H∞ control for networked control systems with limited information

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Journal of the Franklin Institute 349 (2012) 1915–1929 www.elsevier.com/locate/jfranklin

HN control for networked control systems with limited information Tao Liua,b, Hao Zhanga,b, Qijun Chena,b,n, Huaicheng Yanc a

Department of Electronics and Information Engineering, Tongji University, Shanghai 201804, PR China b The Key Laboratory of Embedded System and Service Computing, Ministry of Education, PR China c School of Information Science and Engineering, East China University of Science and Technology, Shanghai 200237, PR China Received 30 September 2011; received in revised form 16 December 2011; accepted 27 February 2012 Available online 17 March 2012

Abstract In this paper, H1 control problems are investigated for a class of networked control systems. An improved networked control system model is proposed and the effects of random packet dropout, delay and sensor fault are considered simultaneously. The packet dropout process is modeled as a Markov chain, and the delays are bounded and occurred in a random way in this paper. The fault for each sensor is governed by an individual random variable satisfying a certain probabilistic distribution. The resulting closed-loop system is transformed to a Markovian switching system. Sufficient conditions for stochastic stability of the closed-loop system are given in terms of linear matrix inequality. A mode-independent controller is designed such that the closed-loop system is stochastically stable and achieves the given H1 disturbance attenuation level. Finally, the simulation of the inverted pendulum control is given to illustrate the effectiveness of the proposed method. & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction In the past few years, the networked control systems (NCS) received increasing attention due to its advantages, such as lower cost, easier installation and maintenance, etc. [1,2]. The fundamental purpose of NCS is to provide access to the control information of the distributed control devices. The sensors, actuators and the controllers become independent nodes on a n Corresponding author at: Department of Electronics and Information Engineering, Tongji University, Shanghai 201804, PR China. Tel.: þ86 21 69589338; fax: þ86 21 69589241. E-mail address: [email protected] (Q. Chen).

0016-0032/$32.00 & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2012.02.017

1916

T. Liu et al. / Journal of the Franklin Institute 349 (2012) 1915–1929

real-time network [17]. However, the network brings many challenges due to the limitation of NCS. Packet dropout, networked-induced delay and sensor fault are three main problems in NCS and make only limited control information transmitted successfully to the actuator. Packet dropout and random delay in NCS are two major causes for the deterioration of system stability. Therefore, various approaches for the packet dropout and random delay issues have been investigated in the existing literature. The controller design and stabilization problems are studied for the NCS with random communication delays in [3–10,18,19], etc. Another important issue in NCS control problem is packet dropout, such as [3,11–18,20,21], etc. Most of the NCS models are presented by using Bernoulli random binary distributed sequence methods or Markov chain. In [4], a new controller design problem is studied for NCS with random communication delays. The random delays are modeled as a linear function of the stochastic variable satisfying Bernoulli random binary distribution. The stabilization problem is addressed for a class of NCS in the discrete-time domain with random delays in [6]. The sensor-to-controller and controller-to-actuator delays are modeled as two Markov chains. The stabilization problem for a class of NCS with packet-loss is investigated in [20]. The packet-loss models are considered as arbitrary packet-loss process and Markovian packet-loss process. On another research front line, sensor fault is one of the most important issues of NCS. The distributed sensors in NCS by the effects of aging, disturbance, temperature, etc. would cause fault and influence the system performance. Most of the previous literature only consider the sensor fault governed by a Bernoulli random binary distribution. Therefore, each sensor has identical fault rate, such as [22,23]. In the current research, it is assumed that each sensor has different fault rate at any sample time independent of the others. The linear minimum variance unbiased state estimation for systems with multiple sensors with different characteristics is considered in [24]. The reliable controller designed for NCS against both probabilistic sensors and actuators faults is investigated in [25,26]. The measurement missing phenomenon is assumed to occur in random way and the missing probability for each sensor is governed by an individual random variable satisfying a certain probabilistic distribution in [27]. However, most of the current work consider one or two of the network condition. With the motivation of the above reasons, it is natural to consider the stabilization and controller design problems for NCS with packet-dropout, random delay and sensor fault simultaneously, which is much closer to real situation. To the best of the authors’ knowledge, there are only few work investigated this problem. This research area has not been fully investigated and thus remains a challenge. In this paper, the H1 control for NCS with limited information is considered. The main contribution can be summarized in the following:



 

An improved NCS model with packet dropout, random delay and sensor fault is proposed. The packet dropout process is modeled as a finite state Markov chain and the resulting closed-loop system is a Markovian switching system. The sensor fault is governed by a matrix variable, which each sensor has different fault rate. The multiple Lyapunov function is applied for analysis and synthesis. The sufficient condition for stochastic stability of the closed-loop system is given in terms of linear matrix inequality (LMI). The state feedback controller is designed by using a cone complementary linearization approach to ensure that the closed-loop system is stochastically stable and achieves the disturbance attenuation level.

