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Advances in Mathematical Physics

Volume 2013 (2013), Article ID 836743, 11 pages

http://dx.doi.org/10.1155/2013/836743

## Fault Tolerant Control for Interval Fractional-Order Systems with Sensor Failures

^{1}Information Engineering College, Henan University of Science and Technology, Luoyang 471023, China^{2}Anhui University of Technology, Maanshan 243002, China

Received 27 March 2013; Revised 26 July 2013; Accepted 26 July 2013

Academic Editor: Changpin Li

Copyright © 2013 Xiaona Song and Hao Shen. 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 problem of robust fault tolerant control for continuous-time fractional-order (FO) systems with interval parameters and sensor faults of has been investigated. By establishing sensor fault model and state observer, an observer-based FO output feedback controller is developed such that the closed-loop FO system is asymptotically stable, not only when all sensor components are working well, but also in the presence of sensor components failures. Finally, numerical simulation examples are given to illustrate the application of the proposed design method.

#### 1. Introduction

Fault tolerant control research and their application to a wide range of industrial and commercial processes have been the subjects of intensive investigations over the past two decades [1, 2]. Since unexpected faults or failures may result in substantial damage, much effort has been devoted to the fault tolerant control for various systems, such as active fault tolerant control for T-S fuzzy systems [3], reliable controller design for linear systems [4], robust satisfactory fault tolerant control of discrete-time systems [5], fault tolerant controller design for singular systems [6], and observer based fault-tolerant control for networked control systems [7]. On the other hand, fractional-order (FO) systems have attracted increasing interests, mainly due to the fact that many real-world physical systems are better characterized by FO differential equation [8–13]. The stability analysis of FO systems has been widely investigated, and there have been many stability results related to the continuous-time FO systems [14–20] and discrete-time FO systems [21]. In particular, in terms of linear matrix inequality, the stability condition has been given for continuous-time FO systems of order in [18] and of order in [20]. For FO-LTI systems with interval parameters, the stability and the controllability problems have been addressed for the first time in [22] and [23], respectively.

Recently, for the FO controller design problem, many authors have done some valuable works [24–26] and applied them to control a variety of dynamical processes, including integer-order and FO systems, so as to enhance the robustness and performance of the control systems. While for interval FO systems, in [27, 28], authors have investigated the stabilization problem of and , respectively. However, the above papers dealt with state feedback control design that requires all state variables to be available. In many cases, this condition is too restrictive. So it is meaningful to control the FO systems via output feedback controller design method, and the observer-based output feedback controller design method is one of the available choices. Moreover, to the best of our knowledge, few results have been obtained for observer-based FO output feedback controller design of FO systems with interval parameters; and sensor faults, which motivates this present study.

This paper investigates the observer-based FO output feedback controller design for the FO systems with interval parameters and sensor faults, the purpose is to design the observer-based FO output feedback control law such that the resulting closed-loop FO system is stable for the order and , respectively. Explicit expression of the desired observer-based FO output feedback controller is given.

*Notations.* Throughout this paper, for real symmetric matrices and , the notation (resp., ) means that the matrix is positive semidefinite (resp., positive definite). The notation represents the transpose of the matrix . denotes the identity matrix. In symmetric block matrices, “” is used as an ellipsis for terms induced by symmetry. Matrices, if not explicitly stated, are assumed to have appropriate dimensions. denotes the expression . stands for the Kronecker products.

#### 2. Preliminaries and Problem Formulation

In this paper, we adopt the following Caputo definition for fractional derivative, which allows utilization of initial values of classical integer-order derivatives with known physical interpretations [12, 29]: where is an integer satisfying , is a continuous function, and is the Euler gamma function given by

Considering the following fractional-order (FO) LTI systems with interval parameters: where is the time fractional derivative order. is the state, is the control input, and is the measured output. The system matrices are known real constant matrices with appropriate dimensions; , are interval uncertainties in the sense that where , satisfy for all , and , satisfy for all .

To deal with the uncertain interval, we introduce the following notations: It can be seen that all elements of and are nonnegative, so we can definewhere and denote the column vectors with the th element being 1 and all the others being 0. Also, denote Here, for and , we have the following lemma.

Lemma 1 (see [27, 28]). *Let
**
Then .*

To investigate the fault tolerant control problem in the event of sensor failures, the fault model should be established first.

Letting sensor faults function matrix as [30] where , and , are known real constants, .

Now, introducing the following matrices where , , , , then one can obtain Now, we consider the following observer-based FO output feedback controller for the FO system (3)-(4): where and are constant matrices to be determined.

Define the observer error as From (3), (4), (14)–(16), and Lemma 1, the following dynamic equations of state and error can be obtained:

Combining (18) and (19) yields the following augmented FO systems: where Our objective is to find a systematic way to determine and , given , , , , , , , and such that the closed-loop system is stable. Note that here is in the range of to , which is never covered in the literature in terms of observer-based FO output feedback stabilization problem.

#### 3. Main Results

In this section, we give a solution to the stability analysis and the observer-based fractional-order (FO) output feedback control problems formulated in the previous part. We first give the following results which will be used in the proof of our main results.

