research article design of robust output feedback ...research article design of robust output...
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Research ArticleDesign of Robust Output Feedback Guaranteed Cost Control fora Class of Nonlinear Discrete-Time Systems
Yan Zhang, Yali Dong, and Tianrui Li
School of Science, Tianjin Polytechnic University, Tianjin 300387, China
Correspondence should be addressed to Yali Dong; [email protected]
Received 17 May 2014; Revised 31 August 2014; Accepted 31 August 2014; Published 10 September 2014
Academic Editor: Yurong Liu
Copyright Β© 2014 Yan Zhang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper investigates static output feedback guaranteed cost control for a class of nonlinear discrete-time systems where thedelay in state vector is inconsistent with the delay in nonlinear perturbations. Based on the output measurement, the controller isdesigned to ensure the robust exponentially stability of the closed-loop system and guarantee the performance of system to achievean adequate level. By using the Lyapunov-Krasovskii functional method, some sufficient conditions for the existence of robustoutput feedback guaranteed cost controller are established in terms of linear matrix inequality. A numerical example is provided toshow the effectiveness of the results obtained.
1. Introduction
In control theory and practice, one of the most importantopen problems is the static output feedback (SOF) problem.The main principle of the SOF control is to utilize themeasured output to excite the plant. Since the controllercan be easily implemented in practice, the SOF controlhas attracted a lot of attention over the past few decadesand has been applied to many areas such as economic,communication, and biological systems [1, 2]. The goal ofdesign SOF controller is to ensure asymptotically stable orexponential stable of the original system [3]. However, inmany practical systems, controller designed is to not onlyensure asymptotically or exponentially stable of the systembut also guarantee the performance of system to achieve anadequate level. One method of dealing with this problem isthe guaranteed cost control first introduced by Chang andPeng [4]. This method has the advantage of providing anupper bound on a given performance index and thus thesystem performance degradation is guaranteed to be nomorethan this bound. Based on this idea, a lot of significant results
have been addressed for continuous-time systems in [5β7]and for discrete-time systems in [8].
It is well known that time-delays as well as parameteruncertainties frequently lead to instability of systems. More-over, the existing of time-delays and uncertainties make thesystem more complex [9, 10].
In the past studies for guaranteed cost control, almostmost of the articles considered linear systems [11, 12]. How-ever, in majority dynamic systems, the nonlinear perturba-tions appear more and more frequently. Therefore, we notonly deal with the time-varying delays and uncertainties, butalso deal with the nonlinearities. Difficulties then arise whenone attempts to derive exponential stabilization conditions.Hence in this case, the methods in linear systems [11, 12]can not be directly applied to nonlinear systems. This callsfor a fresh look at the problem with an improved Lyapunov-Krasovskii functionals and a new set of LMI conditions. Inthis paper, we aim to design robust static output feedbackguaranteed cost controller for a class of nonlinear discrete-time systems with time-varying delays. By constructing aset of improved Lyapunov-Krasovskii functionals, a newcriterion for the existence of robust static output feedback
Hindawi Publishing CorporationInternational Journal of Engineering MathematicsVolume 2014, Article ID 628041, 9 pageshttp://dx.doi.org/10.1155/2014/628041
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2 International Journal of Engineering Mathematics
guaranteed cost controller is established and described interms of linear matrix inequality. A numerical example isprovided to show the effectiveness of the results obtained.
Notations. In this paper, a matrix π΄ is symmetric if π΄ = π΄π.πmax(π΄)(πmin(π΄)) denotes the maximum (minimum) valueof the real parts of eigenvalues of π΄. The symmetric terms ina matrix are denoted by β. π > 0 (resp., π β₯ 0), for π βπ πΓπ, means that thematrix is real symmetric positive definite
(resp., positive semidefinite). π+ denotes the set of all realnonnegative integers.