T. Liu et al. / Journal of the Franklin Institute 349 (2012) 1915–1929

1917

The rest of this paper is organized as follows. In Section 2, a Markovian switching system is modeled for a class of NCS. The main result is given in Section 3. In Section 4, a numerical example is provided to illustrate the effectiveness of the proposed method. The conclusion is made in Section 5. Notation: Throughout the paper, Rn and Rmn denote, respectively, the n-dimensional Euclidean space and the set of all m  n real matrices. For a vector x 2 Rl , the 2-norm of x is defined as jxj ¼ ðxT xÞ1=2 , and for a matrix Q 2 Rmn , JQJ and lmin ðQÞ are defined as the largest singular value and the minimum eigenvalue of Q, respectively. XðÞ stands for the mathematical expectation. In symmetric block matrices, the symbol n is used as an ellipsis for terms induced by symmetry. The superscript ‘‘T’’ denotes the transposition of vectors or matrix. 2. Problem statement In this section, we consider the NCS with random packet dropout, delay and sensor fault. The plant modeled as a continuous-time linear time-invariant system is described by _ ¼ AxðtÞ þ BuðkTÞ þ EoðtÞ, xðtÞ zðtÞ ¼ CxðtÞ þ DoðtÞ,

t 2 ½kT, ðk þ 1ÞTÞ,

n

ð1Þ m

where xðtÞ 2 R is the state of the plant, uðkTÞ 2 R is the discrete-time control input signal, zðtÞ 2 Rq is the regulated output, and oðtÞ 2 Rp is the disturbance input. A, B, C, D, E are known system matrices with appropriate dimensions. Here we consider that u(kT) is a static sampled-data controller with a constant gain K. We assume that the pair (A, B) is stabilizable. Following network setup of system (1) will be considered in this paper: There exist network between sensors–controller and controller–actuators. The sensors, controller and actuators are all clock-driven. We assume that datum are transmitted in a single packet at each time step. Then the continuous-time plant can be discretized as xðk þ 1Þ ¼ GxðkÞ þ HuðkÞ þ F oðkÞ, zðkÞ ¼ CxðkÞ þ DoðkÞ,

ð2Þ RT

RT

where G ¼ GðTÞ ¼ eAT , H ¼ HðTÞ ¼ 0 eAs ds B, F ¼ F ðTÞ ¼ 0 eAs ds E and T is the sampling period. Similar to [18], the system (1) with packet dropouts can be modeled as a discrete-time system with Markovian jump parameters, which is much closer to real situation. Define z ¼ fi1 ,i2 ,i3 , . . .g, a subsequence of time step 1; 2,3, . . ., which contains the information of packet losses, denotes the sequence of time points of successful data transmission from the sensor to actuator. For example, if the datum are successfully transmitted to the actuator at time step 1; 3,6; 7, . . ., we can define z ¼ fi1 ,i3 ,i6 ,i7 , . . .g. Then the definition of Markovian packet dropout process is given below. Definition 1. Packet dropout process is defined as Zk ¼ fikþ1 ik ,ik 2 zg,

ð3Þ

which takes values from d ¼ f1; 2, . . . ,sg, s called maximum packet dropout upper bound. Further explanation on Eq. (3), Zk ¼ 1 denotes no dropout, Zk ¼ 2 stands for 1 dropout between two consecutive successful packets, etc.

T. Liu et al. / Journal of the Franklin Institute 349 (2012) 1915–1929

1918

Definition 2. Stochastic packet dropout process (3) is said to be Markovian if it is governed by a discrete-time homogeneous Markov chain on a probability space ðO,F ,PÞ, and takes values in d with the following transition probabilities: lij ¼ PrðZkþ1 ¼ jjZk ¼ iÞZ0 for i,j 2 d, P where sj ¼ 1 lij ¼ 1 for i 2 d. The transition probability matrix is defined by 2 3 l11 l12    l1s 6 l21 l22    l2s 7 6 7 L¼6 7: ^ & ^ 5 4 ^ ls1 ls2    lss

ð4Þ

Fig. 1 shows timing sequence of NCS with packet dropouts. Let solid lines denote successful packets from sensor to actuator and dashed lines stand for lost packets. Then we can discretize system (1) with stochastic packet dropout by time-varying sampling periods T^ ¼ Zk T, as follows: xðk þ Zk Þ ¼ GðZk ÞxðkÞ þ HðZk ÞuðkÞ þ F ðZk ÞoðkÞ, zðkÞ ¼ CxðkÞ þ DoðkÞ,

