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Journal of Control Science and Engineering
Volume 2008, Article ID 189064, 9 pages
http://dx.doi.org/10.1155/2008/189064
Research Article

Stability Guaranteed Active Fault-Tolerant Control of Networked Control Systems

Centre de Recherche en Automatique de Nancy (CRAN-UMR 7039), Nancy Université, CNRS, BP 239, Vandoeuvre-lès-Nancy Cedex 54506, France

Received 4 May 2007; Revised 15 October 2007; Accepted 9 December 2007

Academic Editor: Jakob Stoustrup

Copyright © 2008 Shanbin Li et al. 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.

Abstract

The stability guaranteed active fault-tolerant control against actuators failures and plant uncertainties in networked control systems (NCSs) is addressed. A detailed design procedure is formulated as a convex optimization problem which can be efficiently solved by existing software. An illustrative example is given to show the efficiency of the proposed method for network-based control for uncertain systems.

1. Introduction

Fault-tolerant control (FTC) techniques against actuator faults can be classified into two groups [1]: passive and active approaches. In passive FTC systems, a single controller with fixed structure/parameters is used to deal with all possible failure scenarios which are assumed to be known a priori. Consequently, the passive controller is usually conservative. Furthermore, if a failure out of those considered in the design occurs, the stability and performance of the closed-loop system might not be guaranteed. Such potential limitations of passive approaches provide a strong motivation for the development of methods and strategies for active FTC (AFTC) systems.

In contrast to passive FTC systems, AFTC techniques rely on a real-time fault detection and isolation (FDI) scheme and a controller reconfiguration mechanism. Such techniques allow a flexibility to select different controllers according to different component failures, and therefore better performance of the closed-loop system can be expected. However, this holds true when the FDI process does not make an incorrect or delayed decision [2]. Some preliminary results have been obtained on AFTC which is immune to imperfect FDI process [3, 4]. In [5], the latter issue is further discussed in a classical setting (i.e., point-to-point control) by using the guaranteed cost control approach and online controller switching in order to ensure stability of the closed-loop system at all times. The aim of this paper is to extend the results in [5] to uncertain plants controlled over digital communication networks. In such networks, the information transfer from sensors to controllers and from controllers to actuators is not instantaneous but suffers communication delays. Such communication delays can be highly variable due to their strong dependence on variable network conditions such as congestion and channel quality. These network-induced delays may impact adversely on the stability and performance of the control system [6, 8]. Networked control systems (NCSs) are now pervasive (see, e.g., the recent special issue of the proceedings of the IEEE [7]), and such systems are long-running real-time systems which should function in a correct manner even in the presence of failures. This makes the issue of fault tolerant control in NCS an important one and entails designing strategies to cope with some of the fundamental problems introduced by the network such as bandwidth limitations, quantization and sampling effects, message scheduling and communication delays. Motivated by the above considerations, we address the problem of fault tolerant control in NCSs with time-varying delays. Specifically, we extend the results of [5] for the stabilization of a plant, subject to model uncertainties and actuator faults, which is controlled over a communication network that induces time-varying but bounded delays.

The outline of the paper is as follows. In Section 2, a network-based control model for an uncertain plant subject to actuator failures is proposed and the guaranteed cost control problem is formulated. In Section 3, the detailed procedure for designing the NCSs-based fault-tolerant controller is given. Section 4 presents a design example to illustrate the benefit of the proposed FTC design procedure. Finally, conclusions are given in Section 5. The proof of the main theorem of the paper is reported in the appendix.

2. Problem Statement

Figure 1 shows the basic networked control architecture which is considered in this paper and which consists of a single uncertain plant, with few sensors and actuators, controlled by a digital controller in a centralized structure.

189064.fig.001
Figure 1: Networked control system with actuator failures.

