Journal of Control Science and Engineering

Volume 2008 (2008), Article ID 189064, 9 pages

http://dx.doi.org/10.1155/2008/189064

## 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.

The delays induced by the network in the closed-loop
control system are modeled as time-varying quantities arising from
the communication delays between sensors and controllers at time . 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: where is the state of
the uncertain plant, is the control
input, are all real
constant matrices, and matrix is the
controller gain matrix to be designed. The time-varying matrix represents
norm-bounded *parameter* uncertainties and satisfies the bound where denotes the
identity matrix with appropriate dimension. The constant matrices and characterize
the structure of these parameter uncertainties. The fault indicator matrix is given by with for , where means that the th actuator is
in healthy state, whereas the th actuator is
meant to experience a total failure when . 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 with . Each failure mode is therefore an
element of the set . We also view as a matrix,
that is, as a particular pattern of matrix in (4) depending on the
values of . 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 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 , that is, at least one actuator should be healthy.

We further assume that the time-varying delays lie between the following positive integer bounds and , that is, Given positive definite symmetric matrices and , we consider the quadratic cost function 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 and a positive scalar such that for
all admissible uncertainties and all failure modes the closed-loop
system (1)–(3) is stable with cost function (7) satisfying , then is said to be a
guaranteed cost and 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 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 actuators and results in a bank of controllers. It follows from inequality , 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 , 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 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: where . The cost function associated to system (8) is therefore where and . Under the assumptions made in Section 2, we can state the following result.

Theorem 1. *If there exist a gain matrix , a scalar , symmetric positive-definite matrices , , , and matrices , , , such that the
following matrix inequalities are satisfied: ** with ** then system (8) is asymptotically stable and the cost function
(9) satisfies
the inequality ** where *

*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 where is a given
matrix. Inequality (13) leads to where denotes the
maximum eigenvalue of matrix , , and .

##### 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 In the sequel, we will denote , and We further restrict to the following case in order to obtain a linear matrix inequality (LMI) (see, e.g., [9]): where is a scalar parameter. Pre- and postmultiply (10) by and , respectively; also pre- and postmultiply (11) by and and denote 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 , there exist a scalar , matrices , such that the following matrix inequalities are
satisfied: ** where ** Then, the control law ** is a guaranteed cost networked control law for system (1) and the
corresponding cost function satisfies ** where and .*

*Remark 4. *From (22), we establish the following inequalities: where and are scalars to
be determined. It is worth noting that condition (23) is not an LMI
because of the term . This is also the case for condition (19) which is not an
LMI because of the term . Note that for any matrix we have

Given a prescribed scalar , the design problem of the optimal guaranteed cost controller can be formulated therefore as the following optimization problem: 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 implies the minimization of the cost in (9). By applying a simple one-dimensional search over for a certain , 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 is fault-free, then we can redesign the th row of controller gain matrix to improve the robust performance for the system against actuator failures. We can rewrite the overall control system as where , matrix is obtained by deleting the th row from , is obtained by deleting the th column from , and is obtained by deleting th row and th column from . The cost function associated to system (26) reads as with , , where is obtained by deleting the th row and th column from . With regard to system (26) where is assumed to be known, we have the following result.

Theorem 3. *If there exist a gain matrix , a scalar , symmetric positive-definite matrices , , , and matrices , , , such that the
following matrix inequalities are satisfied: ** 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 to the following case in order to obtain an LMI: where is a scalar parameter. Pre- and postmultiply (28) with and , respectively; also pre- and postmultiply (29) with and , respectively, and denote The Schur complement trick leads to the following controller redesign result.

Theorem 4. *Suppose that for a prescribed scalar , there exist a scalar , matrices , such that the following matrix inequalities are
satisfied: ** where ** Then, the th control law ** is a guaranteed cost networked control law of system (26) and the
corresponding cost function satisfies **where *

Given a prescribed scalar , the redesign problem of the optimal guaranteed cost controller can be formulated as the following convex optimization problem:

#### 4. Illustrative Example

The dynamics are described by the following matrices: and the design parameters are chosen as When , and , by OP1 (25), the cost is obtained as and the
fault-tolerant controller can be designed for step (i): In step (ii), by OP2 (37), the cost and the feedback gains are redesigned as As a result, the two controllers are determined as follows: Figure 2 reports the simulation for two failures scenarios of the actuators. For this simulation, the time-varying norm-bounded uncertain
matrix is taken as which clearly satisfies the bound . 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):

(i)the dotted line
represents the state response for controller-switching sequence N^{°} : is the initial
controller, and is switched-on
at time instant 38, then at time instant
48, at time instant
68;(ii)the solid line
represents the state response for controller-switching sequence N^{°} : is the initial
controller, and is switched-on
at time instant 38, then at time instant
48, 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 is used as a
performance measure for comparison, where 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 , which means that . We conclude that the proposed method for sequence N^{°} 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 Then, for any matrices , , the following holds: ** where *

Lemma 6 (see [12]). *Let be a symmetric matrix and let be given
matrices with appropriate dimensions, then ** holds for all satisfying if and only if
there exists a scalar such that *

*Proof. *Note that , where . Then from system (8) we have Choose the Lyapunov-Krasovskii function candidates as follows: where Taking the forward difference for the Lyapunov functional , we have From (A.4), we
have where . Choose constant matrices , and satisfying
(11); by
Lemma 5, we
have Similarly, Note that So, we have Furthermore, we have Combining (9) and (A.7)–(A.13), we have where By Lemma 6, we
have It is worth observing that matrix is the Schur
complement of in the matrix
of the left-hand side of inequality (10). Therefore, the negative definiteness of matrix resulting from
inequality (10) implies that Summing both sides of the above inequality from to leads to which, from system stability, yields 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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