Research ArticleDiscrete-Time Exponentially Stabilizing Fuzzy Sliding ModeControl via Lyapunovrsquos Method

Radiša C JovanoviT and Zoran M BuIevac

Faculty of Mechanical Engineering University of Belgrade 16 Kraljice Marije 11120 Belgrade 35 Serbia

Correspondence should be addressed to Radisa Z Jovanovic rjovanovicmasbgacrs

Received 30 July 2014 Revised 7 February 2015 Accepted 8 February 2015

Academic Editor RustomM Mamlook

Copyright copy 2015 R Z Jovanovic and Z M Bucevac This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The exponentially stabilizing state feedback control algorithm is developed by Lyapunovrsquos second method leading to the variablestructure system with chattering free sliding modes Linear time-invariant discrete-time second order plant is considered and thecontrol law is obtained by using a simple fuzzy controller The analytical structure of the proposed controller is derived and usedto prove exponential stability of sliding subspace Essentially the control algorithm drives the system from an arbitrary initial stateto a prescribed so-called sliding subspace S in finite time and with prescribed velocity estimate Inside the sliding subspace S thesystem is switched to the slidingmode regime and stays in it foreverThe proposed algorithm is tested on the real system in practiceDC servo motor and simulation and experimental results are given

1 Introduction

It is well-known that the variable structure control systems(VSCS) theory has been existing for several decades Almosteverything published in the area is related to the VSCSwith sliding modes with known features and advantagesUntil the mid-1980s the results concerned exclusively thecontinuous time type of these systemsWith the developmentof computer and digital technology the discrete-time versionof the problem has become more important in the academicresearch and industries Discrete-time sliding mode controlis an attempt to eliminate the problems caused by thediscretization of continuous-time controllers Great attentionwas directed to the existence of sliding mode regime insidethe so-called sliding subspace and finite system state reachingtime to the sliding subspace in this type of systems because itis quite different from its continuous counterpart Althoughdiscrete-time VSCS with sliding mode is characterized bya phenomenon that the actual so-called sliding control isapplied inside sliding subspace (see eg [1ndash3]) the chatteringproblem was raised Many different ways were proposed toreduce it or fully eliminate it [4] On the other side systemconvergence to the sliding subspace conditions was dealt

with where analogy to the continuous systems conditionswas massively exploited

Many of the authors treated special classes of the sys-tems most frequently linear systems systems in companioncanonical form and nonlinear systems with Lyapunovrsquossecond method as a powerful design and analysis technique[5 6] There are no papers with exponential stability ofsliding subspace except that some authors speak about expo-nential changes of in general case vector variable usuallydesignated for example by 119878 and used for sliding subspacedefinition as 119878 = 0 see [7] among others In [8] authors spokeabout exponential state vector changes on the way towardszero origin of a ball

Recently the integration of fuzzy techniques and conven-tional control approaches has been an active research focusFuzzy controllers are inherently nonlinear controllers andthe major advantage of fuzzy control technology over thetraditional control technology is its capability of capturingand utilizing qualitative human experience and knowledgein a quantitative manner through the use of fuzzy setsfuzzy rules and fuzzy logic [9] Based on the differences offuzzy control rules approaches to fuzzy logic control canbe roughly classified into the conventional fuzzy control

Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2015 Article ID 496085 10 pageshttpdxdoiorg1011552015496085

2 Advances in Fuzzy Systems

and Takagi-Sugeno (T-S) model based fuzzy control Theconventional fuzzy control systems are essentially heuristicand model free and various approaches have been developedfor stability analysis The key idea of these approaches isto regard a fuzzy controller as a nonlinear controller andembed the stability andor control design problem of fuzzycontrol systems into conventional nonlinear system stabilitytheory [10 11] T-S models are based on using a set of fuzzyrules to describe a global nonlinear system in terms of aset of local linear models which are smoothly connected byfuzzy membership functions and they provide a basis fordevelopment of systematic approaches to stability analysisand controller design of fuzzy control systems in view ofpowerful conventional control theory and techniques [12ndash14]

Combining fuzzy logic (FL) and sliding mode control(SMC) theory so-called fuzzy sliding mode control (FSMC)has the advantages of both SMC and FLC The analogybetween a simple and a sliding-mode controller with aboundary layer is shown in [15] In [16 17] for example fuzzycontrollers are designed to satisfy the sliding conditions andin these approaches the focus is on the design of slidingsurface via the fuzzy logic theory Some fuzzy control rulesto construct reaching control under the assumption thatequivalent control already exists is used in [18] A fuzzy-model-based controller which guarantees the stability of theclosed loop controlled system is suggested in [19] where theclosed-loop system consists of the TS fuzzy model and theswitching type fuzzy-model-based controller Fuzzy logic hasbeen also utilized to adapt the parameters of SMC to achievea better performance and especially attenuate the chatteringof control input

The above works are related to continuous systems anddiscrete systems were much less considered in the contextof FSMC The design of the fuzzy sliding mode control tomeet the requirement of necessary and sufficient reachingconditions of sliding mode control of discrete nonlinearsystem is considered in [20] A robust controller based onthe sliding mode and the dynamic T-S fuzzy state model fordiscrete systems is developed in [21]

One motivation for this work stems from the fact thatdiscrete-time type of the problem clearly makes place ofmicroprocessor compensator application in the systems Themainmotivation for the paper to be related to the exponentialstability of the FSMC as a higher quality stability propertygoes from the fact that in most of papers only stability ofsliding subspace 119878 was considered

Following Hui and Zakrsquos [3] conditions for chattering freesliding mode in this paper the exponential stabilizing fuzzycontrol algorithm is developed by Lyapunovrsquos secondmethodleading to the variable structure system with chatteringfree sliding modes for linear time-invariant second orderdiscrete-time system That is the exponential stability tobe in exact sense related to the sliding subspace as a setdefined over working point distance from the set duringits approaching to the set In the work essentially theobjective is to push the system state from an arbitrary initialposition to the sliding subspace 119878 in finite time and withestimate of approaching velocity defined by an exponentiallaw The velocity could not be smaller than the exponential

law defines Once the system reaches 119878 it stays there foreverworking in sliding mode regime which is chattering free aswell and with the system state asymptotic approaching to thezero equilibrium state The control law is obtained by usinga simple fuzzy controller and the analytical structure of theproposed controller is derived and used in the proof of thetheorem on exponential stability of sliding subspace

2 Problem Statement Notation andSome Definitions

Linear time-invariant second order discrete-time system isconsidered which is described by its state equation

119909 (119896 + 1) = 119860119909 (119896) + 119887119906 (119896) (1)

where 119896 isin 11987301198730= 0 1 2 119909(119896) is state vector at time 119896

119906(119896) isin 119877 is control vector at time 119896 119860 and 119887 are real constantmatrix and vector of appropriate dimensions respectivelyand (119860 119887) is assumed to be a controllable pair

For the system (1) define a hyperplane

119888119909 = 0 (2)

where 119888 = (1198881 1198882) is a constant nonzero row vector Thehyperplane is the so-called sliding subspace 119878 (in further textonly 119878) Clearly for second order system 119878 is sliding linedescribed by

119878 = 119909 119888119909 = 0 (3)

Also 119888119887 = 0 is assumedThe objective of this paper is to develop variable structure

type of state feedback control law

119906 = 119906 (119909) (4)

based on fuzzy logic which guarantees that the state120594(119896 119909(0) 119906(sdot)) of the system (1) reaches 119878 in finite time andwith velocity whose estimate is defined by an exponential lawOnce 119878has been reached the controller is required to keep thestate within it thereafter which means positive invariance of119878 relative to the systemmotion and what is denoted as slidingmode regime During this regime inside 119878 convergence tozero equilibrium with prescribed mode 120582 can be guaranteedif the 119888 has been appropriately chosen For 119888 choice in generalcase see [22]

More rigorously 119878 is positive invariant relative to the sys-tem (1)motion if and only if119909(0) isin 119878 implies120594(119896 119909(0) 119906(sdot)) isin119878 forall119896 isin 119873

0

Furthermore some other notations and definitions aregiven for the reason of their usage in theorems which arethe main results Real 119899-dimensional state space 119877119899 is withEuclidean norm denoted by sdot For 119878

119889 (119909 119878) = inf (1003817100381710038171003817119909 minus 1199101003817100381710038171003817 119910 isin 119878) (5)

is the distance between 119878 and the point 119909 isin 119877119899

Definition 1 The state 119909 = 0 of the system (1) (2) is stablein 119878 (with respect to 119878) iff forall120576 isin 119877

+ (119877+

=]0 +infin[)exist120575 = 120575(120576) isin 119877

+such that 119909(0) isin 119878 and 119909(0) lt 120575(120576)

Advances in Fuzzy Systems 3

implies that 120594(119896 119909(0) 119906(sdot)) exists 120594(119896 119909(0) 119906(sdot)) lt 120576 and120594(119896 119909(0) 119906(sdot)) isin 119878 forall119896 ge 0

Definition 2 The state 119909 = 0 of the system (1) (2) is attractive(globally) in 119878 (with respect to 119878) iffexistΔ gt 0 (Δ = infin) such that119909(0) isin 119878 and 119909(0) lt Δ implies that 120594(119896 119909(0) 119906(sdot)) existslim120594(119896 119909(0) 119906(sdot)) 119896 rarr +infin = 0 and 120594(119896 119909(0) 119906(sdot)) isin119878 forall119896 isin 119873

0

Definition 3 The state 119909 = 0 of the system (1) (2) is (globally)asymptotically stable in 119878 (with respect to 119878) iff it is both stableand (globally) attractive in 119878

Definition 4 The system (1) (2) is stable in 119878 (with respect to119878) iff its state 119909 = 0 is globally asymptotically stable in 119878

Definition 5 119878 is exponentially (globally) stable relative tothe system (1) (2) iff existΔ isin]0 +infin[ (Δ = +infin) and exist120572 isin

[1 +infin[ and 120573 isin]0 +infin[ such that distance 119889[119909(0) 119878] lt Δ

implies that 120594(119896 119909(0) 119906(sdot)) exists and 119889[120594(119896 119909(0) 119906(sdot)) 119878] le120572119889[119909(0) 119878]119890

minus120573119896 forall119896 gt 0

Remark 6 Previous definitions are valid and in the generalcase for 119909 isin 119877119899

The problem stated in the objective of the paper canbe reformulated as follows the objective of the paper is todevelop the control law (4) such that 119878 is globally exponentiallystable relative to the system (1) (2) and the system (1)(4) is stable in the sliding mode regime with appropriatelyprescribed modes Control law (4) will be developed usingfuzzy logic controller

3 Structure of FLC

The key idea of fuzzy sliding mode control is to integratefuzzy control and sliding mode control in such a way thatthe advantages of both techniques can be used One approachis to design conventional fuzzy control systems and slidingmode controller is used to determine best values for param-eters in fuzzy control rules Thereby stability is guaranteedand robust performance of the closed-loop control systems isimproved In another approach the control design is based onsliding mode techniques while the fuzzy controller is used asa complementary controller Also sliding mode control lawcan be directly substituted by a fuzzy controller

Sliding mode controllers generally involve a discon-tinuous control action which often results in chatteringphenomena due to imperfections in switching devices anddelays Commonly used methods for chattering eliminationare to replace the relay control by a saturation functionand boundary layer technique In some applications of fuzzysliding mode control the continuous switching functionof the boundary layer is replaced with equivalent fuzzyswitching function

In this paper the fuzzy controller is used in the reachingphase The controller should realize nonlinear control lawwhich will guarantee exponential stability of sliding subspace119878 This section introduces the principal structure of theproposed controller

Fuzzycontrol rules

x1 x1N

Gu

uFN uFx2Nx2 c2

c1 G

G

Figure 1 The proposed fuzzy logic controller

1

05

minusL L

120583(middot)

N P

xjN

Figure 2 The input membership functions

The fuzzy logic controller that will be evaluated is one ofthe simplest Figure 1 It employs only two input variables1199091(119896) and 119909

2(119896) Constant 119888

1and 1198882are component of vector

119888 from (2)

Scaling Factors The use of normalized domains requiresa scale transformation which maps the physical values ofthe input variables (119909

1and 119909

2in the present study) into

a normalized domain This is called input normalizationFurthermore output denormalization maps the normalizedvalue of the control output variable (119906FN) into its respectivephysical domain (119906F) The relationships between scalingfactors (119866 119866

119906) and the input and output variables are as

follows

1199091N (119896) = 119866 sdot 119888

1sdot 1199091(119896)

1199092N (119896) = 119866 sdot 119888

2sdot 1199092(119896)

119906F (119896) = 119866119906 sdot 119906FN (119896)

(6)

Fuzzification Module It converts instantaneous value of aprocess state variable into a linguistic value with the helpof the represented fuzzy set The parametric functionaldescription of the triangular shaped membership functionis the most economic one and hence it is considered hereThe membership functions of input variables are shown inFigure 2

Let 119909lowast119895be the one crisp input Then the fuzzified version

of 119909lowast119895after normalization is its degree of membership in

120583119873(119909lowast

119895N) and 120583119875(119909lowast

119895N) where119873 and 119875 are the linguistic valuestaken by119909

119895NHere symbols119873 and119875have commonmeaningsnegative and positive respectively

Assumption 7 The value 119909119895N(119896) satisfies

119909119895N (119896) isin [minus119871 119871] forall119895 = 1 2 forall119896 isin 119873

0 (7)

4 Advances in Fuzzy Systems

1

120583(middot)

minusH H

N PZ

uFN

Figure 3 The output membership functions

According to the previous assumption mathematicaldescription of the input membership functions is respec-tively given by

120583119873(119909119895N) =

minus119909119895N + 119871

2119871 120583119875(119909119895N) =

119909119895N + 119871

2119871 119895 = 1 2

(8)

Remark 8 It is noticed that

120583119873(119909119895N) + 120583119875 (119909119895N) = 1 119895 = 1 2 (9)

The membership functions for the normalized output(119906FN) are singleton and are shown in Figure 3 In Figures2 and 3 119871 and 119867 are two positive constants chosen by thedesigner which can be fixed after being determined

Remark 9 To ensure that Assumption 7 is valid one mustbe careful in choosing the scaling factor 119866 In practicalimplementation the maximum and minimum values ofvariables 119909

1and 119909

2are known and the factor 119866 is chosen so

that it satisfies the following conditions

119866 sdot 119888119895sdot10038161003816100381610038161003816119909119895

10038161003816100381610038161003816max le 119871 forall119895 = 1 2 (10)

Remark 10 For a second order system it is common to definethe row vector 119888 as 119888 = (1198881 1)

Fuzzy Control Rules Using the aforementioned membershipfunctions the following control rules are established for thefuzzy logic control part

1198771 If 1199091N is 119873 and 119909

2N is 119873 then 119906FN is 119873

1198772 If 1199091N is 119873 and 119909

2N is 119875 then 119906FN is 119885

1198773 If 1199091N is 119875 and 119909

2N is 119873 then 119906FN is 119885

1198774 If 1199091N is 119875 and 119909

2N is 119875 then 119906FN is 119875

(11)

Inference EngineThe basic function of the inference engine isto compute the overall value of control output variable basedon the individual contribution of each rule in the rule baseA degree of match for each rule is established by using thedefined membership functions

Here the antecedent of each rule is evaluated by using thetriangular norm (119905-norm) The 119905-norm used in this paper isintersection (AND function) which is mathematically givenas

120583119898(1199091N 1199092N) = 119879 (120583(sdot) (1199091N) 120583(sdot) (1199092N)) (sdot) = 119873 119875

(12)

Then based on this degree of match the clipped fuzzyset representing the value of the control output variable isdetermined via Mamdani inference method Thus outcomesof fuzzy rules are

120583119862119871119898

(119906FN) = min (120583119898 120583119880119898

(119906FN))

119880119898isin 119873119885 119875) 119898 = 1 2 3 4

(13)

Finally the clipped values for the control output of each rulepreviously denoted by 120583

119862119871119898 119898 = 1 2 3 4 are aggregated

thus forming the value of the overall control outputFrom the rule base it may be noted that the control

rules 1198772and 119877

3generate two memberships 120583

2and 120583

3which

have the same output fuzzy set defined by 120583119885(119906FN) For

such situations a combined membership is obtained by usingthe triangular conorm (119905-conorm or 119904-norm) The 119905-conormconsidered for this study is Lukasiewicz OR 119905-conorm

120583119879119862

23= min (1 120583

2+ 1205833) (14)

Defuzzification Module Defuzzification module converts theset of modified control output values into a crisp valueDefuzzification is done using the well-known COS (centerof sum) method According to this method and taking intoaccount the previously analyzed structure of the proposedfuzzy controller and the fact that the output fuzzy sets aresingleton the crisp value of control output is given by

119906FN =1205831sdot (minus119867) + 120583

119879119862

23sdot 0 + 120583

4sdot 119867

1205831+ 120583119879119862

23+ 1205834

(15)

Since for Lukasiewicz OR 119905-conorm it is valid that

120583119879119862

23= 1205832+ 1205833le 1 (16)

(15) can be written as

119906FN =1205831sdot (minus119867) + 1205832 sdot 0 + 1205833 sdot 0 + 1205834 sdot 119867

1205831+ 1205832+ 1205833+ 1205834

(17)

To obtain analytical expression of the proposed con-troller all combinations of input variables must be consid-ered If Assumption 7 holds there are eight input combina-tions (ICrsquos) as shown in Figure 4 The control rules 119877

1ndash1198774in

Advances in Fuzzy Systems 5

1

05

1

05

IC4 IC3

IC2

IC1

IC5

IC6

IC7 IC8

120583(middot)

120583(middot)

minusL

L

minusL L

P

P

N

N

x2N

x2N

x1N

x1N

Figure 4 The regions of the fuzzy controller input values

Table 1 Outcomes of fuzzy rules in IC regions

IC Fuzzy rules1198771

1198772

1198773

1198774

IC1 120583119873(1199091N) 120583

119873(1199091N) 120583

119873(1199092N) 120583

119875(1199092N)

IC2 120583119873(1199091N) 120583

119873(1199091N) 120583

119873(1199092N) 120583

119875(1199092N)

IC3 120583119873(1199092N) 120583

119873(1199091N) 120583

119873(1199092N) 120583

119875(1199091N)

IC4 120583119873(1199092N) 120583

119873(1199091N) 120583

119873(1199092N) 120583

119875(1199091N)

IC5 120583119873(1199092N) 120583

119875(1199092N) 120583

119875(1199091N) 120583

119875(1199091N)

IC6 120583119873(1199092N) 120583

119875(1199092N) 120583

119875(1199091N) 120583

119875(1199091N)

IC7 120583119873(1199091N) 120583

119875(1199092N) 120583

119875(1199091N) 120583

119875(1199092N)

IC8 120583119873(1199091N) 120583

119875(1199092N) 120583

119875(1199091N) 120583

119875(1199092N)

(11) are used to evaluate appropriate control law in each ICregion as in [23] The results of evaluating the fuzzy controlrules 119877

1ndash1198774are given in Table 1

Applying (17) to the results from Table 1 and taking intoaccount (6) and (8) analytical structure of the controller iseasily obtained as follows

119906F (119896) =

119867119866119866119906(11988811199091(119896) + 119888

21199092(119896))

2 (2119871 minus 1198661198881

10038161003816100381610038161199091 (119896)1003816100381610038161003816)

IC1 IC2 IC5 IC6

119867119866119866119906(11988811199091 (119896) + 11988821199092 (119896))

2 (2119871 minus 1198661198882

10038161003816100381610038161199092 (119896)1003816100381610038161003816)

IC3 IC4 IC7 IC8

(18)

Defining function 119892(1199091 1199092)

119892 (1199091 1199092) =

119867119866119866119906

2 (2119871 minus 1198661198881

10038161003816100381610038161199091 (119896)1003816100381610038161003816) IC1 IC2 IC5 IC6

119867119866119866119906

2 (2119871 minus 1198661198882

10038161003816100381610038161199092 (119896)1003816100381610038161003816) IC3 IC4 IC7 IC8

(19)

(18) can be written as

119906F (119896) = 119892 (1199091 1199092) 119888119909 (119896) (20)

It is obvious that the function 119892(1199091 1199092) is a nonlinear

function with the following minimal and maximal values

119892min =119867119866119866119906

4119871 119892max =

119867119866119866119906

2119871 (21)

and 119892(1199091 1199092) gt 0 forall119909

1 1199092isin 119877 One should not forget

that parameter 119866 must satisfy Assumption 7 The range ofthe value of function 119892(119909

1 1199092) is determined by the choice

of parameters 119867 119866 119871 and 119866119906 Without loss of generality it

can be assumed that 119871 = 119867 = 1 and they are commonlynormalized domain boundaries One example of function119892(1199091 1199092) is shown in Figure 5

4 Main Results

In this section the solution of the problem stated in Section 2is provided by the following theorems

6 Advances in Fuzzy Systems

0 005

05

1

102025

03035

04045

05055

06

minus1

minus1

minus05minus05

x1 x2

Figure 5 Example of function 119892(1199091 1199092) for 119871 = 119867 = 1119866 = 119866119906 = 1

and 1198881 = 1198882 = 1

Theorem 11 Applying

119906119900119904(119896) = minus

1

119888119887119888119860119909 (119896) (22)

the so-called one step control to the system (1) outside 119878 isnecessary and sufficient for 119909(119896) to reach the 119878 in one samplingperiod (one step)

ProofNecessity Let us assume that119909(119896) of the system (1) is arbitraryoutside of sliding subspace 119909(119896) notin 119878 and a control pushesit to the sliding subspace in one step (sampling period) Itfollows that 119909(119896 + 1) isin 119878 That fact description of 119878 and thesystem (1) state equation lead to

119888119860119909 (119896) + 119888119887119906 (119896) = 0 (23)

and finally

119906 (119896) = 119906os(119896) = minus

1

119888119887119888119860119909 (119896) (24)

Sufficiency When the system state is outside of slidingsubspace 119909(119896) notin 119878 the action of one step control is assumedState equation of the system (1) and expression of 119878 give

119888119909 (119896 + 1) = 119888119860119909 (119896) + 119888119887119906os(119896) = 119888119860119909 (119896) minus 119888119887

1

119888119887119888119860119909 (119896)

= 0

(25)

which means that 119909(119896 + 1) isin 119878

Remark 12 One step control can be encountered in VSCSpapers in different contexts see for example [3 4 13] amongothers

Theorem 13 Applying

119906119904119897(119896) = minus

1

119888119887119888119860119909 (119896) (26)

the actual so-called sliding control to the system (1) inside 119878 isnecessary and sufficient for 119878 to be positive invariant relative tothe system (1) solution 120594(119896 119909(0) 119906119904119897(119896))

ProofNecessity Positive invariance of 119878 relative to the system (1)solution 120594(119896 119909(0) 119906sl(119896)) is assumed So every time 119909(119896) isin 119878119909(119896 + 1) isin 119878 as well From the previous sentence descriptionof 119878 and state equation of the system (1) follows that

119888119860119909 (119896) + 119888119887119906sl(119896) = 0 (27)

and further

119906sl(119896) = minus

1

119888119887119888119860119909 (119896) (28)

Sufficiency Let us assume relative to system (1) 119909(119896) isin 119878 andcontrol (26) is applied Then state equation of the system (1)and expression for 119878 give

119888119909 (119896 + 1) = 119888119860119909 (119896) + 119888119887119906sl(119896) = 119888119860119909 (119896) minus 119888119887

1

119888119887119888119860119909 (119896)

= 0

(29)

which implies that 119909(119896 + 1) isin 119878

Remark 14 One step control and sliding control formally areidentical

The application of actual sliding control 119906sl(119896) (hence-forth referred to as 119906

sl(119896)) to the system (1) inside 119878 is

the phenomenon of discrete-time VSCS with sliding modesUsing this fact and the intention for developing of the variablestructure type of the controller for the system (1) the control(4) components could be more precisely specified as

119906 [119909 (119896)] =

119906+[119909 (119896)] 119888119909 (119896) gt 0

119906sl[119909 (119896)] 119888119909 (119896) = 0

119906minus[119909 (119896)] 119888119909 (119896) lt 0

(30)

Contrary to the 119906sl(119896) which is applied to the system(1) inside 119878 let us refer to the control which is applied tothe system (1) outside of 119878 as the outer control 119906ot(119896) Thefollowing theorems are basic for solving the stated problemby Lyapunovrsquos second method

Theorem15 Let the system (1) (30) be considered 119878 is globallyexponentially stable relative to the system (1) (30) if there existsscalar function V and numbers 120578

119894isin]0 +infin[ 119894 = 1 2 3 such that

(a) V(119909) isin 119862(1198772)(b) 1205781119889(119909 119878) le V(119909) le 120578

2119889(119909 119878) forall119909 isin (1198772)

(c) ΔV(119909) le minus1205783119889(119909 119878) forall119909 isin (1198772)

where ΔV(119909) is the first forward finite difference of V(119909)

Proof The proof starts with the assumption that sufficientconditions are fulfilled while the exponential stability of 119878should be shown

Advances in Fuzzy Systems 7

Condition (b) of the theorem implies

minus119889 [119909 (119896) 119878] le 120578minus1

2V [119909 (119896)] (31)

which with (c) give

ΔV [119909 (119896)] le minus120578minus121205783V [119909 (119896)] (32)

The last inequality after solving gives at first

V [119909 (119896)] le V [119909 (0)] 119890minus120573119896 120573 = ln1205782

1205782minus 1205783

1205782gt 1205783 forall119909 (0) isin 119877

2

(33)

and further by means of condition (a)

119889 [119909 (119896) 119878] le 120572119889 [119909 (0) 119878] 119890minus120573119896

120572 = 120578minus1

11205782 (34)

From (a) it is clear that 1205781le 1205782which yields 120572 isin [1 +infin[

and above relationship between 1205782and 1205783guarantee that 120573 isin

]0 +infin[ In that way it is shown that all stated by Definition 5is fulfilled that is the exponential stability of 119878 is proved

Remark 16 Previous theorem is valid and in the general casefor 119909 isin 119877119899

Scalar function V defined by

V (119909) = (sign 119888119909) 119888119909 (35)

as Lyapunovrsquos function candidate and outer control law

119906ot(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) isin 119877 119865 (119896) gt 0 forall119896 isin 1198730

(36)

are chosen relative to the system (1) Evidently such sortof 119906ot together with 119906

sl which is unique and already statedrepresents the control of (30) type The following theoremsare basic for solving of the stated problem by Lyapunovrsquossecond method

Theorem 17 119878 is globally exponentially stable relative to thesystem (1) if the following control is applied

119906 (119896) =

119906119900119905(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) = |119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816

119906119904119897(119896) = minus

1

119888119887[119888119860119909 (119896)]

(37)

with 119906119865(119896) defined by (19) (20) and with 119892(119909

1 1199092) lt 1

Proof Previously introduced Lyapunovrsquos function candidatefulfills evidently the conditions (a) of Theorem 15 Let usshow that conditions (b) and (c) are fulfilled Related to thecondition (b) and taking into account that |119888119909| = 120577119889 where120577 is a constant and 119889 is distance between 119909 and hyperplane119888119909 = 0 it follows that

V (119909) = (sign 119888119909) 119888119909 = |119888119909| = 120577119889 (38)

and further

1205781119889 (119909 119878) le V (119909) le 120578

2119889 (119909 119878)

1205781isin ]0 +infin[ le 120577 120578

2isin ]0 +infin[ ge 120577

(39)

State of system (1) at time 119896 is adopted to be out of 119878Then

ΔV [119909 (119896)] = V [119909 (119896 + 1)] minus V [119909 (119896)]

= 119865 (119896)1003816100381610038161003816sign 119888119909 (119896)

1003816100381610038161003816 minus |119888119909 (119896)|

= (|119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816) sdot1003816100381610038161003816sign 119888119909 (119896)

1003816100381610038161003816 minus |119888119909 (119896)|

= (1 minus 119892 (1199091 1199092)) |119888119909 (119896)| minus |119888119909 (119896)|

= minus119892 (1199091 1199092) |119888119909 (119896)| = minus119892 (1199091 1199092) 120577119889 [119909 (119896) 119878]

le minus119892min120577119889 [119909 (119896) 119878] = minus1205783119889 [119909 (119896) 119878]

(40)

in which way the proof is finished

According toTheorem 17

ΔV [119909 (119896)] = minus119892 (1199091 1199092) V [119909 (119896)] (41)

with 119892(1199091 1199092) gt 0 and 119892(119909

1 1199092) lt 1 which leads to the

following equation

V [119909 (119896 + 1)] = (1 minus 119892 (1199091 1199092)) V [119909 (119896)] = 120583 (1199091 1199092) V [119909 (119896)]

120583 (1199091 1199092) isin ]0 1[

(42)

From nonlinearity of the control low it is obvious thatcontrol gain is high when the system state is far fromsliding surfaces and as small as possible in neighborhoodof the sliding subspace Also it may be noted that V[119909(119896)]increments from step to step that is absolute value of |119888119909|are getting smaller over time

