Adaptive Finite-Time Control for a Flexible Hypersonic Vehicle with Actuator Fault

Abstract

The problem of robust fault-tolerant tracking control is investigated. Simulation on the longitudinal model of a flexible air-breathing hypersonic vehicle (FAHV) with actuator faults and uncertainties is conducted. In order to guarantee that the velocity and altitude track their desired commands in finite time with the partial loss of actuator effectiveness, an adaptive fault-tolerant control strategy is presented based on practical finite-time sliding mode method. The adaptive update laws are used to estimate the upper bound of uncertainties and the minimum value of actuator efficiency factor. Finally, simulation results show that the proposed control strategy is effective in rejecting uncertainties even in the presence of actuator faults.

1. Introduction

Air-breathing hypersonic vehicles (AHVs) are intended to be a reliable and cost-effective technology for access to space. Because the slender geometries and light structures cause significant flexible effects and strong coupling between propulsive and aerodynamic forces resulting from the integration of the scramjet engine, AHVs are confronting many complex problems and challenges, involving many different research areas, such as aerodynamics, thermal protection, and communication, and many problems of these fields have been reported [1–3]. Meanwhile, flight control design for AHVs is a hot topic and a challenging task [4, 5].

During the last decades, a kind of flexible hypersonic vehicle model including flexible dynamics has been developed in [6, 7]. Based on this model, there have been several papers discussing the challenges associated with the control of air-breathing hypersonic vehicle (AHV) [8, 9] and many control methods have been employed in the flight control system. In [10], a linear quadratic regulator (LQR) was presented for a linearized FAHV model. In [11–13], sequential loop closure controller was designed for the FAHV based on adaptive dynamic inversion together with backstepping structure. In [14, 15], approximate feedback linearization based on dynamic inversion method was adopted to design controller for the FAHV. In [16, 17], a nonlinear tracking controller was constructed by using a minimax LQR control approach, which provides robust stability and excellent tracking performance with parameter uncertainties.

The approaches mentioned above do not specifically consider possible actuator faults, which deteriorate the control performance, affect stability, and security of the AHVs, and sometimes even lead to catastrophic accidents. Consequently, it is essential that the actuator faults must be taken into account in the controller design. In the current papers, some fault-tolerant control schemes for AHVs have attracted more and more research attention and gained fruitful results, which can be reported in [18–21]. In [18–20], the results mainly concentrate on the reentry attitude control of the AHV. Meanwhile, the fault tolerant control strategies for the longitudinal model of the AHVs are studied. In [21], an observer-based fault-tolerant control approach using both robust control and LMI techniques is designed for a linearized longitudinal AHV model in the presence of parameter uncertainties and actuator faults, but this method was effective only in the neighborhood of the operating point. On the other hand, nonlinear fault-tolerant control design methods have been devoted to the longitudinal AHV model. A finite-time integral sliding mode control method was proposed in [22], which could achieve superior velocity and altitude tracking performance with actuator fault. In [23], the longitudinal AHV model with unknown parameters and uncertain actuator faults is formatted into a parametric strict-feedback form, and then an adaptive fault-tolerant control scheme based on a combination of back-stepping control and dynamic surface control techniques is applied to make the velocity and altitude track the desired value.

However, the aforesaid methods only consider the rigid body of AHVs without flexible effects. A fault-tolerant control scheme for the FAHV was presented in [24], according to the model obtained by approximate linearization in given flight conditions. So, this scheme may not obtain good control performances when flight dynamics undergo great parameter perturbations. To the best of our knowledge, although considerable effort has been made on the control design for the AHVs, the important issue of fault-tolerant control of the FAHV dynamical system has not been fully investigated yet, which remains challenging and motivates us to do this study.

As a typical robust control method, sliding mode control (SMC) scheme is regarded as an effective method to cope with external disturbances and parametric uncertainties [25]. Recently, the SMC method has been widely applied for the fault tolerant control of aircraft system, spacecraft, and so on. In [26], a fault-tolerant sliding mode controller was presented for an aircraft system, which requires the message of the effectiveness factor, while it may be difficult and expensive to obtain the actuator faults online. In [27], a finite-time convergent SMC scheme is developed to solve the problem of fault-tolerant control for a rigid spacecraft. The drawback of this method is that the message of the lower bound of the effectiveness factor and the upper bound of system uncertainties needs to be known in prior.

