2. College of Information and Control, Nanjing University of Information Science & Technology, Nanjing 210044, China
During the hypersonic reentry of near space hypersonic vehicle (NSHV),the vehicle suffers complex flight environment,the wide range of speed,the high flight altitude, and the tremendous changes of gas thermal properties and aerodynamic characteristics. Therefore,the NSHV is a complex nonlinear system with many uncertainties,and the control system design is very challenging^{[1, 2, 3, 4]}. Different nonlinear control methods have been proposed to design flight control system for the NSHV under severe uncertainties,such as sliding mode control^{[5, 6, 7, 8]},fuzzy control^{[8, 9]},and predictive control^{[10, 11]}.
Backstepping control^{[12]} is an effective design method for nonlinear uncertain systems,whether the system uncertainties satisfy the matched condition or not. It shows unique advantages in dealing with nonlinear system problems. By selecting a suitable Lyapunov function,one can construct virtual control laws step by step,such that a stable control system is obtained. Moreover,the global stability of closedloop system can be guaranteed. However, a disadvantage of the traditional adaptive backstepping design method is that the system should be parameterized^{[13, 14, 15]}.
Sliding mode control^{[16, 17]} is another effective method for nonlinear control system. The sliding mode control is designed to drive and confine the system states to stay in the prescribed sliding surface which exhibits the desired dynamics. In the sliding mode,the closedloop system becomes insensitive to the plant parameter variations and external disturbances.
Zhou et al.^{[17]} proposed a robust backstepping sliding mode control method for nonlinear systems to suppress the influence of uncertainties. However,the sliding mode controller was designed based on the upper bounds of the uncertainties. It required that the upper bounds were known in advance. In ^{[18]},by combining the backstepping control,adaptive control,and sliding mode control,a new adaptive backstepping controller was proposed. Note that in ^{[18]},the derivation of virtual control was treated as an uncertainty,which brought conservatism to design the controller. Plestan et al.^{[19]} proposed a new adaptive sliding mode control method,where the adaptive gains were adjusted in a new manner,which ensured that there was no overestimation of the gains with respect to the real a priori unknown value of uncertainties. Moreover,the proposed method guaranteed the system dynamics arrived at the sliding mode surface within a finite time.
Radial basis function neural network (RBFNN) has been proved to own the superior ability to approximate nonlinear continuous function with arbitrary precision^{[20]}. Moreover,it has less computational burden,simpler network structure,and faster convergence than a multilayer perceptron since only the connective weights between the hidden layer and the output layer of the network are adjusted during training. Therefore,the RBFNN is very useful for applications in the control of nonlinear systems^{[21, 22]}. For example,based on RBFNN,a robust control design method was proposed for strictfeedback block nonlinear systems with mismatched uncertainties^{[23]}.
Motivated by the discussion mentioned above,this paper proposes a robust control method for the attitude control of the NSHV. The proposed control method combines the advantages of the adaptive sliding mode control method,RBFNN and backstepping. Firstly,the attitude control model of the NSHV and an assumption are introduced. Then an adaptive backstepping control approach is proposed. In the attitude angle loop,an adaptive control term is designed to suppress the effect of the compound uncertainties. In the angular rate loop,an adaptive sliding mode control is integrated into the backstepping design. The adaptive gains of sliding mode control are dynamically tuned based on the constructed gain dynamics,which can ensure that there is no overestimation of the control gains. The stability of the proposed approach is analyzed using the Lyapunov method. To improve the control performance and increase the robustness of the backstepping control system,the RBFNNs are introduced to estimate the uncertainties in the attitude angle and angular rate loops,respectively. Thus the robust adaptive control approach not only owns a strong robustness against parameter variations and external disturbances,but also can efficiently save the control efforts. And the stability of the closedloop system is proved based on the Lyapunov stability theory. Finally, the simulation results show that the proposed approach attains a satisfactory performance in the presence of parameter variations and external disturbances.
Ⅱ. PROBLEM DESCRIPTIONThe considered attitude control model of the NSHV comes from the hypersonic vehicle of wingedcone configuration^{[24, 25]}. The equations with 6 degrees of freedom and 12 states can be simplified into the affine nonlinear equations as
$ \dot{x}_1=f_s+G_{s1}x_2+\Phi_1, $  (1) 
$ \dot{x}_2=f_f+G_fu+\Phi_2, $  (2) 
where $x_1 = [{\alpha ,\;\beta ,\;\mu }]^{\rm T}$ is the attitude angle vector,$x_2 = [{p ,\;q ,\;r }]^{\rm T}$ is the attitude angular rate vector,$u=g_{f\delta}\delta+M_r$ is the control input,$f_s,f_f\in {\bf R}^3$ are nonlinear vector functions, $G_{s1},G_f\in {\bf R}^{3\times3}$ are the invertible matrices. The concrete expressions of matrices above are specified in ^{[21]}. $\Phi_1$,$\Phi_2$ are compound uncertainties,which contain parameter variations and external disturbances.
