198774= diag (119875minus1

1119876minus1

2119868)

1198776119894119895=

[

[

[

[

0 0 0

0 0 1198796119894119895

0 0 0

0 0 0

]

]

]

]

6119894119895=

[

[

[

[

[

[

[

[

0 0

1198795119894119895

1198793119894119895

0 0

0 0

0 0

0 0

]

]

]

]

]

]

]

]

1198796119894119895= [

0

119884119860119879

1119894+ Φ1198951198621119894

]

1198777= diag (119868 119868 119868) 119877

8= diag (1

2

119875minus1

1

1

2

119876minus1

2119868 119868)

Π1= [

119883 119868

119883 0] Π

2= [

119883 119868

119883 0]

(47)


119860119870119894= (119883minus1

minus 119884)

minus1

(Ω119894minus 119884119860119894119883 minus 119884119861

119894Ψ119894minus Φ119894119862119894119883)119883minus1

119861119870119894= (119883minus1

minus 119884)

minus1

Φ119894 119862119870119894= Ψ119894119883minus1

1 le 119894 le 119903

(48)


[

[

Σ119894119894Σ1119894119894

Σ2119894119894

lowast minus119877111198772119894119894

lowast lowast 1198773119894119894

]

]

lt 0 1 le 119894 le 119903

[

[

[

Σ119894119895Σ1119894119895+ Σ1119895119894

Σ2119894119895


2119894119895


]

]

]

lt 0 1 le 119894 lt 119895 le 119903

(49)

where

Σ119894119895=

[

[

[

[

Θ119894119895+ Π119879

11198751Π1+ (1205913Π1)119879

1198762(1205913Π1) Θ1119894119895+ [

0 0

(1198841198601119894+ Φ1198951198621119894)119883 0

] Θ2119894

lowast minus119880 + 1198772+ 119877119879

20


]

]

]

]

Σ2119894119895=

[

[

[

[

Θ5119894119895

Θ6119894119895

Θ7119894119895

Θ8119894119895

1198693119894

1198694119894119895+ [

0 119883 (1198601119894119884 + 119862

119879

1119894Φ119879

119895)

0 0

] 0 1198695119894

0 1198696119894

0 119879

2119894

]

]

]

]

Σ119894119895= Σ119894119895+ Σ119895119894+

[

[

[

[

[

[

0 (119862119894119883 minus 119862

119895119883)

119879

(Φ119895minus Φ119894)

119879

+ (Ψ119895minus Ψ119894)

119879

(119884119861119894minus 119884119861119895)

119879

0 0 0

lowast 0 0 0 0

lowast lowast 0 0 0



]

]

]

]

]

]

Σ2119894119895= Σ2119894119895+[

[

0 (Ψ119879

119895minus Ψ119879

119894) (119861119879

119894119884 minus 119861

119879

119895119884) + (119883119862

119879

119894minus 119883119862119879

119895) (Φ119879

119895minus Φ119879

119894) 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]

]

(50)


Note that

Π119879

2+ Π2minus 1198752minus 12059111198761le Π119879

2(1198752+ 12059111198761)minus1

Π2 (51)


[

[

Σ119894119894Σ1119894119894

Σ2119894119894

lowast minus119877111198772119894119894


]

]

lt 0 1 le 119894 le 119903

[

[

[

Σ119894119895Σ1119894119895+ Σ1119895119894

Σ2119894119895


2119894119895

lowast lowast 1198773119894119895

]

]

]

lt 0 1 le 119894 lt 119895 le 119903

(52)

where

3119894119895=

[

[

[

[

[

minus119868 0 0 0


2(1198752+ 12059111198761)minus1

Π211986910119894119895

0



119894119895119868

]

]

]

]

]

1198773119894119895=

[

[

[

[

[

[

[

[

[

minus2119868 0 0 0 0 0


2(1198752+ 12059111198761)minus1

Π211986910119894119895

11986910119895119894

0 0




119894119895119868 0


119895119894119868

]

]

]

]

]

]

]

]

]

(53)

Now set = Π

2Πminus1

1 (54)



1 119868 119868 119868 Π

minus119879

1 119868 Πminus119879

2 119868 119868) (55)


[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Π1119894119894

Π2119894119894

11987511986021198941198751198603119894119875119863119894119894

12059111

119864119879

119894119894Γ119894119894

119875119872119894119894

119873119879

119894119894

lowast 1198691

0 0 0 12059112119864119879

1119894Γ1119894119894

0 0


0 0 0 0 Γ2119894

0 0


0 0 0 Γ3119894

0 0


119868 0 0 Γ4119894119894

0 0


0 0 0 0



2119869119879

2119872119894119894

0



119894119894119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 le 119903

[

[

Λ1119894119895

Λ2119894119895

Λ3119894119895

lowast Λ4119894119895

Λ5119894119895


]

]

lt 0 1 le 119894 lt 119895 le 119903

(56)

where

Λ1= 119867119879

(1198751+ 1205912

31198762minus 1minus 119879

1)119867

Π1119894119895= Λ1+ 119875119860119894119895+ 119860119879

119894119895119875

Λ2= 119867119879

(1minus 119879

2)

Π2119894119895= Λ2+ 1198751198601119894119895

1= Π119879

11198771Π1

(57)







119894= 0 and








1(119905) is 120583

1 then

(119905) = [1198601+ Δ1198601(119905)] 119909 (119905) + [119860

11+ Δ11986011(119905)] 119909 (119905 minus 120591

1)

+ [11986021+ Δ11986021(119905)] (119905 minus 120591

2)

+ [11986031+ Δ11986031(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198611+ Δ1198611(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) = [1198621+ Δ1198621(119905)] 119909 (119905) + 119862

11119909 (119905 minus 120591

1) + 11986321119908 (119905)

119911 (119905) = 1198641119909 (119905) + 119864

11119909 (119905 minus 120591

1) + 1198661119906 (119905)

(58)

Plant Rule 2 if 1199091(119905) is 120583

2 then

(119905) = [1198602+ Δ1198602(119905)] 119909 (119905) + [119860

12+ Δ11986012(119905)] 119909 (119905 minus 120591

1)

+ [11986022+ Δ11986022(119905)] (119905 minus 120591

2)

+ [11986032+ Δ11986032(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198612+ Δ1198612(119905)] 119906 (119905) + 119863

12119908 (119905)

119910 (119905) = [1198622+ Δ1198622(119905)] 119909 (119905) + 119862

12119909 (119905 minus 120591

1) + 11986322119908 (119905)

119911 (119905) = 1198642119909 (119905) + 119864

12119909 (119905 minus 120591

1) + 1198662119906 (119905)

(59)

where

1198601= [

005 0


11= [

005 002

minus005 001]

11986021= [

minus0002 0008

0006 0]

11986031= [

minus005 008

006 0] 119861

1= [

001 0

003 006]

1198621= [

minus003 minus01

02 002] 119862

11= [

minus003 minus001

002 002]

11986311= [

003

006] 119863

21= [

minus05

0]

1198641= [minus02 0] 119864

11= [minus002 0]

1198661= [0 minus003]

1198602= [

minus005 0


12= [

minus005 02

minus05 001]

11986022= [

0008 minus0002

minus0003 002]

11986032= [

005 minus02

minus03 02] 119861

2= [

001 minus008

002 005]

1198622= [

003 minus01

02 002]

11986212= [

minus003 minus001

002 02] 119863

12= [

minus001

002]

11986322= [

minus03

0]

1198642= [minus001 0] 119864

12= [minus001 0]

1198662= [0 002]

(60)

and Δ119860119894(119905) Δ119860

1119894(119905) Δ119860

2119894(119905) Δ119860

3119894(119905) Δ119861

119894(119905) Δ119862

119894(119905) (119894 =


1198721= [

minus002

0] 119873

01= [minus02 01]

11987311= [minus02 02]

11987321= [minus002 001] 119873

31= [minus002 001]

11987341= [minus002 001]

11987351= [minus002 001] 119872

2= [

003

0]

11987302= 11987301 119873

12= 11987311

11987322= 11987321 119873

32= 11987331

11987342= 11987341 119873

52= 11987351

(61)



(119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119860119894+ Δ119860119894(119905)] 119909 (119905)

+ [1198601119894+ Δ1198601119894(119905)] 119909 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [119861119894+ Δ119861119894(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119862119894+ Δ119862119894(119905)] 119909 (119905)

+1198621119894119909 (119905 minus 120591

1) + 1198632119894119908 (119905)

119911 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905)) 119864

119894119909 (119905) + 119864

1119894119909 (119905 minus 120591

1) + 119866119894119906 (119905)

(62)

where

ℎ1(1199091(119905)) =

1

3


2

3

+

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

1 for 1199091gt 1

ℎ2(1199091(119905)) =

2

3


1

3

minus

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

0 for 1199091gt 1

(63)






119908 (119905) =

1

119905 + 01

119905 ge 0 (64)


119883 = [

09868 06071

06071 13948]

119884 = [

129348 01814

01814 63384]

Ω1= [



Ω2= [

82076 minus37031

minus58461 53292]

Φ1= [

31848 00758


Φ2= [

20547 01643


1205951= [


57867 minus778562]

1205952= [



(65)


1198601198961= [

873277 minus680397

1399266 minus1072394]

1198601198962= [

minus249856 158880

111983 minus122840]

1198611198961= [

109115 minus03725

165830 19936]

1198611198962= [

14642 minus00212

31940 05922]

1198621198961= [

minus160423 127577

202647 minus148918]

1198621198962= [

142596 minus104929

32304 minus11934]

(66)








x1(t)

x2(t)

0 10 20 30 40 50 60minus02

0

02

04

06

08

1

12

Time (s)

Stat

e res

pons

ex(t)


2(119905) (dashed

line)

0 10 20 30 40 50 60minus100

minus50

0

50

100

150

200

Time (s)

Con

trol i

nput

u(t)

u1(t)

u2(t)


2(119905) (dashed

line)





infin




infincontrol


Acknowledgments


y1(t)

y2(t)

0 10 20 30 40 50 60minus1

0

1

2

3

4

5

Time (s)

Mea

sure

d ou

tput

y(t)


2(119905)

(dashed line)

z1(t)

z2(t)

0 10 20 30 40 50 60Time (s)

minus40

minus20

0

20

40

60

80

100

120

Con

trolle

d ou

tput

z(t)


2(119905)

(dashed line)


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119881 (119905) = 1198811(119905) + 119881

2(119905) + 119881

3(119905) + 119881

4(119905) + 119881

5(119905) + 119881

6(119905)

(23)

where

1198811(119905) = 119890(119905)

119879

119875119890 (119905)

1198812(119905) = int

119905

119905minus1205911

119890(119904)119879

1198751119890 (119904) 119889119904

1198813(119905) = int

119905

119905minus1205912

119890(119904)119879

1198752119890 (119904) 119889119904

1198814(119905) = int

0

minus1205911

int

119905

119905+120573

119890(120572)119879

1198761119890 (120572) 119889120572 119889120573

1198815(119905) = int

119905

119905minus1205913

[int

119905

119904

119890(120579)119879

119889120579]1198762[int

119905

119904

119890 (120579) 119889120579] 119889119904

1198816(119905) = int

1205913

0

119889119904int

119905

119905minus120573

(120579 minus 119905 + 120573) 119890(120572)119879

1198762119890 (120572) 119889120572 119889120573

(24)


(119905) = 1(119905) +

2(119905) +

3(119905) +

4(119905) +

5(119905) +

6(119905)

(25)

where

1(119905) = 2119890(119905)

119879

119875 119890 (119905)

2(119905) = 119890(119905)

119879

1198751119890 (119905) minus 119890(119905 minus 120591

1)119879

1198751119890 (119905 minus 120591

1)

3(119905) = 119890(119905)

119879

1198752119890 (119905) minus 119890(119905 minus 120591

2)119879

1198752119890 (119905 minus 120591

2)

4(119905) = 120591

1119890(119905)119879

1198761119890 (119905) minus int

119905

119905minus1205911

119890(120572)119879

1198761119890 (120572) 119889120572

5(119905) = 2int

119905

119905minus1205913

(120579 minus 119905 + 1205913) 119890(119905)119879

1198762119890 (120579) 119889120579

minus [int

119905

119905minus1205913

119890(120579)119879

119889120579]1198762[int

119905

119905minus1205913

119890 (120579) 119889120579]

6(119905) =

1

2

1205912

3119890(119905)119879

1198762119890 (119905) minus int

119905

119905minus1205913

(120579 minus 119905 + 1205913) 119890(120579)119879

1198762119890 (120579) 119889120579

(26)


2119890(119905)119879

1198762119890 (120579) le 119890(119905)

119879

1198762119890 (119905) + 119890(120579)

119879

1198762119890 (120579) (27)

Therefore

5(119905) le

1

2

1205912

3119890(119905)119879

1198762119890 (119905)

+ int

119905

119905minus1205913

(120579 minus 119905 + 1205913) 119890(120579)119879

1198762119890 (120579) 119889120579

minus [int

119905

119905minus1205913

119890(120579)119879

119889120579]1198762[int

119905

119905minus1205913

119890 (120579) 119889120579]

(28)


5(119905) +

6(119905) le 120591

2

3119890(119905)119879

1198762119890 (119905) minus 120572(119905)

119879

1198762120572 (119905) (29)

where

120572 (119905) = int

119905

119905minus1205913

119890 (120579) 119889120579 (30)

Then there holds

(119905) le 2119890(119905)119879

119875 119890 (119905)

+ 119890(119905)119879

1198751119890 (119905) minus 119890(119905 minus 120591

1)119879

1198751119890 (119905 minus 120591

1)

+ 119890(119905)119879

1198752119890 (119905) minus 119890(119905 minus 120591

2)119879

1198752119890 (119905 minus 120591

2)

+ 1205911119890(119905)119879

1198761119890 (119905) minus int

119905

119905minus1205911

119890(120572)119879

1198761119890 (120572) 119889120572

+ 1205912

3119890(119905)119879

1198762119890 (119905) minus 120572(119905)

119879

1198762120572 (119905)

+ 2119890(119905)119879

1198771int

119905

119905minus1205911

119890 (120572) 119889120572

minus 2119890(119905)119879

1198771[119890 (119905) minus 119890 (119905 minus 120591

1)]

+ 2119890(119905 minus 1205911)119879

1198772int

119905

119905minus1205911

119890 (120572) 119889120572

minus 2119890(119905 minus 1205911)119879

1198772[119890 (119905) minus 119890 (119905 minus 120591

1)]

(31)


2119890(119905)119879

119875 119890 (119905)

= 2119890(119905)119879

119875

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times [119860119894119895+ Δ119860119894119895(119905)] 119890 (119905)

+ [1198601119894119895+ Δ1198601119894(119905)] 119890 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] 119890 (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119890 (119904) 119889119904

le 2

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 119890(119905)

119879

119875119860119894119895120573 (119905)

+

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times [120576minus1

1119894119895119890(119905)119879

119875119872119894119895119872119879

119894119895119875119890 (119905)

+ 1205761119894119895120573(119905)119879

119879

119894119895119894119895120573 (119905)]

(32)

where

119860119894119895= [1198601198941198951198601119894119895

11986021198941198603119894]

120573 (119905) = [119890(119905)119879

119890(119905 minus 1205911)119879

119890(119905 minus 1205912)119879

120572(119905)119879

119890(120572)119879]