The delays induced by the network in the closed-loop control system are modeled as time-varying quantities 𝜏(𝑘)=𝜏sc𝑘 arising from the communication delays between sensors and controllers at time 𝑘. Without loss of generality, we assume that there are no transmission time-delays between the controllers and the actuators. The actuators might be subject to faults during the system operation. Thus, taking into account the potential failures of actuators, the interconnection of the uncertain discrete-time plant and a discrete-time controller through the digital communication link as depicted in Figure 1 can be described by the following dynamical and state-delayed feedback equations: 𝑥(𝑘+1)=𝐴+𝐷Δ(𝑘)𝐸𝑥(𝑘+𝐵𝑢(𝑘),(1)𝑥(0)=𝑥0,(2)𝑢(𝑘)=𝐾𝑥𝑘𝜏(𝑘),(3) where 𝑥(𝑘)𝑛 is the state of the uncertain plant, 𝑢(𝑘)𝑚 is the control input, 𝐴,𝐵,𝐷,𝐸 are all real constant matrices, and matrix 𝐾 is the controller gain matrix to be designed. The time-varying matrix Δ(𝑘) represents norm-bounded parameter uncertainties and satisfies the bound Δ(𝑘)𝑇Δ(𝑘)𝐼 where 𝐼 denotes the identity matrix with appropriate dimension. The constant matrices 𝐷 and 𝐸 characterize the structure of these parameter uncertainties. The fault indicator matrix is given by =diag{𝑙1,,𝑙𝑚}(4) with 𝑙𝑗{0,1} for 𝑗=1,2,,𝑚, where 𝑙𝑗=1 means that the 𝑗th actuator is in healthy state, whereas the 𝑗th actuator is meant to experience a total failure when 𝑙𝑗=0. Having a finite number of actuators, the set of possible related failure modes is also finite and, by abuse of notation, we denote this set by ={1,2,𝑁}(5)with 𝑁=2𝑚1. Each failure mode 𝑖,(𝑖=1,2,𝑁) is therefore an element of the set . We also view 𝑖 as a matrix, that is, as a particular pattern of matrix in (4) depending on the values of 𝑙𝑗(𝑗=1,2,𝑚). Throughout, when is invoked as a matrix, it will mean that matrix varies over the set of matrices in (5). Note that the faulty mode 𝑖 in the NCS architecture of Figure 1 is estimated by the FDI unit. In order to ensure that system (1) should remain controllable, we assume that the set excludes the element diag{0,0,,0}, that is, at least one actuator should be healthy.

We further assume that the time-varying delays 𝜏(𝑘) lie between the following positive integer bounds 𝜏𝑚 and 𝜏𝑀, that is,𝜏𝑚𝜏(𝑘)𝜏𝑀.(6) Given positive definite symmetric matrices 𝑄1 and 𝑄2, we consider the quadratic cost function 𝐽=𝑘=0[𝑥𝑇(𝑘)𝑄1𝑥(𝑘)+𝑢𝑇(𝑘)𝑄2𝑢(𝑘)],(7) and with respect to this cost function, we define the guaranteed cost controller in the event of actuator failures as follows.

Definition 1. If there exists a control law 𝑢(𝑘) and a positive scalar 𝐽 such that for all admissible uncertainties Δ(𝑘) and all failure modes 𝑖 the closed-loop system (1)–(3) is stable with cost function (7) satisfying 𝐽𝐽, then 𝐽 is said to be a guaranteed cost and 𝑢(𝑘) a guaranteed cost controller for the uncertain system (1).

In Section 3, we will proceed through two main steps to design a cost guaranteed active fault tolerant control in the NCS framework. These steps are

(i)construct a fault-tolerant controller (i.e., a robust controller), with structure as given by (3), which achieves the smallest possible value for 𝐽 under all admissible plant uncertainties and all actuator failure modes;(ii)redesign that part of the above controller associated to only one fault-free actuator in order to improve the robust performance without loss of the stability property of the design in step (i); step (ii) repeats for all 𝑚 actuators and results in a bank of 𝑚 controllers. It follows from inequality 𝑚𝑁=2𝑚1, that the cardinality of the bank of controllers (which is equal to the number of actuators) is less than the cardinality of the set of faulty modes. For each faulty mode 𝑖, the controller to be switched on should be the best as ranked with respect to a closed-loop performance index. In this paper, we will not address the switching and reconfiguration mechanisms; we focus on the design of the bank of 𝑚 controllers.

3. AFTC Design for NCS

3.1. Robust Stability

In this subsection, we establish a sufficient condition for the existence of a guaranteed cost network-based controller for the uncertain plant (1). Note that the control law (3) applied to plant (1) results in the following system: 𝑥(𝑘+1)=𝐴1𝑥(𝑘)+𝐵𝐾𝑥(𝑘𝜏(𝑘)),(8) where 𝐴1=𝐴+𝐷Δ(𝑘)𝐸. The cost function associated to system (8) is therefore 𝐽=𝑘=0𝑥𝑇𝑒(𝑘)𝑄𝑥𝑒(𝑘),(9) where 𝑥𝑇𝑒(𝑘)=[𝑥𝑇(𝑘),𝑥𝑇(𝑘𝜏(𝑘))] and 𝑄=diag{𝑄1,𝐾𝑇𝑄2𝐾}. Under the assumptions made in Section 2, we can state the following result.

