A delay system approach to networked control systems with limited communication capacity

Journal of the Franklin Institute 347 (2010) 1334–1352 www.elsevier.com/locate/jfranklin

A delay system approach to networked control systems with limited communication capacity Jianguo Dai College of Communication and Control Engineering, Jiangnan University, 1800 Lihu Rd., Wuxi, Jiangsu 214122, PR China Received 17 April 2009; received in revised form 3 March 2010; accepted 12 June 2010

Abstract This paper addresses the problem of quantized feedback control for networked control systems (NCSs). Firstly, with consideration of the effect of network conditions, such as network-induced delays, data packet dropouts and signal quantization, the sampled-data model of closed-loop feedback system based on the updating sequence of the event-driven holder is formulated, from which a continuous system with two additive delay components in the state is developed. Subsequently, by making use of a novel interval delay system approach, the stability analysis and control synthesis for NCSs with two/one static quantizer are solved accordingly. Finally, two illustrative examples are given to show the effectiveness and advantage of the method. & 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction As is well known, there has been an increasing demand in networked systems for manufacturing automation, industrial process control, robotics, and many other applications [1]. At the same time, there have been considerable research work appeared to address modeling, stability analysis, control and filtering problems for networked control systems (NCSs). Among the reported results on NCSs, to mention a few, stability issue is investigated in [2,3]; stabilizing controllers, via state or output feedback, are designed in [4–7], respectively; performance preserved control is studied in [8–13]; robust H1 filtering is proposed in [14,15]; and H1 output tracking control is investigated in [16,17]. E-mail address: [email protected] 0016-0032/$32.00 & 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2010.06.007

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Different to the classical control theory, the effect of network conditions, such as network-induced delays, data packet dropouts, and signal quantization, should be considered when discussing the analysis and synthesis problems of NCSs. For example, in the occurrence of network-induced delays and data packet dropouts, the data packets on the sensor side will not be totally and orderly received by the physical plant on the sampling sequence. Especially, since the aperture time of the A/D converter is generally much smaller than the sampling period, the sampled state held in one period will be quantized by the event-driven quantizer to being more than one data packet which transmits with different delays. In conclusion, all of these network issues mentioned previously lead to the modeling difficulty of the close-loop feedback system based on the sampling instants on the sensor side, and this is reason why the modeling method in [4,5,7–11] should be improved. It is worth noting that when the zero-order holder (ZOH) on the actuator side is event driven, the every successfully transmitted date packet to the actuator will update the ZOH once, and by limiting the updating intervals, the systems’ stability can be guaranteed in the sense that at least one packet will be transmitted successfully as input signal in the prescriptive time. With consideration of these facts, a new NCSs’ model based on the updating instants of the ZOH is proposed in [12,15,16]. In [12], the H1 quantized feedback control of NCSs is investigated for the case where the communication over the channel was in one direction. However, in many practical cases for NCSs with limited communication capacity, the communication over the channel is being in two directions, both from sensor to controller and from controller to actuator, and thus the method with two quantizers in the network is developed [4,5,9,18]. In addition, as the stability criterion is deduced from a corollary of the time-delay model with two additive delay components, there is much room left for improvement when applied to the NCSs’ analysis. A significant source of conservativeness lies in the candidate of the Lyapunov–Krasovskii functional for the useful delay information is not totally considered in. For example, in [12], the interval delay of 0rZðtÞrk is not sufficiently calculated. To the best of the author’s knowledge, the stability and stabilization of NCSs with the quantization of both state and control signals have not been fully investigated and still remain challenging. In this paper, the quantized feedback control for NCSs with the communication over the channel being in two directions is examined, where the effect of network-induced delays and data packet dropouts and signal quantization is dealt with in a unified framework. Firstly, based on the ZOH’s updating sequence, the closed-loop feedback NCSs’ model is formulated in continuous-time domain, following which new results on stability analysis and control synthesis of NCSs with two static quantizers are proposed by exploiting the novel techniques of interval delay systems. Whereafter, the improved results in the case of NCSs with only one static quantizer on the sampler side are also derived from the developed method. Finally, the simulation examples are presented to illustrate the improvement and applicability of the proposed method. Notation: Rn denotes the n-dimensional Euclidean space, Rnm is the set of n  m real matrices, I is the identity matrix of appropriate dimensions. The notation X 40 (respectively, X Z0), for X 2 Rnn means that the matrix X is a real symmetric positive definite (respectively, positive semi-definite). In symmetric block matrices, we use an asterisk ðÞ to represent a term that is induced by symmetry and diagf  g stands for a block-diagonal matrix. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

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2. System description Consider the NCSs shown in Fig. 1. Suppose the physical plant is given by the following system: ( _ ¼ ðA þ DAðtÞÞxðtÞ þ BuðtÞ; xðtÞ ð1Þ xðt0 Þ ¼ x0 ; where xðtÞ 2 Rn , uðtÞ 2 Rm are the state vector and control input, respectively; x0 2 Rn denotes the initial condition; A, B are some constant matrices of appropriate dimensions; DAðtÞ denotes the parameter uncertainty satisfying the following condition: DAðtÞ ¼ DF ðtÞE;

ð2Þ

where D and E are constant matrices of appropriate dimensions and F(t) is an unknown time-varying matrix, which is Lebesque measurable in t and satisfies F T ðtÞF ðtÞrI. As depicted in Fig. 1, considering the limited capacity of the communication channels and also for reducing the data transmission rate in the network, the state and control signals are quantized before going into the network medium, respectively, by two quantizers as one is on the sampler side denoted as gðÞ and another on the controller side denoted as f ðÞ. At the same time, it is assumed that the sensor and sampler is clock-driven, while the quantizers, controller, ZOH and actuator are event-driven. The sampling period is assumed to be h and the sampling instants are denoted as sk , k ¼ 1; 2; . . ..

Physical Plant

u

x Sensor

Actuator Sampler ZOH Quantizer g

x Networkinduced Delay

Quantizer

f

Network

Medium

Networkinduced Delay

v Controller

Fig. 1. A typical networked control system with two quantizers.