119894119895= [119873119894119895111989421198943119894]

(33)



119890(119905)119879

1198752119890 (119905)

=

1

2

119903

sum

119894=1

119903

sum

119895=1

119903

sum

119906=1

119903

sum

V=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) ℎ

119906(119904 (119905)) ℎV (119904 (119905)) 120573(119905)

119879

times [119860119894119895+119872119894119895119865119894(119905)119894119895]

119879

times 1198752[119860119906V +119872119906V119865119906 (119905) 119906V]

+ [119860119906V +119872119906V119865119906 (119905) 119906V]

119879

times 1198752[119860119894119895+119872119894119895119865119894(119905) 119894119895] 120573 (119905)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times [119860119894119895+119872119894119895119865119894(119905) 119894119895]

119879

1198752[119860119894119895+119872119894119895119865119894(119905) 119894119895]

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times [119860

119879

119894119895(119875minus1

2minus 120576minus1

2119894119895119872119894119895119872119879

119894119895)

minus1

119860119894119895+ 1205762119894119895119879

119894119895119894119895] 120573 (119905)

1205911119890(119905)119879

1198761119890 (119905)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times 119860

119879

119894119895[(12059111198761)minus1

minus 120576minus1

3119894119895119872119894119895119872119879

119894119895]

minus1

119860119894119895

+ 1205763119894119895119879

119894119895119894119895120573 (119905)

(34)


(119905) le 2119890(119905)119879

119875 119890 (119905) + 119890(119905)119879

1198752119890 (119905)

+ 1205911119890(119905)119879

1198761119890 (119905) + 119890(119905)

119879

(1198751+ 1205912

31198762) 119890 (119905)

minus 119890(119905 minus 1205911)119879

1198751119890 (119905 minus 120591

1)

minus 119890(119905 minus 1205912)119879

1198752119890 (119905 minus 120591

2) minus int

119905

119905minus1205911

119890(120572)119879

1198761119890 (120572) 119889120572

minus 120572(119905)119879

1198762120572 (119905) + 2119890(119905)

119879

1198771int

119905

119905minus1205911

119890 (120572) 119889120572

minus 2119890(119905)119879

1198771[119890 (119905) minus 119890 (119905 minus 120591

1)]

+ 2119890(119905 minus 1205911)119879

1198772int

119905

119905minus1205911

119890 (120572) 119889120572 minus 2119890(119905 minus 1205911)119879


1)]

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times [Ξ119894119895+ 119860119894119895(119875minus1

2minus 120576minus1

2119894119895119872119894119895119872119879

119894119895)

minus1

119860119894119895

+119860119894119895(120591minus1

1119876minus1

1minus 120576minus1

3119894119895119872119894119895119872119879

119894119895)

minus1

119860119894119895+ 120575119894119895119879

119894119895119894119895] 120573 (119905)

(35)

where

Ξ119894119895=

[

[

[

[

[

[

Σ1198941198951198751198601119894119895+ 1198771minus 119877119879

211987511986021198941198751198603119894

12059111198771

lowast minus1198751+ 1198772+ 119877119879

20 0 120591

11198772


0 0


0


]

]

]

]

]

]

Σ119894119895= 119875119860119894119895+ 119860119879

119894119895119875 + 1198751+ 1205912

31198762minus 1198771minus 119877119879

1+ 120576minus1

1119894119895119875119872119894119895119872119879

119894119895119875

(36)


Γ119894119895lt 0 (37)





119869 (119905) = int

119905

0

[119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904)] 119889119904 (38)



119869 (119905) = int

119905

0

[119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904) + (119904)] 119889119904 minus 119881 (119905)

le int

119905

0

[119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904) + (119904)] 119889119904

(39)


(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times 2119890(119905)119879

119875 [119860119894119895(119905) 119890 (119905) + 119860

1119894119895(119905) 119890 (119905 minus 120591

1)

+ 1198602119894(119905) 119890 (119905 minus 120591

2)

+1198603119894(119905) 120572 (119905) + 119863

119894119895119908 (119905)]


+ 119890(119905)119879

1198751119890 (119905) minus 119890(119905 minus 120591

1)119879

1198751119890 (119905 minus 120591

1)

+ 119890(119905)119879

1198752119890 (119905) minus 119890(119905 minus 120591

2)119879

1198752119890 (119905 minus 120591

2)

+ 1205911119890(119905)119879

1198761119890 (119905)

minus int

119905

119905minus1205911

119890(119905)119879

1198761119890 (120572) 119889120572 + 120591

2

3119890(119905)119879

1198762119890 (119905)

minus 120572(119905)119879

1198762120572 (119905)

+ 2119890(119905)119879

1198771int

119905

119905minus1205911

119890 (120572) 119889120572 minus 2119890(119905)119879


1)]

+ 2119890(119905 minus 1205911)119879

1198772int

119905

119905minus1205911

119890 (120572) 119889120572 minus 2119890(119905 minus 1205911)119879


1)]

(40)


119911(119904)119879

119911 (119904) =

119903

sum

119894=1

119903

sum

119895=1

119903

sum

119906=1

119903

sum

V=1

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) ℎ

119906(119904 (119904))

times ℎV (119904 (119904)) 120577(119904)119879

119864119879

119894119895119864119906V120577 (119904)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) 120577(119904)

119879

119864119879

119894119895119864119894119895120577 (119904)

(41)


119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904) + (119904)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) 120577(119904)

119879

Π1119894119895120577 (119904)

=

119903

sum

119894=1

ℎ119894(119904 (119904))

2

120577(119904)119879

Ξ1119894119894120577 (119904)

+ 2

119903

sum

119894119895=1119894lt119895

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) 120577(119904)

119879Ξ1119894119895+ Ξ1119895119894

2

120577 (119904)

(42)

where

Ξ1119894119895=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Σ1198941198951198751198601119894119895(119905) + 119877

1minus 119877119879

21198751198602119894(119905) 119875119860

3119894(119905) 119875119863

11989411989512059111198771

119864119879

119894119895119860119879

119894119895(119905) (1198752+ 12059111198761)

lowast 1198691

0 0 0 12059111198772119864119879

1119894119860119879

1119894119895(119905) (1198752+ 12059111198761)


0 0 0 0 119860119879

2119894(119905) (1198752+ 12059111198761)


0 0 0 119860119879

3119894(119905) (1198752+ 12059111198761)


119868 0 0 119863119879

119894119895(1198752+ 12059111198761)


0 0



]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

120577 (119904) = [119890(119905)119879

119890(119905 minus 1205911)119879

119890(119905 minus 1205912)119879

120572(119905)119879

119908(119905)119879

119890(120572)119879]

119879

119864119894119895= [11986411989411989511986411198940 0 0 0]

(43)



have 1199112lt 120574119908

2






119894119895 1 le 119894 le 119895 le 119903 such that


[

[

[

[

[

[

Σ119894119894Σ1119894119894

Σ2119894119894

Σ3119894119894

1198775

lowast minus119877111198772119894119894

0 0

lowast lowast 1198773119894119894

0 1198776119894119894


0


]

]

]

]

]

]

lt 0 1 le 119894 le 119903 (44)


[

[

[

[

[

[

[

[

[

[

Σ119894119895+ Σ119895119894Σ1119894119895+ Σ1119895119894

Σ2119894119895

Σ4119894119895

Σ5119894119895

Σ6119894119895

0


2119894119895

0 0 0 0


0 0 0 6119894119895



0 0



]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 lt 119895 le 119903 (45)

[


minus119868 minus119883] lt 0 (46)

where

Θ119894119895

= [

119860119894119883 + 119883119860

119879

119894+ 119861119894Ψ119895+ Ψ119879

119895119861119879

119894119860119894+ Ω119879

119894

119860119879

119894+ Ω119894

119884119860119894+ 119860119879

119894119884 + Φ

119895119862119894+ 119862119879

119894Φ119879

119895

]

minus1198771minus 119877119879

1

Θ1119894119895= [

1198601119894119883 119860

1119894

0 1198841198601119894+ Φ1198951198621119894

] + 1198771minus 119877119879

2

Θ2119894= [

1198602119894

0

11988411986021198940] Θ

3119894= [

1198603119894

0

11988411986031198940]

Θ4119894119895= [

1198631119894

1198841198631119894+ Φ1198951198632119894

] Θ5119894119895= [

119883119864119879

119894+ Ψ119879

119895119866119879

119894

119864119879

119894

]

Θ6119894119895= [

119883119860119879

119894+ Ψ119879

119895119861119879

119894Ω119879

119895

119860119879

119894119860119879

119894119884 + 119862

119879

119894Φ119879

119895

]

Θ7119894119895= [

119872119894

0

119884119872119894Φ119895119872119894

]

Θ8119894119895= [

119883119873119879

0119894+ Ψ119879

119895119873119879

4119894119883119873119879

5119894

119873119879

0119894119873119879

5119894

] 1198693119894= [

119883119864119879

1119894

119864119879

1119894

]

1198694119894119895= [

1198831198601119894

0

1198601119894

1198601119894119884 + 119862

119879

1119894Φ119879

119895

]

1198695119894= [

119883119873119879

11198940

119873119879

11198940

] 1198696119894= [

119860119879

2119894119860119879

2119894119884

0 0

]

1198697119894= [

119860119879

3119894119860119879

3119894119884

0 0

]

1198698119894119895= [119863119879

1119894119863119879

1119894119884 + 119863

119879

2119894Φ119879

119895]

1198699= Π119879

2+ Π2minus 1198752minus 12059111198761

11986910119894119895

= [

119872119894

0

119884119872119894Φ119895119872119894

]

Σ119894119895=[

[

[

Θ119894119895

Θ1119894119895

Θ2119894

lowast minus119880 + 1198772+ 119877119879

20


]

]

]

Σ1119894119895=[

[

[

Θ3119894Θ4119894119895

12059111198771

0 0 12059111198772

0 0 0

]

]

]

Σ2119894119895=[

[

[

Θ5119894119895

Θ6119894119895

Θ7119894119895

Θ8119894119895

1198693119894

1198694119894119895

0 1198695119894

0 1198696119894

0 119879

2119894

]

]

]

Σ2119894119895=

[

[

[

[

Θ5119894119895+ Θ5119895119894

Θ6119894119895+ Θ6119895119894

Θ7119894119895

Θ7119895119894

Θ8119894119895

Θ8119895119894

1198693119894+ 1198693119895

1198694119894119895+ 1198694119895119894

0 0 1198695119894

1198695119895

0 1198696119894+ 1198696119895

0 0 119879

2119894119879

2119895

]

]

]

]

Σ3119894119895=[

[

[

Π119879

11205913Π119879

1119879119894119895

0 0 0

0 0 0

]

]

]

119879119894119895= [

0

1198841198601119894+ Φ1198951198621119894

]

1198775=[

[

0 0 0

119879111987910

0 0 0

]

]

1198791= [

119909

0

] Σ4119894119895=[

[

[

1198792119894119895

1198793119894119895

1198794119894119895

1198795119894119895

0 0 0 0

0 0 0 0

]

]

]

1198792119894119895= [

(119862119894119909 minus 119862

119895119909)

119879

0

]

1198793119894119895= [

0

Φ119895minus Φ119894

] 1198794119894119895= [

(Ψ119895minus Ψ119894)

119879

0

]

1198795119894119895= [

0

119884119861119894minus 119884119861119895

]

Σ5119894119895=[

[

Π119879

11205913Π119879

11198791198941198950

0 0 0 1198791

0 0 0 0

]

]


Σ6119894119895=[

[

119879119895119894

0 11987941198792

0 11987910 0

0 0 0 0

]

]

11987711= diag (119876

21205742

119868 119881) 1198772119894119895=[

[

0 11986971198940 119879

3119894

0 1198698119894119895

0 0

0 0 0 0

]

]

2119894119895=[

[

0 1198697119894+ 1198697119895

0 0 119879

3119894119879

3119895

0 1198698119894119895+ 1198698119895119894

0 0 0 0

0 0 0 0 0 0

]

]

1198773119894119895=

[

[

[

[

minus119868 0 0 0

lowast minus119869911986910119894119895

0



119894119895119868

]

]

]

]

3119894119895=

[

[

[

[

[

[

[

[

[

minus2119868 0 0 0 0 0

lowast minus2119869911986910119894119895

11986910119895119894

0 0




119894119895119868 0


119895119894119868

]

]

]

]

]

]

]

]

]

1198774= diag (119875minus1

1119876minus1

2119868)

1198776119894119895=

[

[

[

[

0 0 0

0 0 1198796119894119895

0 0 0

0 0 0

]

]

]

]

6119894119895=

[

[

[

[

[

[

[

[

0 0

1198795119894119895

1198793119894119895

0 0

0 0

0 0

0 0

]

]

]

]

]

]

]

]

1198796119894119895= [

0

119884119860119879

1119894+ Φ1198951198621119894

]

1198777= diag (119868 119868 119868) 119877

8= diag (1

2

119875minus1

1

1

2

119876minus1

2119868 119868)

Π1= [

119883 119868

119883 0] Π

2= [

119883 119868

119883 0]

(47)


119860119870119894= (119883minus1

minus 119884)

minus1

(Ω119894minus 119884119860119894119883 minus 119884119861

119894Ψ119894minus Φ119894119862119894119883)119883minus1

119861119870119894= (119883minus1

minus 119884)

minus1

Φ119894 119862119870119894= Ψ119894119883minus1

1 le 119894 le 119903

(48)


[

[

Σ119894119894Σ1119894119894

Σ2119894119894

lowast minus119877111198772119894119894

lowast lowast 1198773119894119894

]

]

lt 0 1 le 119894 le 119903

[

[

[

Σ119894119895Σ1119894119895+ Σ1119895119894

Σ2119894119895


2119894119895


]

]

]

lt 0 1 le 119894 lt 119895 le 119903

(49)

where

Σ119894119895=

[

[

[

[

Θ119894119895+ Π119879

11198751Π1+ (1205913Π1)119879

1198762(1205913Π1) Θ1119894119895+ [

0 0

(1198841198601119894+ Φ1198951198621119894)119883 0

] Θ2119894

lowast minus119880 + 1198772+ 119877119879

20


]

]

]

]

Σ2119894119895=

[

[

[

[

Θ5119894119895

Θ6119894119895

Θ7119894119895

Θ8119894119895

1198693119894

1198694119894119895+ [

0 119883 (1198601119894119884 + 119862

119879

1119894Φ119879

119895)

0 0

] 0 1198695119894

0 1198696119894

0 119879

2119894

]

]

]

]

Σ119894119895= Σ119894119895+ Σ119895119894+

[

[

[

[

[

[

0 (119862119894119883 minus 119862

119895119883)

119879

(Φ119895minus Φ119894)

119879

+ (Ψ119895minus Ψ119894)

119879

(119884119861119894minus 119884119861119895)

119879

0 0 0

lowast 0 0 0 0

lowast lowast 0 0 0



]

]

]

]

]

]

Σ2119894119895= Σ2119894119895+[

[

0 (Ψ119879

119895minus Ψ119879

119894) (119861119879

119894119884 minus 119861

119879

119895119884) + (119883119862

119879

119894minus 119883119862119879

119895) (Φ119879

119895minus Φ119879

119894) 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]

]

(50)


Note that

Π119879

2+ Π2minus 1198752minus 12059111198761le Π119879

2(1198752+ 12059111198761)minus1

Π2 (51)


