Adaptive sliding mode fault-tolerant control for hypersonic vehicle based on radial basis function neural networks

Abstract

In this article, an adaptive sliding mode fault-tolerant control scheme is proposed to address the problem of robust and fast attitude tracking for a hypersonic vehicle in the presence of unknown external disturbances, additive fault and partial loss of effectiveness fault. Firstly, the healthy and faulty models of the vehicle are given. Then, a radial basis function neural network is designed to estimate the unknown additive fault, and the adaptive method is applied to deal with the unknown partial loss of effectiveness fault. Combined with the sliding mode control theory, the fault-tolerant controllers are designed for the outer and inner loops of the faulty system, respectively. The adaptive laws are designed to update parameter estimates to implement the inner-loop controller. Closed-loop stability is analysed and simulation results verify the effectiveness of the proposed fault-tolerant control scheme.

Introduction

Hypersonic vehicle, whose March number is over five, has important application values in military affair and civil area for their high speed, good break-defence capability and well-known attack performance, and the study of hypersonic vehicles has become a hot topic in the aerospace area. Recently, there have been significant research activities in hypersonic flight control. ^{1 –6} In the study by Xu and Shi, ¹ the commonly used hypersonic flight model for controller design and the recent researches on hypersonic flight control were presented. Meanwhile, the potential challenges about hypersonic flight control were discussed for future research. A backstepping controller was designed by Xu and Zhang ³ for the longitudinal dynamics of a generic hypersonic flight vehicle with neural networks, and to avoid the causality contradiction, the hypersonic flight dynamics were transformed into a strict-feedback form from which the virtual control was designed in this article. In the study by Chen et al., ⁶ a robust attitude control scheme was developed for the near-space vehicle with external unknown disturbance and control input saturation.

Due to the complex flight conditions of hypersonic vehicles, they become sensitive to changes in atmospheric conditions and aerodynamic parameters, and the performance of the control system will inevitably be weaken. These significantly increase the possibility of the occurrence of faults, which can be affected by actuators, control surfaces and sensors. In order to improve the safety and reliability, the fault-tolerant control (FTC) for hypersonic vehicles must be considered.

In recent years, some effective FTC approaches have been proposed for longitude dynamics of hypersonic vehicles, such as backstepping control, ⁷ state tracking control ⁸ and output feedback control. ⁹ In the study by Qi et al., ⁷ an adaptive backstepping FTC scheme was proposed for a hypersonic vehicle with parameter uncertainties and actuator fault. The dynamic inversion control technique was introduced to handle the problem of velocity control, and the adaptive backstepping method was applied to guarantee the attitude tracking. In the work of He et al., ⁹ an adaptive output feedback FTC scheme was designed for a hypersonic vehicle in the presence of external disturbances, parameter uncertainties and actuator faults. The feedback linearization technique was applied to design the nominal controller. Based on the nominal controller, a fault-tolerant controller was proposed to compensate for the influence of actuator fault.

There have been some approaches proposed on controlling the attitude of hypersonic vehicles, such as sliding mode control (SMC), ^10,11 adaptive control, ^12,13 control allocation ^14,15 and so on. In the study by Zhao et al., ¹¹ an adaptive sliding mode FTC scheme was presented for near-space vehicle attitude control systems with actuator fault. The additive fault was considered, and the designed fault-tolerant controller ensured that the desired signals were tracked in the presence of faults. In the work of Jiang et al., ¹³ the problem of robust fault-tolerant tracking control for near-space vehicle was investigated. The vehicle attitude dynamics was transformed into Takagi–Sugeno fuzzy models. An adaptive fault-tolerant controller, which was based on the online estimation of actuator faults, was designed to reduce the effect of actuator fault. In the study by Alwi and Edwards, ¹⁵ in order to solve the problem of FTC for a flight vehicle subjected to control surface faults and failures, an online SMC allocation scheme based on the effectiveness level of the actuators was proposed. The control signals were assigned to the remaining healthy control surfaces. An active FTC strategy based on H_∞ control theory was designed in the study by Cieslak et al. ¹⁶ for an aircraft suffering from degraded stabilizer control effectiveness. In a more recent work, ¹⁷ an integral-type sliding mode fault-tolerant controller, which was incorporated with adaptive technique, was designed to accommodate actuator faults of spacecraft with external disturbances.

Among these approaches, the adaptive SMC is recognized as an effective control technique to deal with model uncertainties, parameter fluctuations and external disturbances. It shows many merits, such as robustness to uncertainties and disturbances, ^18,19 fast response ability, simple structure and easy implementation, and has been applied to a wide range of practical systems. ^{20 –22} Adaptive SMC has been addressed in some FTC studies for flight vehicle attitude tracking. ^23,24 In the study by Xiao et al., ²⁴ an adaptive sliding mode FTC approach was developed. The partial loss of effectiveness fault was effectively handled, and the attitude tracking was ensured.

The neural networks ²⁵ or radial basis function neural networks (RBFNNs) ^26,27 were used to estimate unknown nonlinear functions, and the approximation capability has been illustrated in the study by Sanner and Slotine. ²⁸ In the study by Xu et al., ²⁶ the unknown longitudinal dynamics of a hypersonic aircraft caused by fault was approximated by RBFNN. Combined with the dynamic surface control, the designed control scheme ensured good tracking performance in the presence of fault. Although attitude control has been achieved in some studies using RBFNN, few results in attitude dynamics of hypersonic vehicles are found, let alone integrate it into FTC.

Inspired by the above researches, an adaptive sliding mode FTC scheme based on the RBFNN technique is proposed in this article to deal with the attitude tracking problem for a hypersonic vehicle with unknown external disturbances, additive fault and partial loss of effectiveness fault. The main attraction of this study is that the unknown additive fault is handled using RBFNN, and the proposed FTC scheme, which is based on the estimation of the unknown additive fault, handles additive fault and partial loss of effectiveness fault simultaneously. Furthermore, the designed FTC scheme does not require any information about the effectiveness factors of partial loss of effectiveness fault.

This article is organized as follows. In ‘Problem formulation’ section, the healthy and faulty models of a hypersonic vehicle and the control objective are given. The unknown additive fault is estimated using RBFNN in ‘RBFNNs estimator’ section. The main researches are stated in ‘Adaptive sliding mode fault-tolerant controller design’ section; combined with RBFNN and adaptive control laws, the sliding mode fault-tolerant controllers are designed, which ensure the realization of the control objective. In ‘Simulation results’ section, simulation results are given to verify the effectiveness of the proposed approach, followed by some conclusions in the last section. The definitions on the norm of a vector and the norm of a matrix are given in Appendix 1.

