Adaptive Fault-Tolerant Control for Flight Systems with Input Saturation and Model Mismatch

Abstract

A novel scheme for fault-tolerant control is proposed in this paper, in which model reference adaptive control method is incorporated with control allocation to cope with simultaneous actuator failures, input saturation, and model mismatch in the flight system. In order to reduce performance degradation caused by actuator failures, the proposed scheme redistributes the control signal to healthy actuators and updates the weighting matrix based on actuator effectiveness. Because of saturation errors resulting from actuator constraints and model mismatch caused by abnormal changes in the system, the original reference model may not be appropriate. Under this circumstance, an adaptive reference model which can also provide satisfactory performance is designed. Simulations of a flight control example are given to illustrate the effectiveness of the proposed scheme.

1. Introduction

Actuator failures may adversely affect the stability and performance of flight systems. Fault-tolerant control (FTC) has been studied as a method for accommodating failures. In the past decades, a number of approaches have been proposed for FTC, including sliding mode control-based designs [1, 2], variable structure control-based designs [3], learning-based approaches [4], and adaptive control-based designs [5].

The availability of actuator redundancy is an important element of FTC. Control allocation (CA) is an approach that can effectively manage redundancy by distributing virtual control law requirements to redundant actuators in the most efficient manner [6]. There is an extensive literature on CA approaches and applications. Enns presented the linear and quadratic programming approaches in detail in [7]. Härkegård and Glad compared control allocation with optimal control for solving actuator redundancy in [8]. Several CA approaches taking into account actuator limits have been discussed [9–11]. Kishore et al. gave an algorithm that updated the weighting matrix to deal with actuator limits in [9]. However, in the event of failure, only depending on updating the weighting matrix sometimes may be invalid.

In addition, flight systems may experience other disruptive factors such as model mismatch. Model mismatch can be caused by icing or damage to the aircraft. Joshi et al. first considered this problem in [12] and further developed his work in [13]. Although the previous work has achieved satisfactory tracking performance, none of them considered actuator limits.

In this paper, we develop a novel scheme combining CA with model reference adaptive control (MRAC) for flight systems with simultaneous actuator failures, model mismatch, and input saturation. On the basis of the existing literature, this paper proposes an improved control allocation (ICA) method that updates the weighting matrix based on actuator effectiveness. In addition to updating the weighting matrix, this paper employs the positive 𝜇-modification method [14] to impose constraints on actuators. MRAC can guarantee the closed-loop stability of the system and reduce the error caused by actuator failures. Because of input saturation and model mismatch, the original reference model may not be appropriate. We have designed an adaptive reference model that can provide satisfactory performance under these circumstances. Compared with the traditional CA methods, simulation results show that the new scheme provides better tracking performance. The control structure is shown in Figure 1. This paper focuses on the tracking control on the single system. In the recent study, the coordination control of multiagent systems has attracted considerable attention. Several approaches and applications have been developed in [15–17]. We will explore the FTC of multiple flight systems in our future study.

Figure 1 

The overall control structure.

This paper is organized as follows. Section 2 presents the problem statement. Section 3 gives a detailed description of the novel adaptive fault-tolerant control scheme. Simulation results of a flight control example are given in Section 4. Section 5 is concluding remarks.

2. Problem Statement

Consider a linear time-invariant system in the presence of model mismatch and actuator failures described by 𝑥̇(𝑡)=𝐴+𝛿_𝑎𝑥(𝑡)+𝐵_𝑢(𝑡),𝑦(𝑡)(𝑡),Δ𝑢=𝐶𝑥 (1) where 𝑥(𝑡)∈ℜ^𝑛 is the state vector, 𝑢(𝑡)∈ℜ^𝑚 is the actual control input, 𝑦(𝑡)∈ℜ^𝑝 is the output, and 𝐴,𝐵_𝑢,𝐶 are constant matrices of appropriate dimensions. The matrix 𝛿_𝑎∈ℜ^𝑛×𝑛 denotes unknown parameter deviations or model mismatch caused by icing, modeling errors, or damage. The actuator effectiveness matrix Δ∈ℜ^𝑚×𝑚 is a piecewise constant uncertain matrix which can be expressed as Δ=diag(𝛿₁,𝛿₂,…,𝛿_𝑚), where 0≤𝛿_𝑖≤1,𝑖=1,2,…,𝑚, indicates the effectiveness of the 𝑖th actuator. For example, 𝛿_𝑖=1 indicates that the 𝑖th actuator is working well; 𝛿_𝑖=0 implies that the 𝑖th actuator has damaged completely; 0<𝛿_𝑖<1 means that the 𝑖th actuator has partially lost its effectiveness.

