Adaptive neural back-stepping control of flexible air-breathing hypersonic vehicles with parametric uncertainties

Abstract

Control system is significant for making flight safety. In this study, a novel adaptive neural back-stepping controller is exploited for the longitudinal dynamics of a flexible air-breathing hypersonic vehicle. A combined neural network approach and back-stepping scheme is utilized for developing an output-feedback controller that provides robust tracking of the velocity and altitude commands. For each subsystem, only one neural network is employed to approximate the lumped system uncertainty by updating its weight vector adaptively while the problem of possible control singularity is eliminated. The uniformly ultimately boundedness is guaranteed for the closed-loop control system by means of Lyapunov stability theory. The main contribution is that the design complexity is reduced and less neural networks are required. Finally, simulation results illustrate that the proposed control strategy achieves satisfying tracking performance in spite of flexible effects and system uncertainties.

Introduction

An increasing interest of air-breathing hypersonic vehicle (AHV) has been spurred owing to its prospect for a reliable and cost efficient space access.^1,2 As a key issue in making AHV flight feasible and efficient, the flight control design has been studied widely.^3,4 However, because of the noticeable flexible effects caused by the specific slender geometries and light structural weights, ⁵ and the inconstancy of the vehicle characteristics with varying flight conditions, significant uncertainties affect the vehicle dynamic model. Thus the flight control of AHV is still an open challenge. The control system for this special class of vehicle will have to be extremely robust toward the model uncertainties and external disturbances. ⁶

In the past decades, feedback control approaches combining various linear and nonlinear control methodologies have been presented for AHVs. A robust H_∞ controller ⁷ is developed based on linearized model for the longitudinal model of AHVs. The numerical simulation on nonlinear dynamic model validates that this control strategy provides desired tracking performance with excellent robustness. An adaptive sliding mode controller ⁸ with well robustness with respect to parametric uncertainties is designed for the longitudinal dynamics of generic AHVs. To eliminate the undesired high-frequency chattering caused by the discontinuous control law, a continuous high-order sliding mode controller ⁹ is presented for a flexible AHV (FAHV). In that study, a nonlinear disturbance observer is employed to estimate the uncertainties in order to compensate the controller, which improves the sliding mode controller’s robustness. As instantiated by Bu et al., ¹⁰ the altitude subsystem can be rewritten as the strict-feedback form. Back-stepping has been evolved as a very effective methodology for dealing with uncertain system with such strict-feedback form. By assuming that the vehicle inertial parameters and the coefficients in the forces and moment approximations are subject to uncertainty, a robust adaptive controller ¹¹ is designed based on back-stepping. The uniformly ultimately boundedness of that control scheme is guaranteed by the adopted parameter projection estimation. The problem of control input constraint is also considered in that paper. Similar study is also made by Bu et al. ¹² Different from the previous studies, a severer uncertainty condition is considered in previous works.^13–16 The dynamic model of AHV is assumed to be partially or completely unknown in those studies. Neural network (NN) is employed to estimate such unknown dynamics while novel NN-based output-feedback controller with strong robustness is developed for AHVs. ¹⁷ Viewing each subsystem of AHV as a lumped unknown system, only one radial basis function (RBF) NN is needed to approach such unknown system ¹⁶ while the problem of “explosion of differentiation” that is encountered by Bu et al. ¹⁸ is eliminated.

Although the above works provide effective and efficient methodologies for the flight control design of AHVs, the following challenging problems have not been fully investigated. (1) In many existing studies, the flexible effects are not taken into account adequately. (2) The computational burden should be further reduced.

Motivated by the previous results, a novel adaptive neural controller is presented based on back-stepping scheme^19–23 for the longitudinal dynamics of FAHVs to provide robust tracking of velocity and altitude commands. For the altitude subsystem which is decomposed into two newly functional systems, namely the altitude-flight-path-angle (h-γ) subsystem and the pitch-angle-pitch-rate (θ-Q) subsystem. For each subsystem, only one RBF NN is employed to estimate the lumped system uncertainty while the adaptive neural controller is developed based on the output-feedback formulation. By that means, the possible control singularity problem is dealt with and the computational burden reduces greatly. Similar controller is designed for the velocity subsystem. Simulation results are presented to demonstrate the efficacy of the investigated control algorithm.

