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

Abstract

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.

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 0<𝛼<1 in [18] and of order 1≤𝛼<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 [24–26] 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<𝛼<1 and 1≤𝛼<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<𝛼≤1 and 1≤𝛼<2, respectively. Explicit expression of the desired observer-based FO output feedback controller is given.

Notations. Throughout this paper, for real symmetric matrices 𝑋 and 𝑌, the notation 𝑋≥Y (resp., 𝑋>𝑌) means that the matrix 𝑋−𝑌 is positive semidefinite (resp., positive definite). The notation 𝑀^𝑇 represents the transpose of the matrix 𝑀.  𝐼_𝑛×𝑛 denotes the 𝑛×𝑛 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(𝑋) denotes the expression 𝑋+𝑋^𝑇. ⊗ 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]: 𝐷^𝛼𝑓(𝑡)=𝑑^𝛼𝑓(𝑡)𝑑𝑡^𝛼=1Γ(𝑛−𝛼)^𝑡₀𝑓^(𝑛)(𝜏)𝑑𝜏(𝑡−𝜏)^{𝛼+1−𝑛}, (1) where 𝑛 is an integer satisfying 𝑛−1<𝛼≤𝑛, 𝑓(𝑡) is a continuous function, and Γ(𝛼) is the Euler gamma function given by Γ(𝛼)=^∞₀𝜈^𝛼−1𝑒^−𝜈𝑑𝜈. (2)

Considering the following fractional-order (FO) LTI systems with interval parameters: 𝐷^𝛼𝑥(𝑡)(𝑡)(𝑡)𝑦(𝑡)=𝐴𝑥+𝐵𝑢,0<𝛼<2,=𝐶₀𝑥(𝑡),𝑥(𝑡)=𝑥₀, (3)(4)(5) where 𝛼 is the time fractional derivative order. 𝑥(𝑡)∈𝑅^𝑛 is the state, 𝑢(𝑡)∈𝑅^𝑚 is the control input, and 𝑦(𝑡)∈𝑅^𝑠 is the measured output. The system matrices 𝐶 are known real constant matrices with appropriate dimensions; 𝐴, 𝐵 are interval uncertainties in the sense that 𝐴∈𝐴_𝐼=𝐴,𝐴=𝑎_𝑖𝑗∶𝑎_𝑖𝑗≤𝑎_𝑖𝑗≤𝑎_𝑖𝑗,1≤𝑖,𝑗≤𝑛,𝐵∈𝐵_𝐼=𝐵,𝐵=𝑏_𝑖𝑗∶𝑏_𝑖𝑗≤𝑏_𝑖𝑗≤𝑏_𝑖𝑗,1≤𝑖≤𝑛,1≤𝑗≤𝑚, (6) where 𝐴=[𝑎_𝑖𝑗]_𝑛×𝑛, 𝐴=[𝑎_𝑖𝑗]_𝑛×𝑛 satisfy 𝑎_𝑖𝑗≤𝑎_𝑖𝑗 for all 1≤𝑖,𝑗≤𝑛, and 𝐵=[𝑏_𝑖𝑗]_𝑛×𝑚, 𝐵=[𝑏_𝑖𝑗]_𝑛×𝑚 satisfy 𝑏_𝑖𝑗≤𝑏_𝑖𝑗 for all 1≤𝑖≤𝑛,1≤𝑗≤𝑚.

To deal with the uncertain interval, we introduce the following notations: 𝐴₀=12𝐴+𝐴1,Δ𝐴=2𝐴−𝐴=𝜒_{𝑖𝑗𝑛×𝑛},𝐵₀=12𝐵+𝐵1,Δ𝐵=2𝐵−𝐵=ϝ_{𝑖𝑗𝑛×𝑚}. (7) It can be seen that all elements of Δ𝐴 and Δ𝐵 are nonnegative, so we can define𝐷_𝐴=√𝜒₁₁𝑒^𝑛₁⋯√𝜒_1𝑛𝑒^𝑛₁⋯√𝜒_𝑛1𝑒^𝑛_𝑛⋯√𝜒_𝑛𝑛𝑒^𝑛_{𝑛𝑛×𝑛2},𝐸_𝐴=√𝜒₁₁𝑒^𝑛₁⋯√𝜒_1𝑛𝑒^𝑛_𝑛⋯√𝜒_𝑛1𝑒^𝑛₁⋯√𝜒_𝑛𝑛𝑒^𝑛_𝑛^𝑇_𝑛2×𝑛,𝐷_𝐵=√ϝ₁₁𝑒^𝑛₁⋯ϝ_1𝑝𝑒^𝑛₁⋯√ϝ_𝑛1𝑒^𝑛_𝑛⋯√ϝ_𝑛𝑚𝑒^𝑛_{𝑛𝑛×(𝑛𝑚)},𝐸_𝐵=√ϝ₁₁𝑒^𝑚₁⋯ϝ_1𝑝𝑒^𝑚_𝑚⋯√ϝ_𝑛1𝑒^𝑚₁⋯√ϝ_𝑛𝑚𝑒^𝑚_𝑚^𝑇_{(𝑛𝑚)×𝑚}, (8)where 𝑒^𝑛_𝑘∈ℝ^𝑛 and 𝑒^𝑚_𝑘∈ℝ^𝑚 denote the column vectors with the 𝑘th element being 1 and all the others being 0. Also, denote 𝐻_𝐴=diag𝜌₁₁,…,𝜌_1𝑛,…,𝜌_𝑛1,…,𝜌_𝑛𝑛∈ℝ^𝑛2×𝑛2,𝜌_𝑖𝑗,𝐻≤1,𝑖,𝑗=1,…,𝑛_𝐵=diag𝜎₁₁,…,𝜎_1𝑚,…,𝜎_𝑛1,…,𝜎_𝑛𝑚∈ℝ^{(𝑛𝑚)×(𝑛𝑚)},𝜎_𝑖𝑗≤1,𝑖=1,…,𝑛,𝑗=1,…,𝑚. (9) Here, for 𝐴_𝐼 and 𝐵_𝐼, we have the following lemma.

