A Delay Decomposition Approach to H1 Control of ... - Science Direct

European Journal of Control (2009)5:523–533 # 2009 EUCA DOI:10.3166/EJC.15.523–533

A Delay Decomposition Approach to H1 Control of Networked Control Systems Xian-Ming Zhang1,2, and Qing-Long Han1,2, 1 2

Centre for Intelligent and Networked Systems, Central Queensland University Rockhampton QLD 4702, Australia; School of Computing Sciences, Central Queensland University Rockhampton QLD 4702, Australia

This paper is concerned with H1 control of a networked control system (NCS). First, the NCS is modelled as a linear system with an interval time– varying delay. Second, a delay decomposition approach is developed to derive a less conservative bounded real lemma (BRL). Based on this BRL, delay-dependent conditions for the existence of a state feedback controller, which ensures internally asymptotic stability and a prescribed H1 performance level of the closedloop system, are derived in terms of a nonlinear matrix inequality. Third, the nonconvex feasibility problem is converted into a nonlinear minimization problem subject to a set of linear matrix inequalities (LMIs), from which the suitable controller can be designed by an iterative algorithm. Finally, two numerical examples are given to show the effectiveness of the obtained results. Keywords: Networked control systems (NCSs), State feedback, H1 control, A delay decomposition approach

1. Introduction A networked control system (NCS) is a system in which the feedback control loop is closed through a network. In the last decade, NCSs have been attracting much attention due to some significant Correspondence to: X.-M. Zhang, E-mail: [email protected] E-mail: [email protected]

advantages, such as reduced installation and maintenance cost and increased system agility and so on, see for example [7] and references therein. However, network-induced delays and data dropouts, which make the analysis and synthesis of an NCS complicated, are usually inevitable in an NCS because of the message transmission and limit bandwidth. Therefore, there have many researchers to conduct research on this topic and in the existing literature there are some results such as results on stability analysis [10, 11, 13–15, 22], results on state feedback control [8, 16, 18, 20, 21], and results on H1 stabilization and estimation [2, 3, 17, 19]. In this paper, we consider a typical NCS, shown in Fig. 1, where the physical plant is supposed to be a continuous-time system described by 8 _ ¼ A0 xðtÞ þ Bu uðtÞ þ Bw wðtÞ < xðtÞ ð1Þ zðtÞ ¼ C0 xðtÞ þ Du uðtÞ þ Dw wðtÞ : xðt0 Þ ¼ x0 where xðtÞ 2 Rn ; zðtÞ 2 Rl and uðtÞ 2 Rm are the state, the controlled output and the control input, respectively; wðtÞ 2 Rp is the disturbance input belonging to L 2 ½t0 ; 1Þ. The coefficient matrices A0 ; Bw ; Bu ; C0 ; Dw , and Du are known real constant matrices of appropriate dimensions. Assume that the sampler is clock-driven, the controller and the zero-order hold (ZOH) are eventdriven, and also assume that the data are transmitted with a signal packet under the Try-Once-Discard Received 4 April 2008; Accepted 2 December 2008 Recommended by J. Lunze, E.F. Camacho

524

X.-M. Zhang and Q.-L. Han

Fig. 1 A typical networked control system.

(TOD) protocol [13]. With these assumptions, the continuous signal xðtÞ is sampled by the sampler, and then is transmitted with a single packet to the digital controller through the network to generate a control action. This control action is finally transmitted via the network again and is held by the ZOH to drive the physical plant. During the data transmission, datapacket dropouts and the network-induced delays are usually inevitable, which will possibly degrade the stability of the NCS. In the following, by employing a novel timing mechanism from [3], we will model the NCS as a linear system with time-varying delay, where the sampling intervals, the numbers of packet dropout, and the bounds of the network-induced delay are taken into account. More specifically, let the ZOH update at tk ðk ¼ 1; 2; ; 1Þ, then at each updating instant tk , the available state is xðtk k Þ, where k denotes the sum of network-induced delays from the sampler to the controller and from the controller to the ZOH since the last updating instant tk1 . The network-induced delay k is unknown, possibly timevarying or even random variable, but known to be bounded by two scalars m and M , that is 0 m k M < 1