ð5Þ R Zk T

where GðZk Þ ¼ eAZk T , HðZk Þ ¼ 0 eAs ds B and F ðZk Þ ¼ In particular, we design mode-independent controller

R Zk T 0

eAs ds E.

uðkÞ ¼ KPxðktk Þ ð0rtk rtM Þ, where

2

r1 6 60 P¼6 6 ^ 4 0

0 &

 &

& 

& 0

ð6Þ

3 0 7 ^ 7 7: ^ 7 5 rn

P with ri ði ¼ 1, . . . ,nÞ being n unrelated random variables which are also unrelated with oðkÞ. ri denotes fault of each sensor. We assumed that ri has the probabilistic density function F ðri Þ ði ¼ 1, . . . ,nÞ on the interval [0,1]. The mathematical expectation and variance of ri are ai and b2i , respectively. Here, tk represents the sensor-to-actuator delay, tM is the upper bound of the time delay.

Fig. 1. Timing diagram of NCS with packet dropout.

T. Liu et al. / Journal of the Franklin Institute 349 (2012) 1915–1929

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Considering the effect of random packet dropout, delay and sensor fault simultaneously, the closed-loop system can be rewritten as xðk þ Zk Þ ¼ Gi xðkÞ þ Hi KPxðktk Þ þ Fi oðkÞ ¼ Gi xðkÞ þ Hi KðPPÞxðktk Þ þ Hi KPxðktk Þ þ Fi oðkÞ,

ð7Þ

zðkÞ ¼ CxðkÞ þ DoðkÞ, xðkÞ ¼ cðkÞ,

k ¼ tM ,tM þ 1, . . . ,0,

where cðkÞ is the initial state of the system, n X P ¼ XfPg ¼ aj Uj , j¼1

Uj ¼ diagf0, . . . ,0 ,1, 0, . . . ,0 g, |fflfflfflffl{zfflfflfflffl} |fflfflfflffl{zfflfflfflffl} nj

j1

Gi ¼ GZk ¼ i ,

Hi ¼ HZk ¼ i ,

Fi ¼ FZk ¼ i for i ¼ 1, . . . ,s:

Remark 1. Eq. (6) describes the control input with multiple sensors, in which the matrix P denotes the whole missing case and the random variable ri corresponds to the ith sensor. Note that the sensor fault phenomenon has been extensively investigated. However, most model of sensor fault adopt Bernoulli random binary distributed sequence methods in the existing literature, which can only represent completely fault or completely normal of each sensor and all sensors have identical fault case. In this paper, we consider each sensor has individual fault rate, which is much closer to real situation. 3. Main results This section is devoted to stabilize the system (5) with random packet dropout, delay and sensor fault. Since the closed-loop system (7) contains stochastic parameter, we need to introduce the notion of stochastic stability for the problem formulation. Definition 3. The system (7) is said to be stochastically stable if for every finite x0 ¼ xð0Þ, initial mode Z0 ¼ Zð0Þ 2 d, with oðkÞ  0 and the following inequality holds: ( X

1 X

) JxðkÞJ2 jx0 ,Z0 o1:

ð8Þ

k¼0

With this definition, we aim to design the mode-independent controller (6), ensures that the closed-loop system (7) with random packet dropouts, delay and sensor fault is stochastically stable and achieves given disturbance attenuation level. Then the controller design problems are given as follows: (1) The closed-loop system (7) is stochastically stable.

T. Liu et al. / Journal of the Franklin Institute 349 (2012) 1915–1929

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(2) For a given scalar g40 and all nonzero oðkÞ, under the zero initial condition, the regulated output satisfies 1 X

XfzT ðkÞzðkÞgog2

k¼0

1 X

oT ðkÞoðkÞ:

ð9Þ

k¼0

To establish the relationship between the stability properties of system (1) and discretetime system (7), the following lemma will be given below. Lemma 1 (Xie and Xie [21]). The stochastic stability in discrete-time implies the stochastic stability in continuous time. With the above lemma, the stability of system (1) can be converted into the stability of system (7). Then the sufficient condition for stochastic stability of system (7) is given in the following theorem. Theorem 1. For the given feedback gain matrix K, the discrete-time stochastic switching system (7) with random packet dropout, delay and sensor fault is stochastically stable, if there exist positive matrices Pi ,Q, and R with appropriate dimensions satisfying the following LMI: 3 2 G1 n n n n n 7 6 G2 G3 n n n n 7 6 7 6 6 tM ðM1 þ N1 Þ tM ðM2 þ N2 Þ tM R n n n 7 7 6 ð10Þ 7o0, 6 PLi Gi PLi Hi KP 0 PLi n n 7 6 7 6 6 QG i QH i KP 0 0 Q n 7 5 4 tM RðGi IÞ tM RH i KP 0 0 0 tM R where G1 ¼ M1 þ M1T þ N1 þ N1T Pi , G2 ¼ M1 þ M2 N1 þ N2 , G3 ¼ M2 M2T N2 N2T Q þ

