Tracking performance and global stability guaranteed neural control of uncertain hypersonic flight vehicle

Abstract

In this article, a global adaptive neural dynamic surface control with predefined tracking performance is developed for a class of hypersonic flight vehicles, whose accurate dynamics is hard to obtain. The control scheme developed in this paper overcomes the limitations of neural approximation region by employing a switching mechanism which incorporates an additional robust controller outside the neural approximation region to pull the transient state variables back when they overstep the neural approximation region, such that globally uniformly ultimately bounded stability can be guaranteed. Especially, the developed global adaptive neural control also improves the tracking performance by introducing an error transformation mechanism, such that both transient and steady-state performance can be shaped according to the predefined bounds. Simulation studies on the hypersonic flight vehicle validate that the designed controller has good velocity modulation and velocity stability performance.

Introduction

Hypersonic flight vehicles (HFVs) with the unique superiority of its own (such as high speed, quick response and strong penetration power) meet the increasing commercial and military demands for space access very well. HFVs are the focus of recent research in various space powers, but the flight aerodynamic environment is complex, and its model uncertainty increases because of its high speed, which makes the design of guidance and control system difficult.

In the study by Xu and Shi, ¹ the comprehensive review of hypersonic flight dynamics and control approaches is presented. Until now, varieties of algorithms ^{2 –6} have obtained great effects on HFVs. The minimal learning design for adaptive control of hypersonic flight is given in the study by Xu et al., ⁷ whereas in the study by Xu et al., ⁸ the command filter-based adaptive fault tolerant control is analyzed in the case of actuator fault. Furthermore, the disturbance observer-based design can be used to deal with the time-varying disturbances. ^9,10 Since the work condition faced in HFVs is extremely complex, which is always unknown, dynamic and nonlinear, little knowledge about HFVs dynamics parameters is available in real applications. And then, with the emergence of intelligent science, ^{11 –19} there is a new pathway to research on HFVs control.

In terms of learning capacity, neural network (NN) is an excellent universal approximator. More significantly, the universal approximation property of NN is based on the Stone–Weierstrass theorem, which states that on a compact set, a universal approximator can approximate any real continuous function ²⁰ with arbitrary order of accuracy. NN that is used as a universal approximator in the control problem is investigated in the literature. ^{21 –23} Backstepping technique, ^24,25 a highly effective design method for a systematic nonlinear control synthesis, combines with NN control that has been developed in the study by Ge et al. ²⁶

As a matter of fact, only when in a compact set, the NN approximation ability is valid. Consequently, the conventional NN-based control could only ensure the semi-globally uniformly ultimately bounded (GUUB) stability. For enhancing the robustness of HFVs, it is meaningful to develop a global neural controller. For the sake of extending the application range of a neural controller, a robust controller for robot manipulator is developed. ³⁹ In the study by Xu et al., ²⁷ a global neural controller, designed by connecting an additional robust controller outside the neural approximation region to pull the transient states back into the neural approximation region from the outside, was proposed for HFV with unknown dynamics.

From a security point of view, the aforementioned adaptive neural control couldn’t guarantee the transient performance. Because of the learning characteristic of NN, an unexpected transient performance might occur in the early phases of learning process, especially when large initial parameter estimation errors exist. The transient performance of the controller affects the performance of the vehicle system and the safety of flight. In this way, the demand for transient performance of a vehicle system is particularly prominent. The idea of system with prescribed performance ^{16,28 –31} provides a good way to solve this kind of problem. Strikingly, in the study by Na et al., ²⁹ a priori prescribed performance is achieved by introducing a performance function, which is used for the output error transformation. While in the study by Dai et al., ³⁰ an adaptive neural controller achieves predefined transient tracking performance by employing an error transformation mechanism. In the study by Chen et al., ³¹ backstepping technique is embedded into the design of guaranteed transient performance-based attitude control with control input saturation.

Inspired by the aforementioned work, in this design, an NN employed dynamic surface control (DSC) technique ^12,32,33 is devised for HFVs with guaranteed tracking performance and global stability.

This overall structure is as follows: dynamics of HFVs is characterized in ‘Problem formulation’ section. ‘Preliminaries’ section describes the brief notion of the radial basis function (RBF) NNs, furthermore, useful functions and key lemmas are presented as well. The global neural controller is addressed for HFV in ‘Tracking performance guaranteed neural control design’ section. In ‘Simulation studies’ section, the simulation results validate that the design we suggest is credible. Finally, ‘Conclusion’ section sums up the previous design.

Problem formulation

Hypersonic vehicle dynamics

According to the study by Parker et al., ³⁴ the model of HFV is constructed based on five state variables $X_h={\left[V,h,\alpha ,\gamma ,q\right]}^T$ and two control inputs $U_h={\left[{\delta }_e,\Phi \right]}^T$ , where V represents the velocity, h represents the altitude, α represents the attack angle, γ represents the flight path angle (FPA) and q represents the pitch rate, while the control inputs δ_e and Φ represent the elevator deflection and the fuel equivalence ratio, respectively. The motion of the HFV dynamics takes the following form

$\begin{array}{c}\dot{V}=\frac{T\cos \alpha -D}{m}-\frac{\mu \text{sin}\gamma }{r^2}\\ \dot{h}=V\text{sin}\gamma \\ \dot{\gamma }=\frac{L+T\text{sin}\alpha }{mV}-\frac{\left(\mu -V^{2}r\right)\cos \gamma }{V{r}^2}\\ \dot{\alpha }=q-\dot{\gamma }\\ \dot{q}=\frac{M_{yy}}{I_{yy}}\end{array}$

1

The detailed explanation of the parameters values is summarized in Table 1.

Table 1.

HFV model description.

m	Mass
ρ	Density of air
RE	Radius of the Earth
S	Reference area
r	$h+R_E$
$\overline{q}$	$1/2\rho V^2$
T	$\overline{q}S{C}_T$
D	$\overline{q}S{C}_D$
Iyy	Moment of inertia
Myy	$\overline{q}S\overline{c}\left[C_M(\alpha )+C_M({\delta }_e)+C_M(q)\right]$
CL	0.6203α
CD	$0.6450{\alpha }^2+0.0043378\alpha +0.003772$
$C_M({\delta }_e)$	$0.0292({\delta }_e-\alpha )$
$C_M(\alpha )$	$-0.035{\alpha }^2+0.036617\alpha +5.3261\times {10}^{-6}$
$C_M(q)$	$(q\overline{c}/2V)\times (-6.796{\alpha }^2+0.3015\alpha -0.2289)$
CT	0.02576Φ	if Φ < 1
	$0.0224+0.00336\Phi$	otherwise

HFV: hypersonic flight vehicle.