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In Fig. 1, we denote the quantized measurement of x as x, and the control signal as v, and the input signal as u. Then, at the instant sk , we have xðsk Þ ¼ gðxðsk ÞÞ;

ð3Þ

vðsk þ tsc k Þ ¼ Kxðsk Þ;

ð4Þ

uðsk þ tk Þ ¼ f ðvðsk þ tsc k ÞÞ;

ð5Þ

ca where tk ¼ tsc k þ tk is the signal transmission delay from the sampler to the actuator in sc which tk is the network-induced delay from the sampler to the controller and tca k is the delay from the controller to the actuator. Furthermore, K is the state-feedback gain. For the quantizer on the sampler side, gðÞ is defined as gðxÞ ¼ ½g1 ðx1 Þ g2 ðx2 Þ    gn ðxn ÞT , where gj(xj) (j=1,y,n) are chosen as logarithmic quantizers given by 8 1 1 ðjÞ > > uðjÞ if uðjÞ oxj r u ; xj 40; > < l 1 þ dg j l 1dgj l ð6Þ gj ðxj Þ ¼ 0 if xj ¼ 0; > > > : g ðx Þ if x o0; j j j

with dgj ¼ ð1rgj Þ=ð1 þ rgj Þ ð0orgj o1Þ for rgj is a constant given for gj(xj) and called the ðjÞ quantization density. Moreover, the set of quantized levels is defined as Uj ¼ f7uðjÞ l ; ul ¼ ðjÞ ðjÞ rlgj uðjÞ 0 ; l ¼ 71; 72; . . .g [ f7u0 g [ f0g, with u0 40. According to [19], when defining

Dg ¼ diagfDg1 ;Dg2 ; . . . ;Dgn g; with Dgj 2 ½dgj ;dgj ;

j ¼ 1;2; . . . ;n;

then g(x) can be expressed by the sector bound method as gðxÞ ¼ ðI þ Dg Þx:

ð7Þ

For the quantizer on the controller side, f ðÞ is defined as f ðvÞ ¼ ½f1 ðv1 Þ; f2 ðv2 Þ; . . . ; fm ðvm ÞT , with fi ðÞ (i=1,y,m) also chosen as logarithmic quantizers similar as defined in Eq. (6). When defining Df ¼ diagfDf1 ;Df2 ; . . . ;Dfm g;

ð8Þ

f(v) can be expressed as f ðvÞ ¼ ðI þ Df Þv;

ð9Þ

where Dfi 2 ½dfi ; dfi  (i=1,y,m), and dfi ¼ ð1rfi Þ=ð1 þ rfi Þ with rfi being the quantization density of fi. To make our idea more lucid, in this paper we make the assumption that dfi ¼ df and dgj ¼ dg , where df and dg are two constants. Combining Eqs. (3)–(5), (7) and (9), the input signal can be expressed as uðsk þ tk Þ ¼ ðI þ Df ÞKðI þ Dg Þxðsk Þ9ðK þ DðKÞÞxðsk Þ;

ð10Þ

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where DðKÞ ¼ Df K þ KDg þ Df KDg : Now denoting the updating instants of the ZOH as tk (k=1,2,y), it is clear to see that the updating signal at the instant tk has experienced signal transmission delay tk from the sampling instant. Therefore, we have from Eq. (10) that the quantized feedback controller takes the following form: uðtk Þ ¼ ðK þ DðKÞÞxðtk tk Þ:

ð11Þ

Moreover, considering the behavior of the ZOH, the input signal is uðtÞ ¼ ðK þ DðKÞÞxðtk tk Þ; tk rtotkþ1 ; and thus combining Eqs. (1) and (12) we obtain the following closed-loop system: ( _ ¼ ðA þ DAðtÞÞxðtÞ þ BðK þ DðKÞÞxðtk tk Þ; tk rtotkþ1 ; xðtÞ xðt0 Þ ¼ x0 :

ð12Þ

ð13Þ

Remark 1. Network-induced delay always exists when the data transmits through a network [14], and is non-differentiable interval time-varying delay [20] which has been caused considerable attention in very recent years [21–24], so a natural assumption on tk can be made as 0otm rtk rtM ;

ð14Þ

where tm and tM denote the minimum and maximum delay bounds, respectively. Remark 2. The effect of data packet dropouts in the communication channel can be described as the ZOH is not updated during the time interval of this event, which is referred as vacant sampling. Hence, the effect of one packet dropout in the transmission is just a case that one sampling period delay is induced in the updating interval of the ZOH. Since the updating period {tkþ1tk} of ZOH depends on both the signal transmission delays and data packet dropouts, it can be clearly seen that [12] tkþ1 tk ¼ ðskþ1 þ 1Þh þ tkþ1 tk ;

ð15Þ

where skþ1 is the number of accumulated packet dropouts in this period. In a similar way as in [12], let us represent tk tk in Eq. (13) as tk tk ¼ ttm tðtÞ;

ð16Þ

where tðtÞ ¼ ttk þ tk tm , tk rtotkþ1 . Then, it is obviously that 0rtðtÞotkþ1 tk þ ðtk tm Þ ¼ ðskþ1 þ 1Þh þ tkþ1 tk þ ðtk tm Þrðs þ 1Þh þ tM tm 9l;

ð17Þ

where s denotes the maximum number of packet dropouts in the updating periods, satisfying 0rsk rs ðk ¼ 1; . . . ; 1Þ. Combining Eqs. (15) and (17), it is obtained that tkþ1 tk ¼ ðdkþ1 þ 1Þh þ tkþ1 tk rðd þ 1Þh þ tM tm ¼ l:

ð18Þ

Remark 3. From Eq. (18) we can see that, the maximum interval of ZOH’s updating periods is l, which means that at most l seconds one packet is successfully transmitted to

