Robust adaptive multivariable higher-order sliding mode flight control for air-breathing hypersonic vehicle with actuator failures

Abstract

This article proposes an adaptive multivariable higher-order sliding mode control for the longitudinal model of an air-breathing vehicle under system uncertainties and actuator failures. Firstly, a fast finite-time control law is designed for a chain of integrators. Secondly, based on the input/output feedback linearization technique, the system uncertainty and external disturbances are modeled as additive certainty and the actuator failures are modeled as multiplicative uncertainty. By using the proposed fast finite-time control law, a robust multivariable higher-order sliding mode control is designed for the air-breathing hypersonic vehicle with actuator failures. Finally, adaptive laws are proposed for the adaptation of the parameters in the robust multivariable higher-order sliding mode control. Thus, the bounds of the uncertainties are not needed in the control system design. Simulation results show the effectiveness of the proposed robust adaptive multivariable higher-order sliding mode control.

Introduction

Air-breathing hypersonic vehicle (AHV) has been a widely attention-getting technology for the requirement generated by space exploitations and military applications in recent years. The rocket-propelled aircraft systems differ from AHVs, which employ the scramjet engines to get oxygen from the air, and flight at a very high Mach number. Since the dynamics of an AHV is seriously nonlinear, strongly coupled and fast time-varying, the flight control system design is still one of the most difficult challenges in the development of the AHV technology. ¹

In the past decades, many control methods have been extensively investigated based on an AHV longitudinal dynamic model developed by the NASA Langley Research Center. ² By using a genetic algorithm, a linear stochastic robust altitude and velocity tracking control of an AHV was addressed by Marrison and Stengel. ³ In the article by Yang et al., ⁴ based on fast disturbance estimation by using finite-time disturbance observer, a nonlinear dynamic inversion control was designed for an AHV. In the work by Xu et al., ^5,6 based on a longitudinal dynamics of an AHV, neural-network-based adaptive output tracking controllers were investigated. Xu et al. ⁷ proposed a HGO based neural network controller for the AHV, and the tracking error were guaranteed in the semiglobal sense. Sliding mode control (SMC), which is well known as a robust technique to control nonlinear systems operating under uncertainty conditions, is widely used in the flight control system design for an AHV. ^8,9 In the work by Xu et al., ¹⁰ by using a feedback linearization technique, adaptive SMC was investigated for the altitude and velocity control system design for an AHV. In the article by Hu et al., ¹¹ based on the reference model and linearized model, a robust adaptive SMC strategy was studied for the tracking control of a flexible AHV, which can guarantee the global stability of the closed-loop system without the information of the upper bound of the uncertainties and the external disturbance. In the article by Chen et al., ¹² by using the backstepping method, an attitude control system was designed for the AHV where the control input saturation and the transient tracking performance can be guaranteed. The disturbance observer technique is an effective way to handle the uncertainty in flight control systems. ^13,14 In the article by Chen et al., ¹⁵ a terminal sliding mode disturbance observer based controller for the AHV was proposed.

In real applications, due to the hash flight environment of an AHV, some failures may occur in the actuators, that is to say the effectiveness of the aerodynamic control surfaces may be lost partially, which may lead to significant performance degradation or even instability of the closed-loop system. ^16,17 In order to deal with the AHV actuator failures and increase the reliability of the AHV control system, some fault-tolerant control schemes were investigated in the literature, such as neural network based adaptive control ¹⁸ , adaptive backsteping control, ¹⁹ etc. In the article by Jiang et al., ²⁰ T-S fuzzy systems were employed to approximate the nonlinear attitude dynamics and the actuator fault models, which was demonstrated to be superior to some existing fault-tolerant control schemes. Furthermore, this fuzzy system based fault-tolerant control scheme was expanded to the time-varying actuator failures in the article by Shen and Jiang. ²¹ In the article by Sun et al., ²² based on finite-time integral SMC technique, a fault-tolerant control scheme was proposed for a longitudinal model of an AHV in the presence of actuator failures and external disturbances. In the article by Zhao et al., ²³ a dynamic terminal SMC fault-tolerant control scheme was proposed. By introducing an auxiliary system, the states can rapidly and accurately track the desired signals. In the article by Xu et al., ²⁴ compounded adaptive backstepping control and the command filter technique, a fault-tolerant control scheme was designed for the longitudinal dynamics of a generic AHV in the presence of actuator faults and input constraints. By utilizing the parameter adaptation to deal with uncertainties both in the dynamic model and the actuator failures, an output feedback controller was proposed for AHV by He et al. ²⁵

Higher order sliding mode control (HOSMC) is an effective way to handle the control problems for uncertainty nonlinear systems. ^{26 –29} Compared with the traditional sliding mode control (SMC), the signum function acts on the higher order time derivative of the sliding variable. Then, the restriction on the relative degree in the conventional sliding mode control can be removed. At the same time, HOSMC can achieve a better accuracy with respect to discrete sampling time than the traditional SMC. In the article by Tian et al., ³⁰ based on the multiple-time scale principle, a smooth second order sliding mode control (SOSMC) scheme was investigated for a flexible AHV. Finite time convergent HOSMC methods have been recently proposed in articles by Levant, Laghrouche et al. and Defoort et al. ^{31 –35} The algorithm proposed by Levant ³¹ allows the tracking of smooth signals by tuning only one control gain. The quasi-continuous HOSMC algorithm is presented by Levant, ³² which transient features are much better than the previous algorithm by Levant, ³¹ and the control signal is continuous everywhere except the higher order sliding set. An integral sliding mode concept is extended to design a HOSMC algorithm by Levant ³³ and the advantage of the algorithm is that it allows the choosing of transient dynamics, or assigning a transient-time function of initial conditions. In the article by Laghrouche, ³⁴ the algorithm is based on integral SMC and guarantees the establishment of a HOSMC. The advantages of this method are the ease of implementation and the robustness of the system during the entire response. However, two drawbacks appear in this approach: it directly depends on the initial conditions of the system and complex off-line computations are needed before starting the control action. In the article by Defoort, ³⁵ based on geometric homogeneity ³⁷ and integral traditional SMC, a novel HOSMC scheme was proposed. Combining this scheme with adaptive SMC, an adaptive dynamic HOSMC for flexibility was proposed by Zong et al. ³⁶ In spite of real advantages (constructive approach does not require complex off-line computation), this approach suffers the drawback that it is impossible to get arbitrary fast convergence (the restriction inherited from the work by Bhat and Bernstein ³⁷ ), when the initial states are far away from the origin.

In this article, an adaptive multivariable HOSMC scheme for an AHV is proposed. As compared with the existing results, the main contributions of this article are listed as follows.

In order to achieve the quickness of the AHV flight control system under the high-dynamic flight environment, a fast convergent robust HOSMC is proposed firstly by adding the linear terms in the equivalent dynamics. The proposed HOSMC can provide a faster convergence rate when the initial state is far away from the origin than that of the standard homogeneous HOSMC (e.g. the articles by Levant and Defoort et al. ^31,32,33,35 ). Meanwhile, in the sufficiently small vicinity of the origin, the linear terms can be neglected in comparison with the homogeneous terms. Then, higher order sliding mode is established in finite time with fast transient process.

Both additive and multiplicative uncertainties are considered in the AHV input/output linearization model. The additive uncertainty is caused by uncertainty in system dynamics and multiplicative uncertainty is mainly caused by the actuator failures. Meanwhile, adaptation laws are developed for adaptation of the parameters in the proposed HOSMC. As such the upper bounds of the uncertainties are not required to be known in advance.

The remainder of this article is organized as follows: In ‘Hypersonic flight vehicle model’, the dynamic model of the AHV is presented. In ‘Robust multivariable higher-order sliding mode tracking controller’, a robust multivariable HOSMC scheme for the AHV is presented. In ‘Adaptive robust multivariable HOSMC design’, based on the robust multivariable HOSMC, an adaptation law is developed. Some simulation results are presented in ‘Simulation results’. ‘Conclusions’ concludes this article.

Hypersonic flight vehicle model

In this section, a longitudinal dynamic model of an AHV is presented. This dynamic model is taken from the work by Xu et al. ¹⁰ and can be described as follows.

The state equation

$\underset{x}{\underbrace{\left[\begin{array}{l}h\hfill \\ V\hfill \\ \gamma \hfill \\ \alpha \hfill \\ q\hfill \\ \beta \hfill \\ \ddot{\beta }\hfill \end{array}\right]}}=\underset{f(x)}{\underbrace{\left[\begin{array}{c}V\text{sin}\gamma \\ \frac{\rho V^{2}S{C}_T(\beta )\text{cos}\alpha -\rho V^{2}S{C}_D(\alpha )}{2m}-\frac{\mu \text{sin}\gamma }{{(h+R_E)}^2}\\ \frac{\rho V^{2}S{C}_T(\beta )\text{sin}\alpha +\rho V^{2}S{C}_L(\alpha )}{2m}-\frac{(\mu -V^2\gamma )\text{cos}\gamma }{V{(h+R_E)}^2}\\ -\frac{\rho V^{2}S{C}_T(\beta )\text{sin}\alpha +\rho V^{2}S{C}_L(\alpha )}{2m}+\frac{(\mu -V^2\gamma )\text{cos}\gamma }{V{(h+R_E)}^2}+q\\ \frac{\rho V^{2}S\overline{c}}{2I_{yy}}(C_M(\alpha )+C_M(q)-c_e\alpha )\\ \beta \\ -2\varsigma {\omega }_n\beta -{\omega }_n\beta \end{array}\right]}}+\underset{G(x)=[g_1(x),g_2(x)]}{\underbrace{\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ 0& 0\\ 0& \frac{\rho V^{2}S\overline{c}{c}_e}{2I_{yy}}\\ 0& 0\\ {\omega }_n^2& 0\end{array}\right]}}\underset{u}{\underbrace{\left[\begin{array}{l}{\beta }_c\hfill \\ {\delta }_e\hfill \end{array}\right]}}$

1

The output equation

$y=[\begin{array}{cc}V& h\end{array}{]}^{\text{T}}$

2

where the state vector $x=[\begin{array}{c}h V \gamma \alpha q \beta \stackrel{.}{\beta }\end{array}{]}^{\text{T}}$ . h, V, γ, α, q and β represents the altitude, velocity, flight-path angle, angle of attack, pitch rate, and the throttle setting, respectively. The control input vector $u=[\begin{array}{cc}{\beta }_c& {\delta }_e\end{array}{]}^{\text{T}}$ , where ${\delta }_e$ is the elevator deflection, and ${\beta }_c$ is the throttle setting command. The parameters ${\omega }_n$ and ς denote the natural frequency and damping ratio of the β dynamics. $C_M(q)$ and $C_M(\alpha )$ denote the pitch moment coefficients due to pitch rate, and angle of attack respectively. $C_L(\alpha )$ , $C_D(\alpha )$ and $C_T(\beta )$ represent the lift, drag, and thrust coefficients, respectively. The parameters $I_{yy}$ and m represent the moment of inertia, and the mass of the AHV respectively. $R_E$ , μ denote the radius of the Earth, and gravity respectively. The parameters ρ, S, $\overline{c}$ , and $c_e$ denote the air density, the reference area, the mean aerodynamic chord, and the elevator coefficient respectively.

The aerodynamic forces and moment coefficients in equation (1) are calculated as follows (around a trimmed cruising flight condition: $M=15$ , $V=15,060$ ft/s, $h=110,000$ ft, $\gamma =0$ rad, and $q=0$ rad/s)

$\begin{array}{l}{C}_L=0.6203\alpha \hfill \\ C_D=0.6450{\alpha }^2+0.0043378\alpha +0.003772\hfill \\ C_T=\{\begin{array}{cc}0.02576\beta & \hspace{0.17em}\text{if}\hspace{0.17em}\beta <1\\ 0.0224+0.000336\beta & \hspace{0.17em}\text{if}\hspace{0.17em}\beta \ge 1\end{array}\hfill \\ C_M(\alpha )=-0.035{\alpha }^2+0.036617(1+\Delta C_{M\alpha })\alpha +5.3261\times {10}^{-6}\hfill \\ C_M(q)=\frac{\overline{c}}{2V}q(-6.796{\alpha }^2+0.3015\alpha -0.2289)\hfill \end{array}$

3

There exists parameter perturbation Δ to its nominal values

$\begin{array}{l}m=m_0(1+\Delta m)\hfill \\ I_{yy}=I_0(1+\Delta I)\hfill \\ S=S_0(1+\Delta S)\hfill \\ \overline{c}={\overline{c}}_0(1+\Delta \overline{c})\hfill \\ \rho ={\rho }_0(1+\Delta \rho )\hfill \\ c_e=0.0292(1+\Delta c_e)\hfill \end{array}$

4

where the nominal values of m, μ, $I_{yy}$ , ρ, S, $\overline{c}$ , and $R_E$ are given as follows

$\begin{array}{l}{m}_0=9,375 \text{slugs}\hfill \\ I_0=7\times {10}^6\hspace{0.17em}\text{ft}/{\text{s}}^2\hfill \\ S_0=3,603 \hspace{0.17em}\text{ft}\hfill \\ {\overline{c}}_0=80 \hspace{0.17em}\text{ft}\hfill \\ {\rho }_0=0.24325\times {10}^{-4}\hspace{0.17em}\text{slugs}/{\text{ft}}^3\hfill \\ R_E=20,903,500 \hspace{0.17em}\text{ft}\hfill \\ \mu =1.39\times {10}^{16} \hspace{0.17em}{\text{ft}}^3/{\text{s}}^2\hfill \end{array}$

5

The maximum values of the additive uncertainties used in this article are taken as follows

$\begin{array}{l}\left|\Delta m\right|\le 0.20\hfill \\ \left|\Delta I\right|\le 0.20\hfill \\ \left|\Delta S\right|\le 0.10\hfill \\ \left|\Delta \overline{c}\right|\le 0.10\hfill \\ \left|\Delta \rho \right|\le 0.10\hfill \\ \left|\Delta c_e\right|\le 0.20\hfill \\ \left|\Delta C_{M\alpha }\right|\le 0.25\hfill \end{array}$

6

The control objective is that the output V and h can track the prescribed trajectories $V_{\text{ref}}$ and $h_{\text{ref}}$ under system uncertainties and actuator failures.

Robust multivariable higher-order sliding mode tracking controller

Finite time stabilization of a chain of integrators

Consider a chain of integrators as follows

$\begin{array}{l}{z}_i=z_{i+1}\hfill \\ z_r=u\hfill \end{array}.$

7

The following Lemma will be used in the robust HOSMC design.

Lemma 1. (See Bhat and Bernstein ³⁷ ) Let the positive constants $k_1,\dots ,k_r$ be such that the polynomial ${\lambda }^r+k_r{\lambda }^{r-1}+\cdots +k_2\lambda +k_1$ is Hurwitz. There is $\xi \in (0,1)$ such that, for every $\alpha \in (1-\xi ,1)$ , the system given by equation (7) is stabilized at the origin in finite time under the feedback

$u(z)=-k_1\lfloor {z_1\rceil }^{{\alpha }_1}-k_2\lfloor {z_2\rceil }^{{\alpha }_2}-\cdots -k_r\lfloor {z_r\rceil }^{{\alpha }_r}$

8

where $\lfloor {z_i\rceil }^{{\alpha }_i}=|z_i{|}^{{\alpha }_i}\text{sgn}(z_i)$ and ${\alpha }_1\dots {\alpha }_r$ satisfy

${\alpha }_{i-1}=\frac{{\alpha }_i{\alpha }_{i+1}}{2{\alpha }_{i+1}-{\alpha }_i},\hspace{0.17em}\hspace{1em}i=2,\dots ,r$

9

with ${\alpha }_{r+1}=1$ and ${\alpha }_r=\alpha$ .▪

Remark 1. Unfortunately, if the initial state is far away from the origin, the convergence rate by using the control law given by equation (8) is rather slow, since the power functions in equation (8) almost do not grow with large arguments. The feature is the inevitable consequence of the homogeneity.

Theorem 1. Let the positive constants $k_1,\dots ,k_r$ and ${k\prime }_1,\dots ,{k\prime }_r$ be such that the polynomials ${\lambda }^r+k_r{\lambda }^{r-1}+\cdots +k_2\lambda +k_1$ and ${\lambda }^r+{k\prime }_r{\lambda }^{r-1}+\cdots +{k\prime }_2\lambda +{k\prime }_1$ are Hurwitz. There is $\xi \in (0,1)$ such that, for every $\alpha \in (1-\xi ,1)$ , the system given by equation (7) is stabilized at the origin in finite time under the feedback

$u(z)=-k_1\lfloor {z_1\rceil }^{{\alpha }_1}-{k\prime }_1{z}_1-\cdots -k_r\lfloor {z_r\rceil }^{{\alpha }_r}-{k\prime }_r{z}_r$

10

and ${\alpha }_1,\dots ,{\alpha }_r$ satisfy

${\alpha }_{i-1}=\frac{{\alpha }_i{\alpha }_{i+1}}{2{\alpha }_{i+1}-{\alpha }_i},\hspace{1em}i=2,\dots ,r$

11

with ${\alpha }_{r+1}=1$ and ${\alpha }_r=\alpha$ .

Proof. Firstly, the conclusion $0<{\alpha }_i<1$ , $i=1,\dots ,r$ can be proved by using the induction method.

Step 1: when $k=r$ , it can be obtained

${\alpha }_r=\alpha <{\alpha }_{r+1}=1$

12

Step 2: when $k=r-i$ , suppose that

${\alpha }_{r-i}<{\alpha }_{r-i+1}$

13

Step 3: when $k=r-i-1$ , using equation (11) yields

$\begin{array}{l}{\alpha }_{r-i-1}=\frac{{\alpha }_{r-i}{\alpha }_{r-i+1}}{2{\alpha }_{r-i+1}-{\alpha }_{r-i}}=\frac{{\alpha }_{r-i}{\alpha }_{r-i+1}}{{\alpha }_{r-i+1}+({\alpha }_{r-i+1}-{\alpha }_{r-i})}\hfill \end{array}$

14

By using equation (14), it can be concluded that ${\alpha }_{r-i+1}-{\alpha }_{r-i}>0$ . Observing the above equalities, it yields

${\alpha }_{r-i-1}<\frac{{\alpha }_{r-i}{\alpha }_{r-i+1}}{{\alpha }_{r-i+1}}={\alpha }_{r-i}$

15

Thus, it can be concluded that

$0<{\alpha }_1<{\alpha }_2<\cdots <{\alpha }_r<1$

16

It is noted that finite time control law (10) is the combination control law given by equation (8) and linear control terms. Analyzing the dynamics of the systems given by equations (7) and (10), the two cases are considered as follows.

When the system state is far away from the origin (i.e. $\left|z_i\right|>1$ ), the linear control term $-{k\prime }_i{z}_i$ will dominates over the corresponding nonlinear control term $-k_i{\lfloor z_i\rceil }^{{\alpha }_i}$ . Thus, system (7) dynamics can be approximated by

$\{\begin{array}{l}{z}_1=z_2\hfill \\ z_2=z_3\hfill \\ ⋮\hfill \\ z_{r-1}=z_r\hfill \\ z_r=-{k\prime }_1{z}_1-{k\prime }_2{z}_2-\cdots -{k\prime }_r{z}_r\hfill \end{array}$

17

and the polynomial ${\lambda }^r+{k\prime }_r{\lambda }^{r-1}+\cdots +{k\prime }_2\lambda +{k\prime }_1$ is Hurwitz, that yields faster asymptotic convergence of the states $z_i$ driven by the control law given by equation (10) rather than by the control law given by equation (8).

When the system states are close to the origin (i.e. $\left|z_i\right|<1$ ), the nonlinear control term $-k_i{\lfloor z_i\rceil }^{{\alpha }_i}$ will dominate over the linear ones $-{k\prime }_i{z}_i$ . Thus, the dynamics of the system given by equation (7) can be approximated by

$\{\begin{array}{l}{z}_1=z_2\hfill \\ z_2=z_3\hfill \\ ⋮\hfill \\ z_{r-1}=z_r\hfill \\ z_r=-k_1\lfloor {z_1\rceil }^{{\alpha }_1}-k_2\lfloor {z_2\rceil }^{{\alpha }_2}-\cdots -k_r\lfloor {z_r\rceil }^{{\alpha }_r}\hfill \end{array}$

18

and the polynomial ${\lambda }^r+k_r{\lambda }^{r-1}+\cdots +k_2\lambda +k_1$ is Hurwitz, the system given by equation (18) coincides with equation (8) in Lemma 1, which guarantees finite time convergent to the origin.

Hereby, the dynamics converge very quickly in the system given by equation (7) under feedback law given by equation (10). This completes the proof.□

Robust multivariable HOSMC design for the AHV

By using the input/output feedback linearization technique, the AHV longitudinal dynamic models given by equations (1) and (2) can be completely linearized without internal dynamics, which can be expressed as follows ¹⁰

$\left[\begin{array}{l}{V}^{(3)}\hfill \\ h^{(4)}\hfill \end{array}\right]=\underset{A(x)}{\underbrace{\left[\begin{array}{c}{L}_f^{3}V\\ L_f^{4}h\end{array}\right]}}+\underset{B(x)}{\underbrace{\left[\begin{array}{cc}{L}_{g_1}{L}_f^{2}V& L_{g_2}{L}_f^{2}V\\ L_{g_1}{L}_f^{3}h& L_{g_2}{L}_f^{3}h\end{array}\right]}}\left[\begin{array}{c}{\beta }_c\\ {\delta }_e\end{array}\right]$

19

where L is the standard Lie derivative notation.

Consider the system uncertainty and the actuator failures, and the AHV input/output dynamics, equation (19) can be written as

$\left[\begin{array}{l}{V}^{(3)}\hfill \\ h^{(4)}\hfill \end{array}\right]=(\overline{A}(\mathit{x})+{\Delta }_A)+(\overline{B}(\mathit{x})+{\Delta }_B)\Lambda u$

20

where $\overline{A}(\mathit{x})$ is the nominal system matrix and $\overline{\mathit{B}}(\mathit{x})$ is the nominal control gain matrix. ${\Delta }_A$ and ${\Delta }_B$ are the system uncertainties, which are caused by the parameter uncertainties and external disturbances. Here, $\Lambda =\left[\begin{array}{cc}{\Lambda }_1& \hspace{0.17em}\\ \hspace{0.17em}& {\Lambda }_2\end{array}\right]$ which is the unknown effectiveness matrix of the control input, which is caused by the actuator failures. In this article, only partial loss failures in the actuators are considered, i.e. $0<{\Lambda }_1\le 1$ , $0<{\Lambda }_2\le 1$ , which means the actuator will not lose its effectiveness completely.

Assumption 1. There are three constants ${\epsilon }_{A1}$ , ${\epsilon }_{A2}$ , and ${\epsilon }_B$ , such that the uncertainty functions ${\Delta }_A$ , ${\Delta }_B$ , and the effectiveness matrix $\Lambda$ satisfy

$\parallel {\Delta }_A\parallel \le {\epsilon }_{A1}+{\epsilon }_{A2}\parallel x\parallel ,\parallel ({\Delta }_B\Lambda +B(\Lambda -I)){\overline{B}}^{-1}\parallel \le {\epsilon }_B<1$

21

Remark 2. In equation (20) , we assume that the bound of the additive uncertainty ${\Delta }_A$ is dependent on the norm of system state $x$ , and we can find a similar assumption in the work by Li et al ⁹ . Meanwhile, in this article, we assume that the uncertainty of the control gain matrix ${\Delta }_B$ and the control effectiveness matrix Λ can not change the control direction. The term $(\overline{B}(x)+{\Delta }_B)\Lambda u$ in equation (21) can be written as $(\overline{B}+{\Delta }_B\Lambda +B(\Lambda -I))u$ , and ${\Delta }_B\Lambda +B(\Lambda -I)$ can be seen as the total multiplicative uncertainty. If $\parallel ({\Delta }_B\Lambda +B(\Lambda -I)){\overline{B}}^{-1}\parallel \le {\epsilon }_B<1$ holds, the control direction is the same as that of the nominal system, that is to say in this article, the problem of unknown control direction is not considered.

Since the control objective is that the AHV can track velocity reference signal $V_{\text{ref}}$ and altitude reference signal $h_{\text{ref}}$ , we define the following sliding variables

$\sigma =\left[\begin{array}{l}{\sigma }_V\hfill \\ {\sigma }_h\hfill \end{array}\right]=\left[\begin{array}{c}V-V_{\text{ref}}\\ h-h_{\text{ref}}\end{array}\right]$

22

In order to design HOSMC, we define a set of integral sliding surfaces in ${\sigma }_V$ and ${\sigma }_h$ space, passing through the origin to represent a sliding manifold as

$S=\left[\begin{array}{l}{s}_V\hfill \\ s_h\hfill \end{array}\right]=\left[\begin{array}{c}{\sigma }_V^{(2)}+{\displaystyle {\int }_0^t}\left({\displaystyle \sum _{i=0}^2}\right(k_{V,i}\lfloor {{\sigma }_V^{(i)}\rceil }^{{\alpha }_{V,i}}+{k\prime }_{V,i}{\sigma }_V^{(i)}\left)\right)d\tau \\ {\sigma }_h^{(3)}+{\displaystyle {\int }_0^t}\left({\displaystyle \sum _{i=0}^3}\right(k_{h,i}\lfloor {\sigma }_h^{(i)}\rceil {\alpha }_{h,i}+{k\prime }_{h,i}{\sigma }_h^{(i)}\left)\right)d\tau \end{array}\right]$

23

where the parameters ${\alpha }_{V,i}$ , ${\alpha }_{h,i}$ , $k_{V,i}$ , $k_{h,i}$ , ${k\prime }_{V,i}$ , and ${k\prime }_{h,i}$ are selected according to Theorem 1. By using equation (19), the derivative of the sliding manifold equation (23) can be expressed as

$S=Ѱ(x,\sigma )+B(x)u$

24

where

$\Psi (x,\Sigma )=A-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=0}^r}\left(k{\Sigma }^{\alpha }+k\prime \Sigma \right)$

25

where we adopt the notions that

${\displaystyle \sum _{i=0}^r}\left(k{\Sigma }^{\alpha }+k\prime \Sigma \right)\stackrel{\Delta }{=}\left[\begin{array}{c}{\displaystyle \sum _{i=0}^3}\left(k_{V,i}\lfloor {{\sigma }_V^{(i)}\rceil }^{{\alpha }_{V,i}}+{k\prime }_{V,i}{\sigma }_V^{(i)}\right)\\ {\displaystyle \sum _{i=0}^4}\left(k_{h,i}\lfloor {{\sigma }_h^{(i)}\rceil }^{{\alpha }_{h,i}}+{k\prime }_{h,i}{\sigma }_h^{(i)}\right)\end{array}\right]$

26

$y_{\text{ref}}^{(r)}\stackrel{\Delta }{=}{\left[\begin{array}{cc}{V}_{\text{ref}}^{(3)}& h_{\text{ref}}^{(4)}\end{array}\right]}^{\text{T}}$

27

Theorem 2. For the AHV dynamic model given by equation (1) with sliding variables equation (23) , if the control u is designed as follows

$u=u_n+u_s$

28

where each part is obtained respectively as

$u_n=-{\overline{B}}^{-1}[\overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=0}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )]$

29

$u_s=\{\begin{array}{cc}-(c_1+c_2\Vert x\Vert +c_3\parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel +\eta ){\overline{B}}^{-1}\frac{S}{\Vert S\Vert }& \hspace{0.17em}S\ne 0\\ 0& S=0\end{array}$

30

and the designed parameters are chosen as

$c_1=\frac{{\epsilon }_{A1}}{1-{\epsilon }_B},\hspace{0.17em}{c}_2=\frac{{\epsilon }_{A2}}{1-{\epsilon }_B},\hspace{0.17em}{c}_3=\frac{{\epsilon }_B}{1-{\epsilon }_B},\hspace{0.17em}\eta >0$

31

then, the control laws given by equations (28) to (30) ensures the establishment of higher order sliding modes ${\sigma }_V={\sigma }_V={\ddot{\sigma }}_V=0$ , and ${\sigma }_h={\sigma }_h={\ddot{\sigma }}_h={\sigma }_h^{(3)}=0$ in finite time.

Proof. Consider the following Lyapunov function candidate

$V=\frac{S^{\text{T}}S}{2}$

32

The time derivative of V for $S\ne 0$ is

$\begin{array}{l}V=S^{\text{T}}(Ѱ(x,\Sigma )+B(x)u)\hfill \\ =S^{\text{T}}(A(x)-y_{\text{ref}}^{(r)}+B(x)u+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma ))\hfill \\ =S^{\text{T}}(\overline{A}+{\Delta }_A-y_{\text{ref}}^{(r)}+(\overline{B}+{\Delta }_B\Lambda +B(\Lambda -I))u\hfill \\ +{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma ))\hfill \end{array}$

33

Substituting the control laws given by equations (28) to (30) into equation (33) yields

$\begin{array}{l}V=S^{\text{T}}({\Delta }_A-({\Delta }_B\Lambda +B(\Lambda -I)){\overline{B}}^{-1}(\overline{A}-y_{\text{ref}}^{(r)}\hfill \\ +{\displaystyle \sum _{i=1}^r}(k{\sigma }^{\alpha }+k\prime \sigma ))-(c_1+c_2\Vert \mathbf{x}\Vert +c_3\parallel \overline{A}-y_{\text{ref}}^{(r)}\hfill \\ +{\displaystyle \sum _{i=1}^r}(k{\sigma }^{\alpha }+k\prime \sigma )\parallel +\eta )\cdot (I+({\Delta }_B\Lambda +B(\Lambda -I)){\overline{B}}^{-1})\frac{S}{\Vert S\Vert })\hfill \end{array}$

34

By utilizing equation (21) and equation (31), we can obtain

$\begin{array}{l}V\le \parallel S\parallel \parallel {\Delta }_A\parallel +\parallel S\parallel \parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\sigma }^{\alpha }+k\prime \sigma )\parallel \parallel ({\Delta }_B\Lambda +B(\Lambda -I)){\overline{B}}^{-1}\parallel \hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{1em}-(c_1+c_2\parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel +\eta )\parallel S\parallel \hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{1em}+(c_1+c_2\parallel x\parallel +c_3\parallel \overline{A}-y_{ref}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel +\eta )\cdot \parallel ({\Delta }_B\Lambda +B(\Lambda -I)){\overline{B}}^{-1}\parallel \parallel S\parallel \hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\le ({\epsilon }_{A1}+{\epsilon }_{A2}\parallel x\parallel )\parallel S\parallel +{\epsilon }_B\parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel \parallel S\parallel \hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{1em}-(c_1+c_2\Vert x\Vert +c_3\parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel +\eta )\Vert S\Vert \hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{1em}+{\epsilon }_B(c_1+c_2\Vert S\Vert +c_3\parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel +\eta )\parallel S\parallel \hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=-\Vert S\Vert \left(c_1\left(1-{\epsilon }_B\right)-{\epsilon }_{A1}+\left(c_2\left(1-{\epsilon }_B\right)-{\epsilon }_{A2}\right)\parallel x\parallel \right)\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{1em}-\left(1-{\epsilon }_B\right)\eta \Vert S\Vert -\Vert S\Vert \parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel \cdot \left(c_3(1-{\epsilon }_B)-{\epsilon }_B\right)\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\le -\left(1-{\epsilon }_B\right)\eta \parallel S\parallel \hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=-\left(1-{\epsilon }_B\right)\eta V^{\frac{1}{2}}\hfill \end{array}$

35

Thus, system trajectories evolve on the manifold $S=0$ in finite time and remain there in spite of the uncertainties. In the sliding mode, the closed-loop dynamics obtained can be obtained by $S=0$ . According to the definition of S and Theorem 1, we can conclude that higher-order sliding modes ${\sigma }_V={\sigma }_V={\ddot{\sigma }}_V=0$ and ${\sigma }_h={\sigma }_h={\ddot{\sigma }}_h={\sigma }_h^{(3)}=0$ are established in finite time. This completes this proof.□

Remark 3. From the designed control laws given by equations (28) to (31), it is seen that the switching term $u_s$ is used to compensate the system uncertainties and the actuator failures. Then, the proposed method in this article is a passive fault-tolerant control system.

Adaptive robust multivariable HOSMC design

Under the control laws given by equations (28) to (30), higher-order sliding modes ${\sigma }_V={\sigma }_V={\ddot{\sigma }}_V=0$ and ${\sigma }_h={\sigma }_h={\ddot{\sigma }}_h={\sigma }_h^{(3)}=0$ will be established in finite time. Actually, in an AHV practical flight environment, the bounds of uncertainties may not be obtained easily due to the physical and aerodynamic parameter perturbations and actuator failures, that is to say it is not easy to achieve the upper bound of the terms ${\Delta }_A$ and $({\Delta }_B+B(\Lambda -I)){\overline{B}}^{-1}$ . Then the next phase is to design adaptation laws for the parameters $c_1$ , $c_2$ and $c_3$ in the control law given by equation (30)

$u=u_{\text{n}}+u_{\text{adp}}$

36

where

$u_{\text{n}}=-{\overline{B}}^{-1}[\overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=0}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )]$

37

$u_{\text{adp}}=\{\begin{array}{cc}-({\widehat{c}}_1+{\widehat{c}}_2\Vert x\Vert +{\widehat{c}}_3\parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel +\eta ){\overline{B}}^{-1}\frac{S}{\Vert S\Vert }& \hspace{0.17em}S\ne 0\\ 0& S=0\end{array}$

38

where ${\widehat{c}}_1$ , ${\widehat{c}}_2$ , and ${\widehat{c}}_3$ are the estimated values of $c_1$ , $c_2$ , and $c_3$ , $\eta >0$ is a designed positive constant. The adaptive gains ${\widehat{c}}_1$ , ${\widehat{c}}_2$ , and ${\widehat{c}}_3$ in equation (38) are designed as

${\widehat{c}}_1=\{\begin{array}{cc}\frac{1}{{\gamma }_1}& \hspace{0.17em}\hspace{0.17em}S\ne 0\\ 0& \hspace{0.17em}\hspace{0.17em}S=0\end{array}$

39

${\widehat{c}}_2=\{\begin{array}{cc}\frac{1}{{\gamma }_2}\parallel \mathit{x}\parallel & \hspace{0.17em}\hspace{0.17em}S\ne 0\\ 0& \hspace{0.17em}\hspace{0.17em}S=0\end{array}$

40

${\widehat{c}}_3=\{\begin{array}{cc}\frac{1}{{\gamma }_3}\parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel & \hspace{0.17em}\hspace{0.17em}S\ne 0\\ 0& \hspace{0.17em}\hspace{0.17em}S=0\end{array}$

41

where ${\gamma }_1$ , ${\gamma }_2$ and ${\gamma }_3$ are denoted as adaptation gains, and ${\widehat{c}}_1(0)={\widehat{c}}_2(0)={\widehat{c}}_3(0)=0$ .

Theorem 3. Consider the AHV dynamic model given by equation (1). If the the control law, and the adaptive law are designed as equations (36) to (38) and (39) to (41), respectively. Then, higher-order sliding modes: ${\sigma }_V={\sigma }_V={\ddot{\sigma }}_V=0$ and ${\sigma }_h={\sigma }_h={\ddot{\sigma }}_h={\sigma }_h^{(3)}=0$ can be established in finite time.

Proof. Define parameter errors as

${\tilde{c}}_1\stackrel{\Delta }{=}{\widehat{c}}_1-\frac{{\epsilon }_{A1}}{1-{\epsilon }_B},\hspace{1em}{\tilde{c}}_2\stackrel{\Delta }{=}{\widehat{c}}_2-\frac{{\epsilon }_{A2}}{1-{\epsilon }_B},{\tilde{c}}_3\stackrel{\Delta }{=}{\widehat{c}}_3-\frac{{\epsilon }_B}{1-{\epsilon }_B}$

42

where ${\epsilon }_{A1}$ , ${\epsilon }_{A2}$ and ${\epsilon }_B$ are defined in inequalities in equation (21).

Consider a Lyapunov function candidate as

$V=\parallel S\parallel +\frac{1-{\epsilon }_B}{2}{\gamma }_1{\tilde{c}}_1^2+\frac{1-{\epsilon }_B}{2}{\gamma }_2{\tilde{c}}_2^2+\frac{1-{\epsilon }_B}{2}{\gamma }_3{\tilde{c}}_3^2$

43

Noting that ${\tilde{c}}_i={\widehat{c}}_i\hspace{0.17em}\hspace{0.17em}(i=1,2,3)$ , and using control equations (36) to (41), the time derivative of V is

$V=\frac{S^{\text{T}}S}{\parallel S\parallel }+{\gamma }_1(1-{\epsilon }_B){\tilde{c}}_1{\widehat{c}}_1+{\gamma }_2(1-{\epsilon }_B){\tilde{c}}_2{\widehat{c}}_2+{\gamma }_3(1-{\epsilon }_B){\tilde{c}}_3{\widehat{c}}_3=\frac{S^{\text{T}}}{\parallel S\parallel }({\Delta }_A-({\Delta }_B\Lambda +B(\Lambda -I)){\overline{B}}^{-1}(\overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma ))-({\widehat{c}}_1+{\widehat{c}}_2\parallel x\parallel +c_3\parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel +\eta )\cdot (\mathbf{I}+({\Delta }_B\Lambda +B(\Lambda -I)){\overline{B}}^{-1})\frac{S}{\Vert S\Vert })+(1-{\epsilon }_B){\widehat{c}}_1-{\epsilon }_{A1}+(1-{\epsilon }_B)\Vert x\Vert {\widehat{c}}_2-{\epsilon }_{A2}\Vert x\Vert +(1-{\epsilon }_B){\widehat{c}}_3\parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel -{\epsilon }_B\parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel$

44

Substituting equation (21) into the above equation (44), it yields

$V\le \parallel {\Delta }_A\parallel +\parallel ({\Delta }_B\Lambda +B(\Lambda -I)){\overline{B}}^{-1}\parallel \parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel -({\widehat{c}}_1+{\widehat{c}}_2\Vert x\Vert +{\widehat{c}}_3\parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel +\eta )+({\widehat{c}}_1+{\widehat{c}}_2\parallel x\parallel +{\widehat{c}}_3\parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel +\eta )\cdot \parallel ({\Delta }_B\Lambda +B(\Lambda -I)){\overline{B}}^{-1}\parallel +(1-{\epsilon }_B){\widehat{c}}_1-{\epsilon }_{A1}+(1-{\epsilon }_B){\widehat{c}}_2\Vert x\Vert \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} -{\epsilon }_{A2}\parallel x\parallel +(1-{\epsilon }_B){\widehat{c}}_3\parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel -{\epsilon }_B\parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\le {\epsilon }_{A1}+{\epsilon }_{A2}\Vert x\Vert +{\epsilon }_B\parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel -({\widehat{c}}_1+{\widehat{c}}_2\parallel x\parallel +{\widehat{c}}_3\parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel +\eta )\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} +{\epsilon }_B({\widehat{c}}_1+{\widehat{c}}_2\Vert x\Vert +{\widehat{c}}_3\parallel \overline{A}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel +\eta )+(1-{\epsilon }_B){\widehat{c}}_1-{\epsilon }_{A1}+(1-{\epsilon }_B){\widehat{c}}_2\Vert x\Vert -{\epsilon }_{A2}\Vert x\Vert \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} +(1-{\epsilon }_B){\widehat{c}}_3\parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel -{\epsilon }_B\parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=-\left(1-{\epsilon }_B\right)\eta$

45

The inequality in equation (45) indicates that S will approach $\parallel S\parallel =0$ in finite time $t_f\le t_0+\frac{V(t_0)-V(t_f)}{(1-{\epsilon }_B)\eta }$ . In sliding mode, the closed-loop dynamics obtained can be obtained by $\stackrel{.}{S}=0$ . According to the definition of S and Theorem 1, higher-order sliding modes ${\sigma }_V={\stackrel{.}{\sigma }}_V={\ddot{\sigma }}_V=0$ and ${\sigma }_h={\stackrel{.}{\sigma }}_h={\ddot{\sigma }}_h={\sigma }_h^{(3)}=0$ will be established in finite time. Since the value of V is bounded, one can see that ${\tilde{c}}_i$ (and hence ${\widehat{c}}_i$ ) are all bounded. According to equation (40), the solution of ${\widehat{c}}_2$ is

${\widehat{c}}_2(t)=\frac{1}{{\gamma }_2}{\displaystyle {\int }_{t_0}^{t_f}}\Vert x(\tau )\Vert d\tau +{\widehat{c}}_2(t_0),\hspace{1em}\Vert S\Vert \ne 0$

46

Since ${\widehat{c}}_2$ are bounded, and the integrand is nonnegative, the state variable x of equation (1) must be bounded for $t_0\le t<t_f$ . This completes the proof.□

Remark 4. The methodology is applicable where $\parallel S\parallel =0$ is reachable. However, in real sliding mode control, $\parallel S\parallel$ cannot become exactly zero in finite time due to nonlinearities and switching delays. Thus the adaptive parameters ${\widehat{c}}_1$ , ${\widehat{c}}_2$ , and ${\widehat{c}}_3$ may increase boundlessly. A simple way of overcoming this difficulty is to modify the adaptive tuning law by using the dead zone technique as shown by Utkin ³⁸

$u_{\text{adp}}=\{\begin{array}{cc}-({\widehat{c}}_1+{\widehat{c}}_2\Vert x\Vert +{\widehat{c}}_3\parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel +\eta ){\overline{B}}^{-1}\frac{S}{\Vert S\Vert }& \hspace{0.17em}\Vert S\Vert >\Phi \\ -({\widehat{c}}_1+{\widehat{c}}_2\Vert x\Vert +{\widehat{c}}_3\parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel +\eta ){\overline{B}}^{-1}\frac{S}{\Phi }& \Vert S\Vert \le \Phi \end{array}$

47

${\widehat{c}}_1=\{\begin{array}{cc}\frac{1}{{\gamma }_1}& \hspace{0.17em}\hspace{0.17em}\Vert S\Vert >\Phi \\ 0& \hspace{0.17em}\hspace{0.17em}\Vert S\Vert \le \Phi \end{array}$

48

${\widehat{c}}_2=\{\begin{array}{cc}\frac{1}{{\gamma }_2}\parallel x\parallel & \hspace{0.17em}\hspace{0.17em}\Vert S\Vert >\Phi \\ 0& \hspace{0.17em}\hspace{0.17em}\Vert S\Vert \le \Phi \end{array}$

49

${\widehat{c}}_3=\{\begin{array}{cc}\frac{1}{{\gamma }_3}\parallel \overline{A}-y_{\text{ref}}^{(r)}+{\displaystyle \sum _{i=1}^r}(k{\Sigma }^{\alpha }+k\prime \Sigma )\parallel & \hspace{0.17em}\hspace{0.17em}\Vert S\Vert >\Phi \\ 0& \hspace{0.17em}\hspace{0.17em}\Vert S\Vert \le \Phi \end{array}$

50

Simulation results

In this section, simulation studies are carried out to confirm the effectiveness of the proposed control scheme. The velocity command $V_c$ is a step change of 300 ft/s from 15,060 ft/s. The altitude command $h_c$ is a step change of 2000 ft from 110,000 ft. The tracking reference commands $V_{\text{ref}}$ and $h_{\text{ref}}$ are generated by the following filters

$\frac{V_{\text{ref}}}{V_c}=\frac{{\omega }_{V_1}{\omega }_{V_2}^2}{(s+{\omega }_{V_1})(s^2+2{\varsigma }_V{\omega }_{V_2}+{\omega }_{V_2}^2)}$

51

$\frac{h_{\text{ref}}}{h_c}=\frac{{\omega }_{h_1}{\omega }_{h_2}^2}{(s+{\omega }_{h_1})(s^2+2{\varsigma }_h{\omega }_{h_2}+{\omega }_{h_2}^2)}$

52

where ${\omega }_{V_1}=0.05$ , ${\omega }_{V_2}=0.5$ , ${\varsigma }_V=0.7$ , ${\omega }_{h_1}=0.05$ , ${\omega }_{h_2}=0.5$ , ${\varsigma }_h=0.7$ . To demonstrate the robustness of the proposed control method, the parameters uncertainties are selected as: $\Delta m=0.2$ , $\Delta I=-0.2$ , $\Delta S=-0.1$ , $\Delta \overline{c}=-0.1$ , $\Delta c_e=0.2$ , $\Delta C_{M\alpha }=-0.25$ . The parameters in the proposed controller equations (37) and (47) to (50) are as follows: $k_{V\mathrm{,0}}=0.2$ , $k_{V,1}=0.5$ , $k_{V\mathrm{,2}}=0.5$ , ${k\prime }_{V\mathrm{,0}}=2$ , ${k\prime }_{V,1}=3$ , ${k\prime }_{V\mathrm{,2}}=3$ , $k_{h\mathrm{,0}}=2.4$ , $k_{h,1}=5$ , $k_{h\mathrm{,2}}=3.5$ , $k_{h\mathrm{,3}}=1$ , ${k\prime }_{h\mathrm{,0}}=4.8$ , ${k\prime }_{h,1}=10$ , ${k\prime }_{h\mathrm{,2}}=7$ , ${k\prime }_{h\mathrm{,3}}=2$ , $\Phi =10$ , ${\gamma }_1={\gamma }_2=2$ , ${\gamma }_3=100$ . For a comparison study, the multivariable traditional SMC proposed by Li et al. ⁹ is employed in the controller design for the AHV (equation (1)).

In these simulation studies, two cases of actuator failures are considered as follows:

Case I: The constant actuator failures

$\left[\begin{array}{c}{\beta }_c\\ {\delta }_e\end{array}\right]=\left[\begin{array}{cc}{\Lambda }_1& 0\\ 0& {\Lambda }_2\end{array}\right]u$

53

where

${\Lambda }_1={\Lambda }_2=\{\begin{array}{cc}1& 0\le t<40\\ 0.5& t\ge 40\end{array}$

54

The simulation results are shown in Figures 1 to 5. From Figures 1 and 2, we can see that by using both the proposed HOSMC and the traditional SMC, the AHV output V and h can track the reference signals. The tracking error of the proposed HOSMC is smaller than that of the traditional SMC, and the transient process of the proposed HOSMC is also faster than that of the traditional SMC. Even in the moment t = 40 s, the constant actuator failures occurrence, the proposed HOSMC method can ensure the steady state tracking error goes to zero quickly. Figure 3 shows the adaption gains ${\widehat{c}}_1$ , ${\widehat{c}}_2$ , and ${\widehat{c}}_3$ , and it is observed that the adaptation gains are bounded. The control input signals ${\delta }_e$ and ${\beta }_c$ when using the proposed HOSMC are shown in Figure 4, and the control signals are continuous and the chattering phenomenon is not observed. Figure 5 shows the control input signals of the traditional SMC.

Figure 1.

The response curves of (a) the actual output (V) and (b) the velocity tracking error (Case I).

Figure 2.

The response curves of (a) the actual output (h) and (b) the velocity tracking error (Case I).

Figure 3.

The adaption gains (a) ${\widehat{c}}_1$ , (b) ${\widehat{c}}_2$ , and (c) ${\widehat{c}}_3$ in the proposed HOSMC (Case I).

Figure 4.

The control input (a) ${\delta }_e$ and (b) ${\beta }_c$ when using the proposed HOSMC (Case I).

Figure 5.

The control input (a) ${\delta }_e$ and (b) ${\beta }_c$ when using the traditional SMC (Case I).

Case II: The time-varying actuator failures

$\left[\begin{array}{c}{\beta }_c\\ {\delta }_e\end{array}\right]=\left[\begin{array}{cc}{\Lambda }_1& 0\\ 0& {\Lambda }_2\end{array}\right]u$

55

where

${\Lambda }_1=\{\begin{array}{cc}1& 0\le t<40\\ 0.7+0.2\text{sin}(0.5t)& t\ge 40\end{array}$

56

${\Lambda }_2=\{\begin{array}{cc}1& 0\le t<40\\ 0.6+0.1\text{sin}(0.5t)& t\ge 40\end{array}$

57

The simulation results are displayed in Figures 6 to 10. From Figures 6 and 7, we can see that the proposed HOSMC have a smaller overshoot and a faster transient process. Under the time-varying actuator failures, the proposed HOSMC can have a good performance with $|V-V_{\text{ref}}|<0.4\hspace{0.17em}\text{ft/s}$ and $|h-h_{\text{ref}}|<0.02\hspace{0.17em}\text{ft}$ during this simulation. The traditional SMC can not guarantee a small steady-state tracking error when the time varying actuator failures occur. Figure 8 shows the adaption gains ${\widehat{c}}_1$ , ${\widehat{c}}_2$ , and ${\widehat{c}}_3$ of the proposed HOSMC, which are bounded. The control inputs ${\beta }_c$ and ${\delta }_e$ when using the proposed HOSMC are shown in Figure 9. The control inputs ${\beta }_c$ and ${\delta }_e$ when employing the traditional SMC are shown in Figure 10.

Figure 6.

The response curves of (a) the actual output ( $V$ ) and (b) the velocity tracking error (Case II).

Figure 7.

The response curves of (a) the actual output (h) and (b) the velocity tracking error (Case II).

Figure 8.

The adaption gains (a) ${\widehat{c}}_1$ , (b) ${\widehat{c}}_2$ , and (c) ${\widehat{c}}_3$ in the proposed HOSMC (Case II).

Figure 9.

The control input (a) ${\delta }_e$ and (b) ${\beta }_c$ when using the proposed HOSMC (Case II).

Figure 10.

The control input (a) ${\delta }_e$ and (b) ${\beta }_c$ when using the traditional SMC (Case II).

Conclusions

This article addresses the reference command tracking for the longitudinal dynamics of an AHV with actuator failures. The feedback linearization method is used and the system uncertainties and external disturbances are considered as additive uncertainty and the actuator failures are considered as multiplicative uncertainty. Based on the proposed fast finite-time control law for a chain of integrators, a robust multivariable higher-order sliding mode controller is designed for the AHV. Furthermore, an adaptive law is proposed to estimate the bounds of the additive and multiplicative uncertainty. Simulation results demonstrate that the proposed adaptive multivariable HOSMC can achieve desired tracking performance and good robustness under the actuator failures.