Function V[119909(119896)] is a decreasing function along motionof the system but it will never be zero Obviously the system(1) state never gets into 119878 Moreover from (42) it is obviousthat as the working point approaches the 119878 the value of thefunction 120583(119909

1 1199092) tends to its maximum value

120583max = 120583 (1199091 1199092)max = 1 minus 119892 (1199091 1199092)min (43)

Let 119874120590(119878) be 120590-neighborhood of 119878 and V

119898120590= minV(119909)

119909 is such that 119889(119909 119878) = 120590 To be guaranteed for 119909(119896) toenter 119874

120590(119878) settling time

119896119904= [0 log

120583max

V119898120590

V (0)] + 1 (44)

is necessary to elapse where 119896119904= [0 log

120583max(V119898120590V(0))] is

the biggest integer from the denoted segment After the statereached 119874

120590(119878) one step control 119906os should be applied to the

system (1) in which way finite reaching time in 119878 for 119909(119896) isrealized The previous facts are summarized by the followingtheorem

8 Advances in Fuzzy Systems

Theorem 18 119878 is globally exponentially stable relative to thesystem (1) with finite reaching time in 119878 if the following controlis applied

119906 (119896) =

119906119900119905(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) = |119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816 forall119896 = 0 119896119904minus 1

119906119900119905(119896119904) = 119906119900119904(119896119904) = minus

1

119888119887[119888119860119909 (119896)]

119906119904119897(119896) =minus

1

119888119887[119888119860119909 (119896)] forall119896 = 119896

119904+1 119896119904+ 2

(45)

Proof Compared to the proof of Theorem 17 here the proofis the same

5 Simulation and Experimental Results

The proposed algorithm was tested experimentally on a DCservo motor with the gear and load whose linear mathemat-ical model (without considering the major nonlinear effectsthe speed dependent friction dead zone and backlash) is asfollows

119869 120579 (119905) + 119861 120579 (119905) =120578119892120578119898119896119905119870119892

119877119898

119880 (119905) (46)

where 120579(119905) is the angular position of the load shaft 119869 is thetotal moment of inertia reflecting to the output shaft 119861 isthe viscous friction coefficient and 119880(119905) is the motor inputvoltage 119877

119898 119896119905 120578119898 120578119892 and 119870

119892are respectively the motor

armature resistance the motor torque constant the motorefficiency the gearbox efficiency and the total gear ratio

In the previous model it is assumed that the motorinductance is much smaller than the resistance so it isignored as the case with used servo motor is The experi-ments were performed with the Quanser rotary servo motorSRV02 This model is equipped with the optical encoderand tachometer for motor position and speed measuringrespectively Q8-USB data acquisition board for real-timedata acquisition and control was used with MatlabSimulinksoftware and QUARC real-time control The parametersnumerical values are as follows 119869 = 0021 (kgm2) 119861 =

0084 (Nmsrad) 119877119898= 26 (Ω) 119896

119905= 00077 (NmA) 120578

119898=

069 120578119892= 09 and119870

119892= 70

Choosing 1199091= 120579 and 119909

2= 120579 as state variables and 119906 =

119880(119905) as control at first the continuous time state equationwasobtained and finally after applying an accurate discretizationprocedure with sampling period 119879 = 001 (s) the discrete-time state equation of the system was obtained as follows

119909 (119896 + 1) = [1 008266

0 06746] 119909 (119896) + [

0002653

04979] 119906 (119896) (47)

The above mathematical model of real DC servo motorwas derived as adequate discrete-time state equation In theway the system matrix parameters 119860 and 119887 are availableControl algorithm from Theorem 18 is used to control themotor For simulation and experiment parameter 120590 = 0005

0 01 02 03 04 05

0

01

02

03

04

05

SimulationExperiment

minus02

minus01

x1

(rad

)

t (s)

Figure 6 The state variable 1199091

0 01 02 03 04 05

0123

SimulationExperiment

minus6

minus5

minus4

minus3

minus2

minus1

x2

(rad

s)

t (s)

Figure 7 The state variable 1199092

is adopted which defines very narrow 120590-neighborhood ofsliding subspace 119878 Also sampling time 119879 = 001 (s) andvector 119888 = (80 1) determined so as the system is stable insliding mode regime are adopted which gives 120577 = 119888 =

80006 Fuzzy logic controller is defined by 119871 = 119867 = 1119866 = 002 and 119866

119906= 40 which implies 119892min = 02 and

119892max = 04 Taking into account estimate relationships onone hand between 120577 and 120578

1and 120578

2 and on another hand

between 120577 119892min and 1205783 parameters 1205781= 30 120578

2= 90 and

1205783= 15 are chosen giving parameters of exponential low

120572 = 3 and 120573 = 01823 For initial state 119909(0) = (036 1)119879

after calculation V119898120590

= 04 V(0) = 298 and settling time119896119904= 20 are obtainedThe parallel simulation and experimental results are

shown in Figures 6ndash11 related to the algorithm fromTheorem 18 Figures 6 and 7 show the system state variablesThe control signal that drives the state onto the slidingsubspace is shown in Figure 9 The sliding function anddistance are shown in Figures 8 and 10 respectively From

Advances in Fuzzy Systems 9

0 01 02 03 04 05

0

10

20

30

40

50

cx

minus10

SimulationExperiment

t (s)

Figure 8 The sliding function 119888119909

0 01 02 03 04 05

0

2

4

minus10

minus8

minus6

minus4

minus2

SimulationExperiment

t (s)

u(V

)

Figure 9 The control variable

the figures it could be seen that the system state from initialposition approaches the sliding subspace 119878 but not slowerthan the exponential law defined by the parameters 120572 and120573 At instant 119896

119904= 20 (119905 = 02 sec) the state for the first

time enters the 120590-neighborhood of 119878 If it continues in thesame way it will never reach 119878 To reach it one step controlis applied at 119896

119904driving state in one step to 119878 so the character

of the system approaching 119878 changes at 119896119904 At 119896

119904+ 1 = 21

the system state is in 119878 staying in it thereafter forever Inside119878 the system is stable too so the system state asymptoticallyapproaches the zero state Simulation results fully confirm thetheoretical results and meet the expectations ofTheorems 1517 and 18

The experimental results have demonstrated that theproposed control scheme is valid and effective for the realapplications It is shown that there is no chattering in slidingmode However in practice errors could not be causedonly by computation but also by measurement accuracy(resolution sensitivity) noise existence nonlinearity andinadequate description of the real system by linear model as

0 01 02 03 04 05

002040608

112

minus04

minus02

Exponential law

t (s)

d(xS)

SimulationExperiment

Figure 10 The distance

015 02 025 03 035 04 045 05

0

001

002

003

004

005

minus002

minus001

Exponential law

d(xS)

SimulationExperiment

t (s)

Figure 11 Augmented part of Figure 10

well All this could cause not entering the 120590-neighborhood ofsliding subspace exactly in step 119896

119904than at some later instant

which is the case here in the paper experimental applicationexample After entering the 120590-neighborhood of 119878 at 119896

119904or

not and applying one step control at 119896119904 a little bit of sliding

subspace missing occurs Here that missing is a little bitmore visible at the beginning of reaching the sliding subspacebut very shortly after the like transition period expires itis again invisible like in the mathematical case simulationThis is illustrated by Figure 11 which is an enlarged segmentof Figure 10 for 119905 between 015 and 05 (sec) Evasion of theoccurrence is achieved by appropriate choice of the slidingsubspace and 120590-neighborhood parameters through vector 119888and parameter 120590 as well as the state space area where thelinear model is very adequate

It has already been pointed out that exponential stabilityis a higher quality stability property compared to ldquocommonrdquostability in Lyapunovrsquos sense (in this case it refers to globalasymptotic stability of a set-sliding subspace 119878) For that it is

10 Advances in Fuzzy Systems

natural that the sufficient conditions of Theorem 15 are morerigorous compared to the conditions of ldquocommonrdquo stabilityThe conditions of Theorem 15 are indirectly applicable inthe control of real-world processes through control law ofTheorems 17 and 18 for these control laws were derived fromthe conditions of Theorem 15 Real experiment with controlof DC servomotor is control of real-world process In the wayit is confirmed that the stability conditions ofTheorem 15 areapplicable in the control processes in practice

To resume in both ideal simulation case and in experi-mental application the proposed new developed algorithmin the paper guarantees exponential stability of 119878 and nochattering in sliding mode regime Despite the existence ofall upper stated nonidealities in the case of real applicationby the proper parameters choice the system motion is suchthat distance of working point from 119878 is always under pro-posed exponential envelope as it could be seen in providedexperiment

6 Conclusion

In the paper discrete-time type of VSCS with sliding modesis considered related to the linear time-invariant plant ofsecond orderThe fuzzy exponential stabilizing state feedbackcontrol algorithm so with prescribed approaching to thesliding subspace velocity estimate is developed strictly byLyapunovrsquos second method and with chattering free slidingmode Also application of the proposed ideal cases algo-rithm in practice is considered and discussed through theexperimental example It shows that experiment is quiteconsistent with mathematical ideal cases

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Z Bucevac ldquoLyapunovrsquos method approach in a stabilizingcontrol algorithm design for digital discrete VS systems withsliding modesndashlinear plant caserdquo in Proceedings of the 7thInternational Symposium Modelling Identification and Control( IASTED rsquo88) Grindelwald Switzerland l988

[2] G Bartolini A Ferrara and V I Utkin ldquoAdaptive sliding modecontrol in discrete-time systemsrdquo Automatica vol 31 no 5 pp769ndash773 1995

[3] S Hui and S H Zak ldquoOn discrete-time variable structureslidingmode controlrdquo SystemsampControl Letters vol 38 no 4-5pp 283ndash288 1999

[4] A Bartoszewicz ldquoDiscrete-time quasi-sliding-mode controlstrategiesrdquo IEEE Transactions on Industrial Electronics vol 45no 4 pp 633ndash637 1998

[5] W Gao Y Wang and A Homaifa ldquoDiscrete-time variablestructure control systemsrdquo IEEE Transactions on IndustrialElectronics vol 42 no 2 pp 117ndash122 1995

[6] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[7] Y Zheng and Y Jing ldquoApproximation law for discrete-timevariable structure control systemsrdquo Journal of Control Theoryand Applications vol 4 no 3 pp 291ndash296 2006

[8] G Golo and C Milosavljevic ldquoRobust discrete-time chatteringfree slidingmode controlrdquo SystemsampControl Letters vol 41 no1 pp 19ndash28 2000

[9] H Ying ldquoTheory and application of a novel fuzzy PID controllerusing a simplified Takagi-Sugeno rule schemerdquo InformationSciences vol 123 no 3 pp 281ndash293 2000

[10] G Feng ldquoA survey on analysis and design of model-based fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 5 pp 676ndash697 2006

[11] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[12] H K Lam F H F Leung and P K S Tam ldquoStable androbust fuzzy control for uncertain nonlinear systemsrdquo IEEETransactions on Systems Man and Cybernetics Part A Systemsand Humans vol 30 no 6 pp 825ndash840 2000

[13] R-E Precup M L Tomescu M-B Rdac E M Petriu S PreitlandC-A Dragos ldquoIterative performance improvement of fuzzycontrol systems for three tank systemsrdquo Expert Systems withApplications vol 39 no 9 pp 8288ndash8299 2012

[14] D H Lee Y H Joo and M H Tak ldquoLocal stability analysisof continuous-time Takagi-Sugeno fuzzy systems a fuzzy Lya-punov function approachrdquo Information Sciences vol 257 pp163ndash175 2014

[15] R Palm ldquoRobust control by fuzzy sliding moderdquo Automaticavol 30 no 9 pp 1429ndash1437 1994

[16] C-L Chen andM-H Chang ldquoOptimal design of fuzzy sliding-mode control a comparative studyrdquo Fuzzy Sets and Systems vol93 no 1 pp 37ndash48 1998

[17] J CWu andT S Liu ldquoA sliding-mode approach to fuzzy controldesignrdquo IEEE Transactions on Control Systems Technology vol4 no 2 pp 141ndash151 1996

[18] S-W Kim and J-J Lee ldquoDesign of a fuzzy controller with fuzzysliding surfacerdquo Fuzzy Sets and Systems vol 71 no 3 pp 359ndash367 1995

[19] G Feng S G Cao N W Rees and C K Chak ldquoDesign offuzzy control systems with guaranteed stabilityrdquo Fuzzy Sets andSystems vol 85 no 1 pp 1ndash10 1997

[20] F Qiao QM Zhu AWinfield andCMelhuish ldquoFuzzy slidingmode control for discrete nonlinear systemsrdquo Transactions ofChina Automation Society vol 22 no 2 pp 311ndash315 2003

[21] S H Cho ldquoRobust discrete time tracking control using slidingsurfacesrdquo KSME Journal vol 9 no 1 pp 80ndash90 1995

[22] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992

[23] H Ying W Siler and J J Buckley ldquoFuzzy control theory anonlinear caserdquo Automatica vol 26 no 3 pp 513ndash520 1990

Submit your manuscripts athttpwwwhindawicom

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Applied Computational Intelligence and Soft Computing

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Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

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httpwwwhindawicom Volume 2014

Advances in

Multimedia

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RoboticsJournal of

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Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

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2 Advances in Fuzzy Systems

and Takagi-Sugeno (T-S) model based fuzzy control Theconventional fuzzy control systems are essentially heuristicand model free and various approaches have been developedfor stability analysis The key idea of these approaches isto regard a fuzzy controller as a nonlinear controller andembed the stability andor control design problem of fuzzycontrol systems into conventional nonlinear system stabilitytheory [10 11] T-S models are based on using a set of fuzzyrules to describe a global nonlinear system in terms of aset of local linear models which are smoothly connected byfuzzy membership functions and they provide a basis fordevelopment of systematic approaches to stability analysisand controller design of fuzzy control systems in view ofpowerful conventional control theory and techniques [12ndash14]

Combining fuzzy logic (FL) and sliding mode control(SMC) theory so-called fuzzy sliding mode control (FSMC)has the advantages of both SMC and FLC The analogybetween a simple and a sliding-mode controller with aboundary layer is shown in [15] In [16 17] for example fuzzycontrollers are designed to satisfy the sliding conditions andin these approaches the focus is on the design of slidingsurface via the fuzzy logic theory Some fuzzy control rulesto construct reaching control under the assumption thatequivalent control already exists is used in [18] A fuzzy-model-based controller which guarantees the stability of theclosed loop controlled system is suggested in [19] where theclosed-loop system consists of the TS fuzzy model and theswitching type fuzzy-model-based controller Fuzzy logic hasbeen also utilized to adapt the parameters of SMC to achievea better performance and especially attenuate the chatteringof control input

The above works are related to continuous systems anddiscrete systems were much less considered in the contextof FSMC The design of the fuzzy sliding mode control tomeet the requirement of necessary and sufficient reachingconditions of sliding mode control of discrete nonlinearsystem is considered in [20] A robust controller based onthe sliding mode and the dynamic T-S fuzzy state model fordiscrete systems is developed in [21]

One motivation for this work stems from the fact thatdiscrete-time type of the problem clearly makes place ofmicroprocessor compensator application in the systems Themainmotivation for the paper to be related to the exponentialstability of the FSMC as a higher quality stability propertygoes from the fact that in most of papers only stability ofsliding subspace 119878 was considered

Following Hui and Zakrsquos [3] conditions for chattering freesliding mode in this paper the exponential stabilizing fuzzycontrol algorithm is developed by Lyapunovrsquos secondmethodleading to the variable structure system with chatteringfree sliding modes for linear time-invariant second orderdiscrete-time system That is the exponential stability tobe in exact sense related to the sliding subspace as a setdefined over working point distance from the set duringits approaching to the set In the work essentially theobjective is to push the system state from an arbitrary initialposition to the sliding subspace 119878 in finite time and withestimate of approaching velocity defined by an exponentiallaw The velocity could not be smaller than the exponential

law defines Once the system reaches 119878 it stays there foreverworking in sliding mode regime which is chattering free aswell and with the system state asymptotic approaching to thezero equilibrium state The control law is obtained by usinga simple fuzzy controller and the analytical structure of theproposed controller is derived and used in the proof of thetheorem on exponential stability of sliding subspace

2 Problem Statement Notation andSome Definitions

Linear time-invariant second order discrete-time system isconsidered which is described by its state equation

119909 (119896 + 1) = 119860119909 (119896) + 119887119906 (119896) (1)

where 119896 isin 11987301198730= 0 1 2 119909(119896) is state vector at time 119896

119906(119896) isin 119877 is control vector at time 119896 119860 and 119887 are real constantmatrix and vector of appropriate dimensions respectivelyand (119860 119887) is assumed to be a controllable pair

For the system (1) define a hyperplane

119888119909 = 0 (2)

where 119888 = (1198881 1198882) is a constant nonzero row vector Thehyperplane is the so-called sliding subspace 119878 (in further textonly 119878) Clearly for second order system 119878 is sliding linedescribed by

119878 = 119909 119888119909 = 0 (3)

Also 119888119887 = 0 is assumedThe objective of this paper is to develop variable structure

type of state feedback control law

119906 = 119906 (119909) (4)

based on fuzzy logic which guarantees that the state120594(119896 119909(0) 119906(sdot)) of the system (1) reaches 119878 in finite time andwith velocity whose estimate is defined by an exponential lawOnce 119878has been reached the controller is required to keep thestate within it thereafter which means positive invariance of119878 relative to the systemmotion and what is denoted as slidingmode regime During this regime inside 119878 convergence tozero equilibrium with prescribed mode 120582 can be guaranteedif the 119888 has been appropriately chosen For 119888 choice in generalcase see [22]

More rigorously 119878 is positive invariant relative to the sys-tem (1)motion if and only if119909(0) isin 119878 implies120594(119896 119909(0) 119906(sdot)) isin119878 forall119896 isin 119873

0

Furthermore some other notations and definitions aregiven for the reason of their usage in theorems which arethe main results Real 119899-dimensional state space 119877119899 is withEuclidean norm denoted by sdot For 119878

119889 (119909 119878) = inf (1003817100381710038171003817119909 minus 1199101003817100381710038171003817 119910 isin 119878) (5)

is the distance between 119878 and the point 119909 isin 119877119899

Definition 1 The state 119909 = 0 of the system (1) (2) is stablein 119878 (with respect to 119878) iff forall120576 isin 119877

+ (119877+

=]0 +infin[)exist120575 = 120575(120576) isin 119877

+such that 119909(0) isin 119878 and 119909(0) lt 120575(120576)

Advances in Fuzzy Systems 3

implies that 120594(119896 119909(0) 119906(sdot)) exists 120594(119896 119909(0) 119906(sdot)) lt 120576 and120594(119896 119909(0) 119906(sdot)) isin 119878 forall119896 ge 0

Definition 2 The state 119909 = 0 of the system (1) (2) is attractive(globally) in 119878 (with respect to 119878) iffexistΔ gt 0 (Δ = infin) such that119909(0) isin 119878 and 119909(0) lt Δ implies that 120594(119896 119909(0) 119906(sdot)) existslim120594(119896 119909(0) 119906(sdot)) 119896 rarr +infin = 0 and 120594(119896 119909(0) 119906(sdot)) isin119878 forall119896 isin 119873

0

Definition 3 The state 119909 = 0 of the system (1) (2) is (globally)asymptotically stable in 119878 (with respect to 119878) iff it is both stableand (globally) attractive in 119878

Definition 4 The system (1) (2) is stable in 119878 (with respect to119878) iff its state 119909 = 0 is globally asymptotically stable in 119878

Definition 5 119878 is exponentially (globally) stable relative tothe system (1) (2) iff existΔ isin]0 +infin[ (Δ = +infin) and exist120572 isin

[1 +infin[ and 120573 isin]0 +infin[ such that distance 119889[119909(0) 119878] lt Δ

implies that 120594(119896 119909(0) 119906(sdot)) exists and 119889[120594(119896 119909(0) 119906(sdot)) 119878] le120572119889[119909(0) 119878]119890

minus120573119896 forall119896 gt 0

Remark 6 Previous definitions are valid and in the generalcase for 119909 isin 119877119899

The problem stated in the objective of the paper canbe reformulated as follows the objective of the paper is todevelop the control law (4) such that 119878 is globally exponentiallystable relative to the system (1) (2) and the system (1)(4) is stable in the sliding mode regime with appropriatelyprescribed modes Control law (4) will be developed usingfuzzy logic controller

3 Structure of FLC

The key idea of fuzzy sliding mode control is to integratefuzzy control and sliding mode control in such a way thatthe advantages of both techniques can be used One approachis to design conventional fuzzy control systems and slidingmode controller is used to determine best values for param-eters in fuzzy control rules Thereby stability is guaranteedand robust performance of the closed-loop control systems isimproved In another approach the control design is based onsliding mode techniques while the fuzzy controller is used asa complementary controller Also sliding mode control lawcan be directly substituted by a fuzzy controller

Sliding mode controllers generally involve a discon-tinuous control action which often results in chatteringphenomena due to imperfections in switching devices anddelays Commonly used methods for chattering eliminationare to replace the relay control by a saturation functionand boundary layer technique In some applications of fuzzysliding mode control the continuous switching functionof the boundary layer is replaced with equivalent fuzzyswitching function

In this paper the fuzzy controller is used in the reachingphase The controller should realize nonlinear control lawwhich will guarantee exponential stability of sliding subspace119878 This section introduces the principal structure of theproposed controller

Fuzzycontrol rules

x1 x1N

Gu

uFN uFx2Nx2 c2

c1 G

G

Figure 1 The proposed fuzzy logic controller

1

05

minusL L

120583(middot)

N P

xjN

Figure 2 The input membership functions

The fuzzy logic controller that will be evaluated is one ofthe simplest Figure 1 It employs only two input variables1199091(119896) and 119909

2(119896) Constant 119888

1and 1198882are component of vector

119888 from (2)

Scaling Factors The use of normalized domains requiresa scale transformation which maps the physical values ofthe input variables (119909

1and 119909

2in the present study) into

a normalized domain This is called input normalizationFurthermore output denormalization maps the normalizedvalue of the control output variable (119906FN) into its respectivephysical domain (119906F) The relationships between scalingfactors (119866 119866

119906) and the input and output variables are as

follows

1199091N (119896) = 119866 sdot 119888

1sdot 1199091(119896)

1199092N (119896) = 119866 sdot 119888

2sdot 1199092(119896)

119906F (119896) = 119866119906 sdot 119906FN (119896)

(6)

Fuzzification Module It converts instantaneous value of aprocess state variable into a linguistic value with the helpof the represented fuzzy set The parametric functionaldescription of the triangular shaped membership functionis the most economic one and hence it is considered hereThe membership functions of input variables are shown inFigure 2

Let 119909lowast119895be the one crisp input Then the fuzzified version

of 119909lowast119895after normalization is its degree of membership in

120583119873(119909lowast

119895N) and 120583119875(119909lowast

119895N) where119873 and 119875 are the linguistic valuestaken by119909

119895NHere symbols119873 and119875have commonmeaningsnegative and positive respectively

Assumption 7 The value 119909119895N(119896) satisfies

119909119895N (119896) isin [minus119871 119871] forall119895 = 1 2 forall119896 isin 119873

0 (7)

4 Advances in Fuzzy Systems

1

120583(middot)

minusH H

N PZ

uFN

Figure 3 The output membership functions

According to the previous assumption mathematicaldescription of the input membership functions is respec-tively given by

120583119873(119909119895N) =

minus119909119895N + 119871

2119871 120583119875(119909119895N) =

119909119895N + 119871

2119871 119895 = 1 2

(8)

Remark 8 It is noticed that

120583119873(119909119895N) + 120583119875 (119909119895N) = 1 119895 = 1 2 (9)

The membership functions for the normalized output(119906FN) are singleton and are shown in Figure 3 In Figures2 and 3 119871 and 119867 are two positive constants chosen by thedesigner which can be fixed after being determined

Remark 9 To ensure that Assumption 7 is valid one mustbe careful in choosing the scaling factor 119866 In practicalimplementation the maximum and minimum values ofvariables 119909

1and 119909

2are known and the factor 119866 is chosen so

that it satisfies the following conditions

119866 sdot 119888119895sdot10038161003816100381610038161003816119909119895

10038161003816100381610038161003816max le 119871 forall119895 = 1 2 (10)

Remark 10 For a second order system it is common to definethe row vector 119888 as 119888 = (1198881 1)

Fuzzy Control Rules Using the aforementioned membershipfunctions the following control rules are established for thefuzzy logic control part

1198771 If 1199091N is 119873 and 119909

2N is 119873 then 119906FN is 119873

1198772 If 1199091N is 119873 and 119909

2N is 119875 then 119906FN is 119885

1198773 If 1199091N is 119875 and 119909

2N is 119873 then 119906FN is 119885

1198774 If 1199091N is 119875 and 119909

2N is 119875 then 119906FN is 119875

(11)

Inference EngineThe basic function of the inference engine isto compute the overall value of control output variable basedon the individual contribution of each rule in the rule baseA degree of match for each rule is established by using thedefined membership functions

Here the antecedent of each rule is evaluated by using thetriangular norm (119905-norm) The 119905-norm used in this paper isintersection (AND function) which is mathematically givenas

120583119898(1199091N 1199092N) = 119879 (120583(sdot) (1199091N) 120583(sdot) (1199092N)) (sdot) = 119873 119875

(12)

Then based on this degree of match the clipped fuzzyset representing the value of the control output variable isdetermined via Mamdani inference method Thus outcomesof fuzzy rules are

120583119862119871119898

(119906FN) = min (120583119898 120583119880119898

(119906FN))

119880119898isin 119873119885 119875) 119898 = 1 2 3 4

(13)

Finally the clipped values for the control output of each rulepreviously denoted by 120583

119862119871119898 119898 = 1 2 3 4 are aggregated

thus forming the value of the overall control outputFrom the rule base it may be noted that the control

rules 1198772and 119877

3generate two memberships 120583

2and 120583

3which

have the same output fuzzy set defined by 120583119885(119906FN) For

such situations a combined membership is obtained by usingthe triangular conorm (119905-conorm or 119904-norm) The 119905-conormconsidered for this study is Lukasiewicz OR 119905-conorm

120583119879119862

23= min (1 120583

2+ 1205833) (14)

Defuzzification Module Defuzzification module converts theset of modified control output values into a crisp valueDefuzzification is done using the well-known COS (centerof sum) method According to this method and taking intoaccount the previously analyzed structure of the proposedfuzzy controller and the fact that the output fuzzy sets aresingleton the crisp value of control output is given by

119906FN =1205831sdot (minus119867) + 120583

119879119862

23sdot 0 + 120583

4sdot 119867

1205831+ 120583119879119862

23+ 1205834

(15)

Since for Lukasiewicz OR 119905-conorm it is valid that

120583119879119862

23= 1205832+ 1205833le 1 (16)

(15) can be written as

119906FN =1205831sdot (minus119867) + 1205832 sdot 0 + 1205833 sdot 0 + 1205834 sdot 119867

1205831+ 1205832+ 1205833+ 1205834

(17)

To obtain analytical expression of the proposed con-troller all combinations of input variables must be consid-ered If Assumption 7 holds there are eight input combina-tions (ICrsquos) as shown in Figure 4 The control rules 119877

1ndash1198774in

Advances in Fuzzy Systems 5

1

05

1

05

IC4 IC3

IC2

IC1

IC5

IC6

IC7 IC8

120583(middot)

120583(middot)

minusL

L

minusL L

P

P

N

N

x2N

x2N

x1N

x1N

Figure 4 The regions of the fuzzy controller input values

Table 1 Outcomes of fuzzy rules in IC regions

IC Fuzzy rules1198771

1198772

1198773

1198774

IC1 120583119873(1199091N) 120583

119873(1199091N) 120583

119873(1199092N) 120583

119875(1199092N)

IC2 120583119873(1199091N) 120583

119873(1199091N) 120583

119873(1199092N) 120583

119875(1199092N)

IC3 120583119873(1199092N) 120583

119873(1199091N) 120583

119873(1199092N) 120583

119875(1199091N)

IC4 120583119873(1199092N) 120583

119873(1199091N) 120583

119873(1199092N) 120583

119875(1199091N)

IC5 120583119873(1199092N) 120583

119875(1199092N) 120583

119875(1199091N) 120583

119875(1199091N)

IC6 120583119873(1199092N) 120583

119875(1199092N) 120583

119875(1199091N) 120583

119875(1199091N)

IC7 120583119873(1199091N) 120583

119875(1199092N) 120583

119875(1199091N) 120583

119875(1199092N)

IC8 120583119873(1199091N) 120583

119875(1199092N) 120583

119875(1199091N) 120583

119875(1199092N)

(11) are used to evaluate appropriate control law in each ICregion as in [23] The results of evaluating the fuzzy controlrules 119877

1ndash1198774are given in Table 1

Applying (17) to the results from Table 1 and taking intoaccount (6) and (8) analytical structure of the controller iseasily obtained as follows

