470 Asian Journal of Control, Vol. 9, No. 4, pp. 470-474, December 2007

Manuscript received April 14, 2005; revised January 16, 2006; accepted January 2, 2007.

The authors are with the Institute of Electrical Engineering, Yanshan University, Qinhuangdao, 066004, P.R. China (e-mail: xpguan@ysu.edu.cn).

This work was supported by the China National Distinguished Yong Scholars under Grant 60525303, NSFC under Grant 60404022, and the NSF of Hebei Province, P.R. China under Grants F2005000390, F2006000270.

－Brief Paper－

ROBUST OUTPUT FEEDBACK GUARANTEED COST CONTROL FOR 2-D UNCERTAIN STATE-DELAYED SYSTEMS

Dan Peng, Xinping Guan, and Chengnian Long

ABSTRACT

This paper considers the robust guaranteed cost control problem of two-dimensional (2-D) state-delayed systems. First, the definition of guaranteed cost matrix is proposed and an upper bound of the cost function is given. Then, the guaranteed cost control problem is resolved. Furthermore, the minimum upper bound of the closed-loop cost function is obtained by solving an optimization problem with LMIs’ constraints. A numerical example demonstrates the effectiveness of our results.

KeyWords: 2-D state-delayed systems, guaranteed cost control, dynamic output feedback controller, asymptotically stable, linear matrix inequality (LMI).

I. INTRODUCTION

The problems of robust stability [1,2], H∞ control [3] and guaranteed cost control [4] of 2-D discrete systems have received considerable attention in the past two decades. Since time delays correspond to transportation time or computation time, are intrinsically 2-D, and are frequently the sources of instability and poor performance, the analysis of 2-D time-delay systems is of vital importance in system theory [5].

When controlling a real plant, it is also desirable to design a control system which is not only stable, but also guarantees an adequate level of performance, such as H∞ control [6] and guaranteed cost control for 2-D time-delay systems. For 1-D time-delay systems, guaranteed cost control has obtained profound development in [7-9]. However, for analytical and structural complexity, the above established results mostly have not been extended to 2-D cases.

In this paper, we aim at realizing guaranteed cost control for 2-D state-delayed systems with uncertainties. First, we analyze the guaranteed cost performance. Then, in terms of an LMI approach, a dynamic output feedback controller is given to render the closed-loop system asymptotically stable while guaranteeing a prescribed level of such guaranteed cost

performance, i.e. the robust guaranteed cost control problem is resolved. In addition, an optimization problem of minimizing the upper bound of closed-loop cost function is proposed. Finally, a numerical example shows the effectiveness of our results.

II. ROBUST GUARANTEED COST ANALYSIS

Consider a 2-D autonomous system with state delays and uncertainties given by

1 1( 1 1) ( ) ( 1 )x i j A A x i j+ , + = + ∆ + ,

2 2( ) ( 1)A A x i j+ + ∆ , +

1 1 1( ) ( 1 )d dA A x i j d+ + ∆ + , −

2 2 2( ) ( 1)d dA A x i d j+ + ∆ − , + (1)

where x(i, j) ∈ Rn is the state input, Ak. Akd(k = 1, 2) are constant matrices and ∆AK, ∆Akd(k = 1, 2) are uncertain terms satisfying

1 1 1 11 2 1 1 12

1

( ) ( )

( ) ( 1 2)kd kd kd kd

A H F i j E A L G i j E

A H F i j E k

∆ = , , ∆ = ,

∆ = , = ,

where F1(i, j), G1(i, j), Fkd(i, j)(k = 1, 2) are uncertain matrices satisfying ||(⋅)(i, j)||2 ≤ I.

The boundary conditions are given by:

D. Peng et. al.: Robust Output Feedback Guaranteed Cost Control for 2-D Uncertain State-Delayed Systems 471

1 1{ ( ) } 0 1 0ijx i j i j d d, = ϕ , ∀ ≥ ; = − , − + , , ,

2 2{ ( ) } 0 1 0ijx i j j i d d, = ψ , ∀ ≥ ; = − , − + , , , (2)

ϕ00 = ψ00, and the cost function is expressed as

00 0

Tij ij

i jJ x Rx

∞ ∞

= == ∑ ∑ (3)

where xij = [xT(i + 1, j) xT(i, j + 1)]T, R = diag{R1, R2} and R1, R2 are weighting matrices.

Definition 1. A 2-D state-delayed system (1) with cost function (3) is robustly asymptotically stable with a guaranteed cost matrix P > 0 if there exist matrices Q > 0, Qk > 0 and RK > 0(k = 1, 2) such that

00

0 0R⎡ ⎤

Π + <⎢ ⎥⎣ ⎦

(4)

The following Lemma demonstrates the relationship between the existence of guaranteed cost matrix for system (1) and its asymptotic stability.