Lemma 2 (see [17]). *Let be a real matrix. Then, is asymptotically stable if and only if , where is the spectrum of all eigenvalues of .*

Lemma 3 (see [31]). *Let be a real matrix and . Then , where , if and only if there exists such that
*

Lemma 4 (see [32]). *For any matrices and with appropriate dimensions, the following holds:
**
for any .*

Now, we are in a position to present a solution to the stability analysis and observer-based FO output feedback control problem.

##### 3.1. The Case of

First, we will present a solution to the stability analysis for FO systems (20) with order .

Theorem 5. *Given the controller gain matrix and the observer gain matrix , the system (20) with order and is robustly asymptotically stable for any sensor faults if there exists real symmetric positive definite matrix and scalar constants , , such that
**
where
*

*Proof. *The FO-LTI interval system (3)-(4) is asymptotically stable for any sensor faults if the FO system is asymptotically stable. This is equivalent to that there exists a symmetric positive definite matrix , such that
From Lemma 4, one can have
Substituting (27) and (28) into (26), one can get
Taking (29) into account and using the Schur complement of (24), one obtains
It follows from the above inequality (30) and Lemma 3 that . Therefore, by Lemma 2, the FO-LTI system (20) is asymptotically stable. This completes the proof.

The observer-based FO output feedback control problem for FO systems (20) with order is presented in the following theorem.

Theorem 6. *Given positive scalar constants , the FO system (20) with order and is asymptotically stable if there exist the matrices , , ; the following conditions are satisfied:
**
where
**
Furthermore a desired FO observer-based output feedback controller is given in the form of (14) with parameter as follows:
*

*Proof. *The FO-LTI system (20) is asymptotically stable. It follows from Theorem 5 that this is equivalent to that there exist a symmetric positive definite matrix and positive scalar constants , such that
Introducing the following nonsingular matrix
Let
then, by some calculation, we have
Now, pre- and postmultiplying the inequality in (34) by and , respectively, set ; then we have
Inequality (38) is equivalent to (31) by the Schur complement. This completes the proof.

##### 3.2. The Case of

In this subsection, first we will present a solution to the stability analysis for FO systems (20) with order .

Theorem 7. *Given the controller gain matrix and the observer gain matrix , the FO system (20) with order is robustly asymptotically stable for any sensor faults if there exist two real symmetric positive definite matrices , , two skew-symmetric matrices , , and scalar constants , , such that
**
where*

*Proof. *Suppose that (39)-(40) hold. It follows from (20) that
Note that (13) and Lemma 1 imply
It follows from (43) that
Note that ; it follows from (44) and Lemma 4 that for any real scalars and
Substituting (45) and (46) into (42), one has
Taking (47) into account and using the Schur complement of (39), one obtains
It follows from the above inequality (48) and Theorem 1 of [27] that the uncertain FO-LTI interval system with is asymptotically stable.

The observer-based FO output feedback control problem for FO systems (20) with order is presented in the following theorem.

Theorem 8. *Given positive scalar constants , , The FO system (20) with order is asymptotically stable if there exist the matrices , and symmetric matrix ; the following condition is satisfied:
**
where
**
Furthermore a desired observer-based FO output feedback controller is given in the form of (14) with parameter as follows:
*

*Proof. *The FO-LTI system (20) is asymptotically stable. It follows from Theorem 7 that this is equivalent to that there exist two real symmetric positive definite matrices , , and two skew-symmetric matrices , , such that
By setting , in (52), we can get that if
the FO-LTI system (20) is asymptotically stable. Similar to the proof of Theorem 7, (53) is equivalent to that there exist a symmetric positive definite matrix and positive real scalars and such that
Introducing the following nonsingular matrix
Let
Now, pre- and postmultiplying the inequality in (54) by and , respectively, set ; then we have
Inequality (57) is equivalent to (49) by the Schur complement. This completes the proof.

#### 4. Simulation

Consider the fault tolerant control problem for the fractional-order (FO) systems (3)-(4) of order with the following parameters The purpose is to design a observer-based FO output feedback control law such that the closed-loop system is stable in the event of sensor failure. Now, we choose and the initial state . Then, using Matlab Linear Matrix Inequality (LMI) Control Toolbox to solve the LMI (49), we can obtain the solution as follows: While, for the FO system (3)-(4) with the following parameters: Then, we choose and the initial state . Also using Matlab LMI Control Toolbox to solve the LMI (31), we can obtain the solution as follows: With the observer-based FO output feedback controller, the closed-loop system is stable. The state response of the FO systems of order and are given in Figures 1 and 2, respectively, while the corresponding control input are shown in Figures 3 and 4. The state response of the FO systems of order and are given in Figures 5 and 6, respectively. While Figures 7 and 8 show the corresponding control input .

From these simulation results, it can be seen that the designed observer-based FO output feedback controller ensures the asymptotic stability of the FO systems of in the event of the sensor faults.

#### 5. Conclusion

The problem of fault tolerant control for fractional-order (FO) systems with uncertain interval parameters and sensor faults is studied. By establishing sensor fault model and state observer, an observer-based FO output feedback controller, which stabilizes the FO systems of in the event of some sensor failures, is given. Finally, numerical simulation results show that the proposed method is effective.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 61203047) and the Youth Science Foundation of Henan University of Science and Technology (no. 2012QN006).

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