Observer-based fault-tolerant control for a class of networked control systems with transfer delays

Journal of the Franklin Institute 348 (2011) 763–776 www.elsevier.com/locate/jfranklin

Observer-based fault-tolerant control for a class of networked control systems with transfer delays$ Zehui Maoa,b, Bin Jiangb,, Peng Shic,1 a

Tsinghua National Laboratory for Information Science and Technology, and Department of Automation, Tsinghua University, Beijing 100084, PR China b College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, PR China c Department of Computing and Mathematical Sciences, University of Glamorgan, Pontypridd, CF37 1DL, UK Received 21 December 2006; received in revised form 9 February 2011; accepted 10 February 2011 Available online 22 February 2011

Abstract In this paper, an observer-based fault-tolerant control (FTC) method is proposed for a class of networked control systems (NCSs) with transfer delays. Markov chain is employed to characterize the transfer delays. Then, such kind of networked control systems are modelled as Markovian jump systems. An observer-based FTC scheme using the delayed state information and the estimated fault value is presented to guarantee the stability of the faulty systems. An inverted pendulum example is used to illustrate the efficiency of the proposed method. & 2011 Published by Elsevier Ltd. on behalf of The Franklin Institute.

1. Introduction With the rapid development of communication networks, recently, a great amount of effort has been devoting to the problems of networked control systems. NCSs are control systems in which controller and plant are connected via a communication channel. The defining feature of an NCS is that information (reference input, plant output, control $

Partially supported by NUAA Research Funding (NS2010071, NZ2010003), the National Natural Science Foundation of China (60874051), Natural Science Foundation of Jiangsu Province (BK2007195) and the Engineering and Physical Sciences Research Council, UK (EP/F029195). Corresponding author. E-mail addresses: [email protected] (Z. Mao), [email protected] (B. Jiang), [email protected] (P. Shi). 1 Also with the School of Engineering and Science, Victoria University, Melbourne 8001 Vic, Australia, and School of Mathematics and Statistics, University of South Australia, Mawson Lakes, 5095 SA, Australia. 0016-0032/$32.00 & 2011 Published by Elsevier Ltd. on behalf of The Franklin Institute. doi:10.1016/j.jfranklin.2011.02.004

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input, etc.) is exchanged using a network among control system components (sensors, controller, actuators, etc.). The primary advantages of NCSs are low cost, simple installation and maintenance, increased system agility and reduced system wiring [1]. Many models have been established to represent the NCSs with its specific features, such as discrete model with delay [1], Markovian jump model [2], T–S model [3], hybrid model [4] and so on. Therefore, some method proposed for these models, such as [5–8], can be modified to use in NCSs. Meanwhile, corresponding methods [9–13] for these models are proposed to design and analysis NCSs. The faults in NCSs are similar to that in the conventional systems, which can lead to the performance degradation and even instability. Fault detection and diagnosis (FDD) and fault-tolerant control (FTC) procedures can be designed to guarantee that the system goal is still achieved in spite of the faults (see [14,15] for details). In the past years, there have existed some results about the FDD and FTC for NCSs. For some representative work on this general topic, to name a few, we refer the readers to [16,17] and the references therein. In [18], fault detection method was proposed for a class of continuous-time NCSs with non-ideal network quality of service (QoS), which was described by an integrated index related to the network-induced delay, data dropout and error sequence. In [19], NCSs with random but bounded delays were considered. Using augmented state-space method and improved V-Kiteration algorithm, a class of reliable controllers were designed to make systems asymptotically stable under several stochastic disturbances and actuator failure. Among these literatures, many fruits are about fault detection and passive FTC. For passive FTC, the system is made ‘‘robust’’ against faults by assuming them as uncertainties [20]. In active FTC, a new control system is redesigned with (hopefully) all of the desirable properties of performance and robustness when the faults occur. Thus, it can be seen that the same controller utilized in the passive FTC systems before and after the occurrence of the fault, cannot insure the good performance of both the healthy and faulty systems. To the best of our knowledge, until now, few result has been reported about active FTC for NCSs. This motivates us to study this interesting and challenging problem, which has great potential in practical applications. Based on our previous work [21–23], we present an active FTC method for NCSs with large transfer delays. We employ the multirate sampling technique to model the large random delay NCSs as Markovian jump systems with input delays. Under this model, an observer-based fault diagnosis method is proposed, which can provide accurate estimations of states and faults after faults occur. Based on the fault estimation and delayed state information, an active faulttolerant control is designed to achieve the system stability. Finally, an inverted pendulum example is used to demonstrate the effectiveness of the theoretic results obtained. The rest of this paper is organized as follows. Section 2 gives the system description and the definition of the design problem. An observer-based fault estimation method is derived in Section 3. Section 4 proposes the design method of the FTC with input delays. Application to an inverted pendulum example is presented in Section 5, followed by some concluding remarks in Section 6. 2. System description and problem formulation Consider the NCSs as shown in Fig. 1. The continuous-time, state-space model of the linear time-invariant plant dynamics can be described by the following standard form: _ ¼ AxðtÞ þ BuðtÞ þ Ef ðtÞ xðtÞ

