Tracking error constrained robust adaptive neural prescribed performance control for flexible hypersonic flight vehicle

Abstract

A robust adaptive neural control scheme based on a back-stepping technique is developed for the longitudinal dynamics of a flexible hypersonic flight vehicle, which is able to ensure the state tracking error being confined in the prescribed bounds, in spite of the existing model uncertainties and actuator constraints. Minimal learning parameter technique–based neural networks are used to estimate the model uncertainties; thus, the amount of online updated parameters is largely lessened, and the prior information of the aerodynamic parameters is dispensable. With the utilization of an assistant compensation system, the problem of actuator constraint is overcome. By combining the prescribed performance function and sliding mode differentiator into the neural back-stepping control design procedure, a composite state tracking error constrained adaptive neural control approach is presented, and a new type of adaptive law is constructed. As compared with other adaptive neural control designs for hypersonic flight vehicle, the proposed composite control scheme exhibits not only low-computation property but also strong robustness. Finally, two comparative simulations are performed to demonstrate the robustness of this neural prescribed performance controller.

Introduction

Research effort of hypersonic flight vehicle (HFV) has drawn considerable attention during the past several years, because it can provide large time reductions within both civil and military flight activities. ^{1 –3} The success of the experimental aircraft NASA’s X-43A has affirmed the feasibility of this technique. ³ Unfortunately, the control of HFV is still confronted with a large amount of intractable issues, such as the famous vibrational effects caused by slender geometry and special structures of HFV, intensely coupling between engine system and aerodynamic force and the variation of vehicle characteristics with different flight conditions. ⁴ Thus, in order to keep the flight stability and safety of HFV, transient and steady-state performance characteristics are desirable to be ensured through suitable controllers.

In the literature, due to lack of experimental date on lateral model of HFV, the control problem of HFV mainly focuses on the longitudinal channel. The sum of squares/robust linear matrix inequation method is proposed to design the nonlinear controller for the longitudinal dynamics of HFV with parametric uncertainties. ⁵ Utilizing the Takagi-Sugeno (T-S) fuzzy modelling technique to approximate the nonlinear dynamics of HFV with modelled and unmodelled disturbance, a robust disturbance observer mixed H₂/H_∞ controller is designed for the obtained T-S fuzzy model. ⁶ To provide stable tracking of the velocity and altitude reference trajectories, a high-order extended state observer-enhanced control is adopted to improve the tracking performance. ⁷ By employing the input/output linearization technique to transform the nonlinear model of HFV, a back-stepping technique–based exponential sliding controller is proposed and analysed for the longitudinal dynamics with mismatched uncertainties. ⁸ Moreover, some other sliding mode control schemes, such as super twisting sliding mode control, ⁹ high-order sliding mode control ^10,11 and recursive terminal sliding mode control ¹² , are also proposed to design the control system of HFV in the presence of parametric uncertainties. In addition to the aforementioned control method, numerous significant approaches have been applied to tackle the control problem of HFV, including linear parameter varying control, ¹³ minimax linear quadratic regulator (LQR) control, ¹⁴ back-stepping-based method, ^{3,4,15 –17} predictive control ¹⁸ as well as neural/fuzzy approach. ^{19 –22} Approximation-based adaptive back-stepping control methods have been widely researched for a nonlinear strict-feedback system, ^23,24 and the noteworthy problem of ‘explosion of items’ is elegantly overcome by introducing a dynamic surface control (DSC) technique (low-pass filter) in the control design. ^{25 –28} More specifically, adaptive neural control with the back-stepping technique has also been extensively developed for HFV control. ^29,30 Noting another fact that the operation mode of engine system has rigorous demands on actuators as well as HFV states, the constraints should be considered in the controller design from a practical perspective. ³¹ To ensure relatively satisfying control performance of HFV when physical limitations are in effect, a compensation system is first used to avoid actuator constraint problem. ³¹ After that, other kinds of compensation mechanisms have emerged and evolved to deal with this issue. ^{32 –35} Similarly, command filter–based adaptive back-stepping control is also investigated for HFV in the presence of constraints on system states and actuators. ^36,37 Despite the prominent progress in adaptive neural control methods of HFV, a common characteristic of aforementioned outstanding works is that transient performance is often neglected; only the convergence of the altitude and velocity tracking errors to a residual set are established. ^{38 –40}

Recently, the study of prescribed performance control (PPC) methodology has drawn considerable attention. ^39,41,42 The PPC represents that the tracking error should converge to a predefined bound accompanying with the convergence rate no less than a certain value. ^41,43 By exploiting an output error transformation technique, an adaptive PPC method is first developed for a strict-feedback nonlinear system. ⁴¹ Afterwards, Bu et al. ^42,44 have proposed a control scheme by integrating a PPC technique and neural network (NN) to cope with the output constraint problem of HFV. The control scheme presented in the studies by Bu et al. ^42,44 ensures prescribed transient and steady-state performance in the presence of parameter uncertainties. However, those approaches cannot be utilized to tackle the state constraint problem, and the stability analysis of the constrained state needs to be investigated further. Subsequently, a low-complexity approximation-free PPC scheme has been employed to the control of HFV. ⁴⁵ Although the prescribed performance problem has been tackled, the actuator saturation problem is not considered in it, and there indispensably exist the strict assumptions for nonlinear function with this control scheme. To the best of the author’s knowledge, so far, a composite low-computational neural controller, which is capable of handling actuator and state constraints as well as guaranteeing prescribed performance on transient and steady-state behaviour of the output tracking errors, simultaneously, has not been investigated for HFV.

Motivated by the aforementioned discussion and the control approach, ⁴⁶ a robust adaptive neural PPC scheme is studied for flexible HFV with model uncertainty and actuator constraints. First, on the basis of the functional decomposition, the longitudinal model of HFV can be divided into altitude and velocity subsystems. NNs are employed to estimate the unknown items, thus the prior information of the aerodynamic parameters is no longer needed. Using the PPC technique, an adaptive neural controller is proposed, which is able to ensure the output tracking errors confined in the prescribed bounds, while the issue of state tracking error constraint is also addressed. In order to deal with the problem of ‘increase of NN’ updating parameters’ and ‘explosion of the items’ in a conventional neural back-stepping method, the minimal learning parameter (MLP) technique and ‘first-order sliding mode differentiator (FOSD)’ are used in the control design. Consequently, a low-computational control scheme is obtained. Using an assistant compensation system, the problem of actuator constraint is also eliminated. The contributions of this article are shown as follows:

In contrast to the approach in the study by Bechlioulis and Rovithakis ⁴⁶ , MLP and DSC techniques are incorporated into the controller design, which avoids the problem of increase of NN learning parameters and explosion of the items’, thus deriving a low-computation design.

Compared to previous constraint control schemes, ^36,37 a composite adaptive neural PPC method, which is capable of handling actuator and state constraints as well as guaranteeing prescribed performance on transient and steady-state behaviour of the output tracking errors, is first presented, and a new type of adaptive law is constructed by synthesizing the PPC and MLP technique in the back-stepping design procedures.

Rigorous Lyapunov analysis has been conducted to prove the stability of this control scheme. Meanwhile, comparative simulations are performed to highlight the robustness of the proposed neural prescribed performance controller.