System transformation

The entire model of HFV consists of two parts: velocity subsystem and attitude subsystem. The form of velocity subsystem takes as follows

$\begin{array}{c}\dot{V}=f_v+h_v{u}_v\\ u_v=\Phi \\ y_v=V\end{array}$

2

For the altitude subsystem, the FPA command is selected as follows

${\gamma }_d=\arcsin \left[\frac{-k_h\left(h-h_r\right)-k_i{\displaystyle \int }\left(h-h_r\right)dt+{\dot{h}}_r}{V}\right]$

3

where h_r represents the reference signal, $k_h>0\text{ and }{k}_i>0$ . If the FPA is regulated to track γ_d, the altitude tracking error $\tilde{h}=h-h_r$ converges to zero exponentially. ¹²

Attitude subsystem of HFV is a single-input-single-output system, which can be rewritten as the following form

$\begin{array}{c}{\dot{x}}_1=f_1\left({\overline{x}}_1\right)+h_1{x}_2\\ {\dot{x}}_2=f_2\left({\overline{x}}_2\right)+h_2{x}_3\\ {\dot{x}}_3=f_3\left({\overline{x}}_3\right)+h_{3}u\\ y=x_1\end{array}$

4

where $x_1=\gamma ,x_2=\alpha +\gamma ,x_3=q,{\overline{x}}_1=x_1,{\overline{x}}_2=[x_1,x_2{]}^T\text{ and }{\overline{x}}_3=[x_1,x_2,x_3{]}^T$ . The detailed definition for nonlinear functions is summarized in Table 2.

Table 2.

Definition for nonlinear functions.

h 1	$\overline{q}S{C}_D$
h 2	1
h 3	$0.0292\overline{q}S\overline{c}/I_{yy}$
f 1	$-(\mu -V^{2}r)\cos \gamma /(V{r}^2)-0.6203\overline{q}S\gamma /(mV)$
f 2	0
f 3	$\overline{q}S\overline{c}\left[C_M(\alpha )+C_M(q)-0.0292\alpha \right]/I_{yy}$
fv, Φ > 1	$-(D/m+\mu \text{sin}\gamma /r^2)+0.0224\overline{q}S\cos \alpha /m$
fv, otherwise	$-(D/m+\mu \text{sin}\gamma /r^2)$
hv, Φ > 1	$0.00336\overline{q}S\cos \alpha /m$
hv, otherwise	$0.02576\overline{q}S\cos \alpha /m$

Preliminaries

RBF NNs

In the following controller design process, the RBF NNs ³⁵ are utilized to estimate the unknown dynamics

$\widehat{f}\left(X_{\text{in}},\widehat{W}\right)={\widehat{W}}^{T}S\left(X_{\text{in}}\right)$

5

where $X_{\text{in}}\in {\Omega }_{X_{\text{in}}}\subset R^M$ represents the input vector, the NN’s output $\widehat{f}\in R$ is the estimation of f, $\widehat{W}=[{\widehat{\omega }}_1\mathrm{,...},{\widehat{\omega }}_N{]}^T\in R^N$ is called the adjustable parameter vector, while $S(X_{\text{in}})={\left[s_1(X_{\text{in}})\mathrm{,...},s_N(X_{\text{in}})\right]}^T$ is called the activation function and N represents the number of hidden layer nodes of NN. The component element of $S(X_{\text{in}})$ commonly takes the following form

$s_{\text{i}}\left(X_{\text{in}}\right)=\text{exp}\left[-\frac{{\left(X_{\text{in}}-{\varsigma }_i\right)}^T\left(X_{\text{in}}-{\varsigma }_i\right)}{{\sigma }^2}\right],i=1,\mathrm{...},N$

6

where ${\varsigma }_i=[{\varsigma }_{i1},{\varsigma }_{i2}\mathrm{,...},{\varsigma }_{iM}{]}^T\in R^M$ represents the center of $s_i,$ and σ represents the variance.

The form of RBF NNs estimating the unknown function f on a compact set ${\Omega }_{X_{\text{in}}}\subset R^M$ can be described as follows:

$f\left(X_{\text{in}}\right)=\widehat{f}\left(X_{\text{in}},W^{*}\right)+\epsilon \left(X_{\text{in}}\right),\forall X_{\text{in}}\in {\Omega }_{X_{\text{in}}}$

7

where W* optimal weight vector, and $\epsilon \left(X_{\text{in}}\right)$ is estimation error. Based on the study by Ge et al., ³⁸ W* is given as follows

$W^{*}=\text{arg\hspace{0.17em}}\underset{(W)}{\text{min}}\left[\underset{X_{\text{in}}\in {\Omega }_{X_{\text{in}}}}{\text{sup}}|f(X_{\text{in}})-\widehat{f}(X_{\text{in}},\widehat{W})|\right]$

8

The estimation error is uniformly bounded by

$\left|\epsilon \left(X_{\text{in}}\right)\right|\le {\epsilon }^{*},\forall X_{\text{in}}\in {\Omega }_{X_{\text{in}}}$

9

Predefined transient tracking performance

The predefined performance is achieved by bounding the tracking error to a predefined region, which is determined by sustained decreasing dynamic functions with the time. The expression of predefined tracking performance is as follows

$-{\xi }_{1,i}{\beta }_i\left(t\right)<{\tilde{x}}_i\left(t\right)<{\xi }_{2,i}{\beta }_i\left(t\right)$

10

where ${\xi }_{1,i}\text{ and }{\xi }_{2,i}$ are the positive design constants, and ${\beta }_i\left(t\right)$ is an exponential decay function with ${\lim }_{t\to \infty }{\beta }_i\left(t\right)={\beta }_{i\infty }$ , regarded as a performance function. ³⁶ In this design, ${\beta }_i\left(t\right)$ is selected as follows (Figure 1)

${\beta }_i\left(t\right)=\left({\beta }_{i0}-{\beta }_{i\infty }\right)\text{exp}\left(-{\vartheta }_{i}t\right)+{\beta }_{i\infty }$

11

where ${\beta }_{i0},{\beta }_{i\infty }$ and ϑ_i are the positive constants.

Figure 1.

Tracking error and performance function.

Useful functions and key lemmas

Assumption 1

• The reference signals and their derivatives are all smooth bounded functions.