Assumption 1. The uncertainties of the NSHV satisfy $\\Phi_i\\leq\varphi_i$,where $\varphi_i$ is an unknown constant,$i=1,2$.
The main purpose of this paper is to design a robust adaptive virtual control law $x_{2d}$ and a control law $u$ based on backstepping method,such that the reentry attitude of the NSHV $x_1$ can track the given target $x_{1d}$ asymptotically.
Ⅲ. THE ROBUST ADAPTIVE SLIDING MODE CONTROL DESIGNStep 1. Design the virtual control law to suppress the influence of uncertainty in attitude angle loop (1).
Define the error state vector as
$ z_1=x_1x_{1d}, $  (3) 
$ z_2=x_2x_{2d}. $  (4) 
From (1) and (3),one can obtain
$ \dot{z}_1=f_s+G_{s1}x_2+\Phi_1\dot{x}_{1d}. $  (5) 
Then a virtual control law is designed as
$ x_{2d}=G_{s1}[f_s\hat{\varphi}_1^2z_1/ (\hat{\varphi_1}\z_1\+\varepsilon_1{\rm e}^{a_1t})k_1z_1+\dot{x}_{1d}], $  (6) 
where $\varepsilon_1>0$,$a_1>0$. $\hat{\varphi}_1$ is the estimated value of $\varphi_1$ and the adaptive law is shown as
$ \dot{\hat{\varphi}}_1=\gamma_1\z_1\,\quad \gamma_1>0. $  (7) 
Consider the following Lyapunov function:
$ V_1=\frac{1}{2}z_1^{\rm T}z_1+\frac{1}{2\gamma_1}\tilde{\varphi}_1^2+\frac{\varepsilon_1}{a_1}{\rm e}^{a_1t}. $  (8) 
Differentiating $V_1$ with respect to time and using (6),one can obtain
$\begin{aligned} \dot{V_1}&=z_1^{\rm T}\dot{z}_1+\frac{1}{\gamma_1}\tilde{\varphi}_1\dot{\tilde{\varphi}}_1\varepsilon_1{\rm e}^{a_1t}=\\ &z_1^{\rm T} (f_s+G_{s1}z_2+G_{s1}z_{2d}+\phi_1\dot{x}_{1d})+\frac{1}{\gamma_1}\tilde{\varphi}_1\dot{\tilde{\varphi}}_1 \\ &\varepsilon_1{\rm e}^{a_1t}=\\ &z_1^{\rm T} (G_{s1}z_2k_1z_1+\phi_1\frac{\hat{\varphi}_1^2z_1}{\varphi_1\z_1\ +\varepsilon_1{\rm e}^{a_1t}})+\\ &\hat{\varphi_1}\z_1\\varphi_1\z_1\\varepsilon_1{\rm e}^{a_1t}\leq\\ & z_1^{\rm T} (G_{s1}z_2k_1z_1\frac{\hat{\varphi}_1^2z_1}{\varphi_1\z_1\+\varepsilon_1{\rm e}^{a_1t}})+\\ &\hat{\varphi_1}\z_1\\varepsilon_1{\rm e}^{a_1t}\leq\\ & z_1^{\rm T}G_{s1}z_2k_1z_1^{\rm T}z_1+\varepsilon_1{\rm e}^{a_1t}\varepsilon_1{\rm e}^{a_1t}=\\ &z_1^{\rm T}G_{s1}z_2k_1z_1^{\rm T}z_1. \end{aligned}$ 
If the tracking error $z_2$ converges to zero,$z_1$ is asymptotically stable under the effect of virtual control (6).
Step 2. Design the actual control variable $u$ to make the tracking error of angular velocity loop $z_2$ converge to zero.
In order to well dispel parameter variations and external disturbances,sliding mode control is employed to design the actual control law,such that the system dynamics arrives at the sliding mode surface within a finite time. At the same time,in order to avoid the gain of the sliding mode becomes too large, this paper introduces an adaptive control law to adjust the gain of the sliding control.
According to (2) and (4),the error equation is given as
$ \dot{z}_2=\dot{x}_2\dot{x}_{2d}=f_f+G_fu+\Phi_2\dot{x}_{2d}. $  (9) 
The sliding mode surface is shown as follows:
$ s=z_2+\lambda\int z_2 {\rm d}\sigma, $  (10) 
where $\lambda>0$ decides the bandwidth of the error state $s=[s_1,s_2,s_3]^{\rm T}$.