Theorem 1. If there exist a gain matrix 𝐾, a scalar 𝜖>0, symmetric positive-definite matrices 𝑃1𝑛×𝑛, 𝑅𝑛×𝑛, 𝑆𝑛×𝑛, and matrices 𝑃2𝑛×𝑛, 𝑃3𝑛×𝑛, 𝑊2𝑛×2𝑛, 𝑀2𝑛×𝑛 such that the following matrix inequalities are satisfied: Γ𝑃𝑇0𝐸0x0200b0x0200b𝐵𝐾0x0200b0x0200b𝑀𝑇00x0200b0x0200b𝑅0x0200b+0x0200b𝐾𝑇𝑄2𝐾0𝜖𝐼<0,(10)0x0200b0x0200b𝑊𝑀𝑆0(11) with Γ=𝑃𝑇+0𝐼𝐴𝐼𝐼0𝐼𝐴𝐼𝐼𝑇𝑃+𝜖𝑃𝑇000𝐷𝐷𝑇𝑃+𝜇𝑅+𝑄100𝑃1+𝜏𝑀𝑆+𝜏𝑀+𝑊+𝑀0𝑀0𝑇,𝜏𝜇=1+𝑀𝜏𝑚𝑃,𝑃=10𝑃2𝑃3,(12) then system (8) is asymptotically stable and the cost function (9) satisfies the inequality 𝐽𝑥𝑇(0)𝑃1𝑥(0)+1𝑙=𝜏𝑀𝑥𝑇+(𝑙)Rx(𝑙)0𝜃=𝜏𝑀+11𝑙=1+𝜃𝑦𝑇+(𝑙)Sy(𝑙)𝜏𝑀+1𝜃=𝜏𝑀+21𝑙=𝜃1𝑥𝑇(𝑙)Rx(𝑙),(13) where 𝑦(𝑙)=𝑥(𝑙+1)𝑥(𝑙).

Proof. See the appendix.

Remark 2. The “” sign represents blocks that are readily inferred by symmetry.

Remark 3. Note that the upper bound in (13) depends on the initial condition of system (8). To remove the dependence on the initial condition, we suppose that the initial state of system (8) might be arbitrary but belongs to the set 𝒮={𝑥(𝑙)𝑛𝑥(𝑙)=𝑈𝑣,𝑣𝑇𝑣1,𝑙=𝜏𝑀,𝜏𝑀+1,,𝜏𝑚}, where 𝑈 is a given matrix. Inequality (13) leads to 𝐽𝜆max𝑈𝑇𝑃1𝑈+𝜌1𝜆max𝑈𝑇𝑅𝑈+𝜌2𝜆max𝑈𝑇𝑆𝑈,(14) where 𝜆max() denotes the maximum eigenvalue of matrix (), 𝜌1=𝜇(𝜏𝑀+𝜏𝑚)/2, and 𝜌2=2𝜏𝑀(𝜏𝑀+1).

3.2. Step (i): Controller Design

Now, we derive the guaranteed cost controller in terms of the feasible solutions to a set of linear matrix inequalities.

Using Sherman-Morrison matrix inversion formula, we have 𝑃1=𝑃110𝑃31𝑃2𝑃11𝑃31.(15) In the sequel, we will denote 𝑋=𝑃11,𝑌=𝑃31, and 𝑍=𝑃31𝑃2𝑃11. We further restrict 𝑀 to the following case in order to obtain a linear matrix inequality (LMI) (see, e.g., [9]): 𝑀=𝛿𝑃𝑇0𝐵𝐾,(16) where 𝛿 is a scalar parameter. Pre- and postmultiply (10) by diag{(𝑃1)𝑇,𝑃11,𝐼} and diag{𝑃1,𝑃11,𝐼}, respectively; also pre- and postmultiply (11) by diag{(𝑃1)𝑇,𝑃11} and diag{𝑃1,𝑃11} and denote 𝐿=𝑃11𝑅𝑃11,𝐹=𝐾𝑃11,𝑆=𝑆1,𝑃1𝑇𝑊𝑃1=𝑊1𝑊2𝑊3.(17) Applying the Schur complement and expanding the block matrices, we obtain the following result under the assumptions made in Section 2.

