Sliding Mode Backstepping Control for the Ascent Phase of Near-Space Hypersonic Vehicle Based on a Novel Triple Power Reaching Law

Abstract

This paper presents a novel sliding mode backstepping control scheme for the ascent phase of a near-space hypersonic vehicle (NSHV) base on a triple power reaching law (TPRL). A new model transformation is proposed for NSHV with uncertain parameters subject to uncertainties during ascent phase. To shorten the reaching time and reduce the chattering in sliding mode scheme, TPRL is proposed. Then, based on TPRL, a sliding mode backstepping control scheme is proposed, which is combined with new adaptive laws to further reduce the adverse impact of uncertainties. Simulation results demonstrate that TPRL is effective, and the proposed controller for the ascent phase of NSHV is robust with respect to uncertainties.

1. Introduction

A near-space hypersonic vehicle (NSHV) refers to an aircraft operating in near-space, the flight speed of which can exceed Mach-5 [1,2,3]. Great concerns have been raised regarding the development of NSHV, due to its significant military value and potential economic value [4,5,6]. However, NSHV is of high couplings, strong nonlinearities, and complex uncertainties. Especially in ascent phase, the altitude and speed of NSHV change dramatically, which results in a large consumption of fuel, and the aerodynamic characteristics of NSHV vary greatly [7,8,9]. Therefore, it is a great challenge to design a high-performance flight controller for NSHV in the ascent phase.

The controller design for the ascent phase of NSHV encounters challenges originating from model uncertainties and nonlinearities. Therefore, a controller with robustness and stability is especially important for the ascent phase of NSHV. In recent years, there have been many methods used to study NSHV in the literature, such as H∞ [10], predictive flight control [11,12], fuzzy control [13,14], sliding mode control [15,16], backstepping control [17,18], neural network control [19,20], and adaptive control [21,22].

The sliding mode control provides a systematic approach to modeling imprecision, of which the main advantage is that it is robust subject to model uncertainties and disturbances [23,24]. Yin et al. proposed a high-order sliding mode control scheme for hypersonic vehicles, which combine the advantages of the disturbance observer and gain adaptation to estimate the uncertainties and external disturbance [25]. Guo et al., in 2018, proposed adaptive twisting sliding mode control for hypersonic re-entry vehicles, which adopted an adaptive finite-time observer to estimate the unknown state, guaranteed the stability of system and reduced the chattering phenomenon [26]. In the literature [27], Guo et al. estimated the uncertainty through the observer and proposed a fixed time sliding mode scheme, which reduced the interference of the uncertainty and obtained good tracking effect. However, the impact of external disturbances on the vehicle needs to be further explored. Basin et al. proposed a higher order sliding mode observer using fixed-settling time differentiators, which guarantee the robustness and high-accuracy output tracking in the presence of external disturbances and missile model uncertainties [28]. In the literature [29], Wang et al. proposed a terminal sliding mode controller based on sliding mode disturbance observer for a hypersonic flight vehicles. This approach obtains expected control performance in the presence of uncertainties and external disturbances.

The principle of sliding mode control is that the state of the system approaches the sliding surface under reaching law, so the stability and convergence time of the system are directly affected by the reaching law [30,31]. Many approaches have been proposed to eliminate or attenuate chattering and accelerate convergence time. Zhang et al. developed exponential reaching law in sliding mode control to enable the hypersonic vehicle to track the command input and reach a steady state along the designed sliding surface in the case of mismatched uncertainty [32]. Mozayan et al. proposed a sliding mode controller using a new enhanced exponential reaching law, which can mitigate chattering and approach the sliding surface quickly [33]. In the literature [34], fast terminal sliding-mode control was proposed to ensure that the system states converge within a finite time. An improved double power reaching law was proposed in the literature [35], which can shorten the time to approach the equilibrium point when the system is away from the sliding surface.

As mentioned above, the technique of sliding mode control has good properties, such as its robustness, order reduction, insensitivity to parameter variations, finite-time convergence, disturbance rejection, and good dynamic behavior. However, pure sliding mode control presents drawbacks that include large control authority requirements and chattering [24,33,36]. The performance of pure sliding mode control can be improved by coupling with adaptive control scheme to estimate the parameters of uncertainties and disturbances. In the literature [28], Basin et al. designed a non-recursive higher order sliding mode observer, which was combined with an adaptive algorithm to estimate the control gain to reduce control chattering. Jiao et al. proposed a new adaptive sliding mode control scheme which was combined with type-2 fuzzy approach to make the controller show good robustness in the presence of uncertain parameters. However, this control method was only applied to the aircraft in the cruise phase [37]. Ding et al. in 2020 proposed an improved continuous sliding mode control scheme to ensure the convergence precision, in which nonsingular fast fixed-time is adopted and dual-layer adaptive continuous twisting reaching law is developed [38]. Sagliano et al. proposed an adaptive disturbance-based sliding-mode controller for hypersonic-entry vehicles, which was able to accurately reconstruct disturbances and has good tracking performance [39].

The backstepping approach demonstrates great performance in complicated nonlinear systems control, such as NSHV in the presence of uncertainties and disturbances. In the current years, in order to improve the performance of flight controller, many advanced control methods based on backstepping have been proposed. Zhang et al. proposed an anti-disturbance backstepping control approach for hypersonic vehicle in presence of the large uncertainties and the external disturbances, which obtain the superior tracking performance [40]. In the literature [41], adaptive backstepping controller and dynamic inverse controller were developed for hypersonic vehicle subject to input constraints and aerodynamic uncertainties. Based on backstepping control, Wu et al. developed a nonlinear disturbance observer for a flexible air-breathing hypersonic vehicle [42]. Yu et al. proposed a novel adaptive backstepping control method for hypersonic vehicle with mismatched uncertainties to guarantee asymptotical command tracking [43]. Lu proposed a disturbance observer-based backstepping control scheme for hypersonic flight vehicles, where a nonlinear disturbance observer is conducted to estimate the model uncertainties [44]. However, the literature mentioned above has not focused on the design of hypersonic vehicles in the ascent phase.

In this paper, our scope is to propose a backstepping-based adaptive sliding mode controller adopting triple power reaching law (TPRL), which is able to meet the requirements in tracking control for the ascent phase of NSHV. The contributions of this paper are as follows:

(1)

The proposed TPRL shortens the reaching time and reduces the chattering of sliding mode approach. To shorten the reaching time of system converging to the equilibrium point, the reaching process can be divided by TPRL into three stages, and the reaching speed of each stage can be set, respectively. Then, TPRL can accelerate the speed of convergence effectively. Meanwhile, TPRL has the capability of mitigating the chattering of the sliding mode approach with respect to the traditional reaching law.

(2)

A novel tracking controller for the ascent phase of NSHV is proposed. At present, due to the limitation of data, little research has focused on the tracking control for the ascent phase of hypersonic vehicle in the presence of uncertainties. However, as we know, the ascent mission of NSHV could significantly affect the whole flight performance of the NSHV. In order to improve the tracking accuracy and robustness in the ascent mission of NSHV, a novel backstepping-based adaptive sliding mode controller is developed. Firstly, a transformation model is proposed for NSHV in face of uncertainties. The strict feedback form with uncertain parameters is adopted in the transformed mode, which is more suitable to accommodate uncertainties on NSHV than traditional model transformation. Furthermore, based on the proposed model transformation, a backstepping control scheme is proposed, which is combined with sliding mode control using TPRL. To further attenuate the adverse impact of uncertainties, new adaptive laws are designed. The flight controller proposed improves the tracking performance for NSHV in ascent phase.

This paper is organized as follows. The linearized model of the NSHV is established and a model transformation is proposed in Section 2. In Section 3, a novel TPRL is proposed, and a new adaptive slide mode controller based on backstepping is designed. In Section 4, based on the proposed method, a controller of the NSHV is developed. Numerical simulation results are reported in Section 5. Finally, the summary of key features of the proposed scheme are given in Section 6.

