Gain-Scheduled H∞ Controller Synthesis for 2-D Discrete-Time Linear Parameter-Varying Systems*

Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011

Gain-Scheduled H∞ Controller Synthesis for 2-D Discrete-Time Linear Parameter-Varying Systems ⋆ Jefferson Osowsky Carlos E. de Souza Department of Systems and Control, Laborat´orio Nacional de Computac¸a˜ o Cient´ıfica – LNCC / MCT, Petr´opolis, RJ 25651-075, Brazil. (e-mails: [email protected], [email protected]) Abstract: This paper proposes a gain-scheduled H∞ control design method for 2-dimensional discretetime linear parameter-varying systems. The system is described by a Roesser state-space model with matrices depending affinely on time-varying parameters whose admissible values are assumed to belong to a given convex bounded polyhedral domain. A convex optimization approach in terms of linear matrix inequalities is devised for designing a gain-scheduled H∞ control via static state feedback. The controller design is based on a parameter-dependent Lyapunov function and the controller gain is a matrix fraction of quadratic polynomial matrices on the scheduling parameters. Keywords: Two-dimensional systems, gain-scheduled control, H∞ control, parameter-varying systems, discrete-time systems, parameter-dependent Lyapunov function. 1. INTRODUCTION Two-dimensional (2-D) processes are quite often encountered in nature and engineering and thus 2-D systems find application in a wide range of different fields, such as filtering, image processing, seismographic data processing, gas absorption, water stream heating, thermal processes, etc (see, for instance, Kaczorek [1985], Lu and Antoniou [1992], Du and Xie [2002] and the references therein). Over the past two decades 2-D discrete-time linear systems have been attracting significant interest within the control community and significant advances have been achieved in the theory of control synthesis to solve a variety of problems, as for instance, dynamic output feedback stabilization (Bisiacco ˇ [1985], Du and Xie [1999a]), H∞ control (Sebek [1993], Du et al. [2001], Xie et al. [2002] and the references therein), state feedback robust stabilization (Du and Xie [1999b], Gao et al. [2005]), dynamic output feedback robust stabilization (Du et al. [2001], Xie et al. [2002]), robust H∞ control (Du and Xie [2002]), H2 and mixed H2 /H∞ control (Yang et al. [2006]), linear quadratic Gaussian control (Yang et al. [2007]), and guaranteed cost control (Guan et al. [2001], Dhawan and Ka [2007, 2010, 2011]). In spite of all these developments, little attention has been devoted in the literature to the problem of gain-scheduled control of 2-D linear parameter-varying (LPV) systems. In a recent work of Wu et al. [2009], a method has been proposed for the design of a gain-scheduled H∞ dynamic output feedback control for 2-D discrete-time LPV systems represented by a Fornasini-Marchesini local state-space model. The latter method has the feature that it is based on a parameterdependent Lyapunov function, however it is given in terms of some parameter-dependent linear matrix inequalities (LMIs), and as such the control calculation involves solving an infinite number of LMIs, even when the system matrices and Lyapunov ⋆ This work was supported by CNPq, Brazil, under grants 14.0555/2008-0/GD and 30.3440/2008-2/PQ.

978-3-902661-93-7/11/$20.00 © 2011 IFAC

function are restricted to be affine in the parameters, and in the case of static state feedback. To overcome this difficulty, the authors apply a gridding technique of the parameter value set, similar to that proposed in Scherer [1996]. As a consequence, the controller guarantees stability and H∞ performance only for values of the system parameters in the chosen set of gridding points. To the best of our knowledge, the problem of gainscheduled H∞ control for 2-D systems based on parametric Lyapunov functions is far from being fully resolved, even in the case of LPV systems with affine parameter dependence and for static state feedback. In this paper we consider the problem of gain-scheduled H∞ control for 2-D discrete-time LPV systems described by a Roesser state-space model with matrices that depend affinely on time-varying parameters. The system parameters are assumed to be measured on-line but their trajectory waveforms are not know in advance. It is only assumed that the parameters evolve in a given convex bounded polyhedral domain. This paper develops a method for designing a (static) state feedback gainscheduled H∞ controller based on a parametric Lyapunov function with high order dependence on the scheduling parameters. The proposed design method has the following features: (a) the controller gain is a matrix fraction of polynomial matrices with quadratic dependence on the scheduling parameters; (b) the design method does not involve gridding the parameter value set and is tailored in terms of a finite number of LMIs. Numerical examples are presented to illustrate the method developed in this paper. Notation. Z+ is the set of nonnegative integers, Rn is the ndimensional Euclidean space, Rm×n is the set of m × n real matrices, the k-component of a vector x is denoted by x[ k ] , In is the n × n identity matrix, 0n and 0m×n are the n × n and m × n matrices of zeros, respectively, and diag{· · · } stands for a block-diagonal matrix. For a real matrix S, ST denotes its transpose, Her{S} stands for S + ST and S > 0 means that S is symmetric and positive-definite. For a symmetric block

