Robust static output feedback H∞ control design for linear systems with polytopic uncertainties

Systems & Control Letters 85 (2015) 23–32

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Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle

Robust static output feedback H∞ control design for linear systems with polytopic uncertainties Xiao-Heng Chang a,b , Ju H. Park b,∗ , Jianping Zhou c a

School of Information Science and Engineering, Wuhan University of Science and Technology, 430081 Wuhan, China

b

Department of Electrical Engineering, Yeungnam University, 280 Daehak-Ro, Kyongsan 38541, Republic of Korea

c

School of Computer Science and Technology, Anhui University of Technology, 243032 Ma’anshan, China

article

info

Article history: Received 27 May 2014 Received in revised form 18 May 2015 Accepted 24 August 2015

Keywords: Linear systems Polytopic uncertainties Static output feedback H∞ controllers Linear matrix inequalities (LMIs)

abstract This paper investigates the problem of robust static output feedback H∞ control for linear systems with polytopic uncertainties. A new method is proposed for robust static output feedback H∞ controller design. The proposed design method is applicable for general uncertain systems, without the need to impose any constraints on system matrices. The corresponding design conditions are presented in the form of linear matrix inequalities (LMIs). One of the advantages of the new method lies in its less conservatism. The proposed design method is also applicable to both continuous-time and discrete-time systems. The performance of the method is compared with other methods based on several examples. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Polytopic uncertainty is one of the important types of parametric uncertainty in robust control theory and practical situations where the set of system parameters is a convex polyhedron. The nominal system is located at the center of this polyhedron. Robust analysis and control design problems for linear systems with polytopic uncertainties have been studied extensively in the past decades, and many remarkable results have been obtained [1–8]. On the other hand, in control theory and practice, static output feedback control is very useful and widely adopted in practice since it can be easily implemented with low cost. Robust static output feedback controller design for linear systems with polytopic uncertainties has been investigated by many researchers. In general, the design problem of such controllers can be represented as a bilinear matrix inequality (BMI) problem, which is nonconvex and NP-hard [9,10]. Various numerical algorithms based on the linear matrix inequality (LMI) technique have been widely applied to design robust static output feedback controllers. Two-step methods for designing output feedback controllers have been proposed [11–15]. The first step in these methods is to obtain a state feedback controller gain, while the second step is to obtain an output

∗

Corresponding author. E-mail addresses: [email protected] (X.-H. Chang), [email protected] (J.H. Park), [email protected] (J. Zhou). http://dx.doi.org/10.1016/j.sysconle.2015.08.007 0167-6911/© 2015 Elsevier B.V. All rights reserved.

feedback controller gain. The methods imply that the design is dependent on the state feedback controller gain obtained in the first step. In [16], an iterative LMI (ILMI) approach was presented to obtain the gains of static output feedback controllers. The approach is important for solving the BLMI problem and has been widely used to handle coupling constraints. Robust static output feedback H∞ controller design problems have been extensively discussed in recent years for linear systems with polytopic uncertainties in both the continuous-time and discrete-time contexts using LMI-based convex conditions. A linear parameter-dependent stabilization method for designing static output feedback H∞ controllers was proposed [17]. In another study [18], a descriptor approach with non-strict inequality (semidefinite matrix inequality) was used to study the output feedback H∞ control problem of continuous-time systems with polytopic uncertainties. Another method inserted an equality-constrained condition for the Lyapunov matrix and derived LMI-based conditions for solving the static output feedback controller design problem of linear systems [19], which result can be extended to design H∞ controllers for uncertain linear systems. Another method introduced a slack variable with sub-triangle structure and proposed LMI-based conditions for designing robust static output feedback H∞ controllers for linear systems with time-invariant uncertainties [20]. Sufficient conditions for designing static output feedback H∞ controllers were given based on the properties of the null space of output matrices and by introducing parameterindependent slack variables with a lower-triangular structure [21].

24

X.-H. Chang et al. / Systems & Control Letters 85 (2015) 23–32

Recently, new LMI-based conditions with a line search over a scalar variable for designing robust static output feedback H∞ controllers were proposed [22], where the uncertain output matrix of the considered system is allowed to be of non-full row rank. By assuming that the system input matrix is of full column rank and introducing an additional positive definite matrix, LMI-based design conditions were presented for output feedback H∞ control of linear discrete-time systems [23]. In order to obtain LMI-based conditions for designing static output feedback H∞ controllers, we have to impose some constraints on system matrices, that is, the above mentioned results are limited and cannot be applied to general uncertain systems. In addition, it should be mentioned that the problems of multi-objective output feedback control were considered in [24,25], where the controllers are designed via the technique of linearizing change of variables. However, the technique cannot lead to a solution of output feedback controller for systems with polytopic uncertainty. This is due to the fact that to linearize the matrix inequality, the introduced new variables will have to be vertex-dependent and involve the controller parameters to be sought, which implies that the required controller parameters cannot be computed from the introduced variables [4]. This paper addresses the robust static output feedback H∞ controller design problem for linear systems with polytopic uncertainties. The focus is on designing a static output feedback controller for a general uncertain system, which guarantees a prescribed H∞ performance level for the closed-loop system. A new method is proposed to derive sufficient conditions for robust static output feedback H∞ controller design, which can be described by a set of LMIs. The method can effectively solve the BMI problem in the literature for robust static output feedback H∞ controls, in which the aforementioned constraints imposed on the system matrices have been avoided. Besides wide applications, this new method is less conservative. Numerical examples are provided to illustrate the feasibility of the proposed design method. The remainder of this paper is organized as follows. Section 2 presents a new LMI-based condition for designing robust static output feedback H∞ controllers for uncertain continuous-time systems. Section 3 generalizes the results of Section 2 to the H∞ controller design of discrete-time systems. Section 4 demonstrates examples and compares the proposed results of this paper with previous approaches. Notations. In symmetric block matrices, the symbol (∗) represents a term that is induced by symmetry. I is the identity matrix with appropriate dimensions. diag {· · ·} denotes a diagonal matrix.   L2 [0, ∞) l2 [0, ∞) is the space of square-integrable (summable) vector functions over [0, ∞). diag {· · ·} indicates a block-diagonal matrix. The notation He(A) = A + AT will also be used, and the notation Fm×n indicates that F ∈ Rm×n .