2. Model of NSHV

2.1. Longitudinal Model of NSHV

The three-view drawing of NSHV studied in this paper is shown in Figure 1.

Using the longitudinal force and moment equilibrium of hypersonic vehicle, the longitudinal model of NSHV is obtained as follows [45,46]:

$$\left\{ \begin{array}{c}\dot{V}=\frac{T\cos \alpha -D}{m}-\frac{\mu }{r^2}\sin \gamma \hfill \\ \dot{\gamma }=\frac{L+T\sin \alpha }{mV}-\frac{\mu -V^{2}r}{V{r}^2}\cos \gamma \hfill \\ \dot{\alpha }=q-\dot{\gamma }\hfill \\ \dot{q}=\frac{M_{yy}}{I_{yy}}\hfill \\ \dot{h}=V\sin \gamma \hfill \\ \dot{m}=-\frac{T}{g{I}_{sp}}\hfill \end{array}\right.$$

(1)

where $V$, $h$, $m$, $\gamma$, $\alpha$, and $q$ are velocity, altitude, mass, flight path angle, attack angle, and pitch angle rate, respectively. The $\mu$, $g$, and $r$ are the Earth’s gravity constant, the acceleration of gravity and radial distance from the Earth’s center. The $M_{yy}$, $I_{yy}$ $L$, $D$, $T$, and $I_{sp}$ are pitch moment, rotation inertia, lift, drag, and thrust and fuel ratio, respectively. The $M_{yy}$, $L$, $D$, and $T$ can be expressed as follows:

$$\left\{ \begin{array}{c}{M}_{yy}=\frac{1}{2}\rho V^{2}s\overline{c}\left(C_M^{\alpha }+C_M^q+C_M^{{\delta }_e}\right)\hfill \\ L=\frac{1}{2}\rho V^{2}s{C}_L\hfill \\ D=\frac{1}{2}\rho V^{2}s{C}_D\hfill \\ T=\frac{1}{2}\rho V^{2}s{C}_T\hfill \end{array}\right.$$

(2)

where, $\rho$, $s$, and $\overline{c}$ are air density, wing area, and mean aerodynamic chord, respectively; $C_M^{\alpha }$, $C_M^q$, and $C_M^{{\delta }_e}$ are moment coefficient due to attack angle, moment coefficient due to pitch rate, and moment coefficient due to elevator deflection, respectively. $C_L$, $C_D$, and $C_T$ are lift coefficient, drag coefficient, and engine thrust coefficient, respectively.

In the ascent phase, the thrust coefficient and fuel ratio are set as follows:

$$\left\{ \begin{array}{c}{C}_T=\left\{\begin{array}{c}0.12886\beta (\beta <1)\\ 0.112+0.0168\beta \left(\beta \ge 1\right)\end{array}\right.\hfill \\ I_{sp}=\left\{\begin{array}{c}4700-\frac{h}{100} (M_a<4)\\ 0.112+0.0168\beta \left(M_a\ge 4\right)\end{array}\right.\hfill \end{array}\right.$$

(3)

where $\beta$ and $M_a$ are the state of engine and Mach, respectively. The engine dynamics are modelled as follows:

$$\ddot{\beta }=-2\xi \omega \dot{\beta }-{\omega }^2\beta +{\omega }^2{\beta }_c$$

(4)

where $\beta$, $\omega$, $\xi$ and ${\beta }_c$ are the state of engine, natural frequency, damp ratio, and throttle setting, respectively.

The linearized model of NSHV is developed by repeatedly differentiating $V$ three times and $h$ four times. The output dynamics variables of $V$ and $h$ can be expressed in a form in which control input ${\beta }_c$ and ${\delta }_e$ appear explicitly as follows [32]:

$$\left\{ \begin{array}{c}\stackrel{⃛}{V}=f_V+b_{11}{\beta }_c+b_{12}{\delta }_e\hfill \\ h^{\left(4\right)}=f_h+b_{21}{\beta }_c+b_{22}{\delta }_e\hfill \end{array}\right.$$

(5)

where the detailed expression of the $f_V$, $f_h$, $b_{11}$, $b_{12}$, $b_{21}$, and $b_{22}$ can be found in [32].

In ascent phase, uncertainties are modelled as additive variance $\Delta$ to the nominal value, which is expressed as follows:

$$i=i_0+{\Delta }_i$$

(6)

where $i=m, I_{yy}, \rho , s, \overline{c}, c_e, C_L, C_D, C_T, C_M^{\alpha }, C_M^q,C_M^{{\delta }_e}$, $i$ is the real value, $i_0$ is the nominal value, and ${\Delta }_i$ represents the value of uncertainty.

2.2. Novel Model Transformation

According to Equation (5), traditional model transformation can be expressed as follows:

$$\left\{ \begin{array}{c}{\dot{\xi }}_1={\xi }_2\hfill \\ {\dot{\xi }}_2={\xi }_3\hfill \\ {\dot{\xi }}_3=f_V+b_{11}{\beta }_c+b_{12}{\delta }_e\hfill \\ {\dot{\eta }}_1={\eta }_2\hfill \\ {\dot{\eta }}_2={\eta }_3\hfill \\ {\dot{\eta }}_3={\eta }_4\hfill \\ {\dot{\eta }}_4=f_h+b_{21}{\beta }_c+b_{22}{\delta }_e\hfill \end{array}\right.$$

(7)

where $\xi ={\left[{\xi }_1,{\xi }_2,{\xi }_3\right]}^T={\left[V,\dot{V},\ddot{V}\right]}^T$ and $\eta ={\left[{\eta }_1,{\eta }_2,{\eta }_3,{\eta }_4\right]}^T={\left[h,\dot{h},\ddot{h},\stackrel{⃛}{h}\right]}^T$. It is obvious that the Model (7) is not suitable for the system with uncertainties.

To deal with the drawback of the traditional model (7), we define new states as $x_1={\left[{{\displaystyle \int }}_0^t(V\left(\tau \right)d\tau \hspace{1em}h\right]}^T$, $x_2={\left[\begin{array}{cc}V& \dot{h}\end{array}\right]}^T$, $x_3={\left[\begin{array}{cc}\dot{V}& \ddot{h}\end{array}\right]}^T$, and $x_4={\left[\begin{array}{cc}\ddot{V}& \stackrel{⃛}{h}\end{array}\right]}^T$. Then, the model of NSHV in Equation (5) is written as the following novel model transformation.

$$\left\{ \begin{array}{c}{\dot{x}}_1=x_2+{\phi }_1^T\left(x_1\right){\theta }_1\hfill \\ {\dot{x}}_2=x_3+{\phi }_2^T\left(x_2\right){\theta }_2\hfill \\ {\dot{x}}_3=x_4+{\phi }_3^T\left(x_3\right){\theta }_3\hfill \\ {\dot{x}}_4=f\left(x,t\right)+G\left(x,t\right)u+{\phi }_4^T\left(x_4\right){\theta }_4\hfill \end{array}\right.$$

(8)

where

$$\left\{ \begin{array}{c}f\left(x,t\right)={\left[\begin{array}{cc}{f}_V& f_h\end{array}\right]}^T\hfill \\ G\left(x,t\right)=\left[\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right]\hfill \\ u={\left[\begin{array}{cc}{\beta }_c& {\delta }_e\end{array}\right]}^T\hfill \end{array}\right.$$

(9)

where ${\phi }_i^T\left(x_i\right){\theta }_i, i=1,2,3,4$ represent the uncertainties of each subsystem; ${\phi }_i^T\left(x_i\right)=diag\left[\begin{array}{c}\begin{array}{c}{\phi }_{i,1},{\phi }_{i,2}\end{array}\end{array}\right]$ are known functions, which are assumed to be sufficient smooth; and ${\theta }_i={\left[\begin{array}{c}\begin{array}{c}{\theta }_{i,1},{\theta }_{i,2}\end{array}\end{array}\right]}^T$ are unknown constant parameters.