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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

matrix, the symbol ⋆ stands for the transpose of the blocks outside the main diagonal block. ℓ2 denotes the space of finite energy 2-D discrete-time vector signals ui, j with finite ℓ2 -norm ∞ 2 kuk2 defined by kuk22 = ∑∞ i=0 ∑ j=0 kui, j k , where k · k stands for the Euclidean vector norm. For two polytopes A ⊂ Rn and B ⊂ Rm , the notation A × B means that (A × B) ⊂ R(n+m) is a meta-polytope obtained by the Cartesian product, and V (A ) is the set of all the vertices of A . 2. PROBLEM STATEMENT Consider the class of 2-D discrete-time linear parametervarying systems represented by the following Roesser model:  + x = A(θi, j )xi, j + Bu (θi, j )ui, j + Bw (θi, j )wi, j ,    i, j zi, j = C(θi, j )xi, j + Du(θi, j )ui, j + Dw (θi, j )wi, j , (1)    x = [ (x h )T (x v )T ]T , x+ = [ (x h )T (x v )T ]T i, j

i, j

i, j

with

θi, j p

i, j

i+1, j

i, j+1

h i [1] [ p] T ∈ Rp = θi, j . . . θi, j [k]

(2) p

kHzw k∞ = sup w∈ℓ2

p

k=1 p

[k]

[k]

k=1 p

[k]

[k]

Du (θi, j ) = Du0 + ∑ θi, j Duk , Dw (θi, j ) = Dw0 + ∑ θi, j Dwk k=1

k=1

(5) where ∈ and ∈ are the horizontal and vertical state vectors, respectively, ui, j ∈ Rnu is the control input, wi, j ∈ Rnw is the disturbance input, which is supposed to be an arbitrary signal in ℓ2 , zi, j ∈ Rnz is the controlled output, Ai , Bui , Bwi , Ci , Dui , Dwi , i = 0, . . ., p, are known constant real [k] matrices with appropriate dimensions and θi, j , k = 1, . . . , p are real bounded time-varying parameters. It is assumed that the parameter vector θi, j lies in a given polytopic region Ξ ⊂ R p for all i, j ∈ Z+ . Furthermore, let n = n1 + n2. xi,h j

Rn 1

xi,v j

Rn 2

This paper addresses the problem of gain-scheduled H∞ control of system (1). Specifically, we consider the design of a stabilizing static state feedback control law with a gain that is adjustable on-line as function of the parameter vector θi, j , i.e. ui, j = K(θi, j )xi, j , and such that the resulting closed-loop system as below ( + xi, j = Ac (θi, j )xi, j + Bw (θi, j )wi, j , (6) zi, j = Cc (θi, j )xi, j + Dw (θi, j )wi, j where (7) Ac (θi, j ) = A(θi, j ) + Bu (θi, j )K(θi, j ), Cc (θi, j ) = C(θi, j ) + Du (θi, j )K(θi, j ),



(9)

Attention is focused on developing a control synthesis method based on a parameter-dependent Lyapunov function for the closed-loop system (6) and where the controller gain K(θi, j ) is a matrix fraction of polynomial matrices with quadratic dependence on θi, j , namely K(θi, j ) is assumed to have the following right matrix fraction description: K(θi, j ) = F (θi, j )X −1 (θi, j )

(10)

where F (θi, j ) and X (θi, j ) are polynomial matrices with quadratic dependence on the parameter vector θi, j and X (θi, j ) is nonsingular for all θi, j ∈ Ξ. The problem addressed in this paper is as follows: Given a scalar γ > 0, determine a static state feedback controller with a gain K(θi, j ) as in (10) that ensures the asymptotic stability of system (6) and kHzw k∞ < γ for all trajectories of θi, j in Ξ.