2. Preliminaries
Consider the following control system:
π₯ (π + 1) = [π΄ + Ξπ΄ (π)] π₯ (π)
+ [π΄1+ Ξπ΄1 (π)] π₯ (π β π (π)) + π΅π’ (π)
+πΉπ (π₯ (π)) + πΊπ (π₯ (π β β (π))) ,
π¦ (π) = πΆπ₯ (π) + πΆ1π₯ (π β π (π)) , π β π+,
π₯ (π) = ππ, π = βπ, βπ + 1, . . . , 0,
(1)
where π₯(π) β π π is the state vector, π¦(π) β π π is theobservation output, and π’(π) β π π is the control intput.π΄, π΄1, π΅, πΉ, πΊ, πΆ, and πΆ
1are given constant matrices
with appropriate dimensions. Ξπ΄(π), Ξπ΄1(π) are the time-
varying parameter uncertainties that are assumed to satisfythe following admissible condition:
[Ξπ΄ (π) Ξπ΄1 (π)] = πΞ (π) [π1π2] , Ξπ(π) Ξ (π) β©½ πΌ,
βπ β π+,
(2)
whereπ,π1, andπ
2are some given constant matrices with
appropriate dimensions. The positive integers π(π) and β(π)are time-varying delays satisfying
0 β€ π (π) β€ π, 0 β€ β (π) β€ β, (3)
where π, β are known positive integers. π(π₯(π)) β π π andπ(π₯(π β β(π)) β π
π are unknown nonlinear functions,assumed as
ππ(π₯ (π)) π (π₯ (π)) β©½ π½1π₯
π(π) πΏπ
1πΏ1π₯ (π) ,
ππ(π₯ (π β β (π))) π (π₯ (π β β (π)))
β©½ π½2π₯π(π β β (π)) πΏ
π
2πΏ2π₯ (π β β (π)) ,
(4)
where π½1, π½2
are known positive integers and πΏ1, πΏ2
are known real matrices. The initial condition π =(πβπ, πβπ+1
, . . . , π0) β π (π+1)π with the norm
π = max {βπ₯ (βπ)β , . . . , βπ₯ (0)β} , (5)
where π = max{β, π}.The corresponding cost function is as follows:
π½ =
β
β
π=0
{π₯π(π) ππ₯ (π) + π₯
π(π β π (π)) π1π₯ (π β π (π))
+ π₯π(π β β (π)) π2π₯ (π β β (π)) + π’
π(π) π π’ (π)} ,
(6)
where π, π1, π2, and π are given symmetric positive definite
matrices with appropriate dimensions.Substituting the output feedback controller π’(π) = πΎπ¦(π)
into system (1), we have
π₯ (π + 1) = [π΄ (π) + π΅πΎπΆ] π₯ (π)
+ [π΄1 (π) + π΅πΎπΆ1] π₯ (π β π (π)) + πΉπ (π₯ (π))
+ πΊπ (π₯ (π β β (π))) ,
(7)
where π΄(π) = π΄ + Ξπ΄(π) and π΄1(π) = π΄
1+ Ξπ΄1(π).
The objective of this paper is to design an output feedbackcontroller π’(π) = πΎπ¦(π) for system (1) and cost function(6) such that the resulting closed-loop system is robustexponentially stable with an upper bound for cost function(6).
We first give the following definitions, which will be usedin the next theorems and proofs.
Definition 1. Given πΌ > 0, the closed-loop system (7) is saidto be robust exponentially stable with a decay rate πΌ, if thereexists scalars π > 0 such that for every solution π₯(π, π) of thesystem satisfies the condition:
π₯ (π, π) β€ ππ
βπΌπ π , βπ β π
+. (8)
Definition 2. For system (1) and cost function (6), if thereexist a static output feedback control law π’β(π) and a positiveconstant π½β such that the closed-loop system (7) is robustexponentially stable with a decay rate πΌ and the value (6)satisfies π½ β€ π½β, then π½β is said to be a guaranteed cost indexand π’β(π) is said to be a robust output feedback guaranteedcost control law of the system.
The following lemmas are essential in establishing ourmain results.
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International Journal of Engineering Mathematics 3
Lemma3 (see [11]). For any π₯,π¦ β π π, and positive symmetricdefinite matrixπ β π πΓπ, we have
Β±2π¦ππ₯ β€ π₯
ππβ1π₯ + π¦πππ¦. (9)
Lemma 4 (Schur complement lemma [13]). Given constantmatrixπ,π, andπwith appropriate dimensions satisfyingπ =ππ, π = ππ > 0. Then π + πππβ1π < 0 if and only if π =
(π ππ
π βπ) < 0.
3. Main Results
In this section, by constructing a new set of Lyapunov-Krasovskii functionals, we give a sufficient condition for theexistence of robustly output feedback guaranteed cost controlfor system (1).