Problem formulation

The considered hypersonic aircraft in this study is X-33. ²⁹ The dynamic equations and attitude kinematics of the aircraft in its re-entry mode are given as follows

$J\dot{\omega }=-\Omega J\omega +T+d$

1

$\dot{\gamma }=R(\cdot )\omega$

2

where $J\in R^{3\times 3}$ is the symmetric, positive definite moment of inertia tensor, $\omega =[p,q,r{]}^T$ is the angular rate vector with p, q and r being the roll, pitch and yaw rate, respectively. $T=[T_1,T_2,T_3{]}^T\in R^3$ is the control torque vector and $d\in R^3$ is the external disturbance torque vector, and $\gamma =[\phi ,\beta ,\alpha {]}^T$ is the attitude angle vector with φ, β and α being the bank, sideslip and angle of attack, respectively. The skew symmetric matrix Ω is given by

$\Omega =\left[\begin{array}{ccc}0& -r& q\\ r& 0& -p\\ -q& p& 0\end{array}\right]$

3

and the matrix R(⋅) in the kinematics equation (2) is defined as

$R(\cdot )=\left[\begin{array}{ccc}\text{cos}\alpha & 0& \text{sin}\alpha \\ \text{sin}\alpha & 0& -\text{cos}\alpha \\ 0& 1& 0\end{array}\right]$

4

When additive and partial loss of effectiveness faults appear simultaneously, the dynamic equation (1) can be rewritten as ^30,31

$J\dot{\omega }=-\Omega J\omega +\sigma \left(t\right)T+d+f\left(t,\omega ,\gamma \right)$

5

where σ(t) = diag(σ₁₁, σ₂₂, σ₃₃) and σ_ii (i = 1, 2, 3) represent the effectiveness factors satisfying $0<{\mu }_0\le {\sigma }_{ii}\le 1$ , which indicates the healthy condition of the actuators. σ_ii = 1 indicates that the ith actuator is healthy, $0<{\sigma }_{ii}<1$ implies how much the ith actuator loses its effectiveness and $f(t,\omega ,\gamma )=[f_1,f_2,f_3{]}^T$ represents the additive fault affecting the attitude control system.

Assumption 1

The disturbance torque d in equation (5) is bounded, that is, $\parallel d(t)\parallel \le \overline{d}$ , where $\overline{d}$ is an unknown positive constant. ³²

Assumption 2

The additive fault $f(t,\omega ,\gamma )$ in equation (5) is an unknown nonlinear function vector. ³¹

Assumption 3

The lower bound μ₀ of σ_ii (i = 1, 2, 3) in equation (5) is an unknown positive constant. ³⁰

The control objective is to determine the control torque T in equation (5) such that all closed-loop signals are bounded and the desired attitude angle vector ${\gamma }_d=[{\phi }_d,{\beta }_d,{\alpha }_d{]}^T$ is robustly followed when there exist unknown external disturbances, additive and partial loss of effectiveness faults.

RBFNNs estimator

It has been proved that given any arbitrary ε > 0, a three-layer RBFNN with single hidden layer containing a sufficiently large number of neurons can approach to a continuous nonlinear function f(x) with the approaching error ε. Hence, the nonlinear function f(x) can be represented as ²⁸

$f\left(x\right)=w^{*T}\varphi \left(x\right)+\epsilon$

6

where x ∈ R^p is the input vector, $\varphi (x)=[{\varphi }_1(x),{\varphi }_2(x),\cdots ,{\varphi }_m(x{)]}^T:R^p\to R^m$ is the vector of the radial basis functions, m is the number of hidden layer neurons and ϕ_i(x) is the output of the ith neuron of the hidden layer. $w^{*}\in R^m$ is the optimal weight vector and ε stands for the approaching error.

In general, ϕ_i(x) is defined as the following Gaussian function

${\varphi }_i\left(x\right)=\text{exp}\left(-\frac{\parallel x-b_i{\parallel }^2}{c_i^2}\right),i=1,2,\cdots ,m,$

7

where $b_i=[b_{i1},b_{i2},\cdots ,b_{ip}{]}^T\in R^p$ and c_i ∈ R are the centre and the spread of the ith Gaussian function, respectively.

Therefore, a continuous nonlinear function vector $F(x)=[f_1(x),f_2(x),\cdots ,f_n(x{)]}^T$ can be expressed as

$\begin{array}{l}{f}_i\left(x\right)=w_i^{*T}\varphi \left(x\right)+{\epsilon }_i,i=1,2,\cdots n\\ F\left(x\right)=W^{*T}\varphi \left(x\right)+\epsilon \end{array}$

8

where $W^{*T}=[w_1^{*},w_2^{*},\cdots ,w_n^{*}{]}^T$ and $\epsilon =[{\epsilon }_1,{\epsilon }_2,\cdots ,{\epsilon }_n{]}^T$ .

As a consequence, the unknown additive fault $f(t,\omega ,\gamma )$ in equation (5) can be represented as

$f\left(t,\omega ,\gamma \right)=W^{*T}\varphi \left(t,\omega ,\gamma \right)+\epsilon$

9

where $W^{*}\in R^{m\times 3}$ , $\varphi (x)\in R^m$ and ε ∈ R³ satisfy $\parallel \epsilon \parallel \le \overline{\epsilon }$ , with $\overline{\epsilon }$ being an unknown positive constant. Using the RBFNN, the estimation of $f(t,\omega ,\gamma )$ is designed as

$\widehat{f}\left(t,\omega ,\gamma \right)={\widehat{W}}^T\varphi \left(t,\omega ,\gamma \right)$

10

where $\widehat{W}\in R^{m\times 3}$ is the estimation of the optimal weight matrix W*, W* is a quantity only for theoretical analysis and can be calculated as follows

$W^{*}=\text{arg}{\displaystyle \underset{\widehat{W}\in {\Omega }_f}{{\displaystyle \text{min}}}}\left({\displaystyle \underset{x\in {\Omega }_x}{{\displaystyle \text{sup}}}}|\widehat{f}\left(t,\omega ,\gamma \right)-f\left(t,\omega ,\gamma \right)|\right)$

11

where ${\Omega }_f=\{\widehat{W},\parallel \widehat{W}\parallel \le \delta \}$ is a valid field of $\widehat{W}$ , δ is the upper bound and Ω_x is a compact subset of R^p.

Adaptive sliding mode fault-tolerant controller design

In this section, the two-loop sliding mode FTC scheme, which consists of an outer attitude angle loop and an inner angular rate loop, is designed to achieve attitude tracking. The design of the controllers consists of two steps. Firstly, the outer-loop sliding mode controller is designed to realize attitude tracking by taking the angular rate vector ω as the virtual control input. Then, the inner-loop sliding mode controller is designed so that the command angular rate vector ω_c is tracked. Based on this design, the tracking of the desired attitude angle vector γ_d is achieved.

Outer-loop sliding mode controller design

For the attitude kinematics (2), let the attitude angle tracking error vector be $e_{\gamma }={\gamma }_d-\gamma$ . As in the study by Shtessel and McDuffie, ²⁹ introduce a sliding mode variable

$s_1={\left[s_{11},s_{12},s_{13}\right]}^T=e_{\gamma }+K_{\gamma }{\displaystyle {\int }_0^t}{e}_{\gamma }d\tau$

12

where $K_{\gamma }=\text{diag}(k_1,k_2,k_3)$ and k_i > 0 (i = 1, 2, 3) are design parameters.

