research article design of robust output feedback ...research article design of robust output...

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 © 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 [5–7]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

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 𝐴 is symmetric if 𝐴 = 𝐴𝑇.𝜆max(𝐴)(𝜆min(𝐴)) denotes the maximum (minimum) valueof the real parts of eigenvalues of 𝐴. The symmetric terms ina matrix are denoted by ∗. 𝑋 > 0 (resp., 𝑋 ≥ 0), for 𝑋 ∈𝑅𝑛×𝑛, means that thematrix is real symmetric positive definite

(resp., positive semidefinite). 𝑁+ denotes the set of all realnonnegative integers.

2. Preliminaries

Consider the following control system:

𝑥 (𝑘 + 1) = [𝐴 + Δ𝐴 (𝑘)] 𝑥 (𝑘)

+ [𝐴1+ Δ𝐴1 (𝑘)] 𝑥 (𝑘 − 𝑑 (𝑘)) + 𝐵𝑢 (𝑘)

+𝐹𝑓 (𝑥 (𝑘)) + 𝐺𝑔 (𝑥 (𝑘 − ℎ (𝑘))) ,

𝑦 (𝑘) = 𝐶𝑥 (𝑘) + 𝐶1𝑥 (𝑘 − 𝑑 (𝑘)) , 𝑘 ∈ 𝑁+,

𝑥 (𝑘) = 𝜙𝑘, 𝑘 = −𝜏, −𝜏 + 1, . . . , 0,

(1)

where 𝑥(𝑘) ∈ 𝑅𝑛 is the state vector, 𝑦(𝑘) ∈ 𝑅𝑟 is theobservation output, and 𝑢(𝑘) ∈ 𝑅𝑚 is the control intput.𝐴, 𝐴1, 𝐵, 𝐹, 𝐺, 𝐶, and 𝐶

1are given constant matrices

with appropriate dimensions. Δ𝐴(𝑘), Δ𝐴1(𝑘) are the time-

varying parameter uncertainties that are assumed to satisfythe following admissible condition:

[Δ𝐴 (𝑘) Δ𝐴1 (𝑘)] = 𝑀Δ (𝑘) [𝑁1𝑁2] , Δ𝑇(𝑘) Δ (𝑘) ⩽ 𝐼,

∀𝑘 ∈ 𝑁+,

(2)

where𝑀,𝑁1, and𝑁

2are some given constant matrices with

appropriate dimensions. The positive integers 𝑑(𝑘) and ℎ(𝑘)are time-varying delays satisfying

0 ≤ 𝑑 (𝑘) ≤ 𝑑, 0 ≤ ℎ (𝑘) ≤ ℎ, (3)

where 𝑑, ℎ are known positive integers. 𝑓(𝑥(𝑘)) ∈ 𝑅𝑛 and𝑔(𝑥(𝑘 − ℎ(𝑘)) ∈ 𝑅

𝑛 are unknown nonlinear functions,assumed as

𝑓𝑇(𝑥 (𝑘)) 𝑓 (𝑥 (𝑘)) ⩽ 𝛽1𝑥

𝑇(𝑘) 𝐿𝑇

1𝐿1𝑥 (𝑘) ,

𝑔𝑇(𝑥 (𝑘 − ℎ (𝑘))) 𝑔 (𝑥 (𝑘 − ℎ (𝑘)))

⩽ 𝛽2𝑥𝑇(𝑘 − ℎ (𝑘)) 𝐿

𝑇

2𝐿2𝑥 (𝑘 − ℎ (𝑘)) ,

(4)

where 𝛽1, 𝛽2

are known positive integers and 𝐿1, 𝐿2

are known real matrices. The initial condition 𝜙 =(𝜙−𝜏, 𝜙−𝜏+1

, . . . , 𝜙0) ∈ 𝑅(𝜏+1)𝑛 with the norm

𝜙 = max {‖𝑥 (−𝜏)‖ , . . . , ‖𝑥 (0)‖} , (5)

where 𝜏 = max{ℎ, 𝑑}.The corresponding cost function is as follows:

𝐽 =

∞

∑

𝑘=0

{𝑥𝑇(𝑘) 𝑆𝑥 (𝑘) + 𝑥

𝑇(𝑘 − 𝑑 (𝑘)) 𝑆1𝑥 (𝑘 − 𝑑 (𝑘))

+ 𝑥𝑇(𝑘 − ℎ (𝑘)) 𝑆2𝑥 (𝑘 − ℎ (𝑘)) + 𝑢

𝑇(𝑘) 𝑅𝑢 (𝑘)} ,

(6)

where 𝑆, 𝑆1, 𝑆2, and 𝑅 are given symmetric positive definite

matrices with appropriate dimensions.Substituting the output feedback controller 𝑢(𝑘) = 𝐾𝑦(𝑘)

into system (1), we have

𝑥 (𝑘 + 1) = [𝐴 (𝑘) + 𝐵𝐾𝐶] 𝑥 (𝑘)

+ [𝐴1 (𝑘) + 𝐵𝐾𝐶1] 𝑥 (𝑘 − 𝑑 (𝑘)) + 𝐹𝑓 (𝑥 (𝑘))

+ 𝐺𝑔 (𝑥 (𝑘 − ℎ (𝑘))) ,

(7)

where 𝐴(𝑘) = 𝐴 + Δ𝐴(𝑘) and 𝐴1(𝑘) = 𝐴

1+ Δ𝐴1(𝑘).