The aforementioned references could achieve desired performance through the SMC methodology affected by actuator faults. Although the traditional SMC can guarantee the stability of the system, it adopts a linear switching function. Then the system states and the errors converge to an equilibrium point asymptotically in infinite time. In other words, it means that finite-time convergence is not ensured. Motivated by the above discussions, we propose a novel adaptive sliding mode control scheme for the longitudinal model of the FAHV with uncertainties and actuator faults in this paper. As compared with the existing results, the main contributions are as follows. Firstly, the design method of sliding mode surface based on homogeneous geometry could assure practical finite-time converged tracking of the desired command. Secondly, the upper bounds of aerodynamic uncertainties and the minimum value of actuator efficiency factor are not required in prior. The adaptive law is designed to adjust the control gains dynamically so as to ensure the establishment of sliding mode motion, and the robustness against uncertainties is ensured at the same time. After the uncertainties and actuator faults are compensated using adaptive sliding mode control scheme, the stability of the closed-loop system can be maintained.

The rest of this paper is organized as follows. In Section 2 the FAHV model is introduced and control objective is stated. Section 3 designs the sliding mode surface and the corresponding adaptive finite-time fault tolerant controller was proposed with actuator fault. Simulation results are discussed in Section 4 and the conclusions are provided in Section 5.

2. Problem Statement

The considered FAHV model is derived from [6, 28], and the longitudinal equations of motion of the FAHV are given by 𝑉̇=(𝑇cos𝛼−𝐷)𝑚̇()−𝑔sin𝛾,𝛾=𝐿+𝑇sin𝛼−𝑚𝑉𝑔cos𝛾𝑉,̇̇̇̇𝑀ℎ=𝑉sin𝛾,𝛼=𝑄−𝛾,𝑄=^𝑦𝑦𝐼_𝑦𝑦,𝜂̈_𝑖=−2𝜍_𝑚𝜔_𝑚,𝑖𝜂̇_𝑖−𝜔²_𝑚,𝑖𝜂_𝑖+𝑁_𝑖,𝑖=1,2,3, (1)

where 𝑥=[𝑉,𝛾,ℎ,𝛼,𝑄]^𝑇 is a vector of rigid-body state, which includes the vehicle speed, flight path angle, altitude, angel of attack, and pitch rate, respectively; 𝜂_𝑖, 𝜔_𝑚,𝑖, and 𝜍_𝑚 are the generalized flexible coordinate, natural frequencies, and damping coefficients of the 𝑖th elastic mode. The readers may refer to [7] for a full description of the variables in this model.

Because of coupling in aerodynamic forces of the FAHV model (1), some simplifications must be carried out for the purpose of feedback linearization. The simplification of the model is necessary because we want to obtain a linearized model, and the same simplified process can be found in [29]. An input-output linearization model is developed by repeated differentiation of the outputs 𝑉 and ℎ as follows: 𝑉⃛=𝑓_𝑉+𝑏₁₁𝜙_𝑐+𝑏₁₂𝛿_𝑒,ℎ⁽⁴⁾=𝑓_ℎ+𝑏₂₁𝜙_𝑐+𝑏₂₂𝛿_𝑒, (2)(3)

where 𝜙_𝑐 and 𝛿_𝑒 are control inputs and the specific expressions of 𝑓_𝑉, 𝑓_ℎ, 𝑏₁₁, 𝑏₁₂, 𝑏₂₁, and 𝑏₂₂ are presented in [29, equation (17)].

Compared with [29], the main propose of this study is discussing the fault tolerant controller design for the FAHV to follow a given desired output reference signals 𝑦_𝑑=[𝑉_𝑑,ℎ_𝑑]^𝑇 in the presence of partial loss of actuator effectiveness.

3. Adaptive Finite-Time Fault-Tolerant Controller Design

The specific controller design step includes two parts: sliding mode surface design and sliding mode control design, which can be described as follows.