Assumption 2

• There exists positive constants ${\overline{h}}_{\text{i}}\text{ and }{\underset{\_}{h}}_i,\text{ satisfying }\left|{\underset{\_}{h}}_i\right|\le h_i\le \left|{\overline{h}}_i\right|,i=1,3$ .

Lemma 1

Introduce an extremely important inequality ³⁷ for our design, which holds for any $\upsilon >0\text{ and }\hslash \in R$

$0\le \left|\hslash \right|-\hslash \text{tanh}\left(\frac{\hslash }{\rho }\right)\le \kappa \upsilon$

12

where $\kappa =e^{-(\kappa +1)},thatis,\kappa =0.2785$ .

Define 1

For the given parameter θ_i within the subset ${\Omega }_{\theta }$ , the smooth parameter projection ²⁷ $\text{Proj}({\upsilon }_i)$ is as follows

$\text{Proj}\left({\upsilon }_i\right)=\{\begin{array}{ll}{\upsilon }_i\hfill & \text{if\hspace{0.05em}\hspace{1em}}\left({\widehat{\theta }}_i\in {\Omega }_{\theta }\right)\hfill \\ \hfill & \text{or\hspace{0.05em}\hspace{1em}}\left({\widehat{\theta }}_i={\theta }_i^L,{\upsilon }_i\ge 0\right)\hfill \\ \hfill & \text{or\hspace{0.05em}\hspace{1em}}\left({\widehat{\theta }}_i={\theta }_i^U,{\upsilon }_i\le 0\right)\hfill \\ \hfill & \hfill \\ 0\hfill & \text{if\hspace{0.05em}\hspace{1em}otherwise}\hfill \end{array}$

13

where ${\widehat{\theta }}_i$ is the estimation of θ_i, and ${\theta }_i^U\text{ and }{\theta }_i^L$ are the upper and lower bounds of Ω_θ, respectively.

Definition 2

Defining a compact set Ω_r and the boundaries of which are given as ${0<}^1{r}_i{<}^2{r}_i,i=1,3$ . The set of switching surface functions ²⁷ is shown as follows

$b_k\left(x_k\right)\triangleq \{\begin{array}{ll}1\hfill & \text{if\hspace{0.17em}\hspace{0.05em}}\left|x_k\right|{<}^1{r}_k\hfill \\ \frac{{}^2{r}_k^2-x_k^2}{{}^2{r}_k^2{-}^1{r}_k^2}{e}^{{\left(\frac{x_k^2{-}^1{r}_k^2}{\varpi \left({}^2{r}_k^2{-}^1{r}_k^2\right)}\right)}^{2b}}\hfill & \text{if\hspace{0.05em}}{}^1{r}_k\le \left|x_k\right|{\le }^2{r}_k\hfill \\ 0\hfill & \text{if\hspace{0.17em}\hspace{0.05em}}\left|x_k\right|{>}^2{r}_k\hfill \end{array}$

14

$B_i\left({\overline{x}}_i\right)\triangleq {\displaystyle \prod _{k=1}^i}{b}_k\left(x_k\right)$

15

where $\varpi >0\text{ and }b\ge 1$ .

Tracking performance guaranteed global adaptive neural control

This section investigates global adaptive neural control for HFV altitude subsystem to guarantee the predefined transient and GUUB stability. For the velocity subsystem, a proportion integration differentiation (PID) controller is developed to guarantee the stability. The control block is shown in Figure 2.

Figure 2.

Control strategy.

Controller design

To achieve the tracking control system satisfying the constrained performance, a suitable output error transform function ³⁰ is employed to solve the constrained problem. For this purpose, define a smooth and strictly increasing function ${\Psi }_i(z_i)$ with $z_i\in R,i=1,2,3$ satisfying

$\{\begin{array}{l}-{\xi }_{1,i}<{\Psi }_i\left(z_i\right)<{\xi }_{2,i},\forall z_i\in R_{\infty }\\ \underset{z_i\to +\infty }{\lim }{\Psi }_i\left(z_i\right)={\xi }_{2,i}\\ \underset{z_i\to -\infty }{\lim }{\Psi }_i\left(z_i\right)=-{\xi }_{1,i}\end{array}$

16

Using the properties of ${\Psi }_i\left(z_i\right)$ , the constrained tracking performance (10) can be rewritten by

${\tilde{x}}_i\left(t\right)={\beta }_i\left(t\right){\Psi }_i\left(z_i\right),i=1,2,3$

17

where $z_i\in R$ is defined as a transformed error

$z_i={\Psi }_i^{-1}\left(\frac{{\tilde{x}}_i\left(t\right)}{{\beta }_i\left(t\right)}\right),i=1,2,3$

18

In this article, we choose the smooth and strictly increasing transformed function ${\Psi }_i(z_i)$ as follows

${\Psi }_i\left(z_i\right)=\frac{{\xi }_{2,i}{e}_i^{(z_i+{\overline{\xi }}_i)}-{\xi }_{1,i}{e}_i^{-(z_i+{\overline{\xi }}_i)}}{e_i^{(z_i+{\overline{\xi }}_i)}+e_i^{-(z_i+{\overline{\xi }}_i)}}$

19

where ${\overline{\xi }}_i=\left(\text{ln}{\xi }_{1,i}-\text{ln}{\xi }_{2,i}\right)/2$ . Then, we have

$z_i={\Psi }_i^{-1}\left(\frac{{\tilde{x}}_i\left(t\right)}{{\beta }_i\left(t\right)}\right)=\frac{1}{2}\text{ln}\left(\frac{{\xi }_{1,i}+{\tilde{x}}_i(t)/{\beta }_i(t)}{{\xi }_{2,i}-{\tilde{x}}_i(t)/{\beta }_i(t)}\right)$

20

Differentiating equation (18) with respect to time yields

${\dot{z}}_i={\Upsilon }_i\left({\dot{\tilde{x}}}_i-{\Theta }_i{\tilde{x}}_i\right),i=1,2,3$

21

where ${\Upsilon }_i=\frac{\partial {\Psi }_i^{-1}}{\partial ({\tilde{x}}_i/{\beta }_i\left(t\right))}\frac{1}{{\beta }_i\left(t\right)}\text{ and }{\Theta }_i=\frac{{\dot{\beta }}_i\left(t\right)}{{\beta }_i\left(t\right)}$ . From equations (20) and (21), the following equation can be deducted

${\Upsilon }_i=\frac{1}{2{\beta }_i}\left[\frac{1}{{\xi }_{1,i}+\frac{{\tilde{x}}_i\left(t\right)}{{\beta }_i\left(t\right)}}+\frac{1}{{\xi }_{2,i}-\frac{{\tilde{x}}_i\left(t\right)}{{\beta }_i\left(t\right)}}\right]$

22

According to equation (10), it can be inferred that ${\Upsilon }_i>0$ .