The objective of this paper is to design an output feedbackcontroller 𝑢(𝑘) = 𝐾𝑦(𝑘) 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 𝛼 > 0, the closed-loop system (7) is saidto be robust exponentially stable with a decay rate 𝛼, if thereexists scalars 𝜎 > 0 such that for every solution 𝑥(𝑘, 𝜙) of thesystem satisfies the condition:

𝑥 (𝑘, 𝜙) ≤ 𝜎𝑒

−𝛼𝑘 𝜙 , ∀𝑘 ∈ 𝑁

+. (8)

Definition 2. For system (1) and cost function (6), if thereexist a static output feedback control law 𝑢∗(𝑘) and a positiveconstant 𝐽∗ such that the closed-loop system (7) is robustexponentially stable with a decay rate 𝛼 and the value (6)satisfies 𝐽 ≤ 𝐽∗, then 𝐽∗ is said to be a guaranteed cost indexand 𝑢∗(𝑘) is said to be a robust output feedback guaranteedcost control law of the system.

The following lemmas are essential in establishing ourmain results.

International Journal of Engineering Mathematics 3

Lemma3 (see [11]). For any 𝑥,𝑦 ∈ 𝑅𝑛, and positive symmetricdefinite matrix𝑁 ∈ 𝑅𝑛×𝑛, we have

±2𝑦𝑇𝑥 ≤ 𝑥

𝑇𝑁−1𝑥 + 𝑦𝑇𝑁𝑦. (9)

Lemma 4 (Schur complement lemma [13]). Given constantmatrix𝑋,𝑌, and𝑍with appropriate dimensions satisfying𝑋 =𝑋𝑇, 𝑌 = 𝑌𝑇 > 0. Then 𝑋 + 𝑍𝑇𝑌−1𝑍 < 0 if and only if 𝑆 =

(𝑋 𝑍𝑇

𝑍 −𝑌) < 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 𝛼 > 0, the control 𝑢(𝑘) =𝐾𝑦(𝑘) is a robustly static output feedback guaranteed costcontroller for nonlinear system (1), if there exist symmetricpositive definite matrices 𝑃, 𝑄

1, 𝑄2, 𝑅1, 𝑅2, and 𝐾, arbitrary

matrix 𝑁, and scalars 𝜀1≥ 0, 𝜀

2≥ 0, such that the following

LMI holds:

(

Θ1

0 Θ13

Θ14

Θ15

∗ Θ2

0 0 0

∗ ∗ Θ3

0 0

∗ ∗ ∗ Θ4

0

∗ ∗ ∗ ∗ Θ5

)< 0, (10)

where

Θ1= (

Δ11

Δ12

0

∗ Δ22

Δ23

∗ ∗ Δ33

) , Θ14= (

0 0 0 0

NB NM NF NG0 0 0 0

) ,

Θ13= (

𝐶𝑇𝐾𝑇𝐶𝑇𝐾𝑇

0 0

0 0

) ,

Θ15= (

0 0

0 0

𝐶𝑇

1𝐾𝑇𝐶𝑇

1𝐾𝑇

) , Θ2= diag {Δ

44, Δ55, Δ66} ,

Θ3= diag {−0.5𝑅−1, −𝐼} ,

Θ4= diag {Λ

1, Λ2, −𝐼, −𝐼} ,

Θ5= diag {−0.5𝑒−2𝛼𝑑𝑅−1, −𝑒−2𝛼𝐼} ,

Δ11= − 𝑃 + 𝑑 (𝑄

1+ 𝑅1) + 𝑒𝛼𝑁𝑇

1𝑁1+ ℎ (𝑄

2+ 𝑅2)

+ 𝛽1(𝑒2𝛼+ 𝜀1) 𝐿𝑇

1𝐿1+ 𝑆,

Δ12= 𝑒𝛼𝑁𝐴, Δ

23= 𝑒𝛼𝑑𝑁𝐴1,

Δ33= −𝑄1− 𝑅1+ 𝑒𝛼𝑑𝑁𝑇

2𝑁2+ 𝑒2𝛼𝑑

𝑆1,

Δ22= 𝑃 − (𝑁 + 𝑁

𝑇) ,

Δ44= −𝑄2− 𝑅2+ (𝜀2+ 1) 𝛽

2𝑒2𝛼ℎ

𝐿𝑇

2𝐿2+ 𝑒2𝛼ℎ

𝑆2,

Δ55= −𝜀1𝐼, Δ

66= −𝜀2𝐼, Λ

1= −𝑒−2𝛼𝑑

𝐼,

Λ2= −

1

𝑒𝛼𝑑 + 𝑒𝛼𝐼, 𝑑 = 𝑑 + 1, ℎ = ℎ + 1,

(11)

and the guaranteed cost value is given by 𝐽∗ = 𝜇2‖𝜙‖2, where

𝜇2= 𝜆max (𝑃) + 𝑑 (𝑑 + 1) [𝜆max (𝑄1) + 𝜆max (𝑅1)]

+ ℎ (ℎ + 1) [𝜆max (𝑄2) + 𝜆max (𝑅2)] .(12)

Proof. We first introduce the new variable 𝑧(𝑘) = 𝑒𝛼𝑘𝑥(𝑘).The closed-loop system (7) is reduced to

𝑧 (𝑘 + 1) = [𝐴 (𝑘) + 𝐵𝐾𝐶] 𝑒𝛼𝑧 (𝑘)

+ [𝐴1 (𝑘) + 𝐵𝐾𝐶1] 𝑒

𝛼(𝑑(𝑘)+1)𝑧 (𝑘 − 𝑑 (𝑘))

+ 𝐹𝑒𝛼(𝑘+1)

𝑓 (𝑧 (𝑘)) + 𝐺𝑒𝛼(𝑘+1)

𝑔 (𝑧 (𝑘 − ℎ (𝑘))) ,

(13)

where 𝑓(𝑧(𝑘)) = 𝑓(𝑧(𝑘)/𝑒𝛼𝑘) and 𝑔(𝑧(𝑘 − ℎ(𝑘))) = 𝑔(𝑧(𝑘 −ℎ(𝑘))/𝑒

𝛼(𝑘−ℎ(𝑘))).