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the actuator such that the system’s update and stability can be guaranteed. For the lower bound of signal transmission delay tm is given, we call l the maximum allowable equivalent delay bound (MAEDB) [13]. It is clear that in the same network conditions, the larger the value of MAEDB is, the more allowable the network-induced delays or/and packet dropouts in data transmission are on the premise of system’s stability. Apparently, substituting Eq. (16) into (13) yields the following continuous-time system [25]: ( _ ¼ ðA þ DAðtÞÞxðtÞ þ BðK þ DðKÞÞxðttm tðtÞÞ; xðtÞ ð19Þ xðtÞ ¼ jðt;t0 nÞ9jðtÞ; t 2 ½t0 n;t0 ; _ t0 nÞ ¼ ðA þ DAðtÞÞjðt; t0 nÞ, t 2 ½t0 n; t0 , and where jðt; t0 nÞ is the solution of jðt; n ¼ tm þ l. Remark 4. As the only existing result on stabilization the NCSs with two static quantizers, the guaranteed cost control for such NCSs is addressed in [9], in which the assumption that one data packet is transformed from signal quantization in one sampling period is needed for modeling the closed-loop feedback system. Comparing with what is established in [9], the networked control model formulated here also suits the case where more than one data packet is engendered by the A/D converter in one sampling period by using the updating instant in this paper, and thus is more general than that in [9]. Moreover, the lower bound of network-induced delay which is nonzero is considered in this model. Before stating our main results, the following lemmas are presented, which will play an indispensable role in deriving our criteria. Lemma 1 (Zhang et al. [24] and Li and Yue [26]). C1 , C2 and O are constant matrices of appropriate dimensions and 0rtm rtðtÞrtM . Then ½tðtÞtm C1 þ ½tM tðtÞC2 þ Oo0 holds, if and only if the following inequalities hold: ðtM tm ÞC1 þ Oo0; ðtM tm ÞC2 þ Oo0: Lemma 2 (Gao et al. [12]). Given appropriately dimensioned matrices S1 , S2 , S3 , with S1 ¼ ST1 . Then, S1 þ S3 S2 þ ST2 ST3 o0 holds if for some matrix W 40 S1 þ S3 W 1 ST3 þ ST2 W S2 o0:

3. Main results In this section, we focus on the design of quantized feedback controller for the NCSs with limited communication capacity. First we shall investigate the conditions under which the closed-loop NCSs in Eq. (13) is asymptotically stable.

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Theorem 1. Consider the NCS in Fig. 1. For given positive constants tm , l and the feedback gain matrix K, if there exist matrices P40, R40, Qi 40, Zi 40 (i=1,2,3), and any appropriately dimensioned matrices S, T, U, V, M, N such that for all l=1,2 the following LMIs (linear matrix inequalities) hold: 3 2 Yl X1 þ X2 þ XT2 þ X3 X4 7 6  X5 0 ð20Þ 5o0; 4 1   l Z2 where

2

6 6 6 6 6 X1 ¼ 6 6 6 6 4

PðA þ DAðtÞÞ þ ðA þ DAðtÞÞT P þ Q1 þ Q2 þ Q3 

0 Q1





 

 







0 0

PBðK þ DðKÞÞ 0

0

0

0

 

Q2 

0 0





0 0

X2 ¼ ½S þ V þ M TS NM N UT UV ; X3 ¼ XT31 ðtm Z1 þ lZ2 þ nZ3 þ lRÞX31 ; X31 ¼ ½A þ DAðtÞ 0 0 0 BðK þ DðKÞÞ 0; X4 ¼ ½S T U V ; 1 1 1 X5 ¼ diagft1 m Z1 ; l R; n R; n Z3 g; Y1 ¼ M; Y2 ¼ N;

3 0 7 0 7 7 0 7 7 7; 0 7 7 0 7 5 Q3

ð21Þ

then the closed-loop system in Eq. (13) is asymptotically stable. Proof. Construct a Lyapunov–Krasovskii functional candidate as Z t Z t Z xT ðsÞQ1 xðsÞ ds þ xT ðsÞQ2 xðsÞ ds þ V ðtÞ ¼ xT ðtÞPxðtÞ þ ttm

Z

0

Z

t

_ ds dy þ x_ T ðsÞZ1 xðsÞ

þ tm

Z

Z

tþy

tm

Z

0

Z

t

_ ds dy þ x_ T ðsÞZ2 xðsÞ

tþy

l

xT ðsÞQ3 xðsÞ ds

tn

Z

0

n

Z

t

_ ds dy x_ T ðsÞZ3 xðsÞ

tþy

t

þ n

tl

t

_ ds dy; x_ T ðsÞRxðsÞ

ð22Þ

tþy

where P40, Qi 40, Zi 40 and R40 are matrices to be determined. By the Newton–Leibniz formula, for any appropriately dimensioned matrices S, T, U, V, M, N, we have gj ¼ 0 (j=1,y,6) with   Z t _ ds ; xðsÞ ð23Þ g1 9xT ðtÞS xðtÞxðttm Þ ttm



g2 9xT ðtÞT xðttm Þxðttm tðtÞÞ

Z

ttm

 _ ds ; xðsÞ

ð24Þ

 _ ds ; xðsÞ

ð25Þ

ttm tðtÞ



g3 9x ðtÞU xðttm tðtÞÞxðtnÞ T

Z

ttm tðtÞ

tn

 Z g4 9xT ðtÞV xðtÞxðtnÞ

t tn

 _ ds ; xðsÞ

ð26Þ

J. Dai / Journal of the Franklin Institute 347 (2010) 1334–1352

 Z g5 9x ðtÞM xðtÞxðttðtÞÞ

1341



t

T

_ ds ; xðsÞ

ð27Þ

ttðtÞ

 Z g6 9xT ðtÞN xðttðtÞÞxðtlÞ

ttðtÞ

 _ ds ; xðsÞ

ð28Þ

tl

where xT ðtÞ ¼ ½xT ðtÞ xT ðttm Þ xT ðttðtÞÞ xT ðtlÞ xT ðttm tðtÞÞ xT ðtnÞ. It is easy to see that there exists Z2 40, such that Z t Z t T _ dsrtðtÞxT ðtÞMZ 1 _ ds; 2xT ðtÞM xðsÞ x_ T ðsÞZ2 xðsÞ ð29Þ M xðtÞ þ 2 ttðtÞ