3.1. Sliding Mode Surface Design

Define tracking error variable as follows: 𝑒_𝑉=𝑉−𝑉_𝑑,𝑒_ℎ=ℎ−ℎ_𝑑. (4)(5)

Differentiating (4) and (5) three times, and four times respectively, results in 𝑒_𝑉⃛=𝑓_𝑉−𝑉⃛_𝑑+𝑏₁₁𝜙_𝑐+𝑏₁₂𝛿_𝑒,𝑒_ℎ⁽⁴⁾=𝑓_ℎ−ℎ_𝑑⁽⁴⁾+𝑏₂₁𝜙_𝑐+𝑏₂₂𝛿_𝑒. (6)(7)

Equations (6)-(7) can be expressed in matrix form: 𝑒_𝑉⃛𝑒_ℎ⁽⁴⁾=𝑓_𝑉−𝑉⃛_𝑑𝑓_ℎ−ℎ_𝑑⁽⁴⁾_{𝐹𝐹=1,𝐹2𝑇}+𝑏₁₁𝑏₁₂𝑏₂₁𝑏_22𝐵𝜙_𝑐𝛿_𝑒𝑢+Δ𝐹₁Δ𝐹_{2Δ𝐹=Δ𝐹1,Δ𝐹2𝑇}. (8)

Note that the additional item Δ𝐹 is introduced to represent the flexible effects and coupled uncertainties described in [29, equation (14)].

Introduce new control variable: 𝑈𝑈=¹𝑈₂=𝑏₁₁𝑏₁₂𝑏₂₁𝑏₂₂𝜙_𝑐𝛿_𝑒. (9)

Then (8)-(9) can be rewritten as 𝑒_𝑉⃛𝑒_ℎ⁽⁴⁾=𝐹₁𝐹₂+𝑈₁𝑈₂+Δ𝐹₁Δ𝐹₂. (10)

Assumption 1. The uncertainties discussed in the research are bounded ‖Δ𝐹‖≤𝜐, but the value 𝜐 is unknown in advance.

Assumption 2. The matrix 𝐵 denoted in (8) is nonsingular over the entire flight envelope given in [12], so Assumption 1 is reasonable to be assumed.

Now, according to the definition of HOSM [30, 31], our objective is to design controller which makes the 𝑒_𝑉, 𝑒_ℎ and their derivatives converge to the neighborhood of origin.

Design sliding mode surface as follows:𝑠_𝑉=𝑒_𝑉̈+^𝑡₀𝜆_1𝑉𝑒_𝑉^𝑎1𝑉sign𝑒_𝑉+𝜆_2𝑉𝑒̇_𝑉^𝑎2𝑉sign𝑒_𝑉̇+𝜆_3𝑉𝑒̈_𝑉^𝑎3𝑉sign_𝑉𝐺𝑉𝑒̈,𝑠_ℎ=𝑒_ℎ⃛+^𝑡₀𝜆_1ℎ𝑒_ℎ^𝑎1ℎsign𝑒_ℎ+𝜆_2ℎ𝑒̇_ℎ^𝑎2ℎsign𝑒_ℎ̇+𝜆_3ℎ𝑒̈_ℎ^𝑎3ℎsign𝑒_ℎ̈+𝜆_4ℎ𝑒⃛_ℎ^𝑎4ℎsign_ℎ𝐺ℎ𝑒⃛𝑑𝑠. (11)(12)

The parameters 𝜆_𝑖𝑉(𝑖=1,2,3) and 𝜆_𝑗ℎ(𝑗=1,2,3,4) are some positive constants such that 𝜆_3𝑉𝑠²+𝜆_2𝑉𝑠+𝜆_1𝑉 and 𝜆_4ℎ𝑠³+𝜆_3ℎ𝑠²+𝜆_2ℎ𝑠+𝜆_1ℎ are Hurwitz polynomial. The parameters 𝑎_𝑖𝑉(𝑖=1,2,3) and 𝑎_𝑗ℎ(𝑗=1,2,3,4) are determined by 𝑎_{(𝑖−1)𝑉}=𝑎_𝑖𝑉𝑎_(𝑖+1)𝑉2𝑎_(𝑖+1)𝑉−𝑎_𝑖𝑉{},𝑎,𝑖∈2,3_{(𝑗−1)ℎ}=𝑎_𝑗ℎ𝑎_(𝑗+1)ℎ2𝑎_(𝑗+1)ℎ−𝑎_𝑗ℎ,𝑗∈{2,3,4} (13)

with 𝑎_4𝑉=𝑎_5ℎ=1, 𝑎_3𝑉∈(1−𝜀_𝑉,1), and 𝑎_4ℎ∈(1−𝜀_ℎ,1), where 𝜀_𝑉∈(0,1),𝜀_ℎ∈(0,1).