When the initial errors ${\tilde{x}}_i\left(t\right),i=1,2,3$ satisfies the prescribed performance bound, the predefined tracking performance is guaranteed.

Step 1: Define the FPA error ${\tilde{x}}_1=x_1-x_{1d},\text{ where }{x}_{1d}={\gamma }_d$ . According to equation (4), the derivative of ${\tilde{x}}_1$ is as follows

$\begin{array}{c}{\dot{\tilde{x}}}_1={\dot{x}}_1-{\dot{x}}_{1d}\\ =h_1{x}_2+f_1-{\dot{x}}_{1d}\end{array}$

23

where ${\dot{x}}_{1d}={\dot{\gamma }}_d$ . According to equation (21), we have

${\dot{z}}_1={\Upsilon }_1\left({\dot{\tilde{x}}}_1-{\Theta }_1{\tilde{x}}_1\right)$

24

On the compact set ${\Omega }_{{\overline{x}}_1}$ , NN is utilized to approximate the unknown dynamic f₁ with the following form

$f_1\left({\overline{x}}_1\right)=W_{f_1}^{*T}{S}_{f_1}\left({\overline{x}}_1\right)+{\epsilon }_1$

25

where $W_{f_1}^{*}$ is the optimal weight vector, $S_{f_1}$ is the activation function vector and ε₁ is the estimation error, satisfying $\parallel {\epsilon }_1\parallel <{\epsilon }_1^{*}$ .

The virtual controller $x_{2c}$ of equation (23) can be designed as follows

$\begin{array}{l}{\widehat{h}}_1{x}_{2c}=-k_1{z}_1-{\overline{h}}_1\left|{\tilde{x}}_2\right|\text{sign}\left(z_1\right)+{\dot{x}}_{1d}+{\Theta }_1{\tilde{x}}_1-B_1\left(x_1\right)u_1^N\\ -\left(1-B_1\left(x_1\right)\right)u_1^R\end{array}$

26

where $k_1>0,{\widehat{h}}_1={\widehat{W}}_{h_1}{S}_{h_1}$ on the compact set ${\Omega }_{{\overline{x}}_1}$ , and

$u_1^N={\widehat{f}}_1\left({\overline{x}}_1\right)={\widehat{W}}_{f_1}^T{S}_{f_1}\left({\overline{x}}_2\right)$

27

$u_1^R=f_1^U\left({\overline{x}}_1\right)\text{tanh}\left(\frac{z_1{f}_1^U\left({\overline{x}}_1\right)}{{\upsilon }_1}\right)$

28

where $f_1^U$ is the upper bound of f₁, and υ₁ is the positive parameter.

The adaptation laws of the adjustable parameters ${\widehat{W}}_{f_1}\text{ and }{\widehat{W}}_{h_1}$ are described by

${\dot{\widehat{W}}}_{f_1}={\Gamma }_{f_1}\left(S_{f_1}{z}_1{B}_1\left({\overline{x}}_1\right)-{\delta }_{f_1}{\widehat{W}}_{f_1}\right)$

29

${\dot{\widehat{W}}}_{h_1}=\text{Proj}\left({\Gamma }_{h_1}\left(S_{h_1}{z}_1{x}_{2c}-{\delta }_{h_1}{\widehat{W}}_{h_1}\right)\right)$

30

Introducing a first-order filter and a new state variable $x_{2d}$ , and the filtered signal $x_{2d}$ is obtained by $x_{2c}$ with the following form

${\alpha }_2{\dot{x}}_{2d}+x_{2d}=x_{2c},x_{2d}\left(0\right)=x_{2c}\left(0\right)$

31

where ${\alpha }_2>0$ is the time constant.

Define ${\tilde{x}}_2=x_2-x_{2d}$ , and an error surface e₂ is as follows

$e_2=x_{2d}-x_{2c}$

32

Then, equation (24) can be further derived as follows

${\dot{z}}_1={\Upsilon }_1\left({\dot{\tilde{x}}}_1-{\Theta }_1{\tilde{x}}_1\right)=\Upsilon \mathrm{\_1}\left(h_1{x}_2+f_1-{\dot{x}}_{1d}-{\Theta }_1{\tilde{x}}_1\right)=\Upsilon \mathrm{\_1}\left(h_1\left(x_2-x_{2c}\right)+{\tilde{h}}_1{x}_{2c}+{\widehat{h}}_1{x}_{2c}+f_1-{\dot{x}}_{1d}-{\Theta }_1{\tilde{x}}_1\right)={\Upsilon }_1(h_1{\tilde{x}}_2+h_1{e}_2+{\tilde{h}}_1{x}_{2c}-k_1{z}_1-{\overline{h}}_1|{\tilde{x}}_2|\text{sign}\left(z_1\right)+B_1\left(x_1\right)\left({\tilde{f}}_1+{\epsilon }_1\right)+(1-B_1\left(x_1\right))\left(f_1-u_1^R\right))$

33

Step 2: Define the dynamics of the pitch angle tracking error ${\tilde{x}}_2=x_2-x_{2d}$ . According to equation (4), the derivative of ${\tilde{x}}_2$ is as follows

$\begin{array}{c}{\dot{\tilde{x}}}_2={\dot{x}}_2-{\dot{x}}_{2d}\\ =x_3-{\dot{x}}_{2d}\end{array}$

34

According to equation (21), we have

${\dot{z}}_2={\Upsilon }_2\left({\dot{\tilde{x}}}_2-{\Theta }_2{\tilde{x}}_2\right)$

35

The virtual controller $x_{3c}$ of equation (34) can be designed as follows

$x_{3c}=-k_2{z}_2-\left|{\tilde{x}}_3\right|\text{sign}\left(z_2\right)+{\dot{x}}_{2d}+{\Theta }_2{\tilde{x}}_2$

36

where $k_2>0$ .

Similar to equation (31), the filtered signal $x_{3d}$ is obtained by $x_{3c}$ with the following form

${\alpha }_3{\dot{x}}_{3d}+x_{3d}=x_{3c},x_{3d}\left(0\right)=x_{3c}\left(0\right)$

37

where ${\alpha }_3>0$ is the time constant.

