Multidim Syst Sign Process DOI 10.1007/s11045-014-0301-8

Robust state feedback H∞ control for uncertain 2-D continuous state delayed systems in the Roesser model Imran Ghous · Zhengrong Xiang

Received: 15 April 2014 / Revised: 14 September 2014 / Accepted: 8 November 2014 © Springer Science+Business Media New York 2014

Abstract This paper is concerned with the problem of robust state feedback H∞ stabilization for a class of uncertain two-dimensional (2-D) continuous state delayed systems. The parameter uncertainties are assumed to be norm-bounded. Firstly, a new delay-dependent sufficient condition for the robust asymptotical stability of uncertain 2-D continuous systems with state delay is developed. Secondly, a sufficient condition for H∞ disturbance attenuation performance of the given system is derived. Thirdly, a stabilizing state feedback controller is proposed such that the resulting closed-loop system is robustly asymptotically stable and achieves a prescribed H∞ disturbance attenuation level. All results are developed in terms of linear matrix inequalities. Finally, two examples are provided to validate the effectiveness of the proposed method. Keywords 2-D continuous systems · Time-varying delays · State feedback · H∞ performance · Roesser model · Robust stability 1 Introduction Two-dimensional (2-D) state-space representation has greatly attracted many researchers in recent decades. 2-D system models have applications in many areas such as X-ray image enhancement, geographical data processing, electricity transmission, energy exchange processes, process control and modeling of partial differential equations like Darboux equation used in dynamic processes in gas absorption, water stream heating and air drying (Du and Xie 2002; Kaczorek 1985; Lu 1992; Roesser 1975). Many useful results are available on stability and stabalization issues of 2-D discrete systems (Du and Xie 1999; Hinamoto

I. Ghous · Z. Xiang (B) School of Automation, Nanjing University of Science and Technology, Nanjing 210094, People’s Republic of China e-mail: [email protected] I. Ghous e-mail: [email protected]

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Multidim Syst Sign Process

1997; Paszke et al. 2004; Duan et al. 2013; Lin et al. 2001; Xu et al. 2012). Though the basic consideration of 2-D system theory can often be viewed as a generalization from the conventional one-dimensional (1-D) system theory, there exist deep and substantial differences between the 1-D case and the 2-D case (Fan and Wen 2002; Feng et al. 2012; Xu et al. 2004). In the 2-D case, the system depends on two variables and the available related 2-D preliminaries are insufficient. Furthermore, the initial condition for a 1-D system consists of a single vector, which is obviously bounded, but for a 2-D system, initial conditions are infinite sets which may be unbounded. These factors make the analysis and synthesis of 2-D systems much more complicated and difficult than the 1-D case. Delays are inevitable in almost all practical engineering applications and may lead to performance degradation and instability of the system. The stability and stabilization of one-dimensional (1-D) systems have been widely studied by researchers (He et al. 2008; Li and Gao 2011; Yan et al. 2010). Some free-weighting matrices are introduced to reduce the conservatism of the results in He et al. (2008). In Li and Gao (2011), it was shown that the result can also be improved by a new model transformation without any free-weighting matrices. Stability and stabilization of 2-D discrete systems with time delays have been highlighted by many researchers (Chen 2010; Duan et al. 2013; Huang and Xiang 2013; Paszke et al. 2006). Recently, some work has also been done on the stability and stabilization issues of 2-D continuous systems with and without delays (Benhayoun et al. 2010, 2013; Benzaouia et al. 2011; Galkowski et al. 2002; Hmamed et al. 2010; Lam et al. 2004; Xu et al. 2005; Ghamgui et al. 2011a, b; Shaker and Shaker 2014; Emelianova et al. 2014). Uncertainties are inevitable in practical control systems and can be classified into two categories: disturbance signals and dynamic perturbations. The former includes input and output disturbances, e.g., actuator noise and sensor noise etc. The discrepancy between the mathematical model and the actual dynamics of the system in operation is represented by latter. H∞ technique is well known in literature to minimize the impact of perturbations on system. The theory of H∞ control was introduced in Zames (1981). Many useful results on the problem of H∞ disturbance attenuation performance of 2-D discrete systems have appeared (Xu et al. 2005a, b, 2008, 2013; Ye et al. 2014; Boukili et al. 2014). Recently, Hmamed et al. (2013) solved the H∞ filtering problem of 2-D delay-free continuous systems. Few results have been obtained for 2-D continuous systems with delays (El-Kasri and Hmamed 2013; El-Kasri et al. 2012, 2013). The problem of state feedback H∞ control for uncertain 2-D continuous systems with time-varying delays has not been fully investigated yet and deserves more attention. Thus, in this paper, we focus upon investigating the H∞ control problem of uncertain 2-D continuous systems with time-varying delays. The main contributions in this paper can be summarized as follows: (1) a new stability condition of uncertain 2-D continuous systems with time-varying delays is established; (2) the free-weighting matrix approach is introduced to reduce the conservatism, and the H∞ performance analysis for the uncertain 2D continuous systems is developed; (3) a state feedback H∞ controller design methodology for the underlying systems is formulated in terms of a set of linear matrix inequalities (LMIs). The remainder of this paper is organized as follows. In Sect. 2, problem formulation and some necessary lemmas are given. In Sect. 3, main results are developed. In Sect. 4, two examples are presented to show the effectiveness of the derived results. In Sect. 5, concluding remarks are given. Notation Throughout this paper, the superscript ”T ” denotes the transpose, and the notation X ≥ Y (X > Y ) means that matrix X − Y is positive semi-definite (positive definite, respectively). · denotes the Euclidean norm. I and 0 represent the identity matrix and the zero

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Multidim Syst Sign Process

matrix, respectively. The set of all nonnegative real numbers is represented by R+ . diag {ai } denotes a diagonal matrix with the diagonal elements ai , i = 1, 2, . . . , n. X −1 denotes the inverse of X . The asterisk ∗ in a matrix is used to denote the term that is induced by symmetry. The L 2 -norm of a 2-D signal w(t1 , t2 ) is given by   w2 =

∞

0

∞

w T (t1 , t2 )w(t1 , t2 )dt1 dt2 .

0

We say w(t1 , t2 ) belongs to L 2 {[0, ∞), [0, ∞)}, if w2 < ∞.

2 Problem formulation and preliminaries Consider the following uncertain 2-D continuous state delayed system in Roesser model:   h ∂ x (t1 ,t2 ) ∂t1 ∂ x v (t1 ,t2 ) ∂t2

with

= Ax(t1 , t2 ) + Ad x(t1 − d1 (t1 ), t2 − d2 (t2 )) + Bw(t1 , t2 ) + Eu(t1 , t2 ),

(1a)

z(t1 , t2 ) = C x(t1 , t2 ) + Dw(t1 , t2 ) + Fu(t1 , t2 ),

(1b)

   h x h (t1 , t2 ) x (t1 − d1 (t1 ), t2 ) , x(t , − d (t ), t − d (t )) = 1 1 1 2 2 2 x v (t1 , t2 ) x v (t1 , t2 − d2 (t2 ))          E1 A11 A12 Ad11 Ad12 B1 A= , Ad = , B= , E= , C = C1 C2 , E A21 A22 Ad21 Ad22 B2 2 

x(t1 , t2 ) =

where x h (t1 , t2 ) ∈ R n 1 and x v (t1 , t2 ) ∈ R n 2 are the horizontal and the vertical states respectively, x(t1 , t2 ) is the whole state in R n with n = n 1 + n 2 . w(t1 , t2 ) ∈ R q is the disturbance input which belongs to L 2 {[0, ∞), [0, ∞)}. u(t1 , t2 ) ∈ R m is the controlled input, z(t1 , t2 ) ∈ R p is the controlled output, and t1 and t2 are real numbers in R+ . E, D, F and C are constant matrices with appropriate dimensions. A, Ad and B are uncertain real-valued matrices of appropriate dimensions, and they are assumed to be of the form: A = A + G F(t1 , t2 )W1 , Ad = Ad + G F(t1 , t2 )W2 , B = B + G F(t1 , t2 )W3 , where

 G=

(2)

   Ga , W1 = W1a W1b , W2 = W2a W2b , Gb

and A, Ad , B, G, W1 , W2 and W3 are constant matrices with appropriate dimensions. F(t1 , t2 ) is an unknown matrix representing parameter uncertainty and satisfies F T (t1 , t2 )F(t1 , t2 ) ≤ I. The delays d1 (t1 ) and d2 (t2 ) are assumed to be time-varying differential functions along horizontal and vertical directions respectively, and they satisfy 0 ≤ d1 (t1 ) ≤ dh , 0 ≤ d2 (t2 ) ≤ dv , d˙1 (t1 ) ≤ μ1 < 1, d˙2 (t2 ) ≤ μ2 < 1, where dh , dv , μ1 and μ2 are positive scalars.