[

[

Σ119894119894Σ1119894119894

Σ2119894119894

lowast minus119877111198772119894119894


]

]

lt 0 1 le 119894 le 119903

[

[

[

Σ119894119895Σ1119894119895+ Σ1119895119894

Σ2119894119895


2119894119895

lowast lowast 1198773119894119895

]

]

]

lt 0 1 le 119894 lt 119895 le 119903

(52)

where

3119894119895=

[

[

[

[

[

minus119868 0 0 0


2(1198752+ 12059111198761)minus1

Π211986910119894119895

0



119894119895119868

]

]

]

]

]

1198773119894119895=

[

[

[

[

[

[

[

[

[

minus2119868 0 0 0 0 0


2(1198752+ 12059111198761)minus1

Π211986910119894119895

11986910119895119894

0 0




119894119895119868 0


119895119894119868

]

]

]

]

]

]

]

]

]

(53)

Now set = Π

2Πminus1

1 (54)



1 119868 119868 119868 Π

minus119879

1 119868 Πminus119879

2 119868 119868) (55)


[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Π1119894119894

Π2119894119894

11987511986021198941198751198603119894119875119863119894119894

12059111

119864119879

119894119894Γ119894119894

119875119872119894119894

119873119879

119894119894

lowast 1198691

0 0 0 12059112119864119879

1119894Γ1119894119894

0 0


0 0 0 0 Γ2119894

0 0


0 0 0 Γ3119894

0 0


119868 0 0 Γ4119894119894

0 0


0 0 0 0



2119869119879

2119872119894119894

0



119894119894119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 le 119903

[

[

Λ1119894119895

Λ2119894119895

Λ3119894119895

lowast Λ4119894119895

Λ5119894119895


]

]

lt 0 1 le 119894 lt 119895 le 119903

(56)

where

Λ1= 119867119879

(1198751+ 1205912

31198762minus 1minus 119879

1)119867

Π1119894119895= Λ1+ 119875119860119894119895+ 119860119879

119894119895119875

Λ2= 119867119879

(1minus 119879

2)

Π2119894119895= Λ2+ 1198751198601119894119895

1= Π119879

11198771Π1

(57)







119894= 0 and








1(119905) is 120583

1 then

(119905) = [1198601+ Δ1198601(119905)] 119909 (119905) + [119860

11+ Δ11986011(119905)] 119909 (119905 minus 120591

1)

+ [11986021+ Δ11986021(119905)] (119905 minus 120591

2)

+ [11986031+ Δ11986031(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198611+ Δ1198611(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) = [1198621+ Δ1198621(119905)] 119909 (119905) + 119862

11119909 (119905 minus 120591

1) + 11986321119908 (119905)

119911 (119905) = 1198641119909 (119905) + 119864

11119909 (119905 minus 120591

1) + 1198661119906 (119905)

(58)

Plant Rule 2 if 1199091(119905) is 120583

2 then

(119905) = [1198602+ Δ1198602(119905)] 119909 (119905) + [119860

12+ Δ11986012(119905)] 119909 (119905 minus 120591

1)

+ [11986022+ Δ11986022(119905)] (119905 minus 120591

2)

+ [11986032+ Δ11986032(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198612+ Δ1198612(119905)] 119906 (119905) + 119863

12119908 (119905)

119910 (119905) = [1198622+ Δ1198622(119905)] 119909 (119905) + 119862

12119909 (119905 minus 120591

1) + 11986322119908 (119905)

119911 (119905) = 1198642119909 (119905) + 119864

12119909 (119905 minus 120591

1) + 1198662119906 (119905)

(59)

where

1198601= [

005 0


11= [

005 002

minus005 001]

11986021= [

minus0002 0008

0006 0]

11986031= [

minus005 008

006 0] 119861

1= [

001 0

003 006]

1198621= [

minus003 minus01

02 002] 119862

11= [

minus003 minus001

002 002]

11986311= [

003

006] 119863

21= [

minus05

0]

1198641= [minus02 0] 119864

11= [minus002 0]

1198661= [0 minus003]

1198602= [

minus005 0


12= [

minus005 02

minus05 001]

11986022= [

0008 minus0002

minus0003 002]

11986032= [

005 minus02

minus03 02] 119861

2= [

001 minus008

002 005]

1198622= [

003 minus01

02 002]

11986212= [

minus003 minus001

002 02] 119863

12= [

minus001

002]

11986322= [

minus03

0]

1198642= [minus001 0] 119864

12= [minus001 0]

1198662= [0 002]

(60)

and Δ119860119894(119905) Δ119860

1119894(119905) Δ119860

2119894(119905) Δ119860

3119894(119905) Δ119861

119894(119905) Δ119862

119894(119905) (119894 =


1198721= [

minus002

0] 119873

01= [minus02 01]

11987311= [minus02 02]

11987321= [minus002 001] 119873

31= [minus002 001]

11987341= [minus002 001]

11987351= [minus002 001] 119872

2= [

003

0]

11987302= 11987301 119873

12= 11987311

11987322= 11987321 119873

32= 11987331

11987342= 11987341 119873

52= 11987351

(61)



(119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119860119894+ Δ119860119894(119905)] 119909 (119905)

+ [1198601119894+ Δ1198601119894(119905)] 119909 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [119861119894+ Δ119861119894(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119862119894+ Δ119862119894(119905)] 119909 (119905)

+1198621119894119909 (119905 minus 120591

1) + 1198632119894119908 (119905)

119911 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905)) 119864

119894119909 (119905) + 119864

1119894119909 (119905 minus 120591

1) + 119866119894119906 (119905)

(62)

where

ℎ1(1199091(119905)) =

1

3


2

3

+

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

1 for 1199091gt 1

ℎ2(1199091(119905)) =

2

3


1

3

minus

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

0 for 1199091gt 1

(63)






119908 (119905) =

1

119905 + 01

119905 ge 0 (64)


119883 = [

09868 06071

06071 13948]

119884 = [

129348 01814

01814 63384]

Ω1= [



Ω2= [

82076 minus37031

minus58461 53292]

Φ1= [

31848 00758


Φ2= [

20547 01643


1205951= [


57867 minus778562]

1205952= [



(65)


1198601198961= [

873277 minus680397

1399266 minus1072394]

1198601198962= [

minus249856 158880

111983 minus122840]

1198611198961= [

109115 minus03725

165830 19936]

1198611198962= [

14642 minus00212

31940 05922]

1198621198961= [

minus160423 127577

202647 minus148918]

1198621198962= [

142596 minus104929

32304 minus11934]

(66)








x1(t)

x2(t)

0 10 20 30 40 50 60minus02

0

02

04

06

08

1

12

Time (s)

Stat

e res

pons

ex(t)


2(119905) (dashed

line)

0 10 20 30 40 50 60minus100

minus50

0

50

100

150

200

Time (s)

Con

trol i

nput

u(t)

u1(t)

u2(t)


2(119905) (dashed

line)





infin




infincontrol


Acknowledgments


y1(t)

y2(t)

0 10 20 30 40 50 60minus1

0

1

2

3

4

5

Time (s)

Mea

sure

d ou

tput

y(t)


2(119905)

(dashed line)

z1(t)

z2(t)

0 10 20 30 40 50 60Time (s)

minus40

minus20

0

20

40

60

80

100

120

Con

trolle

d ou

tput

z(t)


2(119905)

(dashed line)


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119890(119905)119879

1198752119890 (119905)

=

1

2

119903

sum

119894=1

119903

sum

119895=1

119903

sum

119906=1

119903

sum

V=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) ℎ

119906(119904 (119905)) ℎV (119904 (119905)) 120573(119905)

119879

times [119860119894119895+119872119894119895119865119894(119905)119894119895]

119879

times 1198752[119860119906V +119872119906V119865119906 (119905) 119906V]

+ [119860119906V +119872119906V119865119906 (119905) 119906V]

119879

times 1198752[119860119894119895+119872119894119895119865119894(119905) 119894119895] 120573 (119905)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times [119860119894119895+119872119894119895119865119894(119905) 119894119895]

119879

1198752[119860119894119895+119872119894119895119865119894(119905) 119894119895]

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times [119860

119879

119894119895(119875minus1

2minus 120576minus1

2119894119895119872119894119895119872119879

119894119895)

minus1

119860119894119895+ 1205762119894119895119879

119894119895119894119895] 120573 (119905)

1205911119890(119905)119879

1198761119890 (119905)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times 119860

119879

119894119895[(12059111198761)minus1

minus 120576minus1

3119894119895119872119894119895119872119879

119894119895]

minus1

119860119894119895

+ 1205763119894119895119879

119894119895119894119895120573 (119905)

(34)


(119905) le 2119890(119905)119879

119875 119890 (119905) + 119890(119905)119879

1198752119890 (119905)

+ 1205911119890(119905)119879

1198761119890 (119905) + 119890(119905)

119879

(1198751+ 1205912

31198762) 119890 (119905)

minus 119890(119905 minus 1205911)119879

1198751119890 (119905 minus 120591

1)

minus 119890(119905 minus 1205912)119879

1198752119890 (119905 minus 120591

2) minus int

119905

119905minus1205911

119890(120572)119879

1198761119890 (120572) 119889120572

minus 120572(119905)119879

1198762120572 (119905) + 2119890(119905)

119879

1198771int

119905

119905minus1205911

119890 (120572) 119889120572

minus 2119890(119905)119879

1198771[119890 (119905) minus 119890 (119905 minus 120591

1)]

+ 2119890(119905 minus 1205911)119879

1198772int

119905

119905minus1205911

119890 (120572) 119889120572 minus 2119890(119905 minus 1205911)119879


1)]

=

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905)) 120573(119905)

119879

times [Ξ119894119895+ 119860119894119895(119875minus1

2minus 120576minus1

2119894119895119872119894119895119872119879

119894119895)

minus1

119860119894119895

+119860119894119895(120591minus1

1119876minus1

1minus 120576minus1

3119894119895119872119894119895119872119879

119894119895)

minus1

119860119894119895+ 120575119894119895119879

119894119895119894119895] 120573 (119905)

(35)

where

Ξ119894119895=

[

[

[

[

[

[

Σ1198941198951198751198601119894119895+ 1198771minus 119877119879

211987511986021198941198751198603119894

12059111198771

lowast minus1198751+ 1198772+ 119877119879

20 0 120591

11198772


0 0


0


]

]

]

]

]

]

Σ119894119895= 119875119860119894119895+ 119860119879

119894119895119875 + 1198751+ 1205912

31198762minus 1198771minus 119877119879

1+ 120576minus1

1119894119895119875119872119894119895119872119879

119894119895119875

(36)


Γ119894119895lt 0 (37)





119869 (119905) = int

119905

0

[119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904)] 119889119904 (38)



119869 (119905) = int

119905

0

[119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904) + (119904)] 119889119904 minus 119881 (119905)

le int

119905

0

[119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904) + (119904)] 119889119904

(39)


(119905) =

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119905)) ℎ

119895(119904 (119905))

times 2119890(119905)119879

119875 [119860119894119895(119905) 119890 (119905) + 119860

1119894119895(119905) 119890 (119905 minus 120591

1)

+ 1198602119894(119905) 119890 (119905 minus 120591

2)

+1198603119894(119905) 120572 (119905) + 119863

119894119895119908 (119905)]


+ 119890(119905)119879

1198751119890 (119905) minus 119890(119905 minus 120591

1)119879

1198751119890 (119905 minus 120591

1)

+ 119890(119905)119879

1198752119890 (119905) minus 119890(119905 minus 120591

2)119879

1198752119890 (119905 minus 120591

2)

+ 1205911119890(119905)119879

1198761119890 (119905)

minus int

119905

119905minus1205911

119890(119905)119879

1198761119890 (120572) 119889120572 + 120591

2

3119890(119905)119879

1198762119890 (119905)

minus 120572(119905)119879

1198762120572 (119905)

+ 2119890(119905)119879

1198771int

119905

119905minus1205911

119890 (120572) 119889120572 minus 2119890(119905)119879


1)]

+ 2119890(119905 minus 1205911)119879

1198772int

119905

119905minus1205911

119890 (120572) 119889120572 minus 2119890(119905 minus 1205911)119879


1)]

(40)


119911(119904)119879

119911 (119904) =

119903

sum

119894=1

119903

sum

119895=1

119903

sum

119906=1

119903

sum

V=1

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) ℎ

119906(119904 (119904))

times ℎV (119904 (119904)) 120577(119904)119879

119864119879

119894119895119864119906V120577 (119904)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) 120577(119904)

119879

119864119879

119894119895119864119894119895120577 (119904)

(41)


119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904) + (119904)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) 120577(119904)

119879

Π1119894119895120577 (119904)

=

119903

sum

119894=1

ℎ119894(119904 (119904))

2

120577(119904)119879

Ξ1119894119894120577 (119904)

+ 2

119903

sum

119894119895=1119894lt119895

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) 120577(119904)

119879Ξ1119894119895+ Ξ1119895119894

2

120577 (119904)

(42)

where

Ξ1119894119895=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Σ1198941198951198751198601119894119895(119905) + 119877

1minus 119877119879

21198751198602119894(119905) 119875119860

3119894(119905) 119875119863

11989411989512059111198771

119864119879

119894119895119860119879

119894119895(119905) (1198752+ 12059111198761)

lowast 1198691

0 0 0 12059111198772119864119879

1119894119860119879

1119894119895(119905) (1198752+ 12059111198761)


0 0 0 0 119860119879

2119894(119905) (1198752+ 12059111198761)


0 0 0 119860119879

3119894(119905) (1198752+ 12059111198761)


119868 0 0 119863119879

119894119895(1198752+ 12059111198761)


0 0



]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

120577 (119904) = [119890(119905)119879

119890(119905 minus 1205911)119879

119890(119905 minus 1205912)119879

120572(119905)119879

119908(119905)119879

119890(120572)119879]

119879

119864119894119895= [11986411989411989511986411198940 0 0 0]

(43)



have 1199112lt 120574119908

2






119894119895 1 le 119894 le 119895 le 119903 such that


[

[

[

[

[

[

Σ119894119894Σ1119894119894

Σ2119894119894

Σ3119894119894

1198775

lowast minus119877111198772119894119894

0 0

lowast lowast 1198773119894119894

0 1198776119894119894


0


]

]

]

]

]

]

lt 0 1 le 119894 le 119903 (44)


[

[

[

[

[

[

[

[

[

[

Σ119894119895+ Σ119895119894Σ1119894119895+ Σ1119895119894

Σ2119894119895

Σ4119894119895

Σ5119894119895

Σ6119894119895

0


2119894119895

0 0 0 0


0 0 0 6119894119895



0 0



]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 lt 119895 le 119903 (45)

[


minus119868 minus119883] lt 0 (46)

where

Θ119894119895

= [

119860119894119883 + 119883119860

119879

119894+ 119861119894Ψ119895+ Ψ119879

119895119861119879

119894119860119894+ Ω119879

119894

119860119879

119894+ Ω119894

119884119860119894+ 119860119879

119894119884 + Φ

119895119862119894+ 119862119879

119894Φ119879

119895

]

minus1198771minus 119877119879

1

Θ1119894119895= [

1198601119894119883 119860

1119894

0 1198841198601119894+ Φ1198951198621119894

] + 1198771minus 119877119879

2

Θ2119894= [

1198602119894

0

11988411986021198940] Θ

3119894= [

1198603119894

0

11988411986031198940]