ð1Þ

Z. Mao et al. / Journal of the Franklin Institute 348 (2011) 763–776

u

Actuator

Plant

Sensor

765

y Net

Net Discrete Controller reconfiguration

Fault Estimation

Observer

Continuous Signal Discrete Signal

Fig. 1. The block of the networked control system with FTC.

yðtÞ ¼ CxðtÞ

ð2Þ

where x 2 Rn is the state vector, u 2 Rm is the control input vector, y 2 Rr is the measurable output vector. The matrices A, B, C, E are real matrices of appropriate dimensions, C is of full row rank. The failure f ðtÞ ¼ bðtTÞf0 2 Rq can be regarded as an additive signal, where the function bðtTÞ is given by ( 0; toT bðtTÞ ¼ : 1; tZT That is, f(t) is zero prior to the failure time ðtoTÞ and is a constant vector f0 2 Rq after the failure occurs ðtZTÞ. For this NCS, we introduce the following assumptions. Assumption 1. The sampling period of the NCS is T, sensor is time-driven, controller and actuator are time-division-driven. The data sampled from the sensor are packed with the time-stamp, thus as this data arrived at the controller, the sensor–controller transfer delay can be known. We use tsc od1 T, d1 2 Z and tca od2 T, d2 2 Z to represent the sensor–controller and controller–actuator delay, respectively, and net delays t9tsc þ tca oðd1 þ d2 ÞT ¼ dT. Assumption 2. The sampling interval ½kT; ðk þ 1ÞT is divided into N. From Assumption 2, we can obtain that the delays tsc belongs to the set j ¼ fsðT=NÞg with s ¼ 0; 1; . . . ; d1 N and ts belongs to the set c ¼ fsðT=NÞg with s ¼ 0; 1; . . . ; dN. Further, we have j  c. Remark 1. We choose the same division for controller and actuator here. To more general case, the divisions can be different, we can also get the set of the delays. For example, the pieces are chosen as p and q to the controller and actuator, respectively. Then the delay sets are defined as fiðT=pÞg, fiðT=pÞ þ jðT=qÞg, where i ¼ 0; 1; . . . ; d1 p, j ¼ 0; 1; . . . ; d2 q.

Denote ts , s ¼ 0; 1; . . . ; d1 N and ts , s ¼ 0; 1; . . . ; dN as the sensor–controller and net delays, respectively. Then, under the Assumptions 1 and 2, considering the sampling period and the affect of delays, in one sampling period, the above plant’s model is

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transformed into (the local model) m X xðk þ 1Þ ¼ Ac xðkÞ þ Bs uðkts Þ þ Ec f ðkÞ

ð3Þ

s¼0

yðkÞ ¼ Cxðkts Þ AT

R Tts

ð4Þ At

where x(k)=x(kT), y(k)=y(kT), f(k)=f(kT), Ac ¼ e , Bs ¼ Tts1 e B dt, Ec ¼ R T AðTtÞ e E dt. It is easy to show that the terms Bs are variable and defined by the upper 0 and lower bound of the integral which is about the time delays. It has been proven that the transfer delay sequences ts and ts are Markov chains, whose probability distribution can be obtained by experiment method [16]. For the net delays ts , the known one-step transition probability from states i to j, where i, j 2 c is given by pij, i.e. pij ¼ ProbðBs ðk þ 1Þ ¼ jjBs ðkÞ ¼ iÞ. Then, the global model of the NCS can be described as m X xðk þ 1Þ ¼ Ac xðkÞ þ Bis ðkÞuðksÞ þ Ec f ðkÞ ð5Þ s¼0

yðkÞ ¼ CxðksÞ

ð6Þ

where Bis represents the sth input delay of the ith model of the Markovian jump system, i ¼ 0; 1; . . . ; dðN þ 1Þ þ 1, m=dN. Now, we formulate the design problem as follows:

 

Under the above Markovian jump model, a fault estimation scheme is presented, which can estimate the NCSs states and faults accurately. The proposed active fault-tolerant controller can stabilize the NCSs using the delayed state and fault estimation information.