Lemma 1 (see [27, 28]). Let 𝐴_𝐽=𝐴=𝐴₀+𝐷_𝐴𝐹_𝐴𝐸_𝐴∣𝐹_𝐴∈𝐻_𝐴,𝐵_𝐽=𝐵=𝐵₀+𝐷_𝐵𝐹_𝐵𝐸_𝐵∣𝐹_𝐵∈𝐻_𝐵. (10) Then 𝐴_𝐼=𝐴_𝐽,𝐵_𝐼=𝐵_𝐽.

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] 𝑊=diag𝑤₁,𝑤₂,…,𝑤_𝑠, (11) where 0≤𝑤_𝑖𝑙≤𝑤_𝑖≤𝑤_𝑖𝑢<∞, and 𝑤_𝑖𝑙<1,𝑤_𝑖𝑢≥1, 𝑖=1,2,…,𝑠 are known real constants, 𝑊≠0.

Now, introducing the following matrices 𝑊₀=diag𝑤₀₁,𝑤₀₂,…,𝑤_0𝑠,𝐽=diag𝑗₁,𝑗₂,…,𝑗_𝑠,|𝐿|=diag𝑙₁,𝑙₂,…,𝑙_𝑠, (12) where 𝑤_0𝑖=(𝑤_𝑖𝑙+𝑤_𝑖𝑢)/2, 𝑙_𝑖=(𝑤_𝑖−𝑤_0𝑖)/𝑤_0𝑖, 𝑗_𝑖=(𝑤_𝑖𝑢−𝑤_𝑖𝑙)/(𝑤_𝑖𝑢+𝑤_𝑖𝑙), (𝑖=1,2,…,𝑠), then one can obtain 𝑊=𝑊₀(𝐼+𝐿),|𝐿|≤𝐽≤𝐼. (13) Now, we consider the following observer-based FO output feedback controller for the FO system (3)-(4): 𝐷^𝛼𝑥(𝑡)=𝐴₀𝑥(𝑡)+𝐵₀𝑢(𝑡)(𝑡)−𝑦(𝑡,𝑦(𝑡)+𝐺𝑊𝑦)=𝐶₀𝑥(𝑡),𝑢(𝑡)=𝐾𝑥(𝑡), (14)(15)(16) where 𝐺 and 𝐾 are constant matrices to be determined.

Define the observer error as 𝑒(t)=𝑥(𝑡)−𝑥(𝑡). (17) From (3), (4), (14)–(16), and Lemma 1, the following dynamic equations of state and error can be obtained: 𝐷^𝛼𝑥(𝑡)=𝐴₀+𝐷_𝐴𝐹_𝐴𝐸_𝐴+𝐵₀+𝐷_𝐵𝐹_𝐵𝐸_𝐵𝐾𝑥(𝑡)−𝐵₀+𝐷_𝐵𝐹_𝐵𝐸_𝐵(𝑡),𝐷𝐾𝑒_𝛼𝑒(𝑡)=𝐷_𝐴𝐹_𝐴𝐸_𝐴+𝐷_𝐵𝐹_𝐵𝐸_𝐵𝐾+𝐺(𝐼−𝑊)𝐶₀𝑥(𝑡)+𝐴₀−𝐺𝐶₀−𝐷_𝐵𝐹_𝐵𝐸_𝐵𝐾𝑒(𝑡). (18)(19)

Combining (18) and (19) yields the following augmented FO systems: 𝐷^𝛼𝑥(𝑡)=𝐴₀+𝐷_𝐴𝐵𝐹_𝐴𝐵𝐸_𝐴𝐵𝑥(𝑡)=𝐴𝑥(𝑡), (20) where 𝑥(𝑡)=𝑥(𝑡)𝑒(𝑡),𝐴=𝐴₀+𝐷_𝐴𝐵𝐹_𝐴𝐵𝐸_𝐴𝐵,𝐴₀=𝐴₀+𝐵₀𝐾−𝐵₀𝐾𝐺(𝐼−𝑊)𝐶₀𝐴₀−𝐺𝐶₀,𝐷_𝐴𝐵=𝐷_𝐴𝐷_𝐵𝐷_𝐴𝐷_𝐵,𝐹_𝐴𝐵=𝐹_𝐴00𝐹_𝐵,𝐸_𝐴𝐵=𝐸_𝐴0𝐸_𝐵𝐾−𝐸_𝐵𝐾. (21) Our objective is to find a systematic way to determine 𝐺 and 𝐾, given 𝛼, 𝐴₀,  𝐵₀,  𝐶₀, 𝐷_𝐴, 𝐷_𝐵, 𝐸_𝐴, and 𝐸_𝐵 such that the closed-loop system is stable. Note that 𝛼 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.