C2 (α), and H (α) are constant matrices of appropriate dimensions and belong to the following uncertainty polytope [1]:

Ω =

=



A(α), B(α), E (α), C1 (α), D(α), F (α), C2 (α), H (α)

 µ µ     αi Ai , Bi , Ei , C1i , Di , Fi , C2i , Hi , αi = 1, αi ≥ 0 . (2) i=1

i= 1

For system (1), the following static output feedback controller is exploited: u(t ) = Ky(t ) = K C2 (α)x(t ) + H (α)w(t ) ,





(3)

where K is the controller gain matrix with appropriate dimensions to be determined later. Combining (1) and (3), the closed-loop system is obtained as x˙ (t ) = A(α)x(t ) + B(α)K C2 (α)x(t ) + H (α)w(t )





+ E (α)w(t ), z (t ) = C1 (α)x(t ) + D(α)K C2 (α)x(t ) + H (α)w(t )





(4)

+ F (α)w(t ). For the continuous-time system (1), the objective of H∞ control is to find an asymptotically stable H∞ controller in the form of (3) such that the following two conditions are satisfied [26]: (1) The closed-loop system (4) is asymptotically stable when w(t ) = 0. (2) The closed-loop system (4) has a prescribed level γ of H∞ noise attenuation, i.e. under the zero initial condition x(0) = ∞ ∞ 0, the inequality 0 z T (t )z (t )dt ≤ γ 2 0 w T (t )w(t )dt is satisfied for any nonzero w(t ) ∈ L2 [0, ∞). Remark 1. In order to obtain LMI-based conditions for designing static output feedback H∞ controllers, previous approaches [19– 22] have to impose some constraints on the system matrices. These works require the output matrix C2 (α) to be fixed; i.e., C2 (α) = C2 (C2 is of full row rank) and H (α) = 0. In [22], a new design approach was given for continuous-time linear polytopic systems, where the output matrix C2 (α) is not required to be of full row rank and fixed. However, the results were obtained based on another constraints of C1T (α)D(α) = 0, F (α) = 0, and H (α) = 0. In our study, the constraints on the system matrices have been avoided, so our results are more suitable for general systems. For convenience of comparison with previous studies, the conditions for static output feedback H∞ controller design are given as follows. Lemma 1 (See [19]). Consider the closed-loop system (4) with C2 (α) = C2 (C2 is of full row rank) and H (α) = 0. For a given scalar γ > 0, if there exist matrices Q > 0, Y , and N satisfying the following conditions

 2. Robust static output feedback H∞ control of continuoustime systems

Ai Q + QATi + Bi YC2 + C2T Y T BTi  EiT C1i Q + Di YC2

Consider a linear continuous-time system with time-invariant polytopic uncertainties described by state-space equations

< 0, i = 1, 2, . . . , µ, C2 Q = NC2 ,

∗ −γ 2 I Fi

 ∗ ∗ −I (5) (6)

then the system is asymptotically stable with the H∞ performance γ .

x˙ (t ) = A(α)x(t ) + B(α)u(t ) + E (α)w(t ), z (t ) = C1 (α)x(t ) + D(α)u(t ) + F (α)w(t ),



(1)

y(t ) = C2 (α)x(t ) + H (α)w(t ), where x(t ) ∈ Rn is the state variable, u(t ) ∈ Rm is the control input, w(t ) ∈ Rf is an arbitrary noise signal in L2 [0, ∞), z (t ) ∈ Rq is the controlled output variable, and y(t ) ∈ Rp is the measurement output. The matrices A(α), B(α), E (α), C1 (α), D(α), F (α),

Lemma 2 (See [22]). Consider the closed-loop system (4) with C1T (α) D(α) = 0, F (α) = 0, and H (α) = 0. Given a scalar γ > 0, for a known scalar parameter β , if there exist matrices V , U, and Qj > 0, j = 1, 2, . . . , µ satisfying the following LMIs:

Ωii < 0, i = 1, 2, . . . , µ, Ωij + Ωji < 0, i < j, i, j = 1, 2, . . . , µ,

(7) (8)

X.-H. Chang et al. / Systems & Control Letters 85 (2015) 23–32

where j

He(Ai Qj + Bi VFp×n )



∗ −I

j  Di VFp×n  T T  Ωij = β V Bi + C2i Qj − UFpj ×n  ET

∗ ∗ −β U − β U T

0 0 0

i

C1i Qj

∗ ∗ ∗ −γ 2 I

0 0

0

 ∗ ∗  ∗ , ∗ −I

with

j

Fp×n =

 T −1 C2 , C2 C2        

C2j ,

Lemma 3. For matrices T , P , U, and A with appropriate dimensions and a scalar β , the following statements are equivalent:



∗ < 0, −β U − β U T

T

β P + UA T

(ii) : T < 0,

(9)

T + A P + PA < 0. T

T

Proof of Lemma 3. Obviously, the matrix inequality (9) implies that the scalar β is nonzero, while pre-multiplying (9) by the full row rank matrix diag {I , β1 I } and post-multiplying by its transpose leads to



∗

T T

P +

1

+

1

− U − UT β β

UA

β  T A

1

1

−I β

T PT

=

∗ 0



0 1 U [A I β

−I]

I 0

(11)

I

and NQ =

Lemma 4 to (11) with Ψ =



A

T

∗

T

0

P

I ] and Q =

, respectively. Then, applying

and W = β1 U gives (10).



Moreover, by using the parameter-dependent Lyapunov function, we can obtain the following lemma easily. Lemma 5. Consider the closed-loop system (4) and given a scalar γ > 0, if there exist matrices Q (α) > 0 and K satisfying the following matrix inequality:





 

E (α) + B(α)KH (α)  C1 (α) + D(α)KC2 (α) Q (α)

A(α) + B(α)KC2 (α) Q (α)

He







< 0,

T



∗

∗ 2

−γ I F (α) + D(α)KH (α)

 ∗ −I

(14) (15)

j

with Ξij given in Box I, where Fp×n are the same as Lemma 2 and

 I,   [ I p×p 0p×(q−p) ],  =  I   ,  0 q×q (p−q)×q

p = q, p < q,

(16)

p > q,

then the system is asymptotically stable with the H∞ performance γ . Furthermore, the controller gain matrix in (3) can be given by K = VU −1 .

(17)

Proof. From (2), (14), and (15), we have µ 

αi2 Ξii +

=

µ  µ 

αi αj (Ξij + Ξji )

i=1 i
µ  µ 

αi αj Ξij

i=1 j=1

U T [0 I ] < 0.

− I ] as NR =



(13)

Υ1 ∗ ∗ Υ2 −γ 2 I ∗    = Υ3 F (α) + D(α)VH (α) −I + He ρ D(α)V Fp×q



Consider the orthogonal   complements   of R = [0

[A

−γ 2 I F (α) + D(α)KH (α)

Ξii < 0, i = 1, 2, . . . , µ, Ξij + Ξji < 0, i < j, i, j = 1, 2, . . . , µ,

  +

∗

Theorem 1. Consider the closed-loop system (4). Given a scalar γ > 0, for known scalar parameters β and ρ , if there exist matrices V , U, and Qj > 0, j = 1, 2, . . . , µ satisfying the following LMIs:

i=1



∗

With Lemmas 3 and 5, we have the following theorem, which provides a new LMI-based condition for designing the static output feedback H∞ controller (3).

Fp×q

Lemma 4 (See [27]). Given matrices Ψ , R and Q with appropriate dimensions, there exists a matrix W such that the following inequality Ψ + RT W T Q + Q T WR < 0 holds if and only if the inequalities NR T Ψ NR < 0 and NQ T Ψ NQ < 0 are satisfied, where NR and NQ denote the orthogonal complement of R and Q , respectively; i.e., RNR = 0 and Q NQ = 0.





   E (α) + B(α)KH (α)T P (α) C1 (α) + D(α)KC2 (α)

(10)

The following lemma is needed for the proof of Lemma 3.





He P (α) A(α) + B(α)KC2 (α)

Denoting Q (α) = P (α), pre- and post-multiplying both sides of the inequality (13) with diag {Q (α), I , I } and its transpose, one can obtain (12). 

In the following, we will develop a new condition for a robust static output feedback H∞ controller design which improves the results in Lemmas 1 and 2. The following technical lemma will be employed to establish our main results.