ð2Þ

In this situation, the state feedback controller can be taken as uðtÞ ¼ uðtk Þ ¼ Kxðtk k Þ;

tk t < tkþ1 ð3Þ

where the gain K is to be determined. Then, the resulting closed-loop system is 8 _ ¼ A0 xðtÞ þ Bw wðtÞ þ Bu Kxðtk k Þ > < xðtÞ zðtÞ ¼ C0 xðtÞ þ Dw wðtÞ þ Du Kxðtk k Þ; > : t 2 ½tk ; tkþ1 Þ: ð4Þ

It is worth pointing out that the above timing mechanism has many advantages; for example, the sampler need not sample periodically, and the distance between two consecutive updating instants, i.e. contains some useful tkþ1 tk ðk ¼ 1; 2; Þ, information, which can be seen from the following kþ1 kþ1 tkþ1 tk ¼ TDropout þ TSampler þ kþ1

ð5Þ

kþ1 denotes the cost time of finite datawhere TDropout kþ1 the corresponding sampacket dropouts and TSampler pling interval from the last updating instant tk to the current updating instant tkþ1 . Clearly, when the network-induced delay kþ1 is sufficiently small and no packet dropouts occur, the distance tkþ1 tk can be regarded as the two successive sampling interval; in the case that kþ1 and TSampler are sufficiently small, then tkþ1 tk can reflect the numbers of packet dropouts; also, when TSampler is sufficiently small and no packet dropouts occur, tkþ1 tk expresses the network-induced delay since the last updating instant tk . For simplicity of presentation, denote max :¼ maxftkþ1 tk jk ¼ 1; 2; g. From the above analysis, the admissible value of max , with which the closedloop system (4) is asymptotically stable with a prescribed H1 disturbance attenuation level, can be regarded as an important performance index of an NCS due to the fact that it takes into account information about the sampling interval, the number of packet dropouts, and the network-induced delay. The larger value of max , the better performance of an NCS, which means a larger endurability on dropouts, and network-induced delays. Therefore, it is of significance in both theory and practice to seek a maximum admissible upper bound of max such that the closed-loop system (4) is asymptotically stable with a prescribed H1 disturbance attenuation level.

H1 Control of NCSs

525

Let dðtÞ ¼ t tk þ k ; tk t < tkþ1 . Then dðtÞ can be considered as a piecewise continuous function satisfying 0 m dðtÞ M < 1; 8t t0

ð6Þ

where m :¼ m and M :¼ max þ M . In this way, the resulting closed-loop system (4) is thus equivalently converted into a linear continuous-time system with an interval time–varying delay described by 8 _ ¼ A0 xðtÞ þ Bu Kxðt dðtÞÞ þ Bw wðtÞ > < xðtÞ zðtÞ ¼ C0 xðtÞ þ Du Kxðt dðtÞÞ þ Dw wðtÞ; > : t 2 ½tk ; tkþ1 Þ:

the LKF was constructed by using both the upper bound M and the lower bound m of the time-varying delay. In order to further reduce conservatism of the results in [9], we now are analyzing the following LKF in [9] Z t ~ xt Þ ¼ xT ðtÞPxðtÞ þ Vðt; xT ðsÞQ1 xðsÞds

Z

þ m

where ðÞ is a continuous vector-valued function on ½t0 M ; t0 . In this paper, we address an H1 control problem: To determine the controller gain K such that the closed-loop system (7) is asymptotically stable with a prescribed H1 disturbance attenuation level , i.e. the closed-loop system (7) with wðtÞ 0 is asymptotically stable; and the closed-loop system (7) is of a prescribed H1 disturbance attenuation level > 0, that is, zðtÞ satisfies jjzjj2 < jjwjj2 for all nonzero wðtÞ 2 L 2 ½t0 ; 1Þ under the initial condition xðÞ 0; 8 2 ½t0 M ; t0 . As is well known, the Lyapunov-Krasovskii functional (LKF) approach is an effective method to deal with the above H1 control problem. One can have some less conservative results by choosing some proper LKFs. In [18], the authors chose an LKF as xt Þ ¼ x ðtÞPxðtÞ þ Vðt;