[k]

Bw (θi, j ) = Bw0 + ∑ θi, j Bwk , C(θi, j ) = C0 + ∑ θi, j Ck , (4) k=1 p

kzk2 : w 6= 0 kwk2

v = 0, subject to zero boundary conditions, namely x0,h j = 0, xi,0 + ∀ i, j ∈ Z .

A(θi, j ) = A0 + ∑ θi, j Ak , Bu (θi, j ) = Bu0 + ∑ θi, j Buk , (3) k=1



(8)

achieves a prescribed performance in the H∞ sense for all [k] trajectories of θi, j . To this end, the parameters θi, j , k = 1, . . . , p are assumed to be measurable on-line, but their trajectories are not available a priori. It is only known that the admissible values of θi, j , for all i, j ∈ Z+ , lie in the polytope Ξ. As an abuse of terminology, the trajectories of θi, j are said to lie in Ξ. To measure the controller performance, let the ℓ2 -induced gain of the closed-loop system in (6) for a given trajectory of θi, j in Ξ (Du and Xie [2002]):

3. PRELIMINARY RESULTS This section presents some basic results needed to derive an LMI method for solving the gain-scheduled H∞ control problem. We first present a characterization of the ℓ2 -induced gain of time-varying 2-D discrete-time systems described by a Roesser model as follows: ( + xi, j = Ai, j xi, j + Bi, j wi, j , (11) zi, j = Ci, j xi, j + Di, j wi, j n nw where xi, j ∈ Rn and x+ i, j ∈ R are as in system (1), wi, j ∈ R , n zi, j ∈ R z , and Ai, j , Bi, j , Ci, j and Di, j are bounded real valued 2-D matrix sequences with appropriate dimensions.

Lemma 1. Given a scalar γ > 0, system (11) is uniformly asymptotically stable and kHzw k∞ < γ if there exist bounded (k) matrices Pi, j ∈ Rnk ×nk , ∀ i, j ∈ Z+ , k = 1, 2, such that (1)

(2)

Pi, j > 0, Pi, j > 0, ∀ i, j ∈ Z+

(12)

T T 2 T (xi,+j )T Pi,+j x+ i, j − xi, j Pi, j xi, j + zi, j zi, j − γ wi, j wi, j < 0, + ∀ x+ i, j , xi, j , wi, j satisfying (11), ∀ i, j ∈ Z

(13)

where

o n o n (1) (2) (1) (2) Pi, j = diag Pi, j , Pi, j , Pi,+j = diag Pi+1, j , Pi, j+1 . (14)

Proof. The stability result follows directly from Theorem 2 (c) in Kurek [1995], whereas the condition kHzw k∞ < γ can be established using similar arguments as in the proof of Theorem 2.4.1 in Du and Xie [2002].  Remark 1. Observe that in the light of the results in Kurek [1995], the function  T
T (2) (1) V (x, i, j) := xTi, j Pi, j xi, j = xi,h j Pi, j xi,h j + xi,v j Pi, j xi,v j

plays a role similar to that of a Lyapunov function for 1-D timevarying systems. In view of this, we shall refer to V (x, i, j) as a Lyapunov function for system (11). 

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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

Next, we present two equivalent formulations of the boundedreal lemma (BRL) for system (11). Lemma 2. Given a scalar γ > 0, system (11) is uniformly asymptotically stable and kHzw k∞ < γ if either of the following equivalent conditions hold: (k)

(a) There exist bounded matrices Pi, j ∈ Rnk ×nk , ∀ i, j ∈ Z+ , k = 1, 2, satisfying the following inequalities:   −Pi, j (Pi,+j Ai, j )T 0 Ci,T j    ⋆ −Pi,+j Pi,+j Bi, j 0   < 0, ∀ i, j ∈ Z+   2 T  γ I D ⋆ ⋆ −  i, j  ⋆ ⋆ ⋆ −I (15) where the matrices Pi, j and Pi,+j are as in (14).