Theorem 5. For a given scalar πΌ > 0, the control π’(π) =πΎπ¦(π) is a robustly static output feedback guaranteed costcontroller for nonlinear system (1), if there exist symmetricpositive definite matrices π, π
1, π2, π 1, π 2, and πΎ, arbitrary
matrix π, and scalars π1β₯ 0, π
2β₯ 0, such that the following
LMI holds:
(
Ξ1
0 Ξ13
Ξ14
Ξ15
β Ξ2
0 0 0
β β Ξ3
0 0
β β β Ξ4
0
β β β β Ξ5
)< 0, (10)
where
Ξ1= (
Ξ11
Ξ12
0
β Ξ22
Ξ23
β β Ξ33
) , Ξ14= (
0 0 0 0
NB NM NF NG0 0 0 0
) ,
Ξ13= (
πΆππΎππΆππΎπ
0 0
0 0
) ,
Ξ15= (
0 0
0 0
πΆπ
1πΎππΆπ
1πΎπ
) , Ξ2= diag {Ξ
44, Ξ55, Ξ66} ,
Ξ3= diag {β0.5π β1, βπΌ} ,
Ξ4= diag {Ξ
1, Ξ2, βπΌ, βπΌ} ,
Ξ5= diag {β0.5πβ2πΌππ β1, βπβ2πΌπΌ} ,
Ξ11= β π + π (π
1+ π 1) + ππΌππ
1π1+ β (π
2+ π 2)
+ π½1(π2πΌ+ π1) πΏπ
1πΏ1+ π,
Ξ12= ππΌππ΄, Ξ
23= ππΌπππ΄1,
Ξ33= βπ1β π 1+ ππΌπππ
2π2+ π2πΌπ
π1,
Ξ22= π β (π + π
π) ,
Ξ44= βπ2β π 2+ (π2+ 1) π½
2π2πΌβ
πΏπ
2πΏ2+ π2πΌβ
π2,
Ξ55= βπ1πΌ, Ξ
66= βπ2πΌ, Ξ
1= βπβ2πΌπ
πΌ,
Ξ2= β
1
ππΌπ + ππΌπΌ, π = π + 1, β = β + 1,
(11)
and the guaranteed cost value is given by π½β = π2βπβ2, where
π2= πmax (π) + π (π + 1) [πmax (π1) + πmax (π 1)]
+ β (β + 1) [πmax (π2) + πmax (π 2)] .(12)
Proof. We first introduce the new variable π§(π) = ππΌππ₯(π).The closed-loop system (7) is reduced to
π§ (π + 1) = [π΄ (π) + π΅πΎπΆ] ππΌπ§ (π)
+ [π΄1 (π) + π΅πΎπΆ1] π
πΌ(π(π)+1)π§ (π β π (π))
+ πΉππΌ(π+1)
π (π§ (π)) + πΊππΌ(π+1)
π (π§ (π β β (π))) ,
(13)
where π(π§(π)) = π(π§(π)/ππΌπ) and π(π§(π β β(π))) = π(π§(π ββ(π))/π
πΌ(πββ(π))).
Associated with (2), the above equality is reduced to
π§ (π + 1) = [π΄ (π) + π΅ (π)πΎπΆ (π)] π§ (π)
+ [π΄1 (π) + π΅ (π)πΎπΆ1 (π)] π§ (π β π (π))
+ πΉ (π) π (π§ (π)) + πΊ (π) π (π§ (π β β (π))) ,
(14)
where
π΄ (π) = ππΌ[π΄ +πΞ (π)π1] , πΆ (π) = π
βπΌππΆ,
π΄1 (π) = π
πΌ(π(π)+1)[π΄1+πΞ (π)π2] ,
π΅ (π) = ππΌ(π+1)
π΅, πΆ1 (π) = π
βπΌ(πβπ(π))πΆ1,
πΉ (π) = ππΌ(π+1)
πΉ, πΊ (π) = ππΌ(π+1)
πΊ,
(15)
ππ
(π§ (π)) π (π§ (π)) β€ π½1πβ2πΌπ
π§π(π) πΏπ
1πΏ1π§ (π) , (16)
ππ(π§ (π β β (π))) π (π§ (π β β (π)))
β€ π½2πβ2πΌ(πββ(π))
π§π(π β β (π)) πΏ
π
2πΏ2π§ (π β β (π)) .