119906F (119896) =

119867119866119866119906(11988811199091(119896) + 119888

21199092(119896))

2 (2119871 minus 1198661198881

10038161003816100381610038161199091 (119896)1003816100381610038161003816)

IC1 IC2 IC5 IC6

119867119866119866119906(11988811199091 (119896) + 11988821199092 (119896))

2 (2119871 minus 1198661198882

10038161003816100381610038161199092 (119896)1003816100381610038161003816)

IC3 IC4 IC7 IC8

(18)

Defining function 119892(1199091 1199092)

119892 (1199091 1199092) =

119867119866119866119906

2 (2119871 minus 1198661198881

10038161003816100381610038161199091 (119896)1003816100381610038161003816) IC1 IC2 IC5 IC6

119867119866119866119906

2 (2119871 minus 1198661198882

10038161003816100381610038161199092 (119896)1003816100381610038161003816) IC3 IC4 IC7 IC8

(19)

(18) can be written as

119906F (119896) = 119892 (1199091 1199092) 119888119909 (119896) (20)

It is obvious that the function 119892(1199091 1199092) is a nonlinear

function with the following minimal and maximal values

119892min =119867119866119866119906

4119871 119892max =

119867119866119866119906

2119871 (21)

and 119892(1199091 1199092) gt 0 forall119909

1 1199092isin 119877 One should not forget

that parameter 119866 must satisfy Assumption 7 The range ofthe value of function 119892(119909

1 1199092) is determined by the choice

of parameters 119867 119866 119871 and 119866119906 Without loss of generality it

can be assumed that 119871 = 119867 = 1 and they are commonlynormalized domain boundaries One example of function119892(1199091 1199092) is shown in Figure 5

4 Main Results

In this section the solution of the problem stated in Section 2is provided by the following theorems

6 Advances in Fuzzy Systems

0 005

05

1

102025

03035

04045

05055

06

minus1

minus1

minus05minus05

x1 x2

Figure 5 Example of function 119892(1199091 1199092) for 119871 = 119867 = 1119866 = 119866119906 = 1

and 1198881 = 1198882 = 1

Theorem 11 Applying

119906119900119904(119896) = minus

1

119888119887119888119860119909 (119896) (22)

the so-called one step control to the system (1) outside 119878 isnecessary and sufficient for 119909(119896) to reach the 119878 in one samplingperiod (one step)

ProofNecessity Let us assume that119909(119896) of the system (1) is arbitraryoutside of sliding subspace 119909(119896) notin 119878 and a control pushesit to the sliding subspace in one step (sampling period) Itfollows that 119909(119896 + 1) isin 119878 That fact description of 119878 and thesystem (1) state equation lead to

119888119860119909 (119896) + 119888119887119906 (119896) = 0 (23)

and finally

119906 (119896) = 119906os(119896) = minus

1

119888119887119888119860119909 (119896) (24)

Sufficiency When the system state is outside of slidingsubspace 119909(119896) notin 119878 the action of one step control is assumedState equation of the system (1) and expression of 119878 give

119888119909 (119896 + 1) = 119888119860119909 (119896) + 119888119887119906os(119896) = 119888119860119909 (119896) minus 119888119887

1

119888119887119888119860119909 (119896)

= 0

(25)

which means that 119909(119896 + 1) isin 119878

Remark 12 One step control can be encountered in VSCSpapers in different contexts see for example [3 4 13] amongothers

Theorem 13 Applying

119906119904119897(119896) = minus

1

119888119887119888119860119909 (119896) (26)

the actual so-called sliding control to the system (1) inside 119878 isnecessary and sufficient for 119878 to be positive invariant relative tothe system (1) solution 120594(119896 119909(0) 119906119904119897(119896))

ProofNecessity Positive invariance of 119878 relative to the system (1)solution 120594(119896 119909(0) 119906sl(119896)) is assumed So every time 119909(119896) isin 119878119909(119896 + 1) isin 119878 as well From the previous sentence descriptionof 119878 and state equation of the system (1) follows that

119888119860119909 (119896) + 119888119887119906sl(119896) = 0 (27)

and further

119906sl(119896) = minus

1

119888119887119888119860119909 (119896) (28)

Sufficiency Let us assume relative to system (1) 119909(119896) isin 119878 andcontrol (26) is applied Then state equation of the system (1)and expression for 119878 give

119888119909 (119896 + 1) = 119888119860119909 (119896) + 119888119887119906sl(119896) = 119888119860119909 (119896) minus 119888119887

1

119888119887119888119860119909 (119896)

= 0

(29)

which implies that 119909(119896 + 1) isin 119878

Remark 14 One step control and sliding control formally areidentical

The application of actual sliding control 119906sl(119896) (hence-forth referred to as 119906

sl(119896)) to the system (1) inside 119878 is

the phenomenon of discrete-time VSCS with sliding modesUsing this fact and the intention for developing of the variablestructure type of the controller for the system (1) the control(4) components could be more precisely specified as

119906 [119909 (119896)] =

119906+[119909 (119896)] 119888119909 (119896) gt 0

119906sl[119909 (119896)] 119888119909 (119896) = 0

119906minus[119909 (119896)] 119888119909 (119896) lt 0

(30)

Contrary to the 119906sl(119896) which is applied to the system(1) inside 119878 let us refer to the control which is applied tothe system (1) outside of 119878 as the outer control 119906ot(119896) Thefollowing theorems are basic for solving the stated problemby Lyapunovrsquos second method

Theorem15 Let the system (1) (30) be considered 119878 is globallyexponentially stable relative to the system (1) (30) if there existsscalar function V and numbers 120578

119894isin]0 +infin[ 119894 = 1 2 3 such that

(a) V(119909) isin 119862(1198772)(b) 1205781119889(119909 119878) le V(119909) le 120578

2119889(119909 119878) forall119909 isin (1198772)

(c) ΔV(119909) le minus1205783119889(119909 119878) forall119909 isin (1198772)

where ΔV(119909) is the first forward finite difference of V(119909)

Proof The proof starts with the assumption that sufficientconditions are fulfilled while the exponential stability of 119878should be shown

Advances in Fuzzy Systems 7

Condition (b) of the theorem implies

minus119889 [119909 (119896) 119878] le 120578minus1

2V [119909 (119896)] (31)

which with (c) give

ΔV [119909 (119896)] le minus120578minus121205783V [119909 (119896)] (32)

The last inequality after solving gives at first

V [119909 (119896)] le V [119909 (0)] 119890minus120573119896 120573 = ln1205782

1205782minus 1205783

1205782gt 1205783 forall119909 (0) isin 119877

2

(33)

and further by means of condition (a)

119889 [119909 (119896) 119878] le 120572119889 [119909 (0) 119878] 119890minus120573119896

120572 = 120578minus1

11205782 (34)

From (a) it is clear that 1205781le 1205782which yields 120572 isin [1 +infin[

and above relationship between 1205782and 1205783guarantee that 120573 isin

]0 +infin[ In that way it is shown that all stated by Definition 5is fulfilled that is the exponential stability of 119878 is proved

Remark 16 Previous theorem is valid and in the general casefor 119909 isin 119877119899

Scalar function V defined by

V (119909) = (sign 119888119909) 119888119909 (35)

as Lyapunovrsquos function candidate and outer control law

119906ot(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) isin 119877 119865 (119896) gt 0 forall119896 isin 1198730

(36)

are chosen relative to the system (1) Evidently such sortof 119906ot together with 119906

sl which is unique and already statedrepresents the control of (30) type The following theoremsare basic for solving of the stated problem by Lyapunovrsquossecond method

Theorem 17 119878 is globally exponentially stable relative to thesystem (1) if the following control is applied

119906 (119896) =

119906119900119905(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) = |119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816

119906119904119897(119896) = minus

1

119888119887[119888119860119909 (119896)]

(37)

with 119906119865(119896) defined by (19) (20) and with 119892(119909

1 1199092) lt 1

Proof Previously introduced Lyapunovrsquos function candidatefulfills evidently the conditions (a) of Theorem 15 Let usshow that conditions (b) and (c) are fulfilled Related to thecondition (b) and taking into account that |119888119909| = 120577119889 where120577 is a constant and 119889 is distance between 119909 and hyperplane119888119909 = 0 it follows that

V (119909) = (sign 119888119909) 119888119909 = |119888119909| = 120577119889 (38)

and further

1205781119889 (119909 119878) le V (119909) le 120578

2119889 (119909 119878)

1205781isin ]0 +infin[ le 120577 120578

2isin ]0 +infin[ ge 120577

(39)

State of system (1) at time 119896 is adopted to be out of 119878Then

ΔV [119909 (119896)] = V [119909 (119896 + 1)] minus V [119909 (119896)]

= 119865 (119896)1003816100381610038161003816sign 119888119909 (119896)

1003816100381610038161003816 minus |119888119909 (119896)|

= (|119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816) sdot1003816100381610038161003816sign 119888119909 (119896)

1003816100381610038161003816 minus |119888119909 (119896)|

= (1 minus 119892 (1199091 1199092)) |119888119909 (119896)| minus |119888119909 (119896)|

= minus119892 (1199091 1199092) |119888119909 (119896)| = minus119892 (1199091 1199092) 120577119889 [119909 (119896) 119878]

le minus119892min120577119889 [119909 (119896) 119878] = minus1205783119889 [119909 (119896) 119878]

(40)

in which way the proof is finished

According toTheorem 17

ΔV [119909 (119896)] = minus119892 (1199091 1199092) V [119909 (119896)] (41)

with 119892(1199091 1199092) gt 0 and 119892(119909

1 1199092) lt 1 which leads to the

following equation

V [119909 (119896 + 1)] = (1 minus 119892 (1199091 1199092)) V [119909 (119896)] = 120583 (1199091 1199092) V [119909 (119896)]

120583 (1199091 1199092) isin ]0 1[

(42)

From nonlinearity of the control low it is obvious thatcontrol gain is high when the system state is far fromsliding surfaces and as small as possible in neighborhoodof the sliding subspace Also it may be noted that V[119909(119896)]increments from step to step that is absolute value of |119888119909|are getting smaller over time

Function V[119909(119896)] is a decreasing function along motionof the system but it will never be zero Obviously the system(1) state never gets into 119878 Moreover from (42) it is obviousthat as the working point approaches the 119878 the value of thefunction 120583(119909

1 1199092) tends to its maximum value

120583max = 120583 (1199091 1199092)max = 1 minus 119892 (1199091 1199092)min (43)

Let 119874120590(119878) be 120590-neighborhood of 119878 and V

119898120590= minV(119909)

119909 is such that 119889(119909 119878) = 120590 To be guaranteed for 119909(119896) toenter 119874

120590(119878) settling time

119896119904= [0 log

120583max

V119898120590

V (0)] + 1 (44)

is necessary to elapse where 119896119904= [0 log

120583max(V119898120590V(0))] is

the biggest integer from the denoted segment After the statereached 119874

120590(119878) one step control 119906os should be applied to the

system (1) in which way finite reaching time in 119878 for 119909(119896) isrealized The previous facts are summarized by the followingtheorem

8 Advances in Fuzzy Systems

Theorem 18 119878 is globally exponentially stable relative to thesystem (1) with finite reaching time in 119878 if the following controlis applied

119906 (119896) =

119906119900119905(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) = |119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816 forall119896 = 0 119896119904minus 1

119906119900119905(119896119904) = 119906119900119904(119896119904) = minus

1

119888119887[119888119860119909 (119896)]

119906119904119897(119896) =minus

1

119888119887[119888119860119909 (119896)] forall119896 = 119896

119904+1 119896119904+ 2

(45)

Proof Compared to the proof of Theorem 17 here the proofis the same

5 Simulation and Experimental Results

The proposed algorithm was tested experimentally on a DCservo motor with the gear and load whose linear mathemat-ical model (without considering the major nonlinear effectsthe speed dependent friction dead zone and backlash) is asfollows

119869 120579 (119905) + 119861 120579 (119905) =120578119892120578119898119896119905119870119892

119877119898

119880 (119905) (46)

where 120579(119905) is the angular position of the load shaft 119869 is thetotal moment of inertia reflecting to the output shaft 119861 isthe viscous friction coefficient and 119880(119905) is the motor inputvoltage 119877

119898 119896119905 120578119898 120578119892 and 119870

119892are respectively the motor

armature resistance the motor torque constant the motorefficiency the gearbox efficiency and the total gear ratio

In the previous model it is assumed that the motorinductance is much smaller than the resistance so it isignored as the case with used servo motor is The experi-ments were performed with the Quanser rotary servo motorSRV02 This model is equipped with the optical encoderand tachometer for motor position and speed measuringrespectively Q8-USB data acquisition board for real-timedata acquisition and control was used with MatlabSimulinksoftware and QUARC real-time control The parametersnumerical values are as follows 119869 = 0021 (kgm2) 119861 =

0084 (Nmsrad) 119877119898= 26 (Ω) 119896

119905= 00077 (NmA) 120578

119898=

069 120578119892= 09 and119870

119892= 70

Choosing 1199091= 120579 and 119909

2= 120579 as state variables and 119906 =

119880(119905) as control at first the continuous time state equationwasobtained and finally after applying an accurate discretizationprocedure with sampling period 119879 = 001 (s) the discrete-time state equation of the system was obtained as follows

119909 (119896 + 1) = [1 008266

0 06746] 119909 (119896) + [

0002653

04979] 119906 (119896) (47)

The above mathematical model of real DC servo motorwas derived as adequate discrete-time state equation In theway the system matrix parameters 119860 and 119887 are availableControl algorithm from Theorem 18 is used to control themotor For simulation and experiment parameter 120590 = 0005

0 01 02 03 04 05

0

01

02

03

04

05

SimulationExperiment

minus02

minus01

x1

(rad

)

t (s)

Figure 6 The state variable 1199091

0 01 02 03 04 05

0123

SimulationExperiment

minus6

minus5

minus4

minus3

minus2

minus1

x2

(rad

s)

t (s)

Figure 7 The state variable 1199092

is adopted which defines very narrow 120590-neighborhood ofsliding subspace 119878 Also sampling time 119879 = 001 (s) andvector 119888 = (80 1) determined so as the system is stable insliding mode regime are adopted which gives 120577 = 119888 =

80006 Fuzzy logic controller is defined by 119871 = 119867 = 1119866 = 002 and 119866

119906= 40 which implies 119892min = 02 and

119892max = 04 Taking into account estimate relationships onone hand between 120577 and 120578

1and 120578

2 and on another hand

between 120577 119892min and 1205783 parameters 1205781= 30 120578

2= 90 and

1205783= 15 are chosen giving parameters of exponential low

120572 = 3 and 120573 = 01823 For initial state 119909(0) = (036 1)119879

after calculation V119898120590

= 04 V(0) = 298 and settling time119896119904= 20 are obtainedThe parallel simulation and experimental results are

shown in Figures 6ndash11 related to the algorithm fromTheorem 18 Figures 6 and 7 show the system state variablesThe control signal that drives the state onto the slidingsubspace is shown in Figure 9 The sliding function anddistance are shown in Figures 8 and 10 respectively From

Advances in Fuzzy Systems 9

0 01 02 03 04 05

0

10

20

30

40

50

cx

minus10

SimulationExperiment

t (s)

Figure 8 The sliding function 119888119909

0 01 02 03 04 05

0

2

4

minus10

minus8

minus6

minus4

minus2

SimulationExperiment

t (s)

u(V

)

Figure 9 The control variable

the figures it could be seen that the system state from initialposition approaches the sliding subspace 119878 but not slowerthan the exponential law defined by the parameters 120572 and120573 At instant 119896

119904= 20 (119905 = 02 sec) the state for the first

time enters the 120590-neighborhood of 119878 If it continues in thesame way it will never reach 119878 To reach it one step controlis applied at 119896

119904driving state in one step to 119878 so the character

of the system approaching 119878 changes at 119896119904 At 119896

119904+ 1 = 21

the system state is in 119878 staying in it thereafter forever Inside119878 the system is stable too so the system state asymptoticallyapproaches the zero state Simulation results fully confirm thetheoretical results and meet the expectations ofTheorems 1517 and 18

The experimental results have demonstrated that theproposed control scheme is valid and effective for the realapplications It is shown that there is no chattering in slidingmode However in practice errors could not be causedonly by computation but also by measurement accuracy(resolution sensitivity) noise existence nonlinearity andinadequate description of the real system by linear model as

0 01 02 03 04 05

002040608

112

minus04

minus02

Exponential law

t (s)

d(xS)

SimulationExperiment

Figure 10 The distance

015 02 025 03 035 04 045 05

0

001

002

003

004

005

minus002

minus001

Exponential law

d(xS)

SimulationExperiment

t (s)

Figure 11 Augmented part of Figure 10

well All this could cause not entering the 120590-neighborhood ofsliding subspace exactly in step 119896

119904than at some later instant

which is the case here in the paper experimental applicationexample After entering the 120590-neighborhood of 119878 at 119896

119904or

not and applying one step control at 119896119904 a little bit of sliding

subspace missing occurs Here that missing is a little bitmore visible at the beginning of reaching the sliding subspacebut very shortly after the like transition period expires itis again invisible like in the mathematical case simulationThis is illustrated by Figure 11 which is an enlarged segmentof Figure 10 for 119905 between 015 and 05 (sec) Evasion of theoccurrence is achieved by appropriate choice of the slidingsubspace and 120590-neighborhood parameters through vector 119888and parameter 120590 as well as the state space area where thelinear model is very adequate

It has already been pointed out that exponential stabilityis a higher quality stability property compared to ldquocommonrdquostability in Lyapunovrsquos sense (in this case it refers to globalasymptotic stability of a set-sliding subspace 119878) For that it is

10 Advances in Fuzzy Systems

natural that the sufficient conditions of Theorem 15 are morerigorous compared to the conditions of ldquocommonrdquo stabilityThe conditions of Theorem 15 are indirectly applicable inthe control of real-world processes through control law ofTheorems 17 and 18 for these control laws were derived fromthe conditions of Theorem 15 Real experiment with controlof DC servomotor is control of real-world process In the wayit is confirmed that the stability conditions ofTheorem 15 areapplicable in the control processes in practice

To resume in both ideal simulation case and in experi-mental application the proposed new developed algorithmin the paper guarantees exponential stability of 119878 and nochattering in sliding mode regime Despite the existence ofall upper stated nonidealities in the case of real applicationby the proper parameters choice the system motion is suchthat distance of working point from 119878 is always under pro-posed exponential envelope as it could be seen in providedexperiment

6 Conclusion

In the paper discrete-time type of VSCS with sliding modesis considered related to the linear time-invariant plant ofsecond orderThe fuzzy exponential stabilizing state feedbackcontrol algorithm so with prescribed approaching to thesliding subspace velocity estimate is developed strictly byLyapunovrsquos second method and with chattering free slidingmode Also application of the proposed ideal cases algo-rithm in practice is considered and discussed through theexperimental example It shows that experiment is quiteconsistent with mathematical ideal cases

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Z Bucevac ldquoLyapunovrsquos method approach in a stabilizingcontrol algorithm design for digital discrete VS systems withsliding modesndashlinear plant caserdquo in Proceedings of the 7thInternational Symposium Modelling Identification and Control( IASTED rsquo88) Grindelwald Switzerland l988

[2] G Bartolini A Ferrara and V I Utkin ldquoAdaptive sliding modecontrol in discrete-time systemsrdquo Automatica vol 31 no 5 pp769ndash773 1995

[3] S Hui and S H Zak ldquoOn discrete-time variable structureslidingmode controlrdquo SystemsampControl Letters vol 38 no 4-5pp 283ndash288 1999

[4] A Bartoszewicz ldquoDiscrete-time quasi-sliding-mode controlstrategiesrdquo IEEE Transactions on Industrial Electronics vol 45no 4 pp 633ndash637 1998

[5] W Gao Y Wang and A Homaifa ldquoDiscrete-time variablestructure control systemsrdquo IEEE Transactions on IndustrialElectronics vol 42 no 2 pp 117ndash122 1995

[6] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[7] Y Zheng and Y Jing ldquoApproximation law for discrete-timevariable structure control systemsrdquo Journal of Control Theoryand Applications vol 4 no 3 pp 291ndash296 2006

[8] G Golo and C Milosavljevic ldquoRobust discrete-time chatteringfree slidingmode controlrdquo SystemsampControl Letters vol 41 no1 pp 19ndash28 2000

[9] H Ying ldquoTheory and application of a novel fuzzy PID controllerusing a simplified Takagi-Sugeno rule schemerdquo InformationSciences vol 123 no 3 pp 281ndash293 2000

[10] G Feng ldquoA survey on analysis and design of model-based fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 5 pp 676ndash697 2006

[11] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[12] H K Lam F H F Leung and P K S Tam ldquoStable androbust fuzzy control for uncertain nonlinear systemsrdquo IEEETransactions on Systems Man and Cybernetics Part A Systemsand Humans vol 30 no 6 pp 825ndash840 2000

[13] R-E Precup M L Tomescu M-B Rdac E M Petriu S PreitlandC-A Dragos ldquoIterative performance improvement of fuzzycontrol systems for three tank systemsrdquo Expert Systems withApplications vol 39 no 9 pp 8288ndash8299 2012

[14] D H Lee Y H Joo and M H Tak ldquoLocal stability analysisof continuous-time Takagi-Sugeno fuzzy systems a fuzzy Lya-punov function approachrdquo Information Sciences vol 257 pp163ndash175 2014

[15] R Palm ldquoRobust control by fuzzy sliding moderdquo Automaticavol 30 no 9 pp 1429ndash1437 1994

[16] C-L Chen andM-H Chang ldquoOptimal design of fuzzy sliding-mode control a comparative studyrdquo Fuzzy Sets and Systems vol93 no 1 pp 37ndash48 1998

[17] J CWu andT S Liu ldquoA sliding-mode approach to fuzzy controldesignrdquo IEEE Transactions on Control Systems Technology vol4 no 2 pp 141ndash151 1996

[18] S-W Kim and J-J Lee ldquoDesign of a fuzzy controller with fuzzysliding surfacerdquo Fuzzy Sets and Systems vol 71 no 3 pp 359ndash367 1995

[19] G Feng S G Cao N W Rees and C K Chak ldquoDesign offuzzy control systems with guaranteed stabilityrdquo Fuzzy Sets andSystems vol 85 no 1 pp 1ndash10 1997

[20] F Qiao QM Zhu AWinfield andCMelhuish ldquoFuzzy slidingmode control for discrete nonlinear systemsrdquo Transactions ofChina Automation Society vol 22 no 2 pp 311ndash315 2003

[21] S H Cho ldquoRobust discrete time tracking control using slidingsurfacesrdquo KSME Journal vol 9 no 1 pp 80ndash90 1995

[22] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992

[23] H Ying W Siler and J J Buckley ldquoFuzzy control theory anonlinear caserdquo Automatica vol 26 no 3 pp 513ndash520 1990

Submit your manuscripts athttpwwwhindawicom

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Human-ComputerInteraction

Advances in

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Advances in Fuzzy Systems 3

implies that 120594(119896 119909(0) 119906(sdot)) exists 120594(119896 119909(0) 119906(sdot)) lt 120576 and120594(119896 119909(0) 119906(sdot)) isin 119878 forall119896 ge 0

Definition 2 The state 119909 = 0 of the system (1) (2) is attractive(globally) in 119878 (with respect to 119878) iffexistΔ gt 0 (Δ = infin) such that119909(0) isin 119878 and 119909(0) lt Δ implies that 120594(119896 119909(0) 119906(sdot)) existslim120594(119896 119909(0) 119906(sdot)) 119896 rarr +infin = 0 and 120594(119896 119909(0) 119906(sdot)) isin119878 forall119896 isin 119873

0

Definition 3 The state 119909 = 0 of the system (1) (2) is (globally)asymptotically stable in 119878 (with respect to 119878) iff it is both stableand (globally) attractive in 119878

Definition 4 The system (1) (2) is stable in 119878 (with respect to119878) iff its state 119909 = 0 is globally asymptotically stable in 119878

Definition 5 119878 is exponentially (globally) stable relative tothe system (1) (2) iff existΔ isin]0 +infin[ (Δ = +infin) and exist120572 isin

[1 +infin[ and 120573 isin]0 +infin[ such that distance 119889[119909(0) 119878] lt Δ

implies that 120594(119896 119909(0) 119906(sdot)) exists and 119889[120594(119896 119909(0) 119906(sdot)) 119878] le120572119889[119909(0) 119878]119890

minus120573119896 forall119896 gt 0

Remark 6 Previous definitions are valid and in the generalcase for 119909 isin 119877119899

The problem stated in the objective of the paper canbe reformulated as follows the objective of the paper is todevelop the control law (4) such that 119878 is globally exponentiallystable relative to the system (1) (2) and the system (1)(4) is stable in the sliding mode regime with appropriatelyprescribed modes Control law (4) will be developed usingfuzzy logic controller

3 Structure of FLC

The key idea of fuzzy sliding mode control is to integratefuzzy control and sliding mode control in such a way thatthe advantages of both techniques can be used One approachis to design conventional fuzzy control systems and slidingmode controller is used to determine best values for param-eters in fuzzy control rules Thereby stability is guaranteedand robust performance of the closed-loop control systems isimproved In another approach the control design is based onsliding mode techniques while the fuzzy controller is used asa complementary controller Also sliding mode control lawcan be directly substituted by a fuzzy controller

Sliding mode controllers generally involve a discon-tinuous control action which often results in chatteringphenomena due to imperfections in switching devices anddelays Commonly used methods for chattering eliminationare to replace the relay control by a saturation functionand boundary layer technique In some applications of fuzzysliding mode control the continuous switching functionof the boundary layer is replaced with equivalent fuzzyswitching function

In this paper the fuzzy controller is used in the reachingphase The controller should realize nonlinear control lawwhich will guarantee exponential stability of sliding subspace119878 This section introduces the principal structure of theproposed controller

Fuzzycontrol rules

x1 x1N

Gu

uFN uFx2Nx2 c2

c1 G

G

Figure 1 The proposed fuzzy logic controller

1

05

minusL L

120583(middot)

N P

xjN

Figure 2 The input membership functions

The fuzzy logic controller that will be evaluated is one ofthe simplest Figure 1 It employs only two input variables1199091(119896) and 119909

2(119896) Constant 119888

1and 1198882are component of vector

119888 from (2)

Scaling Factors The use of normalized domains requiresa scale transformation which maps the physical values ofthe input variables (119909

1and 119909

2in the present study) into

a normalized domain This is called input normalizationFurthermore output denormalization maps the normalizedvalue of the control output variable (119906FN) into its respectivephysical domain (119906F) The relationships between scalingfactors (119866 119866

119906) and the input and output variables are as

follows

1199091N (119896) = 119866 sdot 119888

1sdot 1199091(119896)

1199092N (119896) = 119866 sdot 119888

2sdot 1199092(119896)

119906F (119896) = 119866119906 sdot 119906FN (119896)

(6)

Fuzzification Module It converts instantaneous value of aprocess state variable into a linguistic value with the helpof the represented fuzzy set The parametric functionaldescription of the triangular shaped membership functionis the most economic one and hence it is considered hereThe membership functions of input variables are shown inFigure 2

Let 119909lowast119895be the one crisp input Then the fuzzified version

of 119909lowast119895after normalization is its degree of membership in

120583119873(119909lowast

119895N) and 120583119875(119909lowast

119895N) where119873 and 119875 are the linguistic valuestaken by119909

119895NHere symbols119873 and119875have commonmeaningsnegative and positive respectively

Assumption 7 The value 119909119895N(119896) satisfies

119909119895N (119896) isin [minus119871 119871] forall119895 = 1 2 forall119896 isin 119873

0 (7)

4 Advances in Fuzzy Systems

1

120583(middot)

minusH H

N PZ

uFN

Figure 3 The output membership functions

According to the previous assumption mathematicaldescription of the input membership functions is respec-tively given by

120583119873(119909119895N) =

minus119909119895N + 119871

2119871 120583119875(119909119895N) =

119909119895N + 119871

2119871 119895 = 1 2

(8)

Remark 8 It is noticed that

120583119873(119909119895N) + 120583119875 (119909119895N) = 1 119895 = 1 2 (9)

The membership functions for the normalized output(119906FN) are singleton and are shown in Figure 3 In Figures2 and 3 119871 and 119867 are two positive constants chosen by thedesigner which can be fixed after being determined

Remark 9 To ensure that Assumption 7 is valid one mustbe careful in choosing the scaling factor 119866 In practicalimplementation the maximum and minimum values ofvariables 119909

1and 119909

2are known and the factor 119866 is chosen so

that it satisfies the following conditions

119866 sdot 119888119895sdot10038161003816100381610038161003816119909119895

10038161003816100381610038161003816max le 119871 forall119895 = 1 2 (10)

Remark 10 For a second order system it is common to definethe row vector 119888 as 119888 = (1198881 1)