The remainder of this article is organized as follows. Section “FAHV model” presents the longitudinal model of FAHV which is formulated as output-feedback form. The RBF NN is presented briefly in section “RBF NN approximation.” The novel adaptive neural controller is investigated, and the stability of the closed-loop system is proved in section “Adaptive neural controller design for FAHV.” Section “Simulation study” shows the simulation results, and the conclusions are presented in section “Conclusion.”

FAHV model

The longitudinal dynamic model of FAHV

The longitudinal dynamic model considered in this study is developed by Bolender and Doman. ²⁴ This vehicle is assumed to be a flat Earth, and its span is normalized to unit depth. As pronounced by Parker et al., ²⁵ the weak coupling between the rigid body states and the flexible ones can be eliminated. Hence, we obtain the following nonlinear motion equations^26–28

$$\stackrel{·}{V}=\frac{T\cos \alpha -D}{m}-g\sin \gamma$$

(1)

$$\stackrel{·}{h}=V\sin \gamma$$

(2)

$$\stackrel{·}{\gamma }=\frac{L+T\sin \alpha }{\mathrm{mV}}-\frac{g}{V}\cos \gamma$$

(3)

$$\stackrel{·}{\alpha }=Q-\stackrel{·}{\gamma }$$

(4)

$$\stackrel{·}{Q}=\frac{M}{I_{\mathrm{yy}}}$$

(5)

$${\stackrel{··}{\eta }}_1=-2{\zeta }_1{\omega }_1{\stackrel{·}{\eta }}_1-{\omega }_1^2{\eta }_1+N_1$$

(6)

$${\stackrel{··}{\eta }}_2=-2{\zeta }_2{\omega }_2{\stackrel{·}{\eta }}_2-{\omega }_2^2{\eta }_2+N_2$$

(7)

This model contains five rigid body states: velocity V, altitude h, flight-path angle γ, angle of attack α, and pitch rate Q and four flexible states: ${\eta }_1$ , ${\stackrel{·}{\eta }}_1$ , ${\eta }_2$ , and ${\stackrel{·}{\eta }}_2$ which are corresponding to the first two vibrational modes. ${\zeta }_i$ and ω_i denote the respective damping ratio and natural frequency of the mass-normalized generalized coordinates of the flexible structure, where i = 1, 2. I_yy and m represent the moment of inertia and the vehicle mass, respectively. g is the acceleration due to gravity. The thrust T, drag D, lift L, pitching moment M, and two generalized forces N₁, N₂ are defined as follows ²⁵

$$\begin{array}{c}L\approx \overline{q}S{C}_L(\alpha ,{\delta }_e),D\approx \overline{q}S{C}_D(\alpha ,{\delta }_e)\hfill \\ M\approx z_{T}T+\overline{q}S\overline{c}[C_{M,\alpha }\left(\alpha \right)+C_{M,{\delta }_e}\left({\delta }_e\right)]\hfill \\ T\approx C_T^{{\alpha }^3}{\alpha }^3+C_T^{{\alpha }^2}{\alpha }^2+C_T^{\alpha }\alpha +C_T^0\hfill \\ N_1=N_1^{{\alpha }^2}{\alpha }^2+N_1^{\alpha }\alpha +N_1^0\hfill \\ N_2=N_2^{{\alpha }^2}{\alpha }^2+N_2^{\alpha }\alpha +N_2^{{\delta }_e}{\delta }_e+N_2^0\hfill \end{array}$$

(8)

with

$$\begin{array}{c}{C}_L=C_L^{\alpha }\alpha +C_L^{{\delta }_e}{\delta }_e+C_L^0\hfill \\ C_D=C_D^{{\alpha }^2}{\alpha }^2+C_D^{\alpha }\alpha +C_D^{{\delta }_e^2}{\delta }_e^2+C_D^{{\delta }_e}{\delta }_e+C_D^0\hfill \\ C_{M,\alpha }=C_{M,\alpha }^{{\alpha }^2}{\alpha }^2+C_{M,\alpha }^{\alpha }\alpha +C_{M,\alpha }^0\hfill \\ C_T^{{\alpha }^3}={\beta }_1(h,\overline{q})\Phi +{\beta }_2(h,\overline{q})\hfill \\ C_T^{{\alpha }^2}={\beta }_3(h,\overline{q})\Phi +{\beta }_4(h,\overline{q})\hfill \\ C_T^{\alpha }={\beta }_5(h,\overline{q})\Phi +{\beta }_6(h,\overline{q})\hfill \\ C_T^0={\beta }_7(h,\overline{q})\Phi +{\beta }_8(h,\overline{q})\hfill \\ C_{M,{\delta }_e}=c_e{\delta }_e,\overline{q}=\frac{1}{2}\rho V^2,\rho ={\rho }_0\exp \left[\frac{-(h-h_0)}{h_s}\right]\hfill \end{array}$$