(17)
Consider a Lyapunov-Krasovskii functional candidate for theclosed-loop system (14) as
π (π) = π1 (π) + π2 (π) + π3 (π) + π4 (π) + π5 (π) , (18)
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4 International Journal of Engineering Mathematics
where
π1 (π) = π§
π(π) ππ§ (π) ,
π2 (π) =
πβ1
β
π=πβπ(π)
π§π(π) π1π§ (π) +
πβ1
β
π=πββ(π)
π§π(π) π2π§ (π) ,
π3 (π) =
0
β
π=βπ+1
πβ1
β
π=π+π
π§π(π) π1π§ (π) +
0
β
π=ββ+1
πβ1
β
π=π+π
π§π(π) π2π§ (π) ,
π4 (π) =
πβ1
β
π=πβπ(π)
π§π(π) π 1π§ (π) +
πβ1
β
π=πββ(π)
π§π(π) π 2π§ (π) ,
π5 (π) =
0
β
π=βπ+1
πβ1
β
π=π+π
π§π(π) π 1π§ (π) +
0
β
π=ββ+1
πβ1
β
π=π+π
π§π(π) π 2π§ (π) .
(19)
Calculating the difference of π(π) we have
Ξπ1 (π) = π§
π(π + 1) ππ§ (π + 1) β π§
π(π) ππ§ (π) , (20)
Ξπ2 (π)
β€ π§π(π) (π1 + π2) π§ (π) β π§
π(π β π (π)) π1π§ (π β π (π))
β π§π(π β β (π)) π2π§ (π β β (π))
+
π
β
π=πβπ+1
π§π(π) π1π§ (π) +
π
β
π=πββ+1
π§π(π) π2π§ (π) ,
(21)
Ξπ3 (π)
=
0
β
π=βπ+1
{
{
{
π
β
π=π+π+1
π§π(π) π1π§ (π) β
πβ1
β
π=π+π
π§π(π) π1π§ (π)
}
}
}
+
0
β
π=ββ+1
{
{
{
π
β
π=π+π+1
π§π(π) π2π§ (π) β
πβ1
β
π=π+π
π§π(π) π2π§ (π)
}
}
}
= π§π(π) (ππ1 + βπ2) π§ (π)
β
π
β
π=πβπ+1
π§π(π)π1π§ (π) β
π
β
π=πββ+1
π§π(π)π2π§ (π) .
(22)
Combine (21) and (22), we have
Ξπ2 (π) + Ξπ3 (π) β€ π§
π(π) (ππ1 + βπ2) π§ (π)
β π§π(π β π (π)) π1π§ (π β π (π))
β π§π(π β β (π)) π2π§ (π β β (π)) .
(23)
Similarly, we can get
Ξπ4 (π) + Ξπ5 (π) β€ π§
π(π) (ππ 1 + βπ 2) π§ (π)
β π§π(π β π (π)) π 1π§ (π β π (π))
β π§π(π β β (π)) π 2π§ (π β β (π)) .
(24)
Therefore, from (17)β(24), we have
Ξπ (π) β€ π§π(π + 1) ππ§ (π + 1)
+ π§π(π) [π (π1 + π 1) + β (π2 + π 2) β π] π§ (π)
β π§π(π β π (π)) (π1 + π 1) π§ (π β π (π))
β π§π(π β β (π)) (π2 + π 2) π§ (π β β (π)) .
(25)
Multiplying 2π§π(π + 1)π both sides of the identity (14),
β 2π§π(π + 1)ππ§ (π + 1) + 2π§
π(π + 1)
Γ π(π΄ (π) + π΅ (π)πΎπΆ (π)) π§ (π) + 2π§π(π + 1)
Γ π(π΄1 (π) + π΅ (π)πΎπΆ1 (π)) π§ (π β π (π))
+ 2π§π(π + 1)ππΉ (π) π + 2π§
π(π + 1)ππΊ (π) π = 0.
(26)
Note that for any π1β₯ 0, π
2β₯ 0, it follows from (16) to (17)
π1[π½1πβ2πΌπ
π§π(π) πΏπ
1πΏ1π§ (π) β π
π
(π§ (π)) π (π§ (π))] β₯ 0,
π2[π½2πβ2πΌ(πββ(π))
π§π(π β β (π)) πΏ
π
2πΏ2π§ (π β β (π))
βππ(π§ (π β β (π))) π (π§ (π β β (π)))] β₯ 0.