Lemma 1. Suppose P > 0 is a guaranteed cost matrix of system (1) and cost function (3), then the system (1) is asymptotically stable and the cost function (3) satisfies:

0 1 0 1 2 1 0 0 1 0 1( )T TJ x P Q Q Q x x Qx, , , ,≤ − − − +

1 2

0 0

1 1 1 1 2 1T T

j j i ij d i d

x Q x x Q x, , , ,=− =−

+ +∑ ∑ (5)

where

1

1

1 1 2 2

[ (1 ) (2 ) ( )]

[ ( 1) ( 2) ( )]

1 0 1 0

TT T Tj

TT T Ti

x x j x j x j

x x i x i x i

j d d i d d

,

,

= ,, , ∞,

= ,, , , ∞

= − , − + , , ; = − , − + , , .

Remark 1. Since the upper bound of J0 in (5) depends on the boundary conditions (2), it is impossible to compute. To remove the dependence, assume:

(1)

(2) (1) (1)1

(2) (2)1 1 1

2 2

1

1 01 1 0

T

T

ni j i j i j i j

i j i j i j i ji

i j i jj

R MN

MN N NT

j d d N Ni d d

, , , ,

∞, , , ,=

∞, ,=

⎧ ⎫ϕ , ψ ∈ : ϕ = ,⎪ ⎪⎪ ⎪ψ = , < ,⎪ ⎪= ⎨ ⎬⎪ ⎪= − , − + , , ;⎪ ⎪

< , =− , − + , , .⎪ ⎪⎩ ⎭

∑

∑ (6)

Then, we obtain:

0 max 1 max 1( ) ( )T TJ M PM d M Q M≤ λ + λ

2 max 2( )Td M Q M+ λ (7)

where M and P, Qk(k = 1, 2) are any preselected matrices.

III. ROBUST GUARANTEED COST CONTROL VIA DYNAMIC

OUTPUT FEEDBACK

Consider the following 2-D state-delayed systems:

1 1

2 2

1 1 1

2 2 2

1 1

( 1 1) ( ) ( 1 )

( ) ( 1)

( ) ( 1 )

( ) ( 1)

( ) ( 1 )

d d

d d

x i j A A x i j

A A x i j

A A x i j d

A A x i d j

B B u i j

+ , + = + ∆ + ,

+ + ∆ , +

+ + ∆ + , −

+ + ∆ − , +

+ + ∆ + ,

2 2( ) ( 1)B B u i j+ + ∆ , + (8)

( ) ( ) ( ) ( ) ( )y i j C C x i j D D u i j, = + ∆ , + + ∆ , (9)

where u(i, j) ∈ Rp is the control input and y(i, j) ∈ Rq is the measurable output. Bk(k = 1, 2), C and D are constant matrices and ∆Bk(k = 1, 2), ∆C and ∆D are uncertain terms satisfying:

1 2 2 21 2 2 2 22

1 1 1 2 2 2

( ) ( )

( ) ( )

B H F i j E B L G i j E

C K M i j N D K M i j N

∆ = , , ∆ = ,

∆ = , , ∆ = ,

where F2(i, j), G2(i, j), M1(i, j) and M2(i, j) are uncertain terms satisfying ||(⋅)(i, j)||2 ≤ 1.

The boundary conditions are of the form (2), and the cost function of system (8) is given by:

10 0

( )T Tij ij ij ij

i jJ x Rx u Su

∞ ∞

= == +∑ ∑ (10)

where uij = [uT(i + 1, j) uT(i, j + 1)], S = diag{S1, S2} and S1, S2 are weighting matrices.

Introduce a dynamic output feedback controller:

1 2ˆ ˆ ˆ( 1 1) ( 1 ) ( 1)c cx i j A x i j A x i j+ , + = + , + , +

1 1 2 2ˆ ˆ( 1 ) ( 1)c d c dA x i j d A x i d j+ + , − + − , + 1 2( 1 ) ( 1)c cB y i j B y i j+ + , + , + (11)

( ) ( ) ( )c cu i j C x i j D y i j, = , + , (12)

where ˆ( ) cnx i j R, ∈ . Without loss of generality, we assume Dc = 0.