The control law $u$ can be designed as follows:
$ u=G_f^{1}[f_f+\lambda z_2+\rho {\rm sgn} (s)\dot{x}_{2d}], $  (11) 
where ${\rm sgn} (s)=[{\rm sgn} (s_1),{\rm sgn} (s_2),{\rm sgn} (s_3)]^{\rm T}$,
$ {\rm sgn} (s_i)=\left\{\begin{array} {lcr} 1,& s_i<0,\\ 0,& s_i=0,\\ 1,& s_i>0. \end{array}\right. $  (12) 
The gain of control $\rho (t)\in {\bf R}$ is selected as follows:
If $s^{\rm T}{\rm sgn} (s)\neq0$,then $\dot{\rho (t)}=\rho_1s^{\rm T}{\rm sgn} (s),\quad \rho_1>0$; 
If $s^{\rm T}{\rm sgn} (s)=0$,then $\rho (t)=\rho_2\eta^{\rm T}{\rm sgn} (\eta)+\rho_3,\quad \rho_3>0$,  (13) 
where $\rho_2=\rho (t^*)$,$\tau\dot{\eta}+\eta={\rm sgn} (s)$, $\tau>0$. $t^*$ is the arrival time,such that $s=0$ at $t=t^*$.
For the first case $s^{\rm T}{\rm sgn} (s)\neq0$ in (13),a Lyapunov function candidate is chosen as
$ V_2=\frac{1}{2}s^{\rm T}s+\frac{1}{2\gamma_2} (\rho\rho^*)^2, $  (14) 
where $\rho^*>0$ is the upper bound of $\rho$,and $\gamma_2>0$. The time derivative of (14) is shown as follows:
$ \begin{aligned} \dot{V_2}&=s^{\rm T}\dot{s}+\frac{1}{\gamma_2} (\rho\rho^*)\dot{\rho}=\\ &s^{\rm T} (\dot{z_2}+\lambda z_2)+\frac{1}{\gamma_2} (\rho\rho^*)\dot{\rho}=\\ &s^{\rm T} (f_f+G_fu+\Phi_2\dot{x}_{2d}+\lambda z_2)+\frac{1}{\gamma_2} (\rho\rho^*)\dot{\rho}. \end{aligned} $ 
Substituting the control law (11) into the above equation yields
$ \dot{V_2}=s^{\rm T} (\Phi_2\rho {\rm sgn} (s))+\frac{1}{\gamma_2} (\rho\rho^*)\rho_1s^{\rm T}{\rm sgn} (s). $  (15) 
From Assumption 1 and (15),one can obtain
$ \dot{V_2}\leq\varphi_2s^{\rm T} {\rm sgn} (s)\rho s^{\rm T}{\rm sgn} (s)+ \frac{1}{\gamma_2} (\rho\rho^*)\rho_1s^{\rm T}{\rm sgn} (s)=\notag\\ \varphi_2s^{\rm T}{\rm sgn} (s)\rho s^{\rm T}{\rm sgn} (s)+\rho^*s^{\rm T}{\rm sgn} (s)\rho^*s^{\rm T}{\rm sgn} (s)+\notag\\ \frac{1}{\gamma_2} (\rho\rho^*)\rho_1s^{\rm T}{\rm sgn} (s)=\notag\\ (\varphi_2\rho^*)s^{\rm T}{\rm sgn} (s)+ (\rho\rho^*) (\frac{\rho_1}{\gamma_2}1)s^{\rm T}{\rm sgn} (s). $  (16) 
From Lemma 1 in ^{[19]},there exists $\rho^*>0$ such that $\rho (t)\leq\rho^*$ for all $t>0$,and thus (16) can be represented as
$ \dot{V}_2= (\rho^*\varphi_2)s^{\rm T}{\rm sgn} (s) \left(\frac{\rho_1}{\gamma_2}1\right) \rho\rho^*s^{\rm T}{\rm sgn} (s)\leq\notag\\ \beta_1V_2^\frac{1}{2}, $  (17) 
where $\beta_1=\min{\{\sqrt{2} (\rho^*\!\!\varphi_2),\sqrt{2\gamma_2} ( (1/\gamma_2)\rho_1\!\!1)s^{\rm T}{\rm sgn} (s)\}}$,and $\rho^*>\varphi_2$,$\rho_1>\gamma_2$.
Formulation (17) shows that there exists a finite time $t_F\geq \frac{2V_2 (0)^{\frac{1}{2}}}{\beta_1}$ such that $s=0$ for all $t\geq t_F$. Thus,the proposed control law (11) and the adaptive gain given by (13) satisfy the sliding condition.
Next,let us consider the case of $s^{\rm T}{\rm sgn} (s)=0$. According to Theorem 1 in ^{[19]},the control law (11) with the updated law (13) can ensure that the states of system (9) stay on the sliding surface $s=0$.
Based on the above analysis,we can obtain the following theorem.
Theorem 1. Consider the closedloop nonlinear systems (1) and (2),where the virtual control law $x_{2d}$ is given by (6), and the control input $u$ is defined by (11) with the control gain of (13) and the parameter adaptive law of (7). The trajectories of the system are globally exponentially convergent, and the sliding surface $s=0$ is reachable. Meanwhile,the reentry attitude state $x_1 = [{\alpha ,\;\beta ,\;\mu }]^{\rm T} $ of the NSHV asymptotically tracks the given target $x_{1d} = [{\alpha_c ,\;\beta_c ,\;\mu_c }]^{\rm T} $.