2xT ðtÞN

Z

ttðtÞ

ttðtÞ T _ dsr½ltðtÞxT ðtÞNZ 1 xðsÞ 2 N xðtÞ þ

tl

Z

ttðtÞ

_ ds: x_ T ðsÞZ2 xðsÞ

ð30Þ

tl

Then, taking the time derivative of V(t) along the trajectory of Eq. (19) and combining Eqs. (23)–(30) yield _ þ xT ðtÞðQ1 þ Q2 þ Q3 ÞxðtÞxT ðttm ÞQ1 xðttm ÞxT ðtlÞQ2 xðtlÞ V_ ðtÞ ¼ 2xT ðtÞPxðtÞ _ xT ðtnÞQ3 xðtnÞ þ x_ T ðtÞðtm Z1 þ lZ2 þ nZ3 þ lRÞxðtÞ

Z

t

_ ds x_ T ðsÞZ1 xðsÞ ttm

Z

t

_ ds x_ T ðsÞZ2 xðsÞ

 Z

ttðtÞ ttm

Z

ttðtÞ

_ ds x_ T ðsÞZ2 xðsÞ

Z

tl

_ ds x_ T ðsÞRxðsÞ

 ttm tðtÞ

Z

t

_ ds x_ T ðsÞZ3 xðsÞ tn

ttm tðtÞ

_ ds þ 2 x_ T ðsÞRxðsÞ

tn

6 X

gj

j¼1

T 1 T rxT ðtÞfX1 þ X2 þ XT2 þ X3 þ X6 þ tðtÞMZ 1 2 M þ ½ltðtÞNZ 2 N gxðtÞ þ

10 X

Xl ;

l¼7

ð31Þ where T 1 T 1 T 1 T X6 ¼ tm SZ 1 1 S þ lTR T þ nUR U þ nVZ 3 V ; Z t _ T Z11 ½S T xðtÞ þ Z1 xðsÞ _ X7 ¼  ds; ½S T xðtÞ þ Z1 xðsÞ ttm Z ttm _ T R1 ½T T xðtÞ þ RxðsÞ _ X8 ¼  ½T T xðtÞ þ RxðsÞ ds; ttm tðtÞ

Z

ttm tðtÞ

_ T R1 ½U T xðtÞ þ RxðsÞ _ ½U T xðtÞ þ RxðsÞ ds;

X9 ¼  Ztnt X10 ¼  tn

_ T Z31 ½V T xðtÞ þ Z3 xðsÞ _ ds: ½V T xðtÞ þ Z3 xðsÞ

Using Lemma 1, we can see that for the delay interval 0rtðtÞrl, T 1 T X1 þ X2 þ XT2 þ X3 þ X6 þ tðtÞMZ 1 2 M þ ½ltðtÞNZ 2 N o0

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holds if and only if T X1 þ X2 þ XT2 þ X3 þ X6 þ lMZ1 2 M o0

ð32Þ

T X1 þ X2 þ XT2 þ X3 þ X6 þ lNZ 1 2 N o0:

ð33Þ

and

When Zi 40 (i=1,3) and R40, Xl (l=7,y,10) are all non-positive. By the Schur complement, Eq. (20) guarantees Eqs. (32) and (33) holding. Therefore, we have V_ ðtÞo0, and the asymptotic stability of system in Eq. (19) is achieved. The proof is completed. & Remark 5. Comparing with [12], the delay information of the NCSs, especially in Eq. (17), is fully defined and used in the Lyapunov functional stability method, and a novel interval delay system approach is employed accordingly, such that less conservative results are expected via numerical simulation. If there is no quantizer on the controller side and only one quantizer on the sampler side of the NCS shown in Fig. 1, as the case in [12], the quantized feedback controller takes uðtÞ ¼ KðI þ Dg Þxðtk tk Þ; tk rtotkþ1 ;

ð34Þ

and thus, the closed-loop system is reduced to the following one as ( _ ¼ ðA þ DAðtÞÞxðtÞ þ BKðI þ Dg Þxðtk tk Þ; tk rtotkþ1 ; xðtÞ

ð35Þ

xðt0 Þ ¼ x0 : The following result can be concluded directly from Theorem 1.

Corollary 1. Consider the NCS in Fig. 1, but without the quantizer f ðÞ. For given the positive constants tm , l and the feedback gain matrix K, if there exist matrices P40, R40, Qi 40, Zi 40 (i=1,2,3), and any appropriately dimensioned matrices S, T, U, V, M, N such that for all l=1,2 the following LMIs hold: 2 3 U1 þ X2 þ XT2 þ U3 X4 Yl 6 7  X5 0 ð36Þ 4 5o0; 



l1 Z2

where Xa ða ¼ 2; 4; 5Þ and Yl are given in Eq. (21) and 2

6 6 6 6 6 U1 ¼ 6 6 6 6 4

PðA þ DAðtÞÞ þ ðA þ DAðtÞÞT P þ Q1 þ Q2 þ Q3

0

0

0

PBKðI þ Dg Þ

  

Q1  

0 0 

0 0 Q2

0 0 0

 

 

 

 

0 

U3 ¼ UT31 ðtm Z1 þ lZ2 þ nZ3 þ lRÞU31 ;

0

7 7 7 7 7 7; 7 7 0 7 5 Q3 0 0 0

U31 ¼ ½A þ DAðtÞ 0 0 0 BKðI þ Dg Þ 0;

then the closed-loop system in Eq. (35) is asymptotically stable.

3

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Obviously, our main objective is to determine the gain matrix K such that the feedback closed-loop system is asymptotically stable, and the results are given as follows. Proposition 1. Consider the NCS in Fig. 1, where f ðÞ and gðÞ are logarithmic quantizers with quantization densities rf and rg , respectively. Given positive constants tm , l, g, there exists the feedback control rule (12) such that the closed-loop system in Eq. (13) is asymptotically stable if there exist matrices X 40, R40, Q i 40, Z i 40 (i=1,2,3), W 40 and any matrices Y, S, T , U , V , M , N of appropriate dimensions and positive scalars ej (j=1,y,4) for all l=1,2 satisfying XX ZW ; "

ð37Þ

W

YT



gI

# o0;

2

P1 þ P2 þ PT2 6  6 6 6  6 6 6  6 6  4 

ð38Þ

P4 P5

Yl 0

L2 0

L4 0

 

l1 X Z 2 X 

0 L3

0 L5







L6









1

3 L7 7 0 7 7 0 7 7 7o0; 0 7 7 0 7 5 L8

ð39Þ

where 2 6 6 6 6 6 P1 ¼ 6 6 6 6 4

AX þ XAT þ Q 1 þ Q 2 þ Q 3 þ e1 DDT

0

0

0

BY

 