Θ4119894119895= [

1198631119894

1198841198631119894+ Φ1198951198632119894

] Θ5119894119895= [

119883119864119879

119894+ Ψ119879

119895119866119879

119894

119864119879

119894

]

Θ6119894119895= [

119883119860119879

119894+ Ψ119879

119895119861119879

119894Ω119879

119895

119860119879

119894119860119879

119894119884 + 119862

119879

119894Φ119879

119895

]

Θ7119894119895= [

119872119894

0

119884119872119894Φ119895119872119894

]

Θ8119894119895= [

119883119873119879

0119894+ Ψ119879

119895119873119879

4119894119883119873119879

5119894

119873119879

0119894119873119879

5119894

] 1198693119894= [

119883119864119879

1119894

119864119879

1119894

]

1198694119894119895= [

1198831198601119894

0

1198601119894

1198601119894119884 + 119862

119879

1119894Φ119879

119895

]

1198695119894= [

119883119873119879

11198940

119873119879

11198940

] 1198696119894= [

119860119879

2119894119860119879

2119894119884

0 0

]

1198697119894= [

119860119879

3119894119860119879

3119894119884

0 0

]

1198698119894119895= [119863119879

1119894119863119879

1119894119884 + 119863

119879

2119894Φ119879

119895]

1198699= Π119879

2+ Π2minus 1198752minus 12059111198761

11986910119894119895

= [

119872119894

0

119884119872119894Φ119895119872119894

]

Σ119894119895=[

[

[

Θ119894119895

Θ1119894119895

Θ2119894

lowast minus119880 + 1198772+ 119877119879

20


]

]

]

Σ1119894119895=[

[

[

Θ3119894Θ4119894119895

12059111198771

0 0 12059111198772

0 0 0

]

]

]

Σ2119894119895=[

[

[

Θ5119894119895

Θ6119894119895

Θ7119894119895

Θ8119894119895

1198693119894

1198694119894119895

0 1198695119894

0 1198696119894

0 119879

2119894

]

]

]

Σ2119894119895=

[

[

[

[

Θ5119894119895+ Θ5119895119894

Θ6119894119895+ Θ6119895119894

Θ7119894119895

Θ7119895119894

Θ8119894119895

Θ8119895119894

1198693119894+ 1198693119895

1198694119894119895+ 1198694119895119894

0 0 1198695119894

1198695119895

0 1198696119894+ 1198696119895

0 0 119879

2119894119879

2119895

]

]

]

]

Σ3119894119895=[

[

[

Π119879

11205913Π119879

1119879119894119895

0 0 0

0 0 0

]

]

]

119879119894119895= [

0

1198841198601119894+ Φ1198951198621119894

]

1198775=[

[

0 0 0

119879111987910

0 0 0

]

]

1198791= [

119909

0

] Σ4119894119895=[

[

[

1198792119894119895

1198793119894119895

1198794119894119895

1198795119894119895

0 0 0 0

0 0 0 0

]

]

]

1198792119894119895= [

(119862119894119909 minus 119862

119895119909)

119879

0

]

1198793119894119895= [

0

Φ119895minus Φ119894

] 1198794119894119895= [

(Ψ119895minus Ψ119894)

119879

0

]

1198795119894119895= [

0

119884119861119894minus 119884119861119895

]

Σ5119894119895=[

[

Π119879

11205913Π119879

11198791198941198950

0 0 0 1198791

0 0 0 0

]

]


Σ6119894119895=[

[

119879119895119894

0 11987941198792

0 11987910 0

0 0 0 0

]

]

11987711= diag (119876

21205742

119868 119881) 1198772119894119895=[

[

0 11986971198940 119879

3119894

0 1198698119894119895

0 0

0 0 0 0

]

]

2119894119895=[

[

0 1198697119894+ 1198697119895

0 0 119879

3119894119879

3119895

0 1198698119894119895+ 1198698119895119894

0 0 0 0

0 0 0 0 0 0

]

]

1198773119894119895=

[

[

[

[

minus119868 0 0 0

lowast minus119869911986910119894119895

0



119894119895119868

]

]

]

]

3119894119895=

[

[

[

[

[

[

[

[

[

minus2119868 0 0 0 0 0

lowast minus2119869911986910119894119895

11986910119895119894

0 0




119894119895119868 0


119895119894119868

]

]

]

]

]

]

]

]

]

1198774= diag (119875minus1

1119876minus1

2119868)

1198776119894119895=

[

[

[

[

0 0 0

0 0 1198796119894119895

0 0 0

0 0 0

]

]

]

]

6119894119895=

[

[

[

[

[

[

[

[

0 0

1198795119894119895

1198793119894119895

0 0

0 0

0 0

0 0

]

]

]

]

]

]

]

]

1198796119894119895= [

0

119884119860119879

1119894+ Φ1198951198621119894

]

1198777= diag (119868 119868 119868) 119877

8= diag (1

2

119875minus1

1

1

2

119876minus1

2119868 119868)

Π1= [

119883 119868

119883 0] Π

2= [

119883 119868

119883 0]

(47)


119860119870119894= (119883minus1

minus 119884)

minus1

(Ω119894minus 119884119860119894119883 minus 119884119861

119894Ψ119894minus Φ119894119862119894119883)119883minus1

119861119870119894= (119883minus1

minus 119884)

minus1

Φ119894 119862119870119894= Ψ119894119883minus1

1 le 119894 le 119903

(48)


[

[

Σ119894119894Σ1119894119894

Σ2119894119894

lowast minus119877111198772119894119894

lowast lowast 1198773119894119894

]

]

lt 0 1 le 119894 le 119903

[

[

[

Σ119894119895Σ1119894119895+ Σ1119895119894

Σ2119894119895


2119894119895


]

]

]

lt 0 1 le 119894 lt 119895 le 119903

(49)

where

Σ119894119895=

[

[

[

[

Θ119894119895+ Π119879

11198751Π1+ (1205913Π1)119879

1198762(1205913Π1) Θ1119894119895+ [

0 0

(1198841198601119894+ Φ1198951198621119894)119883 0

] Θ2119894

lowast minus119880 + 1198772+ 119877119879

20


]

]

]

]

Σ2119894119895=

[

[

[

[

Θ5119894119895

Θ6119894119895

Θ7119894119895

Θ8119894119895

1198693119894

1198694119894119895+ [

0 119883 (1198601119894119884 + 119862

119879

1119894Φ119879

119895)

0 0

] 0 1198695119894

0 1198696119894

0 119879

2119894

]

]

]

]

Σ119894119895= Σ119894119895+ Σ119895119894+

[

[

[

[

[

[

0 (119862119894119883 minus 119862

119895119883)

119879

(Φ119895minus Φ119894)

119879

+ (Ψ119895minus Ψ119894)

119879

(119884119861119894minus 119884119861119895)

119879

0 0 0

lowast 0 0 0 0

lowast lowast 0 0 0



]

]

]

]

]

]

Σ2119894119895= Σ2119894119895+[

[

0 (Ψ119879

119895minus Ψ119879

119894) (119861119879

119894119884 minus 119861

119879

119895119884) + (119883119862

119879

119894minus 119883119862119879

119895) (Φ119879

119895minus Φ119879

119894) 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]

]

(50)


Note that

Π119879

2+ Π2minus 1198752minus 12059111198761le Π119879

2(1198752+ 12059111198761)minus1

Π2 (51)


[

[

Σ119894119894Σ1119894119894

Σ2119894119894

lowast minus119877111198772119894119894


]

]

lt 0 1 le 119894 le 119903

[

[

[

Σ119894119895Σ1119894119895+ Σ1119895119894

Σ2119894119895


2119894119895

lowast lowast 1198773119894119895

]

]

]

lt 0 1 le 119894 lt 119895 le 119903

(52)

where

3119894119895=

[

[

[

[

[

minus119868 0 0 0


2(1198752+ 12059111198761)minus1

Π211986910119894119895

0



119894119895119868

]

]

]

]

]

1198773119894119895=

[

[

[

[

[

[

[

[

[

minus2119868 0 0 0 0 0


2(1198752+ 12059111198761)minus1

Π211986910119894119895

11986910119895119894

0 0




119894119895119868 0


119895119894119868

]

]

]

]

]

]

]

]

]

(53)

Now set = Π

2Πminus1

1 (54)



1 119868 119868 119868 Π

minus119879

1 119868 Πminus119879

2 119868 119868) (55)


[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Π1119894119894

Π2119894119894

11987511986021198941198751198603119894119875119863119894119894

12059111

119864119879

119894119894Γ119894119894

119875119872119894119894

119873119879

119894119894

lowast 1198691

0 0 0 12059112119864119879

1119894Γ1119894119894

0 0


0 0 0 0 Γ2119894

0 0


0 0 0 Γ3119894

0 0


119868 0 0 Γ4119894119894

0 0


0 0 0 0



2119869119879

2119872119894119894

0



119894119894119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 le 119903

[

[

Λ1119894119895

Λ2119894119895

Λ3119894119895

lowast Λ4119894119895

Λ5119894119895


]

]

lt 0 1 le 119894 lt 119895 le 119903

(56)

where

Λ1= 119867119879

(1198751+ 1205912

31198762minus 1minus 119879

1)119867

Π1119894119895= Λ1+ 119875119860119894119895+ 119860119879

119894119895119875

Λ2= 119867119879

(1minus 119879

2)

Π2119894119895= Λ2+ 1198751198601119894119895

1= Π119879

11198771Π1

(57)







119894= 0 and








1(119905) is 120583

1 then

(119905) = [1198601+ Δ1198601(119905)] 119909 (119905) + [119860

11+ Δ11986011(119905)] 119909 (119905 minus 120591

1)

+ [11986021+ Δ11986021(119905)] (119905 minus 120591

2)

+ [11986031+ Δ11986031(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198611+ Δ1198611(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) = [1198621+ Δ1198621(119905)] 119909 (119905) + 119862

11119909 (119905 minus 120591

1) + 11986321119908 (119905)

119911 (119905) = 1198641119909 (119905) + 119864

11119909 (119905 minus 120591

1) + 1198661119906 (119905)

(58)

Plant Rule 2 if 1199091(119905) is 120583

2 then

(119905) = [1198602+ Δ1198602(119905)] 119909 (119905) + [119860

12+ Δ11986012(119905)] 119909 (119905 minus 120591

1)

+ [11986022+ Δ11986022(119905)] (119905 minus 120591

2)

+ [11986032+ Δ11986032(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198612+ Δ1198612(119905)] 119906 (119905) + 119863

12119908 (119905)

119910 (119905) = [1198622+ Δ1198622(119905)] 119909 (119905) + 119862

12119909 (119905 minus 120591

1) + 11986322119908 (119905)

119911 (119905) = 1198642119909 (119905) + 119864

12119909 (119905 minus 120591

1) + 1198662119906 (119905)

(59)

where

1198601= [

005 0


11= [

005 002

minus005 001]

11986021= [

minus0002 0008

0006 0]

11986031= [

minus005 008

006 0] 119861

1= [

001 0

003 006]

1198621= [

minus003 minus01

02 002] 119862

11= [

minus003 minus001

002 002]

11986311= [

003

006] 119863

21= [

minus05

0]

1198641= [minus02 0] 119864

11= [minus002 0]

1198661= [0 minus003]

1198602= [

minus005 0


12= [

minus005 02

minus05 001]

11986022= [

0008 minus0002

minus0003 002]

11986032= [

005 minus02

minus03 02] 119861

2= [

001 minus008

002 005]

1198622= [

003 minus01

02 002]

11986212= [

minus003 minus001

002 02] 119863

12= [

minus001

002]

11986322= [

minus03

0]

1198642= [minus001 0] 119864

12= [minus001 0]

1198662= [0 002]

(60)

and Δ119860119894(119905) Δ119860

1119894(119905) Δ119860

2119894(119905) Δ119860

3119894(119905) Δ119861

119894(119905) Δ119862

119894(119905) (119894 =


1198721= [

minus002

0] 119873

01= [minus02 01]

11987311= [minus02 02]

11987321= [minus002 001] 119873

31= [minus002 001]

11987341= [minus002 001]

11987351= [minus002 001] 119872

2= [

003

0]

11987302= 11987301 119873

12= 11987311

11987322= 11987321 119873

32= 11987331

11987342= 11987341 119873

52= 11987351

(61)



(119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119860119894+ Δ119860119894(119905)] 119909 (119905)

+ [1198601119894+ Δ1198601119894(119905)] 119909 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [119861119894+ Δ119861119894(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119862119894+ Δ119862119894(119905)] 119909 (119905)

+1198621119894119909 (119905 minus 120591

1) + 1198632119894119908 (119905)

119911 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905)) 119864

119894119909 (119905) + 119864

1119894119909 (119905 minus 120591

1) + 119866119894119906 (119905)

(62)

where

ℎ1(1199091(119905)) =

1

3


2

3

+

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

1 for 1199091gt 1

ℎ2(1199091(119905)) =

2

3


1

3

minus

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

0 for 1199091gt 1

(63)






119908 (119905) =

1

119905 + 01

119905 ge 0 (64)


119883 = [

09868 06071

06071 13948]

119884 = [

129348 01814

01814 63384]

Ω1= [



Ω2= [

82076 minus37031

minus58461 53292]

Φ1= [

31848 00758


Φ2= [

20547 01643


1205951= [


57867 minus778562]

1205952= [



(65)


1198601198961= [

873277 minus680397

1399266 minus1072394]

1198601198962= [

minus249856 158880

111983 minus122840]

1198611198961= [

109115 minus03725

165830 19936]

1198611198962= [

14642 minus00212

31940 05922]

1198621198961= [

minus160423 127577

202647 minus148918]

1198621198962= [

142596 minus104929

32304 minus11934]

(66)








x1(t)

x2(t)

0 10 20 30 40 50 60minus02

0

02

04

06

08

1

12

Time (s)

Stat

e res

pons

ex(t)


2(119905) (dashed

line)

0 10 20 30 40 50 60minus100

minus50

0

50

100

150

200

Time (s)

Con

trol i

nput

u(t)

u1(t)

u2(t)


2(119905) (dashed

line)





infin




infincontrol


Acknowledgments


y1(t)

y2(t)

0 10 20 30 40 50 60minus1

0

1

2

3

4

5

Time (s)

Mea

sure

d ou

tput

y(t)


2(119905)

(dashed line)

z1(t)

z2(t)

0 10 20 30 40 50 60Time (s)

minus40

minus20

0

20

40

60

80

100

120

Con

trolle

d ou

tput

z(t)


2(119905)

(dashed line)


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+ 119890(119905)119879

1198751119890 (119905) minus 119890(119905 minus 120591

1)119879

1198751119890 (119905 minus 120591

1)

+ 119890(119905)119879

1198752119890 (119905) minus 119890(119905 minus 120591

2)119879

1198752119890 (119905 minus 120591

2)

+ 1205911119890(119905)119879

1198761119890 (119905)

minus int

119905

119905minus1205911

119890(119905)119879

1198761119890 (120572) 119889120572 + 120591

2

3119890(119905)119879

1198762119890 (119905)

minus 120572(119905)119879

1198762120572 (119905)