Remark 1. Because the sign function ${\rm sgn} (\cdot)$ may bring the chattering problem,the saturation function^{[26]} ${\rm sat}(s_i/\sigma_i)$ is adopted in this paper to replace the sign function ${\rm sgn} (\cdot)$,which is
$ {\rm sat} (s_i/\sigma_i)=\left\{\begin{array} {lcr} s_i/\sigma_i,& s_i\leq \sigma_i,\\ {\rm sgn} (s_i),& s_i> \sigma_i, \end{array}\right.$ 
where $\sigma_i>0$ is a boundary layer thickness.
Ⅳ. ADAPTIVE BACKSTEPPING SLIDING MODE CONTROL DESIGN BASED ON RBFNNCompared to the traditional sliding mode controller,the sliding mode controller with the designed adaptive gain do not need the prior knowledge of the upper bounds of the compound uncertainties in the preceding section. So it can provide robustness in the presence of parameter variations and external disturbances. However,when the uncertainties are increased,the gains are required to increase for achieving robustness,and control chattering is aggravated. In fact,the control gain cannot be arbitrarily large value because of control input limitation of the NSHV. To deal with this problem,the sliding mode controller with adaptive gain is combined with neural networks to compensate for the effects of uncertainties.
RBFNN has been widely used for approximating nonlinear functions due to its structural simplicity. It comprises an input layer,a hidden layer and an output layer,which is shown in Fig. 1.
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Fig. 1. The structure of RBFNN 
Given an input $x\in {\bf R}^n$,the output of the RBFNN can be represented as
$ y=W^{\rm T}\phi (x,\xi), $  (18) 
where $W=[w_{ij}]_{N\times m}$ is weight matrix which connects the hidden layer to the output layer,$N$ and $m$ are the numbers of neurons in the hidden layer and network output dimension, respectively. $\phi=[\phi_1,\cdots,\phi_N]^{\rm T}$. The Gaussian basis function $\phi_i$ is chosen as
$ \phi (x,\xi)={\rm exp}\left[\frac{ (x\xi_i)^{\rm T} (x\xi_i)}{\eta^2}\right], $  (19) 
where $\xi=[\xi_1^{\rm T},\cdots,\xi_N^{\rm T}]^{\rm T}$ is the center vector of the Gaussian basis function,and $\eta$ is the width of the Gaussian basis function.
It should be noted that the weights and center values of the RBFNN have significant effects on approximation precision,while the influence of the width is little. In order to reduce computational burden,we choose the width value as constant in this paper. It has been proved that the RBFNN can approximate a bounded continuous function $f (x)$ with arbitrary precision on the compact set $\prod_x$ in ^{[21]}. So
$ f (x)=W^{*{\rm T}}\phi (x,\xi^*)+\delta_f (x), $  (20) 
where $\delta_f (x)$ denotes the approximation error,$W^*,\xi^*$ denote the optimal weights and center values,respectively.
Similar to the analysis of the preceding section,the design of control system is given based on backstepping method.
Step 1. Design the virtual control law to suppress the influence of the uncertainty in the attitude angle loop (1).
For the error dynamics (5) of the angular loop,we design the virtual control law as
$ z_{2d}=G_{s1}[f_su_{ns1}k_1z_1+\dot{x}_{1d}], $  (21) 
where $u_{ns1}=u_{nn1}+u_{s1}$,$u_{nn1}$ is the estimation of uncertainty $\Phi_1$,which is
$ u_{nn1}=W_1^{\rm T}\phi (z_1,\xi_1), $  (22) 
and $W_1,\xi_1$ are computed by the following adaptive laws, respectively:
$ \begin{aligned} &\dot{W}_1=\zeta_{11}\phi (z_1,\xi_1)z_1^{\rm T},\\ &\dot{\xi}_1=\zeta_{12} (z_1^{\rm T}W_1^{\rm T}\phi'_{\xi_1} (z_1, \xi_1))^{\rm T}, \end{aligned} $  (23) 
where $\zeta_{11},\zeta_{12}$ are learning rates.