HFV model and some preliminaries

HFV model

The model of HFV used in this study is based on the study by Parker et al. ² The HFV model includes system states (V, h, α, γ, q), flexible states (η₁ and η₂) and control inputs (δ_e and Φ), where V denotes the velocity, h denotes the altitude, α represents the angle of attack, γ denotes the flight path angle, q denotes the pitch rate; δ_e and Φ denote the elevator deflection and the fuel equivalence ratio, respectively ^4,47,48

$\dot{V}=\frac{-D+T\cos \alpha -mg\text{sin}\gamma }{m}$

1

$\dot{h}=V\text{sin}\gamma$

2

$\dot{\gamma }=\frac{L+T\text{sin}\alpha -mg\cos \gamma }{mV}$

3

$\dot{\alpha }=q-\dot{\gamma }$

4

$\dot{q}=\frac{M_{yy}}{I_{yy}}+\frac{{\tilde{\psi }}_1{\ddot{\eta }}_1}{I_{yy}}+\frac{{\tilde{\psi }}_2{\ddot{\eta }}_2}{I_{yy}}$

5

${\ddot{\eta }}_i=-2{\zeta }_i{\omega }_i{\dot{\eta }}_i-{\omega }_i^2{\eta }_i+N_i+{\tilde{\psi }}_{i}q,i=1,2$

6

where m, I_yy and g denote the mass of HFV and the moment of inertia and gravity. T, D, L and M_A denote the thrust of the engine, drag force, lift force and pitching moment, respectively. ^23,47 The related definitions are given as follows

$T=T_{\Phi }\left(\alpha \right)\Phi +T_0\left(\alpha \right)\approx \left[{\beta }_1\Phi +{\beta }_2\right]{\alpha }^3+\left[{\beta }_3\Phi +{\beta }_4\right]{\alpha }^2+\left[{\beta }_5\Phi +{\beta }_6\right]\alpha +\left[{\beta }_7\Phi +{\beta }_8\right]$

$D\approx \overline{\underset{\dot{}}{q}}S\left(C_D^{{\alpha }^2}{\alpha }^2+C_D^{\alpha }\alpha +C_D^0\right),L=C_L\overline{q}S\approx \overline{q}S\left(C_L^0+C_L^{\alpha }\alpha \right),\rho ={\rho }_0\exp \left[-\frac{h-h_0}{h_s}\right]$

$M_{yy}=M_T+M_0\left(\alpha \right)+M_{{\delta }_e}{\delta }_e\approx z_{T}T+\overline{q}S\overline{c}\left(C_M^{{\alpha }^2}{\alpha }^2+C_M^{\alpha }\alpha +C_M^0+C_M^{{\delta }_e}{\delta }_e\right),\overline{q}=\frac{1}{2}\rho V^2$

$N_1=C_{N_1}^{{\alpha }^2}{\alpha }^2+C_{N_1}^{\alpha }\alpha +C_{N_1}^0,N_2=C_{N_2}^{{\alpha }^2}{\alpha }^2+C_{N_2}^{\alpha }\alpha +C_{N_2}^0+C_{N_2}^{{\delta }_e}{\delta }_e$

The explanation of the other parameters can be referred to the study by Parker et al. ²

Model transformation

Altitude subsystem

Consider x₁ = h, x₂ = γ, x₃ = θ, and x₄ = q, where θ = α + γ and $\overline{x}=(x_2,x_3,x_4)$ . Therefore, equations (2) to (5) can be converted into the formulation as shown below

$\begin{array}{l}{\dot{x}}_1=V{x}_2\\ {\dot{x}}_2=f_2\left(X_2\right)+x_3\\ {\dot{x}}_3=x_4\\ {\dot{x}}_4=f_4\left(X_4\right)+u\\ y=x_1,u=-{\delta }_e\end{array}$

7

where y is the output signal of altitude subsystem (7), f₂(X₂) and f₄(X₄) are unknown functions with the formulation $f_2(X_2)=[L+T\text{sin}(\theta -\gamma )-mg\cos \gamma ]/(mV)-\theta$ and $f_4(X_4)=\left[M_{yy}+{\tilde{\psi }}_1{\ddot{\eta }}_1+{\tilde{\psi }}_2{\ddot{\eta }}_2\right]/I_{yy}+{\delta }_e$ .

Velocity subsystem

Velocity subsystem (1) is transformed into equation (8) shown as follows

$\dot{V}=f_V\left(X_V\right)+\Phi$

8

where $f_V(X_V)=(-D+T\cos \alpha -mg\text{sin}\gamma )/m-\Phi$ is an unknown function.

Remark 1

• Since we only consider the cruise phase in this article, γ is quite small so we can take sinγ ≈ γ in equation (2) to simplify the model.

Remark 2

• Considering the fact that the existence of flexible states η₁ and η₂ is hard to be measured directly, we regard η₁ and η₂ as a part of lumped nonlinear function. Thus, the controller design only depends on system states (V, h, α, γ, q).

Prescribed performance

In this section, we will generalize the preliminaries of PPC. ^43,49 To achieve the control objective, the tracking errors $z_i(t),i=1,2,3,4,V$ should be confined in the prescribed bounds shown as follows

$-M_i{\lambda }_i\left(t\right)<z_i\left(t\right)<{\lambda }_i\left(t\right),\text{ if }{z}_i\left(0\right)>0$

$-{\lambda }_i\left(t\right)<z_i\left(t\right)<M_i{\lambda }_i\left(t\right),\text{ if }{z}_i\left(0\right)<0$

9

where $0\le M_i\le 1$ and ${\lambda }_i(t)>0$ named performance function is defined as

${\lambda }_i\left(t\right)=\left({\lambda }_{i0}-{\lambda }_{i\infty }\right)\exp \left(-l_{i}t\right)+{\lambda }_{i\infty }$

10

where λ_i0, λ_i∞ and l_i are design positive constants; ${\lambda }_{i0}\text{=}{\lambda }_i(0)$ while ${\lambda }_{i\infty }={\lim }_{t\to \infty }{\lambda }_i(t)$ ; l_i denotes the minimum speed of convergence and λ_i∞ is the maximum allowed steady error. ^43,49

To transform the constrained tracking error condition (9) into an equivalent unconstrained one, the following transformation is employed. We have

${\mu }_i\left(t\right)=R_i\left(\frac{z_i\left(t\right)}{{\lambda }_i\left(t\right)}\right)$

11

where ${\mu }_i(t)$ is a transformed error and $R_i(·)$ is an increasing transformation function shown as follows ^43,49

$R_i\left(\frac{z_i\left(t\right)}{{\lambda }_i\left(t\right)}\right)=\ln \left(\frac{M_i+\frac{z_i\left(t\right)}{{\lambda }_i\left(t\right)}}{M_i\left(\frac{1-z_i\left(t\right)}{{\lambda }_i\left(t\right)}\right)}\right)\text{ if }{z}_i\left(0\right)>0$

$R_i\left(\frac{z\left(t\right)}{\lambda \left(t\right)}\right)=\ln \left(\frac{M_i\left(\frac{1+z_i\left(t\right)}{{\lambda }_i\left(t\right)}\right)}{M_i-\frac{z_i\left(t\right)}{{\lambda }_i\left(t\right)}}\right)\text{ if }{z}_i\left(0\right)<0$

12

The derivative of equation (11) is

${\dot{\mu }}_i\left(t\right)=r_i\left({\dot{z}}_i\left(t\right)-\frac{{\dot{\lambda }}_i\left(t\right)}{{\lambda }_i\left(t\right)}{z}_i\left(t\right)\right)$

13

where $r_i=\frac{\partial R_i}{\partial \frac{z_i(t)}{{\lambda }_i(t)}}\frac{1}{{\lambda }_i(t)}>0$ . ^43,49

Useful function and key lemmas

Lemma 1

For any ω₀ > 0 and η ∈ R, the following inequality is established ²³

$0\le \left|\eta \right|-\eta \text{tanh}\left(\frac{\eta }{{\omega }_0}\right)\le {\kappa }_0{\omega }_0$

14

where κ₀ is a constant satisfying ${\kappa }_0=e^{-({\kappa }_0+1)}$ , that is, κ₀ = 0.2785.