3. Sliding Mode Backstepping Controller Design

In this section, a triple power reaching law is proposed and a sliding mode backstepping tracking control scheme combined with adaptive laws is developed. The system structure of the proposed controller is shown in Figure 2.

3.1. A Novel Triple Power Reaching Law for Sliding Mode

In order to speed up the convergence and reduce the chattering of the sliding mode controller, a novel triple power reaching law (TPRL) is proposed and given by:

$$\dot{s}=-l_1{\left|s\right|}^{\alpha }sgn\left(s\right)-l_2{\left|s\right|}^{\beta }sgn\left(s\right)-l_3{\left|s\right|}^{\gamma }sgn\left(s\right)$$

(10)

where $s$ is the sliding surface; $sgn\left(s\right)$ is the sign function; $l_1>0$, $l_2>0$, $l_3>0$, $\alpha >1$, $0<\beta <1$; and $\gamma$ is a variable parameter and is given by:

$$\gamma =\left\{ \begin{array}{c}\delta , \left|s\right|>\omega \hfill \\ 1, \left|s\right|\le \omega \hfill \end{array}\right.$$

(11)

where $\delta$ and $\omega$ are constants and $\delta >\alpha$, $\omega >1$.

Theorem 1. 

For the reaching law given as Equation (10), the state of system$s$converges to the equilibrium point in fixed time and the convergence time$T$satisfies the following conditions:$T<t_1+t_2+t_3$, where

$$\left\{ \begin{array}{c}{t}_1=\frac{{\omega }^{1-\gamma }-s_0^{1-\gamma }}{\left(\gamma -1\right)l_3}\hfill \\ t_2=\frac{1-{\alpha }^{1-\alpha }}{\left(\alpha -1\right)l_1}\hfill \\ t_3=\frac{1}{\left(1-\beta \right)l_2}\hfill \end{array}\right.$$

(12)

Proof of Theorem 1. 

(1) Analysis of accessibility and stability

According to Equation (10), we obtain:

$$\begin{array}{cc}s\dot{s}\hfill & =s\left[-l_1{\left|s\right|}^{\alpha }sgn\left(s\right)-l_2{\left|s\right|}^{\beta }sgn\left(s\right)-l_3{\left|s\right|}^{\gamma }sgn\left(s\right)\right]\hfill \\ \hfill & =-l_1{\left|s\right|}^{\alpha +1}-l_2{\left|s\right|}^{\beta +1}-l_3{\left|s\right|}^{\gamma +1}\le 0\hfill \end{array}$$

(13)

Only if $s=0$ can reach $s\dot{s}=0$.

Therefore, the system state $s$ can reach the equilibrium point $s=0$ under the reaching law Equation (10).

(2) Fixed-time convergence

It is supposed that the initial state of the system is $\left|s_0\right|>\omega >1$. The process of system convergence may be divided into 3 stages as follows: $s_0\to s(t_1)=\omega \to s(t_2)=1\to s(t_3)=0$. For the sake of analysis, we assume that $l_1=l_2=l_3$.

(i)

$s_0\to s(t_1)=\omega$

In this stage, because $\gamma =\omega ,\omega >\alpha >\beta$, then ${\left|s\right|}^{\gamma }>{\left|s\right|}^{\alpha }>{\left|s\right|}^{\beta }$, the third term $-l_3{\left|s\right|}^{\gamma }sgn\left(s\right)$ plays a major role in the reaching law. In other words, the reaching speed at this stage is mainly affected by the third term; then, the reaching law Equation (10) can be expressed as:

$$\dot{s}=-l_3{\left|s\right|}^{\gamma }sgn\left(s\right)$$

(14)

Taking the integrate of Equation (14), we have:

$$s^{1-\gamma }=-\left(1-\gamma \right)l_{3}t+s_0^{1-\gamma }$$

(15)

Then, the max reaching time in this stage can be expressed as follows:

$$t_1=\frac{{\omega }^{1-\gamma }-s_0^{1-\gamma }}{\left(\gamma -1\right)l_3}$$

(16)

Thus, the time required for the system to go from $s_0$ to $s(t_1)$ is less than $t_1$.

(ii)

$s(t_1)=\omega \to s(t_2)=1$

In this stage, $\gamma =1,\alpha >\gamma >\beta >0$ and ${\left|s\right|}^{\alpha }>{\left|s\right|}^{\gamma }>{\left|s\right|}^{\beta }$, and the first term $-l_1{\left|s\right|}^{\alpha }sgn\left(s\right)$ plays a major role in the reaching law; then, the reaching law Equation (10) can be expressed as:

$$\dot{s}=-l_1{\left|s\right|}^{\alpha }sgn\left(s\right)$$

(17)

Integrating Equation (17), we obtain:

$$s^{1-\alpha }=-\left(1-\gamma \right)l_{3}t+{\alpha }^{1-\alpha }$$

(18)

Then, the max reaching time in this stage can be expressed as follows:

$$t_2=\frac{1-{\alpha }^{1-\alpha }}{\left(\alpha -1\right)l_1}$$

(19)

Therefore, the time required for the system to go from $s(t_1)$ to $s(t_2)$ is less than $t_2$.

(iii)

$s(t_2)=1\to s(t_3)=0$

In this stage, because $s<0$, ${\left|s\right|}^{\beta }>{\left|s\right|}^{\gamma }>{\left|s\right|}^{\alpha }$, the second term $-l_2{\left|s\right|}^{\beta }sgn\left(s\right)$ plays a major role in the reaching law, and the reaching law Equation (10) can be expressed as:

$$\dot{s}=-l_2{\left|s\right|}^{\beta }sgn\left(s\right)$$

(20)

Integrating Equation (20), we have:

$$s^{1-\beta }=-\left(1-\beta \right)l_{2}t+1$$

(21)

Then, the max reaching time in this stage can be expressed as follows:

$$t_3=\frac{1}{\left(1-\beta \right)l_2}$$

(22)

Therefore, the time required for the system to go from $s(t_2)$ to $s(t_3)$ is less than $t_3$.

Above all, for the reaching law of sliding mode control as Equation (10), and the reaching time $T$ satisfies the following formula:

$$T<t_1+t_2+t_3$$

(23)

The proof is completed. □

According to the analysis above, for the reaching law given by Equation (10), when the initial state of the system is $s_0>\omega$, the convergence process of sliding surface $s$ can be divided into 3 stages, $s_0\to s(t_1)=\omega \to s(t_2)=1\to s(t_3)=0$, and TPRL can speed up the reaching time of each stage. The advantages of TPRL can be conclude as follows:

(1) Compared with traditional reaching law, the TPRL proposed in this paper can shorten the reaching time of system converging to the equilibrium point. Firstly, TPRL can reduce the reaching time when the initial conditions of the system parameters are far from the equilibrium point. According to Equation (10), when the sliding surface $s>\omega$, $\gamma =\delta >\alpha$. Compared with the traditional double power reaching law, which has the form $\dot{s}=-l_1{\left|s\right|}^{\alpha }sgn\left(s\right)-l_2{\left|s\right|}^{\beta }sgn\left(s\right)$, the third term $-l_3{\left|s\right|}^{\gamma }sgn\left(s\right)$ of TPRL is the largest, which plays a major role in the reaching law and speed up the process of convergence. Secondly, TPRL can reduce the reaching time when the condition of the system is close to the equilibrium point. Compared with the traditional double power reaching law, when $s\le \omega$, $\gamma =1$, TPRL can reach the equilibrium point in shorter time, since the third term of TPRL is set as $-l_3\left|s\right|sgn\left(s\right)$ in this stage, which can increase the value of the reaching law, and thus, shorten the reaching time of the system.