It is clear that on the sliding mode surface s₁ = 0, the motion is governed by

$s_1=e_{\gamma }+K_{\gamma }{\displaystyle {\int }_0^t}{e}_{\gamma }d\tau =0$

13

which ensures the asymptotic convergence of the tracking error. The dynamic evolution of s₁ satisfies

${\dot{s}}_1={\dot{\gamma }}_d-R(\cdot )\omega +K_{\gamma }{e}_{\gamma }=0$

14

Choose the Lyapunov function as

$V_1=\frac{1}{2}{s}_1^T{s}_1$

15

The derivative of V₁ is

${\dot{V}}_1=s_1^T{\dot{s}}_1=s_1^T\left({\dot{\gamma }}_d-R(\cdot )\omega +K_{\gamma }{e}_{\gamma }\right)$

16

To ensure the asymptotic stability of the origin of the system (13), the virtual controller ω_c for the outer loop is chosen as

${\omega }_c=R^{-1}\left({\dot{\gamma }}_d+K_{\gamma }{e}_{\gamma }\right)R^{-1}{\rho }_1\text{sign}\left(s_1\right)+R^{-1}{\rho }_2{s}_1$

17

where ρ₁ > 0 is a design parameter and sign(s₁) is a vector-valued sign function defined as

$\begin{array}{c}\text{sign}\left(s_1\right)={\left[\text{sign}\left(s_{11}\right),\text{sign}\left(s_{12}\right),\text{sign}\left(s_{13}\right)\right]}^T\\ \text{sign}\left(x\right)=\{\begin{array}{ll}1,\hfill & \hspace{0.17em}x>0\hfill \\ 0,\hfill & \hspace{0.17em}x=0\hfill \\ -1,\hfill & \hspace{0.17em}x<0\hfill \end{array}\end{array}$

18

Substituting equation (17) into (16) yields

${\dot{V}}_1=-{\rho }_1{s}_1^T\text{sign}\left(s_1\right)=-{\rho }_1{\displaystyle \sum _{i=1}^3}\left|s_{1i}\right|-{\rho }_2\parallel s_1{\parallel }^2$

19

Clearly, the desired sliding mode surface is guaranteed, and s₁ = 0 will be reached in a finite time. The asymptotic tracking of γ_d by γ is guaranteed.

However, the command angular rate vector ω_c is discontinuous, it will chatter on the sliding surface and cannot be accurately tracked in the inner loop. Therefore, the discontinuous function sign(s₁) in equation (17) is changed to the saturation function $\text{sat}(s_1)=[\text{sat}(s_{11}),\text{sat}(s_{12}),\text{sat}(s_{13}{)]}^T$ , then sat(x) is defined as

$\text{sat}\left(x\right)=\{\begin{array}{ll}\frac{x}{{\eta }_1},\hfill & \left|x\right|\hspace{0.17em}\le \hspace{0.17em}{\eta }_1\hfill \\ \text{sign}\left(x\right),\hfill & \left|x\right|\hspace{0.17em}>\hspace{0.17em}{\eta }_1\hfill \end{array}$

20

where η₁ is a small positive constant. As a consequence, the continuous command angular rate vector is shown as

${\omega }_c=R^{-1}\left({\dot{\gamma }}_d+K_{\gamma }{e}_{\gamma }\right)+R^{-1}{\rho }_1\hspace{0.17em}\text{sat}\left(s_1\right)+R^{-1}{\rho }_2{s}_1$

21

Inner-loop sliding mode controller design

In the study by Xiao et al., ²⁴ the adaptive method is adopted to deal with the unknown partial loss of effectiveness fault of spacecraft. However, only this kind of fault is considered, the fault-tolerant capability is limited. On the basis of this study, two kinds of faults occurring simultaneously are considered in the following design of controller. The following task is to design an inner-loop sliding mode controller T to enable that the command angular rate vector ω_c is tracked by the angular rate vector ω asymptotically.

Define the angular rate tracking error $e_{\omega }={\omega }_c-\omega$ and introduce the sliding mode variable

$s_2={\left[s_{21},s_{22},s_{23}\right]}^T=e_{\omega }+K_{\omega }{\displaystyle {\int }_0^t}{e}_{\omega }d\tau$

22

where $K_{\omega }=\text{diag}(k_4,k_5,k_6)$ and $k_i>0(i=4,5,6)$ are design parameters. The sliding mode surface is defined as

$s_2=e_{\omega }+K_{\omega }{\displaystyle {\int }_0^t}{e}_{\omega }d\tau =0$

23

whose derivative is

${\dot{s}}_2={\dot{e}}_{\omega }+K_{\omega }{e}_{\omega }={\dot{\omega }}_c-\dot{\omega }+K_{\omega }{e}_{\omega }$

24

To facilitate the design of controller, equation (5) can be rewritten as

$J\dot{\omega }=-\Omega J\omega +T-\Delta \sigma \left(t\right)T+d+f(\cdot )$

25

where $\Delta \sigma (t)=I_3-\sigma (t)=\text{diag}(1-{\sigma }_{11}\mathrm{,1}-{\sigma }_{22}\mathrm{,1}-{\sigma }_{33})$ satisfying

$\begin{array}{l}\parallel \Delta \sigma \left(t\right)\parallel =\parallel \text{diag}\left(1-{\sigma }_{11}\mathrm{,1}-{\sigma }_{22}\mathrm{,1}-{\sigma }_{33}\right)\parallel \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\text{max}\{1-{\sigma }_{ii},i=1,2,3\}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\le 1-{\mu }_0\end{array}$

26

Substituting equation (25) into equation (24) yields

$\begin{array}{l}{\dot{s}}_2={\dot{\omega }}_c+K_{\omega }{e}_{\omega }\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-J^{-1}\left[-\Omega J\omega +T-\Delta \sigma \left(t\right)T+d+f(\cdot )\right]\end{array}$

27

Recall assumption 2, according to the description of RBFNN in ‘RBFNNs estimator’ section, the additive fault f(⋅) in equation (25) is replaced by

$f(\cdot )=W^{*T}\varphi (\cdot )+\epsilon$

28

Note assumption 1, the external disturbance d in equation (25) and ε in equation (28) satisfy the following condition

$\parallel d+\epsilon \parallel \hspace{0.17em}\le \hspace{0.17em}\parallel d\parallel +\parallel \epsilon \parallel \hspace{0.17em}\le \overline{d}+\overline{\epsilon }=\overline{D}$

29

where $\overline{D}>0$ is the unknown upper bound.

