Mathematical Problems in Engineering

Volume 2013, Article ID 304092, 11 pages

http://dx.doi.org/10.1155/2013/304092

## Adaptive Fault-Tolerant Controller Design for Networked Systems with Actuator Saturation

School of Automation, Shenyang Aerospace University, Shenyang, Liaoning 110136, China

Received 10 January 2013; Accepted 5 March 2013

Academic Editor: Engang Tian

Copyright © 2013 Wei Guan. 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

In this paper, an indirect adaptive fault-tolerant controller design method is proposed for networked systems in the presence of actuator saturation. Based on the on-line estimation of eventual faults, the parameters of controller are being updated automatically to compensate the fault effects on systems. The designs are given in linear matrix inequalities (LMIs) approach, which can guarantee the disturbance tolerance level and adaptive performances of networked systems in the cases of actuator saturation and actuator failures. An example is given to illustrate the efficiency of the design method.

#### 1. Introduction

With the rapid developments in network technologies, more and more communication networks are used in control systems; especially, the stability analysis for systems with time delays have become an active research area. However, networked control systems with actuator saturation and *time delays* are often encountered in many practical systems such as electrical heaters and long transmission lines in pneumatic, hydraulic, and rolling mill systems. Since the existence of time delay and actuator saturation in a physical system often induces instability of poor performance, research on time delay systems with actuator saturation is a topic of great practical and theoretical importance. If the saturation and time delay are ignored in system analysis and design, the performance of the overall system can be degenerated. More seriously, saturation and time delay can cause instability of the overall system. Therefore, over the last several decades, many researchers have considered various control problems of disturbance rejection for linear systems subject to actuator saturation [1–11]. Papers [4, 5] carried out the gain analysis and minimization. Although there are plenty of papers that are devoted to dealing with different problems for systems with actuator saturation, the main difference and difficulty lie in their treatment of saturation nonlinearity. In paper [2], authors gave a method for maximization of an ellipsoid which is invariant under input saturation, but persistent disturbances. The works of [1, 3, 6–8] consider the situation where disturbance are bounded in energy. The works of [1, 6, 7] formulated and solved the problem of stability analysis and design. In [9, 10], authors presented LMI-based methods for regional stability and performance of linear antiwindup compensators for linear control systems. [12] presents a method for the analysis and control design of linear systems in the presence of actuator saturation and disturbances. During the last few years, problems about actuator saturation have been extended to many other fields of automatic control, such as singular systems [13], systems with parameters uncertainty [14], Markovian jump systems [15], decentralized control systems [16], and Hamiltonian systems [17].

Time delays are frequently encountered in almost all networked systems. Since the existence of a delay in a physical system often induces instability of poor performance, research on time-delay systems is a topic of great practical and theoretical importance. During the last decade, the control problem of systems with time delay has received considerable attention. The main methods can be classified into two types: delay-independent ones and delay-dependent ones.

On the other hand, fault tolerant has become a hot research area because of its importance in practical engineering [18–28]. And the design approach can be broadly classified into two types: Passive approach and Active approach. A passive fault-tolerant controller commonly has a simple structure and is easily implemented [18–22]. The system performances in normal and fault modes can be optimized. Some of these active fault-tolerant control methods may readjust controller parameters or change controller structure to compensate the fault effects on systems. Some of these methods include a strategy involving a fast subsystem for fault detection and isolation (FDI) and a supervisory system that chooses the corresponding controller for a particular type of fault. Most of the results in adaptive fault-tolerant control are based on model reference adaptive control (MRAC) [29–31], but the disturbance attenuation performances of systems have not been addressed yet within the MRAC framework. Paper [32] considered the problem of adaptive reliable controller via state feedback and dynamic output feedback, respectively, for linear time-delay systems against actuator faults. However, when actuator saturation problem is considered, the methods of [32] cannot be used.

As we all know, actuator faults and saturation always happen at the same time for networked systems. However, noting all above results, there is no work that deals with this problem. There are only a few papers that considered the problems about systems with actuator saturation and faults [13, 33, 34]. Motivated by the above observations, this paper studies the problem of designing adaptive fault-tolerant controllers for networked systems with actuator saturation. The designs are developed in the framework of LMIs approach, which can guarantee the disturbance tolerance level and adaptive performances of networked systems in the cases of actuator saturation and actuator failures. The difference between this paper and some existing results is that in this paper the fault tolerant and saturation are considered at same time for networked systems.