Theorem 2. Suppose that for a prescribed scalar 𝛿, there exist a scalar 𝜖>0, matrices 𝑋>0,𝑌,𝑍,𝐹,𝐿>0,𝑆>0,𝑊1,𝑊2,𝑊3, such that the following matrix inequalities are satisfied: Ψ1Ψ20Ψ41Ψ3(1𝛿)𝐵𝐹Ψ42𝐿Ψ43Ψ5<0,(18)𝑊1𝑊20𝑊3𝛿𝐵𝐹𝑋𝑆1𝑋0,(19) where Ψ1=𝑍+𝑍𝑇+𝜇𝐿+𝜏𝑀𝑊1,Ψ2=𝑌+𝑋(𝐴𝐼)𝑇𝑍𝑇+𝜏𝑀𝑊2+𝛿(𝐵𝐹)𝑇,Ψ3=𝑌𝑌𝑇+𝜏𝑀𝑊3+𝜖𝐷𝐷𝑇,Ψ41Ψ42Ψ43=𝑋𝐸𝑇𝜏𝑀𝑍𝑇0𝑋𝑍𝑇0𝜏𝑀𝑌𝑇00𝑌𝑇00𝐹𝑇,Ψ005=diag𝜖𝐼,𝜏𝑀𝑆,𝑄21,𝑄11.,𝑋(20) Then, the control law 𝑢(𝑘)=𝐹𝑋1𝑥𝑘𝜏(𝑘)(21) is a guaranteed cost networked control law for system (1) and the corresponding cost function satisfies 𝐽𝜆max𝑈𝑇𝑋1𝑈+𝜌1𝜆max𝑈𝑇𝑋1𝐿𝑋1𝑈+𝜌2𝜆max𝑈𝑇𝑆1𝑈,(22) where 𝜌1=𝜇(𝜏𝑀+𝜏𝑚)/2 and 𝜌2=2𝜏𝑀(𝜏𝑀+1).

Remark 4. From (22), we establish the following inequalities: 𝛼𝐼𝑈𝑇𝑋<0,𝛽𝐼𝑈𝑇𝑋𝐿1𝑋<0,𝛾𝐼𝑈𝑇𝑆<0,(23) where 𝛼,𝛽, and 𝛾 are scalars to be determined. It is worth noting that condition (23) is not an LMI because of the term 𝑋𝐿1𝑋. This is also the case for condition (19) which is not an LMI because of the term 𝑋𝑆1𝑋. Note that for any matrix 𝑋>0 we have 𝑋𝑆1𝑋2𝑋𝑆,𝑋𝐿1𝑋2𝑋𝐿.(24)

Given a prescribed scalar 𝛿, the design problem of the optimal guaranteed cost controller can be formulated therefore as the following optimization problem: OP1:min𝜖,𝑋,𝑌,𝑍,𝐹,𝐿,𝑆,𝑊1,𝑊2,𝑊3𝛼+𝜌1𝛽+𝜌2𝛾s.t.(i)inequality(18),(ii)𝑊1𝑊20𝑊3𝛿𝐵𝐹2𝑋𝑆0,(iii)𝛼𝐼𝑈𝑇𝑋<0,𝛽𝐼𝑈𝑇2𝑋+𝐿<0,𝛾𝐼𝑈𝑇𝑆<0.(25) Clearly, the above optimization problem (25) is a convex optimization problem which can be effectively solved by existing LMI software [10]. Thus, the minimization of 𝛼+𝜌1𝛽+𝜌2𝛾 implies the minimization of the cost in (9). By applying a simple one-dimensional search over 𝛿 for a certain 𝜏𝑀, a global optimum cost can be found.

3.3. Robust Stability with at Least a Fault-Free Actuator

Based on the controller designed in Theorem 2, let us assume that actuator 𝑖 is fault-free, then we can redesign the 𝑖 th row of controller gain matrix 𝐾 to improve the robust performance for the system against actuator failures. We can rewrite the overall control system as 𝑥(𝑘+1)=𝐴1𝐵𝑥(𝑘)+𝑖𝑖𝐾𝑖+𝑏𝑖𝑘𝑖𝑥𝑘𝜏(𝑘),(26) where 𝐴1=𝐴+𝐷Δ(𝑘)𝐸, matrix 𝐾𝑖 is obtained by deleting the 𝑖th row from 𝐾, 𝐵𝑖 is obtained by deleting the 𝑖th column from 𝐵, and 𝑖 is obtained by deleting 𝑖th row and 𝑖th column from . The cost function associated to system (26) reads as 𝐽=𝑘=0𝑥𝑇𝑒(𝑘)𝑄𝑥𝑒(𝑘)(27) with 𝑥𝑇𝑒(𝑘)=[𝑥𝑇(𝑘),𝑥𝑇(𝑘𝜏(𝑘))], 𝑄=diag{𝑄1,𝑘𝑇𝑖𝑄2𝑖𝑘𝑖+𝐾𝑇𝑖𝑄2𝑖𝐾𝑖}, where 𝑄2𝑖 is obtained by deleting the 𝑖th row and 𝑖th column from 𝑄2. With regard to system (26) where 𝐾𝑖 is assumed to be known, we have the following result.