Lemma 2

The FOSD is shown as follows

$\begin{array}{l}{\dot{\varsigma }}_{i1}=-{\overline{\mu }}_{i1}|{\varsigma }_{i1}-l\left(t\right){|}^{0.5}\text{sign}\left({\varsigma }_{i1}-l\left(t\right)\right)+{\varsigma }_{i2}\\ {\dot{\varsigma }}_{i2}=-{\overline{\mu }}_{i2}\text{sign}\left({\varsigma }_{i2}-{\varsigma }_{i1}\right)\end{array}$

15

where ς_i1 and ς_i2 are the system states, ${\overline{\mu }}_1$ and ${\overline{\mu }}_2$ are the positive parameters of FOSD, and l(t) is an input function. ${\dot{\varsigma }}_{i1}$ can estimate $\dot{l}(t)$ to a desirable precision in case the initial errors ${\varsigma }_{i1}(t_0)-l(t_0)$ and ${\dot{\varsigma }}_{i2}(t_0)-\dot{l}(t_0)$ are bounded. ⁵⁰

Controller design and stability analysis

The control objective in this study is to design a composite adaptive neural prescribed performance controller u and Φ to steer system outputs such as h and V to track their corresponding desired reference signal h_d and V_d with their tracking errors confined to the prescribed performance bounds.

Assumption 1

Assume that the system states are measurable, and there is no time delay in the signal transformation.

Assumption 2

Systems (7) and (8) satisfy $1+\partial f_2/x_3\ne 0$ , $1+\partial f_4/u\ne 0$ and $1+\partial f_V/\Phi \ne 0$ .

Obviously, there are ideal weight vectors $W_2^{*}$ , $W_4^{*}$ , and $W_V^{*}$ , such that

$\{\begin{array}{l}{f}_2=W_2^{*T}{\Phi }_2\left(X_2\right)+{\epsilon }_2\left|{\epsilon }_2\right|\le {\epsilon }_{2M}\\ f_4=W_4^{*T}{\Phi }_4\left(X_4\right)+{\epsilon }_4\left|{\epsilon }_4\right|\le {\epsilon }_{4M}\\ f_V=W_V^{*T}{\Phi }_V\left(X_V\right)+{\epsilon }_V\left|{\epsilon }_V\right|\le {\epsilon }_{VM}\end{array}$

16

where ε_i and ε_iM denote the approximation errors and their bounds, respectively. To decrease the computational burden, the MLP scheme is employed to update NN parameters. Those parameters are defined as ${\varphi }_i=\left|\right|W_i^{*}|{|}^2(i=2,4,V)$ . In the following, we replace ${\Phi }_i(\cdot )$ with Φ_i to simplify the expression.

Altitude controller design

To proceed the design process and tackle the actuator saturation problem, ²⁵ the assistant system (17) is constructed to generate ξ₁

${\dot{\xi }}_1=\Delta u-k_{\xi 1}{\xi }_1$

17

where $k_{\xi 1}>0$ is a positive parameter, and $\Delta u=u-u_d$ , u is the actual control input to the altitude subsystem and u_d is the control input to be designed. ²⁵ The relationship between u_d and u can be expressed as follows

$u=\{\begin{array}{l}\text{sign}\left(u_d\right)u_d^{+},if\left|u_d\right|>u_d^{+}\\ u_d\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em},\hspace{1em}\text{else}\end{array}$

18

where $u_d^{+}>0$ denotes the bound of u_d.

The coordinate change (19) is built shown as follows

$\{\begin{array}{l}{z}_1=x_1-y_d\\ z_2=x_2-{\alpha }_1\\ z_3=x_3-{\alpha }_2\\ z_4=x_4-{\alpha }_3-{\xi }_1\end{array}$

19

where α₁, α₂ and α₃ are middle controllers being established at steps 1, 2 and 3, respectively. y_d = h_d is the reference altitude signal. The control scheme for the altitude subsystem is developed via a back-stepping technique, which contains four-step recursive design procedure.

Step 1

The derivative of $z_1=x_1-y_d$ is expressed as follows

${\dot{z}}_1={\dot{x}}_1-{\dot{y}}_d=V{x}_2-{\dot{y}}_d=V\left(z_2+{\alpha }_1\right)-{\dot{y}}_d$

20

Using equations (13) and (20), the derivative of the transformed altitude error μ₁(t) is shown as follows

${\dot{\mu }}_1\left(t\right)=r_1\left({\dot{z}}_1-\frac{{\dot{\lambda }}_1}{{\lambda }_1}{z}_1\right)=r_1\left(V{x}_2-{\dot{h}}_d-\frac{{\dot{\lambda }}_1}{{\lambda }_1}{z}_1\right)$

21

where $r_1=\partial R_1/\partial (z_1/{\lambda }_1)/{\lambda }_1>r_{1\text{min}}>0$ and ${\lambda }_1(t)=({\lambda }_{10}-{\lambda }_{1\infty })\exp (-l_{1}t)+{\lambda }_{1\infty }$ .

The virtual controller α₁ is designed as follows

${\alpha }_1=\frac{\left(-\left(k_1-\frac{{\dot{r}}_1}{2r_1^2}\right){\mu }_1-k_{12}{\displaystyle \int {\mu }_1}dt+{\dot{y}}_d+\frac{{\dot{\lambda }}_1}{{\lambda }_1}{z}_1\right)}{V}$

22

where k₁ and k₁₂ are the positive parameters.

By invoking equations (13) and (22), one has

${\dot{\mu }}_1=r_1\left(V{z}_2-\left(k_1-\frac{{\dot{r}}_1}{2r_1^2}\right){\mu }_1-k_{12}{\displaystyle \int {\mu }_1}dt\right)$

23

Define a positive Lyapunov function

$L_1=\frac{1}{2{\overline{V}}^2{r}_1}{\mu }_1^2+\frac{k_{12}}{2{\overline{V}}^2}{\left({\displaystyle \int {\mu }_{1}dt}\right)}^2$

24

where $\overline{V}$ is the bound of V.

According to equation (23) and lemma 2, the derivative of L₁ is shown as follows

${\dot{L}}_1=\frac{1}{{\overline{V}}^2{r}_1}{\mu }_1{\dot{\mu }}_1-\frac{{\dot{r}}_1}{2{\overline{V}}^2{r}_1^2}{\mu }_1^2-\frac{k_{12}}{{\overline{V}}^2}{\mu }_1{\displaystyle \int {\mu }_{1}dt}=\frac{-k_1}{{\overline{V}}^2}{\mu }_1^2+\frac{V}{{\overline{V}}^2}{\mu }_1{z}_2$

25

Considering the following fact

$V{\mu }_1{z}_2\le \frac{1}{2k_{11}}{\mu }_1^2+\frac{k_{11}}{2}{V}^2{\lambda }_2^2\left(0\right)$

26

Substituting equation (26) into equation (25) results in

${\dot{L}}_1\le -\frac{1}{{\overline{V}}^2}\left(k_1-\frac{1}{2k_{11}}\right){\mu }_1^2+\frac{V^2}{2{\overline{V}}^2}{k}_{11}{\lambda }_2^2\left(0\right)\hspace{0.17em}\le -\frac{1}{{\overline{V}}^2}\left(k_1-\frac{1}{2k_{11}}\right){\mu }_1^2+\frac{1}{2}{k}_{11}{\lambda }_2^2\left(0\right)$

27

where $k_{11}>0$ .