3. Observer-based fault estimation From the analysis of above section, it follows that, due to network, the system output applied to observer/controller contains the transfer delay. That means the observer/ controller only can used the delayed system output to estimate/control the system state. Therefore, the following observer can only estimate the state value that is not the system current value, but some time before. We define a transformation x=N1z, which can transform the systems (5) and (6) into the following form: 3 2 3 2 3 2 2 3 z1 ðk þ 1Þ Bis1 ðkÞ Ac1 Ec1 m X 6 7 6 z ðk þ 1Þ 7 6 A 7 6 B ðkÞ 7 ð7Þ 4 2 5 ¼ 4 c2 5zðkÞ þ 4 is2 5uðksÞ þ 4 Ec2 5f ðkÞ s¼0 z3 ðk þ 1Þ Bis3 ðkÞ Ac3 Ec3 " yðkÞ ¼

y1 ðkÞ y2 ðkÞ

"

# ¼

0 0

Irq 0

# 0 zðksÞ Iq

ð8Þ

where z1 ðkÞ 2 Rnr , z2 ðkÞ 2 Rrq , z3 ðkÞ 2 Rq . Therefore, only z1 ðkÞ needs to be estimated in mean square.

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Assumption 3. RankðCEÞ ¼ q and Ec3 is nonsingular. Remark 2. Since C is of full row rank, the first part in Assumption 1 means that the effects of the faults are independent, which is about the fault detectability. If this condition does not hold, some fault information cannot be reflected in the system outputs. Other new method should be considered, which would be our future work. The time period considered here is between that the data sampled from sensor at time k1 and k have arrived in the controller side after some delays, and the data sampled at time kþ1 have not arrived in. By pre-multiplying 2 3 1 I 0 Ec1 Ec3 6 1 7 ð9Þ 4 0 I Ec2 Ec3 5 0 0 I on Eq. (7), one can obtain 2 3 1 z1 ðk þ 1ÞEc1 Ec3 z3 ðk þ 1Þ 6 z ðk þ 1ÞE E 1 z ðk þ 1Þ 7 4 2 5 c2 c3 3 z3 ðk þ 1Þ 2 3 2 3 2 3 1 1 0 Ac1 Ec1 Ec3 Ac3 Bis1 Ec1 Ec3 Bis3 m X 6 6 B E E 1 B 7 6 7 1 Ac3 7 ¼ 4 Ac2 Ec2 Ec3 5zðkÞ þ 4 is2 c2 c3 is3 5uðksÞ þ 4 0 5f ðkÞ s¼0 Ec3 Ac3 Bis3

ð10Þ

Note that only the state z1(k) needs to be estimated. Define 1 A c1 9Ac1 Ec1 Ec3 Ac3 ; 1 B is1 9Bis1 Ec1 Ec3 Bis3 ;

A c1 9½A c11 A c12 A c13 ; ðnrÞðnrÞ

1 A c2 9Ac2 Ec2 Ec3 Ac3

ð11Þ

1 B is2 9Bis2 Ec2 Ec3 Bis3

ð12Þ

A c2 9½A c21 A c22 A c23  ðnrÞðrqÞ

ð13Þ

ðnrÞq

ðrqÞðnrÞ

ðrqÞðrqÞ

, A c12 2R , A c13 2R , A c21 2R , A c22 2R with A c11 2R A c23 2RðrqÞq . Then, the first and second block rows of Eq. (10) can be written as

,

z1 ðk þ 1Þ ¼ A c11 z1 ðkÞ þ rðk þ 1Þ

ð14Þ

lðk þ 1Þ ¼ A c21 z1 ðkÞ

ð15Þ

where rðk þ 1Þ and lðk þ 1Þ are defined as 1 rðk þ 1Þ9A c12 z2 ðkÞ þ A c13 z3 ðkÞ þ Ec1 Ec3 z3 ðk þ 1Þ þ

m X

B is1 uðkjÞ

ð16Þ

s¼0

1 Zðk þ 1Þ9z2 ðk þ 1ÞEc2 Ec3 z3 ðk þ 1ÞA c22 z2 ðkÞA c23 z2 ðkÞ

m X

B is2 uðkjÞ

ð17Þ

s¼0

When the data sampled from the sensor are packed with the time, the sensor–controller delays are available, which can determine the current mode of the output. In this case,

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the observer to estimate z1 can be designed as z^ 1 ðk þ 1Þ ¼ A c11 z^ 1 ðkÞ þ rðk þ 1Þ þ Li ðkÞ½Zðk þ 1ÞA c21 z^ 1 ðkÞ

ð18Þ

Let us replace the state z2 and z3 to y1 and y2 and assume ðA c11 ; A c21 Þ is an observable pair. Then the observer gain Li (k) can be chosen such that the following matrix inequality: ðA c11 Li A c21 ÞT R i ðA c11 Li A c21 ÞRi o0