0

Z

m Z 0

þ M

t

_ x_ T ðsÞR1 xðsÞdsd

tþ Z t

M


tþ

Z

þ ðM m Þ

m

Z

M

t

_ x_ T ðsÞSxðsÞdsd

tþ

ð10Þ Notice that the second and third terms in (10) can be rewritten as Z

Z

t

_ x_ ðsÞRxðsÞdsd T

tþ

ð9Þ and derived some results, which are conservative because the lower bound m of the time-varying delay was not taken into account. To reduce the conservatism of the results, different LKFs were proposed in [19, 12] and [9]. Compared with the LKFs in [19, 12], the LKF in [9] has some advantage due to the fact that

tm

xT ðsÞðQ1 þ Q2 ÞxðsÞds þ tm

xT ðsÞQ2 xðsÞds

tM

Notice also that the last three terms in (10) can be rearranged as M _ x_ ðsÞ R1 þ R2 xðsÞdsd m m m tþ Z m Z t M _ þðM m Þ x_ T ðsÞ S þ R2 xðsÞdsd M m tþ M Z

Z

0

t

T

Introducing some new matrix variables, one can simplify the LKF (10) as ~ xt Þ ¼ xT ðtÞPxðtÞ þ Vðt;

Z

t

xT ðsÞZ1 xðsÞds

tm

Z

tm

þ

xT ðsÞZ2 xðsÞds

tM t

T

M

Z

0

with ðt0 Þ ¼ x0 ð8Þ

Z

xT ðsÞQ2 xðsÞds tM

The initial condition of the state xðtÞ on ½t0 M ; t0 is supplemented as 2 ½t0 M ; t0

t

þ

ð7Þ

xðÞ ¼ ðÞ;

tm

Z

Z

þ m

0

m

Z

t

_ x_ T ðsÞS1 xðsÞdsd

tþ

þ ðM m Þ

Z

m

M

Z

t

_ x_ T ðsÞS2 xðsÞdsd

tþ

ð11Þ From the above LKF, one can see that the matrices Z2 and S2 in (11) are defined on the whole delay interval ½M ; m , which maybe lead to some conservative

526


result. Recently, Han [5,6] introduced a delay decomposition approach to study the stability and H1 control of linear delay systems, where the timedelay is a constant delay. The idea of the approach is that the delay interval is uniformly divided into multiple segments, and a proper Lyapunov–Krasovskii functional is chosen with different weighted matrices corresponding to different segments in the Lyapunov– Krasovskii functional. Inspired by Han [5,6], we now extend the approach to the system (7), where the timedelay is an interval time-varying delay. Divide the delay interval ½M ; m into N equidistant subintervals, i.e. ½M ; m ¼

N [

½m j; m ðj 1Þ

j¼1

where ¼ ðM m Þ=N with N a positive integer. Then, on each subinterval ½m j; m ðj 1Þ, choose a different weighting matrix to arrive at Z Vj ðt; xt Þ :¼

t m ðj 1Þ

t m j

Z

þ

xT ðsÞQj xðsÞds

m ðj 1Þ m j

Z

t

_ _ xðsÞR j xðsÞdsd

tþ

where j ¼ 1; 2; ; N. Consequently, a new LKF candidate for system (7) is readily constructed as Vðt; xt Þ ¼ xT ðtÞPxðtÞ þ V0 ðt; xt Þ þ