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Multidim Syst Sign Process

The boundary conditions are given as follows: x h (υ1 , t2 ) = f υ1 (t2 ), ∀ − dh ≤ υ1 ≤ 0, 0 ≤ t2 ≤ T2 ;

(3a)

x (υ1 , t2 ) = 0, ∀ − dh ≤ υ1 ≤ 0, t2 ≥ T2 ;

(3b)

h

v

x (t1 , υ2 ) = gυ2 (t1 ), ∀ − dv ≤ υ2 ≤ 0, 0 ≤ t1 ≤ T1 ;

x v (t1 , υ2 ) = 0, ∀ − dv ≤ υ2 ≤ 0, t1 ≥ T1

(3c) (3d)

where T1 < ∞ and T2 < ∞ are positive constants, f υ1 (t2 ) and gυ2 (t1 ) are given vectors. When there is no disturbance and control inputs in system (1), we can get the following system   ∂ x h (t1 ,t2 ) ∂t1 ∂ x v (t1 ,t2 ) ∂t2

= Ax(t1 , t2 ) + Ad x(t1 − d1 (t1 ), t2 − d2 (t2 ))

(4)

Definition 1 2-D system (4) is said to be robustly asymptotically stable if its state x(t1 , t2 ) satisfies lim sup x(t1 , t2 ) = 0 for all admissible uncertainties described by (2) and all t1 +t2 →∞

boundary conditions in (3). Definition 2 (Hmamed et al. 2010) Let V (t1 , t2 ) = V h (x h (t1 , t2 )) + V v (x v (t1 , t2 )) be a Lyapunov functional of system (4), its unidirectional derivative is given by ∂ V h (x h (t1 , t2 )) ∂ V v (x v (t1 , t2 )) + . V˙u (t1 , t2 ) = ∂t1 ∂t2 Lemma 1 2-D system (4) is robustly asymptotically stable if there exist positive constants ci (i = 1, . . . , 4) such that the Lyapunov functional defined in Definition 2 and its unidirectional derivative along the trajectory of the system satisfy

2

2







c1 x h (t1 , t2 ) ≤ V h (x h (t1 , t2 )) ≤ c2 sup x h (t1 + υ1 , t2 ) , (5)

2 c3 x v (t1 , t2 ) ≤ V v (x v (t1 , t2 )) ≤ c4

−dh ≤υ1 ≤0

sup

−dv ≤υ2 ≤0

v

x (t1 , t2 + υ2 ) 2 ,

V˙u (t1 , t2 ) < 0.

(6) (7)

Proof Consider the line t1 +t2 = t on the plane (t1 , t2 ). Integrating inequality V h (x h (t1 , t2 )) and V v (x v (t1 , t2 )) along this line, we get  √  t h h √  t h h V h (x h (t1 , t2 ))ds = 2 V (x (t − τ, τ ))dτ = 2 V (x (τ, t − τ ))dτ , t1 +t2 =t

 t1 +t2 =t

0

0

(8)  t  t √ √ V v (x v (t1 , t2 ))ds = 2 V v (x v (τ, t − τ ))dτ = 2 V v (x v (t − τ, τ ))dτ . 0

0

(9) By virtue of the Leibniz formula, we have 

 t  d t1 +t2 =t V h (x h (t1 , t2 ))ds √ ∂ V h (x h (t − τ, τ )) = 2 dτ + V h (x h (0, t)) , dt ∂t 0 (10)

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Multidim Syst Sign Process

d

 t1 +t2 =t

V v (x v (t1 , t2 ))ds

dt

√ = 2



t 0

 ∂ V v (x v (τ, t − τ )) v v dτ + V (x (t, 0)) . ∂t (11)

Then we get d

 t1 +t2 =t

[V h (x h (t1 ,t2 ))+V v (x v (t1 ,t2 ))]ds

√  ds + 2 V h (x h (0, t)) + V v (x v (t, 0)) . (12)  Denote W (t) = t1 +t2 =t V h (x h (t1 , t2 )) + V v (x v (t1 , t2 )) ds. From (7), we have =



t1 +t2 =t



dt ∂ V h (x h (t1 ,t2 )) ∂t1

+

∂ V v (x v (t1 ,t2 )) ∂t2

 dW (t) √  h h < 2 V (x (0, t)) + V v (x v (t, 0)) . dt

(13)

Note that W (0) = 0. It follows from (13) that  t√   W (t) < 2 V h (x h (0, τ )) + V v (x v (τ, 0)) dτ .

(14)

0

Combining (8)–(9) with (14) yields  t  t   V h (x h (t − τ, τ )) + V v (x v (t − τ, τ )) dτ < V h (x h (0, τ )) + V v (x v (τ, 0)) dτ . 0

0

Considering the boundary conditions in (3), we obtain from (5) and (6) that  t  V h (x h (t − τ, τ )) + V v (x v (t − τ, τ )) dτ lim t→∞ 0  t  V h (x h (0, τ ) + V v (x v (τ, 0)) dτ < lim t→∞ 0   T

2

v

2

h ≤ max{c2 , c4 } sup x (υ1 , τ ) + sup x (τ, υ2 ) dτ 0

−dh ≤υ1 ≤0

−dv ≤υ2 ≤0

< ∞,

(15)

where T = max{T1 , T2 }. From (5), (6) and (15), it can be obtained that   t 

2

2

h lim

x (t − τ, τ ) + x v (t − τ, τ ) dτ < ∞, t→∞ 0

which implies lim

t1 +t2 →∞

Then it is easy to get that





2

2

h

v



sup x (t1 , t2 ) + x (t1 , t2 ) = 0. lim

t1 +t2 →∞

sup x(t1 , t2 ) = 0. Thus according to Definition 1, 2-D

system (4) is robustly asymptotically stable. The proof is completed. Definition 3 2-D system (1) is said to have a prescribed H∞ disturbance attenuation level γ if the following conditions are satisfied:

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Multidim Syst Sign Process

(1) When w(t1 , t2 ) = 0, system (1) is robustly asymptotically stable. (2) Under zero boundary condition, it holds that J0 =

sup

z22

2 0=w(t1 ,t2 )∈L 2 w2

< γ 2. 

Lemma 2 (Boyd et al. 1994) For a given matrix S = square matrices, the following conditions are equivalent:

 S11 S12 , where S11 and S22 are T S S12 22

(i) S < 0; T S −1 S < 0; (ii) S11 < 0, S22 − S12 11 12 −1 T S12 < 0. (iii) S22 < 0, S11 − S12 S22 Lemma 3 (Xie 1996) Let U, V, W and X be real matrices of appropriate dimensions with X satisfying X = X T , then for all V T V ≤ R, X + U V W + W T V T U T < 0, if and only if there exist a scalar ε such that X + εUU T + ε −1 W RW T < 0. In this paper, we are interested in synthesis of H∞ controller for system (1). The following state feedback control law is employed: u(t1 , t2 ) = K x(t1 , t2 ),

(16)

 with K = K 1 K 2 , where K 1 ∈ R m×n 1 and K 2 ∈ R m×n 2 . Then the corresponding closedloop system is: 

∂ x h (t1 ,t2 ) ∂t1 ∂ x v (t1 ,t2 ) ∂t2

 = (A + E K )x(t1 , t2 ) + Ad x(t1 − d1 (t1 ), t2 − d2 (t2 )) + Bw(t1 , t2 ), (17a)

z(t1 , t2 ) = (C + F K )x(t1 , t2 ) + Dw(t1 , t2 ).