Fuzzy Control Rules Using the aforementioned membershipfunctions the following control rules are established for thefuzzy logic control part

1198771 If 1199091N is 119873 and 119909

2N is 119873 then 119906FN is 119873

1198772 If 1199091N is 119873 and 119909

2N is 119875 then 119906FN is 119885

1198773 If 1199091N is 119875 and 119909

2N is 119873 then 119906FN is 119885

1198774 If 1199091N is 119875 and 119909

2N is 119875 then 119906FN is 119875

(11)

Inference EngineThe basic function of the inference engine isto compute the overall value of control output variable basedon the individual contribution of each rule in the rule baseA degree of match for each rule is established by using thedefined membership functions

Here the antecedent of each rule is evaluated by using thetriangular norm (119905-norm) The 119905-norm used in this paper isintersection (AND function) which is mathematically givenas

120583119898(1199091N 1199092N) = 119879 (120583(sdot) (1199091N) 120583(sdot) (1199092N)) (sdot) = 119873 119875

(12)

Then based on this degree of match the clipped fuzzyset representing the value of the control output variable isdetermined via Mamdani inference method Thus outcomesof fuzzy rules are

120583119862119871119898

(119906FN) = min (120583119898 120583119880119898

(119906FN))

119880119898isin 119873119885 119875) 119898 = 1 2 3 4

(13)

Finally the clipped values for the control output of each rulepreviously denoted by 120583

119862119871119898 119898 = 1 2 3 4 are aggregated

thus forming the value of the overall control outputFrom the rule base it may be noted that the control

rules 1198772and 119877

3generate two memberships 120583

2and 120583

3which

have the same output fuzzy set defined by 120583119885(119906FN) For

such situations a combined membership is obtained by usingthe triangular conorm (119905-conorm or 119904-norm) The 119905-conormconsidered for this study is Lukasiewicz OR 119905-conorm

120583119879119862

23= min (1 120583

2+ 1205833) (14)

Defuzzification Module Defuzzification module converts theset of modified control output values into a crisp valueDefuzzification is done using the well-known COS (centerof sum) method According to this method and taking intoaccount the previously analyzed structure of the proposedfuzzy controller and the fact that the output fuzzy sets aresingleton the crisp value of control output is given by

119906FN =1205831sdot (minus119867) + 120583

119879119862

23sdot 0 + 120583

4sdot 119867

1205831+ 120583119879119862

23+ 1205834

(15)

Since for Lukasiewicz OR 119905-conorm it is valid that

120583119879119862

23= 1205832+ 1205833le 1 (16)

(15) can be written as

119906FN =1205831sdot (minus119867) + 1205832 sdot 0 + 1205833 sdot 0 + 1205834 sdot 119867

1205831+ 1205832+ 1205833+ 1205834

(17)

To obtain analytical expression of the proposed con-troller all combinations of input variables must be consid-ered If Assumption 7 holds there are eight input combina-tions (ICrsquos) as shown in Figure 4 The control rules 119877

1ndash1198774in

Advances in Fuzzy Systems 5

1

05

1

05

IC4 IC3

IC2

IC1

IC5

IC6

IC7 IC8

120583(middot)

120583(middot)

minusL

L

minusL L

P

P

N

N

x2N

x2N

x1N

x1N

Figure 4 The regions of the fuzzy controller input values

Table 1 Outcomes of fuzzy rules in IC regions

IC Fuzzy rules1198771

1198772

1198773

1198774

IC1 120583119873(1199091N) 120583

119873(1199091N) 120583

119873(1199092N) 120583

119875(1199092N)

IC2 120583119873(1199091N) 120583

119873(1199091N) 120583

119873(1199092N) 120583

119875(1199092N)

IC3 120583119873(1199092N) 120583

119873(1199091N) 120583

119873(1199092N) 120583

119875(1199091N)

IC4 120583119873(1199092N) 120583

119873(1199091N) 120583

119873(1199092N) 120583

119875(1199091N)

IC5 120583119873(1199092N) 120583

119875(1199092N) 120583

119875(1199091N) 120583

119875(1199091N)

IC6 120583119873(1199092N) 120583

119875(1199092N) 120583

119875(1199091N) 120583

119875(1199091N)

IC7 120583119873(1199091N) 120583

119875(1199092N) 120583

119875(1199091N) 120583

119875(1199092N)

IC8 120583119873(1199091N) 120583

119875(1199092N) 120583

119875(1199091N) 120583

119875(1199092N)

(11) are used to evaluate appropriate control law in each ICregion as in [23] The results of evaluating the fuzzy controlrules 119877

1ndash1198774are given in Table 1

Applying (17) to the results from Table 1 and taking intoaccount (6) and (8) analytical structure of the controller iseasily obtained as follows

119906F (119896) =

119867119866119866119906(11988811199091(119896) + 119888

21199092(119896))

2 (2119871 minus 1198661198881

10038161003816100381610038161199091 (119896)1003816100381610038161003816)

IC1 IC2 IC5 IC6

119867119866119866119906(11988811199091 (119896) + 11988821199092 (119896))

2 (2119871 minus 1198661198882

10038161003816100381610038161199092 (119896)1003816100381610038161003816)

IC3 IC4 IC7 IC8

(18)

Defining function 119892(1199091 1199092)

119892 (1199091 1199092) =

119867119866119866119906

2 (2119871 minus 1198661198881

10038161003816100381610038161199091 (119896)1003816100381610038161003816) IC1 IC2 IC5 IC6

119867119866119866119906

2 (2119871 minus 1198661198882

10038161003816100381610038161199092 (119896)1003816100381610038161003816) IC3 IC4 IC7 IC8

(19)

(18) can be written as

119906F (119896) = 119892 (1199091 1199092) 119888119909 (119896) (20)

It is obvious that the function 119892(1199091 1199092) is a nonlinear

function with the following minimal and maximal values

119892min =119867119866119866119906

4119871 119892max =

119867119866119866119906

2119871 (21)

and 119892(1199091 1199092) gt 0 forall119909

1 1199092isin 119877 One should not forget

that parameter 119866 must satisfy Assumption 7 The range ofthe value of function 119892(119909

1 1199092) is determined by the choice

of parameters 119867 119866 119871 and 119866119906 Without loss of generality it

can be assumed that 119871 = 119867 = 1 and they are commonlynormalized domain boundaries One example of function119892(1199091 1199092) is shown in Figure 5

4 Main Results

In this section the solution of the problem stated in Section 2is provided by the following theorems

6 Advances in Fuzzy Systems

0 005

05

1

102025

03035

04045

05055

06

minus1

minus1

minus05minus05

x1 x2

Figure 5 Example of function 119892(1199091 1199092) for 119871 = 119867 = 1119866 = 119866119906 = 1

and 1198881 = 1198882 = 1

Theorem 11 Applying

119906119900119904(119896) = minus

1

119888119887119888119860119909 (119896) (22)

the so-called one step control to the system (1) outside 119878 isnecessary and sufficient for 119909(119896) to reach the 119878 in one samplingperiod (one step)

ProofNecessity Let us assume that119909(119896) of the system (1) is arbitraryoutside of sliding subspace 119909(119896) notin 119878 and a control pushesit to the sliding subspace in one step (sampling period) Itfollows that 119909(119896 + 1) isin 119878 That fact description of 119878 and thesystem (1) state equation lead to

119888119860119909 (119896) + 119888119887119906 (119896) = 0 (23)

and finally

119906 (119896) = 119906os(119896) = minus

1

119888119887119888119860119909 (119896) (24)

Sufficiency When the system state is outside of slidingsubspace 119909(119896) notin 119878 the action of one step control is assumedState equation of the system (1) and expression of 119878 give

119888119909 (119896 + 1) = 119888119860119909 (119896) + 119888119887119906os(119896) = 119888119860119909 (119896) minus 119888119887

1

119888119887119888119860119909 (119896)

= 0

(25)

which means that 119909(119896 + 1) isin 119878

Remark 12 One step control can be encountered in VSCSpapers in different contexts see for example [3 4 13] amongothers

Theorem 13 Applying

119906119904119897(119896) = minus

1

119888119887119888119860119909 (119896) (26)

the actual so-called sliding control to the system (1) inside 119878 isnecessary and sufficient for 119878 to be positive invariant relative tothe system (1) solution 120594(119896 119909(0) 119906119904119897(119896))

ProofNecessity Positive invariance of 119878 relative to the system (1)solution 120594(119896 119909(0) 119906sl(119896)) is assumed So every time 119909(119896) isin 119878119909(119896 + 1) isin 119878 as well From the previous sentence descriptionof 119878 and state equation of the system (1) follows that

119888119860119909 (119896) + 119888119887119906sl(119896) = 0 (27)

and further

119906sl(119896) = minus

1

119888119887119888119860119909 (119896) (28)

Sufficiency Let us assume relative to system (1) 119909(119896) isin 119878 andcontrol (26) is applied Then state equation of the system (1)and expression for 119878 give

119888119909 (119896 + 1) = 119888119860119909 (119896) + 119888119887119906sl(119896) = 119888119860119909 (119896) minus 119888119887

1

119888119887119888119860119909 (119896)

= 0

(29)

which implies that 119909(119896 + 1) isin 119878

Remark 14 One step control and sliding control formally areidentical

The application of actual sliding control 119906sl(119896) (hence-forth referred to as 119906

sl(119896)) to the system (1) inside 119878 is

the phenomenon of discrete-time VSCS with sliding modesUsing this fact and the intention for developing of the variablestructure type of the controller for the system (1) the control(4) components could be more precisely specified as

119906 [119909 (119896)] =

119906+[119909 (119896)] 119888119909 (119896) gt 0

119906sl[119909 (119896)] 119888119909 (119896) = 0

119906minus[119909 (119896)] 119888119909 (119896) lt 0

(30)

Contrary to the 119906sl(119896) which is applied to the system(1) inside 119878 let us refer to the control which is applied tothe system (1) outside of 119878 as the outer control 119906ot(119896) Thefollowing theorems are basic for solving the stated problemby Lyapunovrsquos second method

Theorem15 Let the system (1) (30) be considered 119878 is globallyexponentially stable relative to the system (1) (30) if there existsscalar function V and numbers 120578

119894isin]0 +infin[ 119894 = 1 2 3 such that

(a) V(119909) isin 119862(1198772)(b) 1205781119889(119909 119878) le V(119909) le 120578

2119889(119909 119878) forall119909 isin (1198772)

(c) ΔV(119909) le minus1205783119889(119909 119878) forall119909 isin (1198772)

where ΔV(119909) is the first forward finite difference of V(119909)

Proof The proof starts with the assumption that sufficientconditions are fulfilled while the exponential stability of 119878should be shown

Advances in Fuzzy Systems 7

Condition (b) of the theorem implies

minus119889 [119909 (119896) 119878] le 120578minus1

2V [119909 (119896)] (31)

which with (c) give

ΔV [119909 (119896)] le minus120578minus121205783V [119909 (119896)] (32)

The last inequality after solving gives at first

V [119909 (119896)] le V [119909 (0)] 119890minus120573119896 120573 = ln1205782

1205782minus 1205783

1205782gt 1205783 forall119909 (0) isin 119877

2

(33)

and further by means of condition (a)

119889 [119909 (119896) 119878] le 120572119889 [119909 (0) 119878] 119890minus120573119896

120572 = 120578minus1

11205782 (34)

From (a) it is clear that 1205781le 1205782which yields 120572 isin [1 +infin[

and above relationship between 1205782and 1205783guarantee that 120573 isin

]0 +infin[ In that way it is shown that all stated by Definition 5is fulfilled that is the exponential stability of 119878 is proved

Remark 16 Previous theorem is valid and in the general casefor 119909 isin 119877119899

Scalar function V defined by

V (119909) = (sign 119888119909) 119888119909 (35)

as Lyapunovrsquos function candidate and outer control law

119906ot(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) isin 119877 119865 (119896) gt 0 forall119896 isin 1198730

(36)

are chosen relative to the system (1) Evidently such sortof 119906ot together with 119906

sl which is unique and already statedrepresents the control of (30) type The following theoremsare basic for solving of the stated problem by Lyapunovrsquossecond method

Theorem 17 119878 is globally exponentially stable relative to thesystem (1) if the following control is applied

119906 (119896) =

119906119900119905(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) = |119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816

119906119904119897(119896) = minus

1

119888119887[119888119860119909 (119896)]

(37)

with 119906119865(119896) defined by (19) (20) and with 119892(119909

1 1199092) lt 1

Proof Previously introduced Lyapunovrsquos function candidatefulfills evidently the conditions (a) of Theorem 15 Let usshow that conditions (b) and (c) are fulfilled Related to thecondition (b) and taking into account that |119888119909| = 120577119889 where120577 is a constant and 119889 is distance between 119909 and hyperplane119888119909 = 0 it follows that

V (119909) = (sign 119888119909) 119888119909 = |119888119909| = 120577119889 (38)

and further

1205781119889 (119909 119878) le V (119909) le 120578

2119889 (119909 119878)

1205781isin ]0 +infin[ le 120577 120578

2isin ]0 +infin[ ge 120577

(39)

State of system (1) at time 119896 is adopted to be out of 119878Then

ΔV [119909 (119896)] = V [119909 (119896 + 1)] minus V [119909 (119896)]

= 119865 (119896)1003816100381610038161003816sign 119888119909 (119896)

1003816100381610038161003816 minus |119888119909 (119896)|

= (|119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816) sdot1003816100381610038161003816sign 119888119909 (119896)

1003816100381610038161003816 minus |119888119909 (119896)|

= (1 minus 119892 (1199091 1199092)) |119888119909 (119896)| minus |119888119909 (119896)|

= minus119892 (1199091 1199092) |119888119909 (119896)| = minus119892 (1199091 1199092) 120577119889 [119909 (119896) 119878]

le minus119892min120577119889 [119909 (119896) 119878] = minus1205783119889 [119909 (119896) 119878]

(40)

in which way the proof is finished

According toTheorem 17

ΔV [119909 (119896)] = minus119892 (1199091 1199092) V [119909 (119896)] (41)

with 119892(1199091 1199092) gt 0 and 119892(119909

1 1199092) lt 1 which leads to the

following equation

V [119909 (119896 + 1)] = (1 minus 119892 (1199091 1199092)) V [119909 (119896)] = 120583 (1199091 1199092) V [119909 (119896)]

120583 (1199091 1199092) isin ]0 1[

(42)

From nonlinearity of the control low it is obvious thatcontrol gain is high when the system state is far fromsliding surfaces and as small as possible in neighborhoodof the sliding subspace Also it may be noted that V[119909(119896)]increments from step to step that is absolute value of |119888119909|are getting smaller over time

Function V[119909(119896)] is a decreasing function along motionof the system but it will never be zero Obviously the system(1) state never gets into 119878 Moreover from (42) it is obviousthat as the working point approaches the 119878 the value of thefunction 120583(119909

1 1199092) tends to its maximum value

120583max = 120583 (1199091 1199092)max = 1 minus 119892 (1199091 1199092)min (43)

Let 119874120590(119878) be 120590-neighborhood of 119878 and V

119898120590= minV(119909)

119909 is such that 119889(119909 119878) = 120590 To be guaranteed for 119909(119896) toenter 119874

120590(119878) settling time

119896119904= [0 log

120583max

V119898120590

V (0)] + 1 (44)

is necessary to elapse where 119896119904= [0 log

120583max(V119898120590V(0))] is

the biggest integer from the denoted segment After the statereached 119874

120590(119878) one step control 119906os should be applied to the

system (1) in which way finite reaching time in 119878 for 119909(119896) isrealized The previous facts are summarized by the followingtheorem

8 Advances in Fuzzy Systems

Theorem 18 119878 is globally exponentially stable relative to thesystem (1) with finite reaching time in 119878 if the following controlis applied

119906 (119896) =

119906119900119905(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) = |119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816 forall119896 = 0 119896119904minus 1

119906119900119905(119896119904) = 119906119900119904(119896119904) = minus

1

119888119887[119888119860119909 (119896)]

119906119904119897(119896) =minus

1

119888119887[119888119860119909 (119896)] forall119896 = 119896

119904+1 119896119904+ 2

(45)

Proof Compared to the proof of Theorem 17 here the proofis the same

5 Simulation and Experimental Results

The proposed algorithm was tested experimentally on a DCservo motor with the gear and load whose linear mathemat-ical model (without considering the major nonlinear effectsthe speed dependent friction dead zone and backlash) is asfollows

119869 120579 (119905) + 119861 120579 (119905) =120578119892120578119898119896119905119870119892

119877119898

119880 (119905) (46)

where 120579(119905) is the angular position of the load shaft 119869 is thetotal moment of inertia reflecting to the output shaft 119861 isthe viscous friction coefficient and 119880(119905) is the motor inputvoltage 119877

119898 119896119905 120578119898 120578119892 and 119870

119892are respectively the motor

armature resistance the motor torque constant the motorefficiency the gearbox efficiency and the total gear ratio

In the previous model it is assumed that the motorinductance is much smaller than the resistance so it isignored as the case with used servo motor is The experi-ments were performed with the Quanser rotary servo motorSRV02 This model is equipped with the optical encoderand tachometer for motor position and speed measuringrespectively Q8-USB data acquisition board for real-timedata acquisition and control was used with MatlabSimulinksoftware and QUARC real-time control The parametersnumerical values are as follows 119869 = 0021 (kgm2) 119861 =

0084 (Nmsrad) 119877119898= 26 (Ω) 119896

119905= 00077 (NmA) 120578

119898=

069 120578119892= 09 and119870

119892= 70

Choosing 1199091= 120579 and 119909

2= 120579 as state variables and 119906 =

119880(119905) as control at first the continuous time state equationwasobtained and finally after applying an accurate discretizationprocedure with sampling period 119879 = 001 (s) the discrete-time state equation of the system was obtained as follows

119909 (119896 + 1) = [1 008266

0 06746] 119909 (119896) + [

0002653

04979] 119906 (119896) (47)

The above mathematical model of real DC servo motorwas derived as adequate discrete-time state equation In theway the system matrix parameters 119860 and 119887 are availableControl algorithm from Theorem 18 is used to control themotor For simulation and experiment parameter 120590 = 0005

0 01 02 03 04 05

0

01

02

03

04

05

SimulationExperiment

minus02

minus01

x1

(rad

)

t (s)

Figure 6 The state variable 1199091

0 01 02 03 04 05

0123

SimulationExperiment

minus6

minus5

minus4

minus3

minus2

minus1

x2

(rad

s)

t (s)

Figure 7 The state variable 1199092

is adopted which defines very narrow 120590-neighborhood ofsliding subspace 119878 Also sampling time 119879 = 001 (s) andvector 119888 = (80 1) determined so as the system is stable insliding mode regime are adopted which gives 120577 = 119888 =

80006 Fuzzy logic controller is defined by 119871 = 119867 = 1119866 = 002 and 119866

119906= 40 which implies 119892min = 02 and

119892max = 04 Taking into account estimate relationships onone hand between 120577 and 120578

1and 120578

2 and on another hand

between 120577 119892min and 1205783 parameters 1205781= 30 120578

2= 90 and

1205783= 15 are chosen giving parameters of exponential low

120572 = 3 and 120573 = 01823 For initial state 119909(0) = (036 1)119879

after calculation V119898120590

= 04 V(0) = 298 and settling time119896119904= 20 are obtainedThe parallel simulation and experimental results are

shown in Figures 6ndash11 related to the algorithm fromTheorem 18 Figures 6 and 7 show the system state variablesThe control signal that drives the state onto the slidingsubspace is shown in Figure 9 The sliding function anddistance are shown in Figures 8 and 10 respectively From

Advances in Fuzzy Systems 9

0 01 02 03 04 05

0

10

20

30

40

50

cx

minus10

SimulationExperiment

t (s)

Figure 8 The sliding function 119888119909

0 01 02 03 04 05

0

2

4

minus10

minus8

minus6

minus4

minus2

SimulationExperiment

t (s)

u(V

)

Figure 9 The control variable

the figures it could be seen that the system state from initialposition approaches the sliding subspace 119878 but not slowerthan the exponential law defined by the parameters 120572 and120573 At instant 119896

119904= 20 (119905 = 02 sec) the state for the first

time enters the 120590-neighborhood of 119878 If it continues in thesame way it will never reach 119878 To reach it one step controlis applied at 119896

119904driving state in one step to 119878 so the character

of the system approaching 119878 changes at 119896119904 At 119896

119904+ 1 = 21

the system state is in 119878 staying in it thereafter forever Inside119878 the system is stable too so the system state asymptoticallyapproaches the zero state Simulation results fully confirm thetheoretical results and meet the expectations ofTheorems 1517 and 18

The experimental results have demonstrated that theproposed control scheme is valid and effective for the realapplications It is shown that there is no chattering in slidingmode However in practice errors could not be causedonly by computation but also by measurement accuracy(resolution sensitivity) noise existence nonlinearity andinadequate description of the real system by linear model as

0 01 02 03 04 05

002040608

112

minus04

minus02

Exponential law

t (s)

d(xS)

SimulationExperiment

Figure 10 The distance

015 02 025 03 035 04 045 05

0

001

002

003

004

005

minus002

minus001

Exponential law

d(xS)

SimulationExperiment

t (s)

Figure 11 Augmented part of Figure 10

well All this could cause not entering the 120590-neighborhood ofsliding subspace exactly in step 119896

119904than at some later instant

which is the case here in the paper experimental applicationexample After entering the 120590-neighborhood of 119878 at 119896

119904or

not and applying one step control at 119896119904 a little bit of sliding

subspace missing occurs Here that missing is a little bitmore visible at the beginning of reaching the sliding subspacebut very shortly after the like transition period expires itis again invisible like in the mathematical case simulationThis is illustrated by Figure 11 which is an enlarged segmentof Figure 10 for 119905 between 015 and 05 (sec) Evasion of theoccurrence is achieved by appropriate choice of the slidingsubspace and 120590-neighborhood parameters through vector 119888and parameter 120590 as well as the state space area where thelinear model is very adequate

It has already been pointed out that exponential stabilityis a higher quality stability property compared to ldquocommonrdquostability in Lyapunovrsquos sense (in this case it refers to globalasymptotic stability of a set-sliding subspace 119878) For that it is

10 Advances in Fuzzy Systems

natural that the sufficient conditions of Theorem 15 are morerigorous compared to the conditions of ldquocommonrdquo stabilityThe conditions of Theorem 15 are indirectly applicable inthe control of real-world processes through control law ofTheorems 17 and 18 for these control laws were derived fromthe conditions of Theorem 15 Real experiment with controlof DC servomotor is control of real-world process In the wayit is confirmed that the stability conditions ofTheorem 15 areapplicable in the control processes in practice

To resume in both ideal simulation case and in experi-mental application the proposed new developed algorithmin the paper guarantees exponential stability of 119878 and nochattering in sliding mode regime Despite the existence ofall upper stated nonidealities in the case of real applicationby the proper parameters choice the system motion is suchthat distance of working point from 119878 is always under pro-posed exponential envelope as it could be seen in providedexperiment

6 Conclusion

In the paper discrete-time type of VSCS with sliding modesis considered related to the linear time-invariant plant ofsecond orderThe fuzzy exponential stabilizing state feedbackcontrol algorithm so with prescribed approaching to thesliding subspace velocity estimate is developed strictly byLyapunovrsquos second method and with chattering free slidingmode Also application of the proposed ideal cases algo-rithm in practice is considered and discussed through theexperimental example It shows that experiment is quiteconsistent with mathematical ideal cases

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Z Bucevac ldquoLyapunovrsquos method approach in a stabilizingcontrol algorithm design for digital discrete VS systems withsliding modesndashlinear plant caserdquo in Proceedings of the 7thInternational Symposium Modelling Identification and Control( IASTED rsquo88) Grindelwald Switzerland l988

[2] G Bartolini A Ferrara and V I Utkin ldquoAdaptive sliding modecontrol in discrete-time systemsrdquo Automatica vol 31 no 5 pp769ndash773 1995

[3] S Hui and S H Zak ldquoOn discrete-time variable structureslidingmode controlrdquo SystemsampControl Letters vol 38 no 4-5pp 283ndash288 1999

[4] A Bartoszewicz ldquoDiscrete-time quasi-sliding-mode controlstrategiesrdquo IEEE Transactions on Industrial Electronics vol 45no 4 pp 633ndash637 1998

[5] W Gao Y Wang and A Homaifa ldquoDiscrete-time variablestructure control systemsrdquo IEEE Transactions on IndustrialElectronics vol 42 no 2 pp 117ndash122 1995

[6] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[7] Y Zheng and Y Jing ldquoApproximation law for discrete-timevariable structure control systemsrdquo Journal of Control Theoryand Applications vol 4 no 3 pp 291ndash296 2006

[8] G Golo and C Milosavljevic ldquoRobust discrete-time chatteringfree slidingmode controlrdquo SystemsampControl Letters vol 41 no1 pp 19ndash28 2000

[9] H Ying ldquoTheory and application of a novel fuzzy PID controllerusing a simplified Takagi-Sugeno rule schemerdquo InformationSciences vol 123 no 3 pp 281ndash293 2000

[10] G Feng ldquoA survey on analysis and design of model-based fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 5 pp 676ndash697 2006

[11] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[12] H K Lam F H F Leung and P K S Tam ldquoStable androbust fuzzy control for uncertain nonlinear systemsrdquo IEEETransactions on Systems Man and Cybernetics Part A Systemsand Humans vol 30 no 6 pp 825ndash840 2000

[13] R-E Precup M L Tomescu M-B Rdac E M Petriu S PreitlandC-A Dragos ldquoIterative performance improvement of fuzzycontrol systems for three tank systemsrdquo Expert Systems withApplications vol 39 no 9 pp 8288ndash8299 2012

[14] D H Lee Y H Joo and M H Tak ldquoLocal stability analysisof continuous-time Takagi-Sugeno fuzzy systems a fuzzy Lya-punov function approachrdquo Information Sciences vol 257 pp163ndash175 2014

[15] R Palm ldquoRobust control by fuzzy sliding moderdquo Automaticavol 30 no 9 pp 1429ndash1437 1994

[16] C-L Chen andM-H Chang ldquoOptimal design of fuzzy sliding-mode control a comparative studyrdquo Fuzzy Sets and Systems vol93 no 1 pp 37ndash48 1998

[17] J CWu andT S Liu ldquoA sliding-mode approach to fuzzy controldesignrdquo IEEE Transactions on Control Systems Technology vol4 no 2 pp 141ndash151 1996

[18] S-W Kim and J-J Lee ldquoDesign of a fuzzy controller with fuzzysliding surfacerdquo Fuzzy Sets and Systems vol 71 no 3 pp 359ndash367 1995

[19] G Feng S G Cao N W Rees and C K Chak ldquoDesign offuzzy control systems with guaranteed stabilityrdquo Fuzzy Sets andSystems vol 85 no 1 pp 1ndash10 1997

[20] F Qiao QM Zhu AWinfield andCMelhuish ldquoFuzzy slidingmode control for discrete nonlinear systemsrdquo Transactions ofChina Automation Society vol 22 no 2 pp 311ndash315 2003

[21] S H Cho ldquoRobust discrete time tracking control using slidingsurfacesrdquo KSME Journal vol 9 no 1 pp 80ndash90 1995

[22] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992

[23] H Ying W Siler and J J Buckley ldquoFuzzy control theory anonlinear caserdquo Automatica vol 26 no 3 pp 513ndash520 1990

Submit your manuscripts athttpwwwhindawicom

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Distributed Sensor Networks

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Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

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Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Applied Computational Intelligence and Soft Computing

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Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

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Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

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ArtificialNeural Systems

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RoboticsJournal of

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Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

4 Advances in Fuzzy Systems

1

120583(middot)

minusH H

N PZ

uFN

Figure 3 The output membership functions

According to the previous assumption mathematicaldescription of the input membership functions is respec-tively given by

120583119873(119909119895N) =

minus119909119895N + 119871

2119871 120583119875(119909119895N) =

119909119895N + 119871

2119871 119895 = 1 2

(8)

Remark 8 It is noticed that

120583119873(119909119895N) + 120583119875 (119909119895N) = 1 119895 = 1 2 (9)

The membership functions for the normalized output(119906FN) are singleton and are shown in Figure 3 In Figures2 and 3 119871 and 119867 are two positive constants chosen by thedesigner which can be fixed after being determined

Remark 9 To ensure that Assumption 7 is valid one mustbe careful in choosing the scaling factor 119866 In practicalimplementation the maximum and minimum values ofvariables 119909

1and 119909

2are known and the factor 119866 is chosen so

that it satisfies the following conditions

119866 sdot 119888119895sdot10038161003816100381610038161003816119909119895

10038161003816100381610038161003816max le 119871 forall119895 = 1 2 (10)

Remark 10 For a second order system it is common to definethe row vector 119888 as 119888 = (1198881 1)

Fuzzy Control Rules Using the aforementioned membershipfunctions the following control rules are established for thefuzzy logic control part