N X

Vj ðt; xt Þ

j¼1

ð12Þ where V0 ðt; xt Þ is a functional defined on ½m ; 0 as Z V0 ðt; xt Þ :¼

t

xT ðsÞQ0 xðsÞds

tm

Z

þ m

0

m

Z

t

system (7) will be asymptotically stable with a prescribed H1 disturbance attenuation level. By employing the cone complementary linearization technique, the nonconvex feasibility problem described by this BRL will be converted into a nonlinear minimization problem subject to a set of linear matrix inequalities (LMIs), which can be easily implemented by an iterative algorithm. Two examples will be finally given to demonstrate the effectiveness of the obtained results. Notation: Throughout this paper, the notations are standard. The superscripts ‘1’ and ‘T’ mean the inverse and transpose of a matrix, respectively; Rn denotes $n$-dimensional Euclidean space; Rn m is the set of all n m real matrices; P > 0 means that P is positive definite; diagf g denotes a block-diagonal matrix; ð ij Þm m means a big matrix of m m block matrix elements ij ; ði; j ¼ 1; 2; ; mÞ; I and 0 denote an identity matrix and a zero matrix of appropriate dimensions, respectively; min ðVÞ stands for the minimum eigenvalue of a symmetric matrix V; L 2 ½t0 ; 1Þ denotes the space of square integrableR vector func1 tions on ½t0 ; 1Þ with norm k k2 ¼ ð t0 k k2 dtÞ1=2 , where k k denotes the Euclidean vector norm; and the symmetric matrix is denoted term in a symmetric X Y X Y . by ?, e.g., ¼ ? Z YT Z

2. H1 Performance Analysis In this section, we will present a new sufficient condition such that the system (7) is asymptotically stable with a prescribed H1 disturbance attenuation level. To begin with, we introduce the following integral inequality. Lemma 1 [4]: For any constant matrix X 2 Rn n ; X ¼ XT > 0, a scalar function h :¼ hðtÞ 0 and a vector-value function x_ : ½t h; t ! Rn such that the following integration is well defined, then


tþ

Apparently, if Qj ! 0þ and R1 ¼ m R0 ¼ R; ðj ¼ 1; 2; ; NÞ, then the LKF (12) immediately reduces to the one in (9). Also, if we set Q0 ¼ Z1 ; Qj ¼ Z2 ; R0 ¼ S1 ; Rj ¼ NS2 ; ðj ¼ 1; 2; ; NÞ, then the LKF (12) becomes (11). Therefore, the LKF (12) includes those in (9) and (11) as special cases. In this paper, we will apply the proposed LKF (12) to deal with the H1 control of the system (7). A new and less conservative bounded real lemma (BRL) will be first derived to ensure that the resulting closed-loop

Z h

t

_ x_ ðsÞXxðsÞds T

th

xðtÞ

T

xðt hÞ X X xðtÞ ? X xðt hÞ ð13Þ

We now state and establish the following result. Proposition 1: For some given scalars > 0; m ; M , and a positive integer N, the system (7) is asymptotic-

H1 Control of NCSs

527

ally stable with a prescribed H1 disturbance attenuation level if there exist n n real matrices P > 0; Qj > 0; Rj > 0 ðj ¼ 0; 1; ; NÞ such that 2 6 6 Cq :¼6 4

F0 þ Fq

m GT1 R0

? ? ?

GT1 7

R0 ?

GT2

_ xt Þ ¼ 2xT ðtÞPxðtÞ _ þ xT ðtÞQ0 xðtÞ Vðt; xT ðt m ÞQ0 xðt m Þ ( ) N X T 2 2 _ Rj xðtÞ þ x_ ðtÞ m R0 þ

3

Z

0 7 7 7 0 5

0 7

m

j¼1 t

_ x_ T ðsÞR0 xðsÞds

tm

? ? I < 0; q 2 f1; 2; ; Ng

þ

N X

xT ðt m ðj 1ÞÞQj xðt m

j¼1

ð14Þ

ðj 1ÞÞ

N X j¼1

where 2

2 I 0 Bw T P ? 0 ðBu KÞT P ? ? ’0 ? ? ? ? ? ? .. .. .. . . . ? ? ? ? ? ?

6 6 6 6 6 6 F0 :¼ 6 6 6 6 6 4

0 0 R0 ’1 ? .. .

0 0 0 R1 ’2 .. .

? ?

? ?

.. .

0 0 0 0 0 .. .