So the closed-loop uncertain system is expressed as:

1 1 1 1( 1 1) ( ) ( 1 )x i j A H F E x i j+ , + = + + ,

2 1 1 1( ) ( 1)A L G E x i j+ + , +

1 1 1 11 1( ) ( 1 )d d d dA H F E x i j d+ + + , −

2 2 2 12 2( ) ( 1)d d d dA H F E x i d j+ + − , + (13)

472 Asian Journal of Control, Vol. 9, No. 4, December 2007

where ( ) [ ( ) ( )]ˆTT Tx i j x i j i jx, = , , and

1 1

2 2

11

1 21

1 1 1 2

1 21

2 1 2 2

00 0

0 00 0

[ 0] ( 1 2)

0 00 0

0 00 0

k k ck

ck ck cck

kd kdkd kd

ckd

TT Tk

k T T T Tc k c

kdkd

c c

c c

A B CA B C A B DC

A HHA A

E NE

C E C N

E kE

H HH B K B K

L LL B K B K

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥⎢ ⎥⎢⎢⎣ ⎦

=+

⎡ ⎤= , = ⎢ ⎥

⎢ ⎥⎣ ⎦

=

= = ,

=

=

1 2 1 21

1 2 1 21

diag{ ( ) ( ) ( ) ( )}

diag{ ( ) ( ) ( ) ( )}

F i j F i j M i j M i jF

G i j G i j M i j M i jG

⎥⎥

= , , , , , , ,

= , , , , , , ,

So, the closed-loop cost function (10) is rewritten as:

1 10 0

[ ( 1 ) ( 1 )Tc

i jJ i j R x i jx

∞ ∞

= == + , + ,∑ ∑

2( 1) ( 1)]Tci j R x i jx+ , + , + (14)

where Rck = diag{Rk, CcT Sk Cc}(k = 1, 2).

The main result is obtained as follows.

Theorem 1. Consider a 2-D state-delayed system (8) and (9) with the cost function (10), the output feedback controller (11) and (12) resolves the robust guaranteed cost control problem if there exist X > 0, Y > 0, Z,

1 11 21ˆ ˆ, 0 0 0 k k k kZ Z T T T Z Z Z Z, > , > , > , , , , , 0 ( 1 2)kZ k> = ,

and a scalar ε > 0 such that:

1 72 4 5

3 6

00 0 0

0 0 00

0 00

U U U U UU U

II

II

⎡ ⎤⎢ ⎥∗⎢ ⎥⎢ ⎥∗ ∗ −⎢ ⎥ <∗ ∗ ∗ −⎢ ⎥

⎢ ⎥∗ ∗ ∗ ∗ −⎢ ⎥

⎢ ⎥∗ ∗ ∗ ∗ ∗ −ε⎣ ⎦

(15)

where (∗ denotes the block of symmetric matrices)

1 2

1 1 1

1 1 1

1

2 2 2

2 2 2

1

1 11 21

1 2

1 2

21 412 4

1 221

1

0 00 0

ˆ

ˆ ˆ ˆ

0 0

d d

d

X I XA Z C ZY A A Y B Z

X ZU

Y

XA Z C ZA A Y B Z

T ZZ

X X T T T

Y Y Z Z Z

Z I Z Z Z

U UU U

XA Z XA ZU

A A

− − +⎡⎢ ∗ − +⎢⎢ ∗ ∗ −⎢=∗ ∗ ∗ −⎢

⎢∗ ∗ ∗ ∗⎢

⎢ ∗ ∗ ∗ ∗⎣

+ ⎤⎥+ ⎥⎥⎥⎥⎥

− − ⎥⎥∗ − ⎦

− = − + + +

− =− + + +

= − + + +

⎡ ⎤ ⎡ ⎤= , =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

=21 2

1 2 1 1 1 2 141

11 2

2 2 1 2 2 1 2

12 2

31 13 3

32

51 535

52 53

15 5

1 2

0 0

0 0

ˆ00

0 0 0 00 0

0 0

0

k

dd d

d d

d d

kk

k

Tk

k T T Tk k

Y A A Y

XH XH Z K Z K XLU

H H L

XL Z K Z K XH XHL H H

U T ZU U

U Z

U U UU U

EU U

YE Z E

⎡⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎦

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

=

⎡ ⎤− −= , =⎢ ⎥

∗ −⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤= ,⎢ ⎥⎢ ⎥⎣ ⎦

13

1 2

61 16 6

62 1

7 71 7

72

0

00

0 00

00

T

T T T

Tkd

k Tkd

kk T

k k

NYN Z N

U EU U

U YE

RU U U

Y R Z SU

⎡ ⎤⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦

⎡ ⎤= ⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤⎡ ⎤= , = ⎢ ⎥⎢ ⎥

⎢ ⎥⎣ ⎦ ⎣ ⎦

= , =

D. Peng et. al.: Robust Output Feedback Guaranteed Cost Control for 2-D Uncertain State-Delayed Systems 473

for k = 1, 2 and cost function (14) satisfies

2

1 max max 11

( ) ( )T Tk k

kJ M XM d M T M

=

⎡ ⎤≤ ε λ + λ⎢ ⎥⎣ ⎦∑ (16)

Remark 2. If LMI (15) is feasible, the system matrices of the controller would be obtained as

112ˆ (ck k k k kA P Z XA Y Z CY XB Z−= − − −

112 12 12

ˆ) ( )T Tk ckd k kdZ DZ P A P Z XA Y P− − −− , = −

11212

ˆ ( 1 2) Tck k cB P Z k C ZP− −= = , , = (17)

The following theorem realizes the optimal control.