(17b)

3 Main results 3.1 Stability analysis To obtain the main results, we first investigate the stability of uncertain 2-D continuous systems with time-varying delays. The following theorem presents a sufficient condition for robust asymptotical stability of system (1) with w(t1 , t2 ) = 0 and u(t1 , t2 ) = 0. Theorem 1 Given positive scalars dh , dv , μ1 and μ2 , system (1) with w(t1 , t2 ) = 0 and u(t1 , t2 ) = 0 is robustly asymptotically stable if there exist positive definite symmetric matrices P1 , Q 1 , R1 , P2 , Q 2 , R2 , Z i (i = 1, . . . , 4) and any matrices N1 j , S1 j , M1 j , N2 j , S2 j , M2 j ( j = 1, . . . , 3), with appropriate dimensions, such that the following LMI holds   ψ  < 0, (18) ∗

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Multidim Syst Sign Process

where ⎡

T

11 P1 A12 + A21 P2 13 P1 Ad12 ⎢ ∗

22 P2 Ad21 24 ⎢ ⎢ ∗ ∗

33 0 ⎢ ψ =⎢ ∗ ∗

44 ⎢ ∗ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

15 0

35 0

55 ∗

⎤ 0

26 ⎥ ⎥ 0 ⎥ ⎥,

46 ⎥ ⎥ 0 ⎦

66

T

T T

11 = N11 + M11 + N11 + M11 + P1 A11 + A11 P1 + Q 1 + R1 , T T T T + M12 + P1 Ad11 , 15 = −S11 − M11 + N13 + M13 ,

13 = −N11 + S11 + N12 T

T T + M21 + P2 A22 + A22 P2 + Q 2 + R2 ,

22 = N21 + M21 + N21 T T T T

24 = −N21 + S21 + N22 + M22 + P2 Ad22 , 26 = −S21 − M21 + N23 + M23 , T T T T

33 = −N12 + S12 − N12 + S12 − (1 − μ1 )Q 1 , 35 = −S12 − M12 − N13 + S13 , T T T T

44 = −N22 + S22 − N22 + S22 − (1 − μ2 )Q 2 , 46 = −S22 − M22 − N23 + S23 , T T T T

55 = −S13 − M13 − S13 − M13 − R1 , 66 = −S23 − M23 − S23 − M23 − R2 , √ √ √ √ √ √ √ √ dh 1 dv 2 dh 3 dv 4 dh 5 dv 6 dh 7 dv 8 , =

= diag {−Z 1 , −Z 3 , −Z 1 , −Z 3 , −Z 2 , −Z 4 , −(Z 1 + Z 2 ), −(Z 3 + Z 4 )} ,  T  T 0 NT 0 T ,  = 0 NT 0 NT 0 NT T , 1 = N11 0 N12 2 13 21 22 23  T  T 0 ST 0 T ,  = 0 ST 0 ST 0 ST T , 3 = S11 0 S12 4 13 21 22 23  T  T 0 MT 0 T ,  = 0 MT 0 MT 0 MT T , 5 = M11 0 M12 6 13 21 22 23 T  7 = (Z 1 + Z 2 )A11 (Z 1 + Z 2 )A12 (Z 1 + Z 2 )Ad11 (Z 1 + Z 2 )Ad12 0 0 , T  8 = (Z 3 + Z 4 )A21 (Z 3 + Z 4 )A22 (Z 3 + Z 4 )Ad21 (Z 3 + Z 4 )Ad22 0 0 . Proof We choose the following Lyapunov–Krasovskii functional candidate for system (1): V (t1 , t2 ) = V h (x h (t1 , t2 )) + V v (x v (t1 , t2 )),

(19)

with V h (x h (t1 , t2 )) = V1h (x h (t1 , t2 ))

4  k=1 hT

=x  V2h (x h (t1 , t2 )) =  V3h (x h (t1 , t2 )) = V4h (x h (t1 , t2 )) = V v (x v (t1 , t2 )) =

Vkh (x h (t1 , t2 )),

(t1 , t2 )P1 x h (t1 , t2 ), 0  t1 x˙ hT (α, t2 )(Z 1 + Z 2 )x˙ h (α, t2 )dαdβ,

−dh t1

t1 +β

x hT (α, t2 )Q 1 x h (α, t2 )dα,

t1 −d1 (t1 )  t1 hT t1 −dh 4 

x

(α, t2 )R1 x h (α, t2 )dα,

Vkv (x v (t1 , t2 )),

k=1

V1v (x v (t1 , t2 )) = x vT (t1 , t2 )P2 x v (t1 , t2 ),

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Multidim Syst Sign Process

V2v (x v (t1 , t2 )) = V3v (x v (t1 , t2 )) = V4v (x v (t1 , t2 )) =

where x˙ h (α, t2 ) =





0



−dv  t2

t2

t2 +β

x vT (t1 , β)Q 2 x v (t1 , β)dβ,

t2 −d2 (t2 )  t2 vT t2 −dv

∂ x h (t1 ,t2 )   ∂t1 t1 =α

x˙ vT (t1 , α)(Z 3 + Z 4 )x˙ v (t1 , α)dαdβ,

x

(t1 , β)R2 x v (t1 , β)dβ.

and x˙ v (t1 , α) =



∂ x v (t1 ,t2 )  .  ∂t2 t2 =α

The unidirectional derivative of V (t1 , t2 ) can be written as

∂ V h (x h (t1 , t2 )) ∂ V v (x v (t1 , t2 )) + , V˙u (t1 , t2 ) = ∂t1 ∂t2

(20)

where ∂ V1h (x h (t1 , t2 )) ∂ V2h (x h (t1 , t2 )) ∂ V3h (x h (t1 , t2 )) ∂ V4h (x h (t1 , t2 )) ∂ V h (x h (t1 , t2 )) = + + + , ∂t1 ∂t1 ∂t1 ∂t1 ∂t1 ∂ V1v (x v (t1 , t2 )) ∂ V2v (x v (t1 , t2 )) ∂ V3v (x v (t1 , t2 )) ∂ V4v (x v (t1 , t2 )) ∂ V v (x v (t1 , t2 )) = + + + . ∂t2 ∂t2 ∂t2 ∂t2 ∂t2 For simplicity, let us denote x h = x h (t1 , t2 ), x v = x v (t1 , t2 ), xdh1 (t1 ) = x h (t1 −d1 (t1 ), t2 ), xdv2 (t2 ) = x v (t1 , t2 − d2 (t2 )), xdh = x h (t1 − dh , t2 ), xdv = x v (t1 , t2 − dv ), x h (α) = x h (α, t2 ), x v (β) = x v (t1 , β). When w(t1 , t2 ) = 0 and u(t1 , t2 ) = 0, by calculating the unidirectional derivative (20), we obtain V˙u (t1 , t2 ) = 2x hT P1 x˙ h + 2x vT P2 x˙ v + dh x˙ hT (Z 1 + Z 2 )x˙ h + dv x˙ vT (Z 3 + Z 4 )x˙ v + x hT Q 1 x h − (1 − d˙1 (t1 ))x hT Q 1 x h + x vT Q 2 x v d1 (t1 )

d1 (t1 )

− (1 − d˙2 (t2 ))xdvT Q 2 xdv2 (t2 ) + x hT R1 x h − xdhT R1 xdh 2 (t2 )  t1 vT v vT v + x R2 x − x d R2 x d − x˙ hT (α)(Z 1 + Z 2 )x˙ h (α)dα  −

t1 −dh

t2 t2 −dv

x˙ vT (β)(Z 3 + Z 4 )x˙ v (β)dβ.