1198771 If 1199091N is 119873 and 119909

2N is 119873 then 119906FN is 119873

1198772 If 1199091N is 119873 and 119909

2N is 119875 then 119906FN is 119885

1198773 If 1199091N is 119875 and 119909

2N is 119873 then 119906FN is 119885

1198774 If 1199091N is 119875 and 119909

2N is 119875 then 119906FN is 119875

(11)

Inference EngineThe basic function of the inference engine isto compute the overall value of control output variable basedon the individual contribution of each rule in the rule baseA degree of match for each rule is established by using thedefined membership functions

Here the antecedent of each rule is evaluated by using thetriangular norm (119905-norm) The 119905-norm used in this paper isintersection (AND function) which is mathematically givenas

120583119898(1199091N 1199092N) = 119879 (120583(sdot) (1199091N) 120583(sdot) (1199092N)) (sdot) = 119873 119875

(12)

Then based on this degree of match the clipped fuzzyset representing the value of the control output variable isdetermined via Mamdani inference method Thus outcomesof fuzzy rules are

120583119862119871119898

(119906FN) = min (120583119898 120583119880119898

(119906FN))

119880119898isin 119873119885 119875) 119898 = 1 2 3 4

(13)

Finally the clipped values for the control output of each rulepreviously denoted by 120583

119862119871119898 119898 = 1 2 3 4 are aggregated

thus forming the value of the overall control outputFrom the rule base it may be noted that the control

rules 1198772and 119877

3generate two memberships 120583

2and 120583

3which

have the same output fuzzy set defined by 120583119885(119906FN) For

such situations a combined membership is obtained by usingthe triangular conorm (119905-conorm or 119904-norm) The 119905-conormconsidered for this study is Lukasiewicz OR 119905-conorm

120583119879119862

23= min (1 120583

2+ 1205833) (14)

Defuzzification Module Defuzzification module converts theset of modified control output values into a crisp valueDefuzzification is done using the well-known COS (centerof sum) method According to this method and taking intoaccount the previously analyzed structure of the proposedfuzzy controller and the fact that the output fuzzy sets aresingleton the crisp value of control output is given by

119906FN =1205831sdot (minus119867) + 120583

119879119862

23sdot 0 + 120583

4sdot 119867

1205831+ 120583119879119862

23+ 1205834

(15)

Since for Lukasiewicz OR 119905-conorm it is valid that

120583119879119862

23= 1205832+ 1205833le 1 (16)

(15) can be written as

119906FN =1205831sdot (minus119867) + 1205832 sdot 0 + 1205833 sdot 0 + 1205834 sdot 119867

1205831+ 1205832+ 1205833+ 1205834

(17)

To obtain analytical expression of the proposed con-troller all combinations of input variables must be consid-ered If Assumption 7 holds there are eight input combina-tions (ICrsquos) as shown in Figure 4 The control rules 119877

1ndash1198774in

Advances in Fuzzy Systems 5

1

05

1

05

IC4 IC3

IC2

IC1

IC5

IC6

IC7 IC8

120583(middot)

120583(middot)

minusL

L

minusL L

P

P

N

N

x2N

x2N

x1N

x1N

Figure 4 The regions of the fuzzy controller input values

Table 1 Outcomes of fuzzy rules in IC regions

IC Fuzzy rules1198771

1198772

1198773

1198774

IC1 120583119873(1199091N) 120583

119873(1199091N) 120583

119873(1199092N) 120583

119875(1199092N)

IC2 120583119873(1199091N) 120583

119873(1199091N) 120583

119873(1199092N) 120583

119875(1199092N)

IC3 120583119873(1199092N) 120583

119873(1199091N) 120583

119873(1199092N) 120583

119875(1199091N)

IC4 120583119873(1199092N) 120583

119873(1199091N) 120583

119873(1199092N) 120583

119875(1199091N)

IC5 120583119873(1199092N) 120583

119875(1199092N) 120583

119875(1199091N) 120583

119875(1199091N)

IC6 120583119873(1199092N) 120583

119875(1199092N) 120583

119875(1199091N) 120583

119875(1199091N)

IC7 120583119873(1199091N) 120583

119875(1199092N) 120583

119875(1199091N) 120583

119875(1199092N)

IC8 120583119873(1199091N) 120583

119875(1199092N) 120583

119875(1199091N) 120583

119875(1199092N)

(11) are used to evaluate appropriate control law in each ICregion as in [23] The results of evaluating the fuzzy controlrules 119877

1ndash1198774are given in Table 1

Applying (17) to the results from Table 1 and taking intoaccount (6) and (8) analytical structure of the controller iseasily obtained as follows

119906F (119896) =

119867119866119866119906(11988811199091(119896) + 119888

21199092(119896))

2 (2119871 minus 1198661198881

10038161003816100381610038161199091 (119896)1003816100381610038161003816)

IC1 IC2 IC5 IC6

119867119866119866119906(11988811199091 (119896) + 11988821199092 (119896))

2 (2119871 minus 1198661198882

10038161003816100381610038161199092 (119896)1003816100381610038161003816)

IC3 IC4 IC7 IC8

(18)

Defining function 119892(1199091 1199092)

119892 (1199091 1199092) =

119867119866119866119906

2 (2119871 minus 1198661198881

10038161003816100381610038161199091 (119896)1003816100381610038161003816) IC1 IC2 IC5 IC6

119867119866119866119906

2 (2119871 minus 1198661198882

10038161003816100381610038161199092 (119896)1003816100381610038161003816) IC3 IC4 IC7 IC8

(19)

(18) can be written as

119906F (119896) = 119892 (1199091 1199092) 119888119909 (119896) (20)

It is obvious that the function 119892(1199091 1199092) is a nonlinear

function with the following minimal and maximal values

119892min =119867119866119866119906

4119871 119892max =

119867119866119866119906

2119871 (21)

and 119892(1199091 1199092) gt 0 forall119909

1 1199092isin 119877 One should not forget

that parameter 119866 must satisfy Assumption 7 The range ofthe value of function 119892(119909

1 1199092) is determined by the choice

of parameters 119867 119866 119871 and 119866119906 Without loss of generality it

can be assumed that 119871 = 119867 = 1 and they are commonlynormalized domain boundaries One example of function119892(1199091 1199092) is shown in Figure 5

4 Main Results

In this section the solution of the problem stated in Section 2is provided by the following theorems

6 Advances in Fuzzy Systems

0 005

05

1

102025

03035

04045

05055

06

minus1

minus1

minus05minus05

x1 x2

Figure 5 Example of function 119892(1199091 1199092) for 119871 = 119867 = 1119866 = 119866119906 = 1

and 1198881 = 1198882 = 1

Theorem 11 Applying

119906119900119904(119896) = minus

1

119888119887119888119860119909 (119896) (22)

the so-called one step control to the system (1) outside 119878 isnecessary and sufficient for 119909(119896) to reach the 119878 in one samplingperiod (one step)

ProofNecessity Let us assume that119909(119896) of the system (1) is arbitraryoutside of sliding subspace 119909(119896) notin 119878 and a control pushesit to the sliding subspace in one step (sampling period) Itfollows that 119909(119896 + 1) isin 119878 That fact description of 119878 and thesystem (1) state equation lead to

119888119860119909 (119896) + 119888119887119906 (119896) = 0 (23)

and finally

119906 (119896) = 119906os(119896) = minus

1

119888119887119888119860119909 (119896) (24)

Sufficiency When the system state is outside of slidingsubspace 119909(119896) notin 119878 the action of one step control is assumedState equation of the system (1) and expression of 119878 give

119888119909 (119896 + 1) = 119888119860119909 (119896) + 119888119887119906os(119896) = 119888119860119909 (119896) minus 119888119887

1

119888119887119888119860119909 (119896)

= 0

(25)

which means that 119909(119896 + 1) isin 119878

Remark 12 One step control can be encountered in VSCSpapers in different contexts see for example [3 4 13] amongothers

Theorem 13 Applying

119906119904119897(119896) = minus

1

119888119887119888119860119909 (119896) (26)

the actual so-called sliding control to the system (1) inside 119878 isnecessary and sufficient for 119878 to be positive invariant relative tothe system (1) solution 120594(119896 119909(0) 119906119904119897(119896))

ProofNecessity Positive invariance of 119878 relative to the system (1)solution 120594(119896 119909(0) 119906sl(119896)) is assumed So every time 119909(119896) isin 119878119909(119896 + 1) isin 119878 as well From the previous sentence descriptionof 119878 and state equation of the system (1) follows that

119888119860119909 (119896) + 119888119887119906sl(119896) = 0 (27)

and further

119906sl(119896) = minus

1

119888119887119888119860119909 (119896) (28)

Sufficiency Let us assume relative to system (1) 119909(119896) isin 119878 andcontrol (26) is applied Then state equation of the system (1)and expression for 119878 give

119888119909 (119896 + 1) = 119888119860119909 (119896) + 119888119887119906sl(119896) = 119888119860119909 (119896) minus 119888119887

1

119888119887119888119860119909 (119896)

= 0

(29)

which implies that 119909(119896 + 1) isin 119878

Remark 14 One step control and sliding control formally areidentical

The application of actual sliding control 119906sl(119896) (hence-forth referred to as 119906

sl(119896)) to the system (1) inside 119878 is

the phenomenon of discrete-time VSCS with sliding modesUsing this fact and the intention for developing of the variablestructure type of the controller for the system (1) the control(4) components could be more precisely specified as

119906 [119909 (119896)] =

119906+[119909 (119896)] 119888119909 (119896) gt 0

119906sl[119909 (119896)] 119888119909 (119896) = 0

119906minus[119909 (119896)] 119888119909 (119896) lt 0

(30)

Contrary to the 119906sl(119896) which is applied to the system(1) inside 119878 let us refer to the control which is applied tothe system (1) outside of 119878 as the outer control 119906ot(119896) Thefollowing theorems are basic for solving the stated problemby Lyapunovrsquos second method

Theorem15 Let the system (1) (30) be considered 119878 is globallyexponentially stable relative to the system (1) (30) if there existsscalar function V and numbers 120578

119894isin]0 +infin[ 119894 = 1 2 3 such that

(a) V(119909) isin 119862(1198772)(b) 1205781119889(119909 119878) le V(119909) le 120578

2119889(119909 119878) forall119909 isin (1198772)

(c) ΔV(119909) le minus1205783119889(119909 119878) forall119909 isin (1198772)

where ΔV(119909) is the first forward finite difference of V(119909)

Proof The proof starts with the assumption that sufficientconditions are fulfilled while the exponential stability of 119878should be shown

Advances in Fuzzy Systems 7

Condition (b) of the theorem implies

minus119889 [119909 (119896) 119878] le 120578minus1

2V [119909 (119896)] (31)

which with (c) give

ΔV [119909 (119896)] le minus120578minus121205783V [119909 (119896)] (32)

The last inequality after solving gives at first

V [119909 (119896)] le V [119909 (0)] 119890minus120573119896 120573 = ln1205782

1205782minus 1205783

1205782gt 1205783 forall119909 (0) isin 119877

2

(33)

and further by means of condition (a)

119889 [119909 (119896) 119878] le 120572119889 [119909 (0) 119878] 119890minus120573119896

120572 = 120578minus1

11205782 (34)

From (a) it is clear that 1205781le 1205782which yields 120572 isin [1 +infin[

and above relationship between 1205782and 1205783guarantee that 120573 isin

]0 +infin[ In that way it is shown that all stated by Definition 5is fulfilled that is the exponential stability of 119878 is proved

Remark 16 Previous theorem is valid and in the general casefor 119909 isin 119877119899

Scalar function V defined by

V (119909) = (sign 119888119909) 119888119909 (35)

as Lyapunovrsquos function candidate and outer control law

119906ot(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) isin 119877 119865 (119896) gt 0 forall119896 isin 1198730

(36)

are chosen relative to the system (1) Evidently such sortof 119906ot together with 119906

sl which is unique and already statedrepresents the control of (30) type The following theoremsare basic for solving of the stated problem by Lyapunovrsquossecond method

Theorem 17 119878 is globally exponentially stable relative to thesystem (1) if the following control is applied

119906 (119896) =

119906119900119905(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) = |119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816

119906119904119897(119896) = minus

1

119888119887[119888119860119909 (119896)]

(37)

with 119906119865(119896) defined by (19) (20) and with 119892(119909

1 1199092) lt 1

Proof Previously introduced Lyapunovrsquos function candidatefulfills evidently the conditions (a) of Theorem 15 Let usshow that conditions (b) and (c) are fulfilled Related to thecondition (b) and taking into account that |119888119909| = 120577119889 where120577 is a constant and 119889 is distance between 119909 and hyperplane119888119909 = 0 it follows that

V (119909) = (sign 119888119909) 119888119909 = |119888119909| = 120577119889 (38)

and further

1205781119889 (119909 119878) le V (119909) le 120578

2119889 (119909 119878)

1205781isin ]0 +infin[ le 120577 120578

2isin ]0 +infin[ ge 120577

(39)

State of system (1) at time 119896 is adopted to be out of 119878Then

ΔV [119909 (119896)] = V [119909 (119896 + 1)] minus V [119909 (119896)]

= 119865 (119896)1003816100381610038161003816sign 119888119909 (119896)

1003816100381610038161003816 minus |119888119909 (119896)|

= (|119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816) sdot1003816100381610038161003816sign 119888119909 (119896)

1003816100381610038161003816 minus |119888119909 (119896)|

= (1 minus 119892 (1199091 1199092)) |119888119909 (119896)| minus |119888119909 (119896)|

= minus119892 (1199091 1199092) |119888119909 (119896)| = minus119892 (1199091 1199092) 120577119889 [119909 (119896) 119878]

le minus119892min120577119889 [119909 (119896) 119878] = minus1205783119889 [119909 (119896) 119878]

(40)

in which way the proof is finished

According toTheorem 17

ΔV [119909 (119896)] = minus119892 (1199091 1199092) V [119909 (119896)] (41)

with 119892(1199091 1199092) gt 0 and 119892(119909

1 1199092) lt 1 which leads to the

following equation

V [119909 (119896 + 1)] = (1 minus 119892 (1199091 1199092)) V [119909 (119896)] = 120583 (1199091 1199092) V [119909 (119896)]

120583 (1199091 1199092) isin ]0 1[

(42)

From nonlinearity of the control low it is obvious thatcontrol gain is high when the system state is far fromsliding surfaces and as small as possible in neighborhoodof the sliding subspace Also it may be noted that V[119909(119896)]increments from step to step that is absolute value of |119888119909|are getting smaller over time

Function V[119909(119896)] is a decreasing function along motionof the system but it will never be zero Obviously the system(1) state never gets into 119878 Moreover from (42) it is obviousthat as the working point approaches the 119878 the value of thefunction 120583(119909

1 1199092) tends to its maximum value

120583max = 120583 (1199091 1199092)max = 1 minus 119892 (1199091 1199092)min (43)

Let 119874120590(119878) be 120590-neighborhood of 119878 and V

119898120590= minV(119909)

119909 is such that 119889(119909 119878) = 120590 To be guaranteed for 119909(119896) toenter 119874

120590(119878) settling time

119896119904= [0 log

120583max

V119898120590

V (0)] + 1 (44)

is necessary to elapse where 119896119904= [0 log

120583max(V119898120590V(0))] is

the biggest integer from the denoted segment After the statereached 119874

120590(119878) one step control 119906os should be applied to the

system (1) in which way finite reaching time in 119878 for 119909(119896) isrealized The previous facts are summarized by the followingtheorem

8 Advances in Fuzzy Systems

Theorem 18 119878 is globally exponentially stable relative to thesystem (1) with finite reaching time in 119878 if the following controlis applied

119906 (119896) =

119906119900119905(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) = |119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816 forall119896 = 0 119896119904minus 1

119906119900119905(119896119904) = 119906119900119904(119896119904) = minus

1

119888119887[119888119860119909 (119896)]

119906119904119897(119896) =minus

1

119888119887[119888119860119909 (119896)] forall119896 = 119896

119904+1 119896119904+ 2

(45)

Proof Compared to the proof of Theorem 17 here the proofis the same

5 Simulation and Experimental Results

The proposed algorithm was tested experimentally on a DCservo motor with the gear and load whose linear mathemat-ical model (without considering the major nonlinear effectsthe speed dependent friction dead zone and backlash) is asfollows

119869 120579 (119905) + 119861 120579 (119905) =120578119892120578119898119896119905119870119892

119877119898

119880 (119905) (46)

where 120579(119905) is the angular position of the load shaft 119869 is thetotal moment of inertia reflecting to the output shaft 119861 isthe viscous friction coefficient and 119880(119905) is the motor inputvoltage 119877

119898 119896119905 120578119898 120578119892 and 119870

119892are respectively the motor

armature resistance the motor torque constant the motorefficiency the gearbox efficiency and the total gear ratio

In the previous model it is assumed that the motorinductance is much smaller than the resistance so it isignored as the case with used servo motor is The experi-ments were performed with the Quanser rotary servo motorSRV02 This model is equipped with the optical encoderand tachometer for motor position and speed measuringrespectively Q8-USB data acquisition board for real-timedata acquisition and control was used with MatlabSimulinksoftware and QUARC real-time control The parametersnumerical values are as follows 119869 = 0021 (kgm2) 119861 =

0084 (Nmsrad) 119877119898= 26 (Ω) 119896

119905= 00077 (NmA) 120578

119898=

069 120578119892= 09 and119870

119892= 70

Choosing 1199091= 120579 and 119909

2= 120579 as state variables and 119906 =

119880(119905) as control at first the continuous time state equationwasobtained and finally after applying an accurate discretizationprocedure with sampling period 119879 = 001 (s) the discrete-time state equation of the system was obtained as follows

119909 (119896 + 1) = [1 008266

0 06746] 119909 (119896) + [

0002653

04979] 119906 (119896) (47)

The above mathematical model of real DC servo motorwas derived as adequate discrete-time state equation In theway the system matrix parameters 119860 and 119887 are availableControl algorithm from Theorem 18 is used to control themotor For simulation and experiment parameter 120590 = 0005

0 01 02 03 04 05

0

01

02

03

04

05

SimulationExperiment

minus02

minus01

x1

(rad

)

t (s)

Figure 6 The state variable 1199091

0 01 02 03 04 05

0123

SimulationExperiment

minus6

minus5

minus4

minus3

minus2

minus1

x2

(rad

s)

t (s)

Figure 7 The state variable 1199092

is adopted which defines very narrow 120590-neighborhood ofsliding subspace 119878 Also sampling time 119879 = 001 (s) andvector 119888 = (80 1) determined so as the system is stable insliding mode regime are adopted which gives 120577 = 119888 =

80006 Fuzzy logic controller is defined by 119871 = 119867 = 1119866 = 002 and 119866

119906= 40 which implies 119892min = 02 and

119892max = 04 Taking into account estimate relationships onone hand between 120577 and 120578

1and 120578

2 and on another hand

between 120577 119892min and 1205783 parameters 1205781= 30 120578

2= 90 and

1205783= 15 are chosen giving parameters of exponential low

120572 = 3 and 120573 = 01823 For initial state 119909(0) = (036 1)119879

after calculation V119898120590

= 04 V(0) = 298 and settling time119896119904= 20 are obtainedThe parallel simulation and experimental results are

shown in Figures 6ndash11 related to the algorithm fromTheorem 18 Figures 6 and 7 show the system state variablesThe control signal that drives the state onto the slidingsubspace is shown in Figure 9 The sliding function anddistance are shown in Figures 8 and 10 respectively From

Advances in Fuzzy Systems 9

0 01 02 03 04 05

0

10

20

30

40

50

cx

minus10

SimulationExperiment

t (s)

Figure 8 The sliding function 119888119909

0 01 02 03 04 05

0

2

4

minus10

minus8

minus6

minus4

minus2

SimulationExperiment

t (s)

u(V

)

Figure 9 The control variable

the figures it could be seen that the system state from initialposition approaches the sliding subspace 119878 but not slowerthan the exponential law defined by the parameters 120572 and120573 At instant 119896

119904= 20 (119905 = 02 sec) the state for the first

time enters the 120590-neighborhood of 119878 If it continues in thesame way it will never reach 119878 To reach it one step controlis applied at 119896

119904driving state in one step to 119878 so the character

of the system approaching 119878 changes at 119896119904 At 119896

119904+ 1 = 21

the system state is in 119878 staying in it thereafter forever Inside119878 the system is stable too so the system state asymptoticallyapproaches the zero state Simulation results fully confirm thetheoretical results and meet the expectations ofTheorems 1517 and 18

The experimental results have demonstrated that theproposed control scheme is valid and effective for the realapplications It is shown that there is no chattering in slidingmode However in practice errors could not be causedonly by computation but also by measurement accuracy(resolution sensitivity) noise existence nonlinearity andinadequate description of the real system by linear model as

0 01 02 03 04 05

002040608

112

minus04

minus02

Exponential law

t (s)

d(xS)

SimulationExperiment

Figure 10 The distance

015 02 025 03 035 04 045 05

0

001

002

003

004

005

minus002

minus001

Exponential law

d(xS)

SimulationExperiment

t (s)

Figure 11 Augmented part of Figure 10

well All this could cause not entering the 120590-neighborhood ofsliding subspace exactly in step 119896

119904than at some later instant

which is the case here in the paper experimental applicationexample After entering the 120590-neighborhood of 119878 at 119896

119904or

not and applying one step control at 119896119904 a little bit of sliding

subspace missing occurs Here that missing is a little bitmore visible at the beginning of reaching the sliding subspacebut very shortly after the like transition period expires itis again invisible like in the mathematical case simulationThis is illustrated by Figure 11 which is an enlarged segmentof Figure 10 for 119905 between 015 and 05 (sec) Evasion of theoccurrence is achieved by appropriate choice of the slidingsubspace and 120590-neighborhood parameters through vector 119888and parameter 120590 as well as the state space area where thelinear model is very adequate

It has already been pointed out that exponential stabilityis a higher quality stability property compared to ldquocommonrdquostability in Lyapunovrsquos sense (in this case it refers to globalasymptotic stability of a set-sliding subspace 119878) For that it is

10 Advances in Fuzzy Systems

natural that the sufficient conditions of Theorem 15 are morerigorous compared to the conditions of ldquocommonrdquo stabilityThe conditions of Theorem 15 are indirectly applicable inthe control of real-world processes through control law ofTheorems 17 and 18 for these control laws were derived fromthe conditions of Theorem 15 Real experiment with controlof DC servomotor is control of real-world process In the wayit is confirmed that the stability conditions ofTheorem 15 areapplicable in the control processes in practice

To resume in both ideal simulation case and in experi-mental application the proposed new developed algorithmin the paper guarantees exponential stability of 119878 and nochattering in sliding mode regime Despite the existence ofall upper stated nonidealities in the case of real applicationby the proper parameters choice the system motion is suchthat distance of working point from 119878 is always under pro-posed exponential envelope as it could be seen in providedexperiment

6 Conclusion

In the paper discrete-time type of VSCS with sliding modesis considered related to the linear time-invariant plant ofsecond orderThe fuzzy exponential stabilizing state feedbackcontrol algorithm so with prescribed approaching to thesliding subspace velocity estimate is developed strictly byLyapunovrsquos second method and with chattering free slidingmode Also application of the proposed ideal cases algo-rithm in practice is considered and discussed through theexperimental example It shows that experiment is quiteconsistent with mathematical ideal cases

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Z Bucevac ldquoLyapunovrsquos method approach in a stabilizingcontrol algorithm design for digital discrete VS systems withsliding modesndashlinear plant caserdquo in Proceedings of the 7thInternational Symposium Modelling Identification and Control( IASTED rsquo88) Grindelwald Switzerland l988

[2] G Bartolini A Ferrara and V I Utkin ldquoAdaptive sliding modecontrol in discrete-time systemsrdquo Automatica vol 31 no 5 pp769ndash773 1995

[3] S Hui and S H Zak ldquoOn discrete-time variable structureslidingmode controlrdquo SystemsampControl Letters vol 38 no 4-5pp 283ndash288 1999

[4] A Bartoszewicz ldquoDiscrete-time quasi-sliding-mode controlstrategiesrdquo IEEE Transactions on Industrial Electronics vol 45no 4 pp 633ndash637 1998

[5] W Gao Y Wang and A Homaifa ldquoDiscrete-time variablestructure control systemsrdquo IEEE Transactions on IndustrialElectronics vol 42 no 2 pp 117ndash122 1995

[6] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[7] Y Zheng and Y Jing ldquoApproximation law for discrete-timevariable structure control systemsrdquo Journal of Control Theoryand Applications vol 4 no 3 pp 291ndash296 2006

[8] G Golo and C Milosavljevic ldquoRobust discrete-time chatteringfree slidingmode controlrdquo SystemsampControl Letters vol 41 no1 pp 19ndash28 2000

[9] H Ying ldquoTheory and application of a novel fuzzy PID controllerusing a simplified Takagi-Sugeno rule schemerdquo InformationSciences vol 123 no 3 pp 281ndash293 2000

[10] G Feng ldquoA survey on analysis and design of model-based fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 5 pp 676ndash697 2006

[11] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[12] H K Lam F H F Leung and P K S Tam ldquoStable androbust fuzzy control for uncertain nonlinear systemsrdquo IEEETransactions on Systems Man and Cybernetics Part A Systemsand Humans vol 30 no 6 pp 825ndash840 2000

[13] R-E Precup M L Tomescu M-B Rdac E M Petriu S PreitlandC-A Dragos ldquoIterative performance improvement of fuzzycontrol systems for three tank systemsrdquo Expert Systems withApplications vol 39 no 9 pp 8288ndash8299 2012

[14] D H Lee Y H Joo and M H Tak ldquoLocal stability analysisof continuous-time Takagi-Sugeno fuzzy systems a fuzzy Lya-punov function approachrdquo Information Sciences vol 257 pp163ndash175 2014

[15] R Palm ldquoRobust control by fuzzy sliding moderdquo Automaticavol 30 no 9 pp 1429ndash1437 1994

[16] C-L Chen andM-H Chang ldquoOptimal design of fuzzy sliding-mode control a comparative studyrdquo Fuzzy Sets and Systems vol93 no 1 pp 37ndash48 1998

[17] J CWu andT S Liu ldquoA sliding-mode approach to fuzzy controldesignrdquo IEEE Transactions on Control Systems Technology vol4 no 2 pp 141ndash151 1996

[18] S-W Kim and J-J Lee ldquoDesign of a fuzzy controller with fuzzysliding surfacerdquo Fuzzy Sets and Systems vol 71 no 3 pp 359ndash367 1995

[19] G Feng S G Cao N W Rees and C K Chak ldquoDesign offuzzy control systems with guaranteed stabilityrdquo Fuzzy Sets andSystems vol 85 no 1 pp 1ndash10 1997

[20] F Qiao QM Zhu AWinfield andCMelhuish ldquoFuzzy slidingmode control for discrete nonlinear systemsrdquo Transactions ofChina Automation Society vol 22 no 2 pp 311ndash315 2003

[21] S H Cho ldquoRobust discrete time tracking control using slidingsurfacesrdquo KSME Journal vol 9 no 1 pp 80ndash90 1995

[22] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992

[23] H Ying W Siler and J J Buckley ldquoFuzzy control theory anonlinear caserdquo Automatica vol 26 no 3 pp 513ndash520 1990

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Electrical and Computer Engineering

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RoboticsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

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Advances in Fuzzy Systems 5

1

05

1

05

IC4 IC3

IC2

IC1

IC5

IC6

IC7 IC8

120583(middot)

120583(middot)

minusL

L

minusL L

P

P

N

N

x2N

x2N

x1N

x1N

Figure 4 The regions of the fuzzy controller input values

Table 1 Outcomes of fuzzy rules in IC regions

IC Fuzzy rules1198771

1198772

1198773

1198774

IC1 120583119873(1199091N) 120583

119873(1199091N) 120583

119873(1199092N) 120583

119875(1199092N)

IC2 120583119873(1199091N) 120583

119873(1199091N) 120583

119873(1199092N) 120583

119875(1199092N)

IC3 120583119873(1199092N) 120583

119873(1199091N) 120583

119873(1199092N) 120583

119875(1199091N)

IC4 120583119873(1199092N) 120583

119873(1199091N) 120583

119873(1199092N) 120583

119875(1199091N)

IC5 120583119873(1199092N) 120583

119875(1199092N) 120583

119875(1199091N) 120583

119875(1199091N)

IC6 120583119873(1199092N) 120583

119875(1199092N) 120583

119875(1199091N) 120583

119875(1199091N)

IC7 120583119873(1199091N) 120583

119875(1199092N) 120583

119875(1199091N) 120583

119875(1199092N)

IC8 120583119873(1199091N) 120583

119875(1199092N) 120583

119875(1199091N) 120583

119875(1199092N)

(11) are used to evaluate appropriate control law in each ICregion as in [23] The results of evaluating the fuzzy controlrules 119877