(27)
Substituting (26)-(27) into (25), we have
Ξπ (π)
β€ π§π(π + 1) ππ§ (π + 1)
+ π§π(π) [π (π1 + π 1) + β (π2 + π 2) β π] π§ (π)
β π§π(π β π (π)) (π1 + π 1) π§ (π β π (π))
β π§π(π β β (π)) (π2 + π 2) π§ (π β β (π))
β 2π§π(π + 1)ππ§ (π + 1) + 2π§
π(π + 1)
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International Journal of Engineering Mathematics 5
Γ π(π΄ (π) + π΅ (π)πΎπΆ (π)) π§ (π) + 2π§π(π + 1)
Γ π (π΄1 (π) + π΅ (π)πΎπΆ1 (π)) π§ (π β π (π))
+ 2π§π(π + 1)ππΉ (π) π (π§ (π))
+ 2π§π(π + 1)ππΊ (π) π (π§ (π β β (π)))
+ π1[π½1πβ2πΌπ
π§π(π) πΏπ
1πΏ1π§ (π)βπ
π
(π§ (π)) π (π§ (π))]
+ π2[π½2πβ2πΌ(πββ(π))
π§π(π β β (π)) πΏ
π
2πΏ2π§ (π β β (π))
β ππ(π§ (π β β (π))) π (π§ (π β β (π)))] .
(28)
Dealing with partial idem in (28) using Lemma 3, it follows
2π§π(π + 1)π (π΄ (π) + π΅ (π)πΎπΆ (π)) π§ (π)
= 2π§π(π + 1) π
πΌ[ππ΄ + ππΞ (π)π1] π§ (π)
+ 2π§π(π + 1) π
πΌππ΅πΎπΆπ§ (π)
β€ 2π§π(π + 1) π
πΌππ΄π§ (π)
+ ππΌπ§π(π + 1)πππ
ππππ§ (π + 1)
+ ππΌπ§π(π)ππ
1π1π§ (π)
+ π§π(π + 1) π
2πΌππ΅π΅ππππ§ (π + 1)
+ π§π(π) πΆππΎππΎπΆπ§ (π) ,
2π§π(π + 1)π (π΄1 (π) + π΅ (π)πΎπΆ1 (π)) π§ (π β π (π))
= π§π(π + 1) π
πΌ(π(π)+1)(ππ΄1+ π΄π
1ππ) π§ (π β π (π))
+ 2ππΌ(π(π)+1)
π§π(π + 1)ππΞ (π)π2π§ (π β π (π))
+ 2π§π(π + 1) π
πΌπ(π)ππ΅ππΌπΎπΆ1π§ (π β π (π))
β€ π§π(π + 1) π
πΌπ(ππ΄1+ π΄π
1ππ) π§ (π β π (π))
+ ππΌππ§π(π + 1)πππ
ππππ§ (π + 1)
+ ππΌππ§π(π β π (π))π
π
2π2π§ (π β π (π))
+ π§π(π + 1) π
2πΌπππ΅π΅ππππ§ (π + 1)
+ π2πΌπ§π(π β π (π)) πΆ
π
1πΎππΎπΆ1π§ (π β π (π)) ,
2π§π(π + 1)ππΉ (π) π (π§ (π))
= 2π§π(π + 1)ππΉπ
πΌ(π+1)π (π§ (π))
β€ π§π(π + 1)ππΉπΉ
ππππ§ (π + 1)
+ π2πΌ(π+1)
ππ
(π§ (π)) π (π§ (π))
β€ π§π(π + 1)ππΉπΉ
ππππ§ (π + 1)
+ π½1π2πΌπ§π(π) πΏπ
1πΏ1π§ (π) .
(29)
Similarly, we have
2π§π(π + 1)ππΊ (π) π (π§ (π β β (π)))
β€ π§π(π + 1)ππΊπΊ
ππππ§ (π + 1)
+ π½2π2πΌβ
π§π(π β β (π)) πΏ
π
2πΏ2π§ (π β β (π)) .