Assumption 1. Each element of the control input 𝑢(𝑡) is bounded by a constant such that 𝑢_𝑖≤𝑢_max𝑖∀𝑡>0,𝑖=1,2,…,𝑚. (2)

Assumption 2. There is sufficient actuator redundancy in the system so that the input matrix 𝐵_𝑢 can be factorized into a full column rank matrix 𝐵_𝑣∈ℜ^𝑛×𝑝 and a full row rank matrix 𝐵∈ℜ^𝑝×𝑚; that is, 𝐵_𝑢=𝐵_𝑣𝐵. (3)

Assumption 3. (𝐴,𝐵_𝑢Δ) is controllable.

Assumption 4. For the nominal system with Δ=𝐼 and 𝛿_𝑎=0, there exist gains 𝐾₁∈ℜ^𝑛×𝑝 and 𝐾₂∈ℜ^𝑝×𝑝 satisfying the following matching conditions: 𝐴+𝐵_𝑣𝐾^𝑇₁=𝐴_𝑚;𝐵_𝑣𝐾₂=𝐵_𝑚, (4) where 𝐴_𝑚∈ℜ^𝑛×𝑛 is Hurwitz and 𝐵_𝑚∈ℜ^𝑛×𝑝.

Based on Assumption 1, the static actuator model can be defined as follows [14]: 𝑢_𝑖(𝑡)=𝑢_max𝑖𝑢sat^𝑐𝑖(𝑡)𝑢_max𝑖=𝑢_𝑐𝑖(𝑡),𝑢_𝑐𝑖(𝑡)≤𝑢_max𝑖,𝑢_max𝑖sgn𝑢_𝑐𝑖(𝑡),𝑢_𝑐𝑖(𝑡)>𝑢_max𝑖,𝑖=1,…,𝑚, (5) where 𝑢_𝑐𝑖(𝑡) denotes the 𝑖th element of the commanded control input 𝑢_𝑐(𝑡). Based on the positive 𝜇-modification method [14], the commanded control input 𝑢_𝑐(𝑡) can be obtained by 𝑢_𝑐(𝑡)=𝑢_𝑎(𝑡)+𝜇𝑢_𝑐(𝑡), (6) where 𝜇∈ℜ^𝑚×𝑚 is a diagonal matrix of design constants 𝜇₁,…,𝜇_𝑚≥0, 𝑢_𝑎(𝑡) is the adaptive control input, and 𝑢_𝑐(𝑡)=[𝑢_𝑐1(𝑡),…,𝑢_𝑐𝑚(𝑡)]^𝑇, 𝑢_𝑐𝑖(𝑡)=𝑢^𝛿𝑖_max𝑖sat(𝑢_𝑐𝑖(𝑡)/𝑢^𝛿𝑖_max𝑖)−𝑢_𝑐𝑖(𝑡),𝑖=1,…,𝑚.

In addition, the saturation error can be defined as 𝑢(𝑡)=𝑢(𝑡)−𝑢_𝑐(𝑡) (7) which may lead to performance degradation or even system instability.

Based on Assumption 4, we can define the conventional reference model as 𝑥̇_𝑚(𝑡)=𝐴_𝑚𝑥_𝑚(𝑡)+𝐵_𝑚𝑟(𝑡), (8) where 𝑥_𝑚(𝑡)∈ℜ^𝑛 is the reference model state and 𝑟(𝑡)∈ℜ^𝑝 is a bounded reference input. Optimal and robust control theory can be employed to design the gain matrices 𝐾₁, 𝐾₂. However, due to model mismatch 𝛿_𝑎, (𝐴+𝛿_𝑎+𝐵_𝑣𝐾^𝑇₁) will not have the same structure as 𝐴_𝑚 for any 𝐾₁. Therefore, stability and asymptotic tracking performance can no longer be guaranteed.

In response, this paper designs an adaptive reference model while taking saturation error 𝑢 and model mismatch 𝛿_𝑎 into consideration. The control objective is to develop a novel adaptive fault-tolerant control scheme, for the faulty system given by (1) with simultaneous actuator failures, model mismatch, and input saturation, to achieve the desired tracking performance.