’N ?

0 0 0 0 0 .. .

ðt m jÞ

3

RN ’Nþ1

7 7 7 7 7 7 7 7 7 7 7 5

ij ÞðN þ 4Þ ðN þ 4Þ

þð

T ij ÞðN þ 4Þ ðN þ 4Þ

7 :¼ R1 þ R2 þ þ RN G1 :¼ ½ Bw

Bu K

A0

0

j¼1

G2 :¼ ½ Dw

Du K

C0

0

0 :

ð16Þ

ð18Þ ð19Þ

with 8 < PA0 þ AT0 P þ Q0 R0 ; j ¼ 0; ’j :¼ Qj Qj1 Rj Rj1 ; j ¼ 1; ; N; : j ¼ N þ 1: QN RN ; ð20Þ

ij

8 Rq ; i ¼ j ¼ 2; > > < i ¼ 2; j ¼ q þ 3; q þ 4; Rq ; :¼ > Rq ; i ¼ q þ 3; j ¼ q þ 4; > : 0; otherwise:

tm ðj1Þ tm j

_ x_ ðsÞRj xðsÞds: ð22Þ For convenience, introduce a new vector ðtÞ :¼ ½wT ðtÞxT ðt dðtÞÞxT ðtÞxT ðt m ÞxT

ð17Þ 0

Z N X

T

ð15Þ

Fq :¼ ð

xT ðt m jÞQj x

ð21Þ

Proof: Taking the derivative of Vðt; xt Þ in (12) with respect to t along the trajectory of (7), we have, for t 2 ½tk ; tkþ1 Þ,

ðt m Þ xT ðt m NÞT Then, rewrite (7) as _ ¼ G1 ðtÞ xðtÞ zðtÞ ¼ G2 ðtÞ:

ð23Þ

Substituting (23) into (22), after simple algebraic manipulation, we have _ xt Þ þ zT ðtÞzðtÞ 2 wT ðtÞwðtÞ Vðt; n ¼ T ðtÞ GT3 PG1 þ GT1 PG3 þ 4 þ m2 GT1 R0 G1 Z t o _ x_ T ðsÞR0 xðsÞds þ 2 GT1 7G1 þ GT2 G2 zðtÞ m Z N X j¼1

tm

tm ðj1Þ tm j

_ x_ T ðsÞRj xðsÞds ð24Þ

where 7; G1 and G2 are defined in (17)–(19), respectively, and G3 :¼ ½ 0 0 I 0 0 4 :¼ diagf 2 I; 0; Q0 ; Q1 Q0 ; ; QN QN1 ; QN g We now establish the relationships between xðt dðtÞÞ and xðt m Þ; xðt m Þ; ; xðt m NÞ by using Lemma 1. Since dðtÞ is a continuous function, for any t 2 ½tk ; tkþ1 Þ, there should exist a positive integer

528


_ xt Þ þ zT ðtÞzðtÞ 2 wT ðtÞwðtÞ Vðt; T ðtÞ F0 þ Fq þ m2 GT1 R0 G1 þ 2 GT1 71 þ GT2 G2 zðtÞ; t 2 ½tk ; tkþ1 Þ

q; ð1 q NÞ such that dðtÞ 2 ½m þ ðq 1Þ; m þ q. In this situation, we have Z Z ¼

tm ðq1Þ

tm q tdðtÞ

tm q

Z

ð29Þ

_ x_ T ðsÞRq xðsÞds

tm ðq1Þ


where F0 ; Fq are defined in (15) and (16), respectively. First, we consider the asymptotic stability of the system (7) with wðtÞ 0. In this situation, we have n ~0 þ F ~ q þ 2 G ~T ~ _ xt Þ T ðtÞ F Vðt; m 1 R 0 G1 o ~ 1 jðtÞ; t 2 ½tk ; tkþ1 Þ ~ T 7G þ2 G 1


tdðtÞ

Z

½m þ q dðtÞ

tdðtÞ

tm q

Z

½dðtÞ m ðq 1Þ


tm ðq1Þ


ð30Þ

tdðtÞ

ð25Þ Applying Lemma 1 to the last two integral terms in (25), respectively, and after simple manipulations, we have Z

tm ðq1Þ

tm q


3T 2 2Rq xðt dðtÞÞ 6 7 6 4 xðt m ðq 1ÞÞ 5 4 ? ? xðt m qÞ 2 3 xðt dðtÞÞ 6 7 4 xðt m ðq 1ÞÞ 5 2

Rq Rq ?