The required torque command T is designed as

$T=T_n+T_f$

30

where T_n is the normal controller, T_f is the fault-tolerant controller compensating for the influences of external disturbances and faults and the two terms are developed as

$T_n=J{\dot{\omega }}_c+J{K}_{\omega }{e}_{\omega }+\Omega J\omega$

31

$T_f=\widehat{D}\text{sign}\left(s_2\right)-{\widehat{W}}^T\varphi (\cdot )+\left(k_0+k_1\right)s_2$

32

where k₁ is defined as

$k_1={\widehat{\mu }}_1\frac{\parallel T_n+\widehat{D}\text{sign}\left(s_2\right)-{\widehat{W}}^T\phi (\cdot )\parallel }{\parallel s_2\parallel }$

33

and the adaptive laws are designed as

$\dot{\widehat{D}}=c\parallel s_2\parallel$

34

$\dot{\widehat{W}}=-P\varphi (\cdot )s_2^T$

35

${\dot{\widehat{\mu }}}_1=\lambda \parallel T_n+\widehat{D}\text{sign}\left(s_2\right)-{\widehat{W}}^T\phi (\cdot )\parallel \parallel s_2\parallel$

36

where $\widehat{D}$ , $\widehat{W}$ and ${\widehat{\mu }}_1$ are the estimates of $\overline{D}$ , W* and 1/μ₀, respectively. k₀, c and λ are positive constants and $P\in R^{m\times m}$ is a symmetric positive definite matrix.

Choose the following positive definite Lyapunov function

$V_2=\frac{1}{2}{s}_2^{T}J{s}_2+\frac{1}{2c}{\tilde{D}}^2+\frac{1}{2}\text{tr}\left\{{\tilde{W}}^T{P}^{-1}\tilde{W}\right\}+\frac{{\mu }_0}{2\lambda }{\tilde{\mu }}_1^2$

37

where $\tilde{D}=\widehat{D}-\overline{D}$ , ${\tilde{W}}^T={\widehat{W}}^T-W^{*T}$ and ${\tilde{\mu }}_1={\widehat{\mu }}_1-{\mu }_1={\widehat{\mu }}_1-1/{\mu }_0$ are the estimating errors.

The time derivative of V₂ is

${\dot{V}}_2=s_2^{T}J{\dot{s}}_2+\frac{1}{c}\tilde{D}\dot{\tilde{D}}+\text{tr}\left\{{\tilde{W}}^T{P}^{-1}\dot{\tilde{W}}\right\}+\frac{{\mu }_0}{\lambda }{\tilde{\mu }}_1{\dot{\tilde{\mu }}}_1$

38

Let $T_{f1}=T_n+\widehat{D}\text{sign}\left(s_2\right)-{\widehat{W}}^T\varphi (\cdot )$ , and substitute equations (27) and (28) and (30) to (36) into equation (38), we have

$\begin{array}{l}{\dot{V}}_2=s_2^T\left[J{\dot{\omega }}_c+J{K}_{\omega }{e}_{\omega }+\Omega J\omega -T+\Delta \sigma \left(t\right)T-d-\left(W^{*T}\varphi +\epsilon \right)\right]+\frac{1}{c}\tilde{D}\left(c\parallel s_2\parallel \right)-\text{tr}\left\{{\tilde{W}}^T{P}^{-1}P\varphi s_2^T\right\}+\frac{{\mu }_0}{\lambda }\left({\widehat{\mu }}_1-{\mu }_1\right)\left(\lambda \parallel T_{f1}\parallel \parallel s_2\parallel \right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=s_2^T\left[{\widehat{W}}^T\varphi -\widehat{D}\text{sign}\left(s_2\right)-k_0{s}_2-{\widehat{\mu }}_1\parallel T_{f1}\parallel \frac{s_2}{\parallel s_2\parallel }+\Delta \sigma \left(t\right)T-\left(d+\epsilon \right)-W^{*T}\varphi \right]+\tilde{D}\parallel s_2\parallel -\text{tr}\left\{{\tilde{W}}^T\varphi s_2^T\right\}+{\mu }_0\left({\widehat{\mu }}_1-{\mu }_1\right)\parallel T_{f1}\parallel \parallel s_2\parallel \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\le s_2^T\left({\widehat{W}}^T\varphi -W^{*T}\varphi \right)-\widehat{D}\parallel s_2\parallel -k_0\parallel s_2{\parallel }^2-{\widehat{\mu }}_1\parallel T_{f1}\parallel \parallel s_2\parallel +s_2^T\Delta \sigma \left(t\right)T-s_2^T\left(d+\epsilon \right)+\left(\widehat{D}-\overline{D}\right)\parallel s_2\parallel -\text{tr}\left\{{\tilde{W}}^T\varphi s_2^T\right\}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\mu }_0\left({\widehat{\mu }}_1-{\mu }_1\right)\parallel T_{f1}\parallel \parallel s_2\parallel \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=s_2^T{\tilde{W}}^T\varphi -\text{tr}\left\{{\tilde{W}}^T\varphi s_2^T\right\}-\widehat{D}\parallel s_2\parallel -s_2^T\left(d+\epsilon \right)+\left(\widehat{D}-\overline{D}\right)\parallel s_2\parallel -k_0\parallel s_2{\parallel }^2-{\widehat{\mu }}_1\parallel T_{f1}\parallel \parallel s_2\parallel +s_2^T\Delta \sigma \left(t\right)T+{\mu }_0\left({\widehat{\mu }}_1-{\mu }_1\right)\parallel T_{f1}\parallel \parallel s_2\parallel \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\le -k_0\parallel s_2{\parallel }^2-{\widehat{\mu }}_1\parallel T_{f1}\parallel \parallel s_2\parallel +s_2^T\Delta \sigma \left(t\right)\left(T_{f1}+k_0{s}_2+{\widehat{\mu }}_1\parallel T_{f1}\parallel \frac{s_2}{\parallel s_2\parallel }\right)+{\mu }_0\left({\widehat{\mu }}_1-{\mu }_1\right)\parallel T_{f1}\parallel \parallel s_2\parallel \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\le -k_0\parallel s_2{\parallel }^2-{\widehat{\mu }}_1\parallel T_{f1}\parallel \parallel s_2\parallel +\parallel s_2^T\parallel \parallel \Delta \sigma \left(t\right)\parallel \left(\parallel T_{f1}\parallel +k_0\parallel s_2\parallel +{\widehat{\mu }}_1\parallel T_{f1}\parallel \right)+{\mu }_0\left({\widehat{\mu }}_1-{\mu }_1\right)\parallel T_{f1}\parallel \parallel s_2\parallel \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\le -k_0\parallel s_2{\parallel }^2-{\widehat{\mu }}_1\parallel T_{f1}\parallel \parallel s_2\parallel +\left(1-{\mu }_0\right)\parallel s_2^T\parallel \left(\parallel T_{f1}\parallel +k_0\parallel s_2\parallel +{\widehat{\mu }}_1\parallel T_{f1}\parallel \right)+{\mu }_0\left({\widehat{\mu }}_1-{\mu }_1\right)\parallel T_{f1}\parallel \parallel s_2\parallel \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=-{\mu }_0{k}_0\parallel s_2{\parallel }^2+\left[\left(1-{\mu }_0\right)\left(1+{\widehat{\mu }}_1\right)-{\widehat{\mu }}_1\right]\parallel T_{f1}\parallel \parallel s_2^T\parallel +{\mu }_0\left({\widehat{\mu }}_1-{\mu }_1\right)\parallel T_{f1}\parallel \parallel s_2\parallel \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=-{\mu }_0{k}_0\parallel s_2{\parallel }^2-{\mu }_0\parallel T_{f1}\parallel \parallel s_2\parallel \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\le -{\mu }_0{k}_0\parallel s_2{\parallel }^2\le 0\end{array}$