ð19Þ Pm

has positive definite solutions Ri 40, where R i ¼ l¼1 pij Ri . From above discussion, we can let z^ ¼ ½^z 1 z2 z3 T , where ½z2 z3 T ¼ C 1 ½y2 y3 T , where z^ 1 is obtained in Eq. (18). Denote z~ 1 ¼ z^ 1 z1 , then z~ 1 ðk þ 1Þ ¼ ðA c11 Li A c21 Þ~z 1 ðkÞ

ð20Þ

Choose the Lyapunov function V ðkÞ ¼ z~ T1 ðkÞRi z~ 1 ðkÞ, then one has EfV ðk þ 1Þj~z 1 ðkÞ; rðkÞgV ðkÞ ¼ z~ T1 ðkÞððA c11 Li A c21 ÞT R i ðA c11 Li A c21 ÞRi Þ~z 1 ðkÞo0 ð21Þ We can conclude that the estimation error z~ 1 ðkÞ will converge to zero in the mean square if the inequality (19) is satisfied. It should be mentioned that the choice of Li and Ri relates to constant matrix Ac, which implies that, for the proposed Markovian jump system, the observer gain Li and the corresponding matrix Ri are also invariable denoted by L and R. ^ Define ex ðkÞ ¼ xðkÞxðkÞ, then we further have 2 2 3 3 ex1 ðkÞ z~1 ðkÞ 6 7 6 7 ex ðk þ 1Þ ¼ N 1 z~ ðk þ 1Þ ¼ ðA c11 Li A c21 ÞN 1 4 0 5 ¼ ðA c11 Li A c21 Þ4 0 5 0 0 ð22Þ The third block row in Eq. (10) can be written as y2 ðk þ 1Þ ¼ Ac31 z1 ðkÞ þ Ac32 y1 ðkÞ þ Ac33 y2 ðkÞ þ

m X

Bis3 uðksÞ þ Ec3 f ðkÞ

ð23Þ

the system fault can be estimated as follows: " # m X 1 f^ ðkÞ ¼ E y2 ðk þ 1ÞAc31 z^ 1 ðkÞAc32 y1 ðkÞAc33 y2 ðkÞ Bis3 uðksÞ

ð24Þ

s¼0

c3

s¼0

and the fault estimation error 1 1 ef ðkÞ ¼ f ðkÞf^ ðkÞ ¼ Ec3 Ac31 z~ 1 ðkÞ ¼ Ec3 Ac31 Nex1 ðkÞ

ð25Þ

which will converge to zero in the mean square as long as z~ 1 ðkÞ converges to zero in the mean square. Remark 3. From Eq. (24), it can be seen that the faulty signal at time instant k can be estimated only after the measurements from time instant ðk þ 1Þ becomes available. It means that there is a one-step delay in the fault estimation, whose effect on the dynamic response can be neglected for practical application [22]. On the other hand, we can avoid such problem via setting a new vector containing the y2 ðk þ 1Þ, as in [24].

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4. Fault-tolerant control with input delays In this section, the delayed estimates of states and faults provided by the observer in Section 3 are used to design the FTC law, so as to maintain the system performance. The main results are summarized in the following theorem. Theorem 1. Systems (5) and (6) can be stabilized by the feedback control of the form 1 þ ^ ^ B Ec f ðkÞ uðksÞ ¼ Ki xðksÞ ð26Þ g is ^ where xðksÞ ¼ N 1 z^ ðksÞ is the estimation value of xðksÞ, z^ ðksÞ given by Eq. (18), g is the number of the non-zero Bis in ith submode of the Markovian jump system, Ki are obtained by the following Riccati equation, i ¼ 0; 1; . . . ; dðN þ 1Þ þ 1, s ¼ 0; 1; . . . ; m. m X 4ATic P i Aic Pi þ ðm þ 1Þ KiT BTis P i Bis Ki o0 ð27Þ s¼0 Pn where Aic ¼ Ac Bi0 Ki , Pi are symmetric positive definite matrices, and P i ¼ j¼1 pij Pj . Proof. Applying the control (26) to (5) results in the closed-loop dynamics m m X X Bis Ki xðksÞ þ Bis Ki ex ðksÞ þ Ec ef ðkÞ xðk þ 1Þ ¼ Aic xðkÞ ¼ Aic xðkÞ

s¼0

s¼0

m X

m X

Bis Ki xðksÞ þ

s¼0

1 Bis Ki ex ðksÞEc Ec3 Ac31 Nex1 ðkÞ

ð28Þ

s¼0

where ex ðksÞ ¼ ½eTx1 ðksÞ 0 0T . Consider the Lyapunov–Krasovskii function V ðkÞ ¼ V1 ðkÞ þ V2 ðkÞ þ V3 ðkÞ

ð29Þ

where V1 ðkÞ ¼ xT ðkÞPi xðkÞ þ ðm þ 1Þ

m X k1 X

ðxT ðjÞKiT BTij P i Bij Ki xðjÞÞ

s¼0 j¼ks

V2 ðkÞ ¼ V3 ðkÞ ¼

m X

eTx ðksÞHi ex ðksÞ s¼0 eTf ðkÞHi ef ðkÞ

ð30Þ

where Hi 40, G40, Pi are defined by Eq. (27), and P i ¼ Then, we obtain

Pn

j¼1

pij Pj .