Q 1 

0 0

0 0

0 0







Q 2

0

 

 

 

 

0 

P2 ¼ ½S þ V þ M T S N M N U T U V ;

P4 ¼ ½S T 1 1 1 1 1 1 1 1 P5 ¼ diagftm X Z 1 X ;l X R X ;n X R X ;n X Z 3 X g; 2 3 2 2 3 0 0 XE T ea B XAT þ e1 DDT

6 6 6 6 L2 ¼ 6 6 6 6 4

0 0 0 T

Y B 0

T

7 7 7 7 7H; 7 7 7 5

6 6 6 6 L4 ¼ 6 6 6 6 4

0

0 7 7 7 0 7 7; 0 7 7 7 0 5

0

0

0 0 0

6 0 6 6 6 0 L7 ¼ 6 6 0 6 6 T 4Y 0

0 0 0 gdg X 0

3

0

7 7 7 7 7 7; 0 7 7 0 7 5 Q 3 0 0

U V ; 0

3

7 7 7 7 7; 0 7 7 7 gdg X 5 0 0

0

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1344

2 6 6 L3 ¼ 6 6 4

3

T t1 m Z 1 þ e1 DD

e1 DDT

e1 DDT

e1 DDT

 

l1 Z 2 þ e1 DDT 

e1 DDT n1 Z 3 þ e1 DDT

e1 DDT e1 DDT







l1 R þ e1 DDT

L5 ¼ H T ½0 ea B; L6 ¼ diagfe1 I;ea Ig;

H ¼ ½I I I I;

7 7 7; 7 5

Y1 ¼ M ; ð40Þ

L8 ¼ diagfe2 I;e3 gI;e4 gIg; Y 2 ¼ N ; ea ¼ e2 d2f þ e3 þ e4 d2f : Furthermore, the gain matrix K in Eq. (12) is given by K=YX1. Proof. By Schur complement, Eq. (20) is equivalent to 2 3 X1 þ X2 þ XT2 X4 Yl G1 6 7  X5 0 0 7 6 6 7o0; l¼6   l1 Z2 0 7 4 5    G2

ð41Þ

where G1 ¼ XT31 ½Z1 Z2 Z3 R;

1 1 1 G2 ¼ diagft1 m Z1 ;l Z2 ;n Z3 ;l Rg:

ð42Þ

Rewrite Eq. (41) in the following form that l ¼ L þ LTE F T ðtÞLD þ LTD F ðtÞLE þ KT Df LB þ LTB Df K þ TgT K T LB þ LTB KT g þ TgT K T Df LB þ LTB Df KT g ; where

2

O1 þ X2 þ XT2 6  6 L¼6 6  4  with

2 6 6 6 6 6 O1 ¼ 6 6 6 6 4

X4 X5

Yl 0

 

l1 Z2 

ð43Þ 3 O2 7 0 7 7; 0 7 5 G2

ð44Þ

PA þ AT P þ Q1 þ Q2 þ Q3

0

0

0

PBK

 

Q1 

0 0

0 0

0 0

 

 

 

Q2 

0 0









 T

O2 ¼ ½A 0 0 0 BK 0 ½Z1 Z2 Z3 R; and LE ¼ ½E 0 0 0 0 0 0 0 0 0 0 0 0 0 0;

0

3

7 7 7 7 7 7; 0 7 7 0 7 5 Q3 0 0

J. Dai / Journal of the Franklin Institute 347 (2010) 1334–1352

LD ¼ ½DT P 0 0 0 0 0 0 0 0 0 0 DT Z1 DT Z2 DT Z3 DT R; K ¼ ½0 0 0 0 K 0 0 0 0 0 0 0 0 0 0; LB ¼ ½BT P 0 0 0 0 0 0 0 0 0 0 BT Z1 BT Z2 BT Z3 BT R; Tg ¼ ½0 0 0 0 Dg 0 0 0 0 0 0 0 0 0 0:

1345

ð45Þ

It is easy to see from Eq. (43) that there exist scalars el 40 (l=1,2,3,4) such that T T 2 1 T T 1 T T lrL þ e1 LTD LD þ e1 1 LE LE þ e2 LB Df LB þ e2 K K þ e3 LB LB þ e3 Tg K KT g T T þe4 LTB D2f LB þ e1 4 Tg K KT g :

ð46Þ

Using Schur complement, Eq. (38) leads to Y T Y rgW :

ð47Þ

As K=YX1, from Eqs. (37) and (47), we have K T KrgI:

ð48Þ

Combining Eqs. (46) and (48), we derive 2 2 T T 1 T lrL þ e1 LTD LD þ e1 1 LE LE þ ðe2 df þ e3 þ e4 df ÞLB LB þ e2 K K 2 1 2 T þ ðe1 3 gdg þ e4 gdg ÞI I T T 1 T T 9L þ e1 LTD LD þ e1 1 LE LE þ ea LB LB þ e2 K K þ eb I I ;

e2 d2f

e4 d2f ,

2 e1 3 gdg

ð49Þ

2 e1 4 gdg ,

þ e3 þ eb ¼ þ I ¼ ½0 0 0 0 I 0 0 0 0 0 0 0 0 0 0. with ea ¼ From Eq. (49), we can see that (41) holds if 2 3 LTE ea LTB KT L þ e1 LTD LD þ eb I T I 6 7 6  e1 I 0 0 7 6 7o0 ð50Þ 6   ea I 0 7 4 5    e2 I holds. Define J ¼ diagfJ1 ;J2 ;J3 g; where J1 ¼ diagfP1 ;P1 ;P1 ;P1 ;P1 ;P1 g; J3 ¼

J2 ¼ diagfP1 ;P1 ;P1 ;P1 ;P1 g;

1 diagfZ11 ;Z21 ;Z 1 3 ;R ;I;I;Ig:

ð51Þ

Pre- and post-multiplying both sides of Eq. (50) with J, together with the change of matrix variables defined by X ¼ P1 ; Q i ¼ P1 Qi P1 ; Z i ¼ Zi1 ; R ¼ R1 ; ½S T U V Y l  ¼ J1 ½S T U V Yl J2 ;

Y ¼ KP1 ;

then Eq. (39) is obtained by Schur complement, and thus this proof is completed.