+ 2119890(119905)119879

1198771int

119905

119905minus1205911

119890 (120572) 119889120572 minus 2119890(119905)119879


1)]

+ 2119890(119905 minus 1205911)119879

1198772int

119905

119905minus1205911

119890 (120572) 119889120572 minus 2119890(119905 minus 1205911)119879


1)]

(40)


119911(119904)119879

119911 (119904) =

119903

sum

119894=1

119903

sum

119895=1

119903

sum

119906=1

119903

sum

V=1

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) ℎ

119906(119904 (119904))

times ℎV (119904 (119904)) 120577(119904)119879

119864119879

119894119895119864119906V120577 (119904)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) 120577(119904)

119879

119864119879

119894119895119864119894119895120577 (119904)

(41)


119911(119904)119879

119911 (119904) minus 1205742

119908(119904)119879

119908 (119904) + (119904)

le

119903

sum

119894=1

119903

sum

119895=1

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) 120577(119904)

119879

Π1119894119895120577 (119904)

=

119903

sum

119894=1

ℎ119894(119904 (119904))

2

120577(119904)119879

Ξ1119894119894120577 (119904)

+ 2

119903

sum

119894119895=1119894lt119895

ℎ119894(119904 (119904)) ℎ

119895(119904 (119904)) 120577(119904)

119879Ξ1119894119895+ Ξ1119895119894

2

120577 (119904)

(42)

where

Ξ1119894119895=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Σ1198941198951198751198601119894119895(119905) + 119877

1minus 119877119879

21198751198602119894(119905) 119875119860

3119894(119905) 119875119863

11989411989512059111198771

119864119879

119894119895119860119879

119894119895(119905) (1198752+ 12059111198761)

lowast 1198691

0 0 0 12059111198772119864119879

1119894119860119879

1119894119895(119905) (1198752+ 12059111198761)


0 0 0 0 119860119879

2119894(119905) (1198752+ 12059111198761)


0 0 0 119860119879

3119894(119905) (1198752+ 12059111198761)


119868 0 0 119863119879

119894119895(1198752+ 12059111198761)


0 0



]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

120577 (119904) = [119890(119905)119879

119890(119905 minus 1205911)119879

119890(119905 minus 1205912)119879

120572(119905)119879

119908(119905)119879

119890(120572)119879]

119879

119864119894119895= [11986411989411989511986411198940 0 0 0]

(43)



have 1199112lt 120574119908

2






119894119895 1 le 119894 le 119895 le 119903 such that


[

[

[

[

[

[

Σ119894119894Σ1119894119894

Σ2119894119894

Σ3119894119894

1198775

lowast minus119877111198772119894119894

0 0

lowast lowast 1198773119894119894

0 1198776119894119894


0


]

]

]

]

]

]

lt 0 1 le 119894 le 119903 (44)


[

[

[

[

[

[

[

[

[

[

Σ119894119895+ Σ119895119894Σ1119894119895+ Σ1119895119894

Σ2119894119895

Σ4119894119895

Σ5119894119895

Σ6119894119895

0


2119894119895

0 0 0 0


0 0 0 6119894119895



0 0



]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 lt 119895 le 119903 (45)

[


minus119868 minus119883] lt 0 (46)

where

Θ119894119895

= [

119860119894119883 + 119883119860

119879

119894+ 119861119894Ψ119895+ Ψ119879

119895119861119879

119894119860119894+ Ω119879

119894

119860119879

119894+ Ω119894

119884119860119894+ 119860119879

119894119884 + Φ

119895119862119894+ 119862119879

119894Φ119879

119895

]

minus1198771minus 119877119879

1

Θ1119894119895= [

1198601119894119883 119860

1119894

0 1198841198601119894+ Φ1198951198621119894

] + 1198771minus 119877119879

2

Θ2119894= [

1198602119894

0

11988411986021198940] Θ

3119894= [

1198603119894

0

11988411986031198940]

Θ4119894119895= [

1198631119894

1198841198631119894+ Φ1198951198632119894

] Θ5119894119895= [

119883119864119879

119894+ Ψ119879

119895119866119879

119894

119864119879

119894

]

Θ6119894119895= [

119883119860119879

119894+ Ψ119879

119895119861119879

119894Ω119879

119895

119860119879

119894119860119879

119894119884 + 119862

119879

119894Φ119879

119895

]

Θ7119894119895= [

119872119894

0

119884119872119894Φ119895119872119894

]

Θ8119894119895= [

119883119873119879

0119894+ Ψ119879

119895119873119879

4119894119883119873119879

5119894

119873119879

0119894119873119879

5119894

] 1198693119894= [

119883119864119879

1119894

119864119879

1119894

]

1198694119894119895= [

1198831198601119894

0

1198601119894

1198601119894119884 + 119862

119879

1119894Φ119879

119895

]

1198695119894= [

119883119873119879

11198940

119873119879

11198940

] 1198696119894= [

119860119879

2119894119860119879

2119894119884

0 0

]

1198697119894= [

119860119879

3119894119860119879

3119894119884

0 0

]

1198698119894119895= [119863119879

1119894119863119879

1119894119884 + 119863

119879

2119894Φ119879

119895]

1198699= Π119879

2+ Π2minus 1198752minus 12059111198761

11986910119894119895

= [

119872119894

0

119884119872119894Φ119895119872119894

]

Σ119894119895=[

[

[

Θ119894119895

Θ1119894119895

Θ2119894

lowast minus119880 + 1198772+ 119877119879

20


]

]

]

Σ1119894119895=[

[

[

Θ3119894Θ4119894119895

12059111198771

0 0 12059111198772

0 0 0

]

]

]

Σ2119894119895=[

[

[

Θ5119894119895

Θ6119894119895

Θ7119894119895

Θ8119894119895

1198693119894

1198694119894119895

0 1198695119894

0 1198696119894

0 119879

2119894

]

]

]

Σ2119894119895=

[

[

[

[

Θ5119894119895+ Θ5119895119894

Θ6119894119895+ Θ6119895119894

Θ7119894119895

Θ7119895119894

Θ8119894119895

Θ8119895119894

1198693119894+ 1198693119895

1198694119894119895+ 1198694119895119894

0 0 1198695119894

1198695119895

0 1198696119894+ 1198696119895

0 0 119879

2119894119879

2119895

]

]

]

]

Σ3119894119895=[

[

[

Π119879

11205913Π119879

1119879119894119895

0 0 0

0 0 0

]

]

]

119879119894119895= [

0

1198841198601119894+ Φ1198951198621119894

]

1198775=[

[

0 0 0

119879111987910

0 0 0

]

]

1198791= [

119909

0

] Σ4119894119895=[

[

[

1198792119894119895

1198793119894119895

1198794119894119895

1198795119894119895

0 0 0 0

0 0 0 0

]

]

]

1198792119894119895= [

(119862119894119909 minus 119862

119895119909)

119879

0

]

1198793119894119895= [

0

Φ119895minus Φ119894

] 1198794119894119895= [

(Ψ119895minus Ψ119894)

119879

0

]

1198795119894119895= [

0

119884119861119894minus 119884119861119895

]

Σ5119894119895=[

[

Π119879

11205913Π119879

11198791198941198950

0 0 0 1198791

0 0 0 0

]

]


Σ6119894119895=[

[

119879119895119894

0 11987941198792

0 11987910 0

0 0 0 0

]

]

11987711= diag (119876

21205742

119868 119881) 1198772119894119895=[

[

0 11986971198940 119879

3119894

0 1198698119894119895

0 0

0 0 0 0

]

]

2119894119895=[

[

0 1198697119894+ 1198697119895

0 0 119879

3119894119879

3119895

0 1198698119894119895+ 1198698119895119894

0 0 0 0

0 0 0 0 0 0

]

]

1198773119894119895=

[

[

[

[

minus119868 0 0 0

lowast minus119869911986910119894119895

0



119894119895119868

]

]

]

]

3119894119895=

[

[

[

[

[

[

[

[

[

minus2119868 0 0 0 0 0

lowast minus2119869911986910119894119895

11986910119895119894

0 0




119894119895119868 0


119895119894119868

]

]

]

]

]

]

]

]

]

1198774= diag (119875minus1

1119876minus1

2119868)

1198776119894119895=

[

[

[

[

0 0 0

0 0 1198796119894119895

0 0 0

0 0 0

]

]

]

]

6119894119895=

[

[

[

[

[

[

[

[

0 0

1198795119894119895

1198793119894119895

0 0

0 0

0 0

0 0

]

]

]

]

]

]

]

]

1198796119894119895= [

0

119884119860119879

1119894+ Φ1198951198621119894

]

1198777= diag (119868 119868 119868) 119877

8= diag (1

2

119875minus1

1

1

2

119876minus1

2119868 119868)

Π1= [

119883 119868

119883 0] Π

2= [

119883 119868

119883 0]

(47)


119860119870119894= (119883minus1

minus 119884)

minus1

(Ω119894minus 119884119860119894119883 minus 119884119861

119894Ψ119894minus Φ119894119862119894119883)119883minus1

119861119870119894= (119883minus1

minus 119884)

minus1

Φ119894 119862119870119894= Ψ119894119883minus1

1 le 119894 le 119903

(48)


[

[

Σ119894119894Σ1119894119894

Σ2119894119894

lowast minus119877111198772119894119894

lowast lowast 1198773119894119894

]

]

lt 0 1 le 119894 le 119903

[

[

[

Σ119894119895Σ1119894119895+ Σ1119895119894

Σ2119894119895


2119894119895


]

]

]

lt 0 1 le 119894 lt 119895 le 119903

(49)

where

Σ119894119895=

[

[

[

[

Θ119894119895+ Π119879

11198751Π1+ (1205913Π1)119879

1198762(1205913Π1) Θ1119894119895+ [

0 0

(1198841198601119894+ Φ1198951198621119894)119883 0

] Θ2119894

lowast minus119880 + 1198772+ 119877119879

20


]

]

]

]

Σ2119894119895=

[

[

[

[

Θ5119894119895

Θ6119894119895

Θ7119894119895

Θ8119894119895

1198693119894

1198694119894119895+ [

0 119883 (1198601119894119884 + 119862

119879

1119894Φ119879

119895)

0 0

] 0 1198695119894

0 1198696119894

0 119879

2119894

]

]

]

]

Σ119894119895= Σ119894119895+ Σ119895119894+

[

[

[

[

[

[

0 (119862119894119883 minus 119862

119895119883)

119879

(Φ119895minus Φ119894)

119879

+ (Ψ119895minus Ψ119894)

119879

(119884119861119894minus 119884119861119895)

119879

0 0 0

lowast 0 0 0 0

lowast lowast 0 0 0



]

]

]

]

]

]

Σ2119894119895= Σ2119894119895+[

[

0 (Ψ119879

119895minus Ψ119879

119894) (119861119879

119894119884 minus 119861

119879

119895119884) + (119883119862

119879

119894minus 119883119862119879

119895) (Φ119879

119895minus Φ119879

119894) 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]

]

(50)


Note that

Π119879

2+ Π2minus 1198752minus 12059111198761le Π119879

2(1198752+ 12059111198761)minus1

Π2 (51)


[

[

Σ119894119894Σ1119894119894

Σ2119894119894

lowast minus119877111198772119894119894


]

]

lt 0 1 le 119894 le 119903

[

[

[

Σ119894119895Σ1119894119895+ Σ1119895119894

Σ2119894119895


2119894119895

lowast lowast 1198773119894119895

]

]

]

lt 0 1 le 119894 lt 119895 le 119903

(52)

where

3119894119895=

[

[

[

[

[

minus119868 0 0 0


2(1198752+ 12059111198761)minus1

Π211986910119894119895

0



119894119895119868

]

]

]

]

]

1198773119894119895=

[

[

[

[

[

[

[

[

[

minus2119868 0 0 0 0 0


2(1198752+ 12059111198761)minus1

Π211986910119894119895

11986910119895119894

0 0




119894119895119868 0


119895119894119868

]

]

]

]

]

]

]

]

]

(53)

Now set = Π

2Πminus1

1 (54)



1 119868 119868 119868 Π

minus119879

1 119868 Πminus119879

2 119868 119868) (55)


[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Π1119894119894

Π2119894119894

11987511986021198941198751198603119894119875119863119894119894

12059111

119864119879

119894119894Γ119894119894

119875119872119894119894

119873119879

119894119894

lowast 1198691

0 0 0 12059112119864119879

1119894Γ1119894119894

0 0


0 0 0 0 Γ2119894

0 0


0 0 0 Γ3119894

0 0


119868 0 0 Γ4119894119894

0 0


0 0 0 0



2119869119879

2119872119894119894

0



119894119894119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 le 119903

[

[

Λ1119894119895

Λ2119894119895

Λ3119894119895

lowast Λ4119894119895

Λ5119894119895


]

]

lt 0 1 le 119894 lt 119895 le 119903

(56)

where

Λ1= 119867119879

(1198751+ 1205912

31198762minus 1minus 119879

1)119867

Π1119894119895= Λ1+ 119875119860119894119895+ 119860119879

119894119895119875

Λ2= 119867119879

(1minus 119879

2)

Π2119894119895= Λ2+ 1198751198601119894119895

1= Π119879

11198771Π1

(57)







119894= 0 and








1(119905) is 120583

1 then

(119905) = [1198601+ Δ1198601(119905)] 119909 (119905) + [119860

11+ Δ11986011(119905)] 119909 (119905 minus 120591

1)

+ [11986021+ Δ11986021(119905)] (119905 minus 120591

2)

+ [11986031+ Δ11986031(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198611+ Δ1198611(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) = [1198621+ Δ1198621(119905)] 119909 (119905) + 119862

11119909 (119905 minus 120591

1) + 11986321119908 (119905)

119911 (119905) = 1198641119909 (119905) + 119864

11119909 (119905 minus 120591

1) + 1198661119906 (119905)

(58)

Plant Rule 2 if 1199091(119905) is 120583

2 then

(119905) = [1198602+ Δ1198602(119905)] 119909 (119905) + [119860

12+ Δ11986012(119905)] 119909 (119905 minus 120591

1)

+ [11986022+ Δ11986022(119905)] (119905 minus 120591

2)

+ [11986032+ Δ11986032(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198612+ Δ1198612(119905)] 119906 (119905) + 119863

12119908 (119905)

119910 (119905) = [1198622+ Δ1198622(119905)] 119909 (119905) + 119862

12119909 (119905 minus 120591

1) + 11986322119908 (119905)

119911 (119905) = 1198642119909 (119905) + 119864

12119909 (119905 minus 120591

1) + 1198662119906 (119905)

(59)

where

1198601= [

005 0


11= [

005 002

minus005 001]

11986021= [

minus0002 0008

0006 0]

11986031= [

minus005 008

006 0] 119861

1= [

001 0

003 006]

1198621= [

minus003 minus01

02 002] 119862

11= [

minus003 minus001

002 002]

11986311= [

003

006] 119863

21= [

minus05

0]

1198641= [minus02 0] 119864

11= [minus002 0]

1198661= [0 minus003]

1198602= [

minus005 0


12= [

minus005 02

minus05 001]

11986022= [

0008 minus0002

minus0003 002]

11986032= [

005 minus02

minus03 02] 119861

2= [

001 minus008

002 005]

1198622= [

003 minus01

02 002]

11986212= [

minus003 minus001

002 02] 119863

12= [

minus001

002]

11986322= [

minus03

0]