EfV1 ðK þ 1ÞjxðkÞ; rðkÞgV1 ðkÞ ¼ xT ðk þ 1ÞP i xðk þ 1ÞxT ðkÞPi xðkÞ þ ðm þ 1Þ

m X

ðxT ðkÞKiT BTij P i Bij Ki xðkÞ

s¼0

xT ðksÞKiT BTis P i Bis Ki xðksÞÞ ! m X ¼ xT ðkÞ ATic P i Aic P þ ðm þ 1Þ KiT BTis P i Bis Ki xðkÞ þ

m X s¼0

! xT ðksÞKiT BTis P i

s¼0 m X s¼0

!

Bis Ki xðksÞ

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m X

ðm þ 1Þ

! xT ðksÞKiT BTis P i Bis Ki xðksÞ

s¼0

þ

m X

!

m X

eTx ðksÞKiT BTis P i

s¼0

! þ eTf ðkÞEcT P i Ec ef ðkÞ

Bis Ki ex ðksÞ

s¼0

m X Bis Ki xðksÞ þ 2xT ðkÞATic P i Bis Ki ex ðksÞ s¼1 s¼0 ! ! m m X X T T T T T þ2x ðkÞAic P i Ec ef ðkÞ2 x ðksÞKi Bis P i Bis Ki ex ðksÞ

2xT ðkÞATic P i

m X

2

m X

! T

x

ðksÞKiT BTis

s¼0

s¼0

P i Ec ef ðkÞ þ 2

s¼0

m X

!

eTx ðksÞKiT BTis

P i Ec ef ðkÞ

ð31Þ

s¼0

It is also easy to show that !T ! ! n n n X X X T xi P xi rn xi Pxi i¼0

i¼0

ð32Þ

i¼0

Then, using Eq. (32) and definition ex ðksÞ ¼ ½eTx1 ðksÞ 0 0T , the following inequalities are obtained: EfV1 ðK þ 1ÞjxðkÞ; rðkÞgV1 ðkÞ

rx ðkÞ T

ATic P i Aic Pi

þ ðm þ 1Þ

m X

! KiT BTis P i Bis Ki

s¼0 m X

þm

x

T

!

ðksÞKiT BTis P i Bis Ki xðksÞ

xðkÞ m X

ðm þ 1Þ

s¼0

! T

x

ðksÞKiT BTis P i Bis Ki xðksÞ

s¼0 m X

þðm þ 1Þ

! eTx1 ðksÞIeT KiT BTis P i Bis Ki Ie ex1 ðksÞ

þ eTf ðkÞEcT P i Ec ef ðkÞ

s¼0

2xT ðkÞATic P i

m X

Bis Ki xðksÞ þ 2xT ðkÞATic P i

s¼1 T

þ2x 2

m X

ðkÞATic P i Ec ef ðkÞ2

m X

! x

T

ðksÞKiT BTis

m X

!s¼0 T

x

ðksÞKiT BTis

s¼0

Pi

Bis Ki Ie ex1 ðksÞ m X

! Bis Ki Ie ex1 ðksÞ

s¼0

P i Ec ef ðkÞ þ 2

s¼0

m X

eTx1 ðksÞIeT KiT BTis

! P i Ec ef ðkÞ

ð33Þ

s¼0

where Ie ¼ ½I nr 0rq 0q T . Further, we obtain that EfV2 ðK þ 1Þjex ðkÞ; rðkÞgV2 ðkÞ m m X X eTx ðks þ 1ÞH i ex ðks þ 1Þ eTx ðksÞHi ex ðksÞ ¼ s¼0