Define ${\tilde{x}}_3=x_3-x_{3d}$ , and an error surface e₃ is as follows

$e_3=x_{3d}-x_{3c}$

38

Then, equation (35) can be further derived as follows

${\dot{z}}_2={\Upsilon }_2\left({\dot{\tilde{x}}}_2-{\Theta }_2{\tilde{x}}_2\right)={\Upsilon }_2\left(x_3-x_{3d}+x_{3d}-x_{3c}+x_{3c}-{\dot{x}}_{2d}-{\Theta }_2{\tilde{x}}_2\right)={\Upsilon }_2\left({\tilde{x}}_3+e_3-k_2{z}_2-\left|{\tilde{x}}_3\right|\text{sign}\left(z_2\right)\right)$

39

Step 3: Define the dynamics of the pitch rate tracking error ${\tilde{x}}_3=x_3-x_{3d}$ . According to equation (4), the derivative of ${\tilde{x}}_3$ is as follows

$\begin{array}{c}{\dot{\tilde{x}}}_3={\dot{x}}_3-{\dot{x}}_{3d}\\ =h_{3}u+f_3-{\dot{x}}_{3d}\end{array}$

40

According to equation (21), we have

${\dot{z}}_3={\Upsilon }_3\left({\dot{\tilde{x}}}_3-{\Theta }_3{\tilde{x}}_3\right)$

41

Similar to step 1, we have

$\begin{array}{c}{\dot{z}}_3={\Upsilon }_3\left({\dot{\tilde{x}}}_3-{\Theta }_3{\tilde{x}}_3\right)\\ ={\Upsilon }_3\left(h_{3}u+f_3-{\dot{x}}_{3d}-{\Theta }_3{\tilde{x}}_3\right)\end{array}$

42

On the compact set ${\Omega }_{{\overline{x}}_3}$ , NN is utilized to approximate the unknown dynamic f₃ with the following form

$f_3\left({\overline{x}}_3\right)=W_{f_3}^{*T}{S}_{f_3}\left({\overline{x}}_3\right)+{\epsilon }_3$

43

where $W_{f_3}^{*}$ is the optimal weight vector, $S_{f_3}$ is the activation function vector and ε₃ is the estimation error, satisfying $\parallel {\epsilon }_3\parallel <{\epsilon }_3^{*}$ .

The control input u is designed as follows

$\begin{array}{l}\widehat{h}\left({\overline{x}}_3\right)u=-k_3{z}_3+{\Theta }_3{\tilde{x}}_3+{\dot{x}}_{3d}-B_3\left({\overline{x}}_3\right)u_2^N\\ -\left(1-B_3\left({\overline{x}}_3\right)\right)u_3^R\end{array}$

44

where $k_3>0,{\widehat{h}}_3={\widehat{W}}_{h_3}{S}_{h_3}$ on the compact set ${\Omega }_{{\overline{x}}_3}$ and

$u_3^N={\widehat{f}}_3\left({\overline{x}}_3\right)={\widehat{W}}_{f_3}^T{S}_{f_3}\left({\overline{x}}_3\right)$

45

$u_3^R=f_3^U\left({\overline{x}}_3\right)\text{tanh}\left(\frac{z_3{f}_3^U\left({\overline{x}}_3\right)}{{\upsilon }_3}\right)$

46

where $f_3^U$ is the upper bound of f₃, and υ₃ is the positive parameter.

The adaptation laws of the adjustable parameters ${\widehat{W}}_{f_3}\text{ and }{\widehat{W}}_{h_3}$ are described by

${\dot{\widehat{W}}}_{f_3}={\Gamma }_{f_3}\left(S_{f_3}{z}_3{B}_3\left({\overline{x}}_3\right)-{\delta }_{f_3}{\widehat{W}}_{f_3}\right)$

47

${\dot{\widehat{W}}}_{h_3}=\text{Proj}\left({\Gamma }_{h_3}\left(S_{h_3}{z}_{3}u-{\delta }_{h_3}{\widehat{W}}_{h_3}\right)\right)$

48

Equation (41) can be further transformed as follows

${\dot{z}}_3={\Upsilon }_3\left({\dot{\tilde{x}}}_3-{\Theta }_3{\tilde{x}}_3\right)={\Upsilon }_3\left(\left({\tilde{h}}_3+{\widehat{h}}_3\right)u+f_3-{\dot{x}}_{3d}-{\Theta }_3{\tilde{x}}_3\right)={\Upsilon }_3\left({\tilde{h}}_{3}u-k_3{z}_3+B_3\left({\overline{x}}_3\right)\left({\tilde{f}}_3+{\epsilon }_3\right)+(1-B_3\left({\overline{x}}_3\right))\left(f_3-u_3^R\right)\right)$

49

With the aid of the switching function $B_i({\overline{x}}_i),i=1,3$ , an adaptive neural controller and robust controller cooperate with each other to realize global stability.

Stability analysis

To perform the analysis of stability and convergence, let us select the Lyapunov function candidate

$L=L_1+L_2+L_3$

50

where

$L_1=\frac{1}{2}\left({\Upsilon }_1^{-1}{z}_1^2+{\tilde{W}}_{f_1}^T{\Gamma }_{f_1}^{-1}{\tilde{W}}_{f_1}+{\tilde{W}}_{h_1}^T{\Gamma }_{h_1}^{-1}{\tilde{W}}_{h_1}+e_2^2\right)$

51

$L_2=\frac{1}{2}\left({\Upsilon }_2^{-1}{z}_2^2+e_3^2\right)$

52

$L_3=\frac{1}{2}\left({\Upsilon }_3^{-1}{z}_3^2+{\tilde{W}}_{f_3}^T{\Gamma }_{f_3}^{-1}{\tilde{W}}_{f_3}+{\tilde{W}}_{h_3}^T{\Gamma }_{h_3}^{-1}{\tilde{W}}_{h_3}\right)$

53

According to the definition of $e_i,i=2,3$ in equations (32) and (38), the following equation can be obtained as follows

$\begin{array}{c}{\dot{e}}_i={\dot{x}}_{id}-{\dot{x}}_{ic}=-\frac{1}{{\alpha }_i}{e}_i+D_i(•)\end{array}$

54

where $D_i(•)=-{\dot{x}}_{ic},i=2,3$ .