(30)
Adding the following relation to inequality (28)
[π§π(π) ππ§ (π) + π§
π(π β π (π)) π1π§ (π β π (π))
+ π§π(π β β (π)) π2π§ (π β β (π)) + π’
π(π) π π’ (π)]
β [π§π(π) ππ§ (π) + π§
π(π β π (π)) π1π§ (π β π (π))
+ π§π(π β β (π)) π2π§ (π β β (π)) + π’
π(π) π π’ (π)] = 0,
(31)
where π = πβ2πΌππ, π1= πβ2πΌ(πβπ(π))
π1, and π
2= πβ2πΌ(πββ(π))
π2,
and using
π’π(π) π π’ (π)
β€ π§π(π) πΆππΎππ πΎπΆπ§ (π)
+ 2ππΌππ§π(π) πΆππΎππ πΎπΆ1π§ (π β π (π))
+ π2πΌπ
π§π(π β π (π)) πΆ
π
1πΎππ πΎπΆ1π§ (π β π (π))
β€ 2π§π(π) πΆππΎππ πΎπΆπ§ (π)
+ 2π2πΌπ
π§π(π β π (π)) πΆ
π
1πΎππ πΎπΆ1π§ (π β π (π)) ,
(32)
we can get
Ξπ (π)
β€ ππ(π)Ξ©π (π)
β [π§π(π) ππ§ (π) + π§
π(π β π (π)) π1π§ (π β π (π))
+ π§π(π β β (π)) π2π§ (π β β (π)) + π’
π(π) π π’ (π)] ,
(33)
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6 International Journal of Engineering Mathematics
where π(π) = [π§π(π) π§π(π + 1) π§π(π β π(π)) π§π(π ββ(π)) π
π
(π§(π)) ππ(π§(π β β(π)))]
π
and
Ξ© =(
(
Ξ©11
Ξ12
0 0 0 0
β Ξ©22
Ξ23
0 0 0
β β Ξ©33
0 0 0
β β β Ξ44
0 0
β β β β Ξ55
0
β β β β β Ξ66
)
)
,
Ξ©11= Ξ11+ πΆππΎπ(πΌ + π + π
π)πΎπΆ,
Ξ©22= Ξ22+ π2πΌπ
ππ΅π΅πππ+ ππΌπππ
πππ
+ ππΌππππ
πππ+ ππΉπΉ
πππ+ ππΊπΊ
πππ,
Ξ©33= Ξ33+ π2πΌπΆπ
1πΎππΎπΆ1+ 2π2πΌπ
πΆπ
1πΎππ πΎπΆ1.
(34)
By Lemma 4, the condition Ξ© < 0 is equivalent to LMI(10). Therefore, from (33) it follows that
Ξπ (π) < 0, (35)
which implies that π(π) β€ π(0), βπ β π+.We can easily get
π1βπ§(π)β
2β€ π (π) β€ π2
π§π
2, (36)
where βπ§πβ = max{βπ§(π β π)β, . . . , βπ§(π)β}, π
1= πmin(π),
π2= πmax (π) + π (π + 1) [πmax (π1) + πmax (π 1)]
+ β (β + 1) [πmax (π2) + πmax (π 2)] .(37)
From (35) and (36), we can get
π1βπ§(π)β
2β€ π2
π
2
, βπ§ (π)β β€ βπ2
π1
π. (38)
Using the relation π§(π) = ππΌππ₯(π), we can get
βπ₯ (π)β β€ βπ2
π1
πβπΌπ π
, βπ β π+. (39)
Therefore, the closed-loop system (7) is exponentially stable.Next we will find the guaranteed cost value, from (33), we canget
Ξπ (π)
β€ β [π₯π(π) ππ₯ (π) + π₯
π(π β π (π)) π1π₯ (π β π (π))
+ π₯π(π β β (π)) π2π₯ (π β β (π)) + π’
π(π) π π’ (π)] .
(40)
Summing up both sides of (40) from 0 to π β 1, we can getπβ1
β
π=0
[π₯π(π) ππ₯ (π) + π₯
π(π β π (π)) π1π₯ (π β π (π))
+ π₯π(π β β (π)) π2π₯ (π β β (π)) + π’
π(π) π π’ (π)]
β€ π (0) β π (π) .