¼ and

m X

s¼0 T

T

ex1 ðksÞ ððA c11 Li A c21 Þ H i ðA c11 Li A c21 ÞHi Þex1 ðksÞ

s¼0

EfV3 ðK þ 1Þjef ðkÞ; rðkÞgV3 ðkÞ

ð34Þ

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¼ eTf ðk þ 1ÞHi ef ðk þ 1ÞeTf ðkÞHi ef ðkÞ 1 1 1 1 ¼ eTx1 ðk þ 1ÞðEc3 Ac31 NÞT Hi ðEc3 Ac31 NÞex1 ðk þ 1ÞeTx1 ðkÞðEc3 Ac31 NÞT Hi ðEc3 Ac31 NÞex1 ðkÞ 1 1 ¼ eTx1 ðkÞðEc3 Ac31 NÞT ððA c11 Li A c21 ÞT H i ðA c11 Li A c21 ÞHi ÞðEc3 Ac31 NÞex1 ðkÞ

¼ eTf ðkÞððA c11 Li A c21 ÞT H i ðA c11 Li A c21 ÞHi Þef ðkÞ

ð35Þ T

Define xðkÞ ¼ ½xðkÞ xðk1Þ . . . xðkdNÞ ex1 ðk1Þ . . . ex1 ðkdNÞ ef ðkÞ , Kd ¼ diagfKi g, P d ¼ diagfP i g, Id ¼ diagfIe g, B~ d ¼ ½Bi0 Bi1 . . . BiðdNÞ , Bd ¼ diagfBi0 ; Bi1 ; . . . ; BiðdNÞ g. Considering Eqs. (29), (33)–(35), one can further obtain that EfV ðK þ 1ÞjxðkÞ; ex ðkÞ; ef ðkÞ; rðkÞgV ðkÞ ¼ EfV1 ðK þ 1ÞjxðkÞ; rðkÞgV1 ðkÞ þEfV2 ðK þ 1Þjex ðkÞ; rðkÞgV2 ðkÞ þ EfV3 ðK þ 1Þjef ðkÞ; rðkÞgV3 ðkÞ 3 2 ATic P i B~ d Ki ATic P i B~ d Ki Ie ATic P i Ec G11 7 6 6  K T BT P d Bd Kd K T BT P d Bd Kd Id K T B~ T P i Ec 7 7 6 d i d d d d T rx ðkÞ6 7xðkÞ T T ~T 7 6  B  G I K P E 33 i c 5 4 e i d    G44 9xT ðkÞMxðkÞ where G11 ¼ ATic P i Aic Pi þ ðm þ 1Þ

m X

KiT BTis P i Bis Ki

s¼0

G33 ¼ ðm þ 1ÞIdT KdT BTd P d Bd Kd Id þ diagfðA c11 Li A c21 ÞT H i ðA c11 Li A c21 ÞHi g G44 ¼ ðA c11 Li A c21 ÞT H i ðA c11 Li A c21 ÞHi þ EcT P i Ec If Mo0, then EfV ðK þ 1ÞjxðkÞ; rðkÞgV ðkÞo0, which means the system (28) is stable. It is well known that for any a40 and real vectors a and b that 2aT braaT a þ ð1=aÞbT b. Setting a ¼ 1, we have the following inequality: EfV ðK þ 1ÞjxðkÞ; ex ðkÞ; ef ðkÞ; rðkÞgV ðkÞ m m X X c2s jxðksÞj2 þ c3s jex1 ðksÞj2 þ c4 jef ðkÞj2 rc1 jxðkÞj2 þ "

s¼0

pffiffiffi 1=2  3Aic P i xðkÞ þ

s¼0 m X

1 1=2 pffiffiffi Bis Ki P i xðksÞ 3

#T

s¼0 # m X pffiffiffi 1 1=2 1=2 pffiffiffi Bis Ki P i xðksÞ  3Aic P i xðkÞ þ 3 s¼0 " #T m X pffiffiffi 1 1=2 1=2 pffiffiffi Bis Ki P i xðksÞ  3Ec P i ef ðkÞ þ 3 s¼0 " # m X pffiffiffi 1 1=2 1=2 pffiffiffi Bis Ki P i xðksÞ  3Ec P i ef ðkÞ þ 3 s¼0

"

ð36Þ

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772

where c1 ¼ lmax ðG1 Þ;

c2s ¼ lmax ðG2s Þ; c3s ¼ lmax ðG3s Þ;

G1 ¼ 4ATic P i Aic Pi þ ðm þ 1Þ

m X

KiT BTis P i Bis Ki þ



 1 1 þ ATic P i Aic ; m1 m2

s¼0   1 1 T T G2s ¼  þ K B P i Bis Ki ; 3 m3 i is   1 T G3s ¼ m þ 1 þ m1 þ m2 þ K T BT P i Bis Ki þ A Li H i A Li Hi m4 i is

T

c4 ¼ lmax ðG4 Þ; G4 ¼ A Li H i A Li Hi þ ðm3 þ m4 þ 4ÞEcT P i Ec A Li ¼ A c11 Li A c21