Based on the homogeneity theory provided in [32], it is easily shown that 𝑒_𝑉, 𝑒_𝑉̇, 𝑒_𝑉̈ and 𝑒_ℎ, 𝑒_ℎ̇, 𝑒_ℎ̈, 𝑒_ℎ⃛ will converge to the neighborhood of origin in finite time if it is satisfied that 𝑠_𝑉, 𝑠_ℎ converge to the neighborhood of origin in finite time.

3.2. Adaptive Sliding Mode Controller Design

Now, let us consider the situation in which the actuator experiences partial loss of effectiveness fault. Then, differentiating (11) and (12), we obtain 𝑠_𝑉̇𝑠_ℎ̇=𝐹₁𝐹₂+𝐺_{𝑉ℎ𝐺𝐺}𝐺_{=𝑉,𝐺ℎ𝑇}+𝐸₁00𝐸_2𝐸𝑈₁𝑈₂+Δ𝐹₁Δ𝐹₂, (14)

where 𝐸=diag(𝐸₁,𝐸₂)∈𝑅^2×2 is a matrix characterizing the health condition of the actuators with 0≤𝐸_𝑖≤1(𝑖=1,2). Note that the case 𝐸_𝑖=1 means that the 𝑖th actuator is totally healthy, the case 𝐸_𝑖=0 implies that the 𝑖th actuator completely fails, and the case 0<𝐸_𝑖<1 corresponds to the case in which the 𝑖th actuator partially loses its effectiveness, but it still has effect all the time. In this sense, the matrix 𝐸 becomes uncertain and even time varying but remains positive definite. In this study, an assumption 0<𝐸_𝑖≤1 is given.

The control objective is to design the control inputs for 𝜙_𝑐 and 𝛿_𝑒 such that all of the closed-loop signals are bounded and the velocity 𝑉 and altitude ℎ track desired command trajectories 𝑉_𝑑 and ℎ_𝑑 in the presence of flexible uncertainties and loss of effective actuator faults. That is to say, the velocity sliding mode surface 𝑠_𝑉 and altitude sliding mode surface 𝑠_ℎ converge to an arbitrary small set containing the origin in finite time 𝑇₀, which is ‖𝑠_𝑉‖≤𝛿_𝑉 and ‖𝑠_ℎ‖≤𝛿_ℎ for 𝑡≥𝑇₀, where 𝛿_𝑉 and 𝛿_ℎ are arbitrary small positive constant numbers.

Let 𝐸_min=min_𝑖=1,2𝐸_𝑖 and denote 𝜇=1−𝐸_min and then 𝜇<1. Selecting 𝜃=1/(1−𝜇), then the main result of the paper is formulated in the following theorem.

Theorem 3. Consider the nonlinear sliding mode dynamic system (14) with Assumptions 1 and 2, if the control 𝑈=[𝑈₁,𝑈₂]^𝑇 is designed as 𝑈=−𝐹−𝐺−𝑘⋅sig^𝜏(𝑠)−𝜎𝑠‖𝑠‖−𝜐𝑠‖𝑠‖, (15)

with the adaptive gains 𝜎=−𝜓+𝜃𝜓,𝜓=‖𝐹‖+‖𝐺‖+𝑘⋅sig^𝜏(𝑠+̇)𝜐,𝜃=𝑝₀−𝜀₀‖𝑠‖,̇𝜃+𝜓𝜐=𝑝₁−𝜀₁𝜐+‖𝑠‖, (16)(17)(18)

where 𝑠=[𝑠_𝑉,𝑠_ℎ]^𝑇, 𝑘=[𝑘_𝑉,𝑘_ℎ]^𝑇, and 0<𝜏<1, and define the function sig^𝜏(⋅)=sign(⋅)|⋅|^𝜏, 𝑝₀, 𝑝₁, 𝜀₀ and 𝜀₁ are positive control constants, and the initial values 𝜎(0), 𝜐(0) are chosen as positive constants. Then, the system trajectory will converge to the neighborhood of 𝑠_𝑉=𝑠_ℎ=0 in finite time despite of the uncertainties Δ𝐹 and actuator faults 𝐸.