Step 2

The differentiation of z₂ is obtained as follows

${\dot{z}}_2={\dot{x}}_2-{\dot{\alpha }}_1=f_2+z_3+{\alpha }_2-{\dot{\alpha }}_1$

28

Using equations (13) and (28), the derivative of μ₂(t) is shown as follows

${\dot{\mu }}_2\left(t\right)=r_2\left({\dot{z}}_2-\frac{{\dot{\lambda }}_2}{{\lambda }_2}{z}_2\right)=r_2\left(f_2+z_3+{\alpha }_2-{\dot{\alpha }}_1-\frac{{\dot{\lambda }}_2}{{\lambda }_2}{z}_2\right)$

29

where $r_2=\partial R_2/\partial (z_2/{\lambda }_2)/{\lambda }_2>0$ and ${\lambda }_2(t)=({\lambda }_{20}-{\lambda }_{2\infty })\exp (-l_{2}t)+{\lambda }_{2\infty }$ .

To avoid the complex computation of ${\dot{\alpha }}_1$ , an FOSD (30), based on lemma 2, is applied to approximate it

$\begin{array}{l}{\dot{\varsigma }}_{11}=-{\mu }_{11}|{\varsigma }_{11}-{\alpha }_1{|}^{0.5}\text{sign}\left({\varsigma }_{11}-{\alpha }_1\right)+{\varsigma }_{12}\\ {\dot{\varsigma }}_{12}=-{\mu }_{12}\text{sign}\left({\varsigma }_{12}-{\varsigma }_{11}\right)\end{array}$

30

where ς₁₁ and ς₁₂ are the states of the system (30), and μ₁₁ and μ₁₂ are the positive design constants.

According to equation (30), we have

${\dot{\alpha }}_1={\dot{\varsigma }}_{11}+{\tau }_1$

31

where τ₁ denotes the estimate error. Obviously, we have that $\left|{\tau }_1\right|\le {\overline{\tau }}_1$ with ${\overline{\tau }}_1>0$ .

The controller r₂ is designed as follows

${\alpha }_2=-\frac{k_2}{r_2}{\mu }_2+{\dot{\varsigma }}_{11}-\frac{1}{2}{\mu }_2{\widehat{\varphi }}_2{\Phi }_2^T{\Phi }_2-{\widehat{d}}_2\text{tanh}\left(\frac{r_2{\mu }_2}{w_2}\right)-\frac{{\dot{\lambda }}_2}{{\lambda }_2}{z}_2$

32

where k₂ and w₂ are the positive design parameters. $d_2={\epsilon }_2-{\tau }_1$ is bounded with its supremum satisfied with $d_{2M}\ge {\overline{\tau }}_1+{\epsilon }_{2M}$ . ${\widehat{\varphi }}_2$ and ${\widehat{d}}_2$ denote the estimations of ϕ₂ and d_2M, respectively.

The structure of adaptive control laws is expressed as follows

${\dot{\widehat{\varphi }}}_2=\frac{{\rho }_{21}}{2}\left(r_2{\mu }_2^2{\Phi }_2^T{\Phi }_2-2{\sigma }_{21}{\widehat{\varphi }}_2\right)$

33

${\dot{\widehat{d}}}_2={\rho }_{22}\left[r_2{\mu }_2\text{tanh}\left(\frac{r_2{\mu }_2}{w_2}\right)-{\sigma }_{22}{\widehat{d}}_2\right]$

34

By substituting equation (32) into equation (29), it can be rewritten as follows

$\begin{array}{l}{\dot{\mu }}_2=r_2\left(f_2+z_3-\frac{k_2}{r_2}{\mu }_2+{\dot{\varsigma }}_{11}-\frac{1}{2}{\mu }_2{\widehat{\varphi }}_2{\Phi }_2^T{\Phi }_2-{\widehat{d}}_2\text{tanh}\left(\frac{r_2{\mu }_2}{w_2}\right)-{\tau }_1\right)\\ \hspace{1em}\hspace{0.17em}=r_2\left(W_2^{*T}{\Phi }_2+z_3-\frac{k_2}{r_2}{\mu }_2+{\dot{\varsigma }}_{11}-\frac{1}{2}{\mu }_2{\widehat{\varphi }}_2{\Phi }_2^T{\Phi }_2+d_2-{\widehat{d}}_2\text{tanh}\left(\frac{r_2{\mu }_2}{w_2}\right)\right)\end{array}$

35

Choosing the candidate Lyapunov function

$L_2=\frac{1}{2}{\mu }_2^2+\frac{{\tilde{\varphi }}_2^2}{2{\rho }_{21}}+\frac{{\tilde{d}}_2^2}{2{\rho }_{22}}$

36

where ${\tilde{\varphi }}_2={\varphi }_2-{\widehat{\varphi }}_2$ and ${\tilde{d}}_2=d_{2M}-{\widehat{d}}_2$ .

Using equations (33) to (35), the derivative of L₂ is shown as follows

$\begin{array}{l}{\dot{L}}_2={\mu }_2{\dot{\mu }}_2-\frac{1}{{\rho }_{21}}{\tilde{\varphi }}_2{\dot{\widehat{\varphi }}}_2-\frac{1}{{\rho }_{22}}{\tilde{d}}_2{\dot{\widehat{d}}}_2\\ \hspace{1em}\hspace{0.17em}\le r_2{\mu }_2{W}_2^{*T}{\Phi }_2+r_2{\mu }_2{z}_3-k_2{\mu }_2^2-\frac{1}{2}{r}_2{\mu }_2^2{\varphi }_2{\Phi }_2^T{\Phi }_2+r_2{\mu }_2{d}_2-r_2{\mu }_2{d}_{2M}\text{tanh}\left(\frac{r_2{\mu }_2}{w_2}\right)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}+{\sigma }_{21}{\tilde{\varphi }}_2{\widehat{\varphi }}_2+{\sigma }_{22}{\tilde{d}}_2{\widehat{d}}_2\end{array}$

37

Next by considering the following facts

${\sigma }_{21}{\tilde{\varphi }}_2{\widehat{\varphi }}_2=\frac{{\sigma }_{21}}{2}\left({\varphi }_2^2-{\tilde{\varphi }}_2^2-{\widehat{\varphi }}_2^2\right)\le \frac{{\sigma }_{21}}{2}\left({\varphi }_2^2-{\tilde{\varphi }}_2^2\right),r_2{\mu }_2{W}_2^{*T}{\Phi }_2\le \frac{1}{2}{r}_2{\mu }_2^2{\varphi }_2{\Phi }_2^T{\Phi }_2+\frac{1}{2}{r}_2$

38

${\sigma }_{22}{\tilde{d}}_2{\widehat{d}}_2=\frac{1}{2}{\sigma }_{22}\left(d_{2M}^2-{\tilde{d}}_2^2-{\widehat{d}}_2^2\right)\le \frac{1}{2}{\sigma }_{22}{d}_{2M}^2-\frac{1}{2}{\sigma }_{22}{\tilde{d}}_2^2,r_2{\mu }_2{z}_3\le \frac{1}{2k_{22}}{\mu }_2^2+\frac{{\lambda }_3^2\left(0\right)r_2^2{k}_{22}}{2}$

39

$r_2{\mu }_2{d}_2-r_2{\mu }_2{d}_{2M}\text{tanh}\left(\frac{r_2{\mu }_2}{w_2}\right)\le \left|r_2{\mu }_2\right|d_{2M}-r_2{\mu }_2{d}_{2M}\text{tanh}\left(\frac{r_2{\mu }_2}{w_2}\right)\le {\kappa }_0{d}_{2M}{w}_2$

40

we have

$\begin{array}{l}{\dot{L}}_2\le -\left(k_2-\frac{1}{2k_{22}}\right){\mu }_2^2-\frac{1}{2}{\sigma }_{21}{\tilde{\varphi }}_2^2-\frac{1}{2}{\sigma }_{22}{\tilde{d}}_2^2+\frac{1}{2}{\sigma }_{21}{\varphi }_2^2+\frac{1}{2}{\sigma }_{22}{d}_{2M}^2\\ \hspace{1em}+{\kappa }_0{d}_{2M}{w}_2+\frac{1}{2}{r}_2+\frac{{\lambda }_3^2\left(0\right)r_2^2{k}_{22}}{2}\end{array}$

41

Step 3

The differentiation of z₃ is obtained as follows

${\dot{z}}_3={\dot{x}}_3-{\dot{\alpha }}_2=z_4+{\alpha }_3-{\dot{\alpha }}_2$

42

The derivative of μ₃(t) is shown as follows

${\dot{\mu }}_3\left(t\right)=r_3\left({\dot{z}}_3-\frac{{\dot{\lambda }}_3}{{\lambda }_3}{z}_3\right)=r_3\left(z_4+{\alpha }_3-{\dot{\alpha }}_2-\frac{{\dot{\lambda }}_3}{{\lambda }_3}{z}_3\right)$

43

where $r_3=\partial R_3/\partial (z_3/{\lambda }_3)/{\lambda }_3>0$ and ${\lambda }_3(t)=({\lambda }_{30}-{\lambda }_{3\infty })\exp (-l_{3}t)+{\lambda }_{3\infty }$ .