3. Adaptive Fault-Tolerant Control

In order to achieve the desired control objective, we propose an adaptive fault-tolerant control scheme in this section. The scheme mainly incorporates an improved control allocation method and an adaptive virtual controller design.

3.1. Improved Control Allocation Method

Substituting (6) and (7) into (1) yields the following closed-loop system dynamics: 𝑥̇(𝑡)=𝐴+𝛿_𝑎𝑥(𝑡)+𝐵_𝑓𝑢_𝑎(𝑡)+𝐵_𝑓𝑢_𝑎(𝑡), (9) where 𝐵_𝑓=𝐵_𝑢Δ𝑢_𝑎(𝑡)=𝜇𝑢_𝑐(𝑡)+𝑢(𝑡). (10)

Based on Assumption 2, we can get 𝐵_𝑓𝑢_𝑎(𝑡)=𝐵_𝑣𝐵Δ𝑢_𝑎(𝑡)=𝐵_𝑣𝑣(𝑡), (11) where 𝑣(𝑡)∈ℜ^𝑝 is a virtual control input that represents the total control effort produced by actuators.

As distinct from conventional CA problem, the matrix 𝐵_𝑓 incorporates fault information. Generally, CA can be formulated as an optimization problem as follows [1]: min_{𝑢𝑎(𝑡)}𝑢^𝑇_𝑎𝑊𝑢_𝑎,subjectto(11), (12) where 𝑊=diag(𝑤₁,𝑤₂,…,𝑤_𝑚) is a positive definite weighting matrix and the scalar 𝑤_𝑖 represents the weight of the 𝑖th actuator.

In contrast to the preceding CA results, the weighting matrix 𝑊 can be updated online based on actuator effectiveness in this paper. Suppose that the 𝑖th actuator has partially lost effectiveness; that is, Δ=diag(1,…,1,𝛿_𝑖,1,…,1), 0<𝛿_𝑖<1. Let the 𝑖th diagonal element of 𝑊 be perturbed from 𝑤_𝑖 to 𝜅²_𝑖𝑤_𝑖, where the scalar 𝜅_𝑖 can be obtained by 𝜅_𝑖=1/𝛿_𝑖. Then the new weighting matrix becomes 𝑊=𝜗𝑊𝜗, where 𝜗=diag(1,…,1,𝜅_𝑖,1,…,1). Consequently, the solution to (11) can be formulated as 𝑢_𝑎(𝑡)=𝑊⁻¹𝐵^𝑇_𝑓𝐵_𝑓𝑊⁻¹𝐵^𝑇_𝑓⁻¹𝑣(𝑡). (13)

Usually the initial value of the weighting matrix 𝑊 is selected based on the priority of actuators. Once actuator failures occur, 𝑊 will be updated. As 𝛿_𝑖→0, 𝜅_𝑖→∞, the associated component 𝑢_𝑖 in (12) is therefore weighted heavily since 𝑤_𝑖 becomes large. If the 𝑖th actuator has damaged completely, the value of 𝑤_𝑖 will be infinity. Equation (13) is able to distribute the virtual control signal to the remaining physical actuators based on actuator effectiveness. However, due to the failure, only depending on the adaption of 𝑊 to avoid saturation sometimes may be invalid. Therefore, this paper employs a positive 𝜇-modification method to ensure that all control inputs lie within the bounds.

The following lemma gives the algorithm of the positive 𝜇-modification method.

Lemma 5 (see [14]). If 𝜇₁,…,𝜇_𝑚≥0, then the solution to (6) is given by a convex combination of 𝑢_𝑎(𝑡) and 𝑢^𝛿_maxsat(𝑢_𝑎(𝑡)/𝑢^𝛿_max) for all 𝑡>0: 𝑢_𝑐(𝑡)=𝐼_𝑞+𝜇⁻¹𝑢_𝑎(𝑡)+𝜇𝑢^𝛿_maxsat𝑢^𝛿_max⁻¹𝑢_𝑎(𝑡) (14) which can also be represented component-wise in the following form: 𝑢_𝑐𝑖(𝑡)=𝑢_𝑎𝑖(𝑡),𝑢_𝑎𝑖(𝑡)≤𝑢^𝛿𝑖_max𝑖,11+𝜇_𝑖𝑢_𝑎𝑖(𝑡)+𝜇_𝑖𝑢^𝛿𝑖_max𝑖,𝑢_𝑎𝑖(𝑡)>𝑢^𝛿𝑖_max𝑖,11+𝜇_𝑖𝑢_𝑎𝑖(𝑡)−𝜇_𝑖𝑢^𝛿𝑖_max𝑖,𝑢_𝑎𝑖(𝑡)<−𝑢^𝛿𝑖_max𝑖,𝑖=1,…,𝑚. (15)

Remark 6. Detailed discussion of the above lemma can be found in [14].