39

Therefore, V₂ is bounded and it is easy to find that $s_2\in L_{\infty }$ from equation (37). As ${\displaystyle {\int }_0^{\infty }}{\dot{V}}_2\text{d}t=V_2(\infty )-V_2(0)$ , $\underset{t\to \infty }{{\displaystyle \text{lim}}}\hspace{0.17em}{V}_2(t)=V_2(\infty )$ exists for some finite $V_2(\infty )\in R^{+}$ and $V_2(0)<\infty$ . ${\displaystyle {\int }_0^{\infty }}\parallel s_2{\parallel }^2\text{d}t<\infty$ can be obtained based on equation (39). To sum up, $s_2\in L_2\cap L_{\infty }$ . From Barbalat’s lemma, ³³ it is guaranteed that the sliding surface s₂ converges to zero asymptotically. Then, $\underset{t\to \infty }{{\displaystyle \text{lim}}}{e}_{\omega }={\omega }_c-\omega =0$ is obtained based on equation (22).

Similarly, the discontinuous sign function sign(s₂) in equation (32) is replaced by the saturation function $\text{sat}(s_2)=[\text{sat}(s_{21}),\text{sat}(s_{22}),\text{sat}(s_{23}{)]}^T$ . Moreover, in order to avoid singularity of control torque, when the value of ∥ s₂ ∥ is very close to zero, the boundary layer function $s_2/(\parallel s_2\parallel +{\eta }_2)$ is applied to approximate the nonlinear function $s_2/\parallel s_2\parallel$ in equation (32), where η₂ is a very small positive constant. Then, the fault-tolerant controller T_f can be described by

$T_f=\widehat{D}\text{sat}\left(s_2\right)-{\widehat{W}}^T\varphi (\cdot )+k_0{s}_2+k_1{s}_2$

40

where

$k_1{s}_2=\{\begin{array}{ll}{\widehat{\mu }}_1\parallel T_n+\widehat{D}\text{sat}\left(s_2\right)-{\widehat{W}}^T\varphi (\cdot )\parallel \frac{s_2}{\parallel s_2\parallel },\hfill & \parallel s_2\parallel \hspace{0.17em}>\hspace{0.17em}{\eta }_2\hfill \\ {\widehat{\mu }}_1\parallel T_n+\widehat{D}\text{sat}\left(s_2\right)-{\widehat{W}}^T\varphi (\cdot )\parallel \frac{s_2}{\parallel s_2\parallel +{\eta }_2},\hfill & \parallel s_2\parallel \hspace{0.17em}\le \hspace{0.17em}{\eta }_2\hfill \end{array}$

41

And the adaptive law (36) is replaced by

${\dot{\widehat{\mu }}}_1=\lambda \parallel T_n+\widehat{D}\text{sat}\left(s_2\right)-{\widehat{W}}^T\varphi (\cdot )\parallel \parallel s_2\parallel$

42

It is of interest to determine how the inner-loop stabilization properties are altered when the fault-tolerant controller (40) and the adaptive law (42) are applied to the control system. The impact of this substitution on inner-loop stability will be addressed. To make the analysis precise, the parameter projection method ^34,35 is firstly applied to guarantee the boundedness of estimated parameters and avoid parameter drift.

Suppose ${\Omega }_{\overline{D}}$ and ${\Omega }_{{\mu }_1}$ are two convex sets, then ${\Omega }_{\overline{D}}=\left\{\overline{D}\right|{\overline{D}}_{\text{min}}\le \overline{D}\le {\overline{D}}_{\text{max}}\}$ and ${\Omega }_{{\mu }_1}=\left\{{\mu }_1\right|{\mu }_1{ }_{\text{min}}\le {\mu }_1\le {\mu }_1{ }_{\text{max}}\}$ . The values of ${\overline{D}}_{\text{min}}$ , ${\overline{D}}_{\text{max}}$ , μ_{1 min} and μ_{1 max} represent the lower and upper bound of $\overline{D}$ and μ₁, respectively. Let ${\Omega }_{{\delta }_{\overline{D}}}=\left\{\overline{D}\right|{\overline{D}}_{\text{min}}-{\delta }_{\overline{D}}\le \overline{D}\le {\overline{D}}_{\text{max}}+{\delta }_{\overline{D}}\}$ and ${\Omega }_{{\delta }_{{\mu }_1}}=\left\{{\mu }_1\right|{\mu }_1{ }_{\text{min}}-{\delta }_{{\mu }_1}\le {\mu }_1\le {\mu }_1{ }_{\text{max}}+{\delta }_{{\mu }_1}\}$ , where ${\delta }_{\overline{D}}$ and ${\delta }_{{\mu }_1}$ are arbitrarily small positive real numbers. Define the projection $Proj(\widehat{D},{\Phi }_1)$ by

$\text{Proj}\left(\widehat{D},{\Phi }_1\right)=\{\begin{array}{ll}{\phi }_1,\hfill & \text{if}\hspace{0.17em}{D}_{\text{min}}\le \hspace{0.17em}\hspace{0.17em}D\le \hspace{0.17em}\hspace{0.17em}{D}_{\text{max}}\hspace{0.17em}\text{or} \hfill \\ \hfill & \text{if}\hspace{0.17em}D>\hspace{0.17em}{D}_{\text{max}}\hspace{0.17em}\text{and}\hspace{0.17em}{\phi }_1\le 0\hspace{0.17em}\text{or} \hfill \\ \hfill & \text{if}\hspace{0.17em}D<\hspace{0.17em}{D}_{\text{min}}\hspace{0.17em}\text{and}\hspace{0.17em}{\phi }_1\ge 0 \hfill \\ {\overline{\phi }}_1,\hfill & \text{if}\hspace{0.17em}D>\hspace{0.17em}{D}_{\text{max}}\hspace{0.17em}\text{and}\hspace{0.17em}{\phi }_1>0 \hfill \\ {\stackrel{⌣}{\phi }}_1,\hfill & \text{if}\hspace{0.17em}D<\hspace{0.17em}{D}_{\text{min}}\hspace{0.17em}\text{and}\hspace{0.17em}{\phi }_1<0 \hfill \end{array}$