3 Rq 7 0 5 Rq

xðt m qÞ ð26Þ For j 6¼ q, we also have the following by Lemma 1 Z

tm ðj1Þ

tm j

_ x_ T ðsÞRj xðsÞds

xðt m ðj 1ÞÞ

T

xðt m jÞ xðt m ðj 1ÞÞ

Rj ?

Rj Rj

xðt m jÞ ð27Þ and

where

ðtÞ :¼ ½ xT ðt dðtÞÞ

xT ðtÞ

xT ðt m Þ 2 0 ðBu KÞT P 0 6 ’0 R0 6? 6 6? ? ’1 6 6 ~ 0 :¼ 6 ? ? ? F 6 6. .. . .. 6 .. . 6 6 4? ? ?

xðt m Þ xT ðt m NÞT 0

0

0 R1

0 0

’2 .. . ?

0 .. .. . . ’N

0

3

7 7 7 7 7 7 0 7 7 .. 7 . 7 7 7 RN 5 ’Nþ1 0 0

? ? ? ? ? ~ ~ ~ Fq :¼ ð ij ÞðN þ 3Þ ðN þ 3Þ þ ð ij ÞTðN þ 3Þ ðN þ 3Þ 8 Rq ; i ¼ j ¼ 1; > > > < R ; i ¼ 1; j ¼ q þ 2; q þ 3; q ~ij :¼ > R q ; i ¼ q þ 2; j ¼ q þ 3; > > : 0; otherwise: ~ G1 :¼ ½Bu K A0 0 0 Notice that (14) implies 2 3 ~ q m G ~T7 ~ T R0 G ~0 þ F F 1 1 4 ? R0 0 5 < 0; ? ? 7

ðq ¼ 1; 2; ; NÞ

which leads to

Z

t

tm

_ x_ T ðsÞR0 xðsÞds

T

xðtÞ xðt m Þ R0 R0 ?

R0

2 ~T ~ ~0 þ F ~ q þ 2 G ~T ~ Vq :¼ F m 1 R0 G1 þ G1 7 G1 < 0;

xðtÞ xðt m Þ

ð28Þ Substituting (26), (27) and (28) into (24) yields

ðq ¼ 1; 2; ; NÞ: Considering all the possibility of q in the set f1; 2; ; Ng and denoting :¼ minfmin ð Vq Þjq ¼ 1; 2; ; Ng

H1 Control of NCSs

529

_ xt Þ xT ðtÞxðtÞ < 0 for we have from (30) that Vðt; xðtÞ 6¼ 0, from which we can conclude that the system (7) with wðtÞ 0 is asymptotically stable. Next, we consider the H1 performance jjzjj2 < jjwjj2 is satisfied for all nonzero wðtÞ 2 L 2 ½t0 ; 1Þ under the zero initial condition xðÞ ¼ 0; 8 2 ½t0 M ; t0 . In fact, applying Schur complement to (14) yields F0 þ Fq þ m2 GT1 R0 G1 þ 2 GT1 7G1 þ GT2 G2 < 0; ðq ¼ 1; 2; ; NÞ From (29), we have, for t 2 ½tk ; tkþ1 Þ _ xt Þ þ zT ðtÞzðtÞ 2 wT ðtÞwðtÞ < 0 Vðt;