ð37Þ

The constants m1 , m2 , m3 and m4 are appropriately chosen such that G1 o0, G2 o0, G3 o0 and G4 o0. From Eq. (31), if ex=0 and ef=0, according to Theorem 1, it is easily obtained that EfV1 ðK þ 1ÞjxðkÞ; rðkÞgV1 ðkÞo0. Since ex and ef are uniformly bounded, x(k) in Eq. (28) is uniformly bounded as well. Thus the system (28) is input-state stable with considering ex and ef as inputs. &

Remark 4. The considered fault in this paper is time-invariant, thus as soon as the fault estimation scheme detects and estimates the fault using the delayed output and estimation value of state, the fault estimation can be used to design the controller without considering the effect of the delays.

Remark 5. Compare with some existing results on this issue, the good futures of the results obtained in this paper are in two aspects: (1) It proposes an active fault-tolerant control design method for networked control systems, which contains a fault estimation scheme. (2) The NCSs are modelled as Markovian jump systems, so this paper also provides a new active FTC method for Markovian jump systems.

5. An illustrative example In this section, a well-known inverted pendulum system is used to illustrate a potential application field of our approach. The continues plant model is described as 2 3 2 3 0 1 0 1 1 6 0:9756 7 6 0 4:978 0:7187 0 7 6 7 6 7 _ ¼6 xðtÞ 7uðtÞ 7xðtÞ þ 6 40 4 0:6423 5 0 0 15 0 3:7335 7:8959 0 0:7317 2

1 6 yðtÞ ¼ 4 0 0

0 1

0 0

3 0 7 0 5xðtÞ

0

1

0

Z. Mao et al. / Journal of the Franklin Institute 348 (2011) 763–776

773

We choose the sampling period T ¼ 0:01 s, the division of the sampling interval N=2, the max delay tmax o2T, then it is easily obtained the delays set j ¼ f0; 0:005; 0:01; 0:015g. The fault distribution matrix in Eqs. (1) and (2) is defined as ( 0; to4 s E ¼ B; f ðtÞ ¼ 0:4; 4rtr30 s According to the modelling method in Section 2, we obtain seven submodes of the Markovian jump system. Due to the large number of the submodes, only two of them are illustrated here. Mode 0 (t ¼ 0Þ: 3 2 2 3 0:01 1 0:0010 0:0003 0:01 6 0 0:9514 0:007012 6 0:0095 7 0 7 6 7 6 7 _ ¼6 xðtÞ 7xðtÞ þ 6 7uðtÞ 4 0 0:0002 4 0:0064 5 1 0:01 5 0 2

1 6 yðtÞ ¼ 4 0 0

0:03643

0:07884

0 1

0 0

3 0 7 0 5xðtÞ

0

1

0

Mode 3 (t ¼ 0:01; 0:005): 2 1 0:0010 6 0 0:9514 6 _ ¼6 xðtÞ 4 0 0:0002

0:0003 0:007012 1

0 0:03643 0:07884 2 3 0:005012 6 0:004812 7 6 7 þ6 7uðt1Þ 4 0:003203 5

1

0:0069

3 2 3 0:01 0:005012 6 0:004812 7 0 7 7 6 7 7xðtÞ þ 6 7uðt2Þ 4 0:003203 5 0:01 5 1

0:00355

0:00355 2

1

6 yðtÞ ¼ 4 0 0

3

0

0

0

1 0

0 1

7 0 5xðts1 Þ 0

E ¼ ½0:01 0:0095 0:0064 0:0069T In fact, rankðCEÞ ¼ 1. According to the fault estimation and fault-tolerant control method proposed in this paper, the state estimation is given by Eq. (18), then the actuator faults are estimated by Eq. (24) and compensated by Theorem 1. In Fig. 2, the actuator fault estimation with satisfactory accuracy is shown. Fig. 3 depicts the output trajectories of the closed-loop system. It can be seen that the dynamic system outputs (states) converge to zero.

Z. Mao et al. / Journal of the Franklin Institute 348 (2011) 763–776

774

loss of actuator and its estimation 0.9 estimation value actual value

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 0

5

10

15 sec

20

25

30

Fig. 2. Estimation of actuator faults.

output y2 trajectory

output y1 trajectory 0.15

1.8 y1 with FTC y1 without FTC

1.6 1.4 1.2

0

1

−0.05

0.8

−0.1

0.6

−0.15

0.4

−0.2

0.2

−0.25

0

−0.3

−0.2

−0.35 0

5

10

y2 with FTC y2 without FTC

0.1 0.05

15 sec

20

25

30

0

5

10

15 sec

output y3 trajectory 0.1 y3 with FTC y3 without FTC

0.08 0.06 0.04 0.02 0 −0.02 −0.04 0

5

10

15 sec

20

25

30

Fig. 3. Output trajectories: (a) output y1 , (b) output y2 and (c) output y3 .