Associated with (2), the above equality is reduced to

𝑧 (𝑘 + 1) = [𝐴 (𝑘) + 𝐵 (𝑘)𝐾𝐶 (𝑘)] 𝑧 (𝑘)

+ [𝐴1 (𝑘) + 𝐵 (𝑘)𝐾𝐶1 (𝑘)] 𝑧 (𝑘 − 𝑑 (𝑘))

+ 𝐹 (𝑘) 𝑓 (𝑧 (𝑘)) + 𝐺 (𝑘) 𝑔 (𝑧 (𝑘 − ℎ (𝑘))) ,

(14)

where

𝐴 (𝑘) = 𝑒𝛼[𝐴 +𝑀Δ (𝑘)𝑁1] , 𝐶 (𝑘) = 𝑒

−𝛼𝑘𝐶,

𝐴1 (𝑘) = 𝑒

𝛼(𝑑(𝑘)+1)[𝐴1+𝑀Δ (𝑘)𝑁2] ,

𝐵 (𝑘) = 𝑒𝛼(𝑘+1)

𝐵, 𝐶1 (𝑘) = 𝑒

−𝛼(𝑘−𝑑(𝑘))𝐶1,

𝐹 (𝑘) = 𝑒𝛼(𝑘+1)

𝐹, 𝐺 (𝑘) = 𝑒𝛼(𝑘+1)

𝐺,

(15)

𝑓𝑇

(𝑧 (𝑘)) 𝑓 (𝑧 (𝑘)) ≤ 𝛽1𝑒−2𝛼𝑘

𝑧𝑇(𝑘) 𝐿𝑇

1𝐿1𝑧 (𝑘) , (16)

𝑔𝑇(𝑧 (𝑘 − ℎ (𝑘))) 𝑔 (𝑧 (𝑘 − ℎ (𝑘)))

≤ 𝛽2𝑒−2𝛼(𝑘−ℎ(𝑘))

𝑧𝑇(𝑘 − ℎ (𝑘)) 𝐿

𝑇

2𝐿2𝑧 (𝑘 − ℎ (𝑘)) .

(17)

Consider a Lyapunov-Krasovskii functional candidate for theclosed-loop system (14) as

𝑉 (𝑘) = 𝑉1 (𝑘) + 𝑉2 (𝑘) + 𝑉3 (𝑘) + 𝑉4 (𝑘) + 𝑉5 (𝑘) , (18)


where

𝑉1 (𝑘) = 𝑧

𝑇(𝑘) 𝑃𝑧 (𝑘) ,

𝑉2 (𝑘) =

𝑘−1

∑

𝑖=𝑘−𝑑(𝑘)

𝑧𝑇(𝑖) 𝑄1𝑧 (𝑖) +

𝑘−1

∑

𝑖=𝑘−ℎ(𝑘)

𝑧𝑇(𝑖) 𝑄2𝑧 (𝑖) ,

𝑉3 (𝑘) =

0

∑

𝑗=−𝑑+1

𝑘−1

∑

𝑖=𝑘+𝑗


0

∑

𝑗=−ℎ+1

𝑘−1

∑

𝑖=𝑘+𝑗


𝑉4 (𝑘) =

𝑘−1

∑

𝑖=𝑘−𝑑(𝑘)

𝑧𝑇(𝑖) 𝑅1𝑧 (𝑖) +

𝑘−1

∑

𝑖=𝑘−ℎ(𝑘)

𝑧𝑇(𝑖) 𝑅2𝑧 (𝑖) ,

𝑉5 (𝑘) =

0

∑

𝑗=−𝑑+1

𝑘−1

∑

𝑖=𝑘+𝑗

𝑧𝑇(𝑖) 𝑅1𝑧 (𝑖) +

0

∑

𝑗=−ℎ+1

𝑘−1

∑

𝑖=𝑘+𝑗

𝑧𝑇(𝑖) 𝑅2𝑧 (𝑖) .

(19)

Calculating the difference of 𝑉(𝑘) we have

Δ𝑉1 (𝑘) = 𝑧

𝑇(𝑘 + 1) 𝑃𝑧 (𝑘 + 1) − 𝑧

𝑇(𝑘) 𝑃𝑧 (𝑘) , (20)

Δ𝑉2 (𝑘)

≤ 𝑧𝑇(𝑘) (𝑄1 + 𝑄2) 𝑧 (𝑘) − 𝑧

𝑇(𝑘 − 𝑑 (𝑘)) 𝑄1𝑧 (𝑘 − 𝑑 (𝑘))

− 𝑧𝑇(𝑘 − ℎ (𝑘)) 𝑄2𝑧 (𝑘 − ℎ (𝑘))

+

𝑘

∑

𝑖=𝑘−𝑑+1


𝑘

∑

𝑖=𝑘−ℎ+1


(21)

Δ𝑉3 (𝑘)

=

0

∑

𝑗=−𝑑+1

{

{

{

𝑘

∑

𝑖=𝑘+𝑗+1

𝑧𝑇(𝑖) 𝑄1𝑧 (𝑖) −

𝑘−1

∑

𝑖=𝑘+𝑗

𝑧𝑇(𝑖) 𝑄1𝑧 (𝑖)

}

}

}

+

0

∑

𝑗=−ℎ+1

{

{

{

𝑘

∑

𝑖=𝑘+𝑗+1

𝑧𝑇(𝑖) 𝑄2𝑧 (𝑖) −

𝑘−1

∑

𝑖=𝑘+𝑗

𝑧𝑇(𝑖) 𝑄2𝑧 (𝑖)

}

}

}

= 𝑧𝑇(𝑘) (𝑑𝑄1 + ℎ𝑄2) 𝑧 (𝑘)

−

𝑘

∑

𝑗=𝑘−𝑑+1

𝑧𝑇(𝑗)𝑄1𝑧 (𝑗) −

𝑘

∑

𝑗=𝑘−ℎ+1

𝑧𝑇(𝑗)𝑄2𝑧 (𝑗) .