1198642= [minus001 0] 119864

12= [minus001 0]

1198662= [0 002]

(60)

and Δ119860119894(119905) Δ119860

1119894(119905) Δ119860

2119894(119905) Δ119860

3119894(119905) Δ119861

119894(119905) Δ119862

119894(119905) (119894 =


1198721= [

minus002

0] 119873

01= [minus02 01]

11987311= [minus02 02]

11987321= [minus002 001] 119873

31= [minus002 001]

11987341= [minus002 001]

11987351= [minus002 001] 119872

2= [

003

0]

11987302= 11987301 119873

12= 11987311

11987322= 11987321 119873

32= 11987331

11987342= 11987341 119873

52= 11987351

(61)



(119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119860119894+ Δ119860119894(119905)] 119909 (119905)

+ [1198601119894+ Δ1198601119894(119905)] 119909 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [119861119894+ Δ119861119894(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119862119894+ Δ119862119894(119905)] 119909 (119905)

+1198621119894119909 (119905 minus 120591

1) + 1198632119894119908 (119905)

119911 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905)) 119864

119894119909 (119905) + 119864

1119894119909 (119905 minus 120591

1) + 119866119894119906 (119905)

(62)

where

ℎ1(1199091(119905)) =

1

3


2

3

+

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

1 for 1199091gt 1

ℎ2(1199091(119905)) =

2

3


1

3

minus

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

0 for 1199091gt 1

(63)






119908 (119905) =

1

119905 + 01

119905 ge 0 (64)


119883 = [

09868 06071

06071 13948]

119884 = [

129348 01814

01814 63384]

Ω1= [



Ω2= [

82076 minus37031

minus58461 53292]

Φ1= [

31848 00758


Φ2= [

20547 01643


1205951= [


57867 minus778562]

1205952= [



(65)


1198601198961= [

873277 minus680397

1399266 minus1072394]

1198601198962= [

minus249856 158880

111983 minus122840]

1198611198961= [

109115 minus03725

165830 19936]

1198611198962= [

14642 minus00212

31940 05922]

1198621198961= [

minus160423 127577

202647 minus148918]

1198621198962= [

142596 minus104929

32304 minus11934]

(66)








x1(t)

x2(t)

0 10 20 30 40 50 60minus02

0

02

04

06

08

1

12

Time (s)

Stat

e res

pons

ex(t)


2(119905) (dashed

line)

0 10 20 30 40 50 60minus100

minus50

0

50

100

150

200

Time (s)

Con

trol i

nput

u(t)

u1(t)

u2(t)


2(119905) (dashed

line)





infin




infincontrol


Acknowledgments


y1(t)

y2(t)

0 10 20 30 40 50 60minus1

0

1

2

3

4

5

Time (s)

Mea

sure

d ou

tput

y(t)


2(119905)

(dashed line)

z1(t)

z2(t)

0 10 20 30 40 50 60Time (s)

minus40

minus20

0

20

40

60

80

100

120

Con

trolle

d ou

tput

z(t)


2(119905)

(dashed line)


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










[

[

[

[

[

[

[

[

[

[

Σ119894119895+ Σ119895119894Σ1119894119895+ Σ1119895119894

Σ2119894119895

Σ4119894119895

Σ5119894119895

Σ6119894119895

0


2119894119895

0 0 0 0


0 0 0 6119894119895



0 0



]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 lt 119895 le 119903 (45)

[


minus119868 minus119883] lt 0 (46)

where

Θ119894119895

= [

119860119894119883 + 119883119860

119879

119894+ 119861119894Ψ119895+ Ψ119879

119895119861119879

119894119860119894+ Ω119879

119894

119860119879

119894+ Ω119894

119884119860119894+ 119860119879

119894119884 + Φ

119895119862119894+ 119862119879

119894Φ119879

119895

]

minus1198771minus 119877119879

1

Θ1119894119895= [

1198601119894119883 119860

1119894

0 1198841198601119894+ Φ1198951198621119894

] + 1198771minus 119877119879

2

Θ2119894= [

1198602119894

0

11988411986021198940] Θ

3119894= [

1198603119894

0

11988411986031198940]

Θ4119894119895= [

1198631119894

1198841198631119894+ Φ1198951198632119894

] Θ5119894119895= [

119883119864119879

119894+ Ψ119879

119895119866119879

119894

119864119879

119894

]

Θ6119894119895= [

119883119860119879

119894+ Ψ119879

119895119861119879

119894Ω119879

119895

119860119879

119894119860119879

119894119884 + 119862

119879

119894Φ119879

119895

]

Θ7119894119895= [

119872119894

0

119884119872119894Φ119895119872119894

]

Θ8119894119895= [

119883119873119879

0119894+ Ψ119879

119895119873119879

4119894119883119873119879

5119894

119873119879

0119894119873119879

5119894

] 1198693119894= [

119883119864119879

1119894

119864119879

1119894

]

1198694119894119895= [

1198831198601119894

0

1198601119894

1198601119894119884 + 119862

119879

1119894Φ119879

119895

]

1198695119894= [

119883119873119879

11198940

119873119879

11198940

] 1198696119894= [

119860119879

2119894119860119879

2119894119884

0 0

]

1198697119894= [

119860119879

3119894119860119879

3119894119884

0 0

]

1198698119894119895= [119863119879

1119894119863119879

1119894119884 + 119863

119879

2119894Φ119879

119895]

1198699= Π119879

2+ Π2minus 1198752minus 12059111198761

11986910119894119895

= [

119872119894

0

119884119872119894Φ119895119872119894

]

Σ119894119895=[

[

[

Θ119894119895

Θ1119894119895

Θ2119894

lowast minus119880 + 1198772+ 119877119879

20


]

]

]

Σ1119894119895=[

[

[

Θ3119894Θ4119894119895

12059111198771

0 0 12059111198772

0 0 0

]

]

]

Σ2119894119895=[

[

[

Θ5119894119895

Θ6119894119895

Θ7119894119895

Θ8119894119895

1198693119894

1198694119894119895

0 1198695119894

0 1198696119894

0 119879

2119894

]

]

]

Σ2119894119895=

[

[

[

[

Θ5119894119895+ Θ5119895119894

Θ6119894119895+ Θ6119895119894

Θ7119894119895

Θ7119895119894

Θ8119894119895

Θ8119895119894

1198693119894+ 1198693119895

1198694119894119895+ 1198694119895119894

0 0 1198695119894

1198695119895

0 1198696119894+ 1198696119895

0 0 119879

2119894119879

2119895

]

]

]

]

Σ3119894119895=[

[

[

Π119879

11205913Π119879

1119879119894119895

0 0 0

0 0 0

]

]

]

119879119894119895= [

0

1198841198601119894+ Φ1198951198621119894

]

1198775=[

[

0 0 0

119879111987910

0 0 0

]

]

1198791= [

119909

0

] Σ4119894119895=[

[

[

1198792119894119895

1198793119894119895

1198794119894119895

1198795119894119895

0 0 0 0

0 0 0 0

]

]

]

1198792119894119895= [

(119862119894119909 minus 119862

119895119909)

119879

0

]

1198793119894119895= [

0

Φ119895minus Φ119894

] 1198794119894119895= [

(Ψ119895minus Ψ119894)

119879

0

]

1198795119894119895= [

0

119884119861119894minus 119884119861119895

]

Σ5119894119895=[

[

Π119879

11205913Π119879

11198791198941198950

0 0 0 1198791

0 0 0 0

]

]


Σ6119894119895=[

[

119879119895119894

0 11987941198792

0 11987910 0

0 0 0 0

]

]

11987711= diag (119876

21205742

119868 119881) 1198772119894119895=[

[

0 11986971198940 119879

3119894

0 1198698119894119895

0 0

0 0 0 0

]

]

2119894119895=[

[

0 1198697119894+ 1198697119895

0 0 119879

3119894119879

3119895

0 1198698119894119895+ 1198698119895119894

0 0 0 0

0 0 0 0 0 0

]

]

1198773119894119895=

[

[

[

[

minus119868 0 0 0

lowast minus119869911986910119894119895

0



119894119895119868

]

]

]

]

3119894119895=

[

[

[

[

[

[

[

[

[

minus2119868 0 0 0 0 0

lowast minus2119869911986910119894119895

11986910119895119894

0 0




119894119895119868 0


119895119894119868

]

]

]

]

]

]

]

]

]

1198774= diag (119875minus1

1119876minus1

2119868)

1198776119894119895=

[

[

[

[

0 0 0

0 0 1198796119894119895

0 0 0

0 0 0

]

]

]

]

6119894119895=

[

[

[

[

[

[

[

[

0 0

1198795119894119895

1198793119894119895

0 0

0 0

0 0

0 0

]

]

]

]

]

]

]

]

1198796119894119895= [

0

119884119860119879

1119894+ Φ1198951198621119894

]

1198777= diag (119868 119868 119868) 119877

8= diag (1

2

119875minus1

1

1

2

119876minus1

2119868 119868)

Π1= [

119883 119868

119883 0] Π

2= [

119883 119868

119883 0]

(47)


119860119870119894= (119883minus1

minus 119884)

minus1

(Ω119894minus 119884119860119894119883 minus 119884119861

119894Ψ119894minus Φ119894119862119894119883)119883minus1

119861119870119894= (119883minus1

minus 119884)

minus1

Φ119894 119862119870119894= Ψ119894119883minus1

1 le 119894 le 119903

(48)


[

[

Σ119894119894Σ1119894119894

Σ2119894119894

lowast minus119877111198772119894119894

lowast lowast 1198773119894119894

]

]

lt 0 1 le 119894 le 119903

[

[

[

Σ119894119895Σ1119894119895+ Σ1119895119894

Σ2119894119895


2119894119895


]

]

]

lt 0 1 le 119894 lt 119895 le 119903

(49)

where

Σ119894119895=

[

[

[

[

Θ119894119895+ Π119879

11198751Π1+ (1205913Π1)119879

1198762(1205913Π1) Θ1119894119895+ [

0 0

(1198841198601119894+ Φ1198951198621119894)119883 0

] Θ2119894

lowast minus119880 + 1198772+ 119877119879

20


]

]

]

]

Σ2119894119895=

[

[

[

[

Θ5119894119895

Θ6119894119895

Θ7119894119895

Θ8119894119895

1198693119894

1198694119894119895+ [

0 119883 (1198601119894119884 + 119862

119879

1119894Φ119879

119895)

0 0

] 0 1198695119894

0 1198696119894

0 119879

2119894

]

]

]

]

Σ119894119895= Σ119894119895+ Σ119895119894+

[

[

[

[

[

[

0 (119862119894119883 minus 119862

119895119883)

119879

(Φ119895minus Φ119894)

119879

+ (Ψ119895minus Ψ119894)

119879

(119884119861119894minus 119884119861119895)

119879

0 0 0

lowast 0 0 0 0

lowast lowast 0 0 0



]

]

]

]

]

]

Σ2119894119895= Σ2119894119895+[

[

0 (Ψ119879

119895minus Ψ119879

119894) (119861119879

119894119884 minus 119861

119879

119895119884) + (119883119862

119879

119894minus 119883119862119879

119895) (Φ119879

119895minus Φ119879

119894) 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]

]

(50)


Note that

Π119879

2+ Π2minus 1198752minus 12059111198761le Π119879

2(1198752+ 12059111198761)minus1

Π2 (51)


[

[

Σ119894119894Σ1119894119894

Σ2119894119894

lowast minus119877111198772119894119894


]

]

lt 0 1 le 119894 le 119903

[

[

[

Σ119894119895Σ1119894119895+ Σ1119895119894

Σ2119894119895


2119894119895

lowast lowast 1198773119894119895

]

]

]

lt 0 1 le 119894 lt 119895 le 119903

(52)

where

3119894119895=

[

[

[

[

[

minus119868 0 0 0


2(1198752+ 12059111198761)minus1

Π211986910119894119895

0



119894119895119868

]

]

]

]

]

1198773119894119895=

[

[

[

[

[

[

[

[

[

minus2119868 0 0 0 0 0


2(1198752+ 12059111198761)minus1

Π211986910119894119895

11986910119895119894

0 0




119894119895119868 0


119895119894119868

]

]

]

]

]

]

]

]

]

(53)

Now set = Π

2Πminus1

1 (54)



1 119868 119868 119868 Π

minus119879

1 119868 Πminus119879

2 119868 119868) (55)


[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Π1119894119894

Π2119894119894

11987511986021198941198751198603119894119875119863119894119894

12059111

119864119879

119894119894Γ119894119894

119875119872119894119894

119873119879

119894119894

lowast 1198691

0 0 0 12059112119864119879

1119894Γ1119894119894

0 0


0 0 0 0 Γ2119894

0 0


0 0 0 Γ3119894

0 0


119868 0 0 Γ4119894119894

0 0


0 0 0 0



2119869119879

2119872119894119894

0



119894119894119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 le 119903

[

[

Λ1119894119895

Λ2119894119895

Λ3119894119895

lowast Λ4119894119895

Λ5119894119895


]

]

lt 0 1 le 119894 lt 119895 le 119903

(56)

where

Λ1= 119867119879

(1198751+ 1205912

31198762minus 1minus 119879

1)119867

Π1119894119895= Λ1+ 119875119860119894119895+ 119860119879

119894119895119875

Λ2= 119867119879

(1minus 119879

2)

Π2119894119895= Λ2+ 1198751198601119894119895

1= Π119879

11198771Π1

(57)







119894= 0 and








1(119905) is 120583

1 then

(119905) = [1198601+ Δ1198601(119905)] 119909 (119905) + [119860

11+ Δ11986011(119905)] 119909 (119905 minus 120591

1)

+ [11986021+ Δ11986021(119905)] (119905 minus 120591

2)

+ [11986031+ Δ11986031(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198611+ Δ1198611(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) = [1198621+ Δ1198621(119905)] 119909 (119905) + 119862

11119909 (119905 minus 120591

1) + 11986321119908 (119905)

119911 (119905) = 1198641119909 (119905) + 119864

11119909 (119905 minus 120591

1) + 1198661119906 (119905)

(58)

Plant Rule 2 if 1199091(119905) is 120583

2 then

(119905) = [1198602+ Δ1198602(119905)] 119909 (119905) + [119860

12+ Δ11986012(119905)] 119909 (119905 minus 120591

1)

+ [11986022+ Δ11986022(119905)] (119905 minus 120591

2)

+ [11986032+ Δ11986032(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198612+ Δ1198612(119905)] 119906 (119905) + 119863

12119908 (119905)

119910 (119905) = [1198622+ Δ1198622(119905)] 119909 (119905) + 119862

12119909 (119905 minus 120591

1) + 11986322119908 (119905)

119911 (119905) = 1198642119909 (119905) + 119864

12119909 (119905 minus 120591

1) + 1198662119906 (119905)

(59)

where

1198601= [

005 0


11= [

005 002

minus005 001]

11986021= [

minus0002 0008

0006 0]

11986031= [

minus005 008

006 0] 119861

1= [

001 0

003 006]

1198621= [

minus003 minus01

02 002] 119862

11= [

minus003 minus001

002 002]

11986311= [

003

006] 119863

21= [

minus05

0]

1198641= [minus02 0] 119864

11= [minus002 0]

1198661= [0 minus003]

1198602= [

minus005 0


12= [

minus005 02

minus05 001]