(2) TPRL can attenuate the chattering of the sliding mode approach more effectively than traditional reaching law. In order to reduce chattering, this paper tries to present a new reaching law TPRL, which is continuous when $s\to 0$, $\underset{s\to 0}{\lim }(-l_1{\left|s\right|}^{\alpha }sgn\left(s\right)-l_2{\left|s\right|}^{\beta }sgn\left(s\right)-l_3{\left|s\right|}^{\gamma }sgn\left(s\right))=\left\{\begin{array}{c}0, s\to 0^{+}\\ 0, s\to 0^{-}\end{array}\right.$, where $s$ is the sliding surface. For the traditional exponential reaching law given as $\dot{s}=-l_{1}sgn\left(s\right)-l_{2}s$, the discontinuous functions $sgn\left(s\right)$ may cause some undesired effects, such as the chattering phenomenon due to $\underset{s\to 0}{\lim }[-l_{1}sgn\left(s\right)-l_{2}s]=\left\{\begin{array}{c}{l}_1, s\to 0^{+}\\ -l_1, s\to 0^{-}\end{array}\right.$.

3.2. Sliding Mode Backstepping Controller Design

This subsection presents a sliding mode backstepping control scheme for a class of high-order nonlinear system in the presence of uncertainties. The aim is to design a controller to guarantee that the output of the system will track the command signal.

A high order nonlinear system with uncertainties is expressed as follows:

$$\left\{ \begin{array}{c}{\dot{x}}_1=x_2+{\phi }_1^T\left(x_1\right){\theta }_1\hfill \\ {\dot{x}}_2=x_3+{\phi }_2^T\left(x_2\right){\theta }_2\hfill \\ ⋮\hfill \\ {\dot{x}}_i=x_{i+1}+{\phi }_i^T\left(x_i\right){\theta }_i\hfill \\ ⋮\hfill \\ {\dot{x}}_n=f\left(x,t\right)+G\left(x,t\right)u+{\phi }_n^T\left(x_n\right){\theta }_n\hfill \\ y=x_1\hfill \end{array}\right.$$

(24)

where $x_i={\left[\begin{array}{c}\begin{array}{c}{x}_{i,1},x_{i,2}\end{array},\cdots ,x_{i,m}\end{array}\right]}^T\in {ℝ}^m,i=1,2\cdots n$ are states; $f\left(x,t\right)$ and $G\left(x,t\right)$ are given nonlinear functions; $u\in {ℝ}^m$ is control input; $\mathit{y}\in {ℝ}^m$ is the system output; ${\phi }_i^T\left(x_i\right)=diag\left[\begin{array}{c}\begin{array}{c}{\phi }_{i,1},{\phi }_{i,2}\end{array},\cdots ,{\phi }_{i,m}\end{array}\right],i=1,2\cdots n$ are known functions, which are assumed to be sufficient smooth; and ${\theta }_i={\left[\begin{array}{c}\begin{array}{c}{\theta }_{i,1},{\theta }_{i,2}\end{array},\cdots ,{\theta }_{i,m}\end{array}\right]}^T\in {ℝ}^m,i=1,2\cdots n$ are unknown constant parameters, where ${\theta }_i$ is defined as follows:

$${\theta }_i={\widehat{\theta }}_i+{\tilde{\theta }}_i$$

(25)

where ${\widehat{\theta }}_i$ is the estimation of ${\theta }_i$, ${\tilde{\theta }}_i$ is estimation error.

The design steps for controller are as follows.

Defining a command signal for the first subsystem as ${\mu }_1=x_{1\mathrm{d}}$, the tracking error for the first subsystem is defined as follows:

$$z_1=x_1-{\mu }_1$$

(26)

Taking the derivative of Equation (26), we obtain:

$${\dot{z}}_1={\dot{x}}_1-{\dot{\mu }}_1=x_2+{\phi }_1^T\left(x_1\right){\theta }_1-{\dot{\mu }}_1$$

(27)

To stabilize the first subsystem, the virtual control law is defined as follows:

$${\mu }_2=-k_1{z}_1+{\dot{\mu }}_1-{\phi }_1^T\left(x_1\right){\widehat{\theta }}_1$$

(28)

where $k_1$ is a positive definite matrix, $k_i,i=2,3\cdots n$, in later steps are also positive definite matrixes. The tracking error for the second subsystem is defined as follows:

$$z_2=x_2-{\mu }_2$$

(29)

Substituting Equations (28) and (29) into Equation (27), we get:

$${\dot{z}}_1=-k_1{z}_1+z_2-{\phi }_1^T\left(x_1\right){\widehat{\theta }}_1+{\phi }_1^T\left(x_1\right){\theta }_1=-k_1{z}_1+z_2+{\phi }_1^T\left(x_1\right){\tilde{\theta }}_1$$

(30)

A Lyapunov function candidate for the first subsystem is chosen as follows:

$$V_1=\frac{1}{2}{z}_1^T{z}_1+\frac{1}{2}{\tilde{\theta }}_1^T{\Gamma }_1^{-1}{\tilde{\theta }}_1$$

(31)

where ${\Gamma }_1$ is a symmetric positive definite matrix, ${\Gamma }_i,i=2,3\cdots n$, in later steps are also symmetric positive definite matrixes. Taking the derivative of Equation (31) along Equation (30), we have:

$$\begin{array}{c}{\dot{V}}_1=z_1^T\left(-k_1{z}_1+z_2+{\phi }_1^T\left(x_1\right){\tilde{\theta }}_1\right)+{\tilde{\theta }}_1^T{\Gamma }_1^{-1}\left({\dot{\theta }}_1-\dot{{\widehat{\theta }}_1}\right)\hfill \\ =-k_1{z}_1^T{z}_1+z_1^T{z}_2+z_1^T{\phi }_1^T\left(x_1\right){\tilde{\theta }}_1+{\tilde{\theta }}_1^T{\Gamma }_1^{-1}\left({\dot{\theta }}_1-\dot{{\widehat{\theta }}_1}\right)\hfill \\ =-k_1{z}_1^T{z}_1+z_1^T{z}_2+{\tilde{\theta }}_1^T\left({\phi }_1\left(x_1\right)z_1-{\Gamma }_1^{-1}\dot{{\widehat{\theta }}_1}\right)\hfill \end{array}$$

(32)

A Lyapunov function candidate for the 2nd subsystem is chosen as follows:

$$V_2=V_1+\frac{1}{2}{z}_2^T{z}_2+\frac{1}{2}{\tilde{\theta }}_2^T{\Gamma }_2^{-1}{\tilde{\theta }}_2$$

(33)

Taking the derivative of Equation (33), we obtain:

$$\begin{array}{c}{\dot{V}}_2={\dot{V}}_1+z_2^T\left(-k_2{z}_2-z_1+z_3+{\phi }_2^T\left(x_2\right){\tilde{\theta }}_2\right)+{\tilde{\theta }}_2^T{\Gamma }_2^{-1}\left({\dot{\theta }}_2-\dot{{\widehat{\theta }}_2}\right)\hfill \\ ={\dot{V}}_1+z_2^T\left(-k_2{z}_2-z_1+z_3\right)+z_2^T{\phi }_2^T\left(x_2\right){\tilde{\theta }}_2+{\tilde{\theta }}_2^T{\Gamma }_2^{-1}\left({\dot{\theta }}_2-\dot{{\widehat{\theta }}_2}\right)\hfill \\ =-k_1{z}_1^T{z}_1-k_2{z}_2^T{z}_2+z_2^T{z}_3+{\tilde{\theta }}_1^T\left({\phi }_1\left(x_1\right)z_1-{\Gamma }_1^{-1}\dot{{\widehat{\theta }}_1}\right)+{\tilde{\theta }}_2^T\left({\phi }_2\left(x_2\right)z_2-{\Gamma }_2^{-1}\dot{{\widehat{\theta }}_2}\right)\hfill \\ ={\displaystyle {\displaystyle \sum }_{i=1}^2}[-k_i{z}_i^T{z}_i+{\tilde{\theta }}_i^T\left({\phi }_i\left(x_i\right)z_i-{\Gamma }_i^{-1}\dot{{\widehat{\theta }}_i}\right)]+z_2^T{z}_3\hfill \end{array}$$

(34)

where $z_3=x_3-{\mu }_3$ is the tracking error for the 2nd subsystem and ${\mu }_3=-k_2{z}_2+{\dot{\mu }}_2-z_1-{\phi }_2^T\left(x_2\right){\widehat{\theta }}_2$ is the virtual control law.