(i) :





−1

then the system is asymptotically stable with the H∞ performance γ .



Proof. For the system (4), a well-known H∞ performance analysis condition (Bounded Real Lemma [28]) is that there exist matrices P (α) > 0 and K such that

< 0.

C2 (α) is fixed, i.e., C2 (α) = C2 and C2 is of full row rank, C2 (α) is fixed, i.e., C2 (α) = C2 and C2 is of non-full row rank, C2 (α) is non-fixed,

C2 ,

25



 ∗ −I (12)

then the system is asymptotically stable with the H∞ performance γ .

Υ4

H (α) − UH (α)

β V T DT (α) − ρ U Fp×q

∗ ∗ ∗ −β U − β U T

< 0,

     (18)

 where Υ1 = He A(α)Q (α) + B(α)V Fp×n (α) , Υ2 = E T (α) + H T (α) V T BT (α), Υ3 = C1 (α)Q (α) + ρ FqT×p V T BT (α) + D(α)V Fp×n (α), µ Υ4 = β V T BT (α) + C2 (α)Q (α) − U Fp×n (α), Q (α) = j=1 αj Qj , µ j and Fp×n (α) = j=1 αj Fp×n . 

If the inequalities (14) and (15) are satisfied, it implies that the matrix U is nonsingular. Applying Lemma 3 with A = U −1 C2 (α)Q (α) − U Fp×n (α) H (α) − UH (α) −ρ U Fp×q , and

 Υ1 ∗ −γ 2 I T = Υ2 Υ3 F (α) + D(α)VH (α)   B(α)V 0 P = , D(α)V

 ∗   ∗  , −I + He ρ D(α)V Fp×q

26

X.-H. Chang et al. / Systems & Control Letters 85 (2015) 23–32



j



He Ai Qj + Bi V Fp×n



 EiT + HjT V T BTi  Ξij =  C1i Qj + ρ FqT×p V T BTi + Di V Fpj×n β V T BTi + C2i Qj − U Fpj×n

∗ −γ 2 I Fi + Di VHj Hi − UHj

∗ ∗ −I + He(ρ Di V Fp×q ) β V T DTi − ρ U Fp×q

 ∗  ∗    ∗ −β U − β U T

Box I.

the inequality in (18) leads to     He A(α)Q (α) + B(α)V Fp×n (α) ∗ ∗    E T (α) + H T (α)V T BT (α) −γ 2 I ∗ C1 (α)Q (α) + ρ FqT×p V T BT (α) + D(α)V Fp×n (α) F (α) + D(α)VH (α) Θ   B(α)V   + He 0  D(α)V    −1 ×U C2 (α)Q (α) − U Fp×n (α) H (α) − UH (α) −ρ U Fp×q  < 0,

(19)

where Θ = −I + He ρ D(α)V Fp×q . Obviously, (19) can be rewritten as   A(α)Q (α) + Q (α)AT (α) ∗ ∗ T 2  E (α) −γ I ∗ C1 (α)Q (α) F (α) −I    B(α)V   + He  0  U −1 U Fp×n (α) UH (α) ρ U Fp×q  D(α)V   B(α)V + He  0  D(α)V





× U −1 C2 (α)Q (α) − U Fp×n (α) H (α) − UH (α)

Theorem 2. If the condition in Lemma 1 holds, then the condition in Theorem 1 also holds. Proof. Because Q > 0 and C2 is of full row rank, from (6), we have C2 QC2T + C2 QC2T = NC2 C2T + C2 C2T N T > 0.



Ai Q + QATi + Bi YC2 + C2T Y T BTi

 

EiT

∗ −γ 2 I

+



YC2 C2T

T

BTi

0

Fi



YC2 C2T

T

DTi

 ∗  ∗ −I

T

  T  T  × β NC2 C2T + C2 C2T N T YC2 C2T BTi 0 YC2 C2T DTi 

Ai Q + QATi + Bi YC2 + C2T Y T BTi EiT

 =

∗ −γ 2 I

C1i Q + Di YC2 Fi   T  T T + β YC2 C2T BTi 0 β YC2 C2T DTi



− ρ U Fp×q 



A(α)Q (α) + Q (α)AT (α) = E T (α) C1 (α)Q (α)



∗ −γ 2 I F (α)

 ∗ ∗ −I

B(α)V   + He  0  U −1 C2 (α)Q (α) H (α) 0  < 0. D(α)V





(21)

Based on (19), if the LMIs in (5) are satisfied, then there exists a sufficiently small β > 0 such that the following matrix inequalities hold:

C1i Q + Di YC2

 

this conclusion, a constraint on system matrices is required (see the proof of Theorem 14 in that study for details). However, for the same relaxed conclusion with respect to Lemma 1, our results have avoided this constraint.

  T 1 NC2 C2T + C2 C2T N T β YC2 C2T BTi

×

β



Ai Q + QATi + Bi YC2 + C2T Y T BTi

 =



(20)

Considering (17), Eq. (20) implies that the inequality condition (12) holds. Thus, the proof is completed.  Remark 2. Theorem 1 presents a new condition for designing static output feedback H∞ controllers for uncertain continuoustime systems. Obviously, when β and ρ are set to be known parameters, (14) and (15) become LMIs which can be solved by the Matlab LMI toolbox [29]. It should be mentioned that the introduction of the scalar parameters β and ρ is not necessary to derive our results, but the selection of β and ρ provides extra free dimensions in the solution space for the design condition. In addition, in the design procedure, we can search for optimal values of the scalar parameters to minimize the H∞ performance bound. One way to address the search issue is to first solve the feasibility problem of the LMIs (14) and (15) and to obtain a set of initial scalar parameters. Then, applying a numerical optimization algorithm, such as the program fminsearch in the optimization toolbox of Matlab, a locally convergent solution to the problem is obtained [17,30]. A previous study presented a design condition which is more relaxed than the one in Lemma 1 [22]. However, in order to obtain

EiT

∗ −γ 2 I

 ∗  ∗ −I



0 β YC2 C2T

T

DTi



 ∗  ∗ −I

C1i Q + Di YC2 Fi   T  T T + β YC2 C2T BTi + C2 Q − NC2 0 β YC2 C2T DTi

×

 1 NC2 C2T + C2 C2T N T

β    T  T × β YC2 C2T BTi + C2 Q − NC2 0 β YC2 C2T DTi 

  −1  −T T T T C2 + C2T C2 C2T C2 C2T Y T BTi ∗ ∗ Ai Q + QAi + Bi YC2 C2 C2 C2   = EiT −γ 2 I ∗   −1   T T C1i Q + Di YC2 C2 C2 C2 C2 Fi −I    −1  T T T + β YC2 C2T BTi + C2 Q − NC2 C2T C2 C2T C2 0 β YC2 C2T DTi  1 NC2 C2T + C2 C2T N T β   T  −1  T  × β YC2 C2T BTi + C2 Q − NC2 C2T C2 C2T C2 0 β YC2 C2T DTi ×

< 0,

i = 1, 2, . . . , µ.

By defining V = YC2 C2T , U = NC2 C2T , ρ = 0, and Qj = Q , j = 1, 2, . . . , µ, and by applying the Schur complement to the above matrix inequalities, (14) and (15) in Theorem 1 with Hi = 0, i = 1, 2, . . . , µ can be obtained. 