In order to estimate the derivative of α₂, an FOSD is applied the same with step 2. According to lemma 2, we have

${\dot{\alpha }}_2={\dot{\varsigma }}_{21}+{\tau }_2$

44

where τ₂ denotes the estimate error of FOSD with $\left|{\tau }_2\right|\le {\overline{\tau }}_2$ .

Thus, α₃ is shown as follows

${\alpha }_3=-\frac{k_3}{r_3}{\mu }_3+{\dot{\varsigma }}_{21}-\frac{{\dot{\lambda }}_3}{{\lambda }_3}{z}_3$

45

where k₃ > 0 is a control gain.

By substituting equation (45) into equation (43), we have

${\dot{\mu }}_3=r_3\left(z_4-\frac{k_3}{r_3}{\mu }_3-{\tau }_2\right)$

46

Choosing the following candidate Lyapunov function

$L_3=\frac{1}{2}{\mu }_3^2$

47

Consider the following inequality

$z_4-{\tau }_2\le \left|z_4\right|+{\overline{\tau }}_2\le M_3,r_3{\mu }_3\left(z_4-{\tau }_2\right)\le \frac{1}{2k_{31}}{\mu }_3^2+\frac{k_{31}}{2}{r}_3^2{M}_3^2$

48

where $M_3={\lambda }_4(0)+{\overline{\tau }}_2$ and k₃₁ > 0.

By invoking equation (48), the time derivative of L₃ is obtained as follows

${\dot{L}}_3={\mu }_3{r}_3\left(z_4-k_3{\mu }_3-{\tau }_2\right)\le -\left(k_3-\frac{1}{2k_{31}}\right){\mu }_3^2+\frac{k_{31}}{2}{r}_3^2{M}_3^2$

49

Step 4

The actual controller u will be established. The derivative of z₄ is shown as follows

${\dot{z}}_4=f_4+u_d-{\dot{\alpha }}_3+k_{\xi 1}{\xi }_1$

50

Using equation (13), the derivative of μ₄(t) is developed as follows

${\dot{\mu }}_4\left(t\right)=r_4\left({\dot{z}}_4-\frac{{\dot{\lambda }}_4}{{\lambda }_4}{z}_4\right)=r_4\left(f_4+u_d-{\dot{\alpha }}_3+k_{\xi 1}{\xi }_1-\frac{{\dot{\lambda }}_4}{{\lambda }_4}{z}_4\right)$

51

where $r_4=\partial R_4/\partial (z_4/{\lambda }_4)/{\lambda }_4>0$ and ${\lambda }_4(t)=({\lambda }_{40}-{\lambda }_{4\infty })\exp (-l_{4}t)+{\lambda }_{4\infty }$ .

As done previously, an FOSD is applied to estimate ${\dot{\alpha }}_3$ . Considering lemma 2, we have

${\dot{\alpha }}_3={\dot{\varsigma }}_{31}+{\tau }_3$

52

where τ₃ denotes the estimation error with $\left|{\tau }_3\right|\le {\overline{\tau }}_3$ .

According to MLP and PPC techniques, the controller u is designed as follows

$u_d=-\frac{k_4}{r_4}{\mu }_4+{\dot{\varsigma }}_{41}-\frac{1}{2}{\mu }_4{\widehat{\varphi }}_4{\Phi }_4^T{\Phi }_4-{\widehat{d}}_4\text{tanh}\left(\frac{r_4{\mu }_4}{w_4}\right)-\frac{{\dot{\lambda }}_4}{{\lambda }_4}{z}_4-k_{\xi 1}{\xi }_1$

53

where k₄ and w₄ are the positive control constants. $d_4={\epsilon }_4-{\tau }_3$ is the lump approximation error with $\left|d_4\right|\le d_{4M}$ . ${\widehat{\varphi }}_4$ and ${\widehat{d}}_4$ denote the estimations of ϕ₄ and d_4M, respectively. ${\widehat{\varphi }}_4$ and ${\widehat{d}}_4$ are updated as follows

${\dot{\widehat{\varphi }}}_4=\frac{{\rho }_{41}}{2}\left(r_4{\mu }_4^2{\Phi }_4^T{\Phi }_4-2{\sigma }_{41}{\widehat{\varphi }}_4\right)$

54

${\dot{\widehat{d}}}_4={\rho }_{42}\left(r_4{\mu }_4\text{tanh}\left(\frac{r_4{\mu }_4}{w_4}\right)-{\sigma }_{42}{\widehat{d}}_4\right)$

55

where ρ₄₁ > 0 and ρ₄₂ > 0 are the positive design parameters.

Thus equation (51) can be rewritten as follows

${\dot{\mu }}_4\left(t\right)=r_4\left(W_4^{*T}{\Phi }_4-\frac{k_4}{r_4}{\mu }_4-\frac{1}{2}{\mu }_4{\widehat{\varphi }}_4{\Phi }_4^T{\Phi }_4+d_4-{\widehat{d}}_4\text{tanh}\left(\frac{r_4{\mu }_4}{w_4}\right)\right)$

56

Considering the candidate Lyapunov function

$L_4=\frac{1}{2}{\mu }_4^2+\frac{{\tilde{\varphi }}_4^2}{2{\rho }_{41}}+\frac{{\tilde{d}}_4^2}{2{\rho }_{42}}$

57

where ${\tilde{\varphi }}_4={\varphi }_4-{\widehat{\varphi }}_4$ and ${\tilde{d}}_4=d_{4M}-{\widehat{d}}_4$ .

By invoking equations (54) to (56), results in the time derivative of L₄ are as follows

${\dot{L}}_4={\mu }_4{\dot{\mu }}_4-\frac{1}{{\rho }_{41}}{\tilde{\varphi }}_4{\dot{\widehat{\varphi }}}_4-\frac{1}{{\rho }_{42}}{\tilde{d}}_4{\dot{\widehat{d}}}_4\le -k_4{\mu }_4^2+r_4{\mu }_4{W}_4^{*T}{\Phi }_4-\frac{1}{2}{r}_4{\mu }_4^2{\varphi }_4{\Phi }_4^T{\Phi }_4+r_4{\mu }_4{d}_4-r_4{\mu }_4{\widehat{d}}_4\text{tanh}\left(\frac{r_4{\mu }_4}{w_4}\right)+{\sigma }_{41}{\tilde{\varphi }}_4{\widehat{\varphi }}_4+{\sigma }_{42}{\tilde{d}}_4{\widehat{d}}_4$

58

Following facts can be obtained as follows

${\sigma }_{41}{\tilde{\varphi }}_4{\widehat{\varphi }}_4=\frac{{\sigma }_{41}}{2}\left({\varphi }_4^2-{\tilde{\varphi }}_4^2-{\widehat{\varphi }}_4^2\right)\le \frac{{\sigma }_{41}}{2}\left({\varphi }_4^2-{\tilde{\varphi }}_4^2\right),{\sigma }_{42}{\tilde{d}}_4{\widehat{d}}_4\le \frac{1}{2}{\sigma }_{42}\left(d_{4M}^2-{\tilde{d}}_4^2\right)$