n X

b2j ðHi KUj ÞT PLi Hi KUj

j¼1

þ

n X

b2j ðHi KUj ÞT QH i KUj þ tM

i¼j

P ¼ ½P1    Ps ,

n X

b2j ðHi KUj ÞT RH i KUj ,

j¼1

Li ¼ ½li1 In    lis In T

for i ¼ 1, . . . ,s:

Proof. Construct Lyapunov function candidates with oðkÞ ¼ 0 for closed-loop system (7) as V ðxðkÞ,Zk Þ ¼ V1 ðxðkÞ,Zk Þ þ V2 ðxðkÞ,Zk Þ þ V3 ðxðkÞ,Zk Þ, where V1 ðxðkÞ,Zk Þ ¼ xT ðkÞPi xðkÞ,

ð11Þ

T. Liu et al. / Journal of the Franklin Institute 349 (2012) 1915–1929

V2 ðxðkÞ,Zk Þ ¼

k1 X

1921

xT ðkÞQxðkÞ,

ktk

V3 ðxðkÞ,Zk Þ ¼

k1 X

1 X

eT ðsÞReðsÞ:

l ¼ tk s ¼ kþl

Pi 40,Q40 and R40 are matrices to be determined. The new variable e(k) satisfies the following equation: eðkÞ ¼ xðk þ 1ÞxðkÞ: Note that X½PP ¼ diagf0, . . . ,0g. In the following equation, the free weighting matrix will be used. Then for Zk ¼ i,Zkþ1 ¼ j, we have DV1 ðxðkÞ,Zk Þ ¼ X½V1 ðxðk þ Zk ÞÞjZk ¼ iV1 ðxðkÞ,Zk ¼ iÞ ¼ X½ðGi xðkÞ þ Hi KðPPÞxðktk Þ þ Hi KPxðktk ÞÞT PLi ðGi xðkÞ þHi KðPPÞxðktk Þ þ Hi KPxðktk ÞÞxT ðkÞPi xðkÞ ¼ ðGi xðkÞ þ Hi KPxðktk ÞÞT PLi ðGi xðkÞ þ Hi KPxðktk ÞÞ n X b2j ðHi KUj xðktk ÞÞT PLi Hi KUj xðktk Þ xT ðkÞPi xðkÞ þ "

j¼1

T

þ2x ðk,lÞM xðkÞxðktðkÞÞ

k1 X

# eðlÞ ,

ð12Þ

l ¼ ktk

DV2 ðxðkÞ,Zk Þ ¼ X½V2 ðxðk þ Zk ÞÞjZk ¼ iV2 ðxðkÞ,Zk ¼ iÞ rX½ðGi xðkÞ þ Hi KðPPÞxðktk Þ þ Hi KPxðktk ÞÞT QðGi xðkÞ þHi KðPPÞxðktk Þ þ Hi KPxðktk ÞÞxT ðktk ÞQxðktk Þ ¼ ðGi xðkÞ þ Hi KPxðktk ÞÞT QðGi xðkÞ þ Hi KPxðktk ÞÞ n X b2j ðHi KUj xðktk ÞÞT QH i KUj xðktk Þ xT ðktk ÞQxðktk Þ þ "

j¼1

T

þ2x ðk,lÞN xðkÞxðktðkÞÞ

k1 X

# eðlÞ ,

ð13Þ

l ¼ ktk

DV3 ðxðkÞ,Zk Þ ¼ X½V3 ðxðk þ Zk ÞÞjZk ¼ iV3 ðxðkÞ,Zk ¼ iÞ ¼ tM X½ððGi IÞxðkÞ þ Hi KðPPÞxðktk Þ þ Hi KPxðktk ÞÞT R k1 X eT ðlÞReðlÞ ððGi IÞxðkÞ þ Hi KðPPÞxðktk Þ þ Hi KPxðktk ÞÞ l ¼ ktk T

¼ tM ððGi IÞxðkÞ þ Hi KPxðktk ÞÞ RððGi IÞxðkÞ n X b2j ðHi KUj xðktk ÞÞT RH i KUj xðktk Þ þHi KPxðktk ÞÞ þ tM j¼1



k1 X

eT ðlÞReðlÞ,

l ¼ ktk

where xðk,lÞ ¼ ½xT ðkÞ xT ðktk Þ eT ðlÞT , M ¼ ½M1T M2T 0T , N ¼ ½N1T N2T 0T .