(22)

Combine (21) and (22), we have

Δ𝑉2 (𝑘) + Δ𝑉3 (𝑘) ≤ 𝑧

𝑇(𝑘) (𝑑𝑄1 + ℎ𝑄2) 𝑧 (𝑘)

− 𝑧𝑇(𝑘 − 𝑑 (𝑘)) 𝑄1𝑧 (𝑘 − 𝑑 (𝑘))

− 𝑧𝑇(𝑘 − ℎ (𝑘)) 𝑄2𝑧 (𝑘 − ℎ (𝑘)) .

(23)

Similarly, we can get

Δ𝑉4 (𝑘) + Δ𝑉5 (𝑘) ≤ 𝑧

𝑇(𝑘) (𝑑𝑅1 + ℎ𝑅2) 𝑧 (𝑘)

− 𝑧𝑇(𝑘 − 𝑑 (𝑘)) 𝑅1𝑧 (𝑘 − 𝑑 (𝑘))

− 𝑧𝑇(𝑘 − ℎ (𝑘)) 𝑅2𝑧 (𝑘 − ℎ (𝑘)) .

(24)

Therefore, from (17)–(24), we have

Δ𝑉 (𝑘) ≤ 𝑧𝑇(𝑘 + 1) 𝑃𝑧 (𝑘 + 1)

+ 𝑧𝑇(𝑘) [𝑑 (𝑄1 + 𝑅1) + ℎ (𝑄2 + 𝑅2) − 𝑃] 𝑧 (𝑘)

− 𝑧𝑇(𝑘 − 𝑑 (𝑘)) (𝑄1 + 𝑅1) 𝑧 (𝑘 − 𝑑 (𝑘))

− 𝑧𝑇(𝑘 − ℎ (𝑘)) (𝑄2 + 𝑅2) 𝑧 (𝑘 − ℎ (𝑘)) .

(25)

Multiplying 2𝑧𝑇(𝑘 + 1)𝑁 both sides of the identity (14),

− 2𝑧𝑇(𝑘 + 1)𝑁𝑧 (𝑘 + 1) + 2𝑧

𝑇(𝑘 + 1)

× 𝑁(𝐴 (𝑘) + 𝐵 (𝑘)𝐾𝐶 (𝑘)) 𝑧 (𝑘) + 2𝑧𝑇(𝑘 + 1)

× 𝑁(𝐴1 (𝑘) + 𝐵 (𝑘)𝐾𝐶1 (𝑘)) 𝑧 (𝑘 − 𝑑 (𝑘))

+ 2𝑧𝑇(𝑘 + 1)𝑁𝐹 (𝑘) 𝑓 + 2𝑧

𝑇(𝑘 + 1)𝑁𝐺 (𝑘) 𝑔 = 0.

(26)

Note that for any 𝜀1≥ 0, 𝜀

2≥ 0, it follows from (16) to (17)

𝜀1[𝛽1𝑒−2𝛼𝑘


1𝐿1𝑧 (𝑘) − 𝑓

𝑇

(𝑧 (𝑘)) 𝑓 (𝑧 (𝑘))] ≥ 0,

𝜀2[𝛽2𝑒−2𝛼(𝑘−ℎ(𝑘))


𝑇

2𝐿2𝑧 (𝑘 − ℎ (𝑘))

−𝑔𝑇(𝑧 (𝑘 − ℎ (𝑘))) 𝑔 (𝑧 (𝑘 − ℎ (𝑘)))] ≥ 0.

(27)

Substituting (26)-(27) into (25), we have

Δ𝑉 (𝑘)

≤ 𝑧𝑇(𝑘 + 1) 𝑃𝑧 (𝑘 + 1)

+ 𝑧𝑇(𝑘) [𝑑 (𝑄1 + 𝑅1) + ℎ (𝑄2 + 𝑅2) − 𝑃] 𝑧 (𝑘)

− 𝑧𝑇(𝑘 − 𝑑 (𝑘)) (𝑄1 + 𝑅1) 𝑧 (𝑘 − 𝑑 (𝑘))

− 𝑧𝑇(𝑘 − ℎ (𝑘)) (𝑄2 + 𝑅2) 𝑧 (𝑘 − ℎ (𝑘))

− 2𝑧𝑇(𝑘 + 1)𝑁𝑧 (𝑘 + 1) + 2𝑧

𝑇(𝑘 + 1)


× 𝑁(𝐴 (𝑘) + 𝐵 (𝑘)𝐾𝐶 (𝑘)) 𝑧 (𝑘) + 2𝑧𝑇(𝑘 + 1)

× 𝑁 (𝐴1 (𝑘) + 𝐵 (𝑘)𝐾𝐶1 (𝑘)) 𝑧 (𝑘 − 𝑑 (𝑘))

+ 2𝑧𝑇(𝑘 + 1)𝑁𝐹 (𝑘) 𝑓 (𝑧 (𝑘))

+ 2𝑧𝑇(𝑘 + 1)𝑁𝐺 (𝑘) 𝑔 (𝑧 (𝑘 − ℎ (𝑘)))

+ 𝜀1[𝛽1𝑒−2𝛼𝑘


1𝐿1𝑧 (𝑘)−𝑓

𝑇

(𝑧 (𝑘)) 𝑓 (𝑧 (𝑘))]

+ 𝜀2[𝛽2𝑒−2𝛼(𝑘−ℎ(𝑘))


𝑇

2𝐿2𝑧 (𝑘 − ℎ (𝑘))

− 𝑔𝑇(𝑧 (𝑘 − ℎ (𝑘))) 𝑔 (𝑧 (𝑘 − ℎ (𝑘)))] .

