Fault-Tolerant Control for Networked Control Systems with Limited Information in Case of Actuator Fault

Abstract

This paper is concerned with the problem of designing a fault-tolerant controller for uncertain discrete-time networked control systems against actuator possible fault. The step difference between the running step k and the time stamp of the used plant state is modeled as a finite state Markov chain of which the transition probabilities matrix information is limited. By introducing actuator fault indicator matrix, the closed-loop system model is obtained by means of state augmentation technique. The sufficient conditions on the stochastic stability of the closed-loop system are given and the fault-tolerant controller is designed by solving a linear matrix inequality. A numerical example is presented to illustrate the effectiveness of the proposed method.

Academic Editor:Hui Zhang

School of Engineering, Huzhou University, Huzhou, Zhejiang 313000, China

Received 3 January 2015; Revised 17 February 2015; Accepted 25 February 2015; 16 September 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Networked control systems (NCSs) are used in many fields such as remote surgery and unmanned aerial vehicles especially in a number of emerging engineering applications such as arrays of microactuators and even neurobiological and socialeconomical systems [1-3]. Compared with the traditional wiring, the communication channels can simplify the installation and reduce the costs of cables and maintenance of the system. However, the network in the control systems also brings many problems, such as network-induced delay and packet dropout, and makes system analysis more challenging [4, 5]. Network-induced delays can degrade the performance of control systems designed without considering them and even destabilize the system [6, 7].

Because of the complexity caused by network, NCSs are more vulnerable to faults. An effective way to increase the reliability of the NCSs is to introduce fault-tolerant control (FTC). Therefore, the research on fault-tolerant control of NCSs has great theoretical and applied significance; however research on FTC for NCSs is different from that for traditional control systems in many aspects [8, 9]. In [10], a fault estimator was proposed for NCSs with transfer delays, process noise, and model uncertainty. On the basis of the information on fault estimator, a fault-tolerant controller using sliding mode control theory was designed to recover the system performance. In [11], the random packet dropout and the sensor or actuator failure were described as binary random variables; the sufficient condition for asymptotical mean-square stability of the NCSs was derived. By using matrix measure technique, a fault-tolerant controller was designed for NCSs with network-induced delay and model uncertainty in [12]. In [13], a FTC algorithm considering actuator failure of an NCS was presented, and the NCS with data packet dropout was modeled as an asynchronous dynamical system. Based on information scheduling, FTC design methods were proposed for NCSs with communication constrains in [14]. In [15], the problem of fault-tolerant control for NCSs with data packet dropout is studied and the closed-loop system was modeled as Markov jump system. However, elements of transition probabilities matrix are assumed to be completely known and the controller can not be solved by LMIs. To the best of the authors' knowledge, up to now, very limited efforts have been devoted to studying FTC for uncertain NCSs with uncertain transition probability matrices, which motivates our investigation.

Problems of partial sensors inactivation are equal to problems of data pack dropout which can be solved by common technique; we focus on the problems of reliability when actuators are inactivated in this paper.

In this paper, the step difference [figure omitted; refer to PDF] between the running step [figure omitted; refer to PDF] and the time stamp of the used plant state is modeled as a finite state Markov chain. And the information of the transition probabilities matrix is limited; that is, a part of elements of transition probabilities matrix is unknown. The closed-loop system model is obtained by means of state augmentation technique and the mode-dependent fault-tolerant controller is designed which guarantees the stochastic stability of the closed-loop system.

This paper is organized as follows. In Section 2, we formulate the state feedback controller design problem. In Section 3, the sufficient conditions to guarantee the stochastic stability are presented, and the fault-tolerant controller is also given. A simulation example is used to illustrate the effectiveness of the proposed method in Section 4. The conclusion remarks are addressed in Section 5.

2. Problem Formulation

Consider the NCSs setup in Figure 1, in which the controllers are placed in a remote location, and both sensor measurement data and control data are transmitted through network.

Figure 1: Structure of networked control system.