The remainder of this paper is organized as follows. Section 2 introduces notation to be used in the paper, and problem statement is given in it. An adaptive fault-tolerant controller design method is described for networked system in Section 3. In Section 4, an example is given to illustrate the efficiency of the design method. The paper will be concluded in Section 5.

#### 2. Problem Statement and Preliminaries

In this paper, the following LTI plant will be considered: where and is the plant state at time defined by , , is the saturated control input, is the regulated output, and is an exogenous disturbance in , respectively. , , , , , and , are known constant matrices of appropriate dimensions. For simplicity only, we take single delay . The results of this paper can be easily applied to the case of multiple delays.

The following case for *time-varying delay * is considered. That is, is differentiable funcion
where is an upper bound on the derivative of .

In this paper, we formulate the fault-tolerant control problem by using the following model form which is considered in [19, 22]: where represents the signal from the th actuator which has failed in the th fault mode, is an unknown constant, the is the total fault modes, and and represent the lower and upper bounds of , respectively. Denote that where , . Considering the lower and upper bounds and , the following set can be defined:

For convenience, the following uniform actuator fault model is exploited: where is described by . The following definitions and lemmas will be used in the sequel.

*Definition 1 (see [33]). *Consider the following system:
where is parameter vector and is a time-varying parameter vector to be chosen. Let be a given constant, then system (7) is said to be with an adaptive performance index no larger than if for any , there exists a such that the following inequality holds:

*Definition 2. * For a matrix , denote the th row of as , and define

Lemma 3 (see [32]). * If there exists a symmetric matrix with
**
and , such that the following inequalities hold:
**
then inequality
**
holds for all , where and **Let be a set of diagonal matrices whose diagonal elements are either 1 or 0. There are elements in , and one denotes its elements as , , where for with , the diagonal elements of are . Denote . It is easy to see that . Then, one has the following.*

Lemma 4 (see [35]). * For two vectors , . Suppose that . Then,
**
where denotes the convex hull.*

For a networked system, the performance of closed-loop system can be measured by the gain. However, this gain cannot be well defined for closed-loop system, since a sufficiently large disturbance may lead to unstable closed-loop system. For this reason, we need to consider a class of disturbances whose energy is bounded by a given value; that is,

In this paper, we will consider the following problems. The first question is, what is the maximal value of such that the state will be bounded for all for systems with time delay? This question can be referred to as disturbance tolerance level. The system performance can be measured by the restricted gain over . In this paper, gain and will be considered at same time for networked system.

#### 3. Main Results

The dynamics with actuator faults (6) and saturation are described by Rewrite (16) as

The controller structure is given as where is used to estimate , and .

By Lemma 3, the following equality is given, with : for some scalars , , such that , and the following equality holds: where . Denote with , and

*Definition 5. * Let be a positive-definite matrix. Denote
Assuming that is given, we denote .

The following lemma provides a method for choosing of 's, which are Lipschitzian functions in and and thus are useful in controller design method.

Lemma 6 (see [11]). * Let . For each ,
**
and for each , let such that , and define
**Then, 's are functions Lipschitz in and , such that, , , . Moreover, they satisfy relation (19).**By using the functions ’s and controller (18), plant (16) can be written in a quasi-LPV form as follows:
*

Lemma 7 (see [36]). * For any , , , ,, and , the following holds:
**
where .*

*Definition 8. * Firstly, for system (25), consider the following functional
where

Then, the following set can be given:

*Remark 9. * By Definitions 5 and 8, we can draw the conclusion that .

By Lemma 6, we analyze the auxiliary LPV system as follows, of which the closed-loop system comprising of (25) and (18) is a special case, for all :
where , and

Theorem 10. * Let , , and be given constants, then the following two conditions are satisfied.*(I)*The trajectories of the closed-loop system that start from the origin will remain inside the domain for every .*(II)*In normal case, that is, , for ,
**and in actuator failures cases, that is, , for ,
**
where , if there exist matrices , , , , , , , , , , , , , , and symmetric matrixes , , with
**
and , such that the following inequalities hold for all and ; that is, for all :
**
where
**
and also is determined according to the adaptive law
**
where
**
where , , , and () are the adaptive law gains which can be given according to practical applications. Then, the following controller can be given:
*

* Proof. * See appendix.