Moreover,to confront the approximation error $\delta_1=\Phi_1u_{nn1}$,an adaptive robust term is adopted, i.e.,
$ u_{s1}=\frac{\hat{\delta}_1^2z_1}{\hat{\delta_1}\z_1\+c_1{\rm e}^{a_1t}}, $  (24) 
where $\hat\delta_1$ is the estimated value of $\delta_1$ and the adaptive law is shown as
$ \dot{\hat{\delta}}_1=\gamma_1\z_1\,\quad \gamma_1>0. $  (25) 
For convenience,we will rewrite $\Phi_1$ as
$ \Phi_1=W^{*{\rm T}}_1\phi (z_1,\xi_1^*)+\delta_{\Phi_1} (z_1). $  (26) 
The Taylor series expansion of $\phi (z_1,\xi_1^*)$ in (19) at $\xi_1$ can be represented as
$ \phi (z_1,\xi_1^*)=\phi (z_1,\xi_1)+\phi'_\xi\tilde{\xi}_1+o (z_1,\tilde{\xi}_1), $  (27) 
where $\phi'_{\xi_1}\in {\bf R}^{N\times nN}$ is the partial derivative of $\phi (z_1,\xi_1)$ with respect to $\xi_1$, $\tilde{\xi}_1=\xi^*_1\xi_1$,and $o (z_1,\tilde{\xi}_1)$ one higher order term. From (27),one can obtain
$ \o (z_1,\tilde{\xi}_1)\\leq\\phi (z_1,\xi_1)\+\\phi'_{\xi_1}\\\tilde{\xi}_1\. $  (28) 
According to the property of RBFNN,$\o (z_1,\tilde{\xi}_1)\$ is bounded. Then the RBFNN approximation error $\Phi_1$ is as follows:
$ \begin{aligned} \Phi_1u_{nn1}&=W_1^{*{\rm T}}\phi (z_1,\xi_1^*)+\delta_{\Phi_1} (z_1)W_1^{\rm T}\phi (z_1,\xi_1)=\\ &\tilde{W}_1^{\rm T}\phi (z_1,\xi_1)+W_1^{\rm T}\phi'_{\xi_1} (\tilde{\xi}_1)+\tilde{W}_1^{\rm T}\phi'_{\xi_1} (\tilde{\xi}_1)+\\ &W_1^{*{\rm T}}o (z_1,\tilde{\xi}_1)+\delta_{\Phi_1} (z_1), \end{aligned} $  (29) 
where $\tilde{W}_1=W_1^*W_1$. Let $E_1=\tilde{W}_1^{\rm T}\phi'_{\xi_1} (\tilde{\xi}_1)+W_1^{*{\rm T}}o (z_1, \tilde{\xi}_1)+\delta_{\Phi_1} (z_1)$. By the foregoing information,it is known that $E_1$ is bounded with $\E_1\\leq\delta_1$.
Consider the following Lyapunov function:
$ V_1=\frac{1}{2}z_1^{\rm T}z_1+\frac{1}{2\zeta_{11}}{\rm tr} (\tilde{W}_1^{\rm T}\tilde{W}_1) +\notag\\ \quad\frac{1}{2\zeta_{12}}\tilde{\xi}_1^{\rm T}\tilde{\xi}_1+\frac{1}{2\gamma_1}\tilde{\delta}_1^2 +\frac{c_1}{a_1}{\rm e}^{a_1t}, $  (30) 
where $\tilde{W}_1=W_1^*W_1$,$\tilde{\xi}_1=\xi_1^*\xi_1$, $\tilde{\delta}_1=\hat{\delta}_1\delta_1$,and $\dot{\tilde{W}}_1=\dot{W}_1$, $\dot{\tilde{\xi}}_1=\dot{\xi}_1$, $\dot{\tilde{\delta}}_1=\dot{\delta}_1$. The time derivative of (30) is shown as follows:
$ \begin{aligned} \dot{V}_1&=z_1^{\rm T}\dot{z}_1\frac{1}{\zeta_{11}}{\rm tr} (\tilde{W}_1^{\rm T}\dot{W}_1) \frac{1}{\zeta_{12}}\tilde{\xi}_1^{\rm T}\dot{\xi}_1+\\ &\frac{1}{\gamma_1}\tilde{\delta}_1\dot{\tilde\delta}_1 c_1{\rm e}^{a_1t}=\\ &z_1^{\rm T}(f_s+G_{s1}z_2+G_{s1}x_{2d}+\Phi_1\dot{x}_{1d})\frac{1}{\zeta_{12}}\tilde{\xi}_1^{\rm T}\dot{\xi}_1\\ &\frac{1}{\zeta_{11}}{\rm tr} (\tilde{W}_1^{\rm T}\dot{W}_1)+\frac{1}{\gamma_1}\tilde{\delta}_1\dot{\tilde\delta}_1 c_1{\rm e}^{a_1t}=\\ &z_1^{\rm T} (G_{s1}z_2k_1z_1+\Phi_1u_{nn1}u_{ns1})\frac{1}{\zeta_{12}}\tilde{\xi}_1^{\rm T}\dot{\xi}_1\\ &\frac{1}{\zeta_{11}}{\rm tr} (\tilde{W}_1^{\rm T}\dot{W}_1)+\hat{\delta}_1\z_1\\delta_1\z_1\ c_1{\rm e}^{a_1t}=\\ &z_1^{\rm T} (G_{s1}z_2k_1z_1+\tilde{W}_1^{\rm T}\phi (x,\xi_1)+W_1^{\rm T}\phi'_{\xi_1}\tilde{\xi}_1+E_1\\ &u_{ns1}){\rm tr} (\tilde{W}_1^{\rm T}\phi (x,\xi_1)z_1^{\rm T})\tilde{\xi}_1^{\rm T} (z_1^{\rm T}W_1^{\rm T} \phi'_{\xi_1})^{\rm T}+\\ &\hat{\delta}_1\z_1\\delta_1\z_1\ c_1{\rm e}^{a_1t}=\\ &z_1^{\rm T} (G_{s1}z_2k_1z_1+\tilde{W}_1^{\rm T}\phi (x,\xi_1)+E_1u_{ns1})\\ &{\rm tr} (\tilde{W}_1^{\rm T}\phi (x,\xi_1)z_1^{\rm T})+\hat{\delta}_1\z_1\\delta_1\z_1\ c_1{\rm e}^{a_1t}. \end{aligned} $ 