ð52Þ &

Remark 6. The conditions in Proposition 1 are not LMIs because of the nonlinear terms 1 1 XX in Eq. (37) and X Z i X (i=1,2,3), X R X in Eq. (39), respectively. In order to solve this non-convex problem, the inequalities in following are needed.

J. Dai / Journal of the Franklin Institute 347 (2010) 1334–1352

1346

1

By noticing Z i 40 and R40, we have ðIX ÞðIX ÞZ0, ðZ i X ÞZ i ðZ i X ÞZ0 and 1 ðRX ÞR ðRX ÞZ0, which are equivalent to, respectively, XX Z2X I;

1

X Z i X rZ i 2X ;

X R

1

X rR2X :

ð53Þ

By combining the conditions in Proposition 1 and Eq. (53), we readily obtain the following theorem. Theorem 2. Consider the NCS in Fig. 1, where f ðÞ and gðÞ are logarithmic quantizers with quantization densities rf and rg , respectively. Given positive constants tm , l, g, there exists the feedback control rule (12) such that the closed-loop system in Eq. (13) is asymptotically stable if there exist matrices X 40, R40, Q i 40, Z i 40 (i=1,2,3), W 40 and any matrices Y, S, T , U , V , M , N of appropriate dimensions and positive scalars ej (j=1,y,4) for all l=1,2 satisfying Eq. (38) and 2X IW Z0; 2

P1 þ P2 þ PT2 6 6  6 6  6 6 6  6 6  4 

ð54Þ P4 P5

Yl 0

L2 0

L4 0

 

l1 ðZ 2 2X Þ 

0 L3

0 L5







L6









3 L7 7 0 7 7 0 7 7 7o0; 0 7 7 0 7 5 L8

ð55Þ

where Pa ða ¼ 1; 2; 4Þ, Lb ðb ¼ 2; . . . ; 8Þ and Y l are given in Eq. (40) and 1 1 1 P 5 ¼ diagft1 m ðZ 1 2X Þ;l ðR2X Þ;n ðR2X Þ;n ðZ 3 2X Þg:

ð56Þ

Furthermore, the gain matrix K in Eq. (12) is given by K=YX1. Among the existing results dealing with NCSs of limited communication capacity, there are mainly two approaches for studying the problem of quantized feedback control. The first approach considers the NCSs with two quantizers as one is on the sampler side to quantize the state signal and another on the controller side to quantize the control signal, such as in [4,5,9,18]. The second approach investigates the NCSs with one quantizer on the sampler side where only the state signal is quantized, such as in [12,13]. Since the first approach has been studied above, we will investigate the control synthesis for NCSs with one quantizer in following. Proposition 2. Consider the NCS in Fig. 1, but without the quantizer f ðÞ. Given positive constants tm , l, and quantization density of gðÞ, there exists the feedback control rule (34) such that the closed-loop system in Eq. (35) is asymptotically stable if there exist matrices X 40, R40, Q i 40, Z i 40 (i=1,2,3), and a diagonal matrix W 40 and any matrices Y, S, T , U , V , M , N of appropriate dimensions and positive scalar e1 for all

J. Dai / Journal of the Franklin Institute 347 (2010) 1334–1352

l=1,2 satisfying 2 P1 þ P2 þ PT2 6  6 6 6  6 6  4

P4

Yl

L2

P5

0

0

 

l1 X Z 2 X 

0 L3







 where Pa ða ¼ 1; 2; 4; 5Þ, 2 XE T BY 6 0 0 6 6 6 0 0 S4 ¼ 6 6 0 0 6 6 4 0 0 0

0

1

S4

1347

3

0 7 7 7 0 7 7o0; 7 S5 5 S6

ð57Þ

Lb ðb ¼ 2; 3Þ, Y l and the following H are given in Eq. (40), and 3 0 07 7 7 07 7; S5 ¼ H T ½0 BY 0; 07 7 7 X5 0

S6 ¼ diag½e1 I; X W

1

X ; d2 g W ; dg ¼ diagfdg 1 ;dg 2 ; . . . ;dg n g:

Moreover, the control gain matrix K in Eq. (34) is given by K=YX Proof. By Schur complement, Eq. (36) is equivalent to 2 3 U1 þ X2 þ XT2 X4 Yl G4 6 7  X5 0 0 7 6 6 7o0; 6   l1 Z2 0 7 4 5    G2

1

ð58Þ

.

ð59Þ

where G2 is given in Eq. (42), and G4 ¼ UT31 ½Z1 Z2 Z3 R. Obviously, Eq. (59) can be decomposed into the form as L þ FTg þ TgT FT þ LTD F ðtÞLE þ LTE F T ðtÞLD o0;

ð60Þ

LTB K,

and L is given in Eq. (44), Tg, LD, LE, LB are given in Eq. (45). where F ¼ Applying Lemma 2, we can find that the condition (60) holds if and only if there exist a diagonal matrix W40 and a scalar e1 40 such that T L þ FW 1 FT þ TgT WTg þ e1 LTD LD þ e1 1 LE LE o0:

Applying Schur complement to (61), we obtain 2 3 L þ TgT WTg þ e1 LTD LD F LTE 6  W 0 7 4 5o0:   e1 I Define J ¼ diagfJ1 ;J2 ;J 3 g; where J1 and J2 are defined in Eq. (51), and J 3 ¼ diagfZ11 ;Z21 ;Z31 ;R1 ;I;P1 g:

ð61Þ

ð62Þ

1348

J. Dai / Journal of the Franklin Institute 347 (2010) 1334–1352

Pre- and post-multiplying both sides of (62) with J , together with the change of matrix variables defined in Eq. (52) and W ¼ W 1 , then we can obtain Eq. (57) by Schur complement. This completes the proof. & Theorem 3. Consider the NCS in Fig. 1, but without the quantizer f ðÞ. Given positive constants tm , l and quantization density of gðÞ, there exists the feedback control rule (34) such that the closed-loop system in Eq. (35) is asymptotically stable if there exist matrices X 40, R40, Q i 40, Z i 40 (i=1,2,3) and a diagonal matrix W 40 and any matrices Y, S, T , U , V , M , N of appropriate dimensions and positive scalar e1 for all l=1,2 satisfying 2 3 P1 þ P2 þ PT2 P4 Yl L2 S4 6  P5 0 0 0 7 6 7 6 7 1 6 7o0; ðZ 2X Þ 0 0   l ð63Þ 2 6 7 6 7    L3 S5 5 4     S6 where Pa ða ¼ 1; 2; 4Þ, Lb ðb ¼ 2; 3Þ and Y l are given in Eq. (40); Si ði ¼ 4; 5Þ are in Eq. (58); P 5 is in Eq. (56); and S 6 ¼ diagfe1 I;W 2X ;d2 g Wg Moreover, the control gain matrix K in Eq. (34) is given by K=YX1. Remark 7. It is worth pointing out that the nonlinear terms in Propositions 1 and 2 can also be solved by the cone complementarity linearization (CCL) method [27], by which the procedure is very similar as in [12], thus it is omitted here. Comparing with the iterative algorithm needed in the CCL method, our approach based on the matrix inequalities is in easily verifiable condition of LMIs. 4. Illustrative examples In order to show the advantage of the proposed methods, as one is with two quantizers and another with one quantizer, two examples are provided to compare our results with most of the recent related papers. Example 1. For the case of NCSs with two quantizers, we consider the Example 4 in [9], thus the A, B, DAðtÞ in Eq. (1) are     0 2 0 A¼ ; B¼ ; DAðtÞr0:1: 0:5 1 1

As can be seen, the eigenvalues of A are 1 and 2, so the considered open-loop system is not stable. Under the given quantizers’ parameters, our purpose is to determine the feedback controller in the form of Eq. (12) such that the resulting closed-loop system is asymptotically stable. As the same in [9], we choose the quantization densities rg ¼ rf ¼ 0:818 for the quantizers gðÞ and f ðÞ in Fig. 1. By the method suggested in [9], it is found that the closedloop system is asymptotically stable when the MAEDB is 0.2 s, that is, Z ¼ 0:2. While by solving the convex problem in Theorem 2 with same g and tm ¼ 0 (assuming that tm is

J. Dai / Journal of the Franklin Institute 347 (2010) 1334–1352

1349

sufficiently small), we can find the feasible upper bound of l is 0.401, thus the MAEDB obtained here is 0.401 s. It is clear to see that for this example the result using the present method is much better than that in [9]. Moreover, the lower bound of the network-induced delay is allowed to be nonzero. Then, applying Theorem 2 with l ¼ 0:401, the feasible solutions are   1:5124 0:4101 ; X¼ 0:4101 3:3018 Y ¼ ½0:3421 10:0399, and K ¼ ½1:0873 3:1758. Under the initial condition of jðtÞ ¼ ½0:5 0:8T , the simulation result is depicted in Fig. 2, in which the effect of signal quantization is shown with real line and F(t) is chosen as F ðtÞ ¼ sinðtÞ. Example 2. For the case of NCSs with one quantizer, we suppose the physical plant in Fig. 1 is the satellite system depicted in [12], so the parameter matrices in Eqs. (1) and (2) are 2 3 2 3 0 0 1 0 0 6 0 7 6 7 0 0 1 7 6 607 A¼6 7; B ¼ 6 7; D ¼ E ¼ 0: 4 0:3 0:3 0:004 0:004 5 415 0:3

0:3

0:004

0

0:004

If there is no quantizer on the controller side and only one quantizer on the sampler side, as the case in [12], Table 1 provides a detailed comparison of MAEDB for different lower bounds of signal transmission delay, where the quantization density of gðÞ is chosen same as [12]. Table 1 clearly shows that the MAEDB obtained by Theorem 3 in this paper are consistently larger than that in [12], thus the less conservativeness yields in the sense that

0.6 0.4

State Response

0.2 0 −0.2 −0.4

x (t) 1

−0.6

quantized x (t)

−0.8

x (t)

1

2

quantized x (t)

−1 −1.2

2

0

2

4

6

8

10

Time (s)

Fig. 2. System state of the closed-loop NCS without/with two quantizers.

J. Dai / Journal of the Franklin Institute 347 (2010) 1334–1352

1350

Table 1 Calculated MAEDB with different lower bounds of signal transmission delay. tm

0.01

0.03

0.07

0.1

0.15

0.18

[12] Theorem 3

0.301 0.362

0.310 0.347

0.288 0.315

0.260 0.291

0.216 0.253

0.176 0.232

By the definition of this paper, the lower bound of signal transmission delay in [12] is denoted as Zm , and MAEDB in [12] is denoted as k.

0.6 State Response without quantizer

x1(t) x2(t)

0.4

x3(t) x (t)

0.2

4

0 −0.2 −0.4 −0.6 −0.8

0

10

20

30

40

50

Time (s)

Fig. 3. System state of the closed-loop NCS without one quantizer.

more network-induced delays or/and packet dropouts are allowable in the updating periods of ZOH on the premise of system’s stability. For example, when the lower bound of signal transmission delay tm is 0.18 s, Theorem 3 suggests the maximum allowable value of l is 0.232, and 2

0:1332

0:1330

6 0:1330 0:1450 6 X ¼6 4 0:0083 0:0047 0:0092 0:0079

3

0:0083

0:0092

0:0047

0:0079 7 7 7; 0:0019 5

0:0052 0:0019

0:0049

Y ¼ ½0:0022 0:0050 0:0032 0:0005, K ¼ ½0:0166 0:0260 0:8026 0:3447. At the end we illustrate that the closed-loop system is asymptotically stable with above obtained control gain. The state responses of the NCS without or with quantization are depicted in Figs. 3 and 4, respectively, where the initial condition is assumed to be jðtÞ ¼ ½0:1 0:5 0:3 0:2T , and F(t) is chosen as F ðtÞ ¼ sinðtÞ.

J. Dai / Journal of the Franklin Institute 347 (2010) 1334–1352

1351

0.6 State Response with quantizer

quantized x (t) 1

0.4

quantized x (t) 2

quantized x (t) 3

0.2

quantized x4(t)

0 −0.2 −0.4 −0.6 −0.8 −1

0

10

20

30

40

50

Time (s)

Fig. 4. System state of the closed-loop NCS with one quantizer.