⎤ N11 By Newton Leibniz formula, the following equations are true for any matrices N1 = ⎣ N12 ⎦, N ⎤ ⎤ ⎤ ⎤ ⎤ 13 ⎡ ⎡ ⎡ ⎡ ⎡ S11 M11 N21 S21 M21 S1 = ⎣ S12 ⎦, M1 = ⎣ M12 ⎦, N2 = ⎣ N22 ⎦, S2 = ⎣ S22 ⎦ and M2 = ⎣ M22 ⎦ with S13 M13 N23 S23 M23 ⎡

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Multidim Syst Sign Process

appropriate dimensions, such that    t1 x˙ h (α)dα = 0, 2hT N1 x h − xdh1 (t1 ) − t1 −d1 (t1 )   

(21)

2hT S1 xdh1 (t1 ) − xdh −

(22)

  2hT M1 x h − xdh −

t1 −d1 (t1 )

t1 −dh t1

t1 −dh  t2

x˙ h (α)dα = 0,

 x˙ h (α)dα = 0,

(23)

  x˙ v (β)dβ = 0, 2vT N2 x v − xdv2 (t2 ) − t2 −d2 (t2 )    t2 −d2 (t2 )

2vT S2 xdv2 (t2 ) − xdv −

  2vT M2 x v − xdv −

t2 −dv t2

t2 −dv

(24)

x˙ v (β)dβ = 0,

(25)

 x˙ v (β)dβ = 0,

(26)

  T T hT vT v = x vT x vT x where h = x hT xdhT and  . d d2 (t2 ) x d 1 (t1 ) Adding left side of (21)–(26) into (20) gives Q 1 xdh1 (t1 ) V˙u (t1 , t2 ) ≤ 2x hT P1 x˙ h +dh x˙ hT (Z 1 + Z 2 )x˙ h + x hT (Q 1 + R1 )x h −(1−μ1 )xdhT 1 (t1 )  − xdhT R1 xdh −  −

t1 −dh

+ 2 + 2

hT

t1 −d1 (t1 )

x˙ hT (α)Z 1 x˙ h (α)dα −

t1 −d1 (t1 )

t1 −dh



t1

hT



t1

x˙

hT

(α)Z 2 x˙ (α)dα + 2 h



 xdh1 (t1 )

S1

−

xdh



− 

M1 x − h

xdh

−

hT

t1 −d1 (t1 )

t1 −dh

x˙ hT (α)Z 1 x˙ h (α)dα 

N1 x − h

xdh1 (t1 )

−

t1

t1 −d1 (t1 )



 x˙ (α)dα h

x˙ (α)dα h



t1

x˙ (α)dα h

t1 −dh

Q 2 xdv2 (t2 ) + 2x vT P2 x˙ v +dv x˙ vT (Z 3 + Z 4 )x˙ v + x vT (Q 2 + R2 )x v −(1−μ2 )xdvT 2 (t2 ) − xdvT R2 xdv −  −

t2 t2 −dv



t2

t2 −d2 (t2 )

x˙ vT (β)Z 3 x˙ v (β)dβ −



t2 −d2 (t2 ) t2 −dv

x˙ vT (β)Z 3 x˙ v (β)dβ

  x˙ vT (β)Z 4 x˙ v (β)dβ + 2vT N2 x v − xdv2 (t2 ) − 

+ 2vT S2 xdv2 (t2 ) − xdv −   + 2vT M2 x v − xdv −



t2 −d2 (t2 )

t2 −dv t2

t2 −dv



t2

t2 −d2 (t2 )

x˙ v (β)dβ



x˙ v (β)dβ

 x˙ v (β)dβ .

(27)

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Multidim Syst Sign Process

That is ⎡

⎤T ⎡ h ⎤ xh x v ⎢ x ⎥ ⎢ xv ⎥ ⎢ h ⎢ h ⎥ ⎥ ⎢ xd (t ) ⎥ ⎢ xd (t ) ⎥ 1 1 1 1 ⎢ ⎢ ⎥ ⎥ ˙ Vu (t1 , t2 ) ≤ ⎢ v ⎥  ⎢ xv ⎥ ⎢ xd2 (t2 ) ⎥ ⎢ d2 (t2 ) ⎥ ⎣ xh ⎦ ⎣ xh ⎦ d d xdv xdv  t1     − hT N1 + x˙ hT (α)Z 1 Z 1−1 N1T h + Z 1 x˙ hT (α) dα 

t1 −d1 (t1 ) t1 −d1 (t1 ) 



t1 −dh t1



t1 −dh t2



t2 −d2 (t2 ) t2 −d2 (t2 ) 



t2 −dv t2

− − − − −

t2 −dv

where  = ψ + ξ , and



   hT S1 + x˙ hT (α)Z 1 Z 1−1 S1T h + Z 1 x˙ hT (α) dα

   hT M1 + x˙ hT (α)Z 2 Z 2−1 M1T h + Z 2 x˙ hT (α) dα     vT N2 + x˙ vT (β)Z 3 Z 3−1 N2T v + Z 3 x˙ vT (β) dβ    vT S2 + x˙ vT (β)Z 3 Z 3−1 S2T v + Z 3 x˙ vT (β) dβ

    vT M2 + x˙ vT (β)Z 4 Z 4−1 M2T v + Z 4 x˙ vT (β) dβ,

⎡

ξ11 ξ12 ξ13 ⎢ ∗ ξ22 ξ23 ⎢ ⎢ ∗ ∗ ξ33 ξ =⎢ ⎢ ∗ ∗ ∗ ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ T

⎤ ξ14 ξ15 0 ξ24 0 ξ26 ⎥ ⎥ ξ34 ξ35 0 ⎥ ⎥, ξ44 0 ξ46 ⎥ ⎥ ∗ ξ55 0 ⎦ ∗ ∗ ξ66 T

ξ11 = dh A11 (Z 1 + Z 2 )A11 + dv A21 (Z 3 + Z 4 )A21

T T T + dh (N11 Z 1−1 N11 + S11 Z 1−1 S11 + M11 Z 2−1 M11 ), T

T

ξ13 = dh A11 (Z 1 + Z 2 )Ad11 + dv A21 (Z 3 + Z 4 )Ad21

T T T + dh (N11 Z 1−1 N12 + S11 Z 1−1 S12 + M11 Z 2−1 M12 ), T

T

ξ22 = dh A12 (Z 1 + Z 2 )A12 + dv A22 (Z 3 + Z 4 )A22

T T T + dv (N21 Z 3−1 N21 + S21 Z 3−1 S21 + M21 Z 4−1 M21 ), T

T

ξ24 = dh A12 (Z 1 + Z 2 )Ad12 + dv A22 (Z 3 + Z 4 )Ad22

T T T + dv (N21 Z 3−1 N22 + S21 Z 3−1 S22 + M21 Z 4−1 M22 ), T

T

ξ33 = dh Ad11 (Z 1 + Z 2 )Ad11 + dv Ad21 (Z 3 + Z 4 )Ad21 T T T + dh (N12 Z 1−1 N12 + S12 Z 1−1 S12 + M12 Z 2−1 M12 ), T

T

ξ44 = dh Ad12 (Z 1 + Z 2 )Ad12 + dv Ad22 (Z 3 + Z 4 )Ad22 T T T + dv (N22 Z 3−1 N22 + S22 Z 3−1 S22 + M22 Z 4−1 M22 ), T

T

ξ12 = dh A11 (Z 1 + Z 2 )A12 + dv A21 (Z 3 + Z 4 )A22 ,

123

(28)

Multidim Syst Sign Process T T T ξ66 = dv (N23 Z 3−1 N23 + S23 Z 3−1 S23 + M23 Z 4−1 M23 ), T

T

ξ14 = dh A11 (Z 1 + Z 2 )Ad12 + dv A21 (Z 3 + Z 4 )Ad22 ,

T T T ξ15 = dh (N11 Z 1−1 N13 + S11 Z 1−1 S13 + M11 Z 2−1 M13 ), T

T

ξ23 = dh A12 (Z 1 + Z 2 )Ad11 + dv A22 (Z 3 + Z 4 )Ad21 ,

T T T ξ26 = dv (N21 Z 3−1 N23 + S21 Z 3−1 S23 + M21 Z 4−1 M23 ), T

T

ξ34 = dh Ad11 (Z 1 + Z 2 )Ad12 + dv Ad21 (Z 3 + Z 4 )Ad22 , T T T ξ35 = dh (N12 Z 1−1 N13 + S12 Z 1−1 S13 + M12 Z 2−1 M13 ), T T T ξ46 = dv (N22 Z 3−1 N23 + S22 Z 3−1 S23 + M22 Z 4−1 M23 ), T T T + S13 Z 1−1 S13 + M13 Z 2−1 M13 ). ξ55 = dh (N13 Z 1−1 N13

By Lemma 2, we obtain that (18) is equivalent to the following inequality  < 0, which implies V˙u (t1 , t2 ) < 0. Thus according to Lemma 1, system (1) with w(t1 , t2 ) = 0 and u(t1 , t2 ) = 0 is robustly asymptotically stable. This completes the proof. Remark 1 It should be noted that the chosen Lyapunov functional candidate V (t1 , t2 ) is different from that in El-Kasri et al. (2013). When calculating the unidirectional derivative t t of V (t1 , t2 ), we get − t11−dh x˙ hT (α)Z 1 x˙ h (α)dα and − t22−dv x˙ vT (β)Z 3 x˙ v (β)dβ. In order to reduce the conservatism, we separate them into two parts respectively, and treat them using different free-weighting matrices [see (27)]. This method makes our result much less conservative than that in El-Kasri et al. (2013), which can be seen in Example 1 given later. When d1 (t1 ) = 0 and there are no uncertainties, system (1) reduces to the following system: 

∂ x h (t1 ,t2 ) ∂t1 ∂ x v (t1 ,t2 ) ∂t2



 =

A11 A12 A21 A22



   Ad12 v x h (t1 , t2 ) + x (t1 , t2 − d2 (t2 )) Ad22 x v (t1 , t2 )

+Bw(t1 , t2 ) + Eu(t1 , t2 ).