[figure omitted; refer to PDF]

By adding a buffer to the actuator, the delay [figure omitted; refer to PDF] from sensor to controller and the delay [figure omitted; refer to PDF] from controller to actuator can be lumped together, and the new variable is described as [figure omitted; refer to PDF] which is modeled as a Markov chain. And [figure omitted; refer to PDF] denotes the step difference between the running step [figure omitted; refer to PDF] and the time stamp of the used plant state, and it depends on the random time-delay and the data packet drops on the random communication delay and the data packet dropout [16]. Assume that both time-delay and the data packet dropout are bounded, so [figure omitted; refer to PDF] is bounded. The step delay [figure omitted; refer to PDF] takes values in [figure omitted; refer to PDF] and the transition probability matrix of [figure omitted; refer to PDF] is [figure omitted; refer to PDF] . That is, [figure omitted; refer to PDF] jumps from mode [figure omitted; refer to PDF] to [figure omitted; refer to PDF] with probability [figure omitted; refer to PDF] which is defined by [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . The set [figure omitted; refer to PDF] contains [figure omitted; refer to PDF] modes of [figure omitted; refer to PDF] , and the transition probabilities of the jumping process in this paper are considered to be partly accessed; that is, some elements in matrix [figure omitted; refer to PDF] are unknown. For example, for the time-delay [figure omitted; refer to PDF] with 3 modes, the transition probabilities matrix [figure omitted; refer to PDF] may be as follows: [figure omitted; refer to PDF] where " [figure omitted; refer to PDF] " represents the inaccessible elements. For notational clarity, [figure omitted; refer to PDF] , we denote [figure omitted; refer to PDF] with [figure omitted; refer to PDF] Moreover, if [figure omitted; refer to PDF] , it is further described as [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] represents the [figure omitted; refer to PDF] th known element with the index [figure omitted; refer to PDF] in the [figure omitted; refer to PDF] th row of the matrix [figure omitted; refer to PDF] . And [figure omitted; refer to PDF] is described as [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] represents the [figure omitted; refer to PDF] th unknown element with the index [figure omitted; refer to PDF] th in the [figure omitted; refer to PDF] th row of the matrix [figure omitted; refer to PDF] .

Assume that the model of the plant is an uncertain discrete-time system as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is state vector and [figure omitted; refer to PDF] is the control input. [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are all real constant matrices. [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is an uncertain time-varying matrix satisfying the bound [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] denotes the identity matrix with appropriate dimension.

Considering the effect of the random communication delay and the data packet dropout, we describe the state feedback control law as [figure omitted; refer to PDF] The fault indicator matrix [figure omitted; refer to PDF] is given by [figure omitted; refer to PDF] with [figure omitted; refer to PDF] for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] means the [figure omitted; refer to PDF] actuator experiences a total failure, whereas the [figure omitted; refer to PDF] actuator is in healthy state when [figure omitted; refer to PDF] . Since there are [figure omitted; refer to PDF] actuators, the set of possible related failure modes is finite and is denoted by [figure omitted; refer to PDF] with [figure omitted; refer to PDF] elements, where [figure omitted; refer to PDF] [figure omitted; refer to PDF] is a particular pattern of matrix [figure omitted; refer to PDF] .

Consequently, the closed-loop system from (3) and (4) can be expressed as [figure omitted; refer to PDF] At sampling time [figure omitted; refer to PDF] , if we augment the state-variable as [figure omitted; refer to PDF] , the closed-loop system (6) can be written as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] has all elements being zeros except for the [figure omitted; refer to PDF] block being identity.

It can be seen that the closed-loop system (7) is a jump linear system with [figure omitted; refer to PDF] different modes.

It is noticed that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Throughout this paper, we use the following definition.

Definition 1.

System (7) is stochastically stable if for every finite [figure omitted; refer to PDF] and initial mode [figure omitted; refer to PDF] there exists a finite [figure omitted; refer to PDF] such that the following holds: [figure omitted; refer to PDF]

The object of this paper is to construct a fault-tolerant controller with structure as given by (4) which achieves that the closed-loop system (7) is stochastically stable under all actuator failure modes. In the following, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for this paper are denoted as [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively.

To proceed, we will need the following two lemmas.

Lemma 2 (see [17]).

Given matrices [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] of appropriate dimensions and [figure omitted; refer to PDF] is symmetric, [figure omitted; refer to PDF] holds for all [figure omitted; refer to PDF] satisfying [figure omitted; refer to PDF] if and only if there exists a scalar [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .

Lemma 3 (see [18]).

The matrix [figure omitted; refer to PDF] is of full-array rank; then there exist two orthogonal matrices [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , such that [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] are nonzero singular values of [figure omitted; refer to PDF] . If matrix [figure omitted; refer to PDF] has the following structure [figure omitted; refer to PDF] there exists a nonsingular matrix [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

3. Controller Design

With Definition 1, the sufficient conditions on the stochastic stability of the closed-loop system (7) can be obtained.