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 𝐴 be a real matrix. Then, 𝐷^𝛼𝑥(𝑡)=𝐴𝑥(𝑡) is asymptotically stable if and only if |arg(spec(𝐴))|>𝜋𝛼/2, where spec(𝐴) is the spectrum of all eigenvalues of 𝐴.

Lemma 3 (see [31]). Let 𝐴∈𝑅^𝑛×𝑛 be a real matrix and 𝜃=𝜋−𝜋𝛼/2. Then |arg(spec(𝐴))|>𝜋𝛼/2, where 1≤𝛼<2, if and only if there exists 𝑃>0 such that 𝐴𝑃+𝑃𝐴^𝑇sin𝜃𝐴𝑃−𝑃𝐴^𝑇∗cos𝜃𝐴𝑃+𝑃𝐴^𝑇sin𝜃<0. (22)

Lemma 4 (see [32]). For any matrices 𝑋 and 𝑌 with appropriate dimensions, the following holds: 𝑋^𝑇𝑌+𝑌^𝑇1𝑋≤𝜀𝑋^𝑇𝑋+𝜀𝑌^𝑇𝑌, (23) for any 𝜀>0.

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≤𝛼<2

First, we will present a solution to the stability analysis for FO systems (20) with order 1≤𝛼<2.

Theorem 5. Given the controller gain matrix 𝐾 and the observer gain matrix 𝐺, the system (20) with order 1≤𝛼<2 and 𝜃=𝜋−𝜋𝛼/2 is robustly asymptotically stable for any sensor faults if there exists real symmetric positive definite matrix 𝑃 and scalar constants 𝜖_𝑖>0, (𝑖=1,2), such that Γ₁Γ₂Γ₃Γ₄Γ₅∗−𝜖₁𝐼000∗∗−𝜖₂𝐼00∗∗∗−𝜖₁𝐼0∗∗∗∗−𝜖₂𝐼<0, (24) where Γ₁=𝐴₁𝑃+𝑃𝐴^𝑇₁sin𝜃𝐴₁𝑃−𝑃𝐴^𝑇₁∗cos𝜃𝐴₁𝑃+𝑃𝐴^𝑇₁,Γsin𝜃₂=𝜖₁(sin𝜃)𝐺_𝑤𝐽𝜖₁(cos𝜃)𝐺_𝑤𝐽−𝜖₁(cos𝜃)𝐺_𝑤𝐽𝜖₁(sin𝜃)𝐺_𝑤𝐽,Γ₃=𝜖₂(sin𝜃)𝐷_𝐴𝐵𝜖₂(cos𝜃)𝐷_𝐴𝐵−𝜖₂(cos𝜃)𝐷_𝐴𝐵𝜖₂(sin𝜃)𝐷_𝐴𝐵,Γ₄=𝐶₀0𝑃0𝐶₀𝑃^𝑇,Γ₅=𝐸_𝐴𝐵𝑃00𝐸_𝐴𝐵𝑃^𝑇,𝐴₁=𝐴₀+𝐵₀𝐾−𝐵₀𝐾𝐺𝐼−𝑊₀𝐶₀𝐴₀−𝐺𝐶₀,𝐺_𝑤=0−𝐺𝑊₀,𝐶₀=𝐶₀0. (25)