59

$r_4{\mu }_4{d}_4-r_4{\mu }_4{d}_{4M}\text{tanh}\left(\frac{r_4{\mu }_4}{w_4}\right)\le \left|r_4{\mu }_4\right|d_{4M}-r_4{\mu }_4{d}_{4M}\text{tanh}\left(\frac{r_4{\mu }_4}{w_4}\right)\le {\kappa }_0{d}_{4M}{w}_4$

60

$r_4{\mu }_4{W}_4^{*T}\Phi \le \frac{1}{2}{r}_4{\mu }_4^2{\varphi }_4{\Phi }_4^T{\Phi }_4+\frac{1}{2}{r}_4$

61

Thus, equation (58) can be rewritten as follows

${\dot{L}}_4\le -k_4{\mu }_4^2-\frac{1}{2}{\sigma }_{41}{\tilde{\varphi }}_4^2-\frac{1}{2}{\sigma }_{42}{\tilde{d}}_4^2+\frac{1}{2}\left({\sigma }_{42}{d}_{4M}^2+{\sigma }_{41}{\varphi }_4^2+r_4\right)$

62

Theorem 1

Considering the altitude subsystem (7), if the adaptive neural PPC laws are chosen as equations (22), (32), (45), and (53), and updated laws as (33), (34), (54), (55) as well as the parameters satisfied (65). Thus, the boundness of ${\mu }_{i=1,2,3,4}$ , ${\tilde{\varphi }}_{i=2,4}$ and ${\tilde{d}}_{i=2,4}$ can be ensured.

Proof

We select the candidate Lyapunov function shown as follows

$L=L_1+L_2+L_3+L_4$

63

Substitute equations (27), (41), (49) and (62) into the derivative of equation (63), we have

$\dot{L}\le -\frac{1}{{\overline{V}}^2}\left(k_1-\frac{1}{2k_{11}}\right){\mu }_1^2-\left(k_2-\frac{1}{2k_{21}}\right){\mu }_2^2-\frac{1}{2}{\sigma }_{21}{\tilde{\varphi }}_2^2-\frac{1}{2}{\sigma }_{22}{\tilde{d}}_2^2\hspace{1em}-\left(k_3-\frac{1}{2k_{31}}\right){\mu }_3^2-k_4{\mu }_4^2-\frac{1}{2}{\sigma }_{41}{\tilde{\varphi }}_4^2-\frac{1}{2}{\sigma }_{42}{\tilde{d}}_4^2+C_2$

64

where $C_2=\frac{1}{2}\left(k_{11}{\lambda }_2^2\left(0\right)+{\sigma }_{21}{\varphi }_2^2+{\sigma }_{22}{d}_{2M}^2+r_2+{\lambda }_3^2\left(0\right)r_2^2{k}_{22}\text{+}{k}_{31}{r}_3^2{M}_3^2+{\sigma }_{42}{d}_{4M}^2+{\sigma }_{41}{\varphi }_4^2+r_4\right)+{\kappa }_0{d}_{2M}{w}_2+{\kappa }_0{d}_{4M}{w}_4$

The corresponding design parameters should be chosen such that

$k_1-0.5/k_{11}>0,k_2-0.5/k_{21}>0,k_3-0.5/k_{31}>0,k_4>0,{\sigma }_{ij}>0,i=2,4,j=1,2$

65

Define

$\begin{array}{cc}\{\begin{array}{l}{\Omega }_{{\mu }_1}=\left\{{\mu }_1|\left|{\mu }_1\right|\le \sqrt{\frac{C_2}{\frac{\left(\frac{k_1-0.5}{k_{11}}\right)}{{\overline{V}}^2}}}\right\}\\ {\Omega }_{{\mu }_2}=\left\{{\mu }_2|\left|{\mu }_2\right|\le \sqrt{\frac{C_2}{\left(\frac{k_2-0.5}{k_{21}}\right)}}\right\}\\ {\Omega }_{{\mu }_3}=\left\{{\mu }_3|\left|{\mu }_3\right|\le \sqrt{\frac{C_2}{\left(\frac{k_3-0.5}{k_{31}}\right)}}\right\}\\ {\Omega }_{{\mu }_4}=\left\{{\mu }_4|\left|{\mu }_4\right|\le \sqrt{\frac{C_2}{k_4}}\right\}\end{array}& \{\begin{array}{l}{\Omega }_{{\tilde{\varphi }}_2}=\left\{{\tilde{\varphi }}_2|\left|{\tilde{\varphi }}_2\right|\le \sqrt{\frac{2C_2}{{\sigma }_{21}}}\right\}\\ {\Omega }_{{\tilde{\varphi }}_4}=\left\{{\tilde{\varphi }}_4|\left|{\tilde{\varphi }}_4\right|\le \sqrt{\frac{2C_2}{{\sigma }_{41}}}\right\}\\ {\Omega }_{{\tilde{d}}_2}=\left\{{\tilde{d}}_2|\left|{\tilde{d}}_2\right|\le \sqrt{\frac{2C_2}{{\sigma }_{22}}}\right\}\\ {\Omega }_{{\tilde{d}}_4}=\left\{{\tilde{d}}_4|\left|{\tilde{d}}_4\right|\le \sqrt{\frac{2C_2}{{\sigma }_{42}}}\right\}\end{array}\end{array}$

$\dot{L}$ will be negative if ${\mu }_1\notin {\Omega }_{{\mu }_1}$ , ${\mu }_2\notin {\Omega }_{{\mu }_2}$ , ${\mu }_3\notin {\Omega }_{{\mu }_3}$ , ${\mu }_4\notin {\Omega }_{{\mu }_4}$ , ${\tilde{\varphi }}_2\notin {\Omega }_{{\tilde{\varphi }}_2}$ , ${\tilde{\varphi }}_4\notin {\Omega }_{{\tilde{\varphi }}_4}$ , ${\tilde{d}}_2\notin {\Omega }_{{\tilde{d}}_2}$ and ${\tilde{d}}_4\notin {\Omega }_{{\tilde{d}}_4}$ . Therefore, aforementioned errors ${\mu }_{i=1,2,3,4}$ , ${\tilde{\varphi }}_{i=2,4}$ and ${\tilde{d}}_{i=2,4}$ are bounded.

Remark 3

Assumption 2 imposes a controllability condition on systems (7) and (8), which is rational and equivalent to most controllability condition of HFV in the literature. ^17,21,47

Remark 4

By combining sliding mode differentiator, the MLP technique, a composite constrained adaptive neural PPC schemer, is presented, and a new type of adaptive law is constructed simultaneously. The proposed controller is not only able to ensure the state tracking errors confined in the desired performance sets but also owns low-computation since there is only one parameter online to be adjusted for each NNs.

Velocity controller

The velocity tracking error is defined as follows

$z_V=V-V_d-{\xi }_V$

66

where ξ_V is an assistant signal to compensate the saturation effect, and the additional auxiliary system is constructed as follows

${\dot{\xi }}_V=-k_{V\xi }\xi +\Delta \Phi$

67

where $k_{V\xi }>0$ is a designed assistant parameter, and $\Delta \Phi =\Phi -{\Phi }_d$ denotes the error between Φ (actual input) and Φ_d (designed input).