Theorem 3. If there exist a gain matrix 𝑘𝑖, a scalar 𝜖>0, symmetric positive-definite matrices 𝑃1𝑛×𝑛, 𝑅𝑛×𝑛, 𝑆𝑛×𝑛, and matrices 𝑃2𝑛×𝑛, 𝑃3𝑛×𝑛, 𝑊2𝑛×2𝑛, 𝑀2𝑛×𝑛 such that the following matrix inequalities are satisfied: Γ𝑃𝑇0𝐵0x0200b0x0200b0x0200b0x0200b𝑖𝑖𝐾𝑖+𝑏𝑖𝑘𝑖𝐸𝑀𝑇0𝑅0x0200b+𝑘𝑇𝑖𝑄2𝑖𝑘𝑖+𝐾𝑇𝑖𝑄2𝑖𝐾𝑖0𝜖𝐼<0,(28)0x0200b0x0200b𝑊𝑀𝑆0x0200b0x0200b0,(29) then, system (26) is asymptotically stable and the cost function (27) satisfies inequality (13).

Proof. The proof is similar to the proof of Theorem 1.

3.4. Step (ii): Controller Redesign

Proceeding as in step (i), we restrict 𝑀 to the following case in order to obtain an LMI: 𝑀=𝛿𝑃𝑇0𝑏𝑖𝑘𝑖,(30) where 𝛿 is a scalar parameter. Pre- and postmultiply (28) with diag{(𝑃1)𝑇,𝑃11,𝐼} and diag{𝑃1,𝑃11,𝐼}, respectively; also pre- and postmultiply (29) with diag{(𝑃1)𝑇,𝑃11} and diag{𝑃1,𝑃11}, respectively, and denote 𝐿=𝑃11𝑅𝑃11,𝐹=𝑘𝑖𝑃11,𝑆=𝑆1,𝑃1𝑇𝑊𝑃1=𝑊1𝑊2𝑊3.(31) The Schur complement trick leads to the following controller redesign result.

Theorem 4. Suppose that for a prescribed scalar 𝛿, there exist a scalar 𝜖>0, matrices 𝑋>0,𝑌,𝑍,𝐹,𝐿>0,𝑆>0,𝑊1,𝑊2,𝑊3, such that the following matrix inequalities are satisfied: Ψ1Ψ20Ψ41Ψ3𝐵𝑖𝑖𝐾𝑖𝑋+(1𝛿)𝑏𝑖𝐹Ψ42Ψ𝐿43Ψ5<0,(32)𝑊1𝑊20𝑊3𝛿𝑏𝑖𝐹𝑋𝑆1𝑋0,(33) where Ψ1=𝑍+𝑍𝑇+𝜇𝐿+𝜏𝑀𝑊1,Ψ2=𝑌+𝑋(𝐴𝐼)𝑇𝑍𝑇+𝜏𝑀𝑊2𝑏+𝛿𝑖𝐹𝑇,Ψ3=𝑌𝑌𝑇+𝜏𝑀𝑊3+𝜖𝐷𝐷𝑇,Ψ41Ψ42Ψ43=𝑋𝐸𝑇𝜏𝑀𝑍𝑇00𝑋𝑍𝑇0𝜏𝑀𝑌𝑇000𝑌𝑇00(𝐹)𝑇𝑋𝐾𝑇𝑖,Ψ005=diag𝜖𝐼,𝜏𝑀𝑆,𝑄12𝑖,𝑄21𝑖,𝑄11.,𝑋(34) Then, the 𝑖th control law 𝑢𝑖(𝑘)=𝐹𝑋1𝑥𝑘𝜏(𝑘)(35) is a guaranteed cost networked control law of system (26) and the corresponding cost function satisfies 𝐽𝜆max𝑈𝑇𝑋1𝑈+𝜌1𝜆max𝑈𝑇𝑋1𝐿𝑋1𝑈+𝜌2𝜆max𝑈𝑇𝑆1𝑈,(36)where 𝜌1=𝜇(𝜏𝑀+𝜏𝑚)/2and𝜌2=2𝜏𝑀(𝜏𝑀+1).