ð14Þ

T. Liu et al. / Journal of the Franklin Institute 349 (2012) 1915–1929

1922

By linear matrix inequality method, we obtain from Eqs. (12)–(14) that DV ðxðkÞ,Zk Þ ¼ X½V ðxðk þ Zk ÞÞjZk ¼ iV ðxðkÞ,Zk ¼ iÞ ¼ DV1 ðxðkÞ,Zk Þ þ DV2 ðxðkÞ,Zk Þ þ DV3 ðxðkÞ,Zk Þ k1 1 X xT ðk,lÞCi xðk,lÞ, r tM l ¼ kt

ð15Þ

k

where

2

6 Ci ¼ 4

M1 þ M1T þ N1 þ N1T Pi

n

n

M1 þ M2 N1 þ N2 tM ðM1 þ N1 Þ

G3 tM ðM2 þ N2 Þ

n

þYT11 PLi Y11

þ

YT11 QY11

Y11 ¼ ½Gi Hi KP 0,

þ

3 7 5

tM R

YT12 RY12 ,

Y12 ¼ ½Gi I Hi KP 0:

ð16Þ

By the Schur complement lemma, the inequality (10) guarantees Ci o0. Thus, with the above relations the inequality (15) can be rewritten as DV ðxðkÞ,Zk Þ ¼ X½V ðxðk þ Zk ÞÞjZk ¼ iV ðxðkÞ,Zk ¼ iÞ rminflmin ðCi ÞgjxðkÞj2 ¼ mjxðkÞj2 :

ð17Þ

From the previous inequalities, we can obtain

( kþZ ) Xk 2 X½V ðxðk þ Zk ÞjZk ¼ iÞV ðxðkÞ,Zk ¼ iÞrmX jxðkÞj ,

ð18Þ

k¼0

which implies that ( ) 1 X 1 2 jxðkÞj r XfV ðxð0Þ,Z0 Þgr1: X m k¼0

ð19Þ

Therefore, by Definition 3, it can be verified that the closed-loop system (7) is stochastically stable and the proof is completed. & In the following, we are in the position to state and prove the main result. The H1 controller design method is given in the following theorem. Theorem 2. Consider the discrete-time stochastic switching system (7) with random packet dropout, delay and sensor fault, if there exist positive matrices Pi ,Q,R,Xi ,Y ,Z with appropriate dimensions and feedback gain matrix K satisfying matrix inequalities 3 2 ~ n n n n G1 7 6 Y Xi n n n 7 6 21 7 6 6 Y21 0 Y n n 7 ð20Þ 7o0, 6 7 6 t Y 0 0 t Z n 5 4 M 22 M G~ 2 0 0 0 G~ 3 PLi Xi ¼ I,

QY ¼ I,

RZ ¼ I,

ð21Þ

T. Liu et al. / Journal of the Franklin Institute 349 (2012) 1915–1929

where

2

6 6 G~ 1 ¼ 6 6 4

G11

n

n

n

G21 DT C

G22 0

n

n

g2 I þ DT D

n

tM ðM1 þ N1 Þ

tM ðM2 þ N2 Þ

0

tM R

1923

3 7 7 7, 7 5

G11 ¼ M1 þ M1T þ N1 þ N1T Pi þ C T C, G21 ¼ M1 þ M2 N1 þ N2 , G22 ¼ M2 M2T N2 N2T Q, Y21 ¼ ½Gi Hi KP Fi 0, Y22 ¼ ½Gi I Hi KP Fi 0, G~ 2 ¼ ½FT FT tM FT T ,

F ¼ ½fT1 fT2 . . . fTn T ,

fj ¼ ½0 bj Hi KUj 0 0 for j ¼ 1, . . . ,n, G~ 3 ¼ diagfXi , . . . ,Xi ,Y , . . . ,Y ,tM Z, . . . ,tM Z g, |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} n

n

T

P ¼ ½P1    Ps ,

n

Li ¼ ½li1 In    lis In  for

i ¼ 1, . . . ,s:

Then there exists a mode-independent controller which ensures that the closed-loop system (7) is stochastically stable and achieves the given disturbance attenuation performance.