(41)
Letting π β +β, noting that π(π) β 0, we can get
π½ =
β
β
π=0
[π₯π(π) ππ₯ (π) + π₯
π(π β π (π)) π1π₯ (π β π (π))
+ π₯π(π β β (π)) π2π₯ (π β β (π))
+ π’π(π) π π’ (π) ]
β€ π (0) ,
(42)
associated with (36), and we have π½ β€ π2βπβ2= π½β.
Remark 6. When time-delay in state vector keeps consistentwith the delay in nonlinear perturbations and uncertain itemsdisappear, the system (1) induced to
π₯ (π + 1) = π΄π₯ (π) + π΄1π₯ (π β π (π)) + π΅π’ (π)
+ πΉπ (π₯ (π)) + πΊπ (π₯ (π β π (π))) ,
π¦ (π) = πΆπ₯ (π) + πΆ1π₯ (π β π (π)) , π β π+,
π₯ (π) = ππ, π = βπ, βπ + 1, . . . , 0,
(43)
at the same time, the closed-loop system (7), and costfunction (6) are reduced to
π₯ (π + 1) = [π΄ + π΅πΎπΆ] π₯ (π) + [π΄1 + π΅πΎπΆ1] π₯ (π β π (π))
+ πΉπ (π₯ (π)) + πΊπ (π₯ (π β π (π))) ,
π½ =
β
β
π=0
{π₯π(π) ππ₯ (π) + π₯
π(π β π (π)) π1 + π’
π(π) π π’ (π)} .
(44)
Then we give a sufficient condition for the existence of staticoutput feedback control for system (43).
Theorem7. For a given scalar πΌ > 0, the control π’(π) = πΎπ¦(π)is a static output feedback guaranteed cost controller for system(43), if there exist symmetric positive definite matrices π, π
1,
π 1, andπΎ, arbitrary matrixπ, and scalars π
1β₯ 0, π2β₯ 0, such
that the following LMI holds:
(
Ξ10 Ξ13
Ξ14
Ξ15
β Ξ2
0 0 0
β β Ξ3
0 0
β β β Ξ4
0
β β β β Ξ5
)< 0, (45)
-
International Journal of Engineering Mathematics 7
where
Ξ1= (
Ξ
11Ξ12
0
β Ξ22
Ξ23
β β Ξ
33
) , Ξ14= (
0 0 0
ππ΅ ππΉ ππΊ
0 0 0
) ,
Ξ15= (
0 0
0 0
πΆπ
1πΎππΆπ
1πΎπ
) ,
Ξ2= diag {Ξ
44, Ξ
55} , Ξ
4= diag {Ξ
1, Ξ2, βπΌ, βπΌ} ,
Ξ5= diag {β0.5πβ2πΌππ β1, βπβ2πΌπΌ} ,
Ξ
11= βπ + π (π
1+ π 1) + π½1(π2πΌ+ π1) πΏπ
1πΏ1+ π,
Ξ
33= βπ1β π 1+ π2πΌπ
π1+ π½2(π2+ π2πΌπ
) πΏπ
2πΏ2,
Ξ
44= βπ1πΌ, Ξ
55= βπ2πΌ,
(46)
and the guaranteed cost value is given by π½β = π2βπβ2, where
π
2= πmax(π) + π(π + 1)[πmax(π1) + πmax(π 1)].
Proof. Theproof ofTheorem 7 is similar to that ofTheorem 5,which are omitted.
Remark 8. In this paper, we design the controller directlyfrom the LMI without variable transformation [11] whichreduces the amount of calculation. Moreover, based onTheorem 5, one can deduce the criteria for linear discrete-time systems with time-delay and nonlinear discrete-timesystems with constant time-delay.