5. Conclusion Since the data packets on the sensor side are not transmitted to the actuator orderly and totally on the sampling sequence due to the effect of network-induced delays, data packet dropouts and signal quantization, the closed-loop feedback model of NCSs based on the updating instants of ZOH has been formulated, and a quantized feedback controller design procedure has been proposed. Based on the full use of the delay information and utilization of novel techniques for interval delay systems, the existence of admissible network-based controller is of less conservative and in easily verifiable conditions. Two examples are included to show that the proposed criteria improve the existing results significantly. Acknowledgements This work is supported by the Key Research Foundation of Science and Technology of the Ministry of Education of China under Grant 107058. References [1] Y.C. Tian, D. Levy, Compensation for control packet dropout in networked control systems, Information Sciences 178 (5) (2008) 1263–1278. [2] H.C. Yan, X.H. Huang, M. Wang, Delay-dependent stability criteria for a class of networked control systems with multi-input and multi-output, Chaos, Solitons & Fractals 34 (3) (2007) 997–1005. [3] J.W. Cao, S.M. Zhong, Y.Y. Hu, Novel delay-dependent stability conditions for a class of MIMO networked control systems with nonlinear perturbation, Applied Mathematics and Computation 197 (2) (2008) 797–809. [4] D. Yue, J. Lam, Z.D. Wang, Persistent disturbance rejection via state feedback for networked control systems, Chaos, Solitons & Fractal 40 (1) (2009) 382–391. [5] E.G. Tian, D. Yue, C. Peng, Quantized output feedback control for networked control systems, Information Sciences 178 (12) (2008) 2734–2749.

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J. Dai / Journal of the Franklin Institute 347 (2010) 1334–1352

[6] Y.G. Niu, T.G. Jia, X.Y. Wang, F.W. Yang, Output-feedback control design for NCSs subject to quantization and dropout, Information Sciences 179 (21) (2009) 3804–3813. [7] H.G. Zhang, M. Li, J. Yang, Fuzzy model-based robust networked control for a class of nonlinear systems, IEEE Transactions on Systems, Man, and Cybernetics—Part A: Systems and Humans 39 (2) (2009) 437–447. [8] H.G. Zhang, D.D. Yang, T.Y. Chai, Guaranteed cost networked control for T-S fuzzy systems with time delays, IEEE Transactions on Systems, Man, and Cybernetics: Applications and Reviews 37 (2) (2007) 160–172. [9] D. Yue, C. Peng, G.Y. Tang, Guaranteed cost control of linear systems over networks with state and input quantizations, IEEE Proceedings: Control Theory and Applications 153 (6) (2006) 658–664. [10] D. Yue, Q.L. Han, J. Lam, Network-based robust H1 control of systems with uncertainty, Automatica 41 (6) (2005) 999–1007. [11] H.G. Zhang, J. Yang, C.Y. Su, T-S fuzzy-model based robust H1 design for networked control systems with uncertainties, IEEE Transactions on Industrial Informatics 3 (4) (2007) 289–301. [12] H.J. Gao, T.W. Chen, J. Lam, A new delay system approach to network-based control, Automatica 44 (1) (2008) 39–52. [13] C. Peng, Y.C. Tian, Networked H1 control of linear systems with state quantization, Information Sciences 177 (24) (2007) 5763–5774. [14] D. Yue, Q.L. Han, Network-based robust H1 filtering for uncertain linear systems, IEEE Transactions on Signal Procession 54 (11) (2006) 4293–4301. [15] H.J. Gao, T.W. Chen, H1 estimation for uncertain systems with limited communication capacity, IEEE Transaction on Automatic Control 52 (11) (2007) 2070–2084. [16] H.J. Gao, T.W. Chen, Network-based H1 output tracking control, IEEE Transaction on Automatic Control 53 (3) (2008) 655–667. [17] X.C. Jia, D.W. Zhang, X.H. Hao, Fuzzy H1 tracking control for nonlinear networked control systems in T-S fuzzy model, IEEE Transactions on Systems, Man, and Cybernetics—Part B: Cybernetics 39 (4) (2009) 1073–1079. [18] E.G. Tian, D. Yue, X. Zhao, Quantised control design for networked control systems, IET Control Theory Applications 1 (6) (2007) 1693–1699. [19] M.Y. Fu, L.H. Xie, The sector bound approach to quantized feedback control, IEEE Transactions on Automatic Control 50 (11) (2005) 1698–1711. [20] D. Yue, Q.L. Han, C. Peng, State feedback controller design of networked control systems, IEEE Transactions on Circuits and Systems—II: Express Briefs 51 (11) (2004) 640–644. [21] O.M. Kwon, J.H. Park, Delay-range-dependent stabilization of uncertain dynamic systems with interval time-varying delays, Applied Mathematics and Computation 208 (1) (2009) 58–68. [22] L. Li, X.D. Liu, T.Y. Chai, New approaches on H1 control of T-S fuzzy systems with interval time-varying delay, Fuzzy Sets and Systems 160 (12) (2009) 1669–1688. [23] E.G. Tian, D. Yue, Y. J Zhang, Delay-dependent robust H1 control for T-S fuzzy system with interval timevarying delay, Fuzzy Sets and Systems 160 (12) (2009) 1708–1719. [24] Y.J. Zhang, D. Yue, E.G. Tian, New stability criteria of neural networks with interval time-varying delay: a piecewise delay method, Applied Mathematics and Computation 208 (1) (2009) 249–259. [25] H.J. Gao, J.L. Wu, P. Shi, Robust sampled-data H1 control with stochastic sampling, Automatica 45 (7) (2009) 1729–1736. [26] H.J. Li, D. Yue, Synchronization stability of general complex dynamical networks with time-varying delays: a piecewise analysis method, Journal of Computational and Applied Mathematics 232 (2) (2009) 149–158. [27] L.E. Ghaoui, F. Oustry, M. AitRami, A cone complementarity linearization algorithm for static outputfeedback and related problems, IEEE Transactions on Automatic Control 42 (8) (1997) 1171–1176.