Proof. From Theorem 1, it can be verified that under the condition (20), the closed-loop system (7) is stochastically stable with oðkÞ ¼ 0. For nonzero oðkÞ, using the same Lyapunov function candidates as in Theorem 1, the following relations can be obtained. For Zk ¼ i,Zkþ1 ¼ j, we have DV1 ðxðkÞ,Zk Þ ¼ X½V1 ðxðk þ Zk ÞÞjZk ¼ iV1 ðxðkÞ,Zk ¼ iÞ ¼ X½ðGi xðkÞ þ Hi KðPPÞxðktk Þ þ Hi KPxðktk Þ þ Fi oðkÞÞT PLi ðGi xðkÞ þ Hi KðPPÞxðktk Þ þ Hi KPxðktk Þ þ Fi oðkÞÞxT ðkÞPi xðkÞ ¼ ðGi xðkÞ þ Hi KPxðktk Þ þ Fi oðkÞÞT PLi ðGi xðkÞ þ Hi KPxðktk Þ n X b2j ðHi KUj xðktk ÞÞT PLi Hi KUj xðktk ÞxT ðkÞPi xðkÞ þFi oðkÞÞ þ j¼1

"

T

þ2x ðk,lÞM xðkÞxðktðkÞÞ

k1 X

# eðlÞ ,

l ¼ ktk

DV2 ðxðkÞ,Zk Þ ¼ X½V2 ðxðk þ Zk ÞÞjZk ¼ iV2 ðxðkÞ,Zk ¼ iÞ rX½ðGi xðkÞ þ Hi KðPPÞxðktk Þ þ Hi KPxðktk ÞÞT

ð22Þ

T. Liu et al. / Journal of the Franklin Institute 349 (2012) 1915–1929

1924

þFi oðkÞÞT QðGi xðkÞ þ Hi KðPPÞxðktk Þ þHi KPxðktk Þ þ Fi oðkÞÞxT ðktk ÞQxðktk Þ ¼ ðGi xðkÞ þ Hi KPxðktk Þ þ Fi oðkÞÞT QðGi xðkÞ þ Hi KPxðktk Þ n X b2j ðHi KUj xðktk ÞÞT QH i KUj xðktk Þ þFi oðkÞÞ þ j¼1

" T

T

x ðktk ÞQxðktk Þ þ 2x ðk,lÞN xðkÞxðktðkÞÞ

k1 X

# eðlÞ ,

l ¼ ktk

ð23Þ DV3 ðxðkÞ,Zk Þ ¼ X½V3 ðxðk þ Zk ÞÞjZk ¼ iV3 ðxðkÞ,Zk ¼ iÞ ¼ tM X½ððGi IÞxðkÞ þ Hi KðPPÞxðktk Þ þ Hi KPxðktk Þ þFi oðkÞÞT RððGi IÞxðkÞ þ Hi KðPPÞxðktk Þ k1 X eT ðlÞReðlÞ þHi KPxðktk Þ þ Fi oðkÞÞ l ¼ ktk

¼ tM ððGi IÞxðkÞ þ Hi KPxðktk Þ þ Fi oðkÞÞT R ððGi IÞxðkÞ þ Hi KPxðktk Þ þ Fi oðkÞÞ n k1 X X b2j ðHi KUj xðktk ÞÞT RH i KUj xðktk Þ eT ðlÞReðlÞ: þtM j¼1

l ¼ ktk

ð24Þ From Eqs. (22) to (24), we have DV ðxðkÞ,Zk Þ ¼ X½V ðxðk þ Zk ÞÞjZk ¼ iV ðxðkÞ,Zk ¼ iÞ ¼ DV1 ðxðkÞ,Zk Þ þ DV2 ðxðkÞ,Zk Þ þ DV3 ðxðkÞ,Zk Þ þ zT ðkÞzðkÞ g2 oT ðkÞoðkÞzT ðkÞzðkÞ þ g2 oT ðkÞoðkÞ k1 1 X xT ðk,lÞOi xðk,lÞzT ðkÞzðkÞ þ g2 oT ðkÞoðkÞ, r tM l ¼ kt

ð25Þ

k

where

2

6 6 Oi ¼ 6 6 4

G11

n

n

n

G21 DT C

G22 0

n

n

g2 I þ DT D

n

tM ðM1 þ N1 Þ

tM ðM2 þ N2 Þ

0

tM R

þYT21 PLi Y21 þ YT21 QY21 þ YT22 RY22 þ½fT1 fT2 . . . fTn diagfPLi , . . . ,PLi g½fT1 fT2 þ½fT1 fT2 . . . fTn diagfQ, . . . ,Qg½fT1 fT2 . . . þ½fT1 fT2 . . . fTn diagftM R, . . . ,tM Rg½fT1 fT2

. . . fTn T fTn T . . . fTn T :

3 7 7 7 7 5

ð26Þ

Furthermore, using the similar analysis methods as in Theorem 1. Then the following inequality can be obtained: DV ðxðkÞ,Zk Þ ¼ X½V ðxðk þ Zk ÞÞjZk ¼ iV ðxðkÞ,Zk ¼ iÞ