X.-H. Chang et al. / Systems & Control Letters 85 (2015) 23–32

In addition, the previous study [22] claimed that C1T (α)D(α) = 0, F (α) = 0, and H (α) = 0 for the system matrices and the corresponding design is proposed in Lemma 2. The following theorem shows that the proposed design condition in Theorem 1 is more relaxed than that in Lemma 2 by considering this as a system matrix constraint. Theorem 3. If the condition in Lemma 2 holds, then the condition in Theorem 1 also holds. Proof. From (8), for i < j, i, j = 1, 2, . . . , µ, we have Eq. (22) given in Box II, i.e., 

j

He(Ai Qj + Bi VFp×n + Aj Qi + Bj VFpi ×n ) EiT + EjT

          

∗ −2γ 2 I

∗ ∗

∗ ∗

0

−2β U − 2β U T

∗

 β V T BTi + C2i Qj − UFpj ×n +β V

T

BTj

+ C2j Qi −

UFpi ×n



j

Di VFp×n + Dj VFpi ×n

0

C1i Qj + C1j Qi

0

β Di V + β Dj V −2I 0

0

 ∗ ∗      ∗      ∗ 

  

j

He(Ai Qj + Bi VFp×n + Aj Qi + Bj VFpi ×n )

βV

T

BTi

+ C2i Qj −

EiT j UFp×n

1

+ EjT + β V T BTj + C2j Qi − UFpi ×n

∗ −2γ 2 I 0

i,

j =

 ∗  ∗  T −2β U − 2β U

T

(24)

T Because C1T (α)D(α) = 0, then C1i Dj = 0, which implies that

diag {(C1i Qj + C1j Qi )T (C1i Qj + C1j Qi ), 0, 0}

× [C1i Qj + C1j Qi +

+

where K is the controller gain matrix with appropriate dimensions to be determined later. By substituting (27) into (26), the closed-loop system can be described as x(k + 1) = A(α)x(k) + B(α)K C2 (α)x(k)



(28)

(1) The closed-loop system (28) is asymptotically stable when w(k) = 0. (2) The closed-loop system (28) has a prescribed level γ of H∞ noise i.e. under initial condition x(0) = ∞attenuation, ∞the zero T 2 T 0, k=0 z (k)z (k) < γ k=0 w (k)w(k) is satisfied for any nonzero w(k) ∈ l2 [0, ∞). First, an H∞ performance analysis conclusion is described as follows.

−GT (α)P −1 (α)G(α)



∗ ∗ 2  0 −γ I ∗   A(α)G(α) + B(α)KC2 (α)G(α) E (α) + B(α)KH (α) −P (α) C1 (α)G(α) + D(α)KC2 (α)G(α) F (α) + D(α)KH (α) 0

 ∗ ∗  ∗ −I (29)

then the system is asymptotically stable with the H∞ performance γ . Proof. Based on the Bounded Real Lemma [28], the H∞ performance γ of the closed-loop system (28) can be guaranteed by

× [Di VFpj ×n + Dj VFpi ×n 0 β Di V + β Dj V ]   (C1i Qj + C1j Qi + Di VFpj ×n + Dj VFpi ×n )T 1  =  0 2 T (β Di V + β Dj V ) Dj VFpi ×n

(27)

< 0,

  (Di VFpj ×n + Dj VFpi ×n )T 1  +  0 2 (β Di V + β Dj V )T

j Di VFp×n



Lemma 6. Consider the closed-loop system (28). For a scalar γ > 0, if there exist matrices P (α), G(α), and K satisfying the following matrix inequality:

× [Di VFpj ×n + Dj VFpi ×n 0 β Di V + β Dj V ] < 0.

2



For the discrete-time systems (26), the objective of H∞ control is to find an asymptotically stable H∞ controller in the form of (27) such that the following two conditions are satisfied [31]:

+ diag {(C1i Qj + C1j Qi ) (C1i Qj + C1j Qi ), 0, 0} 2   (Di VFpj ×n + Dj VFpi ×n )T 1  +  0  2 T (β Di V + β Dj V )

1

u(k) = Ky(k) = K C2 (α)x(k) + H (α)w(k) ,

+ F (α)w(k).

(23)

Using the Schur complement on (23), for i < j, 1, 2, . . . , µ, it gives

is the controlled output variable, and y(k) ∈ Rp is the measurement output. As with the controller (3), for the system (26), we consider the following static output feedback controller:

 + H (α)w(k) + E (α)w(k),   z (k) = C1 (α)x(k) + D(α)K C2 (α)x(k) + H (α)w(k)

−2I

< 0.

27

0 β Di V + β Dj V ].



0 A(α) + B(α)KC2 (α) C1 (α) + D(α)KC2 (α)

  (25)

By replacing the corresponding part in (24) by (25) and applying the Schur complement to the latter, (15) in Theorem 1 with Fi = 0, Hi = 0, i = 1, 2, . . . , µ, and ρ = 0 can be obtained. From (7), the same analysis process can be applied to (14).  3. Robust static output feedback H∞ control of discrete-time systems Consider a linear discrete-time system with time-invariant polytopic uncertainties described by state-space equations x(k + 1) = A(α)x(k) + B(α)u(k) + E (α)w(k), z (k) = C1 (α)x(k) + D(α)u(k) + F (α)w(k),

−P −1 (α)

(26)

y(k) = C2 (α)x(k) + H (α)w(k), where x(k) ∈ Rn is the state variable, u(k) ∈ Rm is the control input, w(k) ∈ Rf is an arbitrary noise signal in l2 [0, ∞), z (k) ∈ Rq

< 0.

∗ −γ 2 I E (α) + B(α)KH (α) F (α) + D(α)KH (α)

∗ ∗ −P (α) 0

 ∗ ∗  ∗ −I (30)

If (29) is satisfied, it implies that the matrix G(α) is nonsingular. Pre- and post-multiplying (29) by diag {G−T (α), I , I , I } and its transpose, respectively, leads to (30).  In another study [21], an LMI approach to design robust static output feedback H∞ controllers was given. The proposed approach is applicable for linear systems with time-varying polytopic uncertainties, which may simultaneously emerge in system output and input matrices (non-fixed). However, it should be noted that the results of that study cannot be used directly for system output or input matrices with time-invariant polytopic uncertainties. For the time-invariant polytopic uncertainty case, we can obtain the following lemma based on this previous approach [21], except that the system output matrix C2 (α) needs to be fixed (C21 = C22 = · · · = C2µ = C2 ). It is also assumed that the matrix C2 is of full row rank. Consider an invertible matrix T satisfying C2 T = [I

0].

(31)

28

X.-H. Chang et al. / Systems & Control Letters 85 (2015) 23–32

j

He(Ai Qj + Bi VFp×n + Aj Qi + Bj VFpi ×n )



∗ −2I

∗ ∗

∗ ∗

β V T DTi + β V T DTj

−2β U − 2β U T

∗

0 0

0 0

−2γ 2 I

j

Di VFp×n + Dj VFpi ×n

    Ωij + Ωji =     



β V T BTi + C2i Qj − UFpj ×n

+β V T BTj + C2j Qi − UFpi ×n



EiT + EjT C1i Qj + C1j Qi

 ∗ ∗     ∗  <0   ∗  −2I

0

(22)

Box II.