11986022= [

0008 minus0002

minus0003 002]

11986032= [

005 minus02

minus03 02] 119861

2= [

001 minus008

002 005]

1198622= [

003 minus01

02 002]

11986212= [

minus003 minus001

002 02] 119863

12= [

minus001

002]

11986322= [

minus03

0]

1198642= [minus001 0] 119864

12= [minus001 0]

1198662= [0 002]

(60)

and Δ119860119894(119905) Δ119860

1119894(119905) Δ119860

2119894(119905) Δ119860

3119894(119905) Δ119861

119894(119905) Δ119862

119894(119905) (119894 =


1198721= [

minus002

0] 119873

01= [minus02 01]

11987311= [minus02 02]

11987321= [minus002 001] 119873

31= [minus002 001]

11987341= [minus002 001]

11987351= [minus002 001] 119872

2= [

003

0]

11987302= 11987301 119873

12= 11987311

11987322= 11987321 119873

32= 11987331

11987342= 11987341 119873

52= 11987351

(61)



(119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119860119894+ Δ119860119894(119905)] 119909 (119905)

+ [1198601119894+ Δ1198601119894(119905)] 119909 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [119861119894+ Δ119861119894(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119862119894+ Δ119862119894(119905)] 119909 (119905)

+1198621119894119909 (119905 minus 120591

1) + 1198632119894119908 (119905)

119911 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905)) 119864

119894119909 (119905) + 119864

1119894119909 (119905 minus 120591

1) + 119866119894119906 (119905)

(62)

where

ℎ1(1199091(119905)) =

1

3


2

3

+

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

1 for 1199091gt 1

ℎ2(1199091(119905)) =

2

3


1

3

minus

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

0 for 1199091gt 1

(63)






119908 (119905) =

1

119905 + 01

119905 ge 0 (64)


119883 = [

09868 06071

06071 13948]

119884 = [

129348 01814

01814 63384]

Ω1= [



Ω2= [

82076 minus37031

minus58461 53292]

Φ1= [

31848 00758


Φ2= [

20547 01643


1205951= [


57867 minus778562]

1205952= [



(65)


1198601198961= [

873277 minus680397

1399266 minus1072394]

1198601198962= [

minus249856 158880

111983 minus122840]

1198611198961= [

109115 minus03725

165830 19936]

1198611198962= [

14642 minus00212

31940 05922]

1198621198961= [

minus160423 127577

202647 minus148918]

1198621198962= [

142596 minus104929

32304 minus11934]

(66)








x1(t)

x2(t)

0 10 20 30 40 50 60minus02

0

02

04

06

08

1

12

Time (s)

Stat

e res

pons

ex(t)


2(119905) (dashed

line)

0 10 20 30 40 50 60minus100

minus50

0

50

100

150

200

Time (s)

Con

trol i

nput

u(t)

u1(t)

u2(t)


2(119905) (dashed

line)





infin




infincontrol


Acknowledgments


y1(t)

y2(t)

0 10 20 30 40 50 60minus1

0

1

2

3

4

5

Time (s)

Mea

sure

d ou

tput

y(t)


2(119905)

(dashed line)

z1(t)

z2(t)

0 10 20 30 40 50 60Time (s)

minus40

minus20

0

20

40

60

80

100

120

Con

trolle

d ou

tput

z(t)


2(119905)

(dashed line)


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Σ6119894119895=[

[

119879119895119894

0 11987941198792

0 11987910 0

0 0 0 0

]

]

11987711= diag (119876

21205742

119868 119881) 1198772119894119895=[

[

0 11986971198940 119879

3119894

0 1198698119894119895

0 0

0 0 0 0

]

]

2119894119895=[

[

0 1198697119894+ 1198697119895

0 0 119879

3119894119879

3119895

0 1198698119894119895+ 1198698119895119894

0 0 0 0

0 0 0 0 0 0

]

]

1198773119894119895=

[

[

[

[

minus119868 0 0 0

lowast minus119869911986910119894119895

0



119894119895119868

]

]

]

]

3119894119895=

[

[

[

[

[

[

[

[

[

minus2119868 0 0 0 0 0

lowast minus2119869911986910119894119895

11986910119895119894

0 0




119894119895119868 0


119895119894119868

]

]

]

]

]

]

]

]

]

1198774= diag (119875minus1

1119876minus1

2119868)

1198776119894119895=

[

[

[

[

0 0 0

0 0 1198796119894119895

0 0 0

0 0 0

]

]

]

]

6119894119895=

[

[

[

[

[

[

[

[

0 0

1198795119894119895

1198793119894119895

0 0

0 0

0 0

0 0

]

]

]

]

]

]

]

]

1198796119894119895= [

0

119884119860119879

1119894+ Φ1198951198621119894

]

1198777= diag (119868 119868 119868) 119877

8= diag (1

2

119875minus1

1

1

2

119876minus1

2119868 119868)

Π1= [

119883 119868

119883 0] Π

2= [

119883 119868

119883 0]

(47)


119860119870119894= (119883minus1

minus 119884)

minus1

(Ω119894minus 119884119860119894119883 minus 119884119861

119894Ψ119894minus Φ119894119862119894119883)119883minus1

119861119870119894= (119883minus1

minus 119884)

minus1

Φ119894 119862119870119894= Ψ119894119883minus1

1 le 119894 le 119903

(48)


[

[

Σ119894119894Σ1119894119894

Σ2119894119894

lowast minus119877111198772119894119894

lowast lowast 1198773119894119894

]

]

lt 0 1 le 119894 le 119903

[

[

[

Σ119894119895Σ1119894119895+ Σ1119895119894

Σ2119894119895


2119894119895


]

]

]

lt 0 1 le 119894 lt 119895 le 119903

(49)

where

Σ119894119895=

[

[

[

[

Θ119894119895+ Π119879

11198751Π1+ (1205913Π1)119879

1198762(1205913Π1) Θ1119894119895+ [

0 0

(1198841198601119894+ Φ1198951198621119894)119883 0

] Θ2119894

lowast minus119880 + 1198772+ 119877119879

20


]

]

]

]

Σ2119894119895=

[

[

[

[

Θ5119894119895

Θ6119894119895

Θ7119894119895

Θ8119894119895

1198693119894

1198694119894119895+ [

0 119883 (1198601119894119884 + 119862

119879

1119894Φ119879

119895)

0 0

] 0 1198695119894

0 1198696119894

0 119879

2119894

]

]

]

]

Σ119894119895= Σ119894119895+ Σ119895119894+

[

[

[

[

[

[

0 (119862119894119883 minus 119862

119895119883)

119879

(Φ119895minus Φ119894)

119879

+ (Ψ119895minus Ψ119894)

119879

(119884119861119894minus 119884119861119895)

119879

0 0 0

lowast 0 0 0 0

lowast lowast 0 0 0



]

]

]

]

]

]

Σ2119894119895= Σ2119894119895+[

[

0 (Ψ119879

119895minus Ψ119879

119894) (119861119879

119894119884 minus 119861

119879

119895119884) + (119883119862

119879

119894minus 119883119862119879

119895) (Φ119879

119895minus Φ119879

119894) 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]

]

(50)


Note that

Π119879

2+ Π2minus 1198752minus 12059111198761le Π119879

2(1198752+ 12059111198761)minus1

Π2 (51)


[

[

Σ119894119894Σ1119894119894

Σ2119894119894

lowast minus119877111198772119894119894


]

]

lt 0 1 le 119894 le 119903

[

[

[

Σ119894119895Σ1119894119895+ Σ1119895119894

Σ2119894119895


2119894119895

lowast lowast 1198773119894119895

]

]

]

lt 0 1 le 119894 lt 119895 le 119903

(52)

where

3119894119895=

[

[

[

[

[

minus119868 0 0 0


2(1198752+ 12059111198761)minus1

Π211986910119894119895

0



119894119895119868

]

]

]

]

]

1198773119894119895=

[

[

[

[

[

[

[

[

[

minus2119868 0 0 0 0 0


2(1198752+ 12059111198761)minus1

Π211986910119894119895

11986910119895119894

0 0




119894119895119868 0


119895119894119868

]

]

]

]

]

]

]

]

]

(53)

Now set = Π

2Πminus1

1 (54)



1 119868 119868 119868 Π

minus119879

1 119868 Πminus119879

2 119868 119868) (55)


[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Π1119894119894

Π2119894119894

11987511986021198941198751198603119894119875119863119894119894

12059111

119864119879

119894119894Γ119894119894

119875119872119894119894

119873119879

119894119894

lowast 1198691

0 0 0 12059112119864119879

1119894Γ1119894119894

0 0


0 0 0 0 Γ2119894

0 0


0 0 0 Γ3119894

0 0


119868 0 0 Γ4119894119894

0 0


0 0 0 0



2119869119879

2119872119894119894

0



119894119894119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 le 119903

[

[

Λ1119894119895

Λ2119894119895

Λ3119894119895

lowast Λ4119894119895

Λ5119894119895


]

]

lt 0 1 le 119894 lt 119895 le 119903

(56)

where

Λ1= 119867119879

(1198751+ 1205912

31198762minus 1minus 119879

1)119867

Π1119894119895= Λ1+ 119875119860119894119895+ 119860119879

119894119895119875

Λ2= 119867119879

(1minus 119879

2)

Π2119894119895= Λ2+ 1198751198601119894119895

1= Π119879

11198771Π1

(57)







119894= 0 and








1(119905) is 120583

1 then

(119905) = [1198601+ Δ1198601(119905)] 119909 (119905) + [119860

11+ Δ11986011(119905)] 119909 (119905 minus 120591

1)

+ [11986021+ Δ11986021(119905)] (119905 minus 120591

2)

+ [11986031+ Δ11986031(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198611+ Δ1198611(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) = [1198621+ Δ1198621(119905)] 119909 (119905) + 119862

11119909 (119905 minus 120591

1) + 11986321119908 (119905)

119911 (119905) = 1198641119909 (119905) + 119864

11119909 (119905 minus 120591

1) + 1198661119906 (119905)

(58)

Plant Rule 2 if 1199091(119905) is 120583

2 then

(119905) = [1198602+ Δ1198602(119905)] 119909 (119905) + [119860

12+ Δ11986012(119905)] 119909 (119905 minus 120591

1)

+ [11986022+ Δ11986022(119905)] (119905 minus 120591

2)

+ [11986032+ Δ11986032(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198612+ Δ1198612(119905)] 119906 (119905) + 119863

12119908 (119905)

119910 (119905) = [1198622+ Δ1198622(119905)] 119909 (119905) + 119862

12119909 (119905 minus 120591

1) + 11986322119908 (119905)

119911 (119905) = 1198642119909 (119905) + 119864

12119909 (119905 minus 120591

1) + 1198662119906 (119905)

(59)

where

1198601= [

005 0


11= [

005 002

minus005 001]

11986021= [

minus0002 0008

0006 0]

11986031= [

minus005 008

006 0] 119861

1= [

001 0

003 006]

1198621= [

minus003 minus01

02 002] 119862

11= [

minus003 minus001

002 002]

11986311= [

003

006] 119863

21= [

minus05

0]

1198641= [minus02 0] 119864

11= [minus002 0]

1198661= [0 minus003]

1198602= [

minus005 0


12= [

minus005 02

minus05 001]

11986022= [

0008 minus0002

minus0003 002]

11986032= [

005 minus02

minus03 02] 119861

2= [

001 minus008

002 005]

1198622= [

003 minus01

02 002]

11986212= [

minus003 minus001

002 02] 119863

12= [

minus001

002]

11986322= [

minus03

0]

1198642= [minus001 0] 119864

12= [minus001 0]

1198662= [0 002]

(60)

and Δ119860119894(119905) Δ119860

1119894(119905) Δ119860

2119894(119905) Δ119860

3119894(119905) Δ119861

119894(119905) Δ119862

119894(119905) (119894 =


1198721= [

minus002

0] 119873

01= [minus02 01]

11987311= [minus02 02]

11987321= [minus002 001] 119873

31= [minus002 001]

11987341= [minus002 001]

11987351= [minus002 001] 119872

2= [

003

0]

11987302= 11987301 119873

12= 11987311

11987322= 11987321 119873

32= 11987331

11987342= 11987341 119873

52= 11987351

(61)



(119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119860119894+ Δ119860119894(119905)] 119909 (119905)

+ [1198601119894+ Δ1198601119894(119905)] 119909 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [119861119894+ Δ119861119894(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119862119894+ Δ119862119894(119905)] 119909 (119905)

+1198621119894119909 (119905 minus 120591

1) + 1198632119894119908 (119905)

119911 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905)) 119864

119894119909 (119905) + 119864

1119894119909 (119905 minus 120591

1) + 119866119894119906 (119905)

(62)

where

ℎ1(1199091(119905)) =

1

3


2

3

+

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

1 for 1199091gt 1

ℎ2(1199091(119905)) =

2

3


1

3

minus

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

0 for 1199091gt 1

(63)






119908 (119905) =

1

119905 + 01

119905 ge 0 (64)


119883 = [

09868 06071

06071 13948]

119884 = [

129348 01814

01814 63384]

Ω1= [



Ω2= [

82076 minus37031

minus58461 53292]

Φ1= [

31848 00758


Φ2= [

20547 01643


1205951= [


57867 minus778562]

1205952= [



(65)


1198601198961= [

873277 minus680397

1399266 minus1072394]

1198601198962= [

minus249856 158880

111983 minus122840]

1198611198961= [

109115 minus03725

165830 19936]

1198611198962= [

14642 minus00212

31940 05922]

1198621198961= [

minus160423 127577

202647 minus148918]

1198621198962= [

142596 minus104929

32304 minus11934]

(66)








x1(t)

x2(t)

0 10 20 30 40 50 60minus02

0

02

04

06

08

1

12

Time (s)

Stat

e res

pons

ex(t)


2(119905) (dashed

line)

0 10 20 30 40 50 60minus100

minus50

0

50

100

150

200

Time (s)

Con

trol i

nput

u(t)

u1(t)

u2(t)


2(119905) (dashed

line)





infin




infincontrol


Acknowledgments


y1(t)

y2(t)

0 10 20 30 40 50 60minus1

0

1

2

3

4

5

Time (s)

Mea

sure

d ou

tput

y(t)


2(119905)

(dashed line)

z1(t)

z2(t)

0 10 20 30 40 50 60Time (s)

minus40

minus20

0

20

40

60

80

100

120

Con

trolle

d ou

tput

z(t)


2(119905)

(dashed line)


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Note that

Π119879

2+ Π2minus 1198752minus 12059111198761le Π119879

2(1198752+ 12059111198761)minus1

Π2 (51)


[

[

Σ119894119894Σ1119894119894

Σ2119894119894

lowast minus119877111198772119894119894


]

]

lt 0 1 le 119894 le 119903

[

[

[

Σ119894119895Σ1119894119895+ Σ1119895119894

Σ2119894119895


2119894119895

lowast lowast 1198773119894119895

]

]

]

lt 0 1 le 119894 lt 119895 le 119903

(52)

where

3119894119895=

[

[

[

[

[

minus119868 0 0 0


2(1198752+ 12059111198761)minus1

Π211986910119894119895

0



119894119895119868

]

]

]

]

]

1198773119894119895=

[

[

[

[

[

[

[

[

[

minus2119868 0 0 0 0 0


2(1198752+ 12059111198761)minus1

Π211986910119894119895

11986910119895119894

0 0




119894119895119868 0


119895119894119868

]