(28)

Dealing with partial idem in (28) using Lemma 3, it follows

2𝑧𝑇(𝑘 + 1)𝑁 (𝐴 (𝑘) + 𝐵 (𝑘)𝐾𝐶 (𝑘)) 𝑧 (𝑘)

= 2𝑧𝑇(𝑘 + 1) 𝑒

𝛼[𝑁𝐴 + 𝑁𝑀Δ (𝑘)𝑁1] 𝑧 (𝑘)

+ 2𝑧𝑇(𝑘 + 1) 𝑒

𝛼𝑁𝐵𝐾𝐶𝑧 (𝑘)

≤ 2𝑧𝑇(𝑘 + 1) 𝑒

𝛼𝑁𝐴𝑧 (𝑘)

+ 𝑒𝛼𝑧𝑇(𝑘 + 1)𝑁𝑀𝑀

𝑇𝑁𝑇𝑧 (𝑘 + 1)

+ 𝑒𝛼𝑧𝑇(𝑘)𝑁𝑇

1𝑁1𝑧 (𝑘)

+ 𝑧𝑇(𝑘 + 1) 𝑒

2𝛼𝑁𝐵𝐵𝑇𝑁𝑇𝑧 (𝑘 + 1)

+ 𝑧𝑇(𝑘) 𝐶𝑇𝐾𝑇𝐾𝐶𝑧 (𝑘) ,

2𝑧𝑇(𝑘 + 1)𝑁 (𝐴1 (𝑘) + 𝐵 (𝑘)𝐾𝐶1 (𝑘)) 𝑧 (𝑘 − 𝑑 (𝑘))

= 𝑧𝑇(𝑘 + 1) 𝑒

𝛼(𝑑(𝑘)+1)(𝑁𝐴1+ 𝐴𝑇

1𝑁𝑇) 𝑧 (𝑘 − 𝑑 (𝑘))

+ 2𝑒𝛼(𝑑(𝑘)+1)

𝑧𝑇(𝑘 + 1)𝑁𝑀Δ (𝑘)𝑁2𝑧 (𝑘 − 𝑑 (𝑘))

+ 2𝑧𝑇(𝑘 + 1) 𝑒

𝛼𝑑(𝑘)𝑁𝐵𝑒𝛼𝐾𝐶1𝑧 (𝑘 − 𝑑 (𝑘))

≤ 𝑧𝑇(𝑘 + 1) 𝑒

𝛼𝑑(𝑁𝐴1+ 𝐴𝑇

1𝑁𝑇) 𝑧 (𝑘 − 𝑑 (𝑘))

+ 𝑒𝛼𝑑𝑧𝑇(𝑘 + 1)𝑁𝑀𝑀


+ 𝑒𝛼𝑑𝑧𝑇(𝑘 − 𝑑 (𝑘))𝑁

𝑇

2𝑁2𝑧 (𝑘 − 𝑑 (𝑘))

+ 𝑧𝑇(𝑘 + 1) 𝑒

2𝛼𝑑𝑁𝐵𝐵𝑇𝑁𝑇𝑧 (𝑘 + 1)

+ 𝑒2𝛼𝑧𝑇(𝑘 − 𝑑 (𝑘)) 𝐶

𝑇

1𝐾𝑇𝐾𝐶1𝑧 (𝑘 − 𝑑 (𝑘)) ,

2𝑧𝑇(𝑘 + 1)𝑁𝐹 (𝑘) 𝑓 (𝑧 (𝑘))

= 2𝑧𝑇(𝑘 + 1)𝑁𝐹𝑒

𝛼(𝑘+1)𝑓 (𝑧 (𝑘))

≤ 𝑧𝑇(𝑘 + 1)𝑁𝐹𝐹


+ 𝑒2𝛼(𝑘+1)

𝑓𝑇

(𝑧 (𝑘)) 𝑓 (𝑧 (𝑘))

≤ 𝑧𝑇(𝑘 + 1)𝑁𝐹𝐹


+ 𝛽1𝑒2𝛼𝑧𝑇(𝑘) 𝐿𝑇

1𝐿1𝑧 (𝑘) .

(29)

Similarly, we have

2𝑧𝑇(𝑘 + 1)𝑁𝐺 (𝑘) 𝑔 (𝑧 (𝑘 − ℎ (𝑘)))

≤ 𝑧𝑇(𝑘 + 1)𝑁𝐺𝐺


+ 𝛽2𝑒2𝛼ℎ


𝑇

2𝐿2𝑧 (𝑘 − ℎ (𝑘)) .

(30)

Adding the following relation to inequality (28)

[𝑧𝑇(𝑘) 𝑆𝑧 (𝑘) + 𝑧

𝑇(𝑘 − 𝑑 (𝑘)) 𝑆1𝑧 (𝑘 − 𝑑 (𝑘))

+ 𝑧𝑇(𝑘 − ℎ (𝑘)) 𝑆2𝑧 (𝑘 − ℎ (𝑘)) + 𝑢

𝑇(𝑘) 𝑅𝑢 (𝑘)]

− [𝑧𝑇(𝑘) 𝑆𝑧 (𝑘) + 𝑧

𝑇(𝑘 − 𝑑 (𝑘)) 𝑆1𝑧 (𝑘 − 𝑑 (𝑘))

+ 𝑧𝑇(𝑘 − ℎ (𝑘)) 𝑆2𝑧 (𝑘 − ℎ (𝑘)) + 𝑢

𝑇(𝑘) 𝑅𝑢 (𝑘)] = 0,

(31)

where 𝑆 = 𝑒−2𝛼𝑘𝑆, 𝑆1= 𝑒−2𝛼(𝑘−𝑑(𝑘))

𝑆1, and 𝑆

2= 𝑒−2𝛼(𝑘−ℎ(𝑘))