Lemma 7 (See [21]). Consider the closed-loop system (28) with C2 (α) = C2 (C2 is of full low rank) and H (α) = 0. For a given scalar γ > 0, if there exist matrices L1 , Qj , and Sj , j = 1, 2, . . . , µ satisfying the following LMIs:

∆ii < 0, i = 1, 2, . . . , µ, ∆ij + ∆ji < 0, i < j, i, j = 1, 2, . . . , µ,

where

(32)

 −Sj − SjT + T −1 Qj T −T  0 Γij =   Ai TSj + Bi [L1 0] C1i TSj + Di [L1 0]

(33)

with T described in (31) and

where



 −TSj − SjT T T + Qj  0 ∆ij =   Ai TSj + Bi [L1 0] C1i TSj + Di [L1 0]

∗ −γ 2 I Ei Fi

Sj =



∗ ∗ −Qj

∗ ∗ , ∗ −I

0



S11 j S21



0 , j S22

j = 1, 2, . . . , µ,

(34)

Remark 3. For the design condition presented in Lemma 7, in addition to the requirement that the system output matrix C2 (α) be fixed, there is another limitation. Consider the case of C2 = [0 C ], where C is assumed to be a nonsingular   matrix. Then, X C −1

Y

0

, where X and Y

are arbitrary matrices with appropriate dimensions. Combining T and (34) leads to

 TSj =

X C −1

 =

Y 0



S11 j S21

j

XS11 + YS21 C −1 S11

0 j S22 j





YS22 , 0

0

 ∗ ∗ , ∗ −I

j = 1, 2, . . . , µ,

(38)

Proof. From Lemma 6, it is clear that the H∞ performance γ of the closed-loop system (28) can be ensured by (29). Choosing G(α) = TS (α) and noting that

and given T described in (31) and nonsingular S11 , then the system is asymptotically stable with the H∞ performance γ .

the construction of C2 implies that T =

0 , j S22

Ei Fi

∗ ∗ −Qj

then the system is asymptotically stable with the H∞ performance γ .

with Sj =



S11 j S21

∗ −γ 2 I

  T  −1  −1 − S (α) − T −1 Q (α)T −T T Q (α)T −T    × S (α) − T −1 Q (α)T −T ≤ 0, Q (α) > 0,

(39)

implies that

−S T (α)T T Q −1 (α)TS (α) ≤ −S (α) − S T (α) + T −1 Q (α)T −T .

(40)

Then, the inequality (29) is satisfied if the following condition holds:



−S (α) − S T (α) + T −1 Q (α)T −T

 

0 A(α)TS (α) + B(α)KC2 TS (α) C1 (α)TS (α) + D(α)KC2 TS (α)

∗ −γ 2 I E (α) F (α)

∗ ∗ −Q (α) 0

 ∗ ∗  ∗ −I

< 0. j = 1, 2, . . . , µ.

(35)

cannot be achieved. In other words, for (32) and (33), there are no solutions for such a C2 . In fact, in the design condition in Lemma 7, there is a strict system matrix constraint that the matrix C2 TC2T is nonsingular. The details can be found in the proof of Theorem 5. The limitation of Lemma 7 presented in Remark 3 can be avoided using the flexible LMI technique. In the following, we develop another design condition which improves the result in Lemma 7. Lemma 8. Consider the closed-loop system (28) with C2 (α) = C2 (C2 is of full low rank) and H (α) = 0. For a given scalar γ > 0, if there exist matrices L1 , Qj , and Sj , j = 1, 2, . . . , µ satisfying the following LMIs:

Γii < 0, i = 1, 2, . . . , µ, Γij + Γji < 0, i < j, i, j = 1, 2, . . . , µ,

Let Q (α) = S (α) =

From (35), we can see that −TSj − SjT T T + Qj < 0with Qj > 0

(36) (37)

(41)

µ

j =1

µ

αj Qj , Qj > 0, j = 1, 2, . . . , µ and

αj Sj . The inequality (41) can be rewritten as   −Sj − SjT + T −1 Qj T −T ∗ ∗ ∗ µ  µ   0 −γ 2 I ∗ ∗  αi αj   Ai TSj + Bi KC2 TSj  E − Q ∗ i j i =1 j =1 C1i TSj + Di KC2 TSj Fi 0 −I j=1

< 0.

(42)

By considering (38) and C2 T = [I

 C2 TSj = [I

0]

S11 j S21

= [S11 0],

0 j S22

0], we have



j = 1, 2, . . . , µ.

(43)

By Combining (42) and (43) and defining L1 = KS11 , the LMIs (36) and (37) can be obtained.  In a previous study [23], the system input matrix is assumed to be of full column rank, and LMI-based design conditions for output feedback H∞ control of linear discrete-time systems were

X.-H. Chang et al. / Systems & Control Letters 85 (2015) 23–32

presented by introducing an additional positive definite matrix. This approach can be applied to robust output feedback control of linear systems with polytopic uncertainties, and it allows us to obtain the following design results directly.

Proof. First, the inequality conditions (32) and (33) imply that TSj + SjT T T > 0, j = 1, 2, . . . , µ. Since C2 is of full row rank, we have C2 TSj + SjT T T C2T



Lemma 9. Consider the closed-loop system (28) with C2 (α) = C2 (C2 is of full low rank). Given a scalar γ > 0, for a known scalar parameter ν , if there exist matrices V , U, Pj , J1j , J2j , J3j , and Gj , j = 1, 2, . . . , µ satisfying the following LMIs:

Θii < 0, i = 1, 2, . . . , µ, Θij + Θji < 0, i < j, i, j = 1, 2, . . . , µ,

(44) (45)

where Θij given in Box III, with T described in (31), then the system is asymptotically stable with the H∞ performance γ .



= C2 TSj C2T + C2 SjT T T C2T T = S11 [I 0]C2T + C2 [I 0]T S11 T = S11 C2 TC2T + C2 T T C2T S11 > 0,

 C2 TSj = [I

(46)

j

Fp×n =

−1  C2 TC2T [I 0],           [ I 0]T T ,        C2 ,     C2j ,

C2 (α) is fixed, i.e., C2 (α) = C2 , C2 is of full row rank and C2 TC2T is nonsingular , C2 T = [I 0], C2 (α) is fixed, i.e., C2 (α) = C2 , C2 is of full row rank and C2 TC2T is singular , C2 T = [I 0], C2 (α) is fixed, i.e., C2 (α) = C2 and C2 is of non-full row rank, C2 (α) is non-fixed,

then the system is asymptotically stable with the H∞ performance γ . Furthermore, the controller gain matrix in (27) can be given by (17). Proof. From (2) and (40), we have 

−G(α) − GT (α) + P (α)

∗ ∗  0 −γ 2 I ∗   −P (α)  A(α)G(α) + B(α)V Fp×n (α) E (α) + B(α)VH (α)  C1 (α)G(α) + D(α)V Fp×n (α) F (α) + D(α)VH (α) ρ FqT×p V T BT (α) C2 (α)G(α) − U Fp×n (α) H (α) − UH (α) β V T BT (α) < 0,

∗ ∗ ∗ Ω1 Ω2

 ∗ ∗   ∗  ∗ Ω3

(48)

where Ω1 = −I + He ρ D(α)V Fp×q , Ω2 = β V D (α) − ρ U Fp×q , µ µ Ω3 = −β U − β U T , P (α) = j=1 αj Pj , G(α) = j=1 αj Gj , and µ j Fp×n (α) = j=1 αj Fp×n . Then, following the proof of Theorem 1, if the inequality (48) is satisfied, (29) holds. 