(29)

then we have the following result from Theorem 1. Corollary 1 Given positive scalars dv and μ2 , system (29) with w(t1 , t2 ) = 0 and u(t1 , t2 ) = 0 is asymptotically stable if there exist positive definite symmetric matrices P1 , P2 , Z 3 , Z 4 , Q 2 , R2 and any matrices N2 j , S2 j , M2 j ( j = 1, . . . , 3), with appropriate dimensions, such that the following LMI holds

123

Multidim Syst Sign Process

⎡

T P P A

11 P1 A12 + A21 2 1 d12 ⎢ ∗

24 22 ⎢ ⎢ ∗ ∗

44 ⎢ ⎢ ∗ ∗ ∗ ⎢ ⎢ ∗ ∗ ∗ ⎢ ⎢ ∗ ∗ ∗ ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗

√ T (Z + Z ) ⎤ 0 √ 0 3 4 √ 0 √ 0 √dv A21 T

26 √dv N21 √dv S21 √dv M21 √ dv A22 (Z 3 + Z 4 ) ⎥ ⎥ T (Z + Z ) ⎥

46 √dv N22 √dv S22 √dv M22 dv Ad22 3 4 ⎥ ⎥ 0

66 dv N23 dv S23 dv M23 ⎥ < 0, ⎥ 0 0 0 ∗ −Z 3 ⎥ ⎥ ∗ ∗ −Z 3 0 0 ⎥ ⎦ ∗ ∗ ∗ −Z 4 0 ∗ ∗ ∗ ∗ −(Z 3 + Z 4 ) (30)

with T

11 = P1 A11 + A11 P1 .

Proof Corollary 1 can be easily derived from Theorem 1. The detailed proof is omitted here. 3.2 H∞ performance analysis In this subsection, a sufficient condition for H∞ performance of system (1) is established. The following theorem presents a sufficient condition for H∞ disturbance attenuation performance of system (1) with u(t1 , t2 ) = 0 and any non-zero w(t1 , t2 ) ∈ L 2 {[0, ∞), [0, ∞)}. Theorem 2 Given positive scalars dh , dv , μ1 and μ2 , system (1) with u(t1 , t2 ) = 0 is robustly asymptotically stable and has a prescribed H∞ disturbance attenuation performance level γ if there exist positive definite symmetric matrices P1 , Q 1 , R1 , P2 , Q 2 , R2 , Z i (i = 1, . . . , 4), and any matrices N1 j , S1 j , M1 j , N2 j , S2 j , M2 j ( j = 1, . . . , 3), with appropriate dimensions, such that the following LMI holds   ψ  < 0, (31) ∗ where =

√

dh 1

√

dv 2

√

d h 3

√ √ √ √ √ dv 4 dh 5 dv 6 dh 7 dv 8 ϕ1 , ,

= diag {−Z 1 , −Z 3 , −Z 1 , −Z 3 , −Z 2 , −Z 4 , −(Z 1 + Z 2 ), −(Z 3 + Z 4 ), −I } ,  T  T 0 NT 0 0 T ,  = 0 NT 0 NT 0 NT 0 , 1 = N11 0 N12 2 13 21 22 23  T  T 0 ST 0 0 T ,  = 0 ST 0 ST 0 ST 0 T , 3 = S11 0 S12 4 13 21 22 23  T  T 0 MT 0 0 T ,  = 0 MT 0 MT 0 MT 0 T , 5 = M11 0 M12 6 13 21 22 23  T 7 = (Z 1 + Z 2 )A11 (Z 1 + Z 2 )A12 (Z 1 + Z 2 )Ad11 (Z 1 + Z 2 )Ad12 0 0 (Z 1 + Z 2 )B 1 ,  T 8 = (Z 3 + Z 4 )A21 (Z 3 + Z 4 )A22 (Z 3 + Z 4 )Ad21 (Z 3 + Z 4 )Ad22 0 0 (Z 3 + Z 4 )B 2 , ⎤ ⎡ T

11 P1 A12 + A21 P2 13 P1 Ad12 15 0 P1 B 1 ⎢ ∗

22 P2 Ad21 24 0 26 P2 B 2 ⎥ ⎥ ⎢ ⎢ ∗ ∗

33 0

35 0 0 ⎥ ⎥ ⎢ ψ =⎢ ∗ ∗

44 0 46 0 ⎥ ⎥, ⎢ ∗ ⎥ ⎢ ∗ ∗ ∗ ∗

0 0 55 ⎥ ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ 66 0 ⎦ ∗ ∗ ∗ ∗ ∗ ∗ −γ 2 I T  ϕ1 = C1 C2 0 0 0 0 D .

123

Multidim Syst Sign Process

Proof (7) can be directly derived from (31), then by Theorem 1 we can obtain from (31) that system (1) with w(t1 , t2 ) = 0 and u(t1 , t2 ) = 0 is robustly asymptotically stable. Now we shall prove that system (1) with u(t1 , t2 ) = 0 and any non-zero w(t1 , t2 ) ∈ L 2 {[0, ∞), [0, ∞)} has a prescribed H∞ disturbance attenuation level γ . For this purpose, let us introduce (t1 , t2 ) = V˙u (t1 , t2 ) + z T (t1 , t2 )z(t1 , t2 ) − γ 2 w T (t1 , t2 )w(t1 , t2 ).. For simplification, w(t1 , t2 ) is denoted as w in the following equation. Now from (28) it follows that  t1     (t1 , t2 ) ≤ ω T ω − hT N1 + x˙ hT (α)Z 1 Z 1−1 N1T h + Z 1 x˙ hT (α) dα  −





t1 −dh t1



t1 −dh t2



t2 −d2 (t2 ) t2 −d2 (t2 ) 



t2 −dv t2

− − − −

t1 −d1 (t1 ) t1 −d1 (t1 )  hT

t2 −dv



   S1 + x˙ hT (α)Z 1 Z 1−1 S1T h + Z 1 x˙ hT (α) dα

   hT M1 + x˙ hT (α)Z 2 Z 2−1 M1T h + Z 2 x˙ hT (α) dα     vT N2 + x˙ vT (β)Z 3 Z 3−1 N2T v + Z 3 x˙ vT (β) dβ    vT S2 + x˙ vT (β)Z 3 Z 3−1 S2T v + Z 3 x˙ vT (β) dβ

    vT M2 + x˙ vT (β)Z 4 Z 4−1 M2T v + Z 4 x˙ vT (β) dβ,

 T hT x vT w T vT x x with ω = x hT x vT xdhT d 1 (t1 ) d2 (t2 ) d Theorem 2, ⎡ ξ11 ξ12 ξ13 ξ14 ξ15 ⎢ ∗ ξ22 ξ23 ξ24 0 ⎢ ⎢ ∗ ∗ ξ33 ξ34 ξ35 ⎢ ξ =⎢ ⎢ ∗ ∗ ∗ ξ44 0 ⎢ ∗ ∗ ∗ ∗ ξ55 ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

(32)

and  = ψ + ξ , where ψ is given in 0 ξ26 0 ξ46 0 ξ66 ∗

T

T

T

T

⎤ ξ17 ξ27 ⎥ ⎥ ξ37 ⎥ ⎥ ξ47 ⎥ ⎥, 0 ⎥ ⎥ 0 ⎦ ξ77

ξ17 = dh A11 (Z 1 + Z 2 )B 1 + dv A21 (Z 3 + Z 4 )B 2 , ξ27 = dh A12 (Z 1 + Z 2 )B 1 + dv A22 (Z 3 + Z 4 )B 2 , T

T

T

T

ξ37 = dh Ad11 (Z 1 + Z 2 )B 1 + dv Ad21 (Z 3 + Z 4 )B 2 , ξ47 = dh Ad12 (Z 1 + Z 2 )B 1 + dv Ad22 (Z 3 + Z 4 )B 2 , T

T

ξ77 = dh B 1 (Z 1 + Z 2 )B 1 + dv B 2 (Z 3 + Z 4 )B 2 . Applying Lemma 2, it follows from (31) that  < 0,

(33)

123

Multidim Syst Sign Process

which implies (t1 , t2 ) = V˙u (t1 , t2 ) + z T (t1 , t2 )z(t1 , t2 ) − γ 2 w T (t1 , t2 )w(t1 , t2 ) < 0.