(9)

where $\overline{q}$ denotes the dynamical pressure, ρ is the air density, S and $\overline{c}$ represent the reference area and the aerodynamic chord, respectively, $z_T$ denotes the thrust moment arm. The control inputs Ф and δ_e are fuel equivalence ratio and elevator angular deflection. Ф and δ_e do not occur explicitly in the motion equations of FAHV in equations (1)–(7) but appear through T, D, L, M, N₁, and N₂. The value and units of coefficients are the same as Parker et al. ²⁵ and are listed in Tables 2–8 in Appendix 1.

Output-feedback formulation

The control objective pursued in this article is to design output-feedback controller Φ and δ_e such that the velocity V and the altitude h track their reference trajectories V_ref and h_ref from a given set of initial trim conditions to the desired trim values. It is worthy to note that the velocity V is mainly related to Φ and the altitude h is mainly affected by δ_e, since the thrust T affects V, and δ_e has a dominant contribution to h change in equations (1)–(9). Thus it is reasonable to decompose the motion equations (1)–(5) into the velocity subsystem (i.e. equation (1)) and the altitude system (i.e. equations (2)–(5)).

The velocity subsystem equation (1) can be formulated as

$$\{\begin{array}{c}\stackrel{·}{V}=f_V+g_V\Phi +d_V\hfill \\ \multicolumn{1}{c}{y_V=V}\end{array}$$

(10)

where $d_V$ denotes a bounded external disturbance

$$\begin{array}{c}\multicolumn{1}{c}{f_V={\beta }_2(h,\overline{q}){\alpha }^3\cos \alpha /m+{\beta }_4(h,\overline{q}){\alpha }^2\cos \alpha /m}\\ +{\beta }_6(h,\overline{q})\alpha \cos \alpha /m\hfill \\ +{\beta }_8(h,\overline{q})\cos \alpha /m-D/m-g\sin \gamma \hfill \end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{g_V={\beta }_1(h,\overline{q})\alpha \cos {\alpha }^3/m+{\beta }_3(h,\overline{q}){\alpha }^2\cos \alpha /m}\\ \multicolumn{1}{c}{+{\beta }_5(h,\overline{q})\alpha \cos \alpha /m+{\beta }_7(h,\overline{q})\cos \alpha /m}\end{array}$$

The control objective of velocity subsystem is to design an adaptive neural controller Φ to make V→V_ref.

The altitude subsystem equations (2)–(5) can be rewritten as two new subsystems, namely the h-γ subsystem (i.e. equations (2) and (3)) and the θ-Q subsystem (i.e. equations (4) and (5)).

Define the altitude tracking error as $\tilde{h}=h-h_{\mathrm{ref}}$ . The command of γ is generated as ¹⁵

$${\gamma }_d=\arcsin \left[\frac{-k\tilde{h}-k_I\underset{0}{\overset{t}{\int }}\tilde{h}\left(\tau \right)d\tau +{\stackrel{·}{h}}_{\mathrm{ref}}}{V}\right]$$

(11)

if k > 0, k_I > 0, and γ→γ_d, the altitude tracking error $\tilde{h}$ can be regulated to zero exponentially. ¹⁵ That is, h→h_ref. Hence the next design goal is to insure γ→γ_d.

According to the definition of pitch angle θ, we obtain θ =γ + α. Hence the h-γ subsystem can be rewritten as

$$\{\begin{array}{c}\stackrel{·}{\gamma }=f_{\gamma }+g_{\gamma }\theta +d_{\gamma }\hfill \\ \multicolumn{1}{c}{y_{\gamma }=\gamma }\end{array}$$

(12)

where $g_{\gamma }=\overline{q}{\mathrm{SC}}_L^{\alpha }/(\mathrm{mV})$ , $f_{\gamma }=\overline{q}S(C_L^{{\delta }_e}{\delta }_e+C_L^0-C_L^{\alpha }\gamma )/(\mathrm{mV})+T\sin \alpha /(\mathrm{mV})-g\cos \gamma$ , $d_{\gamma }$ is a bounded external disturbance.