X +(θ˘i, j ) = diag{ X1 (θi+1, j ), X2 (θi, j+1 )},

and for notation simplicity the argument θi, j of Ac (θi, j ), Bw (θi, j ), Cc (θi, j ) and Dw (θi, j ) has been omitted. Note that the inequality in (18) is nonconvex in (θi, j , θ˘i, j ) and thus calculating the controller gain directly from this inequality involves solving an infinite number of LMIs. To overcome this difficulty, we shall convexify (18) by introducing appropriate decompositions of the system matrices and Xk (θi, j ), k = 1, 2, and applying the following version of Finsler’s lemma. Lemma 3. Given matrix functions Γ(v)∈ Rr×nσ , ϒ(v)=ϒT (v)∈ Rnσ ×nσ and σ (v) ∈ Rnσ , with v ∈ D ⊆ Rr , then

σ T (v)ϒ(v)σ (v) < 0, ∀ v ∈ D : Γ(v)σ (v) = 0, σ (v) 6= 0 if there exists a matrix R such that

(k)

(b) There exist bounded matrices Xi, j , ∈ Rnk ×nk , ∀ i, j ∈ Z+ , k = 1, 2, satisfying the following inequalities:   −Xi, j Xi,Tj ATi, j 0 Xi,Tj Ci,T j    ⋆  −Xi,+j Bi, j 0   < 0, ∀ i, j ∈ Z+ (16)   2 T ⋆ −γ I Di, j   ⋆ ⋆ ⋆ ⋆ −I where

(1)

(2)

(1)

(2)

Xi, j = diag{ Xi, j , Xi, j }, Xi,+j = diag{ Xi+1, j , Xi, j+1 } (17)

ϒ(v) + Her{R Γ(v) } < 0, ∀ v ∈ D.

where

4. CONTROL DESIGN

where

T  T T , θ˘i, j = θi+1, j θi, j+1

X (θi, j ) = diag{ X1(θi, j ), X2(θi, j )},

  e θi, j ) = In ΩT(θi, j ) T , Ω(θi, j ) = θi, j ⊗ In Ω(   e z (θi, j ) = In ΩT(θi, j ) T , Ωz (θi, j ) = θi, j ⊗ In Ω z z z

(23)

(24) (25) (26) (27)

where ⊗ denotes Kronecker product. Furthermore, let the following general decomposition for the quadratic parameterdependent matrices Xk (θi, j ), k = 1, 2: n o (0) (1) Xk (θi, j ) = Xk + Her Xk Ωk (θi, j ) (2)

In this section we develop an LMI method for designing a gainscheduled H∞ state feedback control law with a gain K(θi, j ) as in (10). The method is based on a parametric version of Lemma 2 (b). More specifically, we shall determine a state feedback gain K(θi, j ) such that for a given scalar γ > 0 there exist real nk × nk quadratic matrix functions Xk (·), k = 1, 2 satisfying the following inequality:   −X (θi, j ) X T (θi, j )ATc 0 X T (θi, j )CcT     0 ⋆ −X +(θ˘i, j ) Bw   < 0,   ⋆ ⋆ −γ 2 I DTw   ⋆ ⋆ ⋆ −I ∀ θi, j ∈ Ξ, θ˘i, j ∈ (Ξ × Ξ)

iT h T  A = AT0 · · · ATp , Bu = BTu0 · · · BTup iT h T  Bw = BTw0 · · · BTw p , C = C0T · · · CTp

h iT h iT Du = DTu0 · · · DTup , Dw = DTw0 · · · DTw p

(a) ⇔ (b): First, note that (15) ensures that Pi, j > 0 and Pi,+j > 0, ∀ i, j ∈ Z+ . Next, let   o n (k) (k) −T (1) (2) Xi, j = Pi, j , k = 1, 2, Xi, j = diag Xi, j , Xi, j , Pre- and post-multiplying (15) by TTi, j and Ti, j , respectively, it can be easily verified that (16) is equivalent to (15). .



The control design method builds on the following representation of the matrices of system (1):  e T(θi, j ) A , Bu (θi, j ) = Ω e T(θi, j )Bu , A(θi, j ) = Ω    e T(θi, j )Bw , C(θi, j ) = Ω e Tz(θi, j ) C, (22) Bw (θi, j ) = Ω    e T(θi, j ) Du , , Dw (θi, j ) = Ω e T (θi, j ) Dw Du (θi, j ) = Ω z z

Proof. Part (a) follows directly from Lemma 1 by considering the system model of (11).

o o n n (1) (2) Xi,+j = diag Xi+1, j , Xi, j+1 , Ti, j = diag Xi, j , Xi,+j , Inw , Inz .

(21)

+ (Ωk (θi, j ))T Xk Ωk (θi, j )

Ωk (θi, j ) = θi, j ⊗ Ink (0)

where Xk

(28)

(29)

(2)

and Xk , k = 1, 2 are symmetric matrices.