Given a prescribed scalar 𝛿, the redesign problem of the optimal guaranteed cost controller can be formulated as the following convex optimization problem: OP2:min𝜖,𝑋,𝑌,𝑍,𝐹,𝐿,𝑆,𝑊1,𝑊2,𝑊3(𝛼+𝜌1𝛽+𝜌2𝛾)s.t.(i)inequality(32),(ii)𝑊1𝑊20𝑊3𝛿𝑏𝑖𝐹2𝑋𝑆0,(iii)𝛼𝐼𝑈𝑇𝑋<0,𝛽𝐼𝑈𝑇2𝑋+𝐿<0,𝛾𝐼𝑈𝑇𝑆<0.(37)

4. Illustrative Example

The dynamics are described by the following matrices: ,,𝐴=0.900.20.5,𝐵=0.20.100.1𝐷=00.10.10,𝐸=0.100.10.1(38) and the design parameters are chosen as 𝑄1=1001,𝑄2=0.1000.1,𝑈=1001.(39) When 𝜏𝑚=1,𝜏𝑀=2, and 𝛿=1, by OP1 (25), the cost is obtained as 𝐽1=61.6653 and the fault-tolerant controller can be designed for step (i): 𝑘1𝑘2=0.0812×1050.1333×1050.1865×1050.3060×105.(40) In step (ii), by OP2 (37), the cost and the feedback gains are redesigned as 𝐽2=39.0026,𝑘1=,𝐽0.87760.28573=49.9616,𝑘2=.0.64940.4161(41) As a result, the two controllers are determined as follows: 𝑘#11𝑘2=0.87760.28570.1865×1050.3060×105,𝑘#21𝑘2=0.0812×1050.1333×105.0.64940.4161(42) Figure 2 reports the simulation for two failures scenarios of the actuators. For this simulation, the time-varying norm-bounded uncertain matrix Δ(𝑘) is taken as Δ𝑘=sin(𝜋𝑘)00cos(𝜋𝑘),(43) which clearly satisfies the bound Δ(𝑘)𝑇Δ(𝑘)𝐼. The left column of Figure 2 is related to failures of actuator 1 and the right column is related to failures of actuator 2. Note also that for these two failures scenarios the system has been disturbed by step disturbances entering additively in the state equations. This is illustrated in the simulation where the step disturbance 1 shown in Figure 2(a) disturbs the state variables at time instant 35 and disappears at time instant 40. The step disturbance 2 shown in Figure 2(b) disturbs the system at time instant 5 and disappears at time instant 10. In Figure 2(c), the solid line represents the failure of actuator 1 which occurs at time instant 15 and disappears at time instant 35, occurs again at time instant 55 and disappears at time instant 65. In Figure 2(d), the solid line represents the failure of actuator 2 which occurs at time instant 35 and disappears at time instant 45, occurs again at time instant 65 and disappears at time instant 80. The delay of fault detection is assumed to be 3 time steps, which is represented by dot-dashed lines as shown in Figures 2(c) and 2(d). Under the above simulation setting, the state responses are shown in Figures 2(e) and 2(f):

fig2
Figure 2: Disturbance, actuator failures, and state response.

(i)the dotted line represents the state response for controller-switching sequence N°1 : #2 is the initial controller, and #1 is switched-on at time instant 38, then #2 at time instant 48, #1 at time instant 68;(ii)the solid line represents the state response for controller-switching sequence N°2 : #1 is the initial controller, and #2 is switched-on at time instant 38, then #1 at time instant 48, #2 at time instant 68;(iii)the dot-dashed line represents the state response under the fault tolerant control (i.e., robust control) of step (i). The trace of matrices (𝑥)(𝑥)𝑇/(simulationtime) is used as a performance measure for comparison, where 𝑥 represents the state trajectory in the different schemes and the simulation time is 80 seconds. After computation, we obtain for the above three control schemes the traces 𝑇𝑟1=0.0279,𝑇𝑟2=0.0338,𝑇𝑟3=0.0499, which means that 𝑇𝑟1<𝑇𝑟2<𝑇𝑟3. We conclude that the proposed method for sequence N°1 is the best control scheme. We also observe that for all possible switching sequences with controllers in the designed controllers bank, the proposed active FTC is able to guarantee at least the closed-loop stability of the overall system.

5. Conclusion

In this paper, the stability guaranteed active fault tolerant control against actuators failure in networked control system with time-varying but bounded delays has been addressed. Plants with norm-bounded parameter uncertainty have been considered, where the uncertainty may appear in the state matrix. Modelling network-induced delays as communication delays between sensors and actuators, linear memoryless state feedback fault-tolerant controllers have been developed through LMI-based methods. A simulation example has been presented to show the potentials of the proposed method for fault-tolerant control in networked control systems.