ð31Þ

Integrating both sides of (31) on t from ½tk ; tkþ1 Þ yields 1 Z tkþ1 X _ xt Þ þ zT ðtÞzðtÞ 2 wT ðtÞwðtÞdt < 0 ½Vðt; tk

k¼0

that is Z

1 t0

(14). Since some nonlinear terms such as ðBu KÞT P exist in the matrix inequalities in (14), it is not easy to directly derive the controller gain K. In the following, we first modify those matrix inequalities in (14) such that only two nonlinear terms are included. Then, we present an approach to solving the gain K by employing the cone complementary linearization algorithm in [1]. To begin with, we have the following Proposition 2: For some given scalars > 0, m , M , and a positive integer N, the closed-loop system (7) is asymptotically stable with a prescribed H1 noise attenuation level if there exist n n real mat > 0; Rj > 0ðj ¼ 0; 1; ; NÞ and m n rices X > 0; Q j real matrix Y such that 2 3 6 0 þ 6q m 3T1 3T1 3T2 6 7 6 ? XR1 0 0 7 0 X 7 4q :¼ 6 6 1 X 0 7 ? ? X 7 4 5 ? ? ? I < 0;

½zT ðtÞzðtÞ 2 wT ðtÞwðtÞdt < Vðt; xt Þjt¼t0

q 2 f1; 2; ; Ng ð32Þ

Vðt; xt Þjt¼1 Under zero initial conditions, xðÞ ¼ 0; 8 2 ½t0 M ; t0 , we have Z 1 Z 1 zT ðtÞzðtÞdt < 2 wT ðtÞwðtÞdt t0

where

t0

which means jjzjj2 < jjwjj2 . This completes the proof. & Remark 1: Proposition 1 presents a new BRL for NCSs based on a new Lyapunov–Krasovskii functional. When controller gain K is given, this proposition can be used to calculate the minimum H1 performance min for prescribed m and M . Also, it can be employed to compute the upper bound of M for given and m . The numerical examples in Section 4 show that Proposition 1 can achieve much less conservative results than those in the literature. However, Proposition 1 depends on the integer N. The larger N, the less conservatism of Proposition 1, but the more CPU time. As a compromise, we can take some small values of N, for example, N ¼ 2 or N ¼ 3, to deal with the H1 control of an NCS in a practical application.

31 :¼ ½ Bw

Bu Y

A0 X

0

0

ð33Þ

32 :¼ ½ Dw

Du Y

C0 X

0

0

ð34Þ

:¼ R1 þ R2 þ þ RN 7 2

2 I 6 ? 6 6 ? 6 6 ? 6 60 :¼ 6 ? 6 6 . 6 .. 6 4 ? ?

0 0 ? ? ? .. . ? ?

Bw T YT Bu T 0 ? ? .. . ? ?

0 0 R0 1 ? .. . ? ?

ð35Þ 0 0 0 R1 2 .. . ? ?

.. .

0 0 0 0 0 .. .

N ?

3 0 0 7 7 0 7 7 0 7 7 0 7 7 .. 7 . 7 7 RN 5 Nþ1 ð36Þ

6q :¼ ðqij ÞðN þ 4Þ ðN þ 4Þ þ ðqij ÞTðN þ 4Þ ðN þ 4Þ

ð37Þ

with

3. H1 Controller Design In this section, we are in a position to solve the controller gain K based on the matrix inequalities in

8 R0 ; j ¼ 0; < A0 X þ XAT0 þ Q 0 j :¼ Qj Qj1 Rj Rj1 ; j ¼ 1; ; N; : j ¼ N þ 1: QN RN ; ð38Þ

530


8 R ; i ¼ j ¼ 2; > > < q Rq ; i ¼ 2; j ¼ q þ 3; q þ 4; qij :¼ > > Rq ; i ¼ q þ 3; j ¼ q þ 4; : 0; otherwise:

Nonlinear Minimization Problem ð39Þ

þ RR0 þ Z 1 Z1 þ Z 2 Z2 Þ Minimize TrðPX þ Q7 ð41Þ

Moreover, the controller gain is given by K ¼ YX1 .

2

Proof: Define a transformation matrix T :¼

1

6 6 Subject to6 4

1

diagfI; P ; ; P ; R1 ; 71 ; Ig: |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} 0

6 0 þ 6q

m 3T1

3T1

3T2

? ?