Proof. The FO-LTI interval system (3)-(4) is asymptotically stable for any sensor faults if the FO system 𝐷^𝛼𝑥(𝑡)=𝐴𝑥(𝑡) is asymptotically stable. This is equivalent to that there exists a symmetric positive definite matrix 𝑃∈ℝ^𝑛×𝑛, such that 𝐴𝑃+𝑃𝐴^𝑇sin𝜃𝐴𝑃−𝑃𝐴^𝑇∗𝐴cos𝜃𝐴𝑃+𝑃^𝑇=sin𝜃𝐴₀𝑃+𝑃𝐴^𝑇₀sin𝜃𝐴₀𝑃−𝑃𝐴^𝑇₀∗cos𝜃𝐴₀𝑃+𝑃𝐴^𝑇₀𝐷sin𝜃+Sym^𝐴𝐵𝐹_𝐴𝐵𝐸_𝐴𝐵𝑃sin𝜃𝐷_𝐴𝐵𝐹_𝐴𝐵𝐸_𝐴𝐵𝑃−𝐷cos𝜃_𝐴𝐵𝐹_𝐴𝐵𝐸_𝐴𝐵𝑃cos𝜃𝐷_𝐴𝐵𝐹_𝐴𝐵𝐸_𝐴𝐵𝑃=sin𝜃𝐴₁𝑃+𝑃𝐴^𝑇₁sin𝜃𝐴₁𝑃−𝑃𝐴^𝑇₁∗cos𝜃𝐴₁𝑃+𝑃𝐴^𝑇₁𝐺sin𝜃+Sym^𝑤𝐿𝐶₀𝑃sin𝜃𝐺_𝑤𝐿𝐶₀𝑃−𝐺cos𝜃_𝑤𝐿𝐶₀𝑃cos𝜃𝐺_𝑤𝐿𝐶₀𝑃𝐷sin𝜃+Sym^𝐴𝐵𝐹_𝐴𝐵𝐸_𝐴𝐵𝑃sin𝜃𝐷_𝐴𝐵𝐹_𝐴𝐵𝐸_𝐴𝐵𝑃−𝐷cos𝜃_𝐴𝐵𝐹_𝐴𝐵𝐸_𝐴𝐵𝑃cos𝜃𝐷_𝐴𝐵𝐹_𝐴𝐵𝐸_𝐴𝐵𝑃sin𝜃<0. (26) From Lemma 4, one can have 𝐺Sym^𝑤𝐿𝐶₀𝑃sin𝜃𝐺_𝑤𝐿𝐶₀𝑃−𝐺cos𝜃_𝑤𝐿𝐶₀𝑃cos𝜃𝐺_𝑤𝐿𝐶₀𝑃()𝐺sin𝜃=Symsin𝜃_𝑤(cos𝜃)𝐺_𝑤(−cos𝜃)𝐺_𝑤(sin𝜃)𝐺_𝑤×𝐿00𝐿𝐶₀0𝑃0𝐶₀𝑃≤𝜖₁(sin𝜃)𝐺_𝑤𝐽(cos𝜃)𝐺_𝑤𝐽(−cos𝜃)𝐺_𝑤𝐽(sin𝜃)𝐺_𝑤𝐽×(sin𝜃)𝐺_𝑤𝐽(cos𝜃)𝐺_𝑤𝐽(−cos𝜃)𝐺_𝑤𝐽(sin𝜃)𝐺_𝑤𝐽^𝑇+𝜖₁⁻¹𝐶₀0𝑃0𝐶₀𝑃^𝑇𝐶₀0𝑃0𝐶₀𝑃,𝐷Sym^𝐴𝐵𝐹_𝐴𝐵𝐸_𝐴𝐵𝑃sin𝜃𝐷_𝐴𝐵𝐹_𝐴𝐵𝐸_𝐴𝐵𝑃−𝐷cos𝜃_𝐴𝐵𝐹_𝐴𝐵𝐸_𝐴𝐵𝑃cos𝜃𝐷_𝐴𝐵𝐹_𝐴𝐵𝐸_𝐴𝐵𝑃()𝐷sin𝜃=Symsin𝜃_𝐴𝐵(cos𝜃)𝐷_𝐴𝐵−(cos𝜃)𝐷_𝐴𝐵(sin𝜃)𝐷_𝐴𝐵𝐹_𝐴𝐵00𝐹_𝐴𝐵×𝐸_𝐴𝐵𝑃00𝐸_𝐴𝐵𝑃≤𝜖₂(sin𝜃)𝐷_𝐴𝐵(cos𝜃)𝐷_𝐴𝐵−(cos𝜃)𝐷_𝐴𝐵(sin𝜃)𝐷_𝐴𝐵×(sin𝜃)𝐷_𝐴𝐵(cos𝜃)𝐷_𝐴𝐵−(cos𝜃)𝐷_𝐴𝐵(sin𝜃)𝐷_𝐴𝐵^𝑇+𝜖₂⁻¹𝐸_𝐴𝐵𝑃00𝐸_𝐴𝐵𝑃^𝑇𝐸_𝐴𝐵𝑃00𝐸_𝐴𝐵𝑃. (27)(28) Substituting (27) and (28) into (26), one can get 𝐴𝑃+𝑃𝐴^𝑇sin𝜃𝐴𝑃−𝑃𝐴^𝑇∗𝐴cos𝜃𝐴𝑃+𝑃^𝑇≤sin𝜃𝐴₁𝑃+𝑃𝐴^𝑇₁sin𝜃𝐴₁𝑃−𝑃𝐴^𝑇₁∗cos𝜃𝐴₁𝑃+𝑃𝐴^𝑇₁sin𝜃+𝜖₁(sin𝜃)𝐺_𝑤𝐽(cos𝜃)𝐺_𝑤𝐽(−cos𝜃)𝐺_𝑤𝐽(sin𝜃)𝐺_𝑤𝐽×(sin𝜃)𝐺_𝑤𝐽(cos𝜃)𝐺_𝑤𝐽(−cos𝜃)𝐺_𝑤𝐽(sin𝜃)𝐺_𝑤𝐽^𝑇+𝜖₂(sin𝜃)𝐷_𝐴𝐵(cos𝜃)𝐷_𝐴𝐵−(cos𝜃)𝐷_𝐴𝐵(sin𝜃)𝐷_𝐴𝐵×(sin𝜃)𝐷_𝐴𝐵(cos𝜃)𝐷_𝐴𝐵−(cos𝜃)𝐷_𝐴𝐵(sin𝜃)𝐷_𝐴𝐵^𝑇+𝜖₁⁻¹𝐶₀0𝑃0𝐶₀𝑃^𝑇𝐶₀0𝑃0𝐶₀𝑃+𝜖₂⁻¹𝐸_𝐴𝐵𝑃00𝐸_𝐴𝐵𝑃^𝑇𝐸_𝐴𝐵𝑃00𝐸_𝐴𝐵𝑃. (29) Taking (29) into account and using the Schur complement of (24), one obtains 𝐴𝑃+𝑃𝐴^𝑇sin𝜃𝐴𝑃−𝑃𝐴^𝑇∗𝐴cos𝜃𝐴𝑃+𝑃^𝑇sin𝜃<0. (30) It follows from the above inequality (30) and Lemma 3 that |arg(spec(𝐴))|>𝛼𝜋/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≤𝛼<2 is presented in the following theorem.