The derivative of z_V is described as follows

${\dot{z}}_V=f_V+{\Phi }_d-{\dot{V}}_d+k_{V\xi }\xi$

68

According to equations (13) and (68), the time derivation of the transformed error μ_V(t) is shown as follows

${\dot{\mu }}_V\left(t\right)=r_V\left({\dot{z}}_V-\frac{{\dot{\lambda }}_V}{{\lambda }_V}z\right)=r_V\left(f_V+{\Phi }_d-{\dot{V}}_d-\frac{{\dot{\lambda }}_V}{{\lambda }_V}{z}_V+k_{V\xi }\xi \right)$

69

where $r_V=\frac{\partial R_V}{\partial (z_V/{\lambda }_V)}\frac{1}{{\lambda }_V}>r_{V\text{min}}>0$ and ${\lambda }_V(t)=({\lambda }_{V0}-{\lambda }_{V\infty })\exp (-l_{V}t)+{\lambda }_{V\infty }$ .

By employing the MLP technique, the controller Φ is designed as follows

${\Phi }_d=-\frac{k_{V1}}{r_V}{\mu }_V-\frac{1}{2}{r}_V{\mu }_V{\widehat{\varphi }}_V{\Phi }_V^T{\Phi }_V-{\widehat{d}}_V\text{tanh}\left(\frac{r_V{\mu }_V}{w_V}\right)+{\dot{V}}_d+\frac{{\dot{\lambda }}_V}{{\lambda }_V}{z}_V-k_{V\xi }\xi$

70

where k_V1 > 0 and w_V > 0 denote designed control gains. ${\widehat{\varphi }}_V$ and ${\widehat{d}}_V$ denote the estimation of ϕ_V and d_VM, respectively. d_V = ε_V is lump approximation error with upper bound d_VM.

Consider the following adaptive laws for ${\widehat{\varphi }}_V$ and ${\widehat{d}}_V$

${\dot{\widehat{\varphi }}}_V=\frac{{\rho }_{V1}}{2}\left(r_V{\mu }_V^2{\Phi }_V^T{\Phi }_V-2{\sigma }_{V1}{\widehat{\varphi }}_V\right)$

71

${\dot{\widehat{d}}}_V={\rho }_{V2}\left[r_V{\mu }_V\text{tanh}\left(\frac{r_V{\mu }_V}{w_V}\right)-{\sigma }_{V2}{\widehat{d}}_V\right]$

72

where ρ_V1, ρ_V2, σ_V1 and σ_V2 denote the positive design parameters.

Theorem 2

Considering velocity subsystem (8), if adaptive neural controller is chosen as equation (70) and updated laws as equations (71) and (72). Then, the boundness of signals μ_V, ${\tilde{\varphi }}_V$ and ${\tilde{d}}_V$ in equation (74) is kept.

Invoking equations (69) and (70) yields

${\dot{\mu }}_V\left(t\right)=r_V\left[-\frac{k_V}{r_V}{\mu }_V+W_V^{*T}{\Phi }_V-\frac{1}{2}{\mu }_V{\widehat{\varphi }}_V{\Phi }_V^T{\Phi }_V+d_V-{\widehat{d}}_V\text{tanh}\left(\frac{r_V{\mu }_V}{w_V}\right)\right]$

73

Considering the following candidate Lyapunov function

$L_V=\frac{1}{2}{\mu }_V^2+\frac{1}{2{\rho }_{V1}}{\tilde{\varphi }}_V^2+\frac{1}{2{\rho }_{V2}}{\tilde{d}}_V^2$

74

where ${\tilde{\varphi }}_V={\varphi }_V-{\widehat{\varphi }}_V$ and ${\tilde{d}}_V=d_{VM}-{\widehat{d}}_V$ .

Based on equations (71) to (73), the derivative of L_V is described as follows

${\dot{L}}_V={\mu }_V{\dot{\mu }}_V-\frac{1}{{\rho }_{V1}}{\tilde{\varphi }}_V{\dot{\widehat{\varphi }}}_V-\frac{1}{{\rho }_{V2}}{\tilde{d}}_V{\dot{\widehat{d}}}_V=-k_V{\mu }_V^2+r_V{\mu }_V{W}_V^{*T}{\Phi }_V-\frac{1}{2}{r}_V{\mu }_V^2{\varphi }_V{\Phi }_V^T{\Phi }_V+r_V\left(d_V{\mu }_V-d_{VM}{\mu }_V\text{tanh}\left(\frac{r_V{\mu }_V}{w_V}\right)\right)+{\sigma }_{V1}{\tilde{\varphi }}_V{\widehat{\varphi }}_V+{\sigma }_{V2}{\tilde{d}}_V{\widehat{d}}_V$

75

Note that the following inequalities hold

$r_V{\mu }_V{W}_V^{*T}{\Phi }_V\le \frac{1}{2}{r}_V{\mu }_V^2{\varphi }_V{\Phi }_V^T{\Phi }_V+\frac{1}{2}{r}_V,{\sigma }_{V2}{\tilde{d}}_V{\widehat{d}}_V\le \frac{{\sigma }_{V2}}{2}\left(d_{VM}^2-{\tilde{d}}_V^2\right)$

76

$r_V{d}_V{\mu }_V-r_V{d}_{VM}{\mu }_V\text{tanh}\left(\frac{r_V{\mu }_V}{w_V}\right)\le d_{VM}\left|r_V{\mu }_V\right|-d_{VM}{r}_V{\mu }_V\text{tanh}\left(\frac{r_V{\mu }_V}{w_V}\right)\le {\kappa }_0{w}_V{d}_{VM}$

77

${\sigma }_{V1}{\tilde{\varphi }}_V{\widehat{\varphi }}_V=\frac{{\sigma }_{V1}}{2}\left({\varphi }_V^2-{\tilde{\varphi }}_V^2-{\widehat{\varphi }}_V^2\right)\le \frac{{\sigma }_{V1}}{2}\left({\varphi }_V^2-{\tilde{\varphi }}_V^2\right)$

78

By considering in equations (76), (77) and (78), equation (75) can be reformulated as follows

${\dot{L}}_V\le -k_V{\mu }_V^2-\frac{{\sigma }_{V1}}{2}{\tilde{\varphi }}_V^2-\frac{{\sigma }_{V2}}{2}{\tilde{d}}_V^2+r_V{\kappa }_0{w}_V{d}_{VM}+\frac{1}{2}\left({\sigma }_{V1}{\varphi }_V^2+{\sigma }_{V2}{d}_{VM}^2+r_V\right)\le -k_V{\mu }_V^2-\frac{{\sigma }_{V1}}{2}{\tilde{\varphi }}_V^2-\frac{{\sigma }_{V2}}{2}{\tilde{d}}_V^2+C_{V2}$

79

where $C_{V2}=r_V{\kappa }_0{w}_V{d}_{VM}+\frac{1}{2}({\sigma }_{V1}{\varphi }_V^2+{\sigma }_{V2}{d}_{VM}^2+r_V)$ .

Define the following compact sets

$\{\begin{array}{l}{\Omega }_{{\mu }_V}=\left\{{\mu }_V|\left|{\mu }_V\right|\le \sqrt{\frac{C_{V_2}}{k_{V1}}}\right\}\\ {\Omega }_{{\tilde{\varphi }}_V}=\left\{{\tilde{\varphi }}_V|\left|{\tilde{\varphi }}_V\right|\le \sqrt{\frac{C_{V2}}{\left(0.5{\sigma }_{V1}\right)}}\right\}\\ {\Omega }_{{\tilde{d}}_V}=\left\{{\tilde{d}}_V|\left|{\tilde{d}}_V\right|\le \sqrt{\frac{C_{V2}}{\left(0.5{\sigma }_{V2}\right)}}\right\}\end{array}$

80

If ${\mu }_V\notin {\Omega }_{{\mu }_V}$ , ${\tilde{\varphi }}_V\notin {\Omega }_{{\tilde{\varphi }}_V}$ , ${\tilde{d}}_V\notin {\Omega }_{{\tilde{d}}_V}$ , ${\dot{L}}_V$ is negative. Therefore, the boundness of signals μ_V, ${\tilde{\varphi }}_V$ and ${\tilde{d}}_V$ in equation (74) is ensured.