Note that in view of (28)-(29), the matrices X (θi, j ) and X +(θ˘i, j ) of (20) and (21) respectively, can be rewritten as  b T (θi, j ) X Ω( b θi, j ),  X (θi, j ) = Ω  T (30)  X +(θ˘i, j ) = Ω e +(θ˘i, j ) X Ω e +(θ˘i, j ) where

(18)

(19) (20) 10170

X=

"

X0 X1 T

X1

#

X2 o n (i) (i) X i = diag X1 , X2 , i = 0, 1, 2 h iT b θi, j ) = I ΩT (θ ) Ω( n i, j

(31) (32) (33)

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

 Ω(θi, j ) = diag Ω1 (θi, j ), Ω2 (θi, j )

  e +(θ˘i, j ) = In (Ω+(θ˘i, j ))T T Ω  ˘ Ω+ i, j(θi, j ) = diag Ω1 (θi+1, j ), Ω2 (θi, j+1 ) .

(34) (35) (36)

Moreover, considering (26), it can be readily verified that b θi, j ) = Π e θi, j ), Π e Ω( e = diag { In , Π } Ω( # " I p ⊗ [ In1 0n1 ×n2 ] . Π= I p ⊗ [ 0n2×n1 In2 ]

(37) (38)

In order to present the control design method, introduce the symmetric matrix function U (θi, j ) as follows:   eTX Π e −Π ⋆ ⋆ ⋆    U1 (θi, j ) −J2T XJ2 ⋆ ⋆   (39) U (θi, j ) =    ⋆  0 BwT J1 −γ 2 I  U2 (θi, j ) 0 Dw −I

where the matrix X is as defined in (31) and (32),   b T (θi, j )X Π e T (θi, j )F , e + Bu Ω U1 (θi, j ) = J1T A Ω u

Then, the matrix function X (·) as defined in (30) is nonsingular over Ξ and the gain-scheduled control law ui, j = K(θi, j )xi, j with a parameter-varying gain K(θi, j ) as follows: ( K(θi, j ) = F (θi, j )X −1 (θi, j ), (46) e Tu (θi, j )F Ω( e θi, j ) F (θi, j ) = Ω ensures that the closed-loop system (6) is uniformly asymptotically stable and kHzw k∞ < γ for all trajectories of θi, j in Ξ. Proof. First, note that if the LMIs in (40) hold then, by convexity it follows that U (θi, j ) + Her{ LG(θi, j , θ˘i, j ) } < 0, ∀ (θi, j , θ˘i, j ) ∈ ( Ξ × Ξa ) (47) In the sequel it will be shown that (47) ensures that the closedloop system (6) with the feedback gain in (46) satisfies (18) with quadratic matrix functions X (·) and X +(·) as defined previously. For the controller gain in (46) and considering (22)-(39), it can be established that (18) is equivalent to ( T η U (θi, j )η < 0, η = Ψ(θi, j , θ˘i, j )ξ , ∀ ξ ∈ Rnξ , (48) ξ 6= 0, ∀ (θi, j , θ˘i, j ) ∈ (Ξ × Ξa) where

e T(θi, j )F, b T (θi, j )X Π e + Du Ω U2 (θi, j ) = C Ω u   T e u (θi, j ) = In ΩT(θi, j ) , Ωu (θi, j ) = θi, j ⊗ In , Ω u u u     I 0 0 I 0 0 J1 = n , J2 = n , 0 I pn 0 pn 0 0 pn I pn  I = diag Inz , 0 pnz

n o e θi, j ), Ω e a (θi, j , θ˘i, j ), In , Ω e z (θi, j ) , Ψ(θi, j , θ˘i, j ) = diag Ω( w   e a (θi, j , θ˘i, j ) = In ΩT (θi, j ) (Ω+(θ˘i, j ))T T . Ω

and F is a real (p+1)nu × (p+1)n matrix to be found. Also, let the meta-polytope Ξa = (Ξ × Ξ) ⊂ R2p .

