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

The problem of robust fault tolerant control for continuous-time fractional-order (FO) systems with interval parameters and sensor faults of 0 < α < 2 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.


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][9][10][11][12][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][15][16][17][18][19][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 0 <  < 1 in [18] and of order 1 ≤  < 2 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][25][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 0 <  < 1 and 1 ≤  < 2, 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 0 <  ≤ 1 and 1 ≤  < 2, respectively.Explicit expression of the desired observer-based FO output feedback controller is given.

Advances in Mathematical Physics
Notations.Throughout this paper, for real symmetric matrices  and , the notation  ≥ Y (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.Sym() denotes the expression  +   .⊗ stands for the Kronecker products.

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  − 1 <  ≤ , () is a continuous function, and Γ() 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 To deal with the uncertain interval, we introduce the following notations: It can be seen that all elements of Δ and Δ are nonnegative, so we can define where    ∈ R  and    ∈ R  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.
To investigate the fault tolerant control problem in the event of sensor failures, the fault model should be established first.

Advances in Mathematical Physics
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 ,  0 ,  0 ,  0 ,   ,   ,   , and   such that the closed-loop system is stable.Note that  here is in the range of 0 to 2, which is never covered in the literature in terms of observer-based FO output feedback stabilization problem.

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.
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 1 ≤  < 2. First, we will present a solution to the stability analysis for FO systems (20) with order 1 ≤  < 2.
Theorem 5. Given the controller gain matrix  and the observer gain matrix , the system (20) with order 1 ≤  < 2 and  =  − /2 is robustly asymptotically stable for any sensor faults if there exists real symmetric positive definite matrix  and scalar constants   > 0, ( = 1, 2), such that where Proof.The FO-LTI interval system (3)-( 4) is asymptotically stable for any sensor faults if the FO system   () = Â() is asymptotically stable.This is equivalent to that there exists a symmetric positive definite matrix  ∈ R × , such that From Lemma Substituting ( 27) and ( 28) into (26) Taking ( 29) into account and using the Schur complement of (24), one obtains It follows from the above inequality (30) and Lemma 3 that | arg(spec( Â))| > /2.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 1 ≤  < 2 is presented in the following theorem.Theorem 6.Given positive scalar constants   ( = 1, 2), the FO system (20) with order 1 ≤  < 2 and  =  − /2 is asymptotically stable if there exist the matrices  1 > 0, , G; 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) Introducing the following nonsingular matrix Let then, by some calculation, we have Now, pre-and postmultiplying the inequality in (34) by diag(, ) and diag(, ), respectively, set  = G; then we have Inequality ( 38) is equivalent to (31) by the Schur complement.This completes the proof.

The Case of 0 < 𝛼 ≤ 1.
In this subsection, first we will present a solution to the stability analysis for FO systems (20) with order 0 <  ≤ 1.
The observer-based FO output feedback control problem for FO systems (20) with order 0 <  ≤ 1 is presented in the following theorem.Theorem 8. Given positive scalar constants  1 ,  2 ( = 1, 2), The FO system (20) with order 0 <  ≤ 1 is asymptotically stable if there exist the matrices G1 ,  and symmetric matrix  1 > 0; 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  1 ∈ R × ,  = 1, 2, and two skew-symmetric matrices By setting  11 =  21 = ,  12 =  22 = 0 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  ∈ R × and positive real scalars  1 and  2 ( = 1, 2) such that Introducing the following nonsingular matrix Let (56) Now, pre-and postmultiplying the inequality in (54) by  2 ⊗ and  2 ⊗ , respectively, set  = G1 ; then we have Inequality (57) is equivalent to (49) by the Schur complement.This completes the proof.

Simulation
Consider the fault tolerant control problem for the fractionalorder (FO) systems ( 3)-( 4) of order 0 <  < 2 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 With the observer-based FO output feedback controller, the closed-loop system is stable.The state response  1 () of the FO systems of order 0 <  ≤ 1 and 1 ≤  < 2 are given in Figures 1 and 2, respectively, while the corresponding control input  1 () are shown in Figures 3 and 4. The state response  2 () of the FO systems of order 0 <  ≤ 1 and 1 ≤  < 2 are given in Figures 5 and 6, respectively.While Figures 7 and 8 show the corresponding control input  2 ().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 0 <  < 2 in the event of the sensor faults.

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 0 <  < 2 in the event of some sensor failures, is given.Finally, numerical simulation results show that the proposed method is effective.