Theorem 4.

The closed-loop system (7) with partly unknown transition probabilities (2) is stochastically stable if there exists matrix [figure omitted; refer to PDF] , [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

Proof.

For the closed-loop system (7), consider the quadratic function which is given by [figure omitted; refer to PDF] Then, [figure omitted; refer to PDF] Hence, if (12) and (13) hold, [figure omitted; refer to PDF] . One has [figure omitted; refer to PDF] where [figure omitted; refer to PDF] ; hence one can get [figure omitted; refer to PDF] . According to Definition 1, system (7) is stochastically stable.

Clearly, no knowledge on [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is needed in (12) and (13), which completes the proof.

Theorem 5.

Consider system (7) with partly unknown transition probabilities (2). If there exist matrices [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and positive scalars [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] then there exists a mode-dependent controller of the form (4) such that the resulting system (7) is stochastically stable. Furthermore, an admissible controller is given by [figure omitted; refer to PDF]

Proof.

According to Theorem 4, we know that the system (7) is stochastically stable with the partly unknown transition probabilities (2) if inequalities (12) and (13) hold. By Schur complement, inequality (12) is equivalent to [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

By Lemma 2, there exists a scalar [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Using Schur complement and Lemma 2 again, one can get [figure omitted; refer to PDF] Similarly, from (13) one can obtain [figure omitted; refer to PDF] Performing a congruence transformation to (23) and (24) by [figure omitted; refer to PDF] , setting [figure omitted; refer to PDF] , one can obtain (25) and (26), respectively. One has [figure omitted; refer to PDF] For the matrix [figure omitted; refer to PDF] of full-column rank, there always exist two orthogonal matrices [figure omitted; refer to PDF] and [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are nonzero singular values of [figure omitted; refer to PDF] . Assume that the matrix [figure omitted; refer to PDF] has the following structure: [figure omitted; refer to PDF] according to Lemma 3, there exists matrix [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , setting [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , one can get [figure omitted; refer to PDF] which implies that [figure omitted; refer to PDF] Thus, (19) is obtained from (29) and (31), which completes the proof.

4. Numerical Example

In this section, a numerical example is given to show the validity and potential of our developed theoretical results. The dynamics are described as follows: [figure omitted; refer to PDF] Assume the time-delay [figure omitted; refer to PDF] takes values from [figure omitted; refer to PDF] and the transition probabilities matrix is [figure omitted; refer to PDF] . When the first actuator experiences a total failure, that is, the fault indicator matrix [figure omitted; refer to PDF] , the fault-tolerant and delay-dependent controller gain is solved from Theorem 5 as follows: [figure omitted; refer to PDF] When the second actuator experiences a total failure while the first actuator works normally, that is, the fault indicator matrix [figure omitted; refer to PDF] , the controller gain is solved as follows: [figure omitted; refer to PDF] When both actuators work normally, that is, the fault indicator matrix [figure omitted; refer to PDF] , the controller gain is solved as [figure omitted; refer to PDF] Zero-input responses of states [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are shown in Figures 2 and 3 when [figure omitted; refer to PDF] .

Figure 2: Zero-input response of [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 3: Zero-input response of [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

The curves of zero-input response states [figure omitted; refer to PDF] , [figure omitted; refer to PDF] show that the NCS with partly unknown transition probabilities is stochastically stable against actuator possible fault.

5. Conclusions

This paper is concerned with the problem of fault-tolerant control for uncertain discrete-time networked systems against actuator possible fault. The time-delay is modeled as a finite state Markov chain and the Markov chain's transition probabilities the information is limited. The closed-loop system is established through the state augmentation technique and the state feedback controller is designed which guarantees the stability of the resulting closed-loop systems. It is shown that the controller design problem under consideration is solvable if a set of LMIs is feasible. Simulation results show that the closed-loop systems are stochastically stable against actuator fault.

Acknowledgments

This study is supported by the National Natural Science Foundation of China under Grant no. 61174029, the National Natural Science Foundation of China under Grant no. 61503136, the Zhejiang Provincial Natural Science Foundation of China under Grant no. LY12F03008, and the Huzhou Natural Science Foundation of China under Grant no. 2014YZ07.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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