43

where

$\begin{array}{l}{\phi }_1=\dot{\widehat{D}}=c\parallel s_2\parallel ,{\overline{\phi }}_1=\left(1+\frac{{\overline{D}}_{\text{max}}-\widehat{D}}{{\delta }_{\overline{D}}}\right){\phi }_1,\\ {\stackrel{⌣}{\phi }}_1=\left(1+\frac{\widehat{D}-{\overline{D}}_{\text{min}}}{{\delta }_{\overline{D}}}\right){\phi }_1\end{array}$

44

and the projection $\text{Proj}({\widehat{\mu }}_1,{\Phi }_2)$ is taken as

$\text{Proj}\left({\widehat{\mu }}_1,{\Phi }_2\right)=\{\begin{array}{ll}{\phi }_2,\hfill & \text{if}\hspace{0.17em}{\mu }_1{ }_{\text{min}}\le \hspace{0.17em}\hspace{0.17em}{\mu }_1\le \hspace{0.17em}{\mu }_1{ }_{\text{max}}\hspace{0.17em}\text{or}\hfill \\ \hfill & \text{if}\hspace{0.17em}\hspace{0.17em}{\widehat{\mu }}_1>{\mu }_1{ }_{\text{max}}\hspace{0.17em}\text{and}\hspace{0.17em}{\phi }_2\le 0\hspace{0.17em}\text{or} \hfill \\ \hfill & \text{if}\hspace{0.17em}\hspace{0.17em}{\widehat{\mu }}_1<{\mu }_1{ }_{\text{min}}\hspace{0.17em}\text{and}\hspace{0.17em}{\phi }_2\ge 0 \hfill \\ {\overline{\phi }}_2,\hfill & \text{if}\hspace{0.17em}\hspace{0.17em}{\mu }_1>{\mu }_1{ }_{\text{max}}\hspace{0.17em}\text{and}\hspace{0.17em}{\phi }_2>0 \hfill \\ {\stackrel{⌣}{\phi }}_2,\hfill & \text{if}\hspace{0.17em}\hspace{0.17em}{\mu }_1<{\mu }_1{ }_{\text{min}}\hspace{0.17em}\text{and}\hspace{0.17em}{\phi }_2<0 \hfill \end{array}$

45

where

$\begin{array}{l}{\phi }_2={\dot{\widehat{\mu }}}_1=\lambda \parallel T_n+\widehat{D}\text{sat}\left(s_2\right)-{\widehat{W}}^T\varphi (\cdot )\parallel \parallel s_2\parallel ,\hfill \\ {\overline{\phi }}_2=\left(1+\frac{{\mu }_1{ }_{\text{max}}-{\widehat{\mu }}_1}{{\delta }_{{\mu }_1}}\right){\phi }_2,{\stackrel{⌣}{\varphi }}_2=\left(1+\frac{{\widehat{\mu }}_1-{\mu }_1{ }_{\text{min}}}{{\delta }_{{\mu }_1}}\right){\phi }_2\hfill \end{array}$

46

By applying the projection operator method, the parameter adaptive laws about $\widehat{D}$ and ${\widehat{\mu }}_1$ are taken as

$\dot{\widehat{D}}=\text{Proj}\left(\widehat{D},{\Phi }_1\right),{\dot{\widehat{\mu }}}_1=\text{Proj}\left({\widehat{\mu }}_1,{\Phi }_2\right)$

47

It has been verified in the study by Khalil ³⁵ that $\widehat{D}(t)\in {\Omega }_{{\delta }_{\overline{D}}}$ and ${\widehat{\mu }}_1(t)\in {\Omega }_{{\delta }_{{\mu }_1}},\forall t\ge 0$ when $\widehat{D}(0)\in {\Omega }_{\overline{D}}$ and ${\widehat{\mu }}_1(0)\in {\Omega }_{{\mu }_1}$ .

Then, in order to analyse the stability, choose the same Lyapunov function as equation (37)

$V_3=\frac{1}{2}{s}_2^{T}J{s}_2+\frac{1}{2c}{\tilde{D}}^2+\frac{1}{2}\text{tr}\left({\tilde{W}}^T{P}^{-1}\tilde{W}\right)+\frac{{\mu }_0}{2\lambda }{\tilde{\mu }}_1^2$

48

Based on the characteristics of sliding mode surface s₂, two cases are considered.

Case 1: $\parallel s_2\parallel >{\eta }_2$ .

Let $T_{f2}=T_n+\widehat{D}\text{sat}(s_2)-{\widehat{W}}^T\varphi (\cdot )$ . According to the proof process of equation (39), it can be obtained that

$\begin{array}{l}{\dot{V}}_3=s_2^T\left[{\widehat{W}}^T\varphi -\widehat{D}\text{sign}\left(s_2\right)-\widehat{D}\left(\text{sat}\left(s_2\right)-\text{sign}\left(s_2\right)\right)-k_0{s}_2-{\widehat{\mu }}_1\parallel T_{f2}\parallel \frac{s_2}{\parallel s_2\parallel }+\Delta \sigma \left(t\right)T-\left(d+\epsilon \right)-W^{*T}\varphi \right]\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\tilde{D}\parallel s_2\parallel -\text{tr}\left\{{\tilde{W}}^T\varphi s_2^T\right\}+{\mu }_0({\widehat{\mu }}_1-{\mu }_1)\parallel T_{f2}\parallel \parallel s_2\parallel \\ \le -k_0\parallel s_2{\parallel }^2-\widehat{D}{s}_2^T\left(\text{sat}\left(s_2\right)-\text{sign}\left(s_2\right)\right)-{\widehat{\mu }}_1\parallel T_{f2}\parallel \parallel s_2\parallel +s_2^T\Delta \sigma \left(t\right)\left(T_{f2}+k_0{s}_2+{\widehat{\mu }}_1\parallel T_{f2}\parallel \frac{s_2}{\parallel s_2\parallel }\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\mu }_0\left({\widehat{\mu }}_1-{\mu }_1\right)\parallel T_{f2}\parallel \parallel s_2\parallel \\ \le -{\mu }_0{k}_0\parallel s_2{\parallel }^2-{\mu }_0\parallel T_{f2}\parallel \parallel s_2\parallel +\widehat{D}\parallel s_2\parallel \parallel \text{sat}\left(s_2\right)-\text{sign}\left(s_2\right)\parallel \\ \le -{\mu }_0{k}_0\parallel s_2{\parallel }^2+\sqrt{3}\widehat{D}\parallel s_2\parallel -{\mu }_0\parallel T_{f2}\parallel \parallel s_2\parallel \end{array}$

49

According to equation (49), it is clear that when $\parallel s_2\parallel >\sqrt{3}\widehat{D}/({\mu }_0{k}_0)$ , ${\dot{V}}_3<0$ . Therefore, it can be inferred that the inner-loop control system is global uniform ultimate bounded stable, ³⁶ and $\underset{t\to \infty }{{\displaystyle \text{lim}}}\parallel s_2(t)\parallel \le \sqrt{3}\widehat{D}/({\mu }_0{k}_0)$ .