Based on the virtual control, equations (26) and (36) together with assumption 1, there is the formula as follows

$\left|D_i(•)\right|\le D_i^U$

55

where $D_i^U$ is the upper bound of $D_i(•),i=2,3$ .

Taking the derivative of L₁, we have

${\dot{L}}_1={\Upsilon }_1^{-1}{z}_1{\dot{z}}_1+\frac{1}{2}{\dot{\Upsilon }}_1^{-1}{z}_1^2+{\tilde{W}}_{f_1}^T{\Gamma }_{f_1}^{-1}{\dot{\tilde{W}}}_{f_1}+{\tilde{W}}_{h_1}^T{\Gamma }_{h_1}^{-1}{\dot{\tilde{W}}}_{h_1}+e_2{\dot{e}}_2=z_1\left(h_1{\tilde{x}}_2+h_1{e}_2+{\tilde{h}}_1{x}_{2c}-k_1{z}_1-{\overline{h}}_1\left|{\tilde{x}}_2\right|\text{sign}\left(z_1\right)\right)+z_1{B}_1\left(x_1\right)\left({\tilde{f}}_1+{\epsilon }_1\right)+z_1(1-B_1\left(x_1\right))\left(f_1-u_1^R\right)-{\tilde{W}}_{f_1}^T{\Gamma }_{f_1}^{-1}{\dot{\widehat{W}}}_{f_1}-{\tilde{W}}_{h_1}^T{\Gamma }_{h_1}^{-1}{\dot{\widehat{W}}}_{h_1}+\frac{1}{2}{\dot{\Upsilon }}_1^{-1}{z}_1^2-e_2\left(\frac{1}{{\alpha }_2}{e}_2-D_2(•)\right)=z_1{h}_1{\tilde{x}}_2+z_1{h}_1{e}_2+z_1{\tilde{h}}_1{x}_{2c}-k_1{z}_1^2-{\overline{h}}_1\left|{\tilde{x}}_2\right|\left|z_1\right|+z_1{B}_1\left(x_1\right)\left({\tilde{f}}_1+{\epsilon }_1\right)+z_1(1-B_1\left(x_1\right))\left(f_1-u_1^R\right)-{\tilde{W}}_{f_1}^T{\Gamma }_{f_1}^{-1}{\dot{\widehat{W}}}_{f_1}-{\tilde{W}}_{h_1}^T{\Gamma }_{h_1}^{-1}{\dot{\widehat{W}}}_{h_1}+\frac{1}{2}{\dot{\Upsilon }}_1^{-1}{z}_1^2-e_2\left(\frac{1}{{\alpha }_2}{e}_2-D_2(•)\right)$

56

where ${\Upsilon }_1^{-1}=-{\dot{\Upsilon }}_1^{-1}/{\Upsilon }_1^2$ , and choose $k_1=k_1^{*}+\frac{1}{2}{\dot{\Upsilon }}_1^{-1}$ . The following inequalities hold

$\begin{array}{c}{\tilde{W}}_{f_1}^T{\widehat{W}}_{f_1}\le \frac{{\delta }_{f_1}}{2}\parallel W_{f_1}^{*}{\parallel }^2-\frac{{\delta }_{f_1}}{2}\parallel {\tilde{W}}_{f_1}^T{\parallel }^2\\ {\tilde{W}}_{h_1}^T{\widehat{W}}_{h_1}\le \frac{{\delta }_{h_1}}{2}\parallel W_{h_1}^{*}{\parallel }^2-\frac{{\delta }_{h_1}}{2}\parallel {\tilde{W}}_{h_1}^T{\parallel }^2\\ z_1{b}_1\left({\overline{x}}_1\right){\epsilon }_1\le \frac{1}{2}{z}_1^2+\frac{1}{2}{\epsilon }_1^{*}{}^2\\ z_1\left(f_1-u_1^R\right)\le \left|z_1{f}_1^U\right|-z_1{f}_1^U\text{tanh}\left(\frac{z_1{f}_1^U}{{\upsilon }_1}\right)\le \kappa {\upsilon }_1\end{array}$

57

Using the Young inequality and adaptive laws (29) and (30), we have

${\dot{L}}_1=z_1{h}_1{\tilde{x}}_2+z_1{h}_1{e}_2+z_1{\tilde{h}}_1{x}_{2c}-k_1^{*}{z}_1^2-{\overline{h}}_1\left|{\tilde{x}}_2\right|\left|z_1\right|+z_1{B}_1\left(x_1\right)\left({\tilde{f}}_1+{\epsilon }_1\right)+z_1(1-B_1\left(x_1\right))\left(f_1-u_1^R\right)-{\tilde{W}}_{f_1}^T{\Gamma }_{f_1}^{-1}{\dot{\widehat{W}}}_{f_1}-{\tilde{W}}_{h_1}^T{\Gamma }_{h_1}^{-1}{\dot{\widehat{W}}}_{h_1}-e_2\left(\frac{1}{{\alpha }_2}{e}_2-D_2(•)\right)\le -k_1^{*}{z}_1^2+{\overline{h}}_1\left|z_1\right|\left|e_2\right|+z_1{B}_1\left(x_1\right){\epsilon }_1+{\delta }_{f_1}{\tilde{W}}_{f_1}^T{\widehat{W}}_{f_1}+{\delta }_{h_1}{\tilde{W}}_{h_1}^T{\widehat{W}}_{h_1}-\frac{1}{{\alpha }_2}{e}_2^2+e_2{D}_2(•)+\kappa {\upsilon }_1\le -\left(k_1^{*}-\frac{{\overline{h}}_1}{2}-\frac{1}{2}\right)z_1^2-\left(\frac{1}{{\alpha }_2}-\frac{{\overline{h}}_1}{2}-\frac{1}{3}\right)e_2^2+{\delta }_{f_1}{\tilde{W}}_{f_1}^T{\widehat{W}}_{f_1}+{\delta }_{h_1}{\tilde{W}}_{h_1}^T{\widehat{W}}_{h_1}+\frac{1}{2}{\epsilon }_1^{*2}+\frac{1}{2}{\left(D_2^U\right)}^2+\kappa {\upsilon }_1\le -\left(k_1^{*}-\frac{{\overline{h}}_1}{2}-\frac{1}{2}\right)z_1^2-\left(\frac{1}{{\alpha }_2}-\frac{{\overline{h}}_1}{2}-\frac{1}{2}\right)e_2^2-\frac{{\delta }_{f_1}}{2}\parallel {\tilde{W}}_{f_1}^T{\parallel }^2-\frac{{\delta }_{h_1}}{2}\parallel {\tilde{W}}_{h_1}^T{\parallel }^2+\frac{{\delta }_{f_1}}{2}\parallel W_{f_1}^{*}{\parallel }^2+\frac{{\delta }_{h_1}}{2}\parallel W_{h_1}^{*}{\parallel }^2+\frac{1}{2}{\epsilon }_1^{*2}+\frac{1}{2}{\left(D_2^U\right)}^2+\kappa {\upsilon }_1\le -{\eta }_1{L}_1+K_{13}$