3.2. Adaptive Virtual Controller Design

The adaptive controller is designed as 𝑣(𝑡)=𝐾₁𝑥(𝑡)+𝐾₂𝑟(𝑡), (16) where 𝐾₁ and 𝐾₂ are the estimates of 𝐾₁ and 𝐾₂, respectively.

Substituting (11) and (16) into (9) yields the closed-loop system 𝑥̇=𝐴+𝐵_𝑣𝐾₁+𝛿_𝑎𝑥+𝐵_𝑣𝐾₂𝑟+𝐵_𝑣𝐾₁𝑥+𝐵_𝑣𝐾₂𝑥+𝐵_𝑣𝐵_𝑓𝑢_𝑎, (17) where 𝐾₁=𝐾₁−𝐾₁, 𝐾₂=𝐾₂−𝐾₂.

In order to compensate for the saturation error 𝑢_𝑎, we design an adaptive compensation controller 𝐾₃∈ℜ^𝑝×𝑝. The initial value of 𝐾₃ is defined as 𝐾₃ and satisfies 𝐾₂𝐾₃=𝐼. (18)

Using (4) and (18), the closed-loop system in (17) can be modified as 𝑥̇=𝐴_𝑚+𝛿_𝑎𝑥+𝐵_𝑚𝑟+𝐾₃𝐵_𝑓𝑢_𝑎+𝐵_𝑣𝐾₁𝑥+𝐵_𝑣𝐾₂𝑟. (19)

The modified closed-loop system dynamics lead to consideration of the following adaptive reference model system: 𝑥̇_𝑚=𝐴_𝑀𝑥_𝑚+𝐵_𝑚𝑟+𝐾₃𝐵_𝑓𝑢_𝑎, (20) where 𝐴_𝑀 is a time-varying matrix incorporating the model mismatch estimate 𝛿_𝑎, denoted as 𝐴_𝑀=𝐴_𝑚+𝛿_𝑎. (21)

Following [18], the system matrix 𝐴_𝑀 can be described as an affine function of a parameter vector 𝜌∈𝑅^𝑛𝜌 that lies in a convex polytope 𝒮 having vertices 𝜌^𝑗,𝑗=1,2,…𝑛_𝑣; that is, 𝐴_𝑀𝜌(𝑡)=𝐴_𝑚+^𝑛𝜌₁𝜌_𝑖(𝑡)𝐴_𝑀𝑖, (22) where 𝐴_𝑀𝑖 are constant matrices and 𝜌_𝑖(𝑡)∈[𝜌_𝑖min,𝜌_𝑖max]. Suppose that 𝐴_𝑀(𝜌^𝑗) denotes the value of the reference model system matrix at vertex 𝜌^𝑗 and there exist positive definite matrices 𝑃=𝑃^𝑇,𝑄=𝑄^𝑇∈ℜ^𝑛×𝑛 satisfying 𝐴_𝑀𝜌^𝑗𝑇𝑃+𝑃𝐴_𝑀𝜌^𝑗<−𝑄,𝑗=1,…,𝑛_𝑣. (23) Based on (23), we can get the following Lyapunov inequality forall𝜌(𝑡)∈𝒮: 𝐴_𝑀𝜌(𝑡)^𝑇𝑃+𝑃𝐴_𝑀𝜌(𝑡)<−𝑄,𝑗=1,…,𝑛_𝑣. (24) Thus the autonomous part of the adaptive reference model in (20) is exponentially stable forall𝜌(𝑡)∈𝒮 [18].

Note that this paper only considers the case that the true value of model mismatch lies within the stable region Figure 6. The model mismatch 𝛿_𝑎 is assumed to satisfy 𝐴_𝑚+𝛿_𝑎^𝑇𝑃+𝑃𝐴_𝑚+𝛿_𝑎<−𝑄. (25)

Subtracting (20) from (19) yields the following error dynamics: 𝑒̇=𝐴_𝑀𝑒+𝐵_𝑣𝐾₁𝑥+𝐵_𝑣𝐾₂𝑟−𝛿_𝑎𝑥−𝐵_𝑚𝐾₃𝐵_𝑓𝑢_𝑎, (26) where 𝑒=𝑥−𝑥_𝑚 and 𝛿_𝑎=𝛿_𝑎−𝛿_𝑎 denote the tracking error and model mismatch estimate error, respectively.