Theorem 6. Given positive scalar constants 𝜂_𝑖(𝑖=1,2), the FO system (20) with order 1≤𝛼<2 and 𝜃=𝜋−𝜋𝛼/2 is asymptotically stable if there exist the matrices 𝑋₁>0, 𝐾, 𝐺; the following conditions are satisfied: Ω₁Ω₂Ω₃Ω₄Ω₅∗−𝜂₁⁻¹𝐼000∗∗−𝜂₂⁻¹𝐼00∗∗∗−𝜂₁𝐼0∗∗∗∗−𝜂₂𝐼<0, (31) where Λ₁=𝐴₀+𝐴^𝑇₀+𝐵₀𝐾+𝐾^𝑇𝐵^𝑇₀−𝐵₀𝐾+𝐶^𝑇₀𝐼−𝑊₀^𝑇𝐺^𝑇𝐺𝐼−𝑊₀𝐶₀−𝐾^𝑇𝐵^𝑇₀𝑋₁𝐴₀+𝐴^𝑇₀𝑋₁−𝐺𝐶₀−𝐶^𝑇₀𝐺^𝑇,Λ₂=𝐴₀−𝐴^𝑇₀+𝐵₀𝐾−𝐾^𝑇𝐵^𝑇₀−𝐵₀𝐾−𝐶^𝑇₀𝐼−𝑊₀^𝑇𝐺^𝑇𝐺𝐼−𝑊₀𝐶₀+𝐾^𝑇𝐵^𝑇₀𝑋₁𝐴₀−𝐴^𝑇₀𝑋₁−𝐺𝐶₀+𝐶^𝑇₀𝐺^𝑇,Ω₁=(sin𝜃)Λ₁(cos𝜃)Λ₂∗(sin𝜃)Λ₁,Λ₃=0−𝐺𝑊₀𝐽,Ω₂=𝜂₁(sin𝜃)Λ₃𝜂₁(cos𝜃)Λ₃−𝜂₁(cos𝜃)Λ₃𝜂₁(sin𝜃)Λ₃,Λ₄=𝐷_𝐴𝐷_𝐵𝑋₁𝐷_𝐴𝑋₁𝐷_𝐵,Ω₃=𝜂₂(sin𝜃)Λ₄𝜂₂(cos𝜃)Λ₄−𝜂₂(cos𝜃)Λ₄𝜂₂(sin𝜃)Λ₄,Λ₅=𝐶^𝑇₀0,Ω₄=Λ₅00Λ₅,Ω₅=𝐸^𝑇_𝐴𝐵00𝐸^𝑇_𝐴𝐵. (6) Furthermore a desired FO observer-based output feedback controller is given in the form of (14) with parameter as follows: 𝐺=𝑋₁⁻¹𝐺. (33)

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 𝑃∈ℝ^𝑛×𝑛 and positive scalar constants 𝜂_𝑖, (𝑖=1,2) such that 𝐴𝑃+𝑃𝐴^𝑇sin𝜃𝐴𝑃−𝑃𝐴^𝑇∗𝐴cos𝜃𝐴𝑃+𝑃^𝑇≤sin𝜃𝐴₁𝑃+𝑃𝐴^𝑇₁sin𝜃𝐴₁𝑃−𝑃𝐴^𝑇₁∗cos𝜃𝐴₁𝑃+𝑃𝐴^𝑇₁sin𝜃+𝜂₁(sin𝜃)𝐺_𝑤𝐽(cos𝜃)𝐺_𝑤𝐽(−cos𝜃)𝐺_𝑤𝐽(sin𝜃)𝐺_𝑤𝐽×(sin𝜃)𝐺_𝑤𝐽(cos𝜃)𝐺_𝑤𝐽(−cos𝜃)𝐺_𝑤𝐽(sin𝜃)𝐺_𝑤𝐽^𝑇+𝜂₂(sin𝜃)𝐷_𝐴𝐵(cos𝜃)𝐷_𝐴𝐵−(cos𝜃)𝐷_𝐴𝐵(sin𝜃)𝐷_𝐴𝐵×(sin𝜃)𝐷_𝐴𝐵(cos𝜃)𝐷_𝐴𝐵−(cos𝜃)𝐷_𝐴𝐵(sin𝜃)𝐷_𝐴𝐵^𝑇+𝜂₁⁻¹𝐶₀0𝑃0𝐶₀𝑃^𝑇𝐶₀0𝑃0𝐶₀𝑃+𝜂₂⁻¹𝐸_𝐴𝐵𝑃00𝐸_𝐴𝐵𝑃^𝑇𝐸_𝐴𝐵𝑃00𝐸_𝐴𝐵𝑃<0. (34) Introducing the following nonsingular matrix 𝑋=𝐼00𝑋₁. (35) Let 𝑃=𝑋⁻¹; (36) then, by some calculation, we have 𝑃>0. (37) Now, pre- and postmultiplying the inequality in (34) by diag(𝑋,𝑋) and diag(𝑋,𝑋), respectively, set 𝑋𝐺=𝐺; then we have Ω₁++𝜂₁Ω₂Ω^𝑇₂+𝜂₂Ω₃Ω^𝑇₃+𝜂₁⁻¹Ω₄Ω^𝑇₄+𝜂₂⁻¹Ω₅Ω^𝑇₅<0. (38) Inequality (38) is equivalent to (31) by the Schur complement. This completes the proof.

3.2. 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<𝛼≤1.