The control objective of h-γ subsystem is to design a virtual controller θ_d based on back-stepping to make γ→γ_d. Thus, the next design objective is to make sure θ→θ_d.

The θ-Q subsystem can be rewritten as

$$\{\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{\theta }=Q}\\ \multicolumn{1}{c}{\stackrel{·}{Q}=f_Q+g_Q{\delta }_e+d_Q}\\ \multicolumn{1}{c}{y_Q=\theta }\end{array}$$

(13)

where $g_Q=\overline{q}S\overline{c}{c}_e/I_{\mathrm{yy}}$ , $f_Q=[z_{T}T+\overline{q}S\overline{c}{C}_{M,\alpha }(\alpha )]/I_{\mathrm{yy}}$ , $d_Q$ means a bounded external disturbance.

The control objective of θ-Q subsystem is to design an adaptive neural controller δ_e to make θ→θ_d.

Remark l

1. By regulating the neural controller Φ, we have V→V_ref.

2. By regulating the neural controller δ_e and the virtual controller θ_d, we obtain θ→θ_d and γ→γ_d, further, h→h_ref.

3. Noting that the altitude subsystem is a complex high-order system, we decompose it into two newly functional systems, namely the altitude-flight-path-angle (h-γ) subsystem (i.e. equation (12)) and the pitch-angle-pitch-rate (θ-Q) subsystem (i.e. equation (13)), all of which are low-order systems. In this way, the design procedure of back-stepping becomes simple.

Assumption 1

g_V, g_γ, g_Q, f_V, f_γ, and f_Q are completely unknown smooth functions. ¹⁶ But g_V and g_γ are strictly positive, and g_Q is strictly negative. It is assumed that there exist constants ${\overline{g}}_V$ , ${\underset{¯}{g}}_V$ , ${\overline{g}}_{\gamma }$ , ${\underset{¯}{g}}_{\gamma }$ , ${\overline{g}}_Q$ , and ${\underset{¯}{g}}_Q$ such that ${\overline{g}}_V$  ≥  $g_V$  ≥  ${\underset{¯}{g}}_V$  > 0, ${\overline{g}}_{\gamma }$  ≥  $g_{\gamma }$  ≥  ${\underset{¯}{g}}_{\gamma }$  > 0, and 0 >  ${\overline{g}}_Q$ ≥  $g_Q$  ≥  ${\underset{¯}{g}}_Q$ .

Assumption 2

There exist positive constants g_VM, g_γM, and g_QM such that $g_{\mathrm{VM}}\ge \left|{\stackrel{·}{g}}_V\right|$ , $g_{\gamma M}\ge \left|{\stackrel{·}{g}}_{\gamma }\right|$ , and $g_{\mathrm{QM}}\ge \left|{\stackrel{·}{g}}_Q\right|$ . ¹⁵

Remark 2

1. According to the discussion by Gao et al.^15,16 and the values of coefficients in Tables 2–8 in Appendix 1, it is easy to note that Assumptions 1 and 2 hold.

2. Assumptions 1 and 2 are only required for analysis purposes in the following stability proof, and the knowledge of the lower and upper bounds for $g_V$ , $|{\stackrel{·}{g}}_V|$ , $g_{\gamma }$ , $|{\stackrel{·}{g}}_{\gamma }|$ , $g_Q$ , and $|{\stackrel{·}{g}}_Q|$ are not needed in the presented controller.

3. Different from the existing studies^7–9 in which the dynamics of FAHV are assumed partly unknown, in this study, we assume that g_V, g_γ, g_Q, f_V, f_γ, and f_Q are completely unknown. In this way, the controller can provide a stronger robustness, since it does not rely on exact dynamic model.