𝑆2,

and using

𝑢𝑇(𝑘) 𝑅𝑢 (𝑘)

≤ 𝑧𝑇(𝑘) 𝐶𝑇𝐾𝑇𝑅𝐾𝐶𝑧 (𝑘)

+ 2𝑒𝛼𝑑𝑧𝑇(𝑘) 𝐶𝑇𝐾𝑇𝑅𝐾𝐶1𝑧 (𝑘 − 𝑑 (𝑘))

+ 𝑒2𝛼𝑑

𝑧𝑇(𝑘 − 𝑑 (𝑘)) 𝐶

𝑇

1𝐾𝑇𝑅𝐾𝐶1𝑧 (𝑘 − 𝑑 (𝑘))

≤ 2𝑧𝑇(𝑘) 𝐶𝑇𝐾𝑇𝑅𝐾𝐶𝑧 (𝑘)

+ 2𝑒2𝛼𝑑

𝑧𝑇(𝑘 − 𝑑 (𝑘)) 𝐶

𝑇

1𝐾𝑇𝑅𝐾𝐶1𝑧 (𝑘 − 𝑑 (𝑘)) ,

(32)

we can get

Δ𝑉 (𝑘)

≤ 𝜉𝑇(𝑘)Ω𝜉 (𝑘)

− [𝑧𝑇(𝑘) 𝑆𝑧 (𝑘) + 𝑧

𝑇(𝑘 − 𝑑 (𝑘)) 𝑆1𝑧 (𝑘 − 𝑑 (𝑘))

+ 𝑧𝑇(𝑘 − ℎ (𝑘)) 𝑆2𝑧 (𝑘 − ℎ (𝑘)) + 𝑢

𝑇(𝑘) 𝑅𝑢 (𝑘)] ,

(33)


where 𝜉(𝑘) = [𝑧𝑇(𝑘) 𝑧𝑇(𝑘 + 1) 𝑧𝑇(𝑘 − 𝑑(𝑘)) 𝑧𝑇(𝑘 −ℎ(𝑘)) 𝑓

𝑇

(𝑧(𝑘)) 𝑔𝑇(𝑧(𝑘 − ℎ(𝑘)))]

𝑇

and

Ω =(

(

Ω11

Δ12

0 0 0 0

∗ Ω22

Δ23

0 0 0

∗ ∗ Ω33

0 0 0

∗ ∗ ∗ Δ44

0 0

∗ ∗ ∗ ∗ Δ55

0

∗ ∗ ∗ ∗ ∗ Δ66

)

)

,

Ω11= Δ11+ 𝐶𝑇𝐾𝑇(𝐼 + 𝑅 + 𝑅

𝑇)𝐾𝐶,

Ω22= Δ22+ 𝑒2𝛼𝑑

𝑁𝐵𝐵𝑇𝑁𝑇+ 𝑒𝛼𝑁𝑀𝑀

𝑇𝑁𝑇

+ 𝑒𝛼𝑑𝑁𝑀𝑀

𝑇𝑁𝑇+ 𝑁𝐹𝐹

𝑇𝑁𝑇+ 𝑁𝐺𝐺

𝑇𝑁𝑇,

Ω33= Δ33+ 𝑒2𝛼𝐶𝑇

1𝐾𝑇𝐾𝐶1+ 2𝑒2𝛼𝑑

𝐶𝑇

1𝐾𝑇𝑅𝐾𝐶1.

(34)

By Lemma 4, the condition Ω < 0 is equivalent to LMI(10). Therefore, from (33) it follows that

Δ𝑉 (𝑘) < 0, (35)

which implies that 𝑉(𝑘) ≤ 𝑉(0), ∀𝑘 ∈ 𝑁+.We can easily get

𝜇1‖𝑧(𝑘)‖

2≤ 𝑉 (𝑘) ≤ 𝜇2

𝑧𝑘

2, (36)

where ‖𝑧𝑘‖ = max{‖𝑧(𝑘 − 𝜏)‖, . . . , ‖𝑧(𝑘)‖}, 𝜇

1= 𝜆min(𝑃),

𝜇2= 𝜆max (𝑃) + 𝑑 (𝑑 + 1) [𝜆max (𝑄1) + 𝜆max (𝑅1)]

+ ℎ (ℎ + 1) [𝜆max (𝑄2) + 𝜆max (𝑅2)] .(37)

From (35) and (36), we can get

𝜇1‖𝑧(𝑘)‖

2≤ 𝜇2

𝜙

2

, ‖𝑧 (𝑘)‖ ≤ √𝜇2

𝜇1

𝜙. (38)

Using the relation 𝑧(𝑘) = 𝑒𝛼𝑘𝑥(𝑘), we can get

‖𝑥 (𝑘)‖ ≤ √𝜇2

𝜇1

𝑒−𝛼𝑘 𝜙

, ∀𝑘 ∈ 𝑁+. (39)

Therefore, the closed-loop system (7) is exponentially stable.Next we will find the guaranteed cost value, from (33), we canget

Δ𝑉 (𝑘)

≤ − [𝑥𝑇(𝑘) 𝑆𝑥 (𝑘) + 𝑥

𝑇(𝑘 − 𝑑 (𝑘)) 𝑆1𝑥 (𝑘 − 𝑑 (𝑘))

+ 𝑥𝑇(𝑘 − ℎ (𝑘)) 𝑆2𝑥 (𝑘 − ℎ (𝑘)) + 𝑢

𝑇(𝑘) 𝑅𝑢 (𝑘)] .