Theorem 7. Given the controller gain matrix 𝐾 and the observer gain matrix 𝐺, the FO system (20) with order 0<𝛼≤1 is robustly asymptotically stable for any sensor faults if there exist two real symmetric positive definite matrices 𝑃_𝑘1∈ℝ^𝑛×𝑛, 𝑘=1,2, two skew-symmetric matrices 𝑃_𝑘2∈ℝ^𝑛×𝑛, 𝑘=1,2, and scalar constants 𝜀_1𝑖𝑗>0, 𝜀_2𝑖𝑗>0(𝑖,𝑗=1,2), such that Σ₁₁Σ₁₂Σ₁₃∗−Σ₂₂0∗∗−Σ₃₃𝑃<0,₁₁𝑃₁₂−𝑃₁₂𝑃₁₁𝑃>0,²¹𝑃₂₂−𝑃₂₂𝑃₂₁>0, (39)(40) where

𝐴₁=𝐴₀+𝐵₀𝐾−𝐵₀𝐾𝐺𝐼−𝑊₀𝐶₀𝐴₀−𝐺𝐶₀,𝐺_𝑤=0−𝐺𝑊₀,𝐶₀=𝐶₀0,Σ₁₁=²²_𝑖=1𝑗=1SymΘ_𝑖𝑗⊗𝐴₁𝑃_𝑖𝑗+𝜀_1𝑖𝑗𝐼₂⊗𝐺_𝑤𝐿𝐺_𝑤𝐿^𝑇+𝜀_2𝑖𝑗𝐼₂⊗𝐷_𝐴𝐵𝐷^𝑇_𝐴𝐵,Σ₁₂=𝐼₂⊗𝐶₀𝑃₁₁^𝑇𝐼₂⊗𝐶₀𝑃₁₂^𝑇𝐼₂⊗𝐶₀𝑃₂₁^𝑇𝐼₂⊗𝐶₀𝑃₂₂^𝑇,Σ₁₃=𝐼₂⊗𝐸_𝐴𝐵𝑃₁₁^𝑇𝐼₂⊗𝐸_𝐴𝐵𝑃₁₂^𝑇𝐼₂⊗𝐸_𝐴𝐵𝑃₂₁^𝑇𝐼₂⊗𝐸_𝐴𝐵𝑃₂₂^𝑇,Σ₂₂=diag𝜀₁₁₁,𝜀₁₁₂,𝜀₁₂₁,𝜀₁₂₂⊗𝐼_2𝑛,Σ₃₃=diag𝜀₂₁₁,𝜀₂₁₂,𝜀₂₂₁,𝜀₂₂₂⊗𝐼_2𝑛,Θ₁₁=𝜋sin2𝛼𝜋−cos2𝛼𝜋cos2𝛼𝜋sin2𝛼,Θ₁₂=𝜋cos2𝛼𝜋sin2𝛼𝜋−sin2𝛼𝜋cos2𝛼,Θ₂₁=𝜋sin2𝛼𝜋cos2𝛼𝜋−cos2𝛼𝜋sin2𝛼,Θ₂₂=𝜋−cos2𝛼𝜋sin2𝛼𝜋−sin2𝛼𝜋−cos2𝛼. (41)

Proof. Suppose that (39)-(40) hold. It follows from (20) that SymΘ²²_{𝑖=1𝑗=1𝑖𝑗}⊗𝐴𝑃_𝑖𝑗=²²_𝑖=1𝑗=1SymΘ_𝑖𝑗⊗𝐴₁𝑃_𝑖𝑗+Θ_𝑖𝑗⊗𝐺_𝑤𝐿𝐶₀𝑃_𝑖𝑗+Θ_𝑖𝑗⊗𝐷_𝐴𝐵𝐹_𝐴𝐵𝐸_𝐴𝐵𝑃_𝑖𝑗. (42) Note that (13) and Lemma 1 imply 𝐿𝐿^𝑇≤𝐽²,𝐹_𝐴𝐵𝐹^𝑇_𝐴𝐵≤𝐼. (43) It follows from (43) that 𝐼₂⊗𝐿𝐼₂⊗𝐿^𝑇≤𝐽²,𝐼₂⊗𝐹_𝐴𝐵𝐼₂⊗𝐹_𝐴𝐵^𝑇≤𝐼. (44) Note that Θ_𝑖𝑗Θ^𝑇_𝑖𝑗=𝐼₂(𝑖,𝑗=1,2); it follows from (44) and Lemma 4 that for any real scalars 𝜀_1𝑖𝑗>0 and 𝜀_2𝑖𝑗>0(𝑖,𝑗=1,2)SymΘ_𝑖𝑗⊗𝐺_𝑤𝐿𝐶₀𝑃_𝑖𝑗=SymΘ_𝑖𝑗⊗𝐺_𝑤𝐼₂⊗𝐿𝐼₂⊗𝐶₀𝑃_𝑖𝑗≤𝜀_1𝑖𝑗Θ_𝑖𝑗⊗𝐺_𝑤𝐼₂⊗𝐿𝐼₂⊗𝐿^𝑇Θ_𝑖𝑗⊗𝐺_𝑤^𝑇+𝜀⁻¹_1𝑖𝑗I₂⊗𝐶₀𝑃_𝑖𝑗^𝑇𝐼₂⊗𝐶₀𝑃_𝑖𝑗≤𝜀_1𝑖𝑗𝐼₂⊗𝐺_𝑤𝐽𝐺_𝑤𝐽^𝑇+𝜀⁻¹_1𝑖𝑗𝐼₂⊗𝐶₀𝑃_𝑖𝑗^𝑇𝐼₂⊗𝐶₀𝑃_𝑖𝑗,SymΘ_𝑖𝑗⊗𝐷_𝐴𝐵𝐹_𝐴𝐵𝐸_𝐴𝐵𝑃_𝑖𝑗=SymΘ_𝑖𝑗⊗𝐷_𝐴𝐵𝐼₂⊗𝐹_𝐴𝐵𝐼₂⊗𝐸_𝐴𝐵𝑃_𝑖𝑗≤𝜀_2𝑖𝑗Θ_𝑖𝑗⊗𝐷_𝐴𝐵𝐼₂⊗𝐹_𝐴𝐵𝐼₂⊗𝐹_𝐴𝐵^𝑇Θ_𝑖𝑗⊗𝐷_𝐴𝐵^𝑇+𝜀⁻¹_2𝑖𝑗𝐼₂⊗𝐸_𝐴𝐵𝑃_𝑖𝑗^𝑇𝐼₂⊗𝐸_𝐴𝐵𝑃_𝑖𝑗≤𝜀_2𝑖𝑗𝐼₂⊗𝐷_𝐴𝐵𝐷^𝑇_𝐴𝐵+𝜀⁻¹_2𝑖𝑗𝐼₂⊗𝐸_𝐴𝐵𝑃_𝑖𝑗^𝑇𝐼₂⊗𝐸_𝐴𝐵𝑃_𝑖𝑗. (45)(46) Substituting (45) and (46) into (42), one has SymΘ²²_{𝑖=1𝑗=1𝑖𝑗}⊗𝐴𝑃_𝑖𝑗≤²²_𝑖=1𝑗=1SymΘ_𝑖𝑗⊗𝐴₁𝑃_𝑖𝑗+𝜀_1𝑖𝑗𝐼₂⊗𝐺_𝑤𝐽𝐺_𝑤𝐽^𝑇+𝜀_2𝑖𝑗𝐼₂⊗𝐷_𝐴𝐵𝐷^𝑇_𝐴𝐵+²²_𝑖=1𝑗=1𝜀⁻¹_1𝑖𝑗𝐼₂⊗𝐶₀𝑃_𝑖𝑗^𝑇𝐼₂⊗𝐶₀𝑃_𝑖𝑗+𝜀⁻¹_2𝑖𝑗𝐼₂⊗𝐸_𝐴𝐵𝑃_𝑖𝑗^𝑇𝐼₂⊗𝐸_𝐴𝐵𝑃_𝑖𝑗. (47) Taking (47) into account and using the Schur complement of (39), one obtains SymΘ²²_{𝑖=1𝑗=1𝑖𝑗}⊗𝐴𝑃_𝑖𝑗<0. (48) It follows from the above inequality (48) and Theorem 1 of [27] that the uncertain FO-LTI interval system with 0<𝛼≤1 is asymptotically stable.