RBF NN approximation

In this article, RBF NN is utilized to approach the system unknown function because of its excellent performance and global approximation. RBF NN can be instantiated as ${\mathit{W}}^T\mathit{h}(\mathit{x})$ with input vector $\mathit{x}\in {\mathbf{R}}^n$ , weight vector $\mathit{W}\in {\mathbf{R}}^m$ , and basis function $\mathit{h}(\mathit{x})\in {\mathbf{R}}^m$ . Where n is the node number, and m is the input number. It has been demonstrated that RBF NN can estimate any continuous function F( x ) over a compact set ${\mathrm{\Omega }}_{\mathit{x}}\subset {\mathbf{R}}^n$ to any arbitrary^29–32

$$\begin{array}{cc}F(\mathit{x})={\mathit{W}}^{*T}\mathit{h}(\mathit{x})+\mu \hfill & \forall \mathit{x}\in \hfill \end{array}{\mathrm{\Omega }}_{\mathit{x}}$$

(14)

where ${\mathit{W}}^{*}$ is the ideal weight vector, $\left|\mu \right|\le {\mu }_M$ is the approach error, the basis function vector $\mathit{h}(\mathit{x})=[h_1(\mathit{x}),h_2(\mathit{x}),\dots ,h_m(\mathit{x}){]}^T\in {\mathbf{R}}^m$ with $h_j(\mathit{x})$ being selected as the Gaussian function

$$h_j(\mathit{x})=\exp \left(\frac{-{\Vert \mathit{x}-{\mathit{a}}_j\Vert }^2}{\left(2b_j^2\right)}\right),\mathrm{j}=1,2,\dots ,m$$

(15)

where ${\mathit{a}}_j=[a_{j1},a_{j2},\dots ,a_{\mathrm{jn}}{]}^T$ and $b_j$ denote the respective center and width of the Gaussian function.

It is worthy to point out that the ideal weight vector ${\mathit{W}}^{*}$ is an artificial quantity required only for analytical purpose. Typically, ${\mathit{W}}^{*}$ is defined as the value of W that minimizes $\left|\mu \right|$ , that is

$${\mathit{W}}^{*}=\arg \underset{\mathit{W}\in {\mathbf{R}}^m}{min}\left\{\underset{\mathit{x}\in {\mathrm{\Omega }}_{\mathit{x}}}{\sup }|F\left(\mathit{x}\right)-{\mathit{W}}^T\mathit{h}(\mathit{x})|\right\}$$

(16)

It is easy to know that ${\mathit{W}}^{*}$ is usually unknown and needs to be estimated in the following design procedure.

Adaptive neural controller design for FAHV

In this section, by assuming that the rigid body states V, h, γ, α, and Q are available for measurement, we investigate an adaptive neural controller for FAHV based on back-stepping design.

Adaptive neural controller design for the velocity subsystem

Design the velocity tracking error and error function

$$e_1=V-V_{\mathrm{ref}}$$

(17)

$$S_1=(\frac{d}{\mathrm{dt}}+{\lambda }_1)\underset{0}{\overset{t}{\int }}{e}_{1}d\tau$$

(18)

where λ₁ > 0 is a design parameter.

Differentiating equation (18) along equations (10) and (17) yields

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{S}}_1={\stackrel{·}{e}}_1+{\lambda }_1{e}_1=g_V\Phi +f_V+d_V}\\ +{\lambda }_1{e}_1-{\stackrel{·}{V}}_{\mathrm{ref}}=g_V(\Phi -F_V)\hfill \end{array}$$

(19)

where $F_V=-(g_V^{-1}{f}_V+g_V^{-1}{\lambda }_1{e}_1-g_V^{-1}{\stackrel{·}{V}}_{\mathrm{ref}}+g_V^{-1}{d}_V)$ is completely unknown, since $f_V$ and $g_V$ are unknown and needs to be approached by the RBF NN equation (14)

$$F_V={\mathit{W}}_1^{*T}{\mathit{h}}_1({\mathit{x}}_1)+{\mu }_1,\left|{\mu }_1\right|\le {\mu }_{1M}$$

(20)

Since ${\mathit{W}}_1^{*}$ is difficult to be obtained, it needs to be estimated online. Thus, we design the following neural controller

$$\Phi =-k_V{S}_1+F_V=-k_V{S}_1+{\hat{\mathit{W}}}_1^T{\mathit{h}}_1({\mathit{x}}_1)$$

(21)

where ${\hat{\mathit{W}}}_1$ denotes the estimation of ${\mathit{W}}_1^{*}$ , and it is updated by the following adaptive law

$${\stackrel{·}{\hat{\mathit{W}}}}_1=-\mathit{\Gamma }1