The following theorem gives the adaption and parameter estimation laws which ensure that 𝑒(𝑡)→0 and all signals remain bounded. We will first assume that the initial conditions satisfy Theorem  4.1 in [14].

Theorem 7. For the system given by (1), the adaptive controller (16) with gain adaption laws 𝐾̇₁=−Γ₁𝐵^𝑇_𝑣𝑃𝑒𝑥^𝑇,𝐾̇₂=−Γ₂𝐵^𝑇_𝑣𝑃𝑒𝑟^𝑇 (27) and parameter estimation laws 𝐾̇₃=Γ₃𝐵^𝑇_𝑚𝑃𝑒𝐵_𝑓𝑢_𝑎^𝑇,̇𝛿_𝑎=Γ_𝑎𝑃𝑒𝑥^𝑇, (28) where Γ₁, Γ₂, Γ₃∈ℜ^𝑝×𝑝 and Γ_𝑎∈ℜ^𝑛×𝑛 are positive definite matrices and 𝑃 is determined by (24), guarantees that all the closed-loop signals remain bounded and that the tracking error 𝑒(𝑡) converges to 0 as 𝑡→∞.

Proof. In order to analyze the closed-loop stability and tracking performance, choose a Lyapunov function as 𝑉=𝑒^𝑇𝑃𝑒+^𝑚_𝑖=1𝛿^𝑇_𝑎𝑖Γ_𝑎⁻¹𝛿_𝑎𝑖+tr𝐾^𝑇₁Γ₁⁻¹𝐾₁+tr𝐾^𝑇₂Γ₂⁻¹𝐾₂+tr𝐾^𝑇₃Γ₃⁻¹𝐾₃, (29) where tr(⋅) denotes the trace of a matrix and 𝛿_𝑎𝑖 denotes the 𝑖th column of 𝛿_𝑎.
Differentiating (29), we can get 𝑉̇=2𝑒^𝑇𝑃𝐴_𝑀𝑒+𝐵_𝑣𝐾₁𝑥+𝐵_𝑣𝐾₂𝑟−𝛿_𝑎𝑥−𝐵_𝑚𝐾₃𝐵_𝑓𝑢_𝑎+^𝑚_𝑖=1𝛿^𝑇_𝑎𝑖Γ_𝑎⁻¹𝛿_𝑎𝑖+tr𝐾^𝑇₁Γ₁⁻¹̇𝐾₁+tr𝐾^𝑇₂Γ₂⁻¹̇𝐾₂+tr𝐾^𝑇₃Γ₃⁻¹̇𝐾₃<−𝑒^𝑇𝑄𝑒−2tr𝛿^𝑇_𝑎𝑃𝑒𝑥^𝑇−Γ_𝑎⁻¹̇𝛿_𝑎+2tr𝐾^𝑇₁𝐵^𝑇_𝑣𝑃𝑒𝑥^𝑇+Γ₁⁻¹̇𝐾₁+2tr𝐾^𝑇₂𝐵^𝑇_𝑣𝑃𝑒𝑟^𝑇+Γ₂⁻¹̇𝐾₂−2tr𝐾^𝑇₃𝐵^𝑇_𝑚𝑃𝑒𝐵_𝑓𝑢^𝑇−Γ₃⁻¹̇𝐾₃. (30)
As 𝐾̇₁=𝐾̇₁, 𝐾̇₂=𝐾̇₂, 𝐾̇₃=𝐾̇₃, and ̇𝛿_𝑎=̇𝛿_𝑎, substituting (27) and (28) into (30) yields 𝑉̇=−𝑒^𝑇𝑄𝑒≤0. (31)
Since 𝑉(𝑡) is a positive definite function and 𝑉̇(𝑡)≤0, the signals 𝑒(𝑡), 𝐾₁(𝑡), 𝐾₂(𝑡), 𝐾₃(𝑡), and 𝛿_𝑎(𝑡) are all bounded. However, due to the modification of the reference model, the stability of the system cannot be directly proven. It is necessary to prove that one of the signals 𝑥(𝑡) or 𝑥_𝑚(𝑡) is bounded.
Then, we define the following candidate Lyapunov function: 𝑊(𝑥)=𝑥^𝑇𝑃_𝑥𝑥. (32) Substituting (4) into (1) yields 𝑥̇=𝐴+𝛿_𝑎𝑥+𝐵_𝑣𝐾^𝑇₁−𝐵_𝑣𝐾^𝑇₁+𝐵_𝑢=𝐴Δ𝑢_𝑚+𝛿_𝑎𝑥−𝐵_𝑣𝐾^𝑇₁+𝐵_𝑢Δ𝑢. (33) Based on (25), 𝐴_𝑚+𝛿_𝑎 is assumed to satisfy the following Lyapunov inequality: 𝐴_𝑚+𝛿_𝑎^𝑇𝑃_𝑥+𝑃_𝑥𝐴_𝑚+𝛿_𝑎<−𝑄_𝑥, (34) where 𝑃_𝑥=𝑃^𝑇_𝑥,𝑄_𝑥=𝑄^𝑇_𝑥∈ℜ^𝑛×𝑛 are both positive definite matrices.
Differentiating (32), we can get 𝑊̇(𝑥)<−𝑥^𝑇𝑄_𝑥𝑥−2𝑥^𝑇𝑃_𝑥𝐵_𝑣𝐾^𝑇₁+2𝑥^𝑇𝑃_𝑥𝐵_𝑢Δ𝑢. (35)
Based on the previous results in [14], we know that there always exists a nonempty annulus region such that 𝑊̇(𝑥)<0 holds forall𝑥 from that region. The boundedness of 𝑥 can be guaranteed as long as the initial conditions satisfy Theorem  4.1 in [14]. Since 𝑥(𝑡) is bounded and 𝑥_𝑚(𝑡)=𝑥(𝑡)−𝑒(𝑡), it is easy to conclude that 𝑥_𝑚(𝑡) is bounded.
Consequently, according to the error dynamics in (26) and the boundedness of the closed-loop signals, we can get 𝑒(̇𝑡)∈𝐿_∞, and thus lim_𝑡→∞𝑒(𝑡)=0. Due to 𝑒(𝑡)∈𝐿₂∩𝐿_∞,𝐾̇₁,𝐾̇₂,𝐾̇₃,̇𝛿_𝑎∈𝐿₂∩𝐿_∞, and 𝑥(𝑡),𝑟(𝑡)∈𝐿_∞, all signals and estimated parameters are bounded, realizing lim_𝑡→∞[𝑥(𝑡)−𝑥_𝑚(𝑡)]=0.
The proof is completed.