The observer-based FO output feedback control problem for FO systems (20) with order 0<𝛼≤1 is presented in the following theorem.

Theorem 8. Given positive scalar constants 𝜇_1𝑖, 𝜇_2𝑖(𝑖=1,2), The FO system (20) with order 0<𝛼≤1 is asymptotically stable if there exist the matrices 𝐺₁, 𝐾 and symmetric matrix 𝑌₁>0; the following condition is satisfied: SymΘ²_𝑖=1𝑖1⊗𝐴₂Ξ₁Ξ₂Ξ₃Ξ₄∗−Ξ₅000∗∗−Ξ₆00∗∗∗−Ξ₇0∗∗∗∗−Ξ₈<0, (49) where 𝐴₂=𝐴₀+𝐵₀𝐾−𝐵₀𝐾𝐺₁𝐼−𝑊₀𝐶₀𝑌₁𝐴₀−𝐺₁𝐶₀,Λ₃=0−𝐺₁𝑊₀𝐽,Λ₄=𝐷_𝐴𝐷_𝐵𝑌₁𝐷_𝐴𝑌₁𝐷_𝐵,Ξ₁=𝐼₂⊗Λ₃𝐼₂⊗Λ₃,Ξ₂=𝐼₂⊗Λ₄𝐼₂⊗Λ₄,Ξ₃=𝐼₂⊗𝐶^𝑇₀𝐼₂⊗𝐶^𝑇₀,Ξ₄=𝐼₂⊗𝐸^𝑇_𝐴𝐵𝐼₂⊗𝐸^𝑇_𝐴𝐵,Ξ₅=diag𝜇⁻¹₁₁⊗𝐼_2𝑛,𝜇⁻¹₁₂⊗𝐼_2𝑛,Ξ₆=diag𝜇⁻¹₂₁⊗𝐼_2𝑛,𝜇⁻¹₂₂⊗𝐼_2𝑛,Ξ₇=diag𝜇₁₁⊗𝐼_2𝑛,𝜇₁₂⊗𝐼_2𝑛,Ξ₈=diag𝜇₂₁⊗𝐼_2𝑛,𝜇₂₂⊗𝐼_2𝑛. (50) Furthermore a desired observer-based FO output feedback controller is given in the form of (14) with parameter as follows: 𝐺=𝑌₁⁻¹𝐺₁. (51)