1ndash1198774are given in Table 1

Applying (17) to the results from Table 1 and taking intoaccount (6) and (8) analytical structure of the controller iseasily obtained as follows

119906F (119896) =

119867119866119866119906(11988811199091(119896) + 119888

21199092(119896))

2 (2119871 minus 1198661198881

10038161003816100381610038161199091 (119896)1003816100381610038161003816)

IC1 IC2 IC5 IC6

119867119866119866119906(11988811199091 (119896) + 11988821199092 (119896))

2 (2119871 minus 1198661198882

10038161003816100381610038161199092 (119896)1003816100381610038161003816)

IC3 IC4 IC7 IC8

(18)

Defining function 119892(1199091 1199092)

119892 (1199091 1199092) =

119867119866119866119906

2 (2119871 minus 1198661198881

10038161003816100381610038161199091 (119896)1003816100381610038161003816) IC1 IC2 IC5 IC6

119867119866119866119906

2 (2119871 minus 1198661198882

10038161003816100381610038161199092 (119896)1003816100381610038161003816) IC3 IC4 IC7 IC8

(19)

(18) can be written as

119906F (119896) = 119892 (1199091 1199092) 119888119909 (119896) (20)

It is obvious that the function 119892(1199091 1199092) is a nonlinear

function with the following minimal and maximal values

119892min =119867119866119866119906

4119871 119892max =

119867119866119866119906

2119871 (21)

and 119892(1199091 1199092) gt 0 forall119909

1 1199092isin 119877 One should not forget

that parameter 119866 must satisfy Assumption 7 The range ofthe value of function 119892(119909

1 1199092) is determined by the choice

of parameters 119867 119866 119871 and 119866119906 Without loss of generality it

can be assumed that 119871 = 119867 = 1 and they are commonlynormalized domain boundaries One example of function119892(1199091 1199092) is shown in Figure 5

4 Main Results

In this section the solution of the problem stated in Section 2is provided by the following theorems

6 Advances in Fuzzy Systems

0 005

05

1

102025

03035

04045

05055

06

minus1

minus1

minus05minus05

x1 x2

Figure 5 Example of function 119892(1199091 1199092) for 119871 = 119867 = 1119866 = 119866119906 = 1

and 1198881 = 1198882 = 1

Theorem 11 Applying

119906119900119904(119896) = minus

1

119888119887119888119860119909 (119896) (22)

the so-called one step control to the system (1) outside 119878 isnecessary and sufficient for 119909(119896) to reach the 119878 in one samplingperiod (one step)

ProofNecessity Let us assume that119909(119896) of the system (1) is arbitraryoutside of sliding subspace 119909(119896) notin 119878 and a control pushesit to the sliding subspace in one step (sampling period) Itfollows that 119909(119896 + 1) isin 119878 That fact description of 119878 and thesystem (1) state equation lead to

119888119860119909 (119896) + 119888119887119906 (119896) = 0 (23)

and finally

119906 (119896) = 119906os(119896) = minus

1

119888119887119888119860119909 (119896) (24)

Sufficiency When the system state is outside of slidingsubspace 119909(119896) notin 119878 the action of one step control is assumedState equation of the system (1) and expression of 119878 give

119888119909 (119896 + 1) = 119888119860119909 (119896) + 119888119887119906os(119896) = 119888119860119909 (119896) minus 119888119887

1

119888119887119888119860119909 (119896)

= 0

(25)

which means that 119909(119896 + 1) isin 119878

Remark 12 One step control can be encountered in VSCSpapers in different contexts see for example [3 4 13] amongothers

Theorem 13 Applying

119906119904119897(119896) = minus

1

119888119887119888119860119909 (119896) (26)

the actual so-called sliding control to the system (1) inside 119878 isnecessary and sufficient for 119878 to be positive invariant relative tothe system (1) solution 120594(119896 119909(0) 119906119904119897(119896))

ProofNecessity Positive invariance of 119878 relative to the system (1)solution 120594(119896 119909(0) 119906sl(119896)) is assumed So every time 119909(119896) isin 119878119909(119896 + 1) isin 119878 as well From the previous sentence descriptionof 119878 and state equation of the system (1) follows that

119888119860119909 (119896) + 119888119887119906sl(119896) = 0 (27)

and further

119906sl(119896) = minus

1

119888119887119888119860119909 (119896) (28)

Sufficiency Let us assume relative to system (1) 119909(119896) isin 119878 andcontrol (26) is applied Then state equation of the system (1)and expression for 119878 give

119888119909 (119896 + 1) = 119888119860119909 (119896) + 119888119887119906sl(119896) = 119888119860119909 (119896) minus 119888119887

1

119888119887119888119860119909 (119896)

= 0

(29)

which implies that 119909(119896 + 1) isin 119878

Remark 14 One step control and sliding control formally areidentical

The application of actual sliding control 119906sl(119896) (hence-forth referred to as 119906

sl(119896)) to the system (1) inside 119878 is

the phenomenon of discrete-time VSCS with sliding modesUsing this fact and the intention for developing of the variablestructure type of the controller for the system (1) the control(4) components could be more precisely specified as

119906 [119909 (119896)] =

119906+[119909 (119896)] 119888119909 (119896) gt 0

119906sl[119909 (119896)] 119888119909 (119896) = 0

119906minus[119909 (119896)] 119888119909 (119896) lt 0

(30)

Contrary to the 119906sl(119896) which is applied to the system(1) inside 119878 let us refer to the control which is applied tothe system (1) outside of 119878 as the outer control 119906ot(119896) Thefollowing theorems are basic for solving the stated problemby Lyapunovrsquos second method

Theorem15 Let the system (1) (30) be considered 119878 is globallyexponentially stable relative to the system (1) (30) if there existsscalar function V and numbers 120578

119894isin]0 +infin[ 119894 = 1 2 3 such that

(a) V(119909) isin 119862(1198772)(b) 1205781119889(119909 119878) le V(119909) le 120578

2119889(119909 119878) forall119909 isin (1198772)

(c) ΔV(119909) le minus1205783119889(119909 119878) forall119909 isin (1198772)

where ΔV(119909) is the first forward finite difference of V(119909)

Proof The proof starts with the assumption that sufficientconditions are fulfilled while the exponential stability of 119878should be shown

Advances in Fuzzy Systems 7

Condition (b) of the theorem implies

minus119889 [119909 (119896) 119878] le 120578minus1

2V [119909 (119896)] (31)

which with (c) give

ΔV [119909 (119896)] le minus120578minus121205783V [119909 (119896)] (32)

The last inequality after solving gives at first

V [119909 (119896)] le V [119909 (0)] 119890minus120573119896 120573 = ln1205782

1205782minus 1205783

1205782gt 1205783 forall119909 (0) isin 119877

2

(33)

and further by means of condition (a)

119889 [119909 (119896) 119878] le 120572119889 [119909 (0) 119878] 119890minus120573119896

120572 = 120578minus1

11205782 (34)

From (a) it is clear that 1205781le 1205782which yields 120572 isin [1 +infin[

and above relationship between 1205782and 1205783guarantee that 120573 isin

]0 +infin[ In that way it is shown that all stated by Definition 5is fulfilled that is the exponential stability of 119878 is proved

Remark 16 Previous theorem is valid and in the general casefor 119909 isin 119877119899

Scalar function V defined by

V (119909) = (sign 119888119909) 119888119909 (35)

as Lyapunovrsquos function candidate and outer control law

119906ot(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) isin 119877 119865 (119896) gt 0 forall119896 isin 1198730

(36)

are chosen relative to the system (1) Evidently such sortof 119906ot together with 119906

sl which is unique and already statedrepresents the control of (30) type The following theoremsare basic for solving of the stated problem by Lyapunovrsquossecond method

Theorem 17 119878 is globally exponentially stable relative to thesystem (1) if the following control is applied

119906 (119896) =

119906119900119905(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) = |119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816

119906119904119897(119896) = minus

1

119888119887[119888119860119909 (119896)]

(37)

with 119906119865(119896) defined by (19) (20) and with 119892(119909

1 1199092) lt 1

Proof Previously introduced Lyapunovrsquos function candidatefulfills evidently the conditions (a) of Theorem 15 Let usshow that conditions (b) and (c) are fulfilled Related to thecondition (b) and taking into account that |119888119909| = 120577119889 where120577 is a constant and 119889 is distance between 119909 and hyperplane119888119909 = 0 it follows that

V (119909) = (sign 119888119909) 119888119909 = |119888119909| = 120577119889 (38)

and further

1205781119889 (119909 119878) le V (119909) le 120578

2119889 (119909 119878)

1205781isin ]0 +infin[ le 120577 120578

2isin ]0 +infin[ ge 120577

(39)

State of system (1) at time 119896 is adopted to be out of 119878Then

ΔV [119909 (119896)] = V [119909 (119896 + 1)] minus V [119909 (119896)]

= 119865 (119896)1003816100381610038161003816sign 119888119909 (119896)

1003816100381610038161003816 minus |119888119909 (119896)|

= (|119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816) sdot1003816100381610038161003816sign 119888119909 (119896)

1003816100381610038161003816 minus |119888119909 (119896)|

= (1 minus 119892 (1199091 1199092)) |119888119909 (119896)| minus |119888119909 (119896)|

= minus119892 (1199091 1199092) |119888119909 (119896)| = minus119892 (1199091 1199092) 120577119889 [119909 (119896) 119878]

le minus119892min120577119889 [119909 (119896) 119878] = minus1205783119889 [119909 (119896) 119878]

(40)

in which way the proof is finished

According toTheorem 17

ΔV [119909 (119896)] = minus119892 (1199091 1199092) V [119909 (119896)] (41)

with 119892(1199091 1199092) gt 0 and 119892(119909

1 1199092) lt 1 which leads to the

following equation

V [119909 (119896 + 1)] = (1 minus 119892 (1199091 1199092)) V [119909 (119896)] = 120583 (1199091 1199092) V [119909 (119896)]

120583 (1199091 1199092) isin ]0 1[

(42)

From nonlinearity of the control low it is obvious thatcontrol gain is high when the system state is far fromsliding surfaces and as small as possible in neighborhoodof the sliding subspace Also it may be noted that V[119909(119896)]increments from step to step that is absolute value of |119888119909|are getting smaller over time

Function V[119909(119896)] is a decreasing function along motionof the system but it will never be zero Obviously the system(1) state never gets into 119878 Moreover from (42) it is obviousthat as the working point approaches the 119878 the value of thefunction 120583(119909

1 1199092) tends to its maximum value

120583max = 120583 (1199091 1199092)max = 1 minus 119892 (1199091 1199092)min (43)

Let 119874120590(119878) be 120590-neighborhood of 119878 and V

119898120590= minV(119909)

119909 is such that 119889(119909 119878) = 120590 To be guaranteed for 119909(119896) toenter 119874

120590(119878) settling time

119896119904= [0 log

120583max

V119898120590

V (0)] + 1 (44)

is necessary to elapse where 119896119904= [0 log

120583max(V119898120590V(0))] is

the biggest integer from the denoted segment After the statereached 119874

120590(119878) one step control 119906os should be applied to the

system (1) in which way finite reaching time in 119878 for 119909(119896) isrealized The previous facts are summarized by the followingtheorem

8 Advances in Fuzzy Systems

Theorem 18 119878 is globally exponentially stable relative to thesystem (1) with finite reaching time in 119878 if the following controlis applied

119906 (119896) =

119906119900119905(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) = |119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816 forall119896 = 0 119896119904minus 1

119906119900119905(119896119904) = 119906119900119904(119896119904) = minus

1

119888119887[119888119860119909 (119896)]

119906119904119897(119896) =minus

1

119888119887[119888119860119909 (119896)] forall119896 = 119896

119904+1 119896119904+ 2

(45)

Proof Compared to the proof of Theorem 17 here the proofis the same

5 Simulation and Experimental Results

The proposed algorithm was tested experimentally on a DCservo motor with the gear and load whose linear mathemat-ical model (without considering the major nonlinear effectsthe speed dependent friction dead zone and backlash) is asfollows

119869 120579 (119905) + 119861 120579 (119905) =120578119892120578119898119896119905119870119892

119877119898

119880 (119905) (46)

where 120579(119905) is the angular position of the load shaft 119869 is thetotal moment of inertia reflecting to the output shaft 119861 isthe viscous friction coefficient and 119880(119905) is the motor inputvoltage 119877

119898 119896119905 120578119898 120578119892 and 119870

119892are respectively the motor

armature resistance the motor torque constant the motorefficiency the gearbox efficiency and the total gear ratio

In the previous model it is assumed that the motorinductance is much smaller than the resistance so it isignored as the case with used servo motor is The experi-ments were performed with the Quanser rotary servo motorSRV02 This model is equipped with the optical encoderand tachometer for motor position and speed measuringrespectively Q8-USB data acquisition board for real-timedata acquisition and control was used with MatlabSimulinksoftware and QUARC real-time control The parametersnumerical values are as follows 119869 = 0021 (kgm2) 119861 =

0084 (Nmsrad) 119877119898= 26 (Ω) 119896

119905= 00077 (NmA) 120578

119898=

069 120578119892= 09 and119870

119892= 70

Choosing 1199091= 120579 and 119909

2= 120579 as state variables and 119906 =

119880(119905) as control at first the continuous time state equationwasobtained and finally after applying an accurate discretizationprocedure with sampling period 119879 = 001 (s) the discrete-time state equation of the system was obtained as follows

119909 (119896 + 1) = [1 008266

0 06746] 119909 (119896) + [

0002653

04979] 119906 (119896) (47)

The above mathematical model of real DC servo motorwas derived as adequate discrete-time state equation In theway the system matrix parameters 119860 and 119887 are availableControl algorithm from Theorem 18 is used to control themotor For simulation and experiment parameter 120590 = 0005

0 01 02 03 04 05

0

01

02

03

04

05

SimulationExperiment

minus02

minus01

x1

(rad

)

t (s)

Figure 6 The state variable 1199091

0 01 02 03 04 05

0123

SimulationExperiment

minus6

minus5

minus4

minus3

minus2

minus1

x2

(rad

s)

t (s)

Figure 7 The state variable 1199092

is adopted which defines very narrow 120590-neighborhood ofsliding subspace 119878 Also sampling time 119879 = 001 (s) andvector 119888 = (80 1) determined so as the system is stable insliding mode regime are adopted which gives 120577 = 119888 =

80006 Fuzzy logic controller is defined by 119871 = 119867 = 1119866 = 002 and 119866

119906= 40 which implies 119892min = 02 and

119892max = 04 Taking into account estimate relationships onone hand between 120577 and 120578

1and 120578

2 and on another hand

between 120577 119892min and 1205783 parameters 1205781= 30 120578

2= 90 and

1205783= 15 are chosen giving parameters of exponential low

120572 = 3 and 120573 = 01823 For initial state 119909(0) = (036 1)119879

after calculation V119898120590

= 04 V(0) = 298 and settling time119896119904= 20 are obtainedThe parallel simulation and experimental results are

shown in Figures 6ndash11 related to the algorithm fromTheorem 18 Figures 6 and 7 show the system state variablesThe control signal that drives the state onto the slidingsubspace is shown in Figure 9 The sliding function anddistance are shown in Figures 8 and 10 respectively From

Advances in Fuzzy Systems 9

0 01 02 03 04 05

0

10

20

30

40

50

cx

minus10

SimulationExperiment

t (s)

Figure 8 The sliding function 119888119909

0 01 02 03 04 05

0

2

4

minus10

minus8

minus6

minus4

minus2

SimulationExperiment

t (s)

u(V

)

Figure 9 The control variable

the figures it could be seen that the system state from initialposition approaches the sliding subspace 119878 but not slowerthan the exponential law defined by the parameters 120572 and120573 At instant 119896

119904= 20 (119905 = 02 sec) the state for the first

time enters the 120590-neighborhood of 119878 If it continues in thesame way it will never reach 119878 To reach it one step controlis applied at 119896

119904driving state in one step to 119878 so the character

of the system approaching 119878 changes at 119896119904 At 119896

119904+ 1 = 21

the system state is in 119878 staying in it thereafter forever Inside119878 the system is stable too so the system state asymptoticallyapproaches the zero state Simulation results fully confirm thetheoretical results and meet the expectations ofTheorems 1517 and 18

The experimental results have demonstrated that theproposed control scheme is valid and effective for the realapplications It is shown that there is no chattering in slidingmode However in practice errors could not be causedonly by computation but also by measurement accuracy(resolution sensitivity) noise existence nonlinearity andinadequate description of the real system by linear model as

0 01 02 03 04 05

002040608

112

minus04

minus02

Exponential law

t (s)

d(xS)

SimulationExperiment

Figure 10 The distance

015 02 025 03 035 04 045 05

0

001

002

003

004

005

minus002

minus001

Exponential law

d(xS)

SimulationExperiment

t (s)

Figure 11 Augmented part of Figure 10

well All this could cause not entering the 120590-neighborhood ofsliding subspace exactly in step 119896

119904than at some later instant

which is the case here in the paper experimental applicationexample After entering the 120590-neighborhood of 119878 at 119896

119904or

not and applying one step control at 119896119904 a little bit of sliding

subspace missing occurs Here that missing is a little bitmore visible at the beginning of reaching the sliding subspacebut very shortly after the like transition period expires itis again invisible like in the mathematical case simulationThis is illustrated by Figure 11 which is an enlarged segmentof Figure 10 for 119905 between 015 and 05 (sec) Evasion of theoccurrence is achieved by appropriate choice of the slidingsubspace and 120590-neighborhood parameters through vector 119888and parameter 120590 as well as the state space area where thelinear model is very adequate

It has already been pointed out that exponential stabilityis a higher quality stability property compared to ldquocommonrdquostability in Lyapunovrsquos sense (in this case it refers to globalasymptotic stability of a set-sliding subspace 119878) For that it is

10 Advances in Fuzzy Systems

natural that the sufficient conditions of Theorem 15 are morerigorous compared to the conditions of ldquocommonrdquo stabilityThe conditions of Theorem 15 are indirectly applicable inthe control of real-world processes through control law ofTheorems 17 and 18 for these control laws were derived fromthe conditions of Theorem 15 Real experiment with controlof DC servomotor is control of real-world process In the wayit is confirmed that the stability conditions ofTheorem 15 areapplicable in the control processes in practice

To resume in both ideal simulation case and in experi-mental application the proposed new developed algorithmin the paper guarantees exponential stability of 119878 and nochattering in sliding mode regime Despite the existence ofall upper stated nonidealities in the case of real applicationby the proper parameters choice the system motion is suchthat distance of working point from 119878 is always under pro-posed exponential envelope as it could be seen in providedexperiment

6 Conclusion

In the paper discrete-time type of VSCS with sliding modesis considered related to the linear time-invariant plant ofsecond orderThe fuzzy exponential stabilizing state feedbackcontrol algorithm so with prescribed approaching to thesliding subspace velocity estimate is developed strictly byLyapunovrsquos second method and with chattering free slidingmode Also application of the proposed ideal cases algo-rithm in practice is considered and discussed through theexperimental example It shows that experiment is quiteconsistent with mathematical ideal cases

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Z Bucevac ldquoLyapunovrsquos method approach in a stabilizingcontrol algorithm design for digital discrete VS systems withsliding modesndashlinear plant caserdquo in Proceedings of the 7thInternational Symposium Modelling Identification and Control( IASTED rsquo88) Grindelwald Switzerland l988

[2] G Bartolini A Ferrara and V I Utkin ldquoAdaptive sliding modecontrol in discrete-time systemsrdquo Automatica vol 31 no 5 pp769ndash773 1995

[3] S Hui and S H Zak ldquoOn discrete-time variable structureslidingmode controlrdquo SystemsampControl Letters vol 38 no 4-5pp 283ndash288 1999

[4] A Bartoszewicz ldquoDiscrete-time quasi-sliding-mode controlstrategiesrdquo IEEE Transactions on Industrial Electronics vol 45no 4 pp 633ndash637 1998

[5] W Gao Y Wang and A Homaifa ldquoDiscrete-time variablestructure control systemsrdquo IEEE Transactions on IndustrialElectronics vol 42 no 2 pp 117ndash122 1995

[6] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[7] Y Zheng and Y Jing ldquoApproximation law for discrete-timevariable structure control systemsrdquo Journal of Control Theoryand Applications vol 4 no 3 pp 291ndash296 2006

[8] G Golo and C Milosavljevic ldquoRobust discrete-time chatteringfree slidingmode controlrdquo SystemsampControl Letters vol 41 no1 pp 19ndash28 2000

[9] H Ying ldquoTheory and application of a novel fuzzy PID controllerusing a simplified Takagi-Sugeno rule schemerdquo InformationSciences vol 123 no 3 pp 281ndash293 2000

[10] G Feng ldquoA survey on analysis and design of model-based fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 5 pp 676ndash697 2006

[11] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[12] H K Lam F H F Leung and P K S Tam ldquoStable androbust fuzzy control for uncertain nonlinear systemsrdquo IEEETransactions on Systems Man and Cybernetics Part A Systemsand Humans vol 30 no 6 pp 825ndash840 2000

[13] R-E Precup M L Tomescu M-B Rdac E M Petriu S PreitlandC-A Dragos ldquoIterative performance improvement of fuzzycontrol systems for three tank systemsrdquo Expert Systems withApplications vol 39 no 9 pp 8288ndash8299 2012

[14] D H Lee Y H Joo and M H Tak ldquoLocal stability analysisof continuous-time Takagi-Sugeno fuzzy systems a fuzzy Lya-punov function approachrdquo Information Sciences vol 257 pp163ndash175 2014

[15] R Palm ldquoRobust control by fuzzy sliding moderdquo Automaticavol 30 no 9 pp 1429ndash1437 1994

[16] C-L Chen andM-H Chang ldquoOptimal design of fuzzy sliding-mode control a comparative studyrdquo Fuzzy Sets and Systems vol93 no 1 pp 37ndash48 1998

[17] J CWu andT S Liu ldquoA sliding-mode approach to fuzzy controldesignrdquo IEEE Transactions on Control Systems Technology vol4 no 2 pp 141ndash151 1996

[18] S-W Kim and J-J Lee ldquoDesign of a fuzzy controller with fuzzysliding surfacerdquo Fuzzy Sets and Systems vol 71 no 3 pp 359ndash367 1995

[19] G Feng S G Cao N W Rees and C K Chak ldquoDesign offuzzy control systems with guaranteed stabilityrdquo Fuzzy Sets andSystems vol 85 no 1 pp 1ndash10 1997

[20] F Qiao QM Zhu AWinfield andCMelhuish ldquoFuzzy slidingmode control for discrete nonlinear systemsrdquo Transactions ofChina Automation Society vol 22 no 2 pp 311ndash315 2003

[21] S H Cho ldquoRobust discrete time tracking control using slidingsurfacesrdquo KSME Journal vol 9 no 1 pp 80ndash90 1995

[22] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992

[23] H Ying W Siler and J J Buckley ldquoFuzzy control theory anonlinear caserdquo Automatica vol 26 no 3 pp 513ndash520 1990

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

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6 Advances in Fuzzy Systems

0 005

05

1

102025

03035

04045

05055

06

minus1

minus1

minus05minus05

x1 x2

Figure 5 Example of function 119892(1199091 1199092) for 119871 = 119867 = 1119866 = 119866119906 = 1

and 1198881 = 1198882 = 1

Theorem 11 Applying

119906119900119904(119896) = minus

1

119888119887119888119860119909 (119896) (22)

the so-called one step control to the system (1) outside 119878 isnecessary and sufficient for 119909(119896) to reach the 119878 in one samplingperiod (one step)

ProofNecessity Let us assume that119909(119896) of the system (1) is arbitraryoutside of sliding subspace 119909(119896) notin 119878 and a control pushesit to the sliding subspace in one step (sampling period) Itfollows that 119909(119896 + 1) isin 119878 That fact description of 119878 and thesystem (1) state equation lead to

119888119860119909 (119896) + 119888119887119906 (119896) = 0 (23)

and finally

119906 (119896) = 119906os(119896) = minus

1

119888119887119888119860119909 (119896) (24)

Sufficiency When the system state is outside of slidingsubspace 119909(119896) notin 119878 the action of one step control is assumedState equation of the system (1) and expression of 119878 give

119888119909 (119896 + 1) = 119888119860119909 (119896) + 119888119887119906os(119896) = 119888119860119909 (119896) minus 119888119887

1

119888119887119888119860119909 (119896)

= 0

(25)

which means that 119909(119896 + 1) isin 119878

Remark 12 One step control can be encountered in VSCSpapers in different contexts see for example [3 4 13] amongothers

Theorem 13 Applying

119906119904119897(119896) = minus

1

119888119887119888119860119909 (119896) (26)

the actual so-called sliding control to the system (1) inside 119878 isnecessary and sufficient for 119878 to be positive invariant relative tothe system (1) solution 120594(119896 119909(0) 119906119904119897(119896))

ProofNecessity Positive invariance of 119878 relative to the system (1)solution 120594(119896 119909(0) 119906sl(119896)) is assumed So every time 119909(119896) isin 119878119909(119896 + 1) isin 119878 as well From the previous sentence descriptionof 119878 and state equation of the system (1) follows that

119888119860119909 (119896) + 119888119887119906sl(119896) = 0 (27)

and further

119906sl(119896) = minus

1

119888119887119888119860119909 (119896) (28)

Sufficiency Let us assume relative to system (1) 119909(119896) isin 119878 andcontrol (26) is applied Then state equation of the system (1)and expression for 119878 give

119888119909 (119896 + 1) = 119888119860119909 (119896) + 119888119887119906sl(119896) = 119888119860119909 (119896) minus 119888119887

1

119888119887119888119860119909 (119896)

= 0

(29)

which implies that 119909(119896 + 1) isin 119878

Remark 14 One step control and sliding control formally areidentical

The application of actual sliding control 119906sl(119896) (hence-forth referred to as 119906

sl(119896)) to the system (1) inside 119878 is

the phenomenon of discrete-time VSCS with sliding modesUsing this fact and the intention for developing of the variablestructure type of the controller for the system (1) the control(4) components could be more precisely specified as

119906 [119909 (119896)] =

119906+[119909 (119896)] 119888119909 (119896) gt 0

119906sl[119909 (119896)] 119888119909 (119896) = 0

119906minus[119909 (119896)] 119888119909 (119896) lt 0

(30)

Contrary to the 119906sl(119896) which is applied to the system(1) inside 119878 let us refer to the control which is applied tothe system (1) outside of 119878 as the outer control 119906ot(119896) Thefollowing theorems are basic for solving the stated problemby Lyapunovrsquos second method

Theorem15 Let the system (1) (30) be considered 119878 is globallyexponentially stable relative to the system (1) (30) if there existsscalar function V and numbers 120578

119894isin]0 +infin[ 119894 = 1 2 3 such that

(a) V(119909) isin 119862(1198772)(b) 1205781119889(119909 119878) le V(119909) le 120578

2119889(119909 119878) forall119909 isin (1198772)

(c) ΔV(119909) le minus1205783119889(119909 119878) forall119909 isin (1198772)

where ΔV(119909) is the first forward finite difference of V(119909)

Proof The proof starts with the assumption that sufficientconditions are fulfilled while the exponential stability of 119878should be shown

Advances in Fuzzy Systems 7

Condition (b) of the theorem implies

minus119889 [119909 (119896) 119878] le 120578minus1

2V [119909 (119896)] (31)

which with (c) give

ΔV [119909 (119896)] le minus120578minus121205783V [119909 (119896)] (32)

The last inequality after solving gives at first

V [119909 (119896)] le V [119909 (0)] 119890minus120573119896 120573 = ln1205782

1205782minus 1205783

1205782gt 1205783 forall119909 (0) isin 119877

2

(33)

and further by means of condition (a)

119889 [119909 (119896) 119878] le 120572119889 [119909 (0) 119878] 119890minus120573119896

120572 = 120578minus1

11205782 (34)

From (a) it is clear that 1205781le 1205782which yields 120572 isin [1 +infin[

and above relationship between 1205782and 1205783guarantee that 120573 isin

]0 +infin[ In that way it is shown that all stated by Definition 5is fulfilled that is the exponential stability of 119878 is proved

Remark 16 Previous theorem is valid and in the general casefor 119909 isin 119877119899

Scalar function V defined by

V (119909) = (sign 119888119909) 119888119909 (35)

as Lyapunovrsquos function candidate and outer control law

119906ot(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) isin 119877 119865 (119896) gt 0 forall119896 isin 1198730

(36)

are chosen relative to the system (1) Evidently such sortof 119906ot together with 119906

sl which is unique and already statedrepresents the control of (30) type The following theoremsare basic for solving of the stated problem by Lyapunovrsquossecond method

Theorem 17 119878 is globally exponentially stable relative to thesystem (1) if the following control is applied

119906 (119896) =

119906119900119905(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) = |119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816

119906119904119897(119896) = minus

1

119888119887[119888119860119909 (119896)]