Proof. The stability analysis of system (14) is performed via constructing the following Lyapunov function: 1𝑊=2𝑠^𝑇𝑠+1−𝜇𝑝₀𝜃−𝜃²+1𝑝₁(𝜐−𝜐)², (19)
where 𝜎=𝜎−𝜎 and 𝜐=𝜐−𝜐. The derivative of (19) is presented 𝑊̇=𝑠^𝑇𝑠̇−1−𝜇𝑝₀𝜃̇1𝜃−𝑝₁𝜐𝜐̇=𝑠^𝑇(𝐹+𝐺+𝐸𝑈+Δ𝐹)−1−𝜇𝑝₀𝜃̇1𝜃−𝑝₁𝜐𝜐̇=𝑠^𝑇𝐹+𝐺+𝑈−(𝐼−𝐸_Δ𝐸)𝑈+Δ𝐹−1−𝜇𝑝₀𝜃̇1𝜃−𝑝₁𝜐𝜐̇=𝑠^𝑇−𝑘⋅sig^𝜏(𝑠)−𝜎𝑠‖𝑠‖−𝜐𝑠‖𝑠‖−(𝐼−Ε_Δ𝐸)−𝑈+Δ𝐹1−𝜇𝑝₀𝜃̇1𝜃−𝑝₁𝜐𝜐̇≤−𝑘_𝑉𝑠_𝑉^𝜏+1−𝑘_ℎ𝑠_ℎ^𝜏+1+𝑠^𝑇−𝜎𝑠‖𝑠‖−𝜐𝑠‖𝑠‖−Δ𝐸𝑈+Δ𝐹−1−𝜇𝑝₀𝜃̇1𝜃−𝑝₁𝜐𝜐̇≤−𝑘_𝑉𝑠_𝑉^𝜏+1−𝑘_ℎ𝑠_ℎ^𝜏+1+𝑠^𝑇−𝜎𝑠‖𝑠‖−−Δ𝐸𝑈1−𝜇𝑝₀𝜃̇𝜃+𝑠^𝑇𝜐𝑠Δ𝐹−‖𝑠‖−1𝑝₁𝜐𝜐̇≤−𝑘_𝑉𝑠_𝑉^𝜏+1−𝑘_ℎ𝑠_ℎ^𝜏+1+𝑠^𝑇−𝜎𝑠‖𝑠‖−−Δ𝐸𝑈1−𝜇𝑝₀𝜃̇𝜃+𝑠^𝑇𝑠Δ𝐹−𝜐‖𝑠‖+𝜐‖𝑠‖−1𝑝₁𝜐𝜐̇. (20)
In view of Assumption 1 and adaptive update laws (18), inequality (20) can be rewritten as 𝑊̇≤−𝑘_𝑉𝑠_𝑉^𝜏+1−𝑘_ℎ𝑠_ℎ^𝜏+1+𝑠^𝑇−𝜎𝑠‖𝑠‖−−Δ𝐸𝑈1−𝜇𝑝₀𝜃̇𝜃+𝜀₁𝜐𝜐≤−𝑘_𝑉𝑠_𝑉^𝜏+1−𝑘_ℎ𝑠_ℎ^𝜏+1−𝜎‖𝑠‖−𝑠^𝑇−Δ𝐸𝑈1−𝜇𝑝₀𝜃̇𝜃+𝜀₁𝜐𝜐≤−𝑘_𝑉𝑠_𝑉^𝜏+1−𝑘_ℎ𝑠_ℎ^𝜏+1−𝜎‖𝑠‖+‖Δ𝐸‖⋅𝑠^𝑇⋅‖𝐹‖+‖𝐺‖+𝑘⋅sig^𝜏(𝑠+𝜐−)𝜎+1−𝜇𝑝₀𝜃̇𝜃+𝜀₁𝜐𝜐. (21)
According to (16), inequality (21) can be rewritten as 𝑊̇≤−𝑘_𝑉𝑠_𝑉^𝜏+1−𝑘_ℎ𝑠_ℎ^𝜏+1+𝜃𝜓‖𝑠‖‖𝑠‖−1−+𝜇⋅𝜃⋅𝜓1−𝜇𝑝₀𝜃̇𝜃+𝜀₁𝜐𝜐≤−𝑘_𝑉𝑠_𝑉^𝜏+1−𝑘_ℎ𝑠_ℎ^𝜏+1+𝜃𝜓‖𝑠‖−1−1−𝜇1−𝜇𝑝₀𝜃̇𝜃+𝜀₁𝜐𝜐≤−𝑘_𝑉𝑠_𝑉^𝜏+1−𝑘_ℎ𝑠_ℎ^𝜏+1+1−𝜇𝜃−1−𝜇𝜃𝜓‖𝑠‖−1−𝜇𝑝₀𝜃̇𝜃+𝜀₁𝜐𝜐≤−𝑘_𝑉𝑠_𝑉^𝜏+1−𝑘_ℎ𝑠_ℎ^𝜏+1+‖𝑠‖−1−𝜇𝜃𝜓1−𝜇𝑝₀𝜃̇𝜃+𝜀₁𝜐𝜐. (22)
According to the adaptive update laws defined in (17), the inequality (22) can be rewritten as 𝑊̇≤−𝑘𝑠_𝑉^𝜏+1+𝑠_ℎ^𝜏+1+1−𝜇𝜀₀𝜃𝜃+𝜀₁𝜐𝜐, (23)
where 𝑘=min_{𝑖=𝑉,ℎ}𝑘_𝑖. In view of Lemma  3.1 in [33], inequality (23) can be written as 𝑊̇≤−𝑘𝑠_𝑉²+𝑠_ℎ^2(𝜏+1)/2+1−𝜇𝜀₀𝜃𝜃+𝜀₁𝜐𝜐1≤−𝑘2𝑠_𝑉²+𝑠_ℎ^2(𝜏+1)/2+1−𝜇𝜀₀𝜃𝜃+𝜀₁𝜐𝜐. (24)