20

25

30

Z. Mao et al. / Journal of the Franklin Institute 348 (2011) 763–776

775

6. Conclusion In this paper, we have investigated the problem of observer-based FTC for a class of NCSs with large delays. The considered NCSs are modelled as Markovian jump systems. Based on this model, an active fault-tolerant controller is designed, which can guarantee that the system state converges to zero in the mean square. Simulation results of an inverted pendulum system are included to verify the effectiveness of the proposed method. Further research work includes two directions: (1) consideration of time-varying faults; (2) extension of the proposed approach to nonlinear NCSs. References [1] W. Zhang, M.S. Branicky, S.M. Philips, Stability of networked control systems, IEEE Control Syst. Mag. 21 (1) (2001) 84–99. [2] Y.Q. Xia, Z. Zhu, M.S. Mahmoud, H2 control for networked control systems with Markovian data losses and delays, ICIC Express Lett. 3 (3(A)) (2009) 271–276. [3] Y. Zheng, H.J. Fang, H.O. Wang, Takagi–Sugeno fuzzy-model-based fault detection for networked control systems with Markov delays, IEEE Trans. Syst. Man Cybern. Part B Cybern. 36 (4) (2006) 924–929. [4] D. Ne ˇsic,´ A.R. Teel, Input-to-state stability of networked control systems, Automatica 40 (12) (2004) 2121–2128. [5] M.S. Mahmoud, P. Shi, Robust Kalman filtering for continuous time-lag systems with Markovian jump parameters, IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 50 (1) (2003) 98–105. [6] M.S. Mahmoud, P. Shi, Robust stability, stabilization and H1 control of time-delay systems with Markovian jump parameters, Int. J. Robust Nonlinear Control 13 (8) (2003) 755–784. [7] M.S. Mahmoud, P. Shi, Y. Shi, Output feedback stabilization and disturbance attenuation of time-delay jumping systems, IMA J. Math. Control Inf. 20 (2) (2003) 179–199. [8] M.S. Mahmoud, Uncertain jumping systems with strong and weak functional delays, Automatica 40 (3) (2004) 501–510. [9] L. Sheng, Y.J. Pan, Stochastic stabilization of sampled-data networked control systems, Int. J. Innovative Comput. Inf. Control 6 (4) (2010) 1949–1972. [10] V. Vesely, T.N. Quang, Robust output networked control system design, ICIC Express Lett. 4 (4) (2010) 1399–1404. [11] X. He, Z.D. Wang, D.H. Zhou, Robust fault detection for networked systems with communication delay and data missing, Automatica 45 (11) (2009) 2634–2639. [12] L. Zhou, G. Lu, Quantized feedback stabilization for networked control systems with nonlinear perturbation, Int. J. Innovative Comput. Inf. Control 6 (6) (2010) 2485–2496. [13] F. Yang, H.J. Fang, Control structure design of networked control systems based on maximum allowable delay bounds, J. Franklin Inst. 346 (6) (2009) 626–635. [14] B. Jiang, F.N. Chowdhury, Parameter fault detection and estimation of a class of nonlinear systems using observers, J. Franklin Inst. 342 (7) (2005) 725–736. [15] H.J. Yang, Y.Q. Xia, P. Shi, Observer-based sliding mode control for a class of discrete systems via delta operator approach, J. Franklin Inst. 347 (7) (2010) 1199–1213. [16] X. He, Z.D. Wang, D.H. Zhou, Robust H1 filtering for networked systems with multiple state delays, Int. J. Control 80 (8) (2007) 1217–1232. [17] Z.H. Mao, B. Jiang, P. Shi, Observer based fault-tolerant control for a class of nonlinear networked control systems, J. Franklin Inst. 347 (6) (2010) 940–956. [18] Z. Gu, D.B. Wang, D. Yue, Fault detection for continuous-time networked control systems with non-ideal QoS, Int. J. Innovative Comput. Inf. Control 6 (8) (2010) 3631–3640. [19] C.X. Yang, Z.H. Guan, J. Huang, Stochastic fault tolerant control of networked control systems, J. Franklin Inst. 346 (10) (2009) 1006–1020. [20] J. Chen, R.J. Patton, Robust Model-Based Fault Diagnosis for Dynamics Systems, Kluwer Academic Publishers, Boston, 1999. [21] Z.H. Mao, B. Jiang, P. Shi, H1 fault detection filter design for networked control systems modelled by discrete Markovian jump systems, IET Control Theory Appl. 1 (5) (2007) 1336–1343.

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