4. Numerical Example
Consider the nonlinear uncertain discrete-time system (1)with the following parameters:
π΄ = (
β0.02 0.12 β0.1
0.1 0.03 0.1
β0.1 0.03 0.2
) ,
π΄1= (
0.02 0.03 0.02
0.01 0.02 β0.01
β0.05 0.02 0.08
) ,
π΅ = (
β0.05 0.1 0.2
0.1 0.32 0.1
0.05 β0.06 0.12
) ,
πΉ = (
0.16 β0.3 0.02
β0.2 0.14 0.2
0.13 0.14 0.1
) ,
πΊ = (
β0.03 0.06 0.02
0.4 0.1 0.1
0.02 β0.03 0.02
) ,
πΆ = (
0.12 0.28 0
β0.06 0.09 0.02
0.2 0.03 0.2
) ,
πΆ1= (
0.04 β0.06 0.07
0.05 0.08 0.07
0.04 0.01 0.05
) ,
π1= (
β0.5 0.2 0.1
0.2 0.3 0.1
β0.8 0.2 0.1
) , π2= (
β0.6 0.3 0.1
β0.1 β0.2 0.2
β0.1 β0.5 0.3
) ,
π = (
0.03 0.02 0.04
β0.05 0.12 0.1
0.03 β0.02 0.02
) ,
π = (
0.6 0.2 0.1
0.2 0.6 0.4
0.1 0.4 0.2
) , π1= (
0.4 β0.1 0.3
β0.1 0.3 0.2
0.3 0.2 0.6
) ,
π2= (
0.5 0.1 0.2
0.1 0.3 0.4
0.2 0.4 0.6
) , π = (
1.12 0.3 β0.1
0.3 1.5 0
β0.1 0 1.6
) ,
πΏ1= (
β0.35 0.2 0.1
0.2 0.3 0.1
0.8 0.2 0.1
) , πΏ2= (
1.2 0.23 2
β1.3 0.4 0.2
1.5 β0.3 1.6
) ,
π (π) = 1.5 + 1.5 sin ππ2, β (π) = 1 + cos ππ
2,
π = 0, 1, 2, . . . .
(47)
Given πΌ = 0.032, π = 3, β = 2, π½1= 0.23, π½
2= 0.25, and the
initial condition
π₯ (β3) = (
β1.6
1.3
1.2
) , π₯ (β2) = (
β1.3
0.8
β1
) ,
π₯ (β1) = (
0.4
β0.5
0.2
) , π₯ (0) = (
1.2
0
β1.2
) ,
(48)
using the LMI Toolbox in MATLAB [14], the LMI (10) inTheorem 5 is satisfied with
π = (
4.1184 0.0455 β0.3818
0.0455 1.6976 β0.2973
β0.3818 β0.2973 10.3666
) ,
π1= (
0.0311 β0.0020 β0.0207
β0.0020 0.0115 β0.0158
β0.0207 β0.0158 0.1978
) ,
-
8 International Journal of Engineering Mathematics
0 5 10 15 20
0
0.5
1
1.5
2
Time (s)
β2
β1.5
β1
β0.5
x
1
x
2
x
3
Figure 1: State trajectories of the closed-loop system.
π2= (
0.0478 β0.0032 β0.0334
β0.0032 0.0163 β0.0254
β0.0334 β0.0254 0.3176
) ,
π 1= (
0.1001 β0.0069 β0.0710
β0.0069 0.0329 β0.0541
β0.0710 β0.0541 0.6715
) ,
π 2= (
0.1046 β0.0077 β0.0783
β0.0077 0.0299 β0.0596
β0.0783 β0.0596 0.7340
) ,
π = (
5.2270 0.1211 0.0022
0.1220 2.2186 β0.4808
0.0003 β0.4800 13.0762
) ,
π1= 7.3374, π
2= 6.7127,
(49)
and the controller parameter
πΎ = (
0.6351 β0.9829 β0.4757
β0.9829 2.7965 0.8749
β0.4757 0.8749 0.9870
) . (50)
Moveover, the solution of the closed-loop system satisfies
βπ₯ (π)β β€ 4.0345πβ0.032π π
,(51)
and the guaranteed cost of the closed-loop system is as follows
π½β= 27.4600
π
2. (52)
Simulation result is presented in Figure 1, which shows theconvergence behavior of the proposed methods.
5. Conclusion
In this paper, the problem of robust output feedback guar-anteed cost control for nonlinear uncertain disctere systemis researched. For all admissible uncertainties, an outputfeedback guaranteed cost controller has been designed suchthat the resulting closed-loop system is robust exponentiallystable and guarantees an adequate level of system perfor-mance. A numerical example has been presented to illustratethe efficiency of the result.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
References
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[2] J. X. Dong and G. H. Yang, βStatic output feedback controlsynthesis for linear systems with time-invariant parametricuncertainties,β IEEE Transactions on Automatic Control, vol. 52,no. 10, pp. 1930β1936, 2007.
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International Journal of Engineering Mathematics 9
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