Inspired by [33] for any positive numbers 𝛿₀>0.5 and 𝛿₁>0.5, inequality (24) can be rewritten as 𝑊̇≤−𝑘12𝑠^𝑇𝑠^(𝜏+1)/2−1−𝜇𝜀₀2𝛿₀−12𝛿₀𝜃^2(𝜏+1)/2−𝜀₁2𝛿₁−12𝛿₁𝜐^2(𝜏+1)/2+1−𝜇𝜀₀2𝛿₀−12𝛿₀𝜃^2(𝜏+1)/2+𝜀₁2𝛿₁−12𝛿₁𝜐^2(𝜏+1)/2+1−𝜇𝜀₀𝜃𝜃+𝜀₁𝜐𝜐. (25)
denote 𝑝₀=𝛿₀𝑘^2/(𝜏+1)𝜀₀2𝛿₀−1,𝑝₁=𝛿₁𝑘^2/(𝜏+1)𝜀₁2𝛿₁.−1 (26)
Then, inequality (25) can be rewritten as 𝑊̇≤−𝑘12𝑠^𝑇𝑠^(𝜏+1)/2+1−𝜇2𝑝₀𝜃^2(𝜏+1)/2+12𝑝₁𝜐^2(𝜏+1)/2+1−𝜇𝜀₀2𝛿₀−12𝛿₀𝜃^2(𝜏+1)/2+𝜀₁2𝛿₁−12𝛿₁𝜐^2(𝜏+1)/2+1−𝜇𝜀₀𝜃𝜃+𝜀₁𝜐𝜐. (27)
According to Lemma  3.2 in [33], when 𝛿₀>0.5, 𝛿₁>0.5, and 0.5<0.5(𝜏+1)<1, the time derivative of the Lyapunov function 𝑊̇ becomes 𝑊̇≤−𝑘12𝑠^𝑇𝑠+1−𝜇2𝑝₀𝜃²+12𝑝₁𝜐^2(𝜏+1)/2+1−𝜇𝜀₀2𝛿₀−12𝛿₀𝜃^2(𝜏+1)/2+𝜀₁2𝛿₁−12𝛿₁𝜐^2(𝜏+1)/2+1−𝜇𝜀₀𝜃𝜃+𝜀₁𝜐𝜐≤−𝑘𝑊^(𝜏+1)/2+1−𝜇𝜀₀2𝛿₀−12𝛿₀𝜃^2(𝜏+1)/2+𝜀₁2𝛿₁−12𝛿₁𝜐^2(𝜏+1)/2+1−𝜇𝜀₀𝜃𝜃+𝜀₁𝜐𝜐. (28)
Note that, for any positive constants 𝛿₀>0.5 and 𝛿₁>0.5, the following inequality holds: 𝜀₁𝜐𝜐=𝜀₁−𝜐²+𝜐𝜐≤𝜀₁−𝜐²+12𝛿₁𝜐²+𝛿₁2𝜐²≤−𝜀₁2𝛿₁−12𝛿₁𝜐²+𝜀₁𝛿₁2𝜐². (29)
Similarly (1−𝜇)𝜀₀𝜃𝜃 satisfies the following inequality: 1−𝜇𝜀₀𝜃𝜃≤−𝜀₀1−𝜇2𝛿₀−12𝛿₀𝜃²+𝜀₀1−𝜇𝛿₀2𝜃². (30)
According to inequality (29), if (𝜀₁(2𝛿₁−1)/2𝛿₁)𝜐²>1, we obtain 𝜀₁2𝛿₁−12𝛿₁𝜐^2(𝜏+1)/2+𝜀₁𝜐𝜐≤𝜀₁2𝛿₁−12𝛿₁𝜐²+𝜀₁𝜐𝜀𝜐≤¹𝛿₁2𝜐². (31)
If (𝜀₁(2𝛿₁−1)/2𝛿)𝜐²≤1, we have 𝜀₁2𝛿₁−12𝛿₁𝜐^2(𝜏+1)/2_{(𝜀1(2𝛿1−1)/2𝛿1)}𝜐²_≤1<𝜀₁2𝛿₁−12𝛿₁𝜐^2(𝜏+1)/2_{(𝜀1(2𝛿1−1)/2𝛿1)}𝜐²_>1. (32)
Therefore, combining (31) and (32) yields 𝜀₁2𝛿₁−12𝛿₁𝜐^2(𝜏+1)/2+𝜀₁𝜐𝜀𝜐≤¹𝛿₁2𝜐². (33)
Similar to (33), the following inequality can be obtained: 1−𝜇𝜀₀2𝛿₀−12𝛿₀𝜃^2(𝜏+1)/2+1−𝜇𝜀₀𝜃𝜃≤1−𝜇𝜀₀𝛿₀2𝜃². (34)
Thus, from (28)–(34), the derivative of the Lyapunov function (28) becomes 𝑊̇≤−𝑘𝑊^(𝜏+1)/2+𝜇₀, (35)