58

where ${\eta }_1=\text{min}\left[2K_{11}\mathrm{,2}{K}_{12},\frac{{\delta }_{f_1}}{\left({\lambda }_{\text{max}}\left({\Gamma }_{f_1}^{-1}\right)\right)},\frac{{\delta }_{h_1}}{\left({\lambda }_{\text{max}}\left({\Gamma }_{h_1}^{-1}\right)\right)}\right],K_{11}=k_1^{*}-\frac{{\overline{h}}_1}{2}-\frac{1}{2}>0,K_{12}=\frac{1}{{\alpha }_2}-\frac{{\overline{h}}_1}{2}-\frac{1}{2}>0,K_{13}=\frac{{\delta }_{f_1}}{2}\Vert W_{f_1}^{*}{\Vert }^2+\frac{{\delta }_{h_1}}{2}\Vert W_{h_1}^{*}{\Vert }^2+\frac{1}{2}{\epsilon }_1^{*2}+\frac{1}{2}{\left(D_2^U\right)}^2+\kappa {\upsilon }_1$ .

Take the derivative of L₂

${\dot{L}}_2={\Upsilon }_2^{-1}{z}_2{\dot{z}}_2+e_3{\dot{e}}_3+\frac{1}{2}{\dot{\Upsilon }}_2^{-1}{z}_2^2=z_2\left(x_3-x_{3d}+x_{3d}-x_{3c}+x_{3c}-{\dot{x}}_{2d}-{\Theta }_2{\tilde{x}}_2\right)+e_3{\dot{e}}_3+\frac{1}{2}{\dot{\Upsilon }}_2^{-1}{z}_2^2=z_2({\tilde{x}}_3+e_3-k_2{z}_2-\left|{\tilde{x}}_3\right|\text{sign}\left(z_2\right))-\frac{1}{{\alpha }_3}{e}_3^2+e_3{D}_3(•)+\frac{1}{2}{\dot{Y}}_2^{-1}{z}_2^2=-k_2{z}_2^2+\frac{1}{2}{\dot{\Upsilon }}_2^{-1}{z}_2^2-\left|z_2\right|\left|{\tilde{x}}_3\right|+z_2{\tilde{x}}_3+z_2{e}_3-\frac{1}{{\alpha }_3}{e}_3^2+e_3{D}_3(•)$

59

where ${\Upsilon }_2^{-1}=-{\dot{\Upsilon }}_2^{-1}/Y_2^2\text{ and }{k}_2=k_2^{*}+\frac{1}{2}{\dot{\Upsilon }}_2^{-1}$ .

${\dot{L}}_2\le -\left(k_2^{*}-\frac{1}{2}\right)z_2^2-\left(\frac{1}{{\alpha }_3}-1\right)e_3^2+\frac{1}{2}{\left(D_3^U\right)}^2\le -{\eta }_2{L}_2+K_{23}$

60

where ${\eta }_2=\text{min}[K_{21},K_{22}],\text{ with }{K}_{21}=k_2^{*}-1/2>0,K_{22}=1/{\alpha }_3-1>0\text{ and }{K}_{23}=1/2{\left(D_3^U\right)}^2$ .

The derivative of L₃ is as follows

${\dot{L}}_3={\Upsilon }_3^{-1}{z}_3{\dot{z}}_3+{\tilde{W}}_{f_3}^T{\Gamma }_{f_3}^{-1}{\dot{\tilde{W}}}_{f_3}+{\tilde{W}}_{h_3}^T{\Gamma }_{h_3}^{-1}{\dot{\tilde{W}}}_{h_3}=z_3\left({\tilde{h}}_{3}u-k_3{z}_3+B_3\left({\overline{x}}_3\right)\left({\tilde{f}}_3+{\epsilon }_3\right)\right)-z_3(1-B_3\left({\overline{x}}_3\right))\left(f_3-u_3^R\right)+{\tilde{W}}_{f_3}^T{\Gamma }_{f_3}^{-1}{\dot{\tilde{W}}}_{f_3}+{\tilde{W}}_{h_3}^T{\Gamma }_{h_3}^{-1}{\dot{\tilde{W}}}_{h_3}+\frac{1}{2}{\dot{\Upsilon }}_3^{-1}{z}_3^2=z_3{\tilde{h}}_{3}u-k_3{z}_3^2+\frac{1}{2}{\dot{\Upsilon }}_3^{-1}{z}_3^2+z_3{B}_3\left({\overline{x}}_3\right)\left({\tilde{f}}_3+{\epsilon }_3\right)-z_3(1-B_3\left({\overline{x}}_3\right))\left(f_3-u_3^R\right)-{\tilde{W}}_{f_3}^T{\Gamma }_{f_3}^{-1}{\dot{\widehat{W}}}_{f_3}-{\tilde{W}}_{h_3}^T{\Gamma }_{h_3}^{-1}{\dot{\widehat{W}}}_{h_3}$

61

where ${\Upsilon }_3^{-1}=-{\dot{\Upsilon }}_3^{-1}/{\Upsilon }_3^2\text{ and }{k}_3=k_3^{*}+\frac{1}{2}{\dot{\Upsilon }}_3^{-1}$ . Using the Young inequality and adaptive laws (47) and (48), we have