Similarly, a Lyapunov function candidate for the $\left(n-1\right)th$ subsystem is chosen as:

$$V_{n-1}=V_{n-2}+\frac{1}{2}{z}_{n-1}^T{z}_{n-1}+\frac{1}{2}{\tilde{\theta }}_{n-1}^T{\Gamma }_{n-1}^{-1}{\tilde{\theta }}_{n-1}$$

(35)

Taking the derivative of Equation (35), we obtain:

$$\begin{array}{c}{\dot{V}}_{n-1}={\dot{V}}_{n-2}+z_{n-1}^T\left(-k_{n-1}{z}_{n-1}-z_{n-2}+z_{n-1}\right)+{\tilde{\theta }}_{n-1}^T(z_{n-1}-{\Gamma }_{n-1}^{-1}{\dot{\widehat{\theta }}}_{n-1})\hfill \\ =-k_1{z}_1^T{z}_1-k_2{z}_2^T{z}_2-\cdots -k_{n-1}{z}_{n-1}^T{z}_{n-1}+z_{n-1}^T{z}_n+{\tilde{\theta }}_1^T\left({\phi }_1\left(x_1\right)z_1-{\Gamma }_1^{-1}\dot{{\widehat{\theta }}_1}\right)\hfill \\ +{\tilde{\theta }}_2^T\left({\phi }_2\left(x_2\right)z_2-{\Gamma }_2^{-1}\dot{{\widehat{\theta }}_2}\right)+\cdots +{\tilde{\theta }}_{n-1}^T({\phi }_{n-1}\left(x_{n-1}\right)z_{n-1}-{\Gamma }_{n-1}^{-1}{\dot{\widehat{\theta }}}_{n-1})\hfill \\ ={\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}[-k_i{z}_i^T{z}_i+{\tilde{\theta }}_i^T({\phi }_i\left(x_i\right)z_i-{\Gamma }_i^{-1}{\dot{\widehat{\theta }}}_i)]+z_{n-1}^T{z}_n\hfill \end{array}$$

(36)

where $z_n=x_n-{\mu }_n$ is the tracking error for the nth subsystem and ${\mu }_n=-k_{n-1}{z}_{n-1}+{\dot{\mu }}_{n-1}-z_{n-2}-{\phi }_n^T\left(x_n\right){\widehat{\theta }}_n$ is the virtual control law.

Defining a sliding function for system as follows:

$$\begin{array}{c}S=c_1{z}_1+c_2{z}_2+\cdots c_i{z}_i+\cdots +z_n\hfill \\ ={\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}{c}_i{z}_i+z_n\hfill \end{array}$$

(37)

where $S\in {ℝ}^m$, $c_i,i=1,2,\cdots n-1$ are positive definite matrixes.

Taking the derivative of Equation (37) along Equation (24), we obtain:

$$\begin{array}{c}\dot{S}={\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}{c}_i{\dot{z}}_i+{\dot{z}}_n={\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}{c}_i{\dot{z}}_i+{\dot{x}}_n-{\dot{\mu }}_n\hfill \\ ={\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}{c}_i{\dot{z}}_i+f\left(x,t\right)+G\left(x,t\right)u+{\phi }_n^T\left(x_n\right){\theta }_n-{\dot{\mu }}_n\hfill \end{array}$$

(38)

A Lyapunov function candidate for the nth subsystem is chosen as:

$$\begin{array}{c}{V}_n=V_{n-1}+\frac{1}{2}{S}^{T}S+\frac{1}{2}{\tilde{\theta }}_n^T{\Gamma }_n^{-1}{\tilde{\theta }}_n\hfill \\ =\frac{1}{2}{\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}{k}_i{z}_i^T{z}_i+\frac{1}{2}{S}^{T}S+\frac{1}{2}{\displaystyle {\displaystyle \sum }_{j=1}^n}{\tilde{\theta }}_j^T{\Gamma }_j^{-1}{\tilde{\theta }}_j\hfill \end{array}$$

(39)

Taking the derivative of Equation (39) along Equations (36) and (38), we obtain:

$$\begin{array}{c}{\dot{V}}_n={\dot{V}}_{n-1}+S^T\dot{S}+{\tilde{\theta }}_n^T{\Gamma }_n^{-1}{\dot{\tilde{\theta }}}_n\hfill \\ ={\dot{V}}_{n-1}+S^T[{\displaystyle \sum _{i=1}^{n-1}}{c}_i{\dot{z}}_i+f(x,t)+G(x,t)u+{\phi }_n^T\left(x_n\right){\theta }_n-{\dot{\mu }}_n]+{\tilde{\theta }}_n^T{\Gamma }_n^{-1}\left({\dot{\theta }}_n-{\dot{\tilde{\theta }}}_n\right)\hfill \\ ={\dot{V}}_{n-1}+S^T[{\displaystyle \sum _{i=1}^{n-1}}{c}_i{\dot{z}}_i+f(x,t)+G(x,t)u+{\phi }_n^T\left(x_n\right){\widehat{\theta }}_n-{\dot{\mu }}_n]+{\tilde{\theta }}_n^T\left[{\phi }_n\left(x_n\right)S-{\Gamma }_n^{-1}{\dot{\widehat{\theta }}}_n\right]\hfill \\ ={\displaystyle \sum _{i=1}^{n-1}}[-k_i{z}_i^T{z}_i+{\tilde{\theta }}_i^T\left({\phi }_i\left(x_i\right)z_i-{\Gamma }_i^{-1}{\dot{\widehat{\theta }}}_i\right)]+z_{n-1}^T{z}_n+S^T[{\displaystyle \sum _{i=1}^{n-1}}{c}_i{\dot{z}}_i+f(x,t)+G(x,t)u\hfill \\ +{\phi }_n^T\left(x_n\right){\widehat{\theta }}_n-{\dot{\mu }}_n]+{\tilde{\theta }}_n^T[{\phi }_n\left(x_n\right)S-{\Gamma }_n^{-1}{\widehat{\theta }}_n]\hfill \end{array}$$

(40)