where 𝜇₀=𝜀₁𝛿₁2𝜐²+1−𝜇𝜀₀𝛿₀2𝜃². (36)
According to Lemma  3.6 in [33], the decrease of 𝑊can drive the sliding mode surfaces 𝑠_𝑉 and𝑠_ℎ to converge to a neighborhood of the sliding surface in finite time. Furthermore, selecting 0<𝛽≤1, inequality (35) can be expressed as 𝑊̇≤−𝛽𝑘𝑊^(𝜏+1)/2−1−𝛽𝑘𝑊^(𝜏+1)/2+𝜇₀. (37)
If −(1−𝛽)𝑘𝑊^(𝜏+1)/2+𝜇₀<0, then 𝑊̇≤−𝛽𝑘𝑊^(𝜏+1)/2. Based on the conclusion from [30], the decrease of 𝑊drives the trajectories of the closed-loop system into 𝑊^(𝜏+1)/2≤𝜇₀/(1−𝛽)𝑘. Therefore, the trajectories of the closed-loop system is bounded in finite time as lim_𝛽→𝛽0𝑠(𝑡)∈‖𝑠‖≤𝜇₀(1−𝛽₀)𝑘_𝜀^1/(𝜏+1), (38)
where 0<𝛽₀<1 and 𝜀 is a small set containing the origin of the closed-loop system. And the time needed to reach (38) is bound as (0)𝑇≤2𝑊^(1−𝜏)/2𝑘𝛽₀(),1−𝜏 (39)
where 𝑊(0) is the initial value of 𝑊. After that, the control objective that the 𝑒_𝑉, 𝑒_𝑉̇, 𝑒_𝑉̈ and 𝑒_ℎ, 𝑒_ℎ̇, 𝑒_ℎ̈, 𝑒_ℎ⃛ converge to the neighborhood of origin is established.
When the control 𝑈=[𝑈₁,𝑈₂]^𝑇 is designed via (9), according to Assumption 2 the actual control variable is calculated as 𝜙_𝑐𝛿_𝑒=𝑏₁₁𝑏₁₂𝑏₂₁𝑏₂₂⁻¹𝑈₁𝑈₂. (40)
It is evident from (40) that the finite-time convergent performance of the proposed adaptive fault tolerant controller can be obtained without the knowledge of the minimum value of actuator effectiveness factor. Meanwhile, the upper bound of uncertainties does not need to be known in advance.