Appendix

Proof of Theorem 1

The following matrix inequalities are essential for the proof of Theorem 1.

Lemma 5 (see [11]). Assume that 𝑎()𝑛𝑎,𝑏()𝑛𝑏,and𝑁()𝑛𝑎×𝑛𝑏. Then, for any matrices 𝑋𝑛𝑎×𝑛𝑎, 𝑌𝑛𝑎×𝑛𝑏,and𝑍𝑛𝑏×𝑛𝑏, the following holds: 2𝑎𝑇𝑎𝑏Nb𝑇𝑌𝑋𝑌𝑁𝑇𝑁𝑇𝑍𝑎𝑏,(A.1) where 𝑌𝑋𝑌𝑇𝑍0.

Lemma 6 (see [12]). Let 𝑌 be a symmetric matrix and let 𝐻,𝐸 be given matrices with appropriate dimensions, then 𝑌+𝐻𝐹𝐸+𝐸𝑇𝐹𝑇𝐻𝑇<0(A.2) holds for all 𝐹 satisfying 𝐹𝑇𝐹𝐼, if and only if there exists a scalar 𝜖>0 such that 𝑌+𝜖𝐻𝐻𝑇+𝜖1𝐸𝑇𝐸<0.(A.3)

Proof. Note that 𝑥(𝑘𝜏(𝑘))=𝑥(𝑘)𝑘1𝑙=𝑘𝜏(𝑘)𝑦(𝑙), where 𝑦(𝑙)=𝑥(𝑙+1)𝑥(𝑙). Then from system (8) we have 𝐴0=1+𝐵𝐾𝐼𝑥(𝑘)𝑦(𝑘)𝐵𝐾𝑘1𝑙=𝑘𝜏𝑘𝑦(𝑙).(A.4) Choose the Lyapunov-Krasovskii function candidates as follows: 𝑉(𝑘)=𝑉1(𝑘)+𝑉2(𝑘)+𝑉3(𝑘),(A.5) where 𝑉1(𝑘)=𝑥𝑇(𝑘)𝑃1𝑉𝑥(𝑘),2(𝑘)=𝑘1𝑙=𝑘𝜏𝑘𝑥𝑇𝑉(𝑙)𝑅𝑥(𝑙),3(𝑘)=1𝜃=𝜏𝑀𝑘1𝑙=𝑘+𝜃𝑦𝑇+(𝑙)𝑆𝑦(𝑙)𝜏𝑚+1𝜃=𝜏𝑀+2𝑘1𝑙=𝑘+𝜃1𝑥𝑇(𝑙)𝑅𝑥(𝑙)(A.6) Taking the forward difference for the Lyapunov functional 𝑉1(𝑘), we haveΔ𝑉1(𝑘)=2𝑥𝑇(𝑘)𝑃1𝑦(𝑘)+𝑦𝑇(𝑘)𝑃1𝑦(𝑘).(A.7) From (A.4), we have 2𝑥𝑇(𝑘)𝑃1𝑦(𝑘)=2𝜂𝑇(𝑘)𝑃𝑇𝑦(𝑘)𝑁𝐵𝐾𝑘1𝑙=𝑘𝜏(𝑘)𝑦(𝑙),(A.8) where 𝐴𝑁=1,+𝐵𝐾𝐼)𝑥(𝑘)𝑦(𝑘𝜂𝑇𝑥(𝑘)=𝑇(𝑘)𝑦𝑇(𝑘). Choose constant matrices 𝑊,𝑀, and 𝑆 satisfying (11); by Lemma 5, we have 2𝑘1𝑙=𝑘𝜏(𝑘)𝜂𝑇(𝑘)𝑃𝑇0𝐵𝐾𝑦(𝑙)𝜏𝑀𝜂𝑇(𝑘)𝑊𝜂(𝑘)+𝑘1𝑙=𝑘𝜏𝑀𝑦𝑇(𝑙)𝑆𝑦(𝑙)+2𝜂𝑇(𝑘){𝑀𝑃𝑇0}.𝐵𝐾𝑥(𝑘)𝑥(𝑘𝜏(𝑘)(A.9) Similarly, Δ𝑉2(𝑘)=𝑥𝑇(𝑘)𝑅𝑥(𝑘)𝑥𝑇+𝑘𝜏(𝑘)𝑅𝑥𝑘𝜏(𝑘)𝑘1𝑙=𝑘+1𝜏(𝑘+1)𝑥𝑇(𝑙)𝑅𝑥(𝑙)𝑘1𝑘𝜏(𝑘)+1𝑥𝑇(𝑙)𝑅𝑥(𝑙).