It is known that $z_1^{\rm T}\tilde{W}_1^{\rm T}\phi (x, \xi_1)={\rm tr} (\tilde{W}_1^{\rm T}\phi (x,\xi_1)z_1^{\rm T})$ and
$ \begin{aligned} & z_1^{\rm T} (E_1u_{ns1})+\hat{\delta}_1\z_1\\delta_1\z_1\ c_1{\rm e}^{a_1t}\leq\\ &~~~~\delta_1\z_1\\frac{\hat{\delta}_1^2\z_1\^2}{\hat{\delta}_1\z_1\+c_1{\rm e}^{a_1t}} +\hat{\delta}_1\z_1\\\ &~~~~\delta_1\z_1\c_1{\rm e}^{a_1t}=\\ &~~~~\frac{\hat{\delta}_1^2\z_1\^2}{\hat{\delta}_1\z_1\+c_1{\rm e}^{a_1t}} +\hat{\delta}_1\z_1\c_1{\rm e}^{a_1t}\leq0.\\ \end{aligned} $ 
So it can be referred that
$ \dot{V}_1\leq z_1^{\rm T}G_{s1}z_2z_1^{\rm T}k_1z_1. $  (31) 
Therefore,if the tracking error $z_2$ of angular velocity loop converges to zero,$z_1$ is asymptotically stable under the effect of virtual control (21). In other words,state $x_1$ tracks the target signal $x_{1d}$ asymptotically.
Step 2. Design the actual control variable $u$ to make the tracking error of angular velocity loop $z_2$ converge to zero. From (2) and (21),one can obtain the error dynamic equation
$ \dot{z}_2=\dot{x}_2\dot{x}_{2d}=f_f+G_fu+\Phi_2\dot{x}_{2d}. $  (32) 
The sliding mode surface is chosen the same as (10). Then,the control law $u$ can be designed as follows:
$ u=G_f^{1}[f_f+\lambda z_2+u_{nn2}+\rho {\rm sgn} (s)\dot{x}_{2d}], $  (33) 
where ${\rm sgn} (s)$ is given by (12),and $\rho (t)\in {\bf R}$ is calculated by (13). $u_{nn2}$ is adopted to estimate the uncertainty $\Phi_2$,and has the following expression:
$ u_{nn2}=W_2^{\rm T}\phi (z_2,\xi_2), $  (34) 
where $W_2$,$\xi_2$ are computed by the following adaptive laws:
$ \begin{aligned} &\dot{W}_2=\zeta_{21}\phi (z_2,\xi_2)s^{\rm T},\\ &\dot{\xi}_2=\zeta_{22} (s^{\rm T}W_2^{\rm T}\phi'_{\xi_2} (z_2, \xi_2))^{\rm T}, \end{aligned} $  (35) 
where $\zeta_{21}>0$,$\zeta_{22}>0$ are learning rates. $\Phi_2$ can be rewritten as
$ \Phi_2=W_2^{*{\rm T}}\phi (z_2,\xi_2^*)+\delta_{\Phi_2}. $  (36) 
And $E_2=W_2^{\rm T}\phi'_{\xi_2}\tilde{\xi}_2+W_2^{*{\rm T}}o (z_2,\tilde{\xi}_2)+\delta_{\Phi_2}$,$\E_2\\leq\delta_2$, $\delta_2>0$.
Theorem 2. Consider the closedloop nonlinear system equations (1) and (2),where the virtual control law $x_{2d}$ is given by (21) and the control input $u$ is defined by (33) with the control gain of (13) and the parameter adaptive laws of (23),(25) and (35). The trajectories of the system are globally exponentially convergent,and the sliding surface $s=0$ is reachable. Meanwhile,the reentry attitude state $x_1$ of the NSHV asymptotically tracks the given target $x_{1d}$.
Proof. The proof consists of two cases; the first case is $s^{\rm T}{\rm sgn} (s)\neq0$ and the second is $s^{\rm T}{\rm sgn} (s)=0$.