(34)

z T (t1 , t2 )z(t1 , t2 ) − γ 2 w T (t1 , t2 )w(t1 , t2 ) < −V˙u (t1 , t2 ).

(35)

That is

It can be obtained that ∞ ∞

 0

<

(z T (t1 , t2 )z(t1 , t2 ) − γ 2 w T (t1 , t2 )w(t1 , t2 ))dt1 dt2

 0∞  ∞ 0

(−V˙u (t1 , t2 ))dt1 dt2 .

(36)

0

Under zero boundary conditions, (36) yields 

∞ ∞

0

 z (t1 , t2 )z(t1 , t2 )dt1 dt2 − T

0

0

∞ ∞

γ 2 w T (t1 , t2 )w(t1 , t2 )dt1 dt2 < 0.

0

That is z22 < γ 2 w22 .

(37)

This completes the proof. 3.3 H∞ controller design In this subsection, we shall focus on the controller design for system (1) to ensure that the closed-loop system (17) is robustly asymptotically stable and has a prescribed H∞ disturbance attenuation level γ . Theorem 3 For system (1), given positive scalars γ , dh , dv , μ1 and μ2 , if there exist positive definite symmetric matrices Hi , Vi (i = 1, . . . , 4), P 1 , P 2 and any matrices Ti , Ui (i = 1, . . . , 9), Y1 and Y2 with appropriate dimensions, and a positive scalar λ such that the following LMI holds: ⎡ ˜ √ ˜ √ ˜ √ ˜ √ ˜ √ ˜ √ ˜ √ ˜ √ ˜ ˜ ⎤ ψ dh 1 dv 2 dh 3 dv 4 dh 5 dv 6 dh 7 dv 8 ϕ˜ 1 λφ˜  ⎢ ∗ −H3 0 0 0 0 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢∗ ∗ −V 0 0 0 0 0 0 0 0 0 ⎥ 3 ⎢ ⎥ ⎢∗ 0 0 0 0 0 0 0 0 ⎥ ∗ ∗ −H3 ⎢ ⎥ ⎢∗ ∗ ∗ ∗ −V3 0 0 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢∗ 0 0 0 0 0 0 ⎥ ∗ ∗ ∗ ∗ −H4 ⎢ ⎥<0 ⎢∗ 0 0 0 √0 0 ⎥ ∗ ∗ ∗ ∗ ∗ −V4 ⎢ ⎥ ⎢∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 0 0 λ√dh G a 0 ⎥ ⎢ ⎥ ⎢∗ 0 λ dv G b 0 ⎥ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 2 ⎢ ⎥ ⎢∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I 0 0 ⎥ ⎢ ⎥ ⎣∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −λI 0 ⎦ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −λI

(38)

123

Multidim Syst Sign Process

where ⎡

η11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ψ˜ = ⎢ ⎢ ∗ ⎢ ∗ ⎢ ⎣ ∗ ∗

η12 η13 Ad12 P 2 η22 Ad21 P 1 η24 ∗ η33 0 ∗ ∗ η44 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

η15 0 η35 0 η55 ∗ ∗

⎤ 0 B1 η26 B2 ⎥ ⎥ 0 0 ⎥ ⎥ η46 0 ⎥ ⎥, 0 0 ⎥ ⎥ η66 0 ⎦ ∗ −γ 2 I

T η11 = (A11 P 1 + E 1 Y1 ) + (P 1 A11 + Y1kT E 1T ) + H1 + H2 + T1 + T3 + T1T + T3T , T + Y1kT E 2T ), η13 = −T1 + T2 + T4T + T6T + Ad11 P 1 , η12 = (A12 P 2 + E 1 Y2 ) + (P 1 A21 T η22 = (A22 P 2 + E 2 Y2 ) + (P 2 A22 + Y2kT E 2T ) + V1 + V2 + U1 + U3 + U1T + U1T ,

η15 = −T2 − T3 + T7T + T9T , η24 = −U1 + U2 + U4T + U6T + Ad22 P 2 , η26 = −U2 − U3 + U7T + U9T , η33 = −T4 + T5 − T4T + T5T − (1 − μ1 )H1 , η35 = −T5 − T6 − T7T + T8T , η44 = −U4 + U5 − U4T + U5T − (1 − μ2 )V, η46 = −U5 − U6 − U7T + U8T , η55 = −T8 − T9 − T8T − T9T − H2 , η66 = −U8 − U9 − U8T − U9T − V2 , 1 = (H3 + H4 − 2P 1 ), 2 = (V3 + V4 − 2P 2 ),   ˜ 1 = TT 0 TT 0 TT 0 0 T ,  ˜ 2 = 0 UT 0 UT 0 UT 0 T ,  1 4 7 1 4 7   ˜ 3 = TT 0 TT 0 TT 0 0 T ,  ˜ 4 = 0 UT 0 UT 0 UT 0 T ,  2 8 2 8 5 5   ˜ 5 = TT 0 TT 0 TT 0 0 T ,  ˜ 6 = 0 UT 0 UT 0 UT 0 T ,  3 6 9 3 6 9  ˜ 7 = A11 P 1 + E 1 Y1 A12 P 2 + E 1 Y2 Ad11 P 1 Ad12 P 2 0 0 B1 T ,   ˜ 8 = A21 P 1 + E 2 Y1 A22 P 2 + E 2 Y2 Ad21 P 1 Ad22 P 2 0 0 B2 T ,   T ϕ˜1 = C1 P 1 + FY1 C2 P 2 + FY2 0 0 0 0 D ,  T  ˜ = W1a P 1 W1b P 2 W2a P 1 W2b P 2 0 0 W3 T , φ˜ = G aT G bT 0 0 0 0 0 ,  then under the following controller u(t1 , t2 ) = [ K 1 K 2 ]x(t1 , t2 ), K 1 = Y1 (P 1 )−1 , K 2 = Y2 (P 2 )−1 , the closed-loop system (17) has a prescribed H∞ disturbance attenuation level γ . Proof By Theorem 2, a sufficient condition for the closed-loop system (17) to have a prescribed H∞ disturbance attenuation level γ is that there exist positive definite symmetric matrices P1 , Q 1 , R1 , P2 , Q 2 , R2 , Z i (i = 1, . . . , 4) and any matrices N1 j , S1 j , M1 j , N2 j , S2 j , M2 j ( j = 1, . . . , 3), with appropriate dimensions, such that    ψ  + φ F(t , t ) + T F T (t , t )φ T < 0, 1 2 1 2 ∗

(39)

123

Multidim Syst Sign Process

where ⎡ 

11 P1 (A12 + E 1 K 2 ) + (A21 + E 2 K 1 )T P2 13 P1 Ad12 ⎢   ⎢ ∗ P2 Ad21 24

22 ⎢ ⎢ ∗  ∗

33 0 ⎢ ψ =⎢ ∗ ∗ ∗

44 ⎢ ⎢ ∗ ∗ ∗ ∗ ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

15 0 0

35 0

55 ∗ ∗

⎤ P1 B1

⎥

26 P2 B2 ⎥ ⎥ 0 0 ⎥ ⎥ ,

46 0 ⎥ ⎥ ⎥ 0 0 ⎥

66 0 ⎦ ∗ −γ 2 I



T T + M11 + P1 (A11 + E 1 K 1 ) + (A11 + E 1 K 1 )T P1 + Q 1 + R1 ,

11 = N11 + M11 + N11 



T T T T + M12 + P1 Ad11 , 24 = −N21 + S21 + N22 + M22 + P2 Ad22 ,

13 = −N11 + S11 + N12 

T T + M21 + P2 (A22 + E 2 K 2 ) + (A22 + E 2 K 2 )T P2 + Q 2 + R2 ,

22 = N21 + M21 + N21    √ √ √ √ √ √  √  √ = dh 1 dv 2 dh 3 dv 4 dh 5 dv 6 dh 7 dv 8 ϕ1 , T     7 = (Z 1 + Z 2 ) A11 (Z 1 + Z 2 ) A12 (Z 1 + Z 2 )Ad11 (Z 1 + Z 2 )Ad12 0 0 (Z 1 + Z 2 )B1 , T     8 = (Z 3 + Z 4 ) A21 (Z 3 + Z 4 ) A22 (Z 3 + Z 4 )Ad21 (Z 3 + Z 4 )Ad22 0 0 (Z 3 + Z 4 )B2 , 





A11 = A11 + E 1 K 1 , A12 = A12 + E 1 K 2 , A21 = A21 + E 2 K 1 , T  φ = G aT P1 G bT P2 0 0 0 0 0 0 0 0 0 0 0 φ1 φ2 0 ,   φ1 = dh G aT (Z 1 + Z 2 ), φ2 = dv G bT (Z 3 + Z 4 ),   = W1a W1b W2a W2b 0 0 W3 0 0 0 0 0 0 0 0 0 .