4. Simulation

To illustrate the efficiency of controller designed previously, a climbing maneuver with longitudinal acceleration for a 100 ft/s velocity change and a 1000 ft altitude change is considered. Simulation studies have been done on the full nonlinear flexible hypersonic vehicle defined in (1). The reference commands have been generated by filtering step reference commands by a second-order prefilter with natural frequency 𝜔_𝑓=0.06 rad/s and damping ratio 𝜁_𝑓=0.95.

The initial trim condition is selected as 𝑉=7710 ft/s and ℎ=85000 ft. Simulation parameters are provided in Table 1.

Table 1 

Simulation parameters setting.

It is assumed that actuator faults are chosen as 𝐸₁𝐸=0.7,𝑡≥100,₂=0.7,𝑡≥100. (41)

The simulation results are provided in Figures 1–4. Figure 1 denotes the response to the 100 ft/s step velocity and 1000 ft step altitude. It has been observed that the velocity and altitude converge to the desired value. The control inputs of 𝜙_𝑐 and 𝛿_𝑒 could be seen in bottom plots of Figure 1.

(a)

(b)

(c)

(d)

Figure 1 

Regulated outputs and control inputs with actuator faults.

(a)

(b)

(c)

(d)

Figure 2 

Other flight states with actuator faults.

(a)

(b)

(c)

(d)

Figure 3 

Sliding mode surface and adaptive parameters 𝜐 and 𝜃 with actuator faults.

(a)

(b)

(c)

Figure 4 

The dynamic response curves of flexile modes with actuator faults.

Figure 2 shows the performance of the angle of attack 𝛼 and the pitch rate 𝑄 at the top, as well as the canard deflection 𝛿_𝑐 and the flight path angle 𝛾 at the bottom.

The velocity and altitude sliding mode surfaces 𝑠_𝑉, 𝑠_ℎ are shown in Figure 3, which are oscillation with small magnitudes when actuator fault occurred. The convergent performance verifies the effectiveness of the proposed control strategy. The adaptive parameters 𝜐 and 𝜃 in control laws of (15)–(18) could be seen in bottom plots of Figure 3, where the convergence of 𝜐 is confirmed. From the simulation results in Figure 3, the approximate equation 𝜃≈3 can be obtained. According to the relationship based on equation 𝜃=1/(1−𝜇), we can solve that 𝜇=0.67 and denote the estimated error as 𝑒_𝜇=𝜇−𝜇≈0.7−0.67=0.03. (42)

The value of 𝑒_𝜇 in our research is in tolerance. Meanwhile, the stability of flexible states is depicted by Figure 4. And it can be seen that the flexible states 𝜂₁, 𝜂₂, and 𝜂₃ converge to constant values, respectively.

In summary, the simulation results demonstrate that, although there are actuator faults and uncertainties in the system, the good tracking performance and satisfactory system responses can be guaranteed.

5. Conclusions and Future Work

In this paper, an effective method has been proposed for linearizing the nonlinear model of the FAHV via feedback, which simplifies the complexity of the controller design process. Furthermore, an adaptive fault-tolerant control scheme based on finite-time sliding mode control technique has been brought forward for the FAHV without any information about the upper bound of uncertainties or the minimum value of actuator effectiveness. Simulation results have been presented to evaluate the validity of the proposed control scheme and to show its robustness to uncertainties and the loss of actuator effectiveness.

Further research work includes two aspects. Firstly, only the loss-of-effectiveness fault has been investigated in this paper; other types of actuator faults such as float failure and actuator faults in FAHV with unknown structure are worth being dealt with. Furthermore, the FAHV model considered in this paper is highly nonlinear and strongly coupled, and a more general active FTC scheme as adaptive fault diagnosis observer in [34, 35] should be investigated in our future study.

Conflict of Interests

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

Acknowledgments

This research was supported in part by National Natural Science Foundation of China (nos. 91016018, 61004073, and 61203119), the Foundation for Key Program of Ministry of Education, China (no. 311012), the Key Program for Basic Research of Tianjin (no. 11JCZDJC25100), and the Key Program of Tianjin Natural Science (no. 12JCZDJC30300), and Aeronautical Science Foundation of China (no. 20125848004) supported by Science and Technology on Aircraft Control Laboratory.