Remark 1

Actually, according to practical experience, $\parallel s_2\parallel >\sqrt{3}\widehat{D}/({\mu }_0{k}_0)$ is sufficient but not necessary for ${\dot{V}}_3<0$ . From equation (49), it can be observed that the inequality $-{\mu }_0\parallel T_{f2}\parallel \parallel s_2\parallel <0$ always stands up, and ∥ s₂ ∥ should not be so large that ${\dot{V}}_3<0$ . Hence, in practical application, when t → ∞, s₂ converges to a relatively smaller range.

Case 2: $\parallel s_2\parallel \le {\eta }_2$ .

Similarly to case 1, by replacing equations (32) and (36) with equations (40) and (42), the derivative of V₃ is

$\begin{array}{l}{\dot{V}}_3=s_2^T\left[{\widehat{W}}^T\varphi -\widehat{D}\text{sign}\left(s_2\right)-\widehat{D}\left(\text{sat}\left(s_2\right)-\text{sign}\left(s_2\right)\right)-k_0{s}_2-{\widehat{\mu }}_1\parallel T_{f2}\parallel \frac{s_2}{\left(\parallel s_2\parallel +{\eta }_2\right)}+\Delta \sigma \left(t\right)T-\left(d+\epsilon \right)-W^{*T}\varphi \right]+\tilde{D}\parallel s_2\parallel \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-\text{tr}\left\{{\tilde{W}}^T\varphi s_2^T\right\}+{\mu }_0({\widehat{\mu }}_1-{\mu }_1)\parallel T_{f2}\parallel \parallel s_2\parallel \\ \le -k_0\parallel s_2{\parallel }^2-\widehat{D}{s}_2^T\left(\text{sat}\left(s_2\right)-\text{sign}\left(s_2\right)\right)-{\widehat{\mu }}_1\parallel T_{f2}\parallel \parallel s_2\parallel \frac{\parallel s_2\parallel }{\parallel s_2\parallel +{\eta }_2}+s_2^T\Delta \sigma \left(t\right)\left(T_{f2}+k_0{s}_2+{\widehat{\mu }}_1\parallel T_{f2}\parallel \frac{s_2}{\parallel s_2\parallel +{\eta }_2}\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\mu }_0({\widehat{\mu }}_1-{\mu }_1)\parallel T_{f2}\parallel \parallel s_2\parallel \\ \le -k_0\parallel s_2{\parallel }^2+\widehat{D}\parallel s_2\parallel \parallel \text{sat}\left(s_2\right)-\text{sign}\left(s_2\right)\parallel -{\widehat{\mu }}_1\parallel T_{f2}\parallel \parallel s_2\parallel \frac{\parallel s_2\parallel }{\parallel s_2\parallel +{\eta }_2}+\left(1-{\mu }_0\right)\parallel s_2^T\parallel \left(\parallel T_{f2}\parallel +k_0\parallel s_2\parallel +{\widehat{\mu }}_1\parallel T_{f2}\parallel \frac{\parallel s_2\parallel }{\parallel s_2\parallel +{\eta }_2}\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\mu }_0\left({\widehat{\mu }}_1-{\mu }_1\right)\parallel T_{f2}\parallel \parallel s_2\parallel \\ \le -{\mu }_0{k}_0\parallel s_2{\parallel }^2+\sqrt{3}\widehat{D}\parallel s_2\parallel -{\mu }_0\parallel T_{f2}\parallel \parallel s_2\parallel +{\mu }_0{\widehat{\mu }}_1\parallel T_{f2}\parallel \parallel s_2\parallel -{\mu }_0{\widehat{\mu }}_1\parallel T_{f2}\parallel \frac{\parallel s_2{\parallel }^2}{\parallel s_2\parallel +{\eta }_2}\end{array}$

50

When $0<{\widehat{\mu }}_1\le 1$ , ${\dot{V}}_3\le -{\mu }_0{k}_0\parallel s_2{\parallel }^2+\sqrt{3}\widehat{D}\parallel s_2\parallel$ . Recall equation (49), with the same analysis method as stated in case 1, the same conclusion can be obtained. The inner-loop control system is global uniform ultimate bounded stable, ³⁶ and it remains valid that $\underset{t\to \infty }{{\displaystyle \text{lim}}}\parallel s_2(t)\parallel \le \sqrt{3}\widehat{D}/({\mu }_0{k}_0)$ .

Remark 2

Similarly, according to equation (50) and actual application, it can also be derived that the value of ∥ s₂ ∥, which is smaller than $\sqrt{3}\widehat{D}/({\mu }_0{k}_0)$ , may also make ${\dot{V}}_3<0$ . Therefore, in practical applications, the actual convergence domain of s₂ is also not that large.

Through the above stability analysis of cases 1 and 2, it can be summarized that when the parameters related to the designed fault-tolerant controller are selected reasonably, there will always be a time t₀ > 0, so that when $\forall t\ge t_0$ , $\parallel s_2(t)\parallel \le \sqrt{3}\widehat{D}/({\mu }_0{k}_0)$ . Moreover, according to the definition of s₂ in equation (22), the angular rate tracking error $e_{\omega }(t)$ is also bounded and it satisfies that $\parallel e_{\omega }(t)={\omega }_c-\omega \parallel \hspace{0.17em}\le {\epsilon }_1,\forall t\ge t_0$ , where ε₁ is a positive constant.

Furthermore, based on the characteristics of two-loop SMC, how the bounded convergence of angular rate affects the attitude tracking of outer loop must be considered. Hence, in the following discussion, the results of stability analysis of inner loop will be applied to analyse the outer-loop stability.

Choose the same Lyapunov function as equation (15)

$V_4=\frac{1}{2}{s}_1^T{s}_1$

51

The derivative of V₄ is

$\begin{array}{l}{\dot{V}}_4=s_1^T{\dot{s}}_1=s_1^T\left({\dot{\gamma }}_d-R(\cdot )\omega +K_{\gamma }{e}_{\gamma }\right)\hfill \\ =s_1^T\left({\dot{\gamma }}_d-R(\cdot )\left({\omega }_c-e_{\omega }\right)+K_{\gamma }{e}_{\gamma }\right)\hfill \end{array}$

52

Substituting equation (21) into equation (52) yields

$\begin{array}{l}{\dot{V}}_4=-{\rho }_1{s}_1^T\text{sign}\left(s_1\right)-{\rho }_2\parallel s_1{\parallel }^2\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\rho }_1{s}_1^T\left(\text{sat}\left(s_1-\text{sign}\left(s_1\right)\right)\right)+s_1^{T}R(\cdot )e_{\omega }\\ \le -{\rho }_1\parallel s_1\parallel -{\rho }_2\parallel s_1{\parallel }^2+\sqrt{3}{\rho }_1\parallel s_1^T\parallel \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\parallel s_1^T\parallel \parallel R(\cdot )e_{\omega }\parallel \end{array}$

53

Recall the definition of R(⋅) in equation (4), it can be concluded that $\parallel R(\cdot )e_{\omega }\parallel =\parallel e_{\omega }\parallel$ . Then,