A22 = A22 + E 2 K 2 ,

According to Lemma 3, a sufficient condition for guaranteeing (39) is that there exists a positive number λ > 0 such that    ψ  + λφφ T + λ−1 T  < 0. ∗

(40)

Applying Lemma 2, we obtain that (40) is equivalent to the following LMI: ⎡ √ √ √ √ √ √ √  √  ψ dh 1 dv 2 dh 3 dv 4 dh 5 dv 6 dh 7 dv 8 ϕ1 λφ  ⎢ ⎢ ∗ −Z 1 0 0 0 0 0 0 0 0 0 0 ⎢ ⎢∗ ∗ −Z 3 0 0 0 0 0 0 0 0 0 ⎢ ⎢∗ ∗ ∗ −Z 1 0 0 0 0 0 0 0 0 ⎢ ⎢∗ ∗ ∗ ∗ −Z 0 0 0 0 0 0 0 3 ⎢ ⎢∗ ∗ ∗ ∗ ∗ −Z 0 0 0 0 0 0 2 ⎢ ⎢∗ ∗ ∗ ∗ ∗ ∗ −Z 4 0 0 0 0 0 ⎢ ⎢∗ ∗ ∗ ∗ ∗ ∗ ∗ ϕ 0 0 λϕ 0 11 12 ⎢ ⎢∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ϕ22 0 λϕ23 0 ⎢ ⎢∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −I 0 0 ⎢ ⎣∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −λI 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −λI

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(41)

123

Multidim Syst Sign Process

with

  T T φ = G aT P1 G bT P2 0 0 0 0 0 ,  = W1a W1b W2a W2b 0 0 W3 ,  ϕ11 = −(Z 1 + Z 2 ), ϕ22 = −(Z 3 + Z 4 ), ϕ12 = dh (Z 1 + Z 2 )G a ,  ϕ23 = dv (Z 3 + Z 4 )G b .

Set H1 = P1−1 Q 1 P1−1 ,

H2 = P1−1 R1 P1−1 , V1 = P2−1 Q 2 P2−1 , V2 = P2−1 R2 P2−1 ,

T1 = P1−1 N11 P1−1 , T2 = P1−1 S11 P1−1 , T3 = P1−1 M11 P1−1 , T4 = P1−1 N12 P1−1 , T5 = P1−1 S12 P1−1 , T6 = P1−1 M12 P1−1 , T7 = P1−1 N13 P1−1 , T8 = P1−1 S13 P1−1 , U1 = P2−1 N21 P2−1 , U2 = P2−1 S21 P2−1 , U3 = P2−1 M21 P2−1 , U4 = P2−1 N22 P2−1 , U5 = P2−1 S22 P2−1 , U6 = P2−1 M22 P2−1 , U7 = P2−1 N23 P2−1 , U8 = P2−1 S23 P2−1 , U9 = P2−1 M23 P2−1 ,

H3 = P1−1 Z 1 P1−1 , V3 = P2−1 Z 3 P2−1 ,

H4 = P1−1 Z 2 P1−1 , V4 = P2−1 Z 4 P2−1 ,

P 1 = P1−1 ,

P 2 = P2−1 , Y1 = K 1 P 1 , Y2 = K 2 P 2 .

Pre- and post-multiplying (41) by  diag P1−1 , P2−1 , P1−1 , P2−1 , P1−1 , P2−1 , I, P1−1 , P2−1 , P1−1 , P2−1 , P1−1 , P2−1 ,  (Z 1 + Z 2 )−1 , (Z 3 + Z 4 )−1 , I, I, I and using the following equations T

T

P 1 (Z 1 + Z 2 )P 1 ≥ 2P 1 − (Z 1 + Z 2 )−1 , P 2 (Z 3 + Z 4 )P 2 ≥ 2P 2 − (Z 3 + Z 4 )−1 , (42) we can obtain that (41) is satisfied if (38) holds. This completes the proof.

4 Illustrate examples In this section, we present two examples to illustrate the effectiveness of the proposed approach. Example 1 Consider the dynamical processes in gas absorption, water stream heating and air drying, which can be described by the following Darboux equation with time delays (Dymkou et al. 2008): ∂ 2 s(x, t) ∂s(x, t) ∂s(x, t) = a1 +a2 + a0 s(x, t)+a3 s(x, t −d2 (t))+a4 f (x, t), ∂ x∂t ∂t ∂x

(43)

where s(x, t) is an unknown function at x(space) ∈ [0, x f ] and t (time) ∈ [0, ∞), a0 , a1 , a2 , a3 and a4 are real coefficients, d2 (t) is the time-varying delay and f (x, t) is the input

123

Multidim Syst Sign Process Table 1 Upper bounds on the allowable time delay for different μ2 Methods

μ2

0

0.1

0.3

0.4

0.6

0.9

El-Kasri et al. (2013)

τ2

2.4601

2.3752

2.1843

2.0749

1.8126

1.1418

Corollary 1

dv

4.0509

4.0027

3.9829

3.9655

3.9359

3.9013

function. Define r (x, t) as follows: r (x, t) =

∂s(x, t) − a2 s(x, t). ∂t

Then (43) can be converted into a Roesser model of the following form:    h     h   ∂ x (t1 ,t2 ) a3 v a4 a1 a0 + a1 a2 x (t1 , t2 ) ∂t1 + x u(t1 , t2 ), (t , t − d (t ))+ = v 1 2 2 2 ∂ x (t1 ,t2 ) 1 a2 0 0 x v (t1 , t2 ) ∂t2

where x h (t1 , t2 ) = r (x, t), x v (t1 , t2 ) = s(x, t) and u(t1 , t2 ) = f (x, t). For specific parameters a0 = 0.2, a1 = −3, a2 = −1, a3 = −0.4 and a4 = 0, Table 1 given below shows the upper bounds on allowable time delay for different μ2 . We can observe that our result is less conservative than that proposed in El-Kasri et al. (2013). Example 2 Consider the 2-D continuous system (1) with parameters as follows:      −1.68 −1.4 −0.01 0 −1.1 0 , A21 = , A22 = , 0.91 0.2 0 −0.6 0 −2.1         −0.3 0 −0.4 0.04 −0.4 0.1 0.01 0 , Ad12 = , Ad21 = , E1 = , 0 −0.6 −0.4 −0.4 0.4 0.4 0 0.02         0.17 0 0.11 0 0 0.04 0 0.01 , E2 = , B1 = , B2 = , −0.2 −0.18 0.1 0.13 0.03 0 0.01 0         −0.01 0 −0.2 0 0.1 0 0.02 0 , C2 = , D= , F= , 0 −0.1 0 −0.9 0 0.1 0 0.04           1 0 0.4 0.2 0.3 0.1 0.1 0 0.2 0.1 , Gb = , W1a = , W1b = , W3 = , 0.3 0.1 0 0.3 0.2 0 0 0 0.2 0       0 0.11 0.12 −0.01 0 sin(0.5π(t1 + t2 )) , , W2b = , F(t1 , t2 ) = 0 sin(0.5π(t1 + t2 )) 0 0.16 0 0  A11 = Ad11 = Ad22 = C1 = Ga = W2a =

2 0

 0 , −0.9



A12 =

d1 (t1 ) = 0.5 + 0.5 sin (0.5πt1 ) , d2 (t2 ) = 1 + sin (0.5πt2 ) ,

where n 1 = 2 and n 2 = 2. Take γ = 1, dh = 1, dv = 2, μ1 = 0.1, and μ2 = 0.2, then by solving (38) in Theorem 3, we obtain λ = 1.1376 and the following matrices: 