T. Liu et al. / Journal of the Franklin Institute 349 (2012) 1915–1929

rzT ðkÞzðkÞ þ g2 oT ðkÞoðkÞ

1925

1 minflmin ðOi ÞgjxðkÞj2 tM

rzT ðkÞzðkÞ þ g2 oT ðkÞoðkÞ:

ð27Þ

Taking expectation and summing up from k ¼ 0 to 1 on both sides of inequality (27), it can be verified that the above inequality equivalents to X½V ðxð1Þ,Z1 ÞV ðxð0Þ,Z0 Þr

1 X

XfzT ðkÞzðkÞg þ g2

k¼0

1 X

oT ðkÞoðkÞ,

ð28Þ

k¼0

which implies that 1 X

XfzT ðkÞzðkÞgrg2

k¼0

1 X

oT ðkÞoðkÞ:

ð29Þ

k¼0

Therefore, it is obtained from inequality (9) that the closed-loop system (7) is stochastically stable and achieves the given disturbance attenuation level. This completes the proof. & Remark 2. It should be pointed out that the sufficient conditions proposed in Theorem 2 are not standard LMI condition anymore. Fortunately, there are various methods to solve the nonlinear matrix inequality. In this paper, it is suggested to use the cone complementarity linearization (CCL) algorithm, which is one of the commonly used methods, to design mode-independent controller from Theorem 2. 4. Numerical example In this section, a numerical example and simulations are used to illustrate the effectiveness of the proposed methods developed in this paper. Consider the simplified model of the inverted pendulum process as follows [28]:       0 1 0 0 _ ¼ xðtÞ xðtÞ þ uðkTÞ þ oðtÞ, 1 0 1 0:1 

   1 0 0 zðtÞ ¼ xðtÞ þ oðtÞ, 1 1 1

t 2 ½kT,ðk þ 1ÞTÞ:

The eigenvalues of A are 1 and 1, and thus this system is unstable. Our purpose is to design a controller such that the closed-loop system is stochastically stable with the given disturbance attenuation level. Suppose that the sampling period is T ¼ 0.02 s, scalar g ¼ 1:2, the upper bound of the packet dropout is s ¼ 3, namely d ¼ f1; 2,3g, the maximal delay bound is tM ¼ 2 and the transition probability matrix is 2 3 0:5 0:5 0 6 7 L ¼ 4 0:3 0:6 0:1 5: 0:1 0:6 0:3

T. Liu et al. / Journal of the Franklin Institute 349 (2012) 1915–1929

1926

In addition, we assume the probabilistic density functions of ri in [0,1] described by 8 8 0:1, r1 ¼ 0, 0:2, r2 ¼ 0, > > < < F ðr1 Þ ¼ 0:2, r1 ¼ 0:5, F ðr2 Þ ¼ 0:2, r2 ¼ 0:5, > > : 0:7, r ¼ 1, : 0:6, r ¼ 1: 1

2

Then the mathematical expectation and variance of r1 and r2 are a1 ¼ 0:71,b21 ¼ 0:12 and a2 ¼ 0:61,b22 ¼ 0:17, respectively. Using the discretize method in this paper, we can obtain the discrete-time model of NCS with packet dropouts, time delay and sensor fault with three operation modes       1:0002 0:0200 1:0008 0:0400 1:0018 0:0600 G1 ¼ , G2 ¼ , G3 ¼ , 0:0200 1:0002 0:0400 1:0008 0:0600 1:0018  H1 ¼  F1 ¼

0:0002

 ,

H2 ¼

 ,



0:0200 0:0000 0:0020



F2 ¼

0:0008 0:0400

0:0001 0:0040

 ,

 ,

 H3 ¼  F3 ¼

0:0018

 ,

0:0600 0:0002 0:0060

 C¼

 ,

D¼

1

0

1

1

  0 1

 ,

:

By using iterative algorithm—CCL from Theorem 2, the mode-independent controller is obtained as K ¼ ½3:6132 3:7836: In order to illustrate the usefulness of the proposed methods, one of the possible realizations of the Markovian packet dropout process and random delay are plotted in Figs. 2 and 3, respectively. In the simulation setup, the initial state is chosen as x0 ¼ ½0 0T . Consider disturbance signal chosen as  sinð0:5kÞ, 50rkr70, oðkÞ ¼ 0 otherwise: 4 3.5

Packet Dropout Process

3 2.5 2 1.5 1 0.5 0 0

0.5

1

1.5

2

2.5 Time (s)

3

3.5

4

4.5

Fig. 2. Simulation of the Markovian packet dropout process.