]

]

]

]

]

]

]

]

(53)

Now set = Π

2Πminus1

1 (54)



1 119868 119868 119868 Π

minus119879

1 119868 Πminus119879

2 119868 119868) (55)


[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Π1119894119894

Π2119894119894

11987511986021198941198751198603119894119875119863119894119894

12059111

119864119879

119894119894Γ119894119894

119875119872119894119894

119873119879

119894119894

lowast 1198691

0 0 0 12059112119864119879

1119894Γ1119894119894

0 0


0 0 0 0 Γ2119894

0 0


0 0 0 Γ3119894

0 0


119868 0 0 Γ4119894119894

0 0


0 0 0 0



2119869119879

2119872119894119894

0



119894119894119868

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

lt 0 1 le 119894 le 119903

[

[

Λ1119894119895

Λ2119894119895

Λ3119894119895

lowast Λ4119894119895

Λ5119894119895


]

]

lt 0 1 le 119894 lt 119895 le 119903

(56)

where

Λ1= 119867119879

(1198751+ 1205912

31198762minus 1minus 119879

1)119867

Π1119894119895= Λ1+ 119875119860119894119895+ 119860119879

119894119895119875

Λ2= 119867119879

(1minus 119879

2)

Π2119894119895= Λ2+ 1198751198601119894119895

1= Π119879

11198771Π1

(57)







119894= 0 and








1(119905) is 120583

1 then

(119905) = [1198601+ Δ1198601(119905)] 119909 (119905) + [119860

11+ Δ11986011(119905)] 119909 (119905 minus 120591

1)

+ [11986021+ Δ11986021(119905)] (119905 minus 120591

2)

+ [11986031+ Δ11986031(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198611+ Δ1198611(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) = [1198621+ Δ1198621(119905)] 119909 (119905) + 119862

11119909 (119905 minus 120591

1) + 11986321119908 (119905)

119911 (119905) = 1198641119909 (119905) + 119864

11119909 (119905 minus 120591

1) + 1198661119906 (119905)

(58)

Plant Rule 2 if 1199091(119905) is 120583

2 then

(119905) = [1198602+ Δ1198602(119905)] 119909 (119905) + [119860

12+ Δ11986012(119905)] 119909 (119905 minus 120591

1)

+ [11986022+ Δ11986022(119905)] (119905 minus 120591

2)

+ [11986032+ Δ11986032(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198612+ Δ1198612(119905)] 119906 (119905) + 119863

12119908 (119905)

119910 (119905) = [1198622+ Δ1198622(119905)] 119909 (119905) + 119862

12119909 (119905 minus 120591

1) + 11986322119908 (119905)

119911 (119905) = 1198642119909 (119905) + 119864

12119909 (119905 minus 120591

1) + 1198662119906 (119905)

(59)

where

1198601= [

005 0


11= [

005 002

minus005 001]

11986021= [

minus0002 0008

0006 0]

11986031= [

minus005 008

006 0] 119861

1= [

001 0

003 006]

1198621= [

minus003 minus01

02 002] 119862

11= [

minus003 minus001

002 002]

11986311= [

003

006] 119863

21= [

minus05

0]

1198641= [minus02 0] 119864

11= [minus002 0]

1198661= [0 minus003]

1198602= [

minus005 0


12= [

minus005 02

minus05 001]

11986022= [

0008 minus0002

minus0003 002]

11986032= [

005 minus02

minus03 02] 119861

2= [

001 minus008

002 005]

1198622= [

003 minus01

02 002]

11986212= [

minus003 minus001

002 02] 119863

12= [

minus001

002]

11986322= [

minus03

0]

1198642= [minus001 0] 119864

12= [minus001 0]

1198662= [0 002]

(60)

and Δ119860119894(119905) Δ119860

1119894(119905) Δ119860

2119894(119905) Δ119860

3119894(119905) Δ119861

119894(119905) Δ119862

119894(119905) (119894 =


1198721= [

minus002

0] 119873

01= [minus02 01]

11987311= [minus02 02]

11987321= [minus002 001] 119873

31= [minus002 001]

11987341= [minus002 001]

11987351= [minus002 001] 119872

2= [

003

0]

11987302= 11987301 119873

12= 11987311

11987322= 11987321 119873

32= 11987331

11987342= 11987341 119873

52= 11987351

(61)



(119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119860119894+ Δ119860119894(119905)] 119909 (119905)

+ [1198601119894+ Δ1198601119894(119905)] 119909 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [119861119894+ Δ119861119894(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119862119894+ Δ119862119894(119905)] 119909 (119905)

+1198621119894119909 (119905 minus 120591

1) + 1198632119894119908 (119905)

119911 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905)) 119864

119894119909 (119905) + 119864

1119894119909 (119905 minus 120591

1) + 119866119894119906 (119905)

(62)

where

ℎ1(1199091(119905)) =

1

3


2

3

+

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

1 for 1199091gt 1

ℎ2(1199091(119905)) =

2

3


1

3

minus

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

0 for 1199091gt 1

(63)






119908 (119905) =

1

119905 + 01

119905 ge 0 (64)


119883 = [

09868 06071

06071 13948]

119884 = [

129348 01814

01814 63384]

Ω1= [



Ω2= [

82076 minus37031

minus58461 53292]

Φ1= [

31848 00758


Φ2= [

20547 01643


1205951= [


57867 minus778562]

1205952= [



(65)


1198601198961= [

873277 minus680397

1399266 minus1072394]

1198601198962= [

minus249856 158880

111983 minus122840]

1198611198961= [

109115 minus03725

165830 19936]

1198611198962= [

14642 minus00212

31940 05922]

1198621198961= [

minus160423 127577

202647 minus148918]

1198621198962= [

142596 minus104929

32304 minus11934]

(66)








x1(t)

x2(t)

0 10 20 30 40 50 60minus02

0

02

04

06

08

1

12

Time (s)

Stat

e res

pons

ex(t)


2(119905) (dashed

line)

0 10 20 30 40 50 60minus100

minus50

0

50

100

150

200

Time (s)

Con

trol i

nput

u(t)

u1(t)

u2(t)


2(119905) (dashed

line)





infin




infincontrol


Acknowledgments


y1(t)

y2(t)

0 10 20 30 40 50 60minus1

0

1

2

3

4

5

Time (s)

Mea

sure

d ou

tput

y(t)


2(119905)

(dashed line)

z1(t)

z2(t)

0 10 20 30 40 50 60Time (s)

minus40

minus20

0

20

40

60

80

100

120

Con

trolle

d ou

tput

z(t)


2(119905)

(dashed line)


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119894= 0 and








1(119905) is 120583

1 then

(119905) = [1198601+ Δ1198601(119905)] 119909 (119905) + [119860

11+ Δ11986011(119905)] 119909 (119905 minus 120591

1)

+ [11986021+ Δ11986021(119905)] (119905 minus 120591

2)

+ [11986031+ Δ11986031(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198611+ Δ1198611(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) = [1198621+ Δ1198621(119905)] 119909 (119905) + 119862

11119909 (119905 minus 120591

1) + 11986321119908 (119905)

119911 (119905) = 1198641119909 (119905) + 119864

11119909 (119905 minus 120591

1) + 1198661119906 (119905)

(58)

Plant Rule 2 if 1199091(119905) is 120583

2 then

(119905) = [1198602+ Δ1198602(119905)] 119909 (119905) + [119860

12+ Δ11986012(119905)] 119909 (119905 minus 120591

1)

+ [11986022+ Δ11986022(119905)] (119905 minus 120591

2)

+ [11986032+ Δ11986032(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [1198612+ Δ1198612(119905)] 119906 (119905) + 119863

12119908 (119905)

119910 (119905) = [1198622+ Δ1198622(119905)] 119909 (119905) + 119862

12119909 (119905 minus 120591

1) + 11986322119908 (119905)

119911 (119905) = 1198642119909 (119905) + 119864

12119909 (119905 minus 120591

1) + 1198662119906 (119905)

(59)

where

1198601= [

005 0


11= [

005 002

minus005 001]

11986021= [

minus0002 0008

0006 0]

11986031= [

minus005 008

006 0] 119861

1= [

001 0

003 006]

1198621= [

minus003 minus01

02 002] 119862

11= [

minus003 minus001

002 002]

11986311= [

003

006] 119863

21= [

minus05

0]

1198641= [minus02 0] 119864

11= [minus002 0]

1198661= [0 minus003]

1198602= [

minus005 0


12= [

minus005 02

minus05 001]

11986022= [

0008 minus0002

minus0003 002]

11986032= [

005 minus02

minus03 02] 119861

2= [

001 minus008

002 005]

1198622= [

003 minus01

02 002]

11986212= [

minus003 minus001

002 02] 119863

12= [

minus001

002]

11986322= [

minus03

0]

1198642= [minus001 0] 119864

12= [minus001 0]

1198662= [0 002]

(60)

and Δ119860119894(119905) Δ119860

1119894(119905) Δ119860

2119894(119905) Δ119860

3119894(119905) Δ119861

119894(119905) Δ119862

119894(119905) (119894 =


1198721= [

minus002

0] 119873

01= [minus02 01]

11987311= [minus02 02]

11987321= [minus002 001] 119873

31= [minus002 001]

11987341= [minus002 001]

11987351= [minus002 001] 119872

2= [

003

0]

11987302= 11987301 119873

12= 11987311

11987322= 11987321 119873

32= 11987331

11987342= 11987341 119873

52= 11987351

(61)



(119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119860119894+ Δ119860119894(119905)] 119909 (119905)

+ [1198601119894+ Δ1198601119894(119905)] 119909 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [119861119894+ Δ119861119894(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119862119894+ Δ119862119894(119905)] 119909 (119905)

+1198621119894119909 (119905 minus 120591

1) + 1198632119894119908 (119905)

119911 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905)) 119864

119894119909 (119905) + 119864

1119894119909 (119905 minus 120591

1) + 119866119894119906 (119905)

(62)

where

ℎ1(1199091(119905)) =

1

3


2

3

+

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

1 for 1199091gt 1

ℎ2(1199091(119905)) =

2

3


1

3

minus

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

0 for 1199091gt 1

(63)






119908 (119905) =

1

119905 + 01

119905 ge 0 (64)


119883 = [

09868 06071

06071 13948]

119884 = [

129348 01814

01814 63384]

Ω1= [



Ω2= [

82076 minus37031

minus58461 53292]

Φ1= [

31848 00758


Φ2= [

20547 01643


1205951= [


57867 minus778562]

1205952= [



(65)


1198601198961= [

873277 minus680397

1399266 minus1072394]

1198601198962= [

minus249856 158880

111983 minus122840]

1198611198961= [

109115 minus03725

165830 19936]

1198611198962= [

14642 minus00212

31940 05922]

1198621198961= [

minus160423 127577

202647 minus148918]

1198621198962= [

142596 minus104929

32304 minus11934]

(66)








x1(t)

x2(t)

0 10 20 30 40 50 60minus02

0

02

04

06

08

1

12

Time (s)

Stat

e res

pons

ex(t)


2(119905) (dashed

line)

0 10 20 30 40 50 60minus100

minus50

0

50

100

150

200

Time (s)

Con

trol i

nput

u(t)

u1(t)

u2(t)


2(119905) (dashed

line)





infin




infincontrol


Acknowledgments


y1(t)

y2(t)

0 10 20 30 40 50 60minus1

0

1

2

3

4

5

Time (s)

Mea

sure

d ou

tput

y(t)


2(119905)

(dashed line)

z1(t)

z2(t)

0 10 20 30 40 50 60Time (s)

minus40

minus20

0

20

40

60

80

100

120

Con

trolle

d ou

tput

z(t)


2(119905)

(dashed line)


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119860119894+ Δ119860119894(119905)] 119909 (119905)

+ [1198601119894+ Δ1198601119894(119905)] 119909 (119905 minus 120591

1)

+ [1198602119894+ Δ1198602119894(119905)] (119905 minus 120591

2)

+ [1198603119894+ Δ1198603119894(119905)] int

119905

119905minus1205913

119909 (119904) 119889119904

+ [119861119894+ Δ119861119894(119905)] 119906 (119905) + 119863

1119894119908 (119905)

119910 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905))

times [119862119894+ Δ119862119894(119905)] 119909 (119905)

+1198621119894119909 (119905 minus 120591

1) + 1198632119894119908 (119905)

119911 (119905) =

2

sum

119894=1

ℎ119894(119904 (119905)) 119864

119894119909 (119905) + 119864

1119894119909 (119905 minus 120591

1) + 119866119894119906 (119905)

(62)

where

ℎ1(1199091(119905)) =

1

3


2

3

+

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

1 for 1199091gt 1

ℎ2(1199091(119905)) =

2

3


1

3

minus

1

3

1199091 for 100381610038161003816

10038161199091

1003816100381610038161003816le 1

0 for 1199091gt 1

(63)






119908 (119905) =

1

119905 + 01

119905 ge 0 (64)


119883 = [

09868 06071

06071 13948]

119884 = [

129348 01814

01814 63384]

Ω1= [



Ω2= [

82076 minus37031

minus58461 53292]

Φ1= [

31848 00758


Φ2= [

20547 01643


1205951= [


57867 minus778562]

1205952= [



(65)


1198601198961= [

873277 minus680397

1399266 minus1072394]

1198601198962= [

minus249856 158880

111983 minus122840]

1198611198961= [

109115 minus03725

165830 19936]

1198611198962= [

14642 minus00212

31940 05922]

1198621198961= [

minus160423 127577

202647 minus148918]

1198621198962= [

142596 minus104929

32304 minus11934]

(66)








x1(t)

x2(t)

0 10 20 30 40 50 60minus02

0

02

04

06

08

1

12

Time (s)

Stat

e res

pons

ex(t)


2(119905) (dashed

line)

0 10 20 30 40 50 60minus100

minus50

0

50

100

150

200

Time (s)

Con

trol i

nput

u(t)

u1(t)

u2(t)


2(119905) (dashed

line)





infin




infincontrol


Acknowledgments


y1(t)

y2(t)

0 10 20 30 40 50 60minus1

0

1

2

3

4

5

Time (s)

Mea

sure

d ou

tput

y(t)


2(119905)

(dashed line)

z1(t)

z2(t)

0 10 20 30 40 50 60Time (s)

minus40

minus20

0

20

40

60

80

100

120

Con

trolle

d ou

tput

z(t)


2(119905)

(dashed line)


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x1(t)

x2(t)

0 10 20 30 40 50 60minus02

0

02

04

06

08

1

12

Time (s)

Stat

e res

pons

ex(t)


2(119905) (dashed

line)

0 10 20 30 40 50 60minus100

minus50

0

50

100

150

200

Time (s)

Con

trol i

nput

u(t)

u1(t)

u2(t)


2(119905) (dashed

line)





infin




infincontrol


Acknowledgments


y1(t)

y2(t)

0 10 20 30 40 50 60minus1

0

1

2

3

4

5

Time (s)

Mea

sure

d ou

tput

y(t)


2(119905)

(dashed line)

z1(t)

z2(t)

0 10 20 30 40 50 60Time (s)

minus40

minus20

0

20

40

60

80

100

120

Con

trolle

d ou

tput

z(t)


2(119905)

(dashed line)


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article robust fuzzy output feedback control for...

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