T

T

In Theorem 4, a significant method is proposed to design static output feedback H∞ controllers for linear uncertain discrete-time systems. The proposed design method overcomes the deficiencies of previous approaches, and it is able to handle the case in which the system output matrices are of non-full row rank and nonfixed. In addition, the proposed method can give less conservative designs than the previous LMI methods. For clarity, the same system matrix C2 as in previous studies is considered [21,23]. Then, we can obtain the following three theorems, in which the relaxation advantage of the proposed method can be shown from a theoretical point of view. Theorem 5. If the condition given in Lemma 7 holds, the condition in Theorem 4 also holds.

0]

S11 j S21

0 j S22



(50)

i.e., C2 TSj − S11 C2 TC2T C2 TC2T



−1

[I 0] = 0,

j = 1, 2, . . . , µ.

(47)

with Πij given in Box IV, where Fp×q is the same as in (16) and

(49)

= [S11 0] = S11 [I 0] = S11 C2 TC2T  −1 × C2 TC2T [I 0], j = 1, 2, . . . , µ,

Theorem 4. Consider the closed-loop system (28). Given a scalar

Πii < 0, i = 1, 2, . . . , µ, Πij + Πji < 0, i < j, i, j = 1, 2, . . . , µ,

j = 1, 2, . . . , µ.

The inequality in (49) implies that the matrix C2 TC2T is also nonsingular. Then, we have

In the following, we give our main results.

γ > 0, for known scalar parameters β and ρ , if there exist matrices V , U, Pj , and Gj , j = 1, 2, . . . , µ satisfying the following LMIs:

29

(51)

Similar to the proof of Theorem 2, the inequalities (38) and (39) in Theorem 4 can respectively be obtained from (25) and (26) by defining Pj = Qj , Gj = TSj , j = 1, 2, . . . , µ, V = L1 C2 TC2T , U = S11 C2 TC2T , Hi = 0, i = 1, 2, . . . , µ, ρ = 0 and applying the Schur complement.  Theorem 6. If the condition given in Lemma 8 holds, the condition in Theorem 4 also holds. Proof. Pre- and post-multiplying (30) and (31) with diag {T , I , I , I } and its transpose, respectively, and defining Pj = Qj , Gj = TSj T T , j = 1, 2, . . . , µ, V = L1 , U = S11 , and Hi = 0, i = 1, 2, . . . , µ, ρ = 0, the proof directly follows the proof of Theorem 2.  Theorem 7. If the condition given in Lemma 9 holds, the condition in Theorem 4 also holds. Proof. Eq. (44) can be rewritten as Eq. (52), given in Box V. Using the matrix inequality congruence property [28,32] on (52) with



I 0  0 0 0

0 I 0 0 0

0 0 I 0 0

0 0 0 I 0

0 0 0 0 I

I 0 0 0 0



0 I  0 , 0 0

for i = 1, 2, . . . , µ, it yields  −Gi − GTi + Pi ∗  0 −γ 2 I   Ai Gi + Bi V [I 0]T T Ei + Bi VHi   C G + D V [I 0]T T F i i + Di VHi  1i i  ν C2 Gi − U [I 0]T T ν(Hi − UHi )

< 0.

∗ ∗ −Pi 0

∗ ∗ ∗ −I

∗ ∗ ∗ ∗

V T BTi

V T DTi

−ν U − ν U T

       (53)

By defining β = ν , one can see that the inequality (53) implies that the inequalities in (46) with ρ = 0 hold.  1

Remark 4. The major contributions of this paper can be summarized as follows: (1) This paper studies the static output feedback control problem of both continuous-time and discrete-time systems with polytopic uncertainties. Novel LMI conditions with a line search

30

X.-H. Chang et al. / Systems & Control Letters 85 (2015) 23–32

 −G − GT + P + J j j 1j j J2j    Ai Gj + Bi V [I 0]T T  C1i Gj + Di V [I 0]T T Θij =  0    0  

∗ 2

−γ I + J3j Ei + Bi VHj Fi + Di VHj

0

∗ ∗ −P j

∗ ∗ ∗ −I

0

0 V T BTi

V T DTi

0

0

0

0

0

0

∗ ∗ ∗ ∗ −ν U − ν U T 

C2 Gj − U [I

∗ ∗ ∗ ∗ ∗

0]T T

T

(Hi − UHj )T

∗ ∗ ∗ ∗ ∗

J1j

−

ν2

−

ν2

J2j



        ∗    J3j − 2 ν

Box III.



−Gj − GTj + Pj

 0  j  Πij =  Ai Gj + Bi V Fp×n  j C1i Gj + Di V Fp×n j C2i Gj − U Fp×n

∗ −γ 2 I Ei + Bi VHj Fi + Di VHj Hi − UHj

∗ ∗ −Pj



ρ FqT×p V T BTi

∗ ∗ ∗   −I + He ρ Di V Fp×q

∗ ∗ ∗ ∗

β V T BTi

β V T DTi − ρ U Fp×q

−β U − β U T

     

Box IV.

For i = 1, 2, . . . , µ,



−Gi − GTi + Pi + J1i

 J2i   Ai Gi + Bi V [I 0]T T  C1i Gi + Di V [I 0]T T   0   0

∗ −γ 2 I + J3i Ei + Bi VHi Fi + Di VHi

0

∗ ∗ −P i

∗ ∗ ∗ −I

0

0 V T BTi

V T DTi

0 0

0 0

0 0

∗ ∗ ∗ ∗ −ν U − ν U T  T ν C2 Gi − U [I 0]T T ν(Hi − UHi )T

 ∗ ∗   ∗   ∗  <0 ∗   ∗  −J3i

∗ ∗ ∗ ∗ ∗ −J1i −J2i

(52)

Box V.

over scalar parameters for designing robust static output feedback controllers are proposed, where constraints imposed on system matrices have been overcome. Therefore, the proposed design can cover a wider range of the feasible region than some existing results. (2) By rigorous theoretical proof, we verify that the design results in [19,21–23] are special cases of the proposed conditions, namely, the new conditions can reduce to those in [19,21–23] under special cases. It is proved that the proposed conditions can give less conservative design. 4. Numerical examples In this section, four numerical examples are presented to illustrate the effectiveness of the proposed method. 4.1. Example 1 The first numerical example is introduced to show that the proposed robust static output feedback H∞ controller design method is more general than previous ones. Consider the continuous-time system (1) with the following system matrices [13]:

−2.98 −0.99 A1 =  0 

0.39

−2.98 −0.99 A2 =  0 

0.39

−0.57 −0.21 0

−5.5550 2.43 −0.21 0 −5.555

0 0.035 0 0 0 0.035 0 0

 −0.034 −0.0011 , 1

−1.89  −0.034 −0.0011 , 1 −1.89

0.032  0  , B1 = B2 =  0  1.6

 





0

0 E1 = E2 =   , 0 1

C11 = C12 = [1 0 0 2], D1 = D2 = 1,

F1 = F2 = 0,

 C21 = C22 =

0 0

0 0

1 0



0 , 1

 H1 = H2 =

0.5 . −1



For this example, it is noted that the results in Lemma 1 [19] and Lemma 2 [22] fail to design static output feedback H∞ controllers. However, the proposed condition in Theorem 1 is feasible to find the static output feedback controller (3) with the H∞ performance γ for this example. Using a Matlab toolbox [29] to solve LMIs (14) and (15) with the optimized values of β = 0.58 and ρ = 0.93, we obtain the minimum H∞ performance γmin = 0.6581. Combining this with (17), an H∞ controller gain matrix in (3) is given as K = [−2.7375

− 0.8618].