${\dot{V}}_4\le -{\rho }_2\parallel s_1{\parallel }^2+\left[\left(\sqrt{3}-1\right){\rho }_1+\parallel e_{\omega }\parallel \right]\parallel s_1\parallel$

54

To make ${\dot{V}}_4<0$ , $\parallel s_1\parallel >[(\sqrt{3}-1){\rho }_1+\parallel e_{\omega }\parallel ]/{\rho }_2$ . According to the stability analysis theory in the study by Ariyur and Krstic, ³⁶ it is concluded that the outer-loop control system is global uniform ultimate bounded stable and $\underset{t\to \infty }{{\displaystyle \text{lim}}}\parallel s_1(t)\parallel \le [(\sqrt{3}-1){\rho }_1+\parallel e_{\omega }\parallel ]/{\rho }_2$ . In the light of the same analysis method as described in inner loop and the definition of s₁ in equation (12), it can be known that there always exists a time t₁ > 0 such that $\parallel e_{\gamma }(t)={\gamma }_d-\gamma \parallel \le {\epsilon }_2$ for all t > t₁, where ε₂ is a positive constant. Moreover, it is easy to see that the size of convergence domain of attitude angle in outer loop is related to the size of convergence domain of angular rate in inner loop.

To sum up, in order to avoid chattering, the modified outer-loop controller (21) and inner-loop controller (40) are used to replace equations (17) and (32), which guarantee the attitude bounded convergence.

Simulation results

In this section, simulation results are given to verify the effectiveness of the designed FTC scheme. The moment of inertia tensor is given in the study by Shtessel and McDuffie ²⁹ and taken as

$J=\left[\begin{array}{ccc}554486& 0& -23002\\ 0& 1136949& 0\\ -23002& 0& 1376852\end{array}\right]\text{kg}\cdot {\text{m}}^2$

It is assumed that the initial attitude angle vector and angular rate vector are ${\gamma }_0{=[0,0,0]}^T\text{rad}$ and ${\omega }_0{=[0.05,0.1,0]}^T\text{rad/s}$ , respectively. The desired attitude angle vector is given by ${\gamma }_d{=[0.1,0,0.2]}^T\text{rad}$ and the external disturbance vector is chosen as $d=[100\text{cos}(0.1t)\mathrm{,80}\text{sin}(t{)\mathrm{,200}]}^T\text{N}\cdot \text{m}$ . The designed controllers parameters are taken as $K_{\gamma }=\text{diag}(0.5,0.5,0.5)$ , $K_{\omega }=\text{diag}(1.5,1.5,1.5)$ , ρ₁ = 0.00001, η₁ = 0.1, η₂ = 0.01, k₀ = 0.1, c = 100, P = 10I₁₃ and λ = 1.

The derivative of ω_c is needed in the inner-loop controller. To avoid the explosion of complexity in calculating the derivative, the following first-order filter is applied to get the differential of ω_c

$\tau {\dot{\overline{\omega }}}_c+{\overline{\omega }}_c={\omega }_c,\hspace{1em}{\overline{\omega }}_c(0)={\omega }_c(0)$

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where ${\overline{\omega }}_c$ is the output of the filter and τ > 0 is a design parameter. In the following simulation, τ is chosen as 1000.

For the purpose of comparison for the effectiveness of FTC, the FTC approach that was developed in the study by Zhao et al. ¹¹ is used to compare with the proposed FTC method in this article. Two cases are simulated.

Case 1: Only additive fault effects on the system.

When t ≥ 20 s, the system undergoes additive time-varying fault, and the details are described as

$f\left(t,\omega ,\gamma \right)=h_1+h_2+h_3$

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where $h_1=[1.8\times {10}^4\text{sin}(t/4)+2\times {10}^4(p^2+\text{exp}(q)),5\times {10}^4\mathrm{,2}\times {10}^4+5\times {10}^4\text{cos}(t-20)-5\times {10}^3\text{exp}(\phi +{\beta }^2{)]}^T$ , $h_2=\text{diag}{(10}^4\mathrm{,1.5}\times {10}^4\mathrm{,1.5}\times {10}^4)\gamma$ and $h_3=\text{diag}(1.5\times {10}^4\mathrm{,5}\times {10}^3,-{10}^4)\omega$ .

The simulation results for this case are shown in Figure 1 to 4. From Figure 1, it is clear that the additive fault $f(t,\omega ,\gamma )$ is estimated accurately using RBFNN. The responses of attitude angle and angular rate are shown in Figures 2 and 3. Compared with the results without FTC, it is observed that the proposed FTC scheme in this article achieves more accurate tracking and better stability performance. From Figures 2 and 3, both FTC schemes can ensure the stability of the control system. However, it is shown that the undesirable chattering occurs in responses of attitude angle and angular rate using the method in the study by Zhao et al. ¹¹ The control input signals are depicted in Figure 4, which implies that the control torques compensate for the effect of additive fault, and the undesired chattering is avoided effectively.

Figure 1.

Additive fault $f(t,\omega ,\gamma )$ and its estimation.

Figure 2.

Attitude angle tracking responses under case 1.

Figure 3.

Angular rate responses under case 1.

Figure 4.

Control torques responses under case 1.

Case 2: Both additive and partial loss of effectiveness faults appear in the system.

When t ≥ 40 s, there exist both additive and partial loss of effectiveness faults, and the effectiveness matrix is given by $\sigma (t)=\text{diag}(0.4,0.8+0.1\text{sin}(t)\mathrm{,0.7})$ , which is unknown for the system. Figures 5 to 7 present the attitude angle tracking responses, angular rate responses and control torques responses, respectively. From Figures 5 and 6, it is clear that the designed fault-tolerant controller has better control performance than the fault-tolerant controller proposed by Zhao et al., ¹¹ and the attitude angle and angular rate cannot satisfactorily track the desired commands using the FTC in the study by Zhao et al. ¹¹ As shown in Figure 7, larger commanded control powers are needed because of the partial loss of torques after 40 s.

Figure 5.

Attitude angle tracking responses under case 2.

Figure 6.

Angular rate responses under case 2.

Figure 7.

Control torques responses under case 2.

Conclusions

In this article, an adaptive sliding mode FTC scheme based on the RBFNN technique has been proposed for a hypersonic vehicle with external disturbances and two kinds of faults. An RBFNN is designed to estimate the unknown additive fault, and the unknown partial loss of effectiveness fault is handled by the adaptive method. Under some reasonable assumptions, the two-loop sliding mode fault-tolerant controllers are designed to guarantee the asymptotical attitude tracking even with the presence of two kinds of faults and external disturbances. The designed fault-tolerant controller does not require any online fault diagnosis to obtain the lower bound on the effectiveness factor of the partial loss of effectiveness fault. The proposed FTC scheme is proved to have excellent control performance. Furthermore, simulation results evaluate the validity of the proposed control scheme.