(A.10) Note that 𝑘𝜏𝑚𝑙=𝑘+1𝜏(𝑘+1)𝑥𝑇=(𝑙)𝑅𝑥(𝑙)𝑘1𝑙=𝑘+1𝜏𝑚𝑥𝑇(𝑙)𝑅𝑥(𝑙)+𝑘𝜏𝑚𝑙=𝑘+1𝜏(𝑘+1)𝑥𝑇(𝑙)𝑅𝑥(𝑙)𝑘1𝑙=𝑘+1𝜏(𝑘)𝑥𝑇(𝑙)𝑅𝑥(𝑙)+𝑘𝜏𝑚𝑙=𝑘+1𝜏𝑀𝑥𝑇(𝑙)𝑅𝑥(𝑙).(A.11) So, we have Δ𝑉2(𝑘)𝑥𝑇(𝑘)𝑅𝑥(𝑘)𝑥𝑇+𝑘𝜏(𝑘)𝑅𝑥𝑘𝜏(𝑘)𝑘𝜏𝑚𝑙=𝑘+1𝜏𝑚𝑥𝑇(𝑙)𝑅𝑥(𝑙).(A.12) Furthermore, we have Δ𝑉3(𝑘)=𝜏𝑀𝑦𝑇(𝑘)𝑆𝑦(𝑘)𝑘1𝑙=𝑘+1𝜏𝑚𝑦𝑇+𝜏(𝑙)𝑆𝑦(𝑙)𝑀𝜏𝑚𝑥𝑇(𝑘)𝑅𝑥(𝑘)𝑘𝜏𝑚𝑙=𝑘+1𝜏𝑚𝑥𝑇(𝑙)𝑅𝑥(𝑙).(A.13) Combining (9) and (A.7)–(A.13), we have Δ𝑉(𝑘)𝜉𝑇Θ(𝑘)0𝜏𝑚,𝜏𝑀+𝐸𝐷Δ(𝑘)𝐸+𝑇Δ𝑇𝐷(𝑘)𝑇𝜉(𝑘)𝑥𝑇𝑒(𝑘)𝑄𝑥𝑒(𝑘),(A.14) where 𝜉𝑇𝜂(𝑘)=𝑇(𝑘)𝑥𝑇,𝐷(𝑘𝜏(𝑘))𝑇=0𝐷𝑇,0,Θ𝑃0𝐸=𝐸00(𝜏𝑚,𝜏𝑀Γ)=0𝑃𝑇0𝐵𝐾𝑀𝑅+𝐾𝑇𝑄2𝐾,Γ0=Γ𝜖𝑃𝑇000𝐷𝐷𝑇𝑃.(A.15) By Lemma 6, we have Δ𝑉(𝑘)𝜉𝑇Θ(𝑘)0𝜏𝑚,𝜏𝑀𝐷𝐷+𝜖𝑇+𝜖1𝐸𝑇𝐸𝜉(𝑘)𝑥𝑇𝑒(𝑘)𝑄𝑥𝑒(𝑘).(A.16) It is worth observing that matrix [Θ0(𝜏𝑚,𝜏𝑀𝐷𝐷)+𝜖𝑇+𝜖1𝐸𝑇𝐸] is the Schur complement of 𝜖𝐼 in the matrix of the left-hand side of inequality (10). Therefore, the negative definiteness of matrix [Θ0(𝜏𝑚,𝜏𝑀𝐷𝐷)+𝜖𝑇+𝜖1𝐸𝑇𝐸] resulting from inequality (10) implies that Δ𝑉(𝑘)𝑥𝑇𝑒(𝑘)𝑄𝑥𝑒(𝑘).(A.17) Summing both sides of the above inequality from 0 to leads to 𝑘=0𝑘0Δ𝑉=𝑉𝑉𝑘=0𝑥𝑇𝑒𝑘𝑄𝑥𝑒𝑘=𝐽(A.18) which, from system stability, yields0𝐽𝑉,(A.19) that is, inequality (13).

Acknowledgments

The authors would like to thank the anonymous reviewers for their constructive comments and suggestions that have improved the quality of the manuscript. This work is supported by the French National Research Agency (Agence Nationale de la Recherche, ANR) project SafeNECS (Safe-Networked Control Systems) under Grant no. ANR-ARA SSIA-NV-15.

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