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 𝑃_𝑘1∈ℝ^𝑛×𝑛, 𝑘=1,2, and two skew-symmetric matrices 𝑃_𝑘2∈ℝ^𝑛×𝑛, 𝑘=1,2, such that SymΘ²²_{𝑖=1𝑗=1𝑖𝑗}⊗𝐴𝑃_𝑖𝑗<0. (52) By setting 𝑃₁₁=𝑃₂₁=𝑄, 𝑃₁₂=𝑃₂₂=0 in (52), we can get that if SymΘ²_𝑖=1𝑖1⊗𝐴𝑄<0, (53) 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 𝑄∈ℝ^𝑛×𝑛 and positive real scalars 𝜇_1𝑖 and 𝜇_2𝑖(𝑖=1,2) such that SymΘ²_𝑖=1𝑖1⊗≤𝐴𝑄_2𝑖=1SymΘ_𝑖1⊗𝐴₁𝑄+𝜇_1𝑖𝐼₂⊗𝐺_𝑤𝐽,𝐼₂⊗𝐺_𝑤𝐽^𝑇+𝜇_2𝑖𝐼₂⊗𝐷_𝐴𝐵𝐼₂⊗𝐷_𝐴𝐵^𝑇+²_𝑖=1𝜇⁻¹_1𝑖𝐼₂⊗𝐶₀𝑄^𝑇𝐼₂⊗𝐶₀𝑄+𝜇⁻¹_2𝑖𝐼₂⊗𝐸_𝐴𝐵𝑄^𝑇𝐼₂⊗𝐸_𝐴𝐵𝑄<0. (54) Introducing the following nonsingular matrix 𝑌=𝐼00𝑌₁. (55) Let 𝑄=𝑌⁻¹>0. (56) Now, pre- and postmultiplying the inequality in (54) by 𝐼₂⊗𝑌 and 𝐼₂⊗𝑌, respectively, set 𝑌𝐺=𝐺₁; then we have SymΘ²_𝑖=1𝑖1⊗𝑌𝐴₁+𝜇_1𝑖𝐼₂⊗Λ₃,𝐼₂⊗Λ₃^𝑇+𝜇_2𝑖𝐼₂⊗𝑋𝐷_𝐴𝐵𝐼₂⊗𝑋𝐷_𝐴𝐵^𝑇+²_𝑖=1𝜇⁻¹_1𝑖𝐼₂⊗𝐶₀^𝑇𝐼₂⊗𝐶₀+𝜇⁻¹_2𝑖𝐼₂⊗𝐸_𝐴𝐵^𝑇𝐼₂⊗𝐸_𝐴𝐵<0. (57) 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 0<𝛼<2 with the following parameters 𝛼=0.8,𝐴₀=,𝐵−100.10.01₀=1.8001,𝐶₀=,𝐷10.101_𝐴=,𝐷0.0010.00100000.0010.001_𝐵=,𝐸0.0020.00200000.0020.002_𝐴=0.00100.001000.00100.001^𝑇,𝐸_𝐵=0.00200.002000.00200.002^𝑇,𝑤_𝑙=0.2000.5,𝑤_𝑢=.1.2001.5 (58) 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 𝜂₁=0.1,𝜂₂=0.1, (59) and the initial state 𝑥₀=−0.10.1^𝑇. Then, using Matlab Linear Matrix Inequality (LMI) Control Toolbox to solve the LMI (49), we can obtain the solution as follows: 𝑌₁=10³×,,.0.0215−0.2798−0.2798−2.7749𝐺=−0.1054−0.03880.0102−0.0002𝐾=−16.2663−0.0800−0.0144−27.9000 (60) While, for the FO system (3)-(4) with the following parameters: 𝛼=1.6,𝐴₀=,𝐵−0.1000.01₀=−500−5,𝐶₀=,𝐷10.101_𝐴=,𝐷0.0010.00100000.0010.001_𝐵=,𝐸0.0020.00200000.0020.002_𝐴=0.00100.001000.00100.001^𝑇,𝐸_𝐵=0.00200.002000.00200.002^𝑇,𝑤_𝑙=0.2000.5,𝑤_𝑢=.1.2001.5 (61) Then, we choose 𝜇₁₁=0.1,𝜇₁₂=0.1,𝜇₂₁=0.1,𝜇₂₂=0.1, (62) and the initial state 𝑥₀=0.1−0.1^𝑇. Also using Matlab LMI Control Toolbox to solve the LMI (31), we can obtain the solution as follows: 𝑋₁=38.63690.78500.7850−434.9420,𝐺=0.01090.00070.00020.0008,𝐾=0.5454−0.00120.00430.5240. (63)(64)(65) With the observer-based FO output feedback controller, the closed-loop system is stable. The state response 𝑥₁(𝑡) of the FO systems of order 0<𝛼≤1 and 1≤𝛼<2 are given in Figures 1 and 2, respectively, while the corresponding control input 𝑢₁(𝑡) are shown in Figures 3 and 4. The state response 𝑥₂(𝑡) of the FO systems of order 0<𝛼≤1 and 1≤𝛼<2 are given in Figures 5 and 6, respectively. While Figures 7 and 8 show the corresponding control input 𝑢₂(𝑡).

Figure 1 

State response 𝑥1(𝑡)  (0<𝛼≤1).

Figure 2 

State response 𝑥1(𝑡)  (1≤𝛼<2).

Figure 3 

Control input 𝑢1(𝑡)  (0<𝛼≤1).

Figure 4 

Control input 𝑢1(𝑡)  (1≤𝛼<2).

Figure 5 

State response 𝑥2(𝑡)  (0<𝛼≤1).

Figure 6 

State response 𝑥2(𝑡)  (1≤𝛼<2).

Figure 7 

Control input 𝑢2(𝑡)  (0<𝛼≤1).

Figure 8 

Control input 𝑢2(𝑡)  (1≤𝛼<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<𝛼<2 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 0<𝛼<2 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).