H1 =  H4 =  V3 =  P2 =

2.8888 0.3881 0.0903 0.0272 1.6155 −1.7948 3.2811 −2.7413

123

     0.3881 0.1337 0.1139 0.3066 −0.2698 , H2 = , H3 = , 3.4798 0.1139 0.4112 −0.2698 2.8044      0.0272 0.7597 −0.7911 0.2616 −0.3046 , V1 = , V2 = , 0.2220 −0.7911 0.9711 −0.3046 0.3772      −1.7948 0.4000 −0.4683 4.6521 −0.4544 , V4 = , P1 = , 2.0840 −0.4683 0.5741 −0.4544 3.4690      −2.7413 −0.3093 0.2873 −0.0010 0.0019 , T1 = , T2 = , 3.4889 0.2712 −2.8117 0.0002 0.0018

Multidim Syst Sign Process

 T3 =  T6 =  T9 =  U3 =  U6 =  U9 =

−0.0908 −0.0268 0.0008 0.0019 0.0899 0.0249 −0.3870 0.4583 −0.0447 0.0490 0.3188 −0.3762

     −0.0239 0.3078 −0.2789 −0.3046 0.2644 , T4 = , T5 = , −0.2228 −0.2627 2.7572 0.2674 −2.7669      −0.0023 0.0009 −0.0049 0.3053 −0.2665 , T7 = , T8 = , −0.0040 −0.0083 0.0543 −0.2671 2.7603      0.0268 −1.3908 1.5193 −0.1793 0.2167 , U1 = , U2 = , 0.2262 1.5572 −1.7796 0.2134 −0.2607      0.4510 0.6912 −0.7881 −0.8877 0.9928 , U4 = , U5 = , −0.5573 −0.7733 0.9215 0.9820 −1.1432      0.0486 0.2946 −0.2885 1.0142 −1.1456 , U7 = , U8 = , −0.0545 −0.3379 0.3453 −1.1376 1.3331      −0.3679 2.0800 −2.7938 11.7936 −14.8657 , Y1 = , Y2 = . 0.4516 −5.3587 13.3063 −42.8410 55.9897

The controller parameters are given as follows:  K1 =

0.3732 −0.7873

 −0.7565 , 3.7326

 K2 =

0.1006 1.0208

 −4.1818 . 16.8499

The trajectories of horizontal and vertical states are depicted in Figs. 1, 2, 3 and 4, respectively, where the boundary conditions are given as: x h (υ1 , t2 ) = [ 0.2 0.2 ]T , ∀ − 1 ≤ υ1 ≤ 0, 0 ≤ t2 ≤ 1; x h (υ1 , t2 ) = 0, ∀ − 1 ≤ υ1 ≤ 0, t2 ≥ 1; x v (t1 , υ2 ) = [ 0.2 0.2 ]T , ∀ − 2 ≤ υ2 ≤ 0, 0 ≤ t1 ≤ 1; x v (t1 , υ2 ) = 0, ∀ − 2 ≤ υ2 ≤ 0, t1 ≥ 1. The disturbance is w(t1 , t2 ) = 0.02e−0.05(t1 +t2 ) . It can be seen that the closed-loop system is asymptotically stable. For a fixed H∞ disturbance attenuation performance level γ = 1, Table 2 given below illustrates the impact of different values of dh , dv , μ1 and μ2 upon the feasibility of LMI (38) presented in Theorem 3 for the same system parameters given in Example 2.

Xh1(t1,t2)

0.4 0.2

0 -0.2 -0.4 4 3

4 3

2

t2

2

1

1 0

0

t1

Fig. 1 The trajectory of horizontal state x1h (t1 , t2 )

123

Multidim Syst Sign Process

Xh2(t1,t2)

0.2 0.1

0 -0.1 -0.2 4 3

4 3

2

t2

2

1

1 0

t1

0

Fig. 2 The trajectory of horizontal state x2h (t1 , t2 )

0.2

Xv1(t1,t2)

0.15 0.1 0.05 0 -0.05 4 3

4 3

2

t2

2

1

1 0

t1

0

Fig. 3 The trajectory of vertical state x1v (t1 , t2 )

Xv2(t1,t2)

0.3 0.2 0.1 0 -0.1 4 4

3 3

2

t2

2

1

1 0

Fig. 4 The trajectory of vertical state x2v (t1 , t2 )

123

0

t1

Multidim Syst Sign Process Table 2 The feasibility of (38) for a fixed γ = 1 and different values of dh = dv and μ1 = μ2 dh

0.5

1

μ1

0.3

Feasibility

Feasible

0.6

0.9

0.3

1.2 0.6

Feasible

0.9

1.5

0.3

0.6

0.9

Feasible

Infeasible

0.3

0.6

0.9

Infeasible

5 Conclusions The robust stability and H∞ control problems of uncertain 2-D continuous systems with timevarying delays have been addressed in this paper. A new delay-dependent robust asymptotical stability criterion is obtained for uncertain 2-D continuous state delayed systems. The freeweighting matrix approach is introduced to reduce the conservatism. A sufficient condition for such systems to have H∞ performance is developed. A state feedback controller design methodology is developed such that the resultant closed-loop system has a prescribed H∞ disturbance attenuation level γ . All results are presented in LMIs, which can be solved by LMI toolbox. Our future work will focus on the finite frequency H∞ control of 2-D continuous systems (Li and Gao 2012; Li et al. 2012; Yang et al. 2008). Acknowledgments This work was supported by the National Natural Science Foundation of China under Grant No. 61273120 and the Postgraduate Innovation Project of Jiangsu Province (Grant No. CXZZ13_0208).

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Multidim Syst Sign Process Xu, H., Zou, Y., & Xu, S. (2005b). Robust H∞ control for a class of uncertain nonlinear two-dimensional systems. International Journal of Innovative Computing, Information and Control, 1(2), 181–191. Xu, H., Zou, Y., Xu, S., & Guo, L. (2008). Robust H∞ control for uncertain two-dimensional discrete systems described by the general model via output feedback controllers. International Journal of Control, Automation, and Systems, 6(5), 785–791. Xu, J., Nan, Y., Zhang, G., Ou, L., & Ni, H. (2013). Delay-dependent H∞ control for uncertain 2-D discrete systems with state delay in the Roesser model. Circuits, Systems, and Signal Processing, 32(3), 1097– 1112. Xu, S., Lam, J., Zou, Y., Lin, Z., & Paszke, W. (2005). Robust H∞ filtering for uncertain 2-D continuous systems. IEEE Transactions on Signal Processing, 53(5), 1731–1738. Yan, H., Zhang, H., & Meng, M. Q.-H. (2010). Delay-range-dependent robust H∞ control for uncertain systems with interval time-varying delays. Neurocomputing, 73(7), 1235–1243. Yang, R., Xie, L., & Zhang, C. (2008). Generalized two-dimensional Kalman–Yakubovich–Popov lemma for discrete Roesser model. IEEE Transactions on Circuits and Systems I: Regular Papers, 55(10), 3223– 3233. Ye, S., Li, J., & Yao, J. (2014). Robust H∞ control for a class of 2-D discrete delayed systems. ISA Transactions. doi:10.1016/j.isatra.2013.12.017. Zames, G. (1981). Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses. IEEE Transactions on Automatic Control, 26(2), 301–320. Imran Ghous received his B.Sc. and M.Sc. degree in Electrical Engineering from University of Engineering & Technology, Taxila, Pakistan, in 2011 and 2013 respectively. He has been appointed as a lecturer at COMSATS Institute of Information Technology, Lahore, Pakistan. He is currently pursuing his Ph.D. degree in Control Theory and Control Engineering at School of Automation, Nanjing University of Science and Technology, China. His research interests include twodimensional systems and switched systems.

Zhengrong Xiang received his Ph.D. degree in Control Theory and Control Engineering at Nanjing University of Science and Technology, Nanjing, China, in 1998. Since 1998 he has been faculty member and he is currently full professor at Nanjing University of Science and Technology. He was appointed as Lecturer in 1998 and Associate Professor in 2001 at Nanjing University of Science and Technology. He is a member of the IEEE, member of the Chinese Association for Artificial Intelligence, and a very active reviewer for many international journals. His main research interests include switched systems, nonlinear control, robust control, and networked control systems.

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