Moreover, as n o η : η = Ψ(θi, j , θ˘i, j )ξ , ∀ ξ ∈ Rnξ , ξ 6= 0 o n = η : G(θi, j , θ˘i, j )η = 0, η 6= 0

where G(·) is as in (42), it follows that (48) is equivalent to

The gain-scheduled H∞ control design method is presented in the next theorem. Theorem 1. Consider system (1) and let Ξ be a given polytope of admissible θi, j . Given a scalar γ > 0, suppose there exist (i) (0) (2) nk ×nk matrices Xk , i = 0, 1, 2, k = 1, 2, with Xk and Xk , k = 1, 2 symmetric, and matrices F and L satisfying the following LMIs: U (α ) + Her{ LG(α , δ ) } < 0, ∀ (α , δ ) ∈ V ( Ξ × Ξa ) (40) where the matrix X is as in (31) and (32), α ∈ R p and T  δ = δ1T δ2T ∈ R2p , δk ∈ R p , k = 1, 2   0 0 0 G1 (α ) G2 (α , δ ) 0 0  G(α , δ ) =  0 0

0

0

(41)

(42)

G3 (α )

G1 (α ) = [ Ω(α ) −I pn ], G3 (α ) = [ Ωz (α ) −I pnz ] # " Ω(α ) −I pn 0 pn G2 (α , δ ) = Ω+(δ ) 0 pn −I pn Ω+(δ ) = diag {Ω1 (δ1 ), Ω2 (δ2 )} .

(43) (44) (45)

η T U (θi, j )η < 0, ∀ η 6= 0 : G(θi, j , θ˘i, j )η = 0, ∀ (θi, j , θ˘i, j ) ∈ (Ξ × Ξa).

(49)

Now, by Lemma 3, (47) ensures that (49) holds, or equivalently, the inequality in (18) is satisfied. Next, (18) implies the nonsingularity of the matrix function X (·) over the polytope Ξ. Finally, in view of Lemma 2 (b), it follows that the closedloop system of (6) with the controller gain in (46) is uniformly asymptotically stable and kHzw k∞ < γ for all trajectories of θi, j in Ξ.  Theorem 1 provides a convex optimization approach for designing a gain-scheduled state feedback H∞ control for the 2-D system in (1). The proposed design is tailored in terms of a finite number of LMIs that are required to be satisfied at the vertices of the polytope (Ξ × Ξ × Ξ) of admissible values of (θi, j , θi+1, j , θi, j+1 ). This is in contrast with the gain-scheduled H∞ control design of Wu et al. [2009] when specialized to static state feedback. Note that the latter design is given in terms of parameterized LMIs that must be satisfied for all (θi, j , θi+1, j , θi, j+1 ) in (Ξ × Ξ × Ξ), and a gridding technique of the parameter value set is applied to these inequalities. This technique is, however, known to, in general, involve increased computational effort as it requires using a fine gridding of the parameter value set in order to ensure asymptotic stability and

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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

a prescribed H∞ performance for all the admissible parameters’ trajectory, specially in the case where the waveforms of the admissible trajectories are not known a prior. Remark 2. Note that, since the conditions in (40) are affine in κ = γ 2 , finding the controller of Theorem 1 with the minimum γ is a convex optimization problem of minimizing κ subject to the LMI constraints in (40).

the dependence of the controller performance on the complexity of the controller gain, the optimal gain-scheduled controller of Theorem 1 has been designed for matrices F (θi, j ) and Xk (θi, j ), k = 1, 2 of various degrees in θi, j . Four designs as below have been considered: (D1) Control design with F (θi, j ) affine in θi, j and Xk (θi, j ), k = 1, 2 independent of θi, j ;

e θi, j ) and Remark 3. In view of the definitions of the matrices Ω( e u (θi, j ), it can be easily verified that, without loss of generality, Ω the matrix F of F (θi, j ) in (46) can be assumed to have the following structure:   F0 0 F= , F0 ∈ Rnu ×n , F2 ∈ R pnu ×pn . F1 F2

(D2) Control design with F (θi, j ) and Xk (θi, j ), k = 1, 2 affine in θi, j ;

As a special case of Theorem 1, we get the following boundedreal lemma for the unforced system of (1) with ui, j ≡ 0 :

Fig. 1 displays the achieved minimum upper-bound γ on kHzw k∞ versus α for the gain-scheduled control designs D1 - D4. Observe that improved performance is obtained when the degrees of the polynomial matrices F (θi, j ) and Xk (θi, j ), k = 1, 2 are increased. Design D4, which is the one with the controller gain of the highest complexity, achieves the best performance, in particular for large α . For instance, for α = 1.3 the minimum values of γ provided by the designs D1, D2 and D3 are respectively, 52.89%, 34.75% and 6.24% larger than that of design D4.