(54)

The open-loop eigenvalues of A1 are −3.2086, −1.8738, and 0.0012 ± 0.3125j, while those of A2 are −2.0127, −0.1051, and −1.4811 ± 0.6239j, respectively. Using this as an output feedback controller, the eigenvalues of A1 + B1 KC21 are −0.1013, −3.2093, and −1.5741 ± 1.2890j, while the eigenvalues of A2 + B2 KC22 are −1.8009 ± 0.7337j and −1.4285 ± 1.3207j. With w(t ) = .5 sin(t )/(t 1 + 1) and initial conditions of zero, the ratios of z (t ) and

γ (t ) =

t 0

z T (i)z (i)di/

t 0

w T (i)w(i)di with the controller gain

(54) are shown in Figs. 1 and 2, respectively. Fig. 2 shows that the ratio tends toward a constant value 0.6268, which is less than the prescribed value of 0.6581.

X.-H. Chang et al. / Systems & Control Letters 85 (2015) 23–32

31

contrast, by using the optimized values of β = 0.11 and ρ = 0.12, Eqs. (14) and (15) yield a minimum of 6.6836, which is clearly much better. This shows that the condition proposed in Theorem 1 is less conservative than that given in Lemma 2 [22]. 4.3. Example 3 The advantage of the design result in Theorem 4 is illustrated by this numerical example. Consider the discrete-time system (28), which belongs to the 2-polytopic convex polyhedron with

Fig. 1. Response of z (t ) with gain (54).

Fig. 2. The ratio of



t 0

z T (i)z (i)di/

t 0

wT (i)w(i)di with gain (54).

4.2. Example 2 In this example, a comparison between Theorem 1 in this paper and Lemma 2 [22] is given by considering a constraint on system matrices. This numerical example is a continuous-time system used in a previous study [22], where the system output matrix C2 (α) is chosen to be non-fixed. It is in the form of (1) with

 −0.9896 A1 = 0.2648

17.41 −0.8512 0

0

 −1.702 A2 = 0.2201

50.72 −1.418 0

0

  −97.78 0 3

B1 =

,

96.15 −11.39 , −30



263.5 −31.99 , −30 0 3

B2 =

This example illustrates that the proposed design method outperforms other methods for robust static output feedback H∞ control. Consider a modified version of the pitch control [33] of F4E described by

,

a11 a21 x˙ (t ) =  0 0



0 1 , 1

  E1 = E2 =

C11 = C12 = 1, F1 = 0,

D1 = 0,

D2 = 0,



F2 = 0,

 C21 =

Lemma 7 [21], Lemma 9 [23], Lemma 8, and Theorem 4 in this paper are applicable for designing the robust static output feedback H∞ controller (27) for this example. By applying those design methods, the minimum values of the H∞ performance γ are given in Table 1. The computation results show that the proposed method in this paper provides a better alternative design for this example. 4.4. Example 4



  −85.09

  −0.2228 1.2665 0.0210 −0.1407 −0.1431 −0.3077 0.4837 −0.1988 0.8809 −0.6714   0.0583  , A1 = −0.5078 0.0185 −0.5140 −0.3025 −0.2847 0.3645 0.0138 0.3732 0.1503  0.3795 0.8853 0.3176 1.3000 −0.6100   −0.0463 1.0000 0.4761 0.0006 −0.1967  0.1390 0.2433 0.5270 −0.2392 −0.4557   1.0720 −0.3483 0.0574 0.2562  , A2 =  0.2463  0.4315 0.0915 −0.1487 −0.0171 −0.3573 0.2005 −0.2659 −1.4680 0.5854 1.0000     −0.4 0.62  −0.46   −1.06      B1 =  −0.1  , B2 =  −0.59  ,  0.23  0.0852 −0.6543 −0.1     −0.05 −0.26  −0.85   0.688      E1 =  −0.82  , E2 = −1.1511 , −0.0623  −1.04  1.12 −0.14 C11 = [−0.088 − 0.312 0.1733 1.167 0.55], C12 = [−0.89 − 0.0812 0.73 − 0.43 − 0.473], D1 = 0.6, D2 = −1.14, F1 = 0.84, F2 = 0.2390, C21 = C22 = [0 0 0 0 2], H1 = H2 = 0.

1 0

0 1



0 , 0

 C22 =

1 0

0 1



0 , 1

H1 = H2 = 0. For this example, Lemma 2 [22] and Theorem 1 in this paper are applicable for designing the static output feedback H∞ controller in the form of (3). The LMIs (7) and (8) with optimized β = 0.11 yield a minimum value of 7.0362 for the H∞ performance γ . In

a12 a22 0 0

a13 a23 −30 0

b1 0 0   0  x(t ) +  u(t ) 30  0  4 4 −10 10







1 0 + 0 0

0 1 0 0

0 0 w(t ), 1 0

(55)



1 0 0

0 1 0

0 0 1

0 0 0 x(t ) + 0 u(t ), 0 1



1 y(t ) = 0

0 1

0 0

0 x(t ). 0

z (t ) =







 

32

X.-H. Chang et al. / Systems & Control Letters 85 (2015) 23–32 Table 1 Comparison of γmin obtained by different methods for Example 3. Methods

Theorem 3 (βopt , ρopt )

Lemma 9 (νopt ) [23]

Lemma 8

Lemma 7 [21]

γmin

8.4041 (0.13, 0.09)

8.9882 (8.01)

9.4059

Infeasible

Table 2 The parameters of the four operating points. Operating point

1

2

3

4

a11 a12 a13 a21 a22 a23 b1

−0.9896

−0.6607

−1.702

−0.5162

17.41 96.15 0.2648 −0.8512 −11.39 −97.78

18.11 84.34 0.08201 −0.6587 −10.81 −272.2

50.72 263.5 0.2201 −1.418 −31.99 −85.09

29.96 178.9 −0.6896 −1.225 −30.38 −175.6

Table 3 Comparison of γmin obtained by different methods for Example 4. Methods

Theorem 1 (βopt , ρopt )

[11]

[18]

γmin

2.29 (0.008, 0.02)