From Theorem 10, we have the following algorithm to optimize the adaptive performance in normal and fault cases and the disturbance tolerance level with considering time delay.

*Algorithm 11. * Suppose that and denote the adaptive performance indexes for the normal case and fault cases of the closed-loop system (30), respectively. Let denote the disturbance tolerance level. Then, , are minimized, and is maximized if the following optimization problem is solvable:
where , , , and , , and are weighting coefficients.

By Definition 2, condition (b) cannot be shown as LMIs directly. However, obviously, , which implies that (b) can be replaced with (b1) as follows:
Equation (44) is equivalent to
for all , where is the th row of . Then, it can be drawn that (45) is equivalent to the following inequalities:
where and is the th row of , .

*Remark 12. *Theorem 10 prevents a condition for the existence of an adaptive fault tolerant controller. In Theorem 10, if set , , , and , , the condition of Theorem 10 is reduced to fixed gains condition. By the following example, we can get that the adaptive controller can guarantee better effect.

#### 4. Examples

*Example 13. * Consider the system of the form (1) with
and the following two possible fault modes.

Fault mode 1: both of the two actuators are normal; that is . Fault mode 2: the first actuator is outage, and the second actuator may be normal or loss of effectiveness, described by , , where denotes the maximal loss of effectiveness for the second actuator.

Let , , and , and let the optimal indexes with fixed controller gains are , , , and . By solving (43), the optimal indexes can be given as , , , and . For getting smaller number for every optimal index, we may revise , and . Then, the indexes can be drawn that , , and . For illustrating the efficiency of the design method, the following simulations is given.

During the following simulation, fault case is considered as follows. At second, the first actuator is outage. Here, we choose .

Firstly, we consider the performance. The disturbance is given as Figures 1 and 2 show the responses curves of the first state in normal and fault case, respectively.

Then, we consider the disturb tolerance problem. The disturbance is given as Figures 3 and 4 show the responses curves of the states in normal case.

#### 5. Conclusions

In this paper, an adaptive fault-tolerant controllers design method was given for networked systems with actuator saturation. The designs were proposed in LMIs approach, which could guarantee the disturbance tolerance ability and adaptive performances of networked systems in the cases of actuator saturation and actuator failures. An example, has been given to illustrate the efficiency of the design method.

#### Appendix

* Proof of Theorem 10. *Item (II) will be proved firstly. By Definition 8, since , then
From the derivative of along the closed-loop system (30), it follows that
where
By Lemma 7, taking , , it follows that
where and , , satisfying .

Furthermore, by (20) it follows that
where
with , .

Then,
where

Let , then we have

In fact, is an unknow constant which denotes the loss of effectiveness of the th actuator. So, from , it follows that . Now, if the adaptive laws are chosen as (40), then

Let , then
where

Furthermore, the problem reduces to

It is obvious from the requirement of and the fact that in (A.13) must be negative and is nonsingular.

Defining
We multiply by and , on the left and the right, respectively. Applying Fisher's lemma to the emerging quadratic term in , denoting , , and choosing , where is a diagonal matrix, we obtain the following: is equivalent to
with
Furthermore, (A.15) can be described by
where
and , , , , are defined in (39).

If we multiply , on the left and on the right, by and , then it follows that . By Lemmas 3 and 7, it is easy to see if conditions (34), (37), and (38) hold, then (A.17) and are satisfied, which implies that . Furthermore, by Lemma 3 and (37), it follows that holds for any , , , and satisfying (40). The proofs for the normal case of closed-loop system (16) are similar and omitted here.

To prove item (I): by the proof of item (II), we have , which implies that

Then, the conclusion can be drawn that trajectories of the closed-loop system that start from the origin will remain inside for every .

#### Acknowledgments

This work was supported in part by Program for the Funds of National Science of China (Grant nos. 61203087 and 61074090), the Liaoning Education Department Plan Project of China (no. L2010426), and the Liaoning Students Innovation Training Plan Project of China (nos. 201210143012 and 2012106).

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