Remark 8. Since the reference model is time varying, it is necessary to ensure that the parameter estimate 𝛿_𝑎 remains within the stable region. If 𝛿_𝑎 exceeds the stable region, a parameter projection method [18] will be utilized to ensure that the adaptive reference model has satisfactory stability and performance.

4. Application Example

To investigate the effectiveness of the proposed adaptive fault-tolerant control scheme, we utilize a flight control example based on the ADMIRE model [19]. A low-speed flight case is considered, where the control surface efficiency is poor.

4.1. System Description

The state space of the model is given by (1), where the nominal values of the system matrices are in the appendix and 𝑥=𝛼𝛽𝑝𝑞𝑟^𝑇,𝑦=𝛼𝛽𝑝^𝑇,𝑢=𝛿_𝑐𝛿_𝑟𝑒𝛿_𝑙𝑒𝛿_𝑟^𝑇. (36) System states 𝛼, 𝛽, 𝑝, 𝑞, and 𝑟 represent the angle of attack, the sideslip angle, the roll rate, the pitch rate, and the yaw rate, respectively. Control inputs 𝛿_𝑐, 𝛿_𝑟𝑒, 𝛿_𝑙𝑒, and 𝛿_𝑟 represent the positions of the canard wings, the right elevons, the left elevons, and the rudder, respectively.

The virtual control input 𝑣 contains the angular accelerations in roll, pitch, and yaw produced by the control surface.