First,let us consider the first case ($s^{\rm T}{\rm sgn} (s)\neq0$). The following Lyapunov function candidate is considered:
$ V_2=\frac{1}{2}s^{\rm T}s+\frac{1}{2\gamma_2} (\rho\rho^*)^2+\frac{1}{2\zeta_{21}}{\rm tr} (\tilde{W}_2^{\rm T}\tilde{W}_2) +\frac{1}{2\zeta_{22}}\tilde{\xi}_2^{\rm T}\tilde{\xi}_2, $  (37) 
where $\rho^*>0$ is the upper bound of $\rho$,and $\gamma_2>0$, $\tilde{W}_2=W_2^*W_2$,$\tilde{\xi}_2=\xi_2^*\xi_2$, $\dot{\tilde{W}}_2=\dot W_2$,$\dot{\tilde{\xi}}_2=\dot{\xi}_2$.
Differentiating (37) with respect to time yields
$ \begin{aligned} \dot{V}_2&=s^{\rm T}\dot{s}+ \frac{1}{\gamma_2} (\rho\rho^*)\dot{\rho} \frac{1}{\zeta_{21}}{\rm tr} (\tilde{W}_2^{\rm T}\dot{W}_2) \frac{1}{2\zeta_{22}}\tilde{\xi}_2^{\rm T}\dot{\xi}_2=\\ &s^{\rm T} (\dot{z}_2+\lambda z_2)+\frac{1}{\gamma_2} (\rho\rho^*)\dot{\rho} {\rm tr} (\tilde{W}_2^{\rm T}\phi (z_2,\xi_2)s^{\rm T})\\ &\tilde{\xi}_2^{\rm T} (s^{\rm T}W_2^{\rm T}\phi'_{\xi_2})^{\rm T}=\\ &s^{\rm T} (f_f+G_fu+\Phi_2\dot{x}_{2d}+\lambda z_2)+\frac{1}{\gamma_2} (\rho\rho^*)\dot{\rho}\\ &{\rm tr} (\tilde{W}_2^{\rm T}\phi (z_2,\xi_2)s^{\rm T}) \tilde{\xi}_2^{\rm T} (s^{\rm T}W_2^{\rm T}\phi'_{\xi_2})^{\rm T}. \end{aligned} $  (38) 
Substitute control law (33) into (38),then
$ \begin{aligned} \dot{V}_2&=s^{\rm T} (\Phi_2u_{nn2}\rho {\rm sgn} (s)) +\frac{1}{\gamma_2} (\rho\rho^*)\dot{\rho}\\ &{\rm tr} (\tilde{W}_2^{\rm T}\phi (z_2,\xi_2)s^{\rm T}) \tilde{\xi}_2^{\rm T} (s^{\rm T}W_2^{\rm T}\phi'_{\xi_2})^{\rm T}=\\ &s^{\rm T} (\tilde{W}_2^{\rm T}\phi (z_2,\xi_2)+E_2\rho {\rm sgn} (s)) +\frac{1}{\gamma_2} (\rho\rho^*)\dot{\rho}\\ &{\rm tr} (\tilde{W}_2^{\rm T}\phi (z_2,\xi_2)s^{\rm T}). \end{aligned} $ 
Because $s^{\rm T}\tilde{W}_2^{\rm T}\phi (z_2,\xi_2)={\rm tr} (\tilde{W}_2^{\rm T}\phi (z_2,\xi_2)s^{\rm T})$ and $s^{\rm T}E_2$ $\leq \delta_2 \s\ \leq \delta_2s^{\rm T}{\rm sgn} (s)$, the following result is obtained
$ \begin{aligned} \dot{V}_2&=s^{\rm T} (E_2\rho {\rm sgn} (s)) +\frac{1}{\gamma_2} (\rho\rho^*)\dot{\rho}\leq\\ & \delta_2 s^{\rm T} {\rm sgn} (s)s^{\rm T}\rho {\rm sgn} (s) +\frac{1}{\gamma_2} (\rho\rho^*)\dot{\rho}=\\ & (\delta_2\rho^*) s^{\rm T} {\rm sgn} (s) + (\rho\rho^*) (\frac{\rho_1}{\gamma_2}1)s^{\rm T} {\rm sgn} (s). \end{aligned} $ 
For all $t>0$,there exists $\rho^*>0$ such that $\rho (t)\leq\rho^*$. Let $\beta_1={\rm min}{\{\sqrt{2} (\rho^*\delta_2),\sqrt{2\gamma_2} (\frac{\rho_1}{\gamma_2}1)s^{\rm T}{\rm sgn} (s)\}}$,and $\rho^*>\delta_2$,$\rho_1>\gamma_2$. So
$ V_2\leq \beta_1 (s^{\rm T}s+ (\rho\rho^*)^2) \leq \beta_1 V_2^{ \frac{1}{2}} \leq 0. $  (39) 
Inequality (39) shows that there exists a finite time $t_F\leq \frac{2V_2 (0)^{1/2}}{\beta_1}$ such that $s=0$ for all $t>t_F$. Thus,the proposed control law (33) and the adaptive gain given by (13) satisfy the sliding condition.
Next,let us consider the case of $s^{\rm T}{\rm sgn} (s)=0$. According to Theorem 1 in ^{[19]},the control law (33) with the adaptive laws (13) and (35) can guarantee the states of system (9) stay on the sliding surface $s=0$ .