Simulations

The model parameters of HFV are the same with the study by Parker et al. ² The initial trim conditions are set as $V_0=7700\hspace{0.17em}\text{ft}/\text{s}$ , $h_0=85000\hspace{0.17em}\text{ft}$ , ${\alpha }_0=1.6325°$ , ${\gamma }_0=0$ , $q_0=0$ , ${\eta }_1=0.97$ , ${\dot{\eta }}_1=0$ , ${\eta }_2=0.7967$ and ${\dot{\eta }}_2=0$ . The parameters of flexible states ⁴⁷ are set as ${\tilde{\psi }}_1=-363.5003$ , ${\tilde{\psi }}_2=328.8029$ , ${\omega }_1=16.0214$ , ${\omega }_2=13.1082$ , ${\zeta }_1=0.05$ and ${\zeta }_1=0.03$ . The performance functions are shown as ${\lambda }_i(t)=({\lambda }_{i0}-{\lambda }_{i\infty })\exp (-l_{i}t)+{\lambda }_{i\infty }\hspace{0.17em}i=1,2,3,4,V$ , where ${\lambda }_{10}=50$ , ${\lambda }_{1\infty }=15$ , ${\lambda }_{20}=0.04$ , ${\lambda }_{2\infty }=0.02$ , ${\lambda }_{30}=0.05$ , ${\lambda }_{3\infty }=0.02$ , ${\lambda }_{40}=0.1$ , ${\lambda }_{4\infty }=0.05$ , ${\lambda }_{V0}=3$ , ${\lambda }_{V\infty }=0.5$ and $l_i=0.08$ ; moreover, M_i is set equal to one. The NN inputs are selected as $X_2=[V,\gamma ,\theta ]$ , $X_4=[V,\gamma ,\theta ,q]$ and $X_V=[V]$ , and the scope of the input variables is defined as $V\in [7700\hspace{0.17em}\text{ft}/\text{s, }7900\hspace{0.17em}\text{ft}/\text{s]}$ , $\gamma \in [-1°,1°]$ , $\theta \in [0,3°]$ and $q\in [-4°/\text{s},4°/\text{s]}$ . The centres including 50 nodes are equally distributed in their scopes. The control parameters are chosen as k₁ = 25.8, k₁₂ = 0.6, k₂ = 0.006, k₃ = 0.01, k₄ = 0.06, k_V = 6, $k_{\xi 1}=1$ and $k_{V\xi }=1$ . Gains for the adaptive laws are selected as ${\rho }_{21}=40$ , ${\rho }_{41}=10$ , ${\rho }_{V1}=2$ , ${\sigma }_{21}=0.01$ , ${\sigma }_{41}=0.05$ , ${\sigma }_{V1}=0.25$ , ${\rho }_{i2}=1$ , ${\sigma }_{i2}=0.1$ and w_i = 1, $i=2,4,V$ . Reference commands are smoothened via several second-order filters shown as follows

$\frac{h_d}{h_{d0}}=\frac{0.01}{s^2+0.19s+0.01},\frac{V_d}{V_{d0}}=\frac{0.01}{s^2+0.19s+0.01}$

81

Simulation 1

In the simulation, the initial tracking errors are assumed to be $z_1(0)=40\hspace{0.17em}\text{ft}$ and ${\text{z}}_V(0)=2\hspace{0.17em}\text{ft}/\text{s}$ . For comparison purposes, the adaptive PPC scheme (named AP) ⁴² is used. In this article, the control parameters of AP are chosen through a trial method to achieve nearly equal tracking performance with robust adaptive prescribed performance controller (RAP). The simulation results are shown in Figures 1 to 6. The superiority of RAP will be further revealed in the next simulation. The output tracking errors for altitude and velocity along with their performance function bounds are presented in Figure 1. Meanwhile, the system states (γ, θ, q) and their accompanying tracking errors (z₂, z₃, z₄) are depicted in Figure 2. The control inputs (δ_e, Φ) are shown in Figure 4. Sure enough, the output tracking performance is satisfactory, and all the state tracking errors are confined in their prescribed bounds, despite there exists some shake of (δ_e, Φ) along with the first 20 s. In addition, the flexible states and auxiliary states are pictured in Figures 3 and 5, respectively. Obviously, the flexible states are both bounded for RAP and AP controller. Compared with AP, the RAP has short convergence time with its flexible states. In all, this simulation has proved the availability of RAP control scheme.

Figure 1.

Altitude and velocity tracking.

Figure 2.

System states and tracking errors.

Figure 3.

Flexible states.

Figure 4.

Control inputs.

Figure 5.

Assistant states.

Figure 6.

Tracking errors and control inputs with AP in case 1.

Simulation 2

To ulteriorly illustrate the robustness of the proposed RAP controller, the coefficient variation of the HFV model is taken into consideration in three different cases as shown below. For comparison purpose, the NNs are hold to be unvaried, while the control parameters, the performance functions and the initial values are kept as before. Thus the superiority of RAP is shown in Figures 6 to 8. First, we apply the AP scheme to control the HFV with coefficients variation like cases 1 and 2. As Figure 6 clearly demonstrates, the output performance of AP controller can be ensured with case 1. Unfortunately, if we increase the coefficient to case 2 without changing the control gains, the tracking errors of AP control approach overstep the prescribed performance bounds λ₁ and λ_V; meanwhile, the closed-loop system becomes unstable as shown in Figure 7. In contrast to AP control scheme, we apply the RAP control scheme to HFV affected by the same coefficient variation with case 1. As shown in Figure 8, the RAP control scheme also guarantees the output error performance. If we increase the coefficient to case 2, the output tracking prescribed performance and stable closed-loop system behaviour of RAP control scheme are still achieved. It must be pointed out that the RAP control scheme operates successfully even though we further soar the coefficient to case 3. Therefore, compared to AP control scheme, the significant increase in robustness of the proposed RAP is achieved.

$\text{Case }1:C=\{\begin{array}{l}{C}_0\left(1+0.2\text{sin}\left(0.05\pi t\right)\right),\hspace{1em}\text{if}\hspace{0.17em}\text{t}\ge \text{60s}\\ C_0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em},\hspace{1em}\text{else}\end{array}$

$\text{Case }2:C=\{\begin{array}{l}{C}_0\left(1+0.3\text{sin}\left(0.05\pi t\right)\right),\hspace{1em}\text{if}\hspace{0.17em}\text{t}\ge \text{60s}\\ C_0\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{0.17em},\hspace{1em}\text{else}\end{array}$

$\text{Case }3:C=\{\begin{array}{l}{C}_0\left(1+0.5\text{sin}\left(0.1\pi t\right)\right),\hspace{1em}\text{if}\hspace{0.17em}\text{t}\ge \text{60s}\\ C_0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em},\hspace{1em}\text{else}\end{array}$

Figure 7.

Tracking errors and control inputs with AP in case 2.

Figure 8.

Tracking errors and control inputs with RAP in cases 1, 2, and 3.

Conclusion

In this study, a guaranteed prescribed performance adaptive neural control scheme has been presented for flexible HFV. Using the MLP, FOSD technique and prescribed performance function, a low-computation state tracking error constrained adaptive neural controller is constructed, wherein the issue of increase of NN learning parameters and explosion of the items’ is removed. With the utilization of an assistant system, the problem of actuator saturation is also eliminated. Compared with other adaptive neural control designs, the proposed controller is not only able to ensure the state tracking errors confined in the desired performance sets but also owns low-computation and better robustness. Finally, two simulations have been performed to demonstrate the robustness of this control scheme.