(37)

with 119906119865(119896) defined by (19) (20) and with 119892(119909

1 1199092) lt 1

Proof Previously introduced Lyapunovrsquos function candidatefulfills evidently the conditions (a) of Theorem 15 Let usshow that conditions (b) and (c) are fulfilled Related to thecondition (b) and taking into account that |119888119909| = 120577119889 where120577 is a constant and 119889 is distance between 119909 and hyperplane119888119909 = 0 it follows that

V (119909) = (sign 119888119909) 119888119909 = |119888119909| = 120577119889 (38)

and further

1205781119889 (119909 119878) le V (119909) le 120578

2119889 (119909 119878)

1205781isin ]0 +infin[ le 120577 120578

2isin ]0 +infin[ ge 120577

(39)

State of system (1) at time 119896 is adopted to be out of 119878Then

ΔV [119909 (119896)] = V [119909 (119896 + 1)] minus V [119909 (119896)]

= 119865 (119896)1003816100381610038161003816sign 119888119909 (119896)

1003816100381610038161003816 minus |119888119909 (119896)|

= (|119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816) sdot1003816100381610038161003816sign 119888119909 (119896)

1003816100381610038161003816 minus |119888119909 (119896)|

= (1 minus 119892 (1199091 1199092)) |119888119909 (119896)| minus |119888119909 (119896)|

= minus119892 (1199091 1199092) |119888119909 (119896)| = minus119892 (1199091 1199092) 120577119889 [119909 (119896) 119878]

le minus119892min120577119889 [119909 (119896) 119878] = minus1205783119889 [119909 (119896) 119878]

(40)

in which way the proof is finished

According toTheorem 17

ΔV [119909 (119896)] = minus119892 (1199091 1199092) V [119909 (119896)] (41)

with 119892(1199091 1199092) gt 0 and 119892(119909

1 1199092) lt 1 which leads to the

following equation

V [119909 (119896 + 1)] = (1 minus 119892 (1199091 1199092)) V [119909 (119896)] = 120583 (1199091 1199092) V [119909 (119896)]

120583 (1199091 1199092) isin ]0 1[

(42)

From nonlinearity of the control low it is obvious thatcontrol gain is high when the system state is far fromsliding surfaces and as small as possible in neighborhoodof the sliding subspace Also it may be noted that V[119909(119896)]increments from step to step that is absolute value of |119888119909|are getting smaller over time

Function V[119909(119896)] is a decreasing function along motionof the system but it will never be zero Obviously the system(1) state never gets into 119878 Moreover from (42) it is obviousthat as the working point approaches the 119878 the value of thefunction 120583(119909

1 1199092) tends to its maximum value

120583max = 120583 (1199091 1199092)max = 1 minus 119892 (1199091 1199092)min (43)

Let 119874120590(119878) be 120590-neighborhood of 119878 and V

119898120590= minV(119909)

119909 is such that 119889(119909 119878) = 120590 To be guaranteed for 119909(119896) toenter 119874

120590(119878) settling time

119896119904= [0 log

120583max

V119898120590

V (0)] + 1 (44)

is necessary to elapse where 119896119904= [0 log

120583max(V119898120590V(0))] is

the biggest integer from the denoted segment After the statereached 119874

120590(119878) one step control 119906os should be applied to the

system (1) in which way finite reaching time in 119878 for 119909(119896) isrealized The previous facts are summarized by the followingtheorem

8 Advances in Fuzzy Systems

Theorem 18 119878 is globally exponentially stable relative to thesystem (1) with finite reaching time in 119878 if the following controlis applied

119906 (119896) =

119906119900119905(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) = |119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816 forall119896 = 0 119896119904minus 1

119906119900119905(119896119904) = 119906119900119904(119896119904) = minus

1

119888119887[119888119860119909 (119896)]

119906119904119897(119896) =minus

1

119888119887[119888119860119909 (119896)] forall119896 = 119896

119904+1 119896119904+ 2

(45)

Proof Compared to the proof of Theorem 17 here the proofis the same

5 Simulation and Experimental Results

The proposed algorithm was tested experimentally on a DCservo motor with the gear and load whose linear mathemat-ical model (without considering the major nonlinear effectsthe speed dependent friction dead zone and backlash) is asfollows

119869 120579 (119905) + 119861 120579 (119905) =120578119892120578119898119896119905119870119892

119877119898

119880 (119905) (46)

where 120579(119905) is the angular position of the load shaft 119869 is thetotal moment of inertia reflecting to the output shaft 119861 isthe viscous friction coefficient and 119880(119905) is the motor inputvoltage 119877

119898 119896119905 120578119898 120578119892 and 119870

119892are respectively the motor

armature resistance the motor torque constant the motorefficiency the gearbox efficiency and the total gear ratio

In the previous model it is assumed that the motorinductance is much smaller than the resistance so it isignored as the case with used servo motor is The experi-ments were performed with the Quanser rotary servo motorSRV02 This model is equipped with the optical encoderand tachometer for motor position and speed measuringrespectively Q8-USB data acquisition board for real-timedata acquisition and control was used with MatlabSimulinksoftware and QUARC real-time control The parametersnumerical values are as follows 119869 = 0021 (kgm2) 119861 =

0084 (Nmsrad) 119877119898= 26 (Ω) 119896

119905= 00077 (NmA) 120578

119898=

069 120578119892= 09 and119870

119892= 70

Choosing 1199091= 120579 and 119909

2= 120579 as state variables and 119906 =

119880(119905) as control at first the continuous time state equationwasobtained and finally after applying an accurate discretizationprocedure with sampling period 119879 = 001 (s) the discrete-time state equation of the system was obtained as follows

119909 (119896 + 1) = [1 008266

0 06746] 119909 (119896) + [

0002653

04979] 119906 (119896) (47)

The above mathematical model of real DC servo motorwas derived as adequate discrete-time state equation In theway the system matrix parameters 119860 and 119887 are availableControl algorithm from Theorem 18 is used to control themotor For simulation and experiment parameter 120590 = 0005

0 01 02 03 04 05

0

01

02

03

04

05

SimulationExperiment

minus02

minus01

x1

(rad

)

t (s)

Figure 6 The state variable 1199091

0 01 02 03 04 05

0123

SimulationExperiment

minus6

minus5

minus4

minus3

minus2

minus1

x2

(rad

s)

t (s)

Figure 7 The state variable 1199092

is adopted which defines very narrow 120590-neighborhood ofsliding subspace 119878 Also sampling time 119879 = 001 (s) andvector 119888 = (80 1) determined so as the system is stable insliding mode regime are adopted which gives 120577 = 119888 =

80006 Fuzzy logic controller is defined by 119871 = 119867 = 1119866 = 002 and 119866

119906= 40 which implies 119892min = 02 and

119892max = 04 Taking into account estimate relationships onone hand between 120577 and 120578

1and 120578

2 and on another hand

between 120577 119892min and 1205783 parameters 1205781= 30 120578

2= 90 and

1205783= 15 are chosen giving parameters of exponential low

120572 = 3 and 120573 = 01823 For initial state 119909(0) = (036 1)119879

after calculation V119898120590

= 04 V(0) = 298 and settling time119896119904= 20 are obtainedThe parallel simulation and experimental results are

shown in Figures 6ndash11 related to the algorithm fromTheorem 18 Figures 6 and 7 show the system state variablesThe control signal that drives the state onto the slidingsubspace is shown in Figure 9 The sliding function anddistance are shown in Figures 8 and 10 respectively From

Advances in Fuzzy Systems 9

0 01 02 03 04 05

0

10

20

30

40

50

cx

minus10

SimulationExperiment

t (s)

Figure 8 The sliding function 119888119909

0 01 02 03 04 05

0

2

4

minus10

minus8

minus6

minus4

minus2

SimulationExperiment

t (s)

u(V

)

Figure 9 The control variable

the figures it could be seen that the system state from initialposition approaches the sliding subspace 119878 but not slowerthan the exponential law defined by the parameters 120572 and120573 At instant 119896

119904= 20 (119905 = 02 sec) the state for the first

time enters the 120590-neighborhood of 119878 If it continues in thesame way it will never reach 119878 To reach it one step controlis applied at 119896

119904driving state in one step to 119878 so the character

of the system approaching 119878 changes at 119896119904 At 119896

119904+ 1 = 21

the system state is in 119878 staying in it thereafter forever Inside119878 the system is stable too so the system state asymptoticallyapproaches the zero state Simulation results fully confirm thetheoretical results and meet the expectations ofTheorems 1517 and 18

The experimental results have demonstrated that theproposed control scheme is valid and effective for the realapplications It is shown that there is no chattering in slidingmode However in practice errors could not be causedonly by computation but also by measurement accuracy(resolution sensitivity) noise existence nonlinearity andinadequate description of the real system by linear model as

0 01 02 03 04 05

002040608

112

minus04

minus02

Exponential law

t (s)

d(xS)

SimulationExperiment

Figure 10 The distance

015 02 025 03 035 04 045 05

0

001

002

003

004

005

minus002

minus001

Exponential law

d(xS)

SimulationExperiment

t (s)

Figure 11 Augmented part of Figure 10

well All this could cause not entering the 120590-neighborhood ofsliding subspace exactly in step 119896

119904than at some later instant

which is the case here in the paper experimental applicationexample After entering the 120590-neighborhood of 119878 at 119896

119904or

not and applying one step control at 119896119904 a little bit of sliding

subspace missing occurs Here that missing is a little bitmore visible at the beginning of reaching the sliding subspacebut very shortly after the like transition period expires itis again invisible like in the mathematical case simulationThis is illustrated by Figure 11 which is an enlarged segmentof Figure 10 for 119905 between 015 and 05 (sec) Evasion of theoccurrence is achieved by appropriate choice of the slidingsubspace and 120590-neighborhood parameters through vector 119888and parameter 120590 as well as the state space area where thelinear model is very adequate

It has already been pointed out that exponential stabilityis a higher quality stability property compared to ldquocommonrdquostability in Lyapunovrsquos sense (in this case it refers to globalasymptotic stability of a set-sliding subspace 119878) For that it is

10 Advances in Fuzzy Systems

natural that the sufficient conditions of Theorem 15 are morerigorous compared to the conditions of ldquocommonrdquo stabilityThe conditions of Theorem 15 are indirectly applicable inthe control of real-world processes through control law ofTheorems 17 and 18 for these control laws were derived fromthe conditions of Theorem 15 Real experiment with controlof DC servomotor is control of real-world process In the wayit is confirmed that the stability conditions ofTheorem 15 areapplicable in the control processes in practice

To resume in both ideal simulation case and in experi-mental application the proposed new developed algorithmin the paper guarantees exponential stability of 119878 and nochattering in sliding mode regime Despite the existence ofall upper stated nonidealities in the case of real applicationby the proper parameters choice the system motion is suchthat distance of working point from 119878 is always under pro-posed exponential envelope as it could be seen in providedexperiment

6 Conclusion

In the paper discrete-time type of VSCS with sliding modesis considered related to the linear time-invariant plant ofsecond orderThe fuzzy exponential stabilizing state feedbackcontrol algorithm so with prescribed approaching to thesliding subspace velocity estimate is developed strictly byLyapunovrsquos second method and with chattering free slidingmode Also application of the proposed ideal cases algo-rithm in practice is considered and discussed through theexperimental example It shows that experiment is quiteconsistent with mathematical ideal cases

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Z Bucevac ldquoLyapunovrsquos method approach in a stabilizingcontrol algorithm design for digital discrete VS systems withsliding modesndashlinear plant caserdquo in Proceedings of the 7thInternational Symposium Modelling Identification and Control( IASTED rsquo88) Grindelwald Switzerland l988

[2] G Bartolini A Ferrara and V I Utkin ldquoAdaptive sliding modecontrol in discrete-time systemsrdquo Automatica vol 31 no 5 pp769ndash773 1995

[3] S Hui and S H Zak ldquoOn discrete-time variable structureslidingmode controlrdquo SystemsampControl Letters vol 38 no 4-5pp 283ndash288 1999

[4] A Bartoszewicz ldquoDiscrete-time quasi-sliding-mode controlstrategiesrdquo IEEE Transactions on Industrial Electronics vol 45no 4 pp 633ndash637 1998

[5] W Gao Y Wang and A Homaifa ldquoDiscrete-time variablestructure control systemsrdquo IEEE Transactions on IndustrialElectronics vol 42 no 2 pp 117ndash122 1995

[6] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[7] Y Zheng and Y Jing ldquoApproximation law for discrete-timevariable structure control systemsrdquo Journal of Control Theoryand Applications vol 4 no 3 pp 291ndash296 2006

[8] G Golo and C Milosavljevic ldquoRobust discrete-time chatteringfree slidingmode controlrdquo SystemsampControl Letters vol 41 no1 pp 19ndash28 2000

[9] H Ying ldquoTheory and application of a novel fuzzy PID controllerusing a simplified Takagi-Sugeno rule schemerdquo InformationSciences vol 123 no 3 pp 281ndash293 2000

[10] G Feng ldquoA survey on analysis and design of model-based fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 5 pp 676ndash697 2006

[11] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[12] H K Lam F H F Leung and P K S Tam ldquoStable androbust fuzzy control for uncertain nonlinear systemsrdquo IEEETransactions on Systems Man and Cybernetics Part A Systemsand Humans vol 30 no 6 pp 825ndash840 2000

[13] R-E Precup M L Tomescu M-B Rdac E M Petriu S PreitlandC-A Dragos ldquoIterative performance improvement of fuzzycontrol systems for three tank systemsrdquo Expert Systems withApplications vol 39 no 9 pp 8288ndash8299 2012

[14] D H Lee Y H Joo and M H Tak ldquoLocal stability analysisof continuous-time Takagi-Sugeno fuzzy systems a fuzzy Lya-punov function approachrdquo Information Sciences vol 257 pp163ndash175 2014

[15] R Palm ldquoRobust control by fuzzy sliding moderdquo Automaticavol 30 no 9 pp 1429ndash1437 1994

[16] C-L Chen andM-H Chang ldquoOptimal design of fuzzy sliding-mode control a comparative studyrdquo Fuzzy Sets and Systems vol93 no 1 pp 37ndash48 1998

[17] J CWu andT S Liu ldquoA sliding-mode approach to fuzzy controldesignrdquo IEEE Transactions on Control Systems Technology vol4 no 2 pp 141ndash151 1996

[18] S-W Kim and J-J Lee ldquoDesign of a fuzzy controller with fuzzysliding surfacerdquo Fuzzy Sets and Systems vol 71 no 3 pp 359ndash367 1995

[19] G Feng S G Cao N W Rees and C K Chak ldquoDesign offuzzy control systems with guaranteed stabilityrdquo Fuzzy Sets andSystems vol 85 no 1 pp 1ndash10 1997

[20] F Qiao QM Zhu AWinfield andCMelhuish ldquoFuzzy slidingmode control for discrete nonlinear systemsrdquo Transactions ofChina Automation Society vol 22 no 2 pp 311ndash315 2003

[21] S H Cho ldquoRobust discrete time tracking control using slidingsurfacesrdquo KSME Journal vol 9 no 1 pp 80ndash90 1995

[22] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992

[23] H Ying W Siler and J J Buckley ldquoFuzzy control theory anonlinear caserdquo Automatica vol 26 no 3 pp 513ndash520 1990

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

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Advances in Fuzzy Systems 7

Condition (b) of the theorem implies

minus119889 [119909 (119896) 119878] le 120578minus1

2V [119909 (119896)] (31)

which with (c) give

ΔV [119909 (119896)] le minus120578minus121205783V [119909 (119896)] (32)

The last inequality after solving gives at first

V [119909 (119896)] le V [119909 (0)] 119890minus120573119896 120573 = ln1205782

1205782minus 1205783

1205782gt 1205783 forall119909 (0) isin 119877

2

(33)

and further by means of condition (a)

119889 [119909 (119896) 119878] le 120572119889 [119909 (0) 119878] 119890minus120573119896

120572 = 120578minus1

11205782 (34)

From (a) it is clear that 1205781le 1205782which yields 120572 isin [1 +infin[

and above relationship between 1205782and 1205783guarantee that 120573 isin

]0 +infin[ In that way it is shown that all stated by Definition 5is fulfilled that is the exponential stability of 119878 is proved

Remark 16 Previous theorem is valid and in the general casefor 119909 isin 119877119899

Scalar function V defined by

V (119909) = (sign 119888119909) 119888119909 (35)

as Lyapunovrsquos function candidate and outer control law

119906ot(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) isin 119877 119865 (119896) gt 0 forall119896 isin 1198730

(36)

are chosen relative to the system (1) Evidently such sortof 119906ot together with 119906

sl which is unique and already statedrepresents the control of (30) type The following theoremsare basic for solving of the stated problem by Lyapunovrsquossecond method

Theorem 17 119878 is globally exponentially stable relative to thesystem (1) if the following control is applied

119906 (119896) =

119906119900119905(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) = |119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816

119906119904119897(119896) = minus

1

119888119887[119888119860119909 (119896)]

(37)

with 119906119865(119896) defined by (19) (20) and with 119892(119909

1 1199092) lt 1

Proof Previously introduced Lyapunovrsquos function candidatefulfills evidently the conditions (a) of Theorem 15 Let usshow that conditions (b) and (c) are fulfilled Related to thecondition (b) and taking into account that |119888119909| = 120577119889 where120577 is a constant and 119889 is distance between 119909 and hyperplane119888119909 = 0 it follows that

V (119909) = (sign 119888119909) 119888119909 = |119888119909| = 120577119889 (38)

and further

1205781119889 (119909 119878) le V (119909) le 120578

2119889 (119909 119878)

1205781isin ]0 +infin[ le 120577 120578

2isin ]0 +infin[ ge 120577

(39)

State of system (1) at time 119896 is adopted to be out of 119878Then

ΔV [119909 (119896)] = V [119909 (119896 + 1)] minus V [119909 (119896)]

= 119865 (119896)1003816100381610038161003816sign 119888119909 (119896)

1003816100381610038161003816 minus |119888119909 (119896)|

= (|119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816) sdot1003816100381610038161003816sign 119888119909 (119896)

1003816100381610038161003816 minus |119888119909 (119896)|

= (1 minus 119892 (1199091 1199092)) |119888119909 (119896)| minus |119888119909 (119896)|

= minus119892 (1199091 1199092) |119888119909 (119896)| = minus119892 (1199091 1199092) 120577119889 [119909 (119896) 119878]

le minus119892min120577119889 [119909 (119896) 119878] = minus1205783119889 [119909 (119896) 119878]

(40)

in which way the proof is finished

According toTheorem 17

ΔV [119909 (119896)] = minus119892 (1199091 1199092) V [119909 (119896)] (41)

with 119892(1199091 1199092) gt 0 and 119892(119909

1 1199092) lt 1 which leads to the

following equation

V [119909 (119896 + 1)] = (1 minus 119892 (1199091 1199092)) V [119909 (119896)] = 120583 (1199091 1199092) V [119909 (119896)]

120583 (1199091 1199092) isin ]0 1[

(42)

From nonlinearity of the control low it is obvious thatcontrol gain is high when the system state is far fromsliding surfaces and as small as possible in neighborhoodof the sliding subspace Also it may be noted that V[119909(119896)]increments from step to step that is absolute value of |119888119909|are getting smaller over time

Function V[119909(119896)] is a decreasing function along motionof the system but it will never be zero Obviously the system(1) state never gets into 119878 Moreover from (42) it is obviousthat as the working point approaches the 119878 the value of thefunction 120583(119909

1 1199092) tends to its maximum value

120583max = 120583 (1199091 1199092)max = 1 minus 119892 (1199091 1199092)min (43)

Let 119874120590(119878) be 120590-neighborhood of 119878 and V

119898120590= minV(119909)

119909 is such that 119889(119909 119878) = 120590 To be guaranteed for 119909(119896) toenter 119874

120590(119878) settling time

119896119904= [0 log

120583max

V119898120590

V (0)] + 1 (44)

is necessary to elapse where 119896119904= [0 log

120583max(V119898120590V(0))] is

the biggest integer from the denoted segment After the statereached 119874

120590(119878) one step control 119906os should be applied to the

system (1) in which way finite reaching time in 119878 for 119909(119896) isrealized The previous facts are summarized by the followingtheorem

8 Advances in Fuzzy Systems

Theorem 18 119878 is globally exponentially stable relative to thesystem (1) with finite reaching time in 119878 if the following controlis applied

119906 (119896) =

119906119900119905(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) = |119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816 forall119896 = 0 119896119904minus 1

119906119900119905(119896119904) = 119906119900119904(119896119904) = minus

1

119888119887[119888119860119909 (119896)]

119906119904119897(119896) =minus

1

119888119887[119888119860119909 (119896)] forall119896 = 119896

119904+1 119896119904+ 2

(45)

Proof Compared to the proof of Theorem 17 here the proofis the same

5 Simulation and Experimental Results

The proposed algorithm was tested experimentally on a DCservo motor with the gear and load whose linear mathemat-ical model (without considering the major nonlinear effectsthe speed dependent friction dead zone and backlash) is asfollows

119869 120579 (119905) + 119861 120579 (119905) =120578119892120578119898119896119905119870119892

119877119898

119880 (119905) (46)

where 120579(119905) is the angular position of the load shaft 119869 is thetotal moment of inertia reflecting to the output shaft 119861 isthe viscous friction coefficient and 119880(119905) is the motor inputvoltage 119877

119898 119896119905 120578119898 120578119892 and 119870

119892are respectively the motor

armature resistance the motor torque constant the motorefficiency the gearbox efficiency and the total gear ratio

In the previous model it is assumed that the motorinductance is much smaller than the resistance so it isignored as the case with used servo motor is The experi-ments were performed with the Quanser rotary servo motorSRV02 This model is equipped with the optical encoderand tachometer for motor position and speed measuringrespectively Q8-USB data acquisition board for real-timedata acquisition and control was used with MatlabSimulinksoftware and QUARC real-time control The parametersnumerical values are as follows 119869 = 0021 (kgm2) 119861 =

0084 (Nmsrad) 119877119898= 26 (Ω) 119896

119905= 00077 (NmA) 120578

119898=

069 120578119892= 09 and119870

119892= 70

Choosing 1199091= 120579 and 119909

2= 120579 as state variables and 119906 =

119880(119905) as control at first the continuous time state equationwasobtained and finally after applying an accurate discretizationprocedure with sampling period 119879 = 001 (s) the discrete-time state equation of the system was obtained as follows

119909 (119896 + 1) = [1 008266

0 06746] 119909 (119896) + [

0002653

04979] 119906 (119896) (47)

The above mathematical model of real DC servo motorwas derived as adequate discrete-time state equation In theway the system matrix parameters 119860 and 119887 are availableControl algorithm from Theorem 18 is used to control themotor For simulation and experiment parameter 120590 = 0005

0 01 02 03 04 05

0

01

02

03

04

05

SimulationExperiment

minus02

minus01

x1

(rad

)

t (s)

Figure 6 The state variable 1199091

0 01 02 03 04 05

0123

SimulationExperiment

minus6

minus5

minus4

minus3

minus2

minus1

x2

(rad

s)

t (s)

Figure 7 The state variable 1199092

is adopted which defines very narrow 120590-neighborhood ofsliding subspace 119878 Also sampling time 119879 = 001 (s) andvector 119888 = (80 1) determined so as the system is stable insliding mode regime are adopted which gives 120577 = 119888 =

80006 Fuzzy logic controller is defined by 119871 = 119867 = 1119866 = 002 and 119866

119906= 40 which implies 119892min = 02 and

119892max = 04 Taking into account estimate relationships onone hand between 120577 and 120578

1and 120578

2 and on another hand

between 120577 119892min and 1205783 parameters 1205781= 30 120578

2= 90 and

1205783= 15 are chosen giving parameters of exponential low

120572 = 3 and 120573 = 01823 For initial state 119909(0) = (036 1)119879

after calculation V119898120590

= 04 V(0) = 298 and settling time119896119904= 20 are obtainedThe parallel simulation and experimental results are

shown in Figures 6ndash11 related to the algorithm fromTheorem 18 Figures 6 and 7 show the system state variablesThe control signal that drives the state onto the slidingsubspace is shown in Figure 9 The sliding function anddistance are shown in Figures 8 and 10 respectively From

Advances in Fuzzy Systems 9

0 01 02 03 04 05

0

10

20

30

40

50

cx

minus10

SimulationExperiment

t (s)

Figure 8 The sliding function 119888119909

0 01 02 03 04 05

0

2

4

minus10

minus8

minus6

minus4

minus2

SimulationExperiment

t (s)

u(V

)

Figure 9 The control variable

the figures it could be seen that the system state from initialposition approaches the sliding subspace 119878 but not slowerthan the exponential law defined by the parameters 120572 and120573 At instant 119896

119904= 20 (119905 = 02 sec) the state for the first

time enters the 120590-neighborhood of 119878 If it continues in thesame way it will never reach 119878 To reach it one step controlis applied at 119896

119904driving state in one step to 119878 so the character

of the system approaching 119878 changes at 119896119904 At 119896

119904+ 1 = 21

the system state is in 119878 staying in it thereafter forever Inside119878 the system is stable too so the system state asymptoticallyapproaches the zero state Simulation results fully confirm thetheoretical results and meet the expectations ofTheorems 1517 and 18

The experimental results have demonstrated that theproposed control scheme is valid and effective for the realapplications It is shown that there is no chattering in slidingmode However in practice errors could not be causedonly by computation but also by measurement accuracy(resolution sensitivity) noise existence nonlinearity andinadequate description of the real system by linear model as

0 01 02 03 04 05

002040608

112

minus04

minus02

Exponential law

t (s)

d(xS)

SimulationExperiment

Figure 10 The distance

015 02 025 03 035 04 045 05

0

001

002

003

004

005

minus002

minus001

Exponential law

d(xS)

SimulationExperiment

t (s)

Figure 11 Augmented part of Figure 10

well All this could cause not entering the 120590-neighborhood ofsliding subspace exactly in step 119896

119904than at some later instant

which is the case here in the paper experimental applicationexample After entering the 120590-neighborhood of 119878 at 119896

119904or

not and applying one step control at 119896119904 a little bit of sliding

subspace missing occurs Here that missing is a little bitmore visible at the beginning of reaching the sliding subspacebut very shortly after the like transition period expires itis again invisible like in the mathematical case simulationThis is illustrated by Figure 11 which is an enlarged segmentof Figure 10 for 119905 between 015 and 05 (sec) Evasion of theoccurrence is achieved by appropriate choice of the slidingsubspace and 120590-neighborhood parameters through vector 119888and parameter 120590 as well as the state space area where thelinear model is very adequate

It has already been pointed out that exponential stabilityis a higher quality stability property compared to ldquocommonrdquostability in Lyapunovrsquos sense (in this case it refers to globalasymptotic stability of a set-sliding subspace 119878) For that it is

10 Advances in Fuzzy Systems

natural that the sufficient conditions of Theorem 15 are morerigorous compared to the conditions of ldquocommonrdquo stabilityThe conditions of Theorem 15 are indirectly applicable inthe control of real-world processes through control law ofTheorems 17 and 18 for these control laws were derived fromthe conditions of Theorem 15 Real experiment with controlof DC servomotor is control of real-world process In the wayit is confirmed that the stability conditions ofTheorem 15 areapplicable in the control processes in practice

To resume in both ideal simulation case and in experi-mental application the proposed new developed algorithmin the paper guarantees exponential stability of 119878 and nochattering in sliding mode regime Despite the existence ofall upper stated nonidealities in the case of real applicationby the proper parameters choice the system motion is suchthat distance of working point from 119878 is always under pro-posed exponential envelope as it could be seen in providedexperiment

6 Conclusion

In the paper discrete-time type of VSCS with sliding modesis considered related to the linear time-invariant plant ofsecond orderThe fuzzy exponential stabilizing state feedbackcontrol algorithm so with prescribed approaching to thesliding subspace velocity estimate is developed strictly byLyapunovrsquos second method and with chattering free slidingmode Also application of the proposed ideal cases algo-rithm in practice is considered and discussed through theexperimental example It shows that experiment is quiteconsistent with mathematical ideal cases

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Z Bucevac ldquoLyapunovrsquos method approach in a stabilizingcontrol algorithm design for digital discrete VS systems withsliding modesndashlinear plant caserdquo in Proceedings of the 7thInternational Symposium Modelling Identification and Control( IASTED rsquo88) Grindelwald Switzerland l988

[2] G Bartolini A Ferrara and V I Utkin ldquoAdaptive sliding modecontrol in discrete-time systemsrdquo Automatica vol 31 no 5 pp769ndash773 1995

[3] S Hui and S H Zak ldquoOn discrete-time variable structureslidingmode controlrdquo SystemsampControl Letters vol 38 no 4-5pp 283ndash288 1999

[4] A Bartoszewicz ldquoDiscrete-time quasi-sliding-mode controlstrategiesrdquo IEEE Transactions on Industrial Electronics vol 45no 4 pp 633ndash637 1998