5

T. Liu et al. / Journal of the Franklin Institute 349 (2012) 1915–1929

1927

3

Time varying delays

2.5

2

1.5

1

0.5

0 0

0.5

1

1.5

2

2.5 Time (s)

3

3.5

4

4.5

5

Fig. 3. Simulation of the random delay process.

0.015 x1 x2

State trajectories

0.01

0.005

0

−0.005

−0.01 0

0.5

1

1.5

2

2.5 Time (s)

3

3.5

4

4.5

5

Fig. 4. Trajectories of states of NCS.

Using our controller by solving Theorem 2, we have tried to control the system with possibly packet dropout, random delay and sensor fault. Fig. 4 shows the state trajectories of the closed-loop system. From these figures, it can be illustrated that the controller we designed can guarantee the stochastic stability of the NCS under the above network conditions. 5. Conclusion In this paper, the problems of stabilization and H1 control have been studied for a class of NCS with packet dropout, random delay and sensor fault. The packet dropout is modeled as a Markovian process and the resulting closed-loop system is a Markovian switching system. The sensor fault is described that each sensor has individual fault rate.

1928

T. Liu et al. / Journal of the Franklin Institute 349 (2012) 1915–1929

Sufficient conditions for stochastic stability of closed-loop system are given in terms of LMI. The controller design method can be used to design a mode-independent controller such that the closed-loop system is stochastically stable and achieves H1 disturbance attenuation level. The effectiveness of the proposed methods has been illustrated by a numerical example. It is worth mentioning that the proposed method enables us to establish the relation between the above network conditions and the stability of the closedloop system. It is well known that NCS subject to quantization have been a hot research area. Hence, how quantization affect the stability of the proposed model in this paper will be investigated in our future research. Acknowledgments This work was partially supported by the National High Technology Research and Development Program of China (863 Program) (No. 2009AA043001), the Program of the International Science and Technology Cooperation (Nos. 2009DFA12520, 2010DFA12210, 1016704500), National Natural Science Foundation of China (Nos. 60904015, 61004028), ‘‘Chen Guang’’ project supported by Shanghai Municipal Education Commission, Shanghai Education Development Foundation (No. 09CG17), Shanghai Leading Academic Discipline Project (No. B004), the Shanghai Pujiang Program (No. 10PJ1402800) and the Fundamental Research Funds for the Central Universities (No. WH1014013). References [1] N. Elia, S.K. Mitter, Stabilization of linear systems with limited information, IEEE Transactions on Automatic Control 46 (9) (2001) 1384–1400. [2] H. Ishii, B.A. Francis, Stabilization with control networks, Automatica 38 (10) (2002) 1745–1751. [3] W. Zhang, L. Yu, Modelling and control of networked control systems with both network-induced delay and packet-dropout, Automatica 44 (12) (2008) 3206–3210. [4] F. Yang, Z. Wang, Y.S. Hung, M. Gani, H1 control for networked systems with random communication delays, IEEE Transactions on Automatic Control 51 (3) (2006) 511–518. [5] J. Dai, A delay approach to networked control systems with limited communication capacity, Journal of the Franklin Institute 347 (7) (2010) 1334–1352. [6] L. Zhang, Y. Shi, T. Chen, B. Huang, A new method for stabilization of networked control systems with random delays, IEEE Transactions on Automatic Control 50 (8) (2005) 1177–1181. [7] D. Huang, S.K. Nguang, State feedback control of uncertain networked control systems with random time delays, IEEE Transactions on Automatic Control 53 (3) (2008) 829–834. [8] Y. Shi, B. Yu, Output feedback stabilization of networked control systems with random delays modeled by Markov chains, IEEE Transactions on Automatic Control 54 (7) (2009) 1668–1674. [9] F. Rasool, S.K. Nguang, Quantised robust H1 output feedback control of discrete-time systems with random communication delays, IET Control Theory & Applications 4 (11) (2010) 2252–2262. [10] E. Tian, D. Yue, Y. Zhang, On improved delay-dependent robust H1 control for systems with interval timevarying delay, Journal of the Franklin Institute 348 (4) (2011) 555–567. [11] J. Wu, T. Chen, Design of networked control systems with packet dropouts, IEEE Transactions on Automatic Control 52 (7) (2007) 1314–1319. [12] Y. Niu, T. Jia, X. Wang, F. Yang, Output-feedback control design for NCSs subject to quantization and dropout, Information Sciences 179 (21) (2009) 3804–3813. [13] W. Zhang, L. Yu, Output feedback stabilization of networked control systems with packet dropouts, IEEE Transactions on Automatic Control 52 (49) (2007) 1705–1710. [14] Y. Niu, D.W.C. Ho, Design of sliding mode control subject to packet losses, IEEE Transactions on Automatic Control 55 (11) (2010) 2623–2628.

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