(40)

Summing up both sides of (40) from 0 to 𝑛 − 1, we can get𝑛−1

∑

𝑘=0

[𝑥𝑇(𝑘) 𝑆𝑥 (𝑘) + 𝑥

𝑇(𝑘 − 𝑑 (𝑘)) 𝑆1𝑥 (𝑘 − 𝑑 (𝑘))

+ 𝑥𝑇(𝑘 − ℎ (𝑘)) 𝑆2𝑥 (𝑘 − ℎ (𝑘)) + 𝑢

𝑇(𝑘) 𝑅𝑢 (𝑘)]

≤ 𝑉 (0) − 𝑉 (𝑛) .

(41)

Letting 𝑛 → +∞, noting that 𝑉(𝑛) → 0, we can get

𝐽 =

∞

∑

𝑘=0

[𝑥𝑇(𝑘) 𝑆𝑥 (𝑘) + 𝑥

𝑇(𝑘 − 𝑑 (𝑘)) 𝑆1𝑥 (𝑘 − 𝑑 (𝑘))

+ 𝑥𝑇(𝑘 − ℎ (𝑘)) 𝑆2𝑥 (𝑘 − ℎ (𝑘))

+ 𝑢𝑇(𝑘) 𝑅𝑢 (𝑘) ]

≤ 𝑉 (0) ,

(42)

associated with (36), and we have 𝐽 ≤ 𝜇2‖𝜙‖2= 𝐽∗.

Remark 6. When time-delay in state vector keeps consistentwith the delay in nonlinear perturbations and uncertain itemsdisappear, the system (1) induced to

𝑥 (𝑘 + 1) = 𝐴𝑥 (𝑘) + 𝐴1𝑥 (𝑘 − 𝑑 (𝑘)) + 𝐵𝑢 (𝑘)

+ 𝐹𝑓 (𝑥 (𝑘)) + 𝐺𝑔 (𝑥 (𝑘 − 𝑑 (𝑘))) ,

𝑦 (𝑘) = 𝐶𝑥 (𝑘) + 𝐶1𝑥 (𝑘 − 𝑑 (𝑘)) , 𝑘 ∈ 𝑁+,

𝑥 (𝑘) = 𝜙𝑘, 𝑘 = −𝑑, −𝑑 + 1, . . . , 0,

(43)

at the same time, the closed-loop system (7), and costfunction (6) are reduced to

𝑥 (𝑘 + 1) = [𝐴 + 𝐵𝐾𝐶] 𝑥 (𝑘) + [𝐴1 + 𝐵𝐾𝐶1] 𝑥 (𝑘 − 𝑑 (𝑘))

+ 𝐹𝑓 (𝑥 (𝑘)) + 𝐺𝑔 (𝑥 (𝑘 − 𝑑 (𝑘))) ,

𝐽 =

∞

∑

𝑘=0

{𝑥𝑇(𝑘) 𝑆𝑥 (𝑘) + 𝑥

𝑇(𝑘 − 𝑑 (𝑘)) 𝑆1 + 𝑢

𝑇(𝑘) 𝑅𝑢 (𝑘)} .

(44)

Then we give a sufficient condition for the existence of staticoutput feedback control for system (43).

Theorem7. For a given scalar 𝛼 > 0, the control 𝑢(𝑘) = 𝐾𝑦(𝑘)is a static output feedback guaranteed cost controller for system(43), if there exist symmetric positive definite matrices 𝑃, 𝑄

1,

𝑅1, and𝐾, arbitrary matrix𝑁, and scalars 𝜀

1≥ 0, 𝜀2≥ 0, such

that the following LMI holds:

(

Γ10 Θ13

Γ14

Γ15

∗ Γ2

0 0 0

∗ ∗ Θ3

0 0

∗ ∗ ∗ Γ4

0

∗ ∗ ∗ ∗ Γ5

)< 0, (45)


where

Γ1= (

Δ

11Δ12

0

∗ Δ22

Δ23

∗ ∗ Δ

33

) , Γ14= (

0 0 0

𝑁𝐵 𝑁𝐹 𝑁𝐺

0 0 0

) ,

Θ15= (

0 0

0 0

𝐶𝑇

1𝐾𝑇𝐶𝑇

1𝐾𝑇

) ,

Γ2= diag {Δ

44, Δ

55} , Γ

4= diag {Λ

1, Λ2, −𝐼, −𝐼} ,

Γ5= diag {−0.5𝑒−2𝛼𝑑𝑅−1, −𝑒−2𝛼𝐼} ,

Δ

11= −𝑃 + 𝑑 (𝑄

1+ 𝑅1) + 𝛽1(𝑒2𝛼+ 𝜀1) 𝐿𝑇

1𝐿1+ 𝑆,

Δ

33= −𝑄1− 𝑅1+ 𝑒2𝛼𝑑

𝑆1+ 𝛽2(𝜀2+ 𝑒2𝛼𝑑

) 𝐿𝑇

2𝐿2,

Δ

44= −𝜀1𝐼, Δ

55= −𝜀2𝐼,

(46)

and the guaranteed cost value is given by 𝐽∗ = 𝜇2‖𝜙‖2, where

𝜇

2= 𝜆max(𝑃) + 𝑑(𝑑 + 1)[𝜆max(𝑄1) + 𝜆max(𝑅1)].