In summary,it can be concluded that the trajectories of the system are globally exponentially convergent,and the sliding surface $s=0$ is reachable. Meanwhile,the reentry attitude state $x_1$ of the NSHV can well track the given command $x_{1d}$.
Remark 2. It is hard to calculate $\dot{x}_{2d}=[\dot{x}_{2d1},\dot{x}_{2d2}$,$\dot{x}_{2d3}]^{\rm T}$. So the robust differentiator^{[27]} is introduced to handle this problem,which is
$ \left\{ \begin{aligned} &\dot{z}_{0i}=\lambda_{0i}z_{0i}x_{2di}^{\frac{1}{2}} {\rm sign} (z_{0i}x_{2di})+z_{1i},\\ &\dot{z}_{1i}=\lambda_{1i} {\rm sign} (z_{1i}\alpha_i),\quad i=1,2,3, \end{aligned} \right. $  (40) 
where $\lambda_{1i}>0$,$\lambda_{0i}>0$. Levant^{[27]} proved that the robust differentiator can estimate the firstorder derivative of the input signal exactly if there is no noise in the input. That is to say,$z_{0i}=\alpha_i$,$z_{1i}=\dot{\alpha}_i$ are obtained within a finite time. When there is some noise in the input,the estimate error can be offset by the robustness terms.
Ⅴ. SIMULATIONSThe initial conditions of simulation are $V_0=3.1$km/s, $H_0=34$km,$\alpha_0=1^\circ$,$\beta_0=0.2^\circ$, $\mu_0=1^\circ$,$p_0=q_0=r_0=0^\circ/{\rm s}$. The command signals are $\alpha_c=1^\circ$,$\beta_c=0^\circ$, $\mu_c=1^\circ$ given by the command filter $2/ (s+2)$. Suppose there exist uncertainties in the aerodynamic parameter. Besides, the disturbance torque in angular velocity loop is defined as $10^5\times [{\rm sin} (t),{\rm cos}(2t),\sin (3t)]^{\rm T}\,{\rm N\cdot m}$.
The parameters of the control system are given by $k_1={\rm diag}\{0.8,0.8,0.8\}$,$\lambda={\rm diag}\{1.2,1.2,1.2\}$, $a_1=2$,$\varepsilon_1=0.5$,$\rho_1=2$,$\rho_2=0.5$, $\rho_3=0.2$,$\tau=0.1$,$\sigma_i=0.02$,$\zeta_{11}=2$, $\zeta_{12}=1$,$\zeta_{21}=1$,$\zeta_{22}=0.5$. The number of hidden layer neurons in the RBFNN is 4.
For comparative simulation,the traditional backstepping method is also employed to design the attitude control system for the NSHV as follows:
$ x_{2d}=G_{s1}^{1}[f_sk_1z_1+\dot{x}_{1d}], $  (41) 
$ u=G_f^{1}[f_f+k_2z_2+G_{s1}^{\rm T}\dot{x}_{2d}]. $  (42) 
The parameters of the control system (41)$\sim$(42) are $k_1={\rm diag}\{5,5,5\}$,$k_2={\rm diag}\{10,10,10\}$. The simulation results are shown in Figs.2 and 3,where subscript $c$ means command signal,subscripts $1,2$ express the simulation result of the traditional backstepping control method and the method proposed in this paper,respectively. Fig.2 shows the changing curve of attitude angle and angular rate. It is observed that the performance of the proposed control method is much better than that of the traditional backstepping control method. Fig.3 shows that large rudder surface deflections occur when using the traditional backstepping control method,it is because a high control gain is necessary to suppress parameter variations and external disturbances. In contrast,the proposed method avoids this problem.
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Fig. 2. Comparison of attitude control performance. 
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Fig. 3. Comparison of control surface deflections 
In this paper,a strongly robust adaptive sliding mode control system is proposed based on backstepping design,the adaptive control technique and the RBFNN for the attitude tracking problem of the NSHV in the presence of parameter variations and external disturbances. Firstly,to relax the requirement for the upper bound of the compound uncertainty in control design,an adaptive control technique is employed. Based on backstepping design,in attitude angular loop,the unknown upper bound is estimated by using an adaptive term,and then a virtual control law is derived. In angular rate loop,sliding mode control with the adaptive gain is considered to design the real control law. Moreover,the constructed gain dynamics can ensure that there is no overestimation of the control gain. Secondly,to further improve the control performance,the RBFNN is employed to estimate the compound uncertainties directly. Moreover,an adaptive control term designed in the first step are used to deal with the approximated errors. Thus,based on the similar design in the first step,the virtual control law in the attitude angular loop and the real control law in the angular rate loop are obtained, respectively. Furthermore,the closedloop stability is proved using Lyapunov theorem. Finally,simulation results show good performance of the proposed control approach for the NSHV attitude control. Moreover,the robustness to parameter variations and external disturbances are successfully accomplished.
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