Lemma 4. Consider the unforced system of (1) with ui, j ≡ 0 and let Ξ be a given polytope of admissible θi, j . Given a scalar (i) γ > 0, suppose there exist nk ×nk matrices Xk , i = 0, 1, 2, k = (0)

(2)

1, 2, with Xk and Xk , k = 1, 2 symmetric, and a matrix L satisfying the following LMIs:   eT X Π e −Π ⋆ ⋆ ⋆    U10 (α ) −J2T XJ2 ⋆ ⋆    + Her{ LG(α , δ ) } < 0,   ⋆  0 BwT J1 −γ 2 I  U20 (α )

0

Dw

(D3) Control design with F (θi, j ) quadratic in θi, j and Xk (θi, j ), k = 1, 2 affine in θi, j ; (D4) Control design with F (θi, j ) and Xk (θi, j ), k = 1, 2 quadratic in θi, j .

2 1.9

−I

∀ (α , δ ) ∈ V ( Ξ × Ξa )

1.8 1.7

(50)

D1

1.6

where b T (α )X Π, e U10 (α ) = J1TA Ω

min

1.5 1.4

γ

b T(α )X Π e U20 (α ) = C Ω

D2 1.3

and the other matrices are the same as in Theorem 1. Then, system (1) with ui, j ≡ 0 is uniformly asymptotically stable and  kHzw k∞ < γ for all trajectories of θi, j in Ξ.

1.2

D3 1.1

D4 1

Note that since Lemma 4 does not involve controller synthesis, it applies to the general case of time-varying parameters, which can be either measured on-line or uncertain.

0.9 0.8 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

α

5. NUMERICAL EXAMPLES

Fig. 1. Minimum upper-bound γ on kHzw k∞ of the closed-loop system for designs D1-D4.

Example 1. Consider the 2-D discrete-time system (1) of a thermal process analysed in Du and Xie [2002] with the addition of a time-varying scheduling parameter θi, j , namely let the system in (1) with matrices as below:   0 1 , A(θi, j ) = 0.6+0.4θi, j 0.1−0.6θi, j     0 1 Bu (θi, j ) = , Bw (θi, j ) = , 0.3+0.2θi, j 0.5

Example 2. This example is aimed to illustrate Lemma 4. To this end, consider the 2-D system of Example 7.1 in Du and Xie [2002] with the addition of a time-varying parameter θi, j to the (2,2) element of the matrix A, namely let the unforced system of (1) with ui, j ≡ 0 and the following matrices:     0.3 0 1 A(θi, j ) = , , Bw (θi, j ) = 1 0.2+θi, j 0

C(θi, j ) = [0 1], Du (θi, j ) = 0, Dw (θi, j ) = 0.5

C(θi, j ) = [ 0 0.5 ], Dw (θi, j ) = 0

where θi, j ∈ R and satisfies |θi, j | ≤ α , ∀ i, j ∈ Z+ ,

and with θi, j satisfying |θi, j | ≤ α , ∀ i, j ∈ Z+ .

Theorem 1 has been applied to design gain-scheduled state feedback H∞ controllers for different values of α . To illustrate

Lemma 4 has been applied to the system as above to find the minimum upper-bound γ on kHzw k∞ for different values of

10172

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

α . The achieved results are shown in Fig. 2. It turns out that Lemma 4 provides a solution for any α ≤ 0.7998. Observe that 0 ≤ α < 0.8 is the range of α for stability of the underlying system in the case where θi, j is a constant parameter. 10

9

8

7

5

γ

min

6

4

3

2

1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

α

Fig. 2. Minimum upper-bound γ on kHzw k∞ .

6. CONCLUDING REMARKS This paper has investigated the problem of gain-scheduled H∞ control of 2-D discrete-time linear parameter-varying systems described by a Roesser model with matrices depending affinely on time-varying parameters, which are supposed to be measured on-line and with admissible values confined to a given convex bounded polyhedral domain. An LMI method is proposed for designing a gain-scheduled H∞ controller based on static state feedback. The design method employs a parameterdependent Lyapunov function and does not require a priori knowledge of the parameters’ trajectory and gridding of the parameter set value. Furthermore, the controller gain is allowed to be a matrix fraction of polynomial matrices with quadratic dependence on the scheduling parameters.

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