[5] W Gao Y Wang and A Homaifa ldquoDiscrete-time variablestructure control systemsrdquo IEEE Transactions on IndustrialElectronics vol 42 no 2 pp 117ndash122 1995

[6] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[7] Y Zheng and Y Jing ldquoApproximation law for discrete-timevariable structure control systemsrdquo Journal of Control Theoryand Applications vol 4 no 3 pp 291ndash296 2006

[8] G Golo and C Milosavljevic ldquoRobust discrete-time chatteringfree slidingmode controlrdquo SystemsampControl Letters vol 41 no1 pp 19ndash28 2000

[9] H Ying ldquoTheory and application of a novel fuzzy PID controllerusing a simplified Takagi-Sugeno rule schemerdquo InformationSciences vol 123 no 3 pp 281ndash293 2000

[10] G Feng ldquoA survey on analysis and design of model-based fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 5 pp 676ndash697 2006

[11] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[12] H K Lam F H F Leung and P K S Tam ldquoStable androbust fuzzy control for uncertain nonlinear systemsrdquo IEEETransactions on Systems Man and Cybernetics Part A Systemsand Humans vol 30 no 6 pp 825ndash840 2000

[13] R-E Precup M L Tomescu M-B Rdac E M Petriu S PreitlandC-A Dragos ldquoIterative performance improvement of fuzzycontrol systems for three tank systemsrdquo Expert Systems withApplications vol 39 no 9 pp 8288ndash8299 2012

[14] D H Lee Y H Joo and M H Tak ldquoLocal stability analysisof continuous-time Takagi-Sugeno fuzzy systems a fuzzy Lya-punov function approachrdquo Information Sciences vol 257 pp163ndash175 2014

[15] R Palm ldquoRobust control by fuzzy sliding moderdquo Automaticavol 30 no 9 pp 1429ndash1437 1994

[16] C-L Chen andM-H Chang ldquoOptimal design of fuzzy sliding-mode control a comparative studyrdquo Fuzzy Sets and Systems vol93 no 1 pp 37ndash48 1998

[17] J CWu andT S Liu ldquoA sliding-mode approach to fuzzy controldesignrdquo IEEE Transactions on Control Systems Technology vol4 no 2 pp 141ndash151 1996

[18] S-W Kim and J-J Lee ldquoDesign of a fuzzy controller with fuzzysliding surfacerdquo Fuzzy Sets and Systems vol 71 no 3 pp 359ndash367 1995

[19] G Feng S G Cao N W Rees and C K Chak ldquoDesign offuzzy control systems with guaranteed stabilityrdquo Fuzzy Sets andSystems vol 85 no 1 pp 1ndash10 1997

[20] F Qiao QM Zhu AWinfield andCMelhuish ldquoFuzzy slidingmode control for discrete nonlinear systemsrdquo Transactions ofChina Automation Society vol 22 no 2 pp 311ndash315 2003

[21] S H Cho ldquoRobust discrete time tracking control using slidingsurfacesrdquo KSME Journal vol 9 no 1 pp 80ndash90 1995

[22] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992

[23] H Ying W Siler and J J Buckley ldquoFuzzy control theory anonlinear caserdquo Automatica vol 26 no 3 pp 513ndash520 1990

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

8 Advances in Fuzzy Systems

Theorem 18 119878 is globally exponentially stable relative to thesystem (1) with finite reaching time in 119878 if the following controlis applied

119906 (119896) =

119906119900119905(119896) = minus

1

119888119887[119888119860119909 (119896) minus 119865 (119896) sign 119888119909 (119896)]

119865 (119896) = |119888119909 (119896)| minus1003816100381610038161003816119906F (119896)

1003816100381610038161003816 forall119896 = 0 119896119904minus 1

119906119900119905(119896119904) = 119906119900119904(119896119904) = minus

1

119888119887[119888119860119909 (119896)]

119906119904119897(119896) =minus

1

119888119887[119888119860119909 (119896)] forall119896 = 119896

119904+1 119896119904+ 2

(45)

Proof Compared to the proof of Theorem 17 here the proofis the same

5 Simulation and Experimental Results

The proposed algorithm was tested experimentally on a DCservo motor with the gear and load whose linear mathemat-ical model (without considering the major nonlinear effectsthe speed dependent friction dead zone and backlash) is asfollows

119869 120579 (119905) + 119861 120579 (119905) =120578119892120578119898119896119905119870119892

119877119898

119880 (119905) (46)

where 120579(119905) is the angular position of the load shaft 119869 is thetotal moment of inertia reflecting to the output shaft 119861 isthe viscous friction coefficient and 119880(119905) is the motor inputvoltage 119877

119898 119896119905 120578119898 120578119892 and 119870

119892are respectively the motor

armature resistance the motor torque constant the motorefficiency the gearbox efficiency and the total gear ratio

In the previous model it is assumed that the motorinductance is much smaller than the resistance so it isignored as the case with used servo motor is The experi-ments were performed with the Quanser rotary servo motorSRV02 This model is equipped with the optical encoderand tachometer for motor position and speed measuringrespectively Q8-USB data acquisition board for real-timedata acquisition and control was used with MatlabSimulinksoftware and QUARC real-time control The parametersnumerical values are as follows 119869 = 0021 (kgm2) 119861 =

0084 (Nmsrad) 119877119898= 26 (Ω) 119896

119905= 00077 (NmA) 120578

119898=

069 120578119892= 09 and119870

119892= 70

Choosing 1199091= 120579 and 119909

2= 120579 as state variables and 119906 =

119880(119905) as control at first the continuous time state equationwasobtained and finally after applying an accurate discretizationprocedure with sampling period 119879 = 001 (s) the discrete-time state equation of the system was obtained as follows

119909 (119896 + 1) = [1 008266

0 06746] 119909 (119896) + [

0002653

04979] 119906 (119896) (47)

The above mathematical model of real DC servo motorwas derived as adequate discrete-time state equation In theway the system matrix parameters 119860 and 119887 are availableControl algorithm from Theorem 18 is used to control themotor For simulation and experiment parameter 120590 = 0005

0 01 02 03 04 05

0

01

02

03

04

05

SimulationExperiment

minus02

minus01

x1

(rad

)

t (s)

Figure 6 The state variable 1199091

0 01 02 03 04 05

0123

SimulationExperiment

minus6

minus5

minus4

minus3

minus2

minus1

x2

(rad

s)

t (s)

Figure 7 The state variable 1199092

is adopted which defines very narrow 120590-neighborhood ofsliding subspace 119878 Also sampling time 119879 = 001 (s) andvector 119888 = (80 1) determined so as the system is stable insliding mode regime are adopted which gives 120577 = 119888 =

80006 Fuzzy logic controller is defined by 119871 = 119867 = 1119866 = 002 and 119866

119906= 40 which implies 119892min = 02 and

119892max = 04 Taking into account estimate relationships onone hand between 120577 and 120578

1and 120578

2 and on another hand

between 120577 119892min and 1205783 parameters 1205781= 30 120578

2= 90 and

1205783= 15 are chosen giving parameters of exponential low

120572 = 3 and 120573 = 01823 For initial state 119909(0) = (036 1)119879

after calculation V119898120590

= 04 V(0) = 298 and settling time119896119904= 20 are obtainedThe parallel simulation and experimental results are

shown in Figures 6ndash11 related to the algorithm fromTheorem 18 Figures 6 and 7 show the system state variablesThe control signal that drives the state onto the slidingsubspace is shown in Figure 9 The sliding function anddistance are shown in Figures 8 and 10 respectively From

Advances in Fuzzy Systems 9

0 01 02 03 04 05

0

10

20

30

40

50

cx

minus10

SimulationExperiment

t (s)

Figure 8 The sliding function 119888119909

0 01 02 03 04 05

0

2

4

minus10

minus8

minus6

minus4

minus2

SimulationExperiment

t (s)

u(V

)

Figure 9 The control variable

the figures it could be seen that the system state from initialposition approaches the sliding subspace 119878 but not slowerthan the exponential law defined by the parameters 120572 and120573 At instant 119896

119904= 20 (119905 = 02 sec) the state for the first

time enters the 120590-neighborhood of 119878 If it continues in thesame way it will never reach 119878 To reach it one step controlis applied at 119896

119904driving state in one step to 119878 so the character

of the system approaching 119878 changes at 119896119904 At 119896

119904+ 1 = 21

the system state is in 119878 staying in it thereafter forever Inside119878 the system is stable too so the system state asymptoticallyapproaches the zero state Simulation results fully confirm thetheoretical results and meet the expectations ofTheorems 1517 and 18

The experimental results have demonstrated that theproposed control scheme is valid and effective for the realapplications It is shown that there is no chattering in slidingmode However in practice errors could not be causedonly by computation but also by measurement accuracy(resolution sensitivity) noise existence nonlinearity andinadequate description of the real system by linear model as

0 01 02 03 04 05

002040608

112

minus04

minus02

Exponential law

t (s)

d(xS)

SimulationExperiment

Figure 10 The distance

015 02 025 03 035 04 045 05

0

001

002

003

004

005

minus002

minus001

Exponential law

d(xS)

SimulationExperiment

t (s)

Figure 11 Augmented part of Figure 10

well All this could cause not entering the 120590-neighborhood ofsliding subspace exactly in step 119896

119904than at some later instant

which is the case here in the paper experimental applicationexample After entering the 120590-neighborhood of 119878 at 119896

119904or

not and applying one step control at 119896119904 a little bit of sliding

subspace missing occurs Here that missing is a little bitmore visible at the beginning of reaching the sliding subspacebut very shortly after the like transition period expires itis again invisible like in the mathematical case simulationThis is illustrated by Figure 11 which is an enlarged segmentof Figure 10 for 119905 between 015 and 05 (sec) Evasion of theoccurrence is achieved by appropriate choice of the slidingsubspace and 120590-neighborhood parameters through vector 119888and parameter 120590 as well as the state space area where thelinear model is very adequate

It has already been pointed out that exponential stabilityis a higher quality stability property compared to ldquocommonrdquostability in Lyapunovrsquos sense (in this case it refers to globalasymptotic stability of a set-sliding subspace 119878) For that it is

10 Advances in Fuzzy Systems

natural that the sufficient conditions of Theorem 15 are morerigorous compared to the conditions of ldquocommonrdquo stabilityThe conditions of Theorem 15 are indirectly applicable inthe control of real-world processes through control law ofTheorems 17 and 18 for these control laws were derived fromthe conditions of Theorem 15 Real experiment with controlof DC servomotor is control of real-world process In the wayit is confirmed that the stability conditions ofTheorem 15 areapplicable in the control processes in practice

To resume in both ideal simulation case and in experi-mental application the proposed new developed algorithmin the paper guarantees exponential stability of 119878 and nochattering in sliding mode regime Despite the existence ofall upper stated nonidealities in the case of real applicationby the proper parameters choice the system motion is suchthat distance of working point from 119878 is always under pro-posed exponential envelope as it could be seen in providedexperiment

6 Conclusion

In the paper discrete-time type of VSCS with sliding modesis considered related to the linear time-invariant plant ofsecond orderThe fuzzy exponential stabilizing state feedbackcontrol algorithm so with prescribed approaching to thesliding subspace velocity estimate is developed strictly byLyapunovrsquos second method and with chattering free slidingmode Also application of the proposed ideal cases algo-rithm in practice is considered and discussed through theexperimental example It shows that experiment is quiteconsistent with mathematical ideal cases

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Z Bucevac ldquoLyapunovrsquos method approach in a stabilizingcontrol algorithm design for digital discrete VS systems withsliding modesndashlinear plant caserdquo in Proceedings of the 7thInternational Symposium Modelling Identification and Control( IASTED rsquo88) Grindelwald Switzerland l988

[2] G Bartolini A Ferrara and V I Utkin ldquoAdaptive sliding modecontrol in discrete-time systemsrdquo Automatica vol 31 no 5 pp769ndash773 1995

[3] S Hui and S H Zak ldquoOn discrete-time variable structureslidingmode controlrdquo SystemsampControl Letters vol 38 no 4-5pp 283ndash288 1999

[4] A Bartoszewicz ldquoDiscrete-time quasi-sliding-mode controlstrategiesrdquo IEEE Transactions on Industrial Electronics vol 45no 4 pp 633ndash637 1998

[5] W Gao Y Wang and A Homaifa ldquoDiscrete-time variablestructure control systemsrdquo IEEE Transactions on IndustrialElectronics vol 42 no 2 pp 117ndash122 1995

[6] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[7] Y Zheng and Y Jing ldquoApproximation law for discrete-timevariable structure control systemsrdquo Journal of Control Theoryand Applications vol 4 no 3 pp 291ndash296 2006

[8] G Golo and C Milosavljevic ldquoRobust discrete-time chatteringfree slidingmode controlrdquo SystemsampControl Letters vol 41 no1 pp 19ndash28 2000

[9] H Ying ldquoTheory and application of a novel fuzzy PID controllerusing a simplified Takagi-Sugeno rule schemerdquo InformationSciences vol 123 no 3 pp 281ndash293 2000

[10] G Feng ldquoA survey on analysis and design of model-based fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 5 pp 676ndash697 2006

[11] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[12] H K Lam F H F Leung and P K S Tam ldquoStable androbust fuzzy control for uncertain nonlinear systemsrdquo IEEETransactions on Systems Man and Cybernetics Part A Systemsand Humans vol 30 no 6 pp 825ndash840 2000

[13] R-E Precup M L Tomescu M-B Rdac E M Petriu S PreitlandC-A Dragos ldquoIterative performance improvement of fuzzycontrol systems for three tank systemsrdquo Expert Systems withApplications vol 39 no 9 pp 8288ndash8299 2012

[14] D H Lee Y H Joo and M H Tak ldquoLocal stability analysisof continuous-time Takagi-Sugeno fuzzy systems a fuzzy Lya-punov function approachrdquo Information Sciences vol 257 pp163ndash175 2014

[15] R Palm ldquoRobust control by fuzzy sliding moderdquo Automaticavol 30 no 9 pp 1429ndash1437 1994

[16] C-L Chen andM-H Chang ldquoOptimal design of fuzzy sliding-mode control a comparative studyrdquo Fuzzy Sets and Systems vol93 no 1 pp 37ndash48 1998

[17] J CWu andT S Liu ldquoA sliding-mode approach to fuzzy controldesignrdquo IEEE Transactions on Control Systems Technology vol4 no 2 pp 141ndash151 1996

[18] S-W Kim and J-J Lee ldquoDesign of a fuzzy controller with fuzzysliding surfacerdquo Fuzzy Sets and Systems vol 71 no 3 pp 359ndash367 1995

[19] G Feng S G Cao N W Rees and C K Chak ldquoDesign offuzzy control systems with guaranteed stabilityrdquo Fuzzy Sets andSystems vol 85 no 1 pp 1ndash10 1997

[20] F Qiao QM Zhu AWinfield andCMelhuish ldquoFuzzy slidingmode control for discrete nonlinear systemsrdquo Transactions ofChina Automation Society vol 22 no 2 pp 311ndash315 2003

[21] S H Cho ldquoRobust discrete time tracking control using slidingsurfacesrdquo KSME Journal vol 9 no 1 pp 80ndash90 1995

[22] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992

[23] H Ying W Siler and J J Buckley ldquoFuzzy control theory anonlinear caserdquo Automatica vol 26 no 3 pp 513ndash520 1990

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Fuzzy Systems 9

0 01 02 03 04 05

0

10

20

30

40

50

cx

minus10

SimulationExperiment

t (s)

Figure 8 The sliding function 119888119909

0 01 02 03 04 05

0

2

4

minus10

minus8

minus6

minus4

minus2

SimulationExperiment

t (s)

u(V

)

Figure 9 The control variable

the figures it could be seen that the system state from initialposition approaches the sliding subspace 119878 but not slowerthan the exponential law defined by the parameters 120572 and120573 At instant 119896

119904= 20 (119905 = 02 sec) the state for the first

time enters the 120590-neighborhood of 119878 If it continues in thesame way it will never reach 119878 To reach it one step controlis applied at 119896

119904driving state in one step to 119878 so the character

of the system approaching 119878 changes at 119896119904 At 119896

119904+ 1 = 21

the system state is in 119878 staying in it thereafter forever Inside119878 the system is stable too so the system state asymptoticallyapproaches the zero state Simulation results fully confirm thetheoretical results and meet the expectations ofTheorems 1517 and 18

The experimental results have demonstrated that theproposed control scheme is valid and effective for the realapplications It is shown that there is no chattering in slidingmode However in practice errors could not be causedonly by computation but also by measurement accuracy(resolution sensitivity) noise existence nonlinearity andinadequate description of the real system by linear model as

0 01 02 03 04 05

002040608

112

minus04

minus02

Exponential law

t (s)

d(xS)

SimulationExperiment

Figure 10 The distance

015 02 025 03 035 04 045 05

0

001

002

003

004

005

minus002

minus001

Exponential law

d(xS)

SimulationExperiment

t (s)

Figure 11 Augmented part of Figure 10

well All this could cause not entering the 120590-neighborhood ofsliding subspace exactly in step 119896

119904than at some later instant

which is the case here in the paper experimental applicationexample After entering the 120590-neighborhood of 119878 at 119896

119904or

not and applying one step control at 119896119904 a little bit of sliding

subspace missing occurs Here that missing is a little bitmore visible at the beginning of reaching the sliding subspacebut very shortly after the like transition period expires itis again invisible like in the mathematical case simulationThis is illustrated by Figure 11 which is an enlarged segmentof Figure 10 for 119905 between 015 and 05 (sec) Evasion of theoccurrence is achieved by appropriate choice of the slidingsubspace and 120590-neighborhood parameters through vector 119888and parameter 120590 as well as the state space area where thelinear model is very adequate

It has already been pointed out that exponential stabilityis a higher quality stability property compared to ldquocommonrdquostability in Lyapunovrsquos sense (in this case it refers to globalasymptotic stability of a set-sliding subspace 119878) For that it is

10 Advances in Fuzzy Systems

natural that the sufficient conditions of Theorem 15 are morerigorous compared to the conditions of ldquocommonrdquo stabilityThe conditions of Theorem 15 are indirectly applicable inthe control of real-world processes through control law ofTheorems 17 and 18 for these control laws were derived fromthe conditions of Theorem 15 Real experiment with controlof DC servomotor is control of real-world process In the wayit is confirmed that the stability conditions ofTheorem 15 areapplicable in the control processes in practice

To resume in both ideal simulation case and in experi-mental application the proposed new developed algorithmin the paper guarantees exponential stability of 119878 and nochattering in sliding mode regime Despite the existence ofall upper stated nonidealities in the case of real applicationby the proper parameters choice the system motion is suchthat distance of working point from 119878 is always under pro-posed exponential envelope as it could be seen in providedexperiment

6 Conclusion

In the paper discrete-time type of VSCS with sliding modesis considered related to the linear time-invariant plant ofsecond orderThe fuzzy exponential stabilizing state feedbackcontrol algorithm so with prescribed approaching to thesliding subspace velocity estimate is developed strictly byLyapunovrsquos second method and with chattering free slidingmode Also application of the proposed ideal cases algo-rithm in practice is considered and discussed through theexperimental example It shows that experiment is quiteconsistent with mathematical ideal cases

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Z Bucevac ldquoLyapunovrsquos method approach in a stabilizingcontrol algorithm design for digital discrete VS systems withsliding modesndashlinear plant caserdquo in Proceedings of the 7thInternational Symposium Modelling Identification and Control( IASTED rsquo88) Grindelwald Switzerland l988

[2] G Bartolini A Ferrara and V I Utkin ldquoAdaptive sliding modecontrol in discrete-time systemsrdquo Automatica vol 31 no 5 pp769ndash773 1995

[3] S Hui and S H Zak ldquoOn discrete-time variable structureslidingmode controlrdquo SystemsampControl Letters vol 38 no 4-5pp 283ndash288 1999

[4] A Bartoszewicz ldquoDiscrete-time quasi-sliding-mode controlstrategiesrdquo IEEE Transactions on Industrial Electronics vol 45no 4 pp 633ndash637 1998

[5] W Gao Y Wang and A Homaifa ldquoDiscrete-time variablestructure control systemsrdquo IEEE Transactions on IndustrialElectronics vol 42 no 2 pp 117ndash122 1995

[6] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[7] Y Zheng and Y Jing ldquoApproximation law for discrete-timevariable structure control systemsrdquo Journal of Control Theoryand Applications vol 4 no 3 pp 291ndash296 2006

[8] G Golo and C Milosavljevic ldquoRobust discrete-time chatteringfree slidingmode controlrdquo SystemsampControl Letters vol 41 no1 pp 19ndash28 2000

[9] H Ying ldquoTheory and application of a novel fuzzy PID controllerusing a simplified Takagi-Sugeno rule schemerdquo InformationSciences vol 123 no 3 pp 281ndash293 2000

[10] G Feng ldquoA survey on analysis and design of model-based fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 5 pp 676ndash697 2006

[11] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[12] H K Lam F H F Leung and P K S Tam ldquoStable androbust fuzzy control for uncertain nonlinear systemsrdquo IEEETransactions on Systems Man and Cybernetics Part A Systemsand Humans vol 30 no 6 pp 825ndash840 2000

[13] R-E Precup M L Tomescu M-B Rdac E M Petriu S PreitlandC-A Dragos ldquoIterative performance improvement of fuzzycontrol systems for three tank systemsrdquo Expert Systems withApplications vol 39 no 9 pp 8288ndash8299 2012

[14] D H Lee Y H Joo and M H Tak ldquoLocal stability analysisof continuous-time Takagi-Sugeno fuzzy systems a fuzzy Lya-punov function approachrdquo Information Sciences vol 257 pp163ndash175 2014

[15] R Palm ldquoRobust control by fuzzy sliding moderdquo Automaticavol 30 no 9 pp 1429ndash1437 1994

[16] C-L Chen andM-H Chang ldquoOptimal design of fuzzy sliding-mode control a comparative studyrdquo Fuzzy Sets and Systems vol93 no 1 pp 37ndash48 1998

[17] J CWu andT S Liu ldquoA sliding-mode approach to fuzzy controldesignrdquo IEEE Transactions on Control Systems Technology vol4 no 2 pp 141ndash151 1996

[18] S-W Kim and J-J Lee ldquoDesign of a fuzzy controller with fuzzysliding surfacerdquo Fuzzy Sets and Systems vol 71 no 3 pp 359ndash367 1995

[19] G Feng S G Cao N W Rees and C K Chak ldquoDesign offuzzy control systems with guaranteed stabilityrdquo Fuzzy Sets andSystems vol 85 no 1 pp 1ndash10 1997

[20] F Qiao QM Zhu AWinfield andCMelhuish ldquoFuzzy slidingmode control for discrete nonlinear systemsrdquo Transactions ofChina Automation Society vol 22 no 2 pp 311ndash315 2003

[21] S H Cho ldquoRobust discrete time tracking control using slidingsurfacesrdquo KSME Journal vol 9 no 1 pp 80ndash90 1995

[22] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992

[23] H Ying W Siler and J J Buckley ldquoFuzzy control theory anonlinear caserdquo Automatica vol 26 no 3 pp 513ndash520 1990

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

10 Advances in Fuzzy Systems

natural that the sufficient conditions of Theorem 15 are morerigorous compared to the conditions of ldquocommonrdquo stabilityThe conditions of Theorem 15 are indirectly applicable inthe control of real-world processes through control law ofTheorems 17 and 18 for these control laws were derived fromthe conditions of Theorem 15 Real experiment with controlof DC servomotor is control of real-world process In the wayit is confirmed that the stability conditions ofTheorem 15 areapplicable in the control processes in practice

To resume in both ideal simulation case and in experi-mental application the proposed new developed algorithmin the paper guarantees exponential stability of 119878 and nochattering in sliding mode regime Despite the existence ofall upper stated nonidealities in the case of real applicationby the proper parameters choice the system motion is suchthat distance of working point from 119878 is always under pro-posed exponential envelope as it could be seen in providedexperiment

6 Conclusion

In the paper discrete-time type of VSCS with sliding modesis considered related to the linear time-invariant plant ofsecond orderThe fuzzy exponential stabilizing state feedbackcontrol algorithm so with prescribed approaching to thesliding subspace velocity estimate is developed strictly byLyapunovrsquos second method and with chattering free slidingmode Also application of the proposed ideal cases algo-rithm in practice is considered and discussed through theexperimental example It shows that experiment is quiteconsistent with mathematical ideal cases

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Z Bucevac ldquoLyapunovrsquos method approach in a stabilizingcontrol algorithm design for digital discrete VS systems withsliding modesndashlinear plant caserdquo in Proceedings of the 7thInternational Symposium Modelling Identification and Control( IASTED rsquo88) Grindelwald Switzerland l988

[2] G Bartolini A Ferrara and V I Utkin ldquoAdaptive sliding modecontrol in discrete-time systemsrdquo Automatica vol 31 no 5 pp769ndash773 1995

[3] S Hui and S H Zak ldquoOn discrete-time variable structureslidingmode controlrdquo SystemsampControl Letters vol 38 no 4-5pp 283ndash288 1999

[4] A Bartoszewicz ldquoDiscrete-time quasi-sliding-mode controlstrategiesrdquo IEEE Transactions on Industrial Electronics vol 45no 4 pp 633ndash637 1998

[5] W Gao Y Wang and A Homaifa ldquoDiscrete-time variablestructure control systemsrdquo IEEE Transactions on IndustrialElectronics vol 42 no 2 pp 117ndash122 1995

[6] X Yu and O Kaynak ldquoSliding-mode control with soft comput-ing a surveyrdquo IEEE Transactions on Industrial Electronics vol56 no 9 pp 3275ndash3285 2009

[7] Y Zheng and Y Jing ldquoApproximation law for discrete-timevariable structure control systemsrdquo Journal of Control Theoryand Applications vol 4 no 3 pp 291ndash296 2006

[8] G Golo and C Milosavljevic ldquoRobust discrete-time chatteringfree slidingmode controlrdquo SystemsampControl Letters vol 41 no1 pp 19ndash28 2000

[9] H Ying ldquoTheory and application of a novel fuzzy PID controllerusing a simplified Takagi-Sugeno rule schemerdquo InformationSciences vol 123 no 3 pp 281ndash293 2000

[10] G Feng ldquoA survey on analysis and design of model-based fuzzycontrol systemsrdquo IEEE Transactions on Fuzzy Systems vol 14no 5 pp 676ndash697 2006

[11] H OWang K Tanaka andM F Griffin ldquoAn approach to fuzzycontrol of nonlinear systems Stability and design issuesrdquo IEEETransactions on Fuzzy Systems vol 4 no 1 pp 14ndash23 1996

[12] H K Lam F H F Leung and P K S Tam ldquoStable androbust fuzzy control for uncertain nonlinear systemsrdquo IEEETransactions on Systems Man and Cybernetics Part A Systemsand Humans vol 30 no 6 pp 825ndash840 2000

[13] R-E Precup M L Tomescu M-B Rdac E M Petriu S PreitlandC-A Dragos ldquoIterative performance improvement of fuzzycontrol systems for three tank systemsrdquo Expert Systems withApplications vol 39 no 9 pp 8288ndash8299 2012

[14] D H Lee Y H Joo and M H Tak ldquoLocal stability analysisof continuous-time Takagi-Sugeno fuzzy systems a fuzzy Lya-punov function approachrdquo Information Sciences vol 257 pp163ndash175 2014

[15] R Palm ldquoRobust control by fuzzy sliding moderdquo Automaticavol 30 no 9 pp 1429ndash1437 1994

[16] C-L Chen andM-H Chang ldquoOptimal design of fuzzy sliding-mode control a comparative studyrdquo Fuzzy Sets and Systems vol93 no 1 pp 37ndash48 1998

[17] J CWu andT S Liu ldquoA sliding-mode approach to fuzzy controldesignrdquo IEEE Transactions on Control Systems Technology vol4 no 2 pp 141ndash151 1996

[18] S-W Kim and J-J Lee ldquoDesign of a fuzzy controller with fuzzysliding surfacerdquo Fuzzy Sets and Systems vol 71 no 3 pp 359ndash367 1995

[19] G Feng S G Cao N W Rees and C K Chak ldquoDesign offuzzy control systems with guaranteed stabilityrdquo Fuzzy Sets andSystems vol 85 no 1 pp 1ndash10 1997

[20] F Qiao QM Zhu AWinfield andCMelhuish ldquoFuzzy slidingmode control for discrete nonlinear systemsrdquo Transactions ofChina Automation Society vol 22 no 2 pp 311ndash315 2003

[21] S H Cho ldquoRobust discrete time tracking control using slidingsurfacesrdquo KSME Journal vol 9 no 1 pp 80ndash90 1995

[22] V I Utkin Sliding Modes in Control and Optimization Com-munications and Control Engineering Series Springer BerlinGermany 1992

[23] H Ying W Siler and J J Buckley ldquoFuzzy control theory anonlinear caserdquo Automatica vol 26 no 3 pp 513ndash520 1990

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014