Z1 ?

0 Z2

0 0

?

?

Nþ3

< 0;

Pre- and post-multiplying Cq ðq ¼ 1; 2; ; NÞ defined in (14) by T T and T , respectively, together with the settings as X :¼ P1 ; Y :¼ KP1 and :¼ P1 Q P1 ; ði ¼ 0; 1; ; NÞ Ri :¼ P1 Ri P1 ; Q i i

3 7 7 7 5

? I ðq ¼ 1; 2; ; NÞ ð42Þ

R P

P Z1

0;

Q P

P Z2

0;

P I

I 0; X

one obtains

ð43Þ

4q ¼ T T Cq T

ð40Þ

Q I

where 4q is defined in (32). So, if (32) holds for q ¼ 1; 2; ;N,then,from(40),Cq < 0; ðq ¼ 1; 2; ; NÞ. Therefore, the closed-loop system (7) is asymptotically stable with a prescribed H1 level from Proposition 1, which completes the proof. & It is clear that the matrix inequality 4q < 0 has 1 only two nonlinear terms, i.e. XR0 X and ¼ "1 X, where 1 X. If we set R0 ¼ "0 X and 7 X7 "0 ; "1 > 0 are two scalars to be tuned, then the problem of solving the matrix inequalities in (32) become a standard convex optimization problem with parameter tuning. However, this linearization approach, as is well known, will lead to more conservative results. In the sequel, utilizing the cone complementary idea [1], we are about to convert the nonconvex feasibility problem of (32) into a nonlinear minimization problem subject to a set of LMIs, which can be effectively solved by employing an modified iterative algorithm. For this goal, introduce two variables Z1 > 0 and 1 1 X Z2 , Z2 > 0 satisfying XR0 X Z1 and X7 which are, respectively, equivalent to

1 R0 X1

X1 Z1 1

0; 1

1 7 X1

X1 Z1 2 1

0:

; R ¼ R Let P ¼ X1 ; Q ¼ 7 and Z 1 ¼ Z1 0 1 ; 1 Z 2 ¼ Z2 . Then, similar to [1], the nonconvex feasibility problem of (32) can be converted into a nonlinear minimization problem subject to LMIs as follows.

I 0; 7

R

I R0

I

0;

Z1

I

0; I Z1 Z2 I 0 I Z2 ð44Þ

An iterative algorithm can be used to solve the above nonlinear minimization problem, which is stated in the following. Algorithm 1: Maximize M for given m and . ini such that Step 1 Choose a sufficiently small initial M so ini . (42)–(44) are feasible. Set M ¼ M Step 2 Find a feasible set ðP 0 ; Q0 ; R0 ; Z 01 ; Z 02 ; 0 0 X0 ; Z01 ; Z02 ; Y0 ; Rj ; Q j eðj ¼ 0; 1; ; NÞÞ satisfying (42)–(44). Set l ¼ 0. Step 3 Solve the following LMI problem for the vari ables ðP; Q; R; Z 1 ; Z 2 ; X; Z1 ; Z2 ; Y; Rj ; Q j ð j ¼ 0; 1; ; NÞÞ: ! þ7 l Q þ Rl R0 P l X þ X l P þ Ql 7 Minimize Tr l þR R þ Z l Z1 þ Zl Z 1 þZ l Z2 þ Zl Z 2 0

1

1

2

2

Subject to (42)–(44). :¼ N Rl . Set where 7 i¼1 i P lþ1 ¼ P, Qlþ1 ¼ Q, Rlþ1 ¼ R, Z 2lþ1 ¼ Z2 , Xlþ1 ¼ X, Z1lþ1 ¼ Z1 , lþ1 R ¼ Rj ðj ¼ 0; 1; ; NÞ. l

Z 1lþ1 ¼ Z 1 , Z2lþ1 ¼ Z2 ,

j

Step 4 If matrix inequality (32) is satisfied and

H1 Control of NCSs

531

! þ7 l Q þ Rl R0 P l X þ Xl P þ Ql 7 10n