Theorem 2. If the following optimization problem

1 1 2 2min{ }d dε + λ + λ + λ

1

1 1

( ) LMI(15)

s t ( ) 0

( ) 0

T

Tk k

k k

i

I M Xii

XM X

I M Tiii

T M T

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

⎧⎪⎪⎪

⎡ ⎤⎪ −λ⎪. . <⎢ ⎥⎨−⎢ ⎥⎪ ⎣ ⎦

⎪−λ⎪ <⎪ −⎪⎩

(18)

is feasible for k = 1, 2, then an optimal guaranteed cost is:

1 1 2 2( )optJ d d= ε λ + λ + λ (19)

Remark 3. The optimal guaranteed cost control problem can be carried out by the algorithm: Step 1. Solve the optimization problem (18) using MINCX in Matlab LMI toolbox and obtain Jopt. Step 2. Compute 12P̂ and P12 from the singular value

decomposition of 12 12ˆTP P I YX= − .

Step 3. Calculate the system matrices Ack, Ackd, Bck(k = 1, 2) and Cc of controller by Eq. (17).

IV. NUMERICAL EXAMPLE

Consider a 2-D uncertain state-delayed system (8)-(9) with (for simplicity, we only present some items):

1 2

1 2

1 2

0 01 0 05 0 05 00 05 0 01 0 05 0 01

0 0 002 0 03 00 008 0 0 0

[0 25 0] [0 5 0 2]

d d

T T

A A

A A

B B

. − . .⎡ ⎤ ⎡ ⎤= , =⎢ ⎥ ⎢ ⎥. . . .⎣ ⎦ ⎣ ⎦

− . − .⎡ ⎤ ⎡ ⎤= , =⎢ ⎥ ⎢ ⎥− .⎣ ⎦ ⎣ ⎦

= , =. . .

1 2

1

[6 25 0 75] 0 5

[0 2] [0 003 0 008]

[0 01 0 08]

T T

T

C D

H H

L

= . . , = .

= , = . .

= . − .

Assume Eq. (6) with 1 00 1

M⎡ ⎤

= ,⎢ ⎥⎣ ⎦

state delays d1 = 5, d2

= 6 and the cost function (10) with

1 2

1 2

0 04 0 0 25 00 0 04 0 0 25

1

R R

S S

. .⎡ ⎤ ⎡ ⎤= , =⎢ ⎥ ⎢ ⎥. .⎣ ⎦ ⎣ ⎦

= =

First, solve LMI (15), then, in terms of the feasible solutions, the control matrices and guaranteed cost upper bound are obtained as follows:

1

2

1

2

1 2

0 2256 1 14360 0136 0 0807

0 1291 2 19790 0079 0 1313

0 0241 0 38800 0015 0 0247

0 0293 0 01790 0019 0 0012

1 8143 0 09830 1157 0 0041

[ 0

c

c

c d

c d

c c

c

A

A

A

A

B B

C

. .⎡ ⎤= ⎢ ⎥. .⎣ ⎦

. − .⎡ ⎤= ⎢ ⎥. − .⎣ ⎦

− . .⎡ ⎤= ⎢ ⎥− . .⎣ ⎦

− . − .⎡ ⎤= ⎢ ⎥− . − .⎣ ⎦

. − .⎡ ⎤ ⎡ ⎤= , =⎢ ⎥ ⎢ ⎥. − .⎣ ⎦ ⎣ ⎦

= − .

1

0065 0 1409]

106 6087J

.

≤ .

Furthermore, the gradual convergence of x1 of the closed-loop system is shown in Fig. 1. It is obvious that the guaranteed cost controller stabilizes 2-D system (8) and (9). The state response for x2 is similar and thus omitted.

For a reasonable comparison, we use an optimal algorithm and obtain the optimal guaranteed cost Jopt = 1.6305. Comparing the above two conditions, it is shown that the latter controller guarantees not only asymptotic stability but also minimization of the guaranteed cost of the closed- loop system.

V. CONCLUSIONS

We consider the method of dynamic output feedback control to resolve the robust guaranteed cost control problem for 2-D state-delayed systems. Sufficient

474 Asian Journal of Control, Vol. 9, No. 4, December 2007

conditions

Fig. 1. Closed-loop state response of x1(i; j).

have been derived to ensure the robust asymptotic stability and guaranteed cost performance via LMI. Moreover, an optimization problem minimizing the upper bound of closed-loop cost function is proposed. A numerical example is given to show the effectiveness of our results.

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