3.08

7.52

The parameters ai, j , i = 1, 2, j = 1, 2, 3 and b1 are given in Table 2 in the four operating points which are assumed here as vertices of a polytopic plant. The minimum values of the H∞ performance γ obtained with Theorem 1 and with the approaches from [11,18] are shown in Table 3. 5. Conclusion In this paper, the problem of robust static output feedback H∞ control for linear uncertain systems was studied. A new method was proposed for designing robust static output feedback H∞ controllers using the parameter-dependent Lyapunov function. In contrast to previous approaches, in which some constraints on the systems matrices have to be imposed in order to obtain LMIbased design conditions, the proposed method is applicable to general uncertain systems. Using this method, design conditions were introduced in terms of LMIs. The proposed method provides more relaxed designs to ensure better H∞ performance for the systems. Numerical examples have shown the effectiveness of the proposed design method. Acknowledgments The authors would like to thank the Chief Editor, Associate Editor, and the Reviewers for their very helpful comments and suggestions for improving this paper. The work of X.-H. Chang was supported in part by the National Natural Science Foundation of China (Grant No. 61104071), by the Program for Liaoning Excellent Talents in University, China (Grant No. LJQ2012095), and by the Open Program of the Key Laboratory of Manufacturing Industrial Integrated Automation, Shenyang University, China (Grant No. 1120211415). Also, the research of J.H. Park was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2013R1A1A2A10005201). References [1] M.C. de Oliveira, J. Bernussou, J.C. Geromel, A new discrete-time robust stability condition, Systems Control Lett. 37 (4) (1999) 261–265. [2] H. Gao, X. Meng, T. Chen, A new design of robust H2 filters for uncertain systems, Systems Control Lett. 57 (7) (2008) 585–593. [3] B. Shen, Z. Wang, H. Shu, G. Wei, Robust H∞ finite-horizon filtering with randomly occurred nonlinearities and quantization effects, Automatica 46 (11) (2010) 1743–1751.

[4] L. Lu, R. Yang, L. Xie, Robust H2 and H∞ control of discrete-time systems with polytopic uncertainties via dynamic output feedback, in: Proceedings of the 2005 American Control Conference, 2005, pp. 4315–4320. [5] P.J. de Oliveira, R.C.L.F. Oliveira, V.J.S. Leite, V.F. Montagner, P.L.D. Peres, H∞ guaranteed cost computation by means of parameter-dependent Lyapunov functions, Automatica 40 (6) (2004) 1053–1061. [6] E. Gershon, U. Shaked, Static H2 and H∞ output-feedback of discrete-time LTI systems with state multiplicative noise, Systems Control Lett. 55 (3) (2006) 232–239. [7] H. Zhang, Y. Shi, A.S. Mehr, Robust static output feedback control and remote PID design for networked motor systems, IEEE Trans. Ind. Electron. 58 (12) (2011) 5396–5405. [8] H. Gao, X. Meng, T. Chen, A parameter-dependent approach to robust H∞ filtering for time-delay systems, IEEE Trans. Automat. Control 53 (10) (2008) 2420–2425. [9] S. Kanev, C. Scherer, M. Verhaegen, B. De Schutter, Robust output-feedback controller design via local BMI optimization, Automatica 40 (7) (2004) 1115–1127. [10] V. Blondel, J.N. Tsitsiklis, NP-hardness of some linear control design problems, SIAM J. Control Optim. 35 (6) (1997) 2118–2127. [11] C.M. Agulhari, R.C.L.F. Oliveira, P.L.D. Peres, LMI relaxations for reduce-dorder robust H∞ control of continuous-time uncertain linear systems, IEEE Trans. Automat. Control 57 (6) (2012) 1532–1537. [12] C.M. Agulhari, R.C.L.F. Oliveira, P.L.D. Peres, Robust H∞ static output-feedback design for time-invariant discrete-time polytopic systems from parameterdependent state-feedback gains, in: Proceedings of the 2010 American Control Conference, 2010, pp. 4677–4682. [13] R.E. Benton, D. Smith, A non-iterative LMI-based algorithm for robust staticoutput-feedback stabilization, Internat. J. Control 72 (14) (1999) 1322–1330. [14] D. Mehdi, E.K. Boukas, O. Bachelier, Static output feedback design for uncertain linear discrete time systems, IMA J. Math. Control Inform. 21 (1) (2004) 1–13. [15] D. Arzelier, D. Peaucelle, S. Salhi, Robust static output feedback stabilization for polytopic uncertain systems: Improving the guaranteed performance bound, in: Proceedings of the 4th IFAC Symposium on Robust Control Design, ROCOND 2003, 2003, pp. 425–430. [16] F. Leibfritz, An LMI-based algorithm for designing suboptimal static H2 /H∞ output-feedback controllers, SIAM J. Control Optim. 39 (6) (2001) 1711–1735. [17] U. Shaked, An LPD approach to robust H2 and H∞ static output feedback design, IEEE Trans. Automat. Control 48 (5) (2003) 866–872. [18] I. Yaesh, U. Shaked, Robust reduced-order output-feedback H∞ control, in: Proceedings of the 6th IFAC Symposium on Robust Control Design, ROCOND 2009, 2009, pp. 155–160. [19] C.A.R. Crusius, A. Trofino, Sufficient LMI conditions for output feedback control problems, IEEE Trans. Automat. Control 44 (5) (1999) 1053–1057. [20] M.C. de Oliveira, J.C. Geromel, J. Bernussou, Extended H2 and H∞ norm characterizations and controller parametrizations for discrete-time systems, Internat. J. Control 75 (9) (2002) 666–679. [21] J. Dong, G.-H. Yang, Robust static output feedback control for linear discretetime systems with time-varying uncertainties, Systems Control Lett. 57 (2) (2008) 123–131. [22] J. Dong, G.-H. Yang, Robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties, Automatica 49 (6) (2013) 1821–1829. [23] X.-H. Chang, G.-H. Yang, New results on output feedback H∞ control for linear discrete-time systems, IEEE Trans. Automat. Control 59 (5) (2014) 1355–1359. [24] G. Hilhorst, G. Pipeleers, J. Swevers, Reduced-order multi-objective H∞ control of an overhead crane test setup, in: Proceedings of the 52nd Conference on Decision and Control, 2013, pp. 770–775. [25] C.W. Scherer, P. Gahinet, M. Chilali, Multiobjective outputfeedback control via LMI optimization, IEEE Trans. Automat. Control 42 (7) (1997) 896–911. [26] S. Xu, J. Lam, Y. Zou, New results on delay-dependent robust H∞ control for systems with time-varying delays, Automatica 42 (2) (2006) 343–348. [27] P. Gahinet, P. Apkarian, A linear matrix inequality approach to H∞ control, Internat. J. Robust Nonlinear Control 4 (4) (1994) 421–448. [28] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequality in System and Control Theory, in: Studies in Applied Mathematics, SIAM, Philadelphia, 1994. [29] P. Gahinet, A. Nemirovski, A.J. Laub, M. Chilali, LMI Control Toolbox, The MathWorks, Natick, MA, 1995. [30] L. Xie, L. Lu, D. Zhang, H. Zhang, Improved robust H2 and H∞ filtering for uncertain discrete-time systems, Automatica 40 (5) (2004) 873–880. [31] S. Xu, T. Chen, Robust H∞ control for uncertain discrete-time systems with time-varying delays via exponential output feedback controllers, Systems Control Lett. 51 (3–4) (2004) 171–183. [32] F. Delmotte, T.M. Guerra, M. Ksantini, Continuous Takagi–Sugeno’s models: Reduction of the number of LMI conditions in various fuzzy control design technics, IEEE Trans. Fuzzy Syst. 15 (3) (2007) 426–438. [33] J. Ackermann, Longitudinal control of fighter aircraft F4E with additional canards, in: K. P. Sondergeld (compiler), A Collection of Plant Models and Design Specifications for Robust Control, DFVLR, Oberpfaffenhoffen, Germany.