The actuator position constraints are given by 𝛿_min=−25^∘−30^∘−30^∘−30^∘𝑇,𝛿_max=25^∘30^∘30^∘30^∘𝑇. (37)

It is assumed that the actuator 𝛿_𝑐 loses 99% of effectiveness at 1.01 s while the remaining actuators are functional throughout and the aircraft experiences damage at 0 s; therefore, a model mismatch exists in the system matrix from the outset. The model mismatch 𝛿_𝑎 in the system is 𝛿_𝑎=0000.500000−0.5000000000000000. (38)

The reference inputs are given in Figure 2.

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(b)

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Figure 2 

The reference inputs.

4.2. Controller Design

The matrices 𝐾₁, 𝐾₂ are designed using the linear quadratic regulator (LQR) method. The reference model is designed based on (4).

The quadratic stable region is obtained as 𝛿_𝑎14 and 𝛿_𝑎25∈[−1.1564,1.1564]. The security threshold is arbitrarily set at 80% of the maximum stable region. The corresponding Lyapunov matrix 𝑃 in (25) is obtained as 𝑃=0.08920.0568−0.0070.08070.1270.0568657.84615.8863−0.1136−44.8585−0.00715.88630.6474−0.0084−2.34450.0807−0.1136−0.00840.16530.11520.127−44.8585−2.34450.115247.7764. (39) Using (18), the matrix 𝐾₃ can be decided. The gains Γ₁, Γ₂, Γ₃, and Γ_𝑎 are selected by trial and error.

The initial value of the weighting matrix 𝑊 is chosen as diag{0.01,1,1,1}. Once actuator failures occur, the weighting matrix will be updated.

4.3. Simulation Results

Case 1. In the first case, all adaption is turned off; that is, the weighting matrix, the gains, and the parameter estimates are not updated. The plant states are shown in Figure 3. Obviously, the stability of the system cannot be guaranteed due to the presence of actuator failures, model mismatch, and input saturation.

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(e)

Figure 3 

The system states of Case 1.

Case 2. We utilize the adaptive fault-tolerant scheme in this case. The controller gain and parameter estimates are generated by the adaptive laws (27)-(28). At 𝑡=1.01 s, the first actuator fails while the remaining actuators continue to be functional, adaptively compensating for the failed first actuator. Unfortunately, actuators 𝛿₂ and 𝛿₄ have exceeded the input constraints as shown in Figure 4. Using the antisaturation algorithm, all of the actuators can be rigorously kept within the bounds, as shown in Figure 5. During the simulation, the parameter estimates 𝛿_𝑎14 and 𝛿_𝑎25 lie in the quadratic stable region throughout and approach the true value of model mismatch adaptively as shown in Figure 7. The plant with the adaptive reference model states and the tracking errors are illustrated in Figures 7 and 8, respectively. Simulation results show that the tracking errors remain small and converge to zero asymptotically.

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Figure 4 

The adaptive control input 𝑢𝑎 of Case 2.

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Figure 5 

The commanded control input 𝑢𝑐 of Case 2.

Figure 6 

The model mismatch estimate of Case 2.

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(e)

Figure 7 

The system states of Case 2.

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Figure 8 

The tracking errors of Case 2.

5. Conclusion

This paper proposed a novel fault-tolerant control scheme for flight systems in the presence of simultaneous actuator failures, model mismatch, and input saturation. The scheme combined CA and MRAC to guarantee the system’s stability and tracking performance. Practical conditions such as model mismatch and input saturation were taken into consideration in this paper. Due to model mismatch and input saturation, the original reference model might no longer be appropriate; therefore, we designed an adaptive reference model that also provides satisfactory performance. Simulation results indicated that mismatch estimation, input compensation, and satisfactory tracking performance could be obtained in the presence of actuator failures, model mismatch, and input saturation. Future investigation will be necessary to achieve adaptive fault estimation and test more complex and realistic models. The FTC of multiple flight systems will be explored in our future study.

Appendix

Consider the following:,,,𝐵𝐴=−0.54320.013700.977800−0.11790.22150−0.96610−10.5128−0.996700.61762.6221−0.00300−0.5057000.7075−0.09390−0.2127𝐶=100000100000100𝐵=0−4.24234.24231.48711.6532−1.2735−1.27350.00240−0.28050.2805−0.8823_𝑣=0_2×3𝐼_3×3,𝐵_𝑢=𝐵_𝑣𝐵. (A.1)

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Key Foundation of China under Grant no. 91216124, no. 91116002, no. 90916003, and no. 91216304, the foundation of National Astronautica Key Lab (2012-865-6) and Aeronautical Science Foundation of China under Grant no. 20100171001.