${\dot{L}}_3=z_3{\tilde{h}}_{3}u-k_3^{*}{z}_3^2+z_3{B}_3\left({\overline{x}}_3\right)\left({\tilde{f}}_3+{\epsilon }_3\right)+z_3(1-B_3\left({\overline{x}}_3\right))\left(f_3-u_3^R\right)-{\tilde{W}}_{f_3}^T{S}_{f_3}{z}_3{B}_3\left({\overline{x}}_3\right)+{\delta }_{f_3}{\tilde{W}}_{f_3}^T{\widehat{W}}_{f_3}-{\tilde{W}}_{h_3}^T{S}_{h_3}{z}_{3}u+{\delta }_{h_3}{\tilde{W}}_{h_3}^T{\widehat{W}}_{h_3}=-k_3^{*}{z}_3^2+z_3{B}_3\left({\overline{x}}_3\right){\epsilon }_3+z_3(1-B_3\left({\overline{x}}_3\right))\left(f_3-u_3^R\right)+{\delta }_{f_3}{\tilde{W}}_{f_3}^T{\widehat{W}}_{f_3}+{\delta }_{h_3}{\tilde{W}}_{h_3}^T{\widehat{W}}_{h_3}\le -k_3^{*}{z}_3^2+\frac{1}{2}{z}_3^2+\frac{1}{2}{\epsilon }_3^2+\kappa {\omega }_3-\frac{{\delta }_{f_3}}{2}\parallel {\tilde{W}}_{f_3}^T{\parallel }^2+\frac{{\delta }_{f_3}}{2}\parallel W_{f_3}^{*}{\parallel }^2-\frac{{\delta }_{h_3}}{2}\parallel {\tilde{W}}_{h_3}^T{\parallel }^2+\frac{{\delta }_{h_3}}{2}\parallel W_{h_3}^{*}{\parallel }^2\le -\left(k_3^{*}-\frac{1}{2}\right)z_3^2-\frac{{\delta }_{f_3}}{2}\parallel {\tilde{W}}_{f_3}^T{\parallel }^2-\frac{{\delta }_{h_3}}{2}\parallel {\tilde{W}}_{h_3}^T{\parallel }^2+\frac{{\delta }_{f_3}}{2}\parallel W_{f_3}^{*}{\parallel }^2+\frac{{\delta }_{h_3}}{2}\parallel W_{h_3}^{*}{\parallel }^2+\frac{1}{2}{\epsilon }_3^{*2}+\kappa {\upsilon }_3\le -{\eta }_3{L}_3+K_{33}$

62

where ${\eta }_1=\text{min}[2K_{31},{\delta }_{f_3}/({\lambda }_{\text{max}}({\Gamma }_{f_3}^{-1})),{\delta }_{h_3}/({\lambda }_{\text{max}}({\Gamma }_{h_3}^{-1}))],K_{31}=k_3^{*}-1/2>0\text{ and }{K}_{33}={\delta }_{f_3}/2\parallel W_{f_3}^{*}{\parallel }^2+{\delta }_{h_3}/2\parallel W_{h_3}^{*}{\parallel }^2+1/2{\epsilon }_3^{*2}+\kappa {\upsilon }_3$ .

Eventually, the derivative of L is as follows

$\dot{L}={\dot{L}}_1+{\dot{L}}_2+{\dot{L}}_3\le -\eta L+K$

63

where $\eta =\text{min[}{\eta }_1,{\eta }_2,{\eta }_3]\text{ and }K=K_{13}+K_{23}+K_{33}$ .

Remark 1

Let $\eta >K/P_0,\text{ then }\dot{L}<0$ on $L(0)=P_0$ . Thus, if $L(0)\le P_0,\text{ then }L(t)\le P_0$ for all $t>0$ . Solving inequality (62), we have

$0\le L\left(t\right)\le \frac{K}{\eta }+\left(L\left(0\right)-\frac{K}{\eta }\right)e^{-\eta t},\forall t\ge 0$

64

and

$\underset{t\to +\infty }{\lim }L\left(t\right)\le \frac{K}{\eta }$

65

Summarizing the above analysis, It is concluded that the constrained tracking control of the HFV can regulate the transformed tracking errors to be arbitrarily small by increasing the control gains $(K_1,K_{2}and{K}_3)$ and decreasing the filter parameters $({\alpha }_{2}and{\alpha }_3)$ .

Simulation studies

The comparison simulation study is carried out to verify the design we identified above is feasible, and a general model of HFV is used for simulation.

The initial cruise condition of the vehicle is set as $V=7850\text{\hspace{0.17em}ft/s},h=85000\text{ft},\gamma =0,\alpha =0\text{ and }q=0$ . The step command is $\Delta V=600\text{ft/s}\text{ and }\Delta h=1000\text{ft}$ .

The FPA command parameters are chosen as $k_h=0.5\text{ and }{k}_i=0.1$ . The filter time constant is chosen as ${\alpha }_i=0.05,i=2,3$ . The boundaries of switching function are set as ${}^1{r}_i{=0.1,}^2{r}_i=0.2,i=1,3$ , while the robust parameters υ₁ and υ₃ are set as $2\times {10}^{-6}\text{ and }{10}^{-3}$ , respectively.

The PID controller is designed for velocity subsystem, and the parameters are chosen as $P_v=0.5,I_v=0.001\text{ and }{D}_v=0.01$ .

$\Phi =P_v{e}_V+I_v{\displaystyle {\int }_0^t}{e}_{V}d\sigma +D_v\frac{d{e}_V}{dt}$

66

where e_V represents the velocity error.

The control gains for the DSC are chosen as $k_1=2,k_2=2\text{ and }{k}_3=5$ . The design parameters for adaptive laws are ${\Gamma }_{f_1}={\Gamma }_{f_3}=2I,{\Gamma }_{h_1}={\Gamma }_{h_3}={10}^{-6},{\delta }_{f_1}={\delta }_{f_3}=0.01\text{ and }{\delta }_{f_1}={\delta }_{f_3}=0.02$ . The number of NN nodes is set as $N_1=N_3=125$ .

The tracking performance results for the altitude and velocity of HFV shown in Figures 2 and 3 are desirable, and Figures 4 to 6 demonstrate that the states are all bounded. Furthermore, Figure 7 shows the tracking errors, while its predefined performance bounded by decaying functions of time is also desirable. In the presence of unknown dynamic, the proposed control scheme for HFV can guarantee predefined tracking performance.

Figure 3.

Tracking performace of altitude subsystem (without predefined performance).

Figure 4.

Tracking performace of altitude subsystem (with predefined performance).

Figure 5.

Tracking performace of altitude.

Figure 6.

Tracking performace of velocity.

Figure 7.

Bounded tracking performace.

Conclusion

In this work, a global adaptive neural controller with guaranteed tracking performance is designed for HFV. The new view of this article is that a smooth switching algorithm is employed, by which adaptive neural controller and robust controller cooperate to ensure the system GUUB. Moreover, the developed global adaptive neural control can also improve the tracking performance by introducing an error transformation mechanism. Simulation results have shown clearly that the superior performance of our design.