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:

𝐴 = (

−0.02 0.12 −0.1

0.1 0.03 0.1

−0.1 0.03 0.2

) ,

𝐴1= (

0.02 0.03 0.02

0.01 0.02 −0.01

−0.05 0.02 0.08

) ,

𝐵 = (

−0.05 0.1 0.2

0.1 0.32 0.1

0.05 −0.06 0.12

) ,

𝐹 = (

0.16 −0.3 0.02

−0.2 0.14 0.2

0.13 0.14 0.1

) ,

𝐺 = (

−0.03 0.06 0.02

0.4 0.1 0.1

0.02 −0.03 0.02

) ,

𝐶 = (

0.12 0.28 0

−0.06 0.09 0.02

0.2 0.03 0.2

) ,

𝐶1= (

0.04 −0.06 0.07

0.05 0.08 0.07

0.04 0.01 0.05

) ,

𝑁1= (

−0.5 0.2 0.1

0.2 0.3 0.1

−0.8 0.2 0.1

) , 𝑁2= (

−0.6 0.3 0.1

−0.1 −0.2 0.2

−0.1 −0.5 0.3

) ,

𝑀 = (

0.03 0.02 0.04

−0.05 0.12 0.1

0.03 −0.02 0.02

) ,

𝑆 = (

0.6 0.2 0.1

0.2 0.6 0.4

0.1 0.4 0.2

) , 𝑆1= (

0.4 −0.1 0.3

−0.1 0.3 0.2

0.3 0.2 0.6

) ,

𝑆2= (

0.5 0.1 0.2

0.1 0.3 0.4

0.2 0.4 0.6

) , 𝑅 = (

1.12 0.3 −0.1

0.3 1.5 0

−0.1 0 1.6

) ,

𝐿1= (

−0.35 0.2 0.1

0.2 0.3 0.1

0.8 0.2 0.1

) , 𝐿2= (

1.2 0.23 2

−1.3 0.4 0.2

1.5 −0.3 1.6

) ,

𝑑 (𝑘) = 1.5 + 1.5 sin 𝑘𝜋2, ℎ (𝑘) = 1 + cos 𝑘𝜋

2,

𝑘 = 0, 1, 2, . . . .

(47)

Given 𝛼 = 0.032, 𝑑 = 3, ℎ = 2, 𝛽1= 0.23, 𝛽

2= 0.25, and the

initial condition

𝑥 (−3) = (

−1.6

1.3

1.2

) , 𝑥 (−2) = (

−1.3

0.8

−1

) ,

𝑥 (−1) = (

0.4

−0.5

0.2

) , 𝑥 (0) = (

1.2

0

−1.2

) ,

(48)

using the LMI Toolbox in MATLAB [14], the LMI (10) inTheorem 5 is satisfied with

𝑃 = (

4.1184 0.0455 −0.3818

0.0455 1.6976 −0.2973

−0.3818 −0.2973 10.3666

) ,

𝑄1= (

0.0311 −0.0020 −0.0207

−0.0020 0.0115 −0.0158

−0.0207 −0.0158 0.1978

) ,


0 5 10 15 20

0

0.5

1

1.5

2

Time (s)

−2

−1.5

−1

−0.5

x

1

x

2

x

3

Figure 1: State trajectories of the closed-loop system.

𝑄2= (

0.0478 −0.0032 −0.0334

−0.0032 0.0163 −0.0254

−0.0334 −0.0254 0.3176

) ,

𝑅1= (

0.1001 −0.0069 −0.0710

−0.0069 0.0329 −0.0541

−0.0710 −0.0541 0.6715

) ,

𝑅2= (

0.1046 −0.0077 −0.0783

−0.0077 0.0299 −0.0596

−0.0783 −0.0596 0.7340

) ,

𝑁 = (

5.2270 0.1211 0.0022

0.1220 2.2186 −0.4808

0.0003 −0.4800 13.0762

) ,

𝜀1= 7.3374, 𝜀

2= 6.7127,

(49)

and the controller parameter

𝐾 = (

0.6351 −0.9829 −0.4757

−0.9829 2.7965 0.8749

−0.4757 0.8749 0.9870

) . (50)

Moveover, the solution of the closed-loop system satisfies

‖𝑥 (𝑘)‖ ≤ 4.0345𝑒−0.032𝑘 𝜙

,(51)

and the guaranteed cost of the closed-loop system is as follows

𝐽∗= 27.4600

𝜙

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

[1] I. Yaesh and U. Shaked, “𝐻∞

optimization with pole con-straints of static output-feedback controllers-a non-smoothoptimization approach,” IEEE Transactions on Control SystemsTechnology, vol. 20, no. 4, pp. 1066–1072, 2012.

[2] J. X. Dong and G. H. Yang, “Static output feedback controlsynthesis for linear systems with time-invariant parametricuncertainties,” IEEE Transactions on Automatic Control, vol. 52,no. 10, pp. 1930–1936, 2007.

[3] D. W. C. Ho and G. Lu, “Robust stabilization for a classof discrete-time non-linear systems via output feedback: theunified LMI approach,” International Journal of Control, vol. 76,no. 2, pp. 105–115, 2003.

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[7] J.Wei, Y. Dong, andY. Su, “Guaranteed cost control of uncertainT-S fuzzy systems via output feedback approach,” WSEASTransactions on Systems, vol. 10, no. 9, pp. 306–317, 2011.

[8] W.-H. Chen, Z.-H. Guan, and X. Lu, “Delay-dependent guar-anteed cost control for uncertain discrete-time systems withdelay,” IEE Proceedings: Control Theory and Applications, vol.150, no. 4, pp. 412–416, 2003.

[9] L. Chen, “Unfragile guaranteed-cost 𝐻∞

control of uncertainstate-delay sampling system,” Procedia Engineering, vol. 29, pp.3359–3363, 2012.

[10] F. Qiu, B. Cui, and Y. Ji, “Further results on robust stability ofneutral system with mixed time-varying delays and nonlinearperturbations,”Nonlinear Analysis. RealWorld Applications, vol.11, no. 2, pp. 895–906, 2010.

[11] M. V. Thuan, V. N. Phat, and H. M. Trinh, “Dynamic out-put feedback guaranteed cost control for linear systems withinterval time-varying delays in states and outputs,” AppliedMathematics and Computation, vol. 218, no. 21, pp. 10697–10707,2012.


[12] J. H. Park, “On dynamic output feedback guaranteed costcontrol of uncertain discrete-delay systems: LMI optimizationapproach,” Journal of OptimizationTheory and Applications, vol.121, no. 1, pp. 147–162, 2004.

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