Substituting Equation (37) into Equation (40), we have:

$$\begin{array}{c}{\dot{V}}_n={\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}[-k_i{z}_i^T{z}_i+{\tilde{\theta }}_i^T({\phi }_i\left(x_i\right)z_i-{\Gamma }_i^{-1}{\dot{\widehat{\theta }}}_i)]+z_{n-1}^T\left(S-{\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}{c}_i{z}_i\right)\hfill \\ +S^T[{\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}{c}_i{\dot{z}}_i+f\left(x,t\right)+G\left(x,t\right)u+{\phi }_n^T\left(x_n\right){\widehat{\theta }}_n-{\dot{\mu }}_n]+{\tilde{\theta }}_n^T[{\phi }_n\left(x_n\right)S-{\Gamma }_n^{-1}\dot{{\widehat{\theta }}_n}]\hfill \\ ={\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}[-k_i{z}_i^T{z}_i+{\tilde{\theta }}_i^T({\phi }_i\left(x_i\right)z_i-{\Gamma }_i^{-1}{\dot{\widehat{\theta }}}_i)]-z_{n-1}^T{\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}{c}_i{z}_i\hfill \\ +S^T[z_{n-1}+{\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}{c}_i{\dot{z}}_i+f\left(x,t\right)+G\left(x,t\right)u+{\phi }_n^T\left(x_n\right){\widehat{\theta }}_n-{\dot{\mu }}_n]+{\tilde{\theta }}_n^T[{\phi }_n\left(x_n\right)S-{\Gamma }_n^{-1}\dot{{\widehat{\theta }}_n}]\hfill \\ =-{\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}{k}_i{z}_i^T{z}_i-z_{n-1}^T{\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}{c}_i{z}_i+S^T[z_{n-1}+{\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}{c}_i{\dot{z}}_i+f\left(x,t\right)+G\left(x,t\right)u+{\phi }_n^T\left(x_n\right){\widehat{\theta }}_n\hfill \\ -{\dot{\mu }}_n]+{\displaystyle {\displaystyle \sum }_{i=1}^{n-1}}{\tilde{\theta }}_i^T[{\phi }_i\left(x_i\right)z_i-{\Gamma }_i^{-1}{\dot{\widehat{\theta }}}_i]+{\tilde{\theta }}_n^T[{\phi }_n\left(x_n\right)S-{\Gamma }_n^{-1}\dot{{\widehat{\theta }}_n}]\hfill \end{array}$$

(41)

In order to ensure the stability of system, the control law is designed as follows:

$$u=G{\left(x,t\right)}^{-1}(-z_{n-1}-{\displaystyle \sum }_{i=1}^{n-1}{c}_i{\dot{z}}_i-f\left(x,t\right)-G\left(x,t\right)-{\phi }_n^T\left(x_n\right){\widehat{\theta }}_n+{\dot{\mu }}_n+u_{sw})$$

(42)

where $u_{sw}\in {ℝ}^m$ is the triple power reaching law proposed as Equation (10) and $u_{sw}$ is designed as:

$$u_{sw,i}=-l_{i,1}{\left|s_i\right|}^{{\alpha }_i}sgn\left(s_i\right)-l_{i,2}{\left|s_i\right|}^{{\beta }_i}sgn\left(s_1\right)-l_{i,1}{\left|s_i\right|}^{{\gamma }_i}sgn\left(s_i\right)$$

(43)

where $i=1, 2\cdots m$. Adaptive laws are designed as follows:

$$\left\{ \begin{array}{c}{\dot{\widehat{\theta }}}_1={\Gamma }_1{\phi }_1\left(x_1\right)z_1\hfill \\ {\dot{\widehat{\theta }}}_2={\Gamma }_2{\phi }_2\left(x_2\right)z_2\hfill \\ \hspace{0.17em}\hspace{0.17em} ⋮\hfill \\ {\dot{\widehat{\theta }}}_{n-1}={\Gamma }_{n-1}{\phi }_{n-1}\left(x_{n-1}\right)z_{n-1}\hfill \\ {\dot{\widehat{\theta }}}_n={\Gamma }_n{\phi }_n\left(x_n\right)S\hfill \end{array}\right.$$

(44)

3.3. Stability Analysis

Theorem 2. 

The closed-loop system consisting of the nonlinear system given by Equation (24) with the controller given by Equation (42), and the adaptive laws given by Equation (44) can be guaranteed to be stable.

Proof of Theorem 2. 

A Lyapunov function candidate is chosen as Equation (39). Substituting Equations (42) and (44) into Equation (41), we obtain:

$$\begin{array}{c}{\dot{V}}_n=-{\displaystyle \sum _{j=1}^{n-1}}{k}_j{z}_j^T{z}_j-z_{n-1}^T{\displaystyle \sum _{i=1}^{n-1}}{c}_i{z}_i-S^T{u}_{sw}\hfill \\ =-{\displaystyle \sum _{j=1}^{n-1}}{k}_j{z}_j^T{z}_j-z_{n-1}^T{\displaystyle \sum _{i=1}^{n-1}}{c}_i{z}_i-{\displaystyle \sum _{i=1}^m}(l_{i,1}{\left|s_i\right|}^{{\alpha }_i+1}+l_{i,2}{\left|s_i\right|}^{{\beta }_{i+1}}+l_{i,3}{\left|s_i\right|}^{{\gamma }_i+1})\hfill \\ =-z^{T}Qz-{\displaystyle \sum _{i=1}^m}(l_{i,1}{\left|s_i\right|}^{{\alpha }_i+1}+l_{i,2}{\left|s_i\right|}^{{\beta }_{i+1}}+l_{i,3}{\left|s_i\right|}^{{\gamma }_{i+1}})\hfill \end{array}$$

(45)

in which

$$Q=\left[\begin{array}{ccccc}{k}_1& 0& \cdots & 0& 0\\ 0& k_2& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & k_{n-2}& 0\\ c_1& c_2& \cdots & c_{\mathrm{n}-2}& k_{n-1}+c_{n-1}\end{array}\right]$$

(46)

and

$$z={\left[\begin{array}{c}\begin{array}{cccc}{z}_1& z_2& \cdots & z_{n-1}\end{array}\end{array}\right]}^T$$

(47)

Because the principal minors in matrix $Q$ are positive, $Q$ is a positive definite matrix. Then, we obtain:

$${\dot{V}}_n=-z^{T}Qz-{\displaystyle \sum }_{i=1}^m(l_{i,1}{\left|s_i\right|}^{{\alpha }_i+1}+l_{i,2}{\left|s_i\right|}^{{\beta }_i+1}+l_{i,3}{\left|s_i\right|}^{{\gamma }_i+1})\le 0$$

(48)

The proof is completed. □

4. Flight Controller Design for NSHV

In this section, the flight controller for the ascent phase of NSHV is developed through Theorems 1 and 2.

4.1. Controller Design

For the given transformation model of Equation (8), the command signals are velocity $V_d$ and altitude $h_d$ and the control object is that the controller can asymptotically track the command signal.

According to Equations (42) and (43), the tracking control law is designed as follows:

$$u=G{\left(x,t\right)}^{-1}(-z_3-{\displaystyle \sum }_{i=1}^3{c}_i{\dot{z}}_i-f\left(x,t\right)-{\phi }_3^T\left(x_3\right){\widehat{\theta }}_3+{\dot{\mu }}_4+u_{sw})$$

(49)

The reaching laws for sliding mode is designed as follows:

$$u_{sw}=\left[\begin{array}{c}-l_{1,1}{\left|s_1\right|}^{{\alpha }_1}sgn\left(s_1\right)-l_{1,2}{\left|s_1\right|}^{{\beta }_1}sgn\left(s_1\right)-l_{1,3}{\left|s_1\right|}^{{\gamma }_1}sgn\left(s_1\right)\\ -l_{2,1}{\left|s_2\right|}^{{\alpha }_2}sgn\left(s_2\right)-l_{2,2}{\left|s_2\right|}^{{\beta }_2}sgn\left(s_2\right)-l_{2,3}{\left|s_2\right|}^{{\gamma }_2}sgn\left(s_2\right)\end{array}\right]$$

(50)

where

$${\gamma }_i=\left\{ \begin{array}{c}{\delta }_i, \left|s_i\right|>{\omega }_i\\ 1, \left|s_i\right|\le {\omega }_i\end{array}\right.,i=1,2$$

(51)

According to Equation (44), adaptive laws are designed as follows:

$$\left\{ \begin{array}{c}{\dot{\widehat{\theta }}}_1={\Gamma }_1{\phi }_1\left(x_1\right)z_1\\ {\dot{\widehat{\theta }}}_2={\Gamma }_2{\phi }_2\left(x_2\right)z_2\\ {\dot{\widehat{\theta }}}_3={\Gamma }_3{\phi }_3\left(x_3\right)z_3\\ {\dot{\widehat{\theta }}}_4={\Gamma }_4{\phi }_4\left(x_4\right)S\end{array}\right.$$

(52)

where $S=c_1{z}_1+c_2{z}_2+c_3{z}_3+z_4$, and the errors are defined as follows:

$$\left\{ \begin{array}{c}{z}_1=x_1-{\mu }_1\\ z_2=x_2-{\mu }_2\\ z_3=x_3-{\mu }_3\\ z_4=x_4-{\mu }_4\end{array}\right.$$

(53)

The virtual control laws are designed as follows:

$$\left\{ \begin{array}{c}{\mu }_1={\left[{{\displaystyle \int }}_0^t{V}_d\left(\tau \right)d\tau \hspace{1em}{h}_d\right]}^T\hfill \\ {\mu }_2=-k_1{z}_1+{\dot{\mu }}_1-{\phi }_1^T\left(x_1\right){\widehat{\theta }}_1\hfill \\ {\mu }_3=-k_2{z}_2+{\dot{\mu }}_2-z_1-{\phi }_2^T\left(x_2\right){\widehat{\theta }}_2\hfill \\ {\mu }_4=-k_3{z}_3+{\dot{\mu }}_3-z_2-{\phi }_3^T\left(x_3\right){\widehat{\theta }}_3\hfill \end{array}\right.$$

(54)

4.2. Stability Analysis

To analyze the stability of system, according to Equation (39), a Lyapunov function candidate for NSHV is chosen as:

$$V_4=\frac{1}{2}{\displaystyle \sum }_{i=1}^3{k}_i{z}_i^T{z}_i+\frac{1}{2}{\displaystyle \sum }_{j=1}^4{\tilde{\theta }}_j^T{\Gamma }_j^{-1}{\tilde{\theta }}_j+\frac{1}{2}{S}^{T}S$$

(55)

According to Theorem 2, two matrixes are defined as follows:

$$\left\{ \begin{array}{c}z={\left[\begin{array}{c}\begin{array}{ccc}{z}_1& z_2& z_3\end{array}\end{array}\right]}^T\hfill \\ Q=\left[\begin{array}{ccc}{k}_1& 0& 0\\ 0& k_2& 0\\ c_1& c_2& k_3+c_3\end{array}\right]\hfill \end{array}\right.$$

(56)

Taking the derivative of Equation (55), according to Equation (48), we have:

$${\dot{V}}_4=-z^{T}Qz-{\displaystyle \sum }_{i=1}^2(l_{i,1}{\left|s_i\right|}^{{\alpha }_i+1}+l_{i,2}{\left|s_i\right|}^{{\beta }_i+1}+l_{i,3}{\left|s_i\right|}^{{\gamma }_i+1})\le 0$$

(57)

5. Simulation Results

For the verification of proposed control method in NSHV applications, simulation is performed by MATLAB/Simulink. This section includes two scenarios, and the NSHV is flying in the ascent phase. Scenario 1 verifies the advantage of the novel reaching law proposed in this paper, and the good tracking performance of the proposed method in this paper for NSHV in the ascent phase is illustrated in Scenario 2.

5.1. Scenario 1: Simulation of Reaching Laws

To illustrate the effectiveness of triple power reaching law proposed in this paper, five different reaching laws are adopted for comparison. The five compared reaching law are expressed as follows:

(a) Triple power reaching law (TPRL):

$$\dot{s}=-l_1{\left|s\right|}^{\alpha }sgn\left(s\right)-l_2{\left|s\right|}^{\beta }sgn\left(s\right)-l_3{\left|s\right|}^{\gamma }sgn\left(s\right)$$

(58)

(b) Double power reaching law (DPRL):

$$\dot{s}=-l_1{\left|s\right|}^{\alpha }sgn\left(s\right)-l_2{\left|s\right|}^{\beta }sgn\left(s\right)$$

(59)

(c) Exponential reaching law (ERL):

$$\dot{s}=-l_{1}sgn\left(s\right)-l_{2}s$$

(60)

(d) Traditional symbolic function reaching law (TRL):

$$\dot{s}=-l_{1}sgn\left(s\right)$$

(61)

(e) Fast power reaching law (FPRL):

$$\dot{s}=-l_1{\left|s\right|}^{\beta }sgn\left(s\right)-l_{2}s$$

(62)

The traditional sliding mode controller (SMC) is adopted in this scenario to track the velocity of the NSHV, and the sliding function is defined as follows [24]:

$$s_v={\left(\frac{d}{dt}+\lambda \right)}^3{{\displaystyle \int }}_0^{t}e\left(\tau \right)dt$$

(63)

The model parameters of the NSHV are presented in

Table 1. Parameters of NSHV.

Parameter	Value	Unites
Mass	100,200	$\mathrm{kg}$
Reference area	389	m
Aerodynamic chord	30	${\mathrm{m}}^2$
Moment of inertia	8,466,900	$\mathrm{kg}·{\mathrm{m}}^2$

.

In this scenario, the initial parameters of the NSHV are set as $V_0=4590\mathrm{m}/\mathrm{s}$, $h_0=33,528\mathrm{m}$, $\gamma =0^{°}$, $\alpha ={2.75}^{°}$, and $q=0^{°}/\mathrm{s}$. the uncertainties are set as 0, and the command signal of step velocity is set as 10 m/s.

The value of parameters for five different reaching laws are given in

Table 2. Parameters of five reaching laws.

TPRL	DPRL	ERL	TRL	FPRL
$l_1=1$	$l_1=1$	$l_1=1$	$l_1=1$	$l_1=1$
$l_2=1$	$l_2=1$	$l_2=1$		$l_2=1$
$l_3=1$	$\alpha =1.3$			$\beta =0.7$
$\alpha =1.3$	$\beta =0.7$
$$\begin{gathered} \beta =0.7 \\ \delta =1.8 \end{gathered}$$
$\omega =2$

.

The simulation results for five different reaching laws are shown in Figure 3, Figure 4 and Figure 5.

As can be seen from Figure 3, in the five reaching laws, the sliding surface under TPRL can converge to the equilibrium point zero in the shortest time. In Figure 4, compared with the other four reaching laws, the rise-time of velocity under TPRL is shortest and the velocity tracking under TPRL reaches a steady state in a shortest time. In Figure 5, the control input under TPRL exhibits no chattering, while that for TRL and ERL exhibit chattering.

According to the simulation results above, the following conclusions can be drawn as follows:

(1) It can be known from Figure 3 and Figure 4 that TPRL proposed in this paper can guarantee system converging to the equilibrium point in shorter time than other reaching laws mentioned above. In other words, considering similar gains, TPRL approach specifies faster reaching speed with respect to other reaching laws.

(2) It can be seen from Figure 5 that TPRL reduces the chattering of sliding mode approach with respect to the traditional reaching laws. When the system state comes close to the equilibrium point, the coefficients of TPRL methods gradually diminish to mitigate the chattering problem, expressed as $\underset{s\to 0}{\lim }(-l_1{\left|s\right|}^{\alpha }sgn\left(s\right)-l_2{\left|s\right|}^{\beta }sgn\left(s\right)-l_3{\left|s\right|}^{\gamma }sgn\left(s\right))=\left\{\begin{array}{c}0,s\to 0^{+}\\ 0,s\to 0^{-}\end{array}\right.$. However, in Scenario 1, when s→0, ERL is expressed as $\underset{s\to 0}{\lim }[-l_{1}sgn\left(s\right)-l_{2}s]=\left\{\begin{array}{c}1,s\to 0^{+}\\ -1,s\to 0^{-}\end{array}\right.$, and it is shown as $\underset{s\to 0}{\lim }[-l_{1}sgn\left(s\right)]=\left\{\begin{array}{c}1,s\to 0^{+}\\ -1,s\to 0^{-}\end{array}\right.$ for TRL. Therefore, TPRL can reduce chattering effectively than traditional reaching law ERL and TRL.

5.2. Scenario 2: Simulation of Controller for NSHV

In this scenario, the NSHV is flying in the ascent phase, and the initial conditions of the NSHV are $V_0=1700 \mathrm{m}/\mathrm{s}$, $h_0=12,000 \mathrm{m}$, $\gamma =0^{°}$, $\alpha =-{1.3}^{°}$, and $q=0^{°}/\mathrm{s}$ and the final velocity and altitude are $V_f=3700 \mathrm{m}/\mathrm{s}$ and $h_f=32,000 \mathrm{m}$, respectivly. It is worth noting that due to limitations in real systems, the elevator deflection ${\delta }_e$ should be limited in the range from $-{20}^{°}$ to ${20}^{°}$ and ${\beta }_c$ should be limited in the range from 0 to 1. The parameters of the NSHV are given in Table 1.

To verify effectiveness of the method proposed in this paper, the command signals are set as $h_c\left(t\right)=h_0+\Delta h\left(t\right)$ and $V_c\left(t\right)=V_0+\Delta V\left(t\right)$, where $\Delta h\left(t\right)$ and $\Delta V\left(t\right)$ are generated by inputting step signals into the following filters:

$$\left\{ \begin{array}{c}\frac{\Delta h\left(s\right)}{h_{step}\left(s\right)}=\frac{{0.03}^4}{{\left(s+0.03\right)}^4}\\ \frac{\Delta V\left(s\right)}{V_{step}\left(s\right)}=\frac{{0.03}^4}{{\left(s+0.03\right)}^4}\end{array}\right.$$

(64)

where the altitude step signal is set as $h_{\mathit{step}}=20,000\mathrm{m}$ and the velocity step signal is set as $V_{\mathit{step}}=2000\mathrm{m}/\mathrm{s}$. Uncertainties and disturbances are considered as follows:

(1)

Parameters of uncertainties are given by Equation (13) and set as 20%, which are presented as follows:

$${\Delta }_i=0.2sin\left(0.02\pi t\right)$$

(65)

where $\begin{array}{c}i=m,I_{yy},\rho ,s,\overline{c},c_e,C_L,C_D,C_T,C_M^{\alpha },C_M^q,C_M^{{\delta }_e}\hfill \end{array}$

(2)

Parameters of disturbances for control input are set as follows:

$$\left\{ \begin{array}{c}{\beta }_c={\beta }_{c0}+0.1sin\left(0.02\pi t\right)\\ {\delta }_e={\delta }_{e0}+0.1sin\left(0.02\pi t\right)\end{array}\right.$$

(66)

The controller is developed in Equation (49) along with the adaptive update laws given by Equation (52). The parameters of the controller are presented in

Table 3. Controller Parameters.

Parameter	Value
$k_1$	$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$k_2$	$\left[\begin{array}{cc}1.2& 0\\ 0& 1.2\end{array}\right]$
$k_3$	$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$c_1$	$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$c_2$	$\left[\begin{array}{cc}1.5& 0\\ 0& 1.5\end{array}\right]$
$c_3$	$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$l_{1,1}$	0.5
$l_{1,2}$	0.3
$l_{1,3}$	0.3
$l_{2,1}$	0.5
$l_{2,2}$	0.3
$l_{2,3}$	0.3
${\alpha }_1$	1.2
${\alpha }_2$	1.2
${\beta }_1$	0.7
${\beta }_2$	0.7
${\delta }_1$	1.8
${\delta }_2$	1.8
${\omega }_1$	2
${\omega }_2$	2
${\Gamma }_1$	$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
${\Gamma }_2$	$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
${\Gamma }_3$	$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
${\Gamma }_4$	$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

.

To explore the effectiveness of proposed method, it is compared with three approaches: (a) the traditional sliding mode control using traditional double power reaching law (SMC); (b) the backstepping sliding mode control using traditional double power reaching law (BSMC), where BSMC is based on the conventional model given Equation (7); and (c) the terminal sliding mode control (TSMC). The simulation results of the NSHV in ascent phase are shown in Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14.

The tracking performance of velocity and altitude are shown in Figure 6, Figure 7, Figure 8 and Figure 9. As can be seen in Figure 6 and Figure 7, compared with SMC, BSMC, and TSMC, the proposed method can reduce the tracking error of velocity more effectively and converge in a shorter time. By contrast, there is a 7.4 m/s maximum tracking error under SMC, 3.5 m/s under BSMC, and 3.8 m/s under TSMC, respectively. The tracking errors of velocity for SMC, BSMC, and TSMC are with larger fluctuations. Furthermore, Figure 8 and Figure 9 show that the altitude tracking error of the proposed method in this paper is the smallest and the controller developed by the proposed method can track stably. In comparison, the tracking errors of altitude under SMC, BSMC, and TSMC are seen in larger fluctuations.

Figure 10, Figure 11 and Figure 12 show the responses of attack angle, pitch angle rate, and mass, respectively. From Figure 10 and Figure 11, it obvious that the responses of attack angle and pitch angle under the proposed method are smooth and within a reasonable range. The responses of mass in Figure 12 indicate that NSHV consumes a large amount of fuel in the ascent phase and the mass changes smoothly. The control input responses of the elevator deflection and the throttle setting are shown in Figure 13 and Figure 14. In Figure 13, the responses of the throttle setting change smoothly and fluctuate within the margin range. In Figure 14, it can be seen that the response of the elevator deflection under the method proposed is with smaller fluctuations than those for SMC, BSMC, and TSMC and fluctuates within the margin range.

According to the simulation results above, the following points can be summarized:

(1)

In the ascent phase of NSHV, sliding mode backstepping controller based on TPRL guarantees a better tracking performance in situations of uncertainties in comparison with SMC, BSMC, and TSMC. Firstly, the tracking errors of both velocity and altitude under the proposed method are smaller than those for SMC, BSMC, and TSMC. Secondly, the attack angle and pitch angle rate change more smoothly under the proposed method. Thirdly, the control inputs curve for throttle setting and elevator deflection are smoother under proposed method, compared with SMC, BSMC, and TSMC.

(2)

It is evident that the proposed controller can reduce the adverse influence of uncertainties more effectively in comparison with SMC, BSMC, and TSMC. BSMC is based on the model transformation given by Equation (7), which is not suited to dealing with uncertainties, while the proposed method is based on the new model transformation given by Equation (8), which can compensate the uncertainties more effectively. In the ascent phase of NSHV, uncertainties are the significant reason for tracking errors. It can be seen that the tracking errors of velocity and altitude converge more effectively under the proposed method, while those for SMC, BSMC, and TSMC are with larger fluctuations.

6. Conclusions

In this paper, a backstepping-based adaptive sliding mode controller using a novel reaching law TPRL is proposed for NSHV in the ascent phase. Firstly, a new model transformation for NSHV is proposed in the presence of uncertainties. Secondly, in order to reduce reaching time and attenuate chattering for sliding mode control, a new reaching law TPRL is proposed. Then, to ensure the tracking accuracy and robustness of system, based on TPRL, a sliding mode backstepping controller is developed. Furthermore, the adaptive laws are designed to reduce the adverse effect of uncertainties. Finally, the simulation results verify the advantages of TPRL and also demonstrate that the proposed controller provides good tracking performance for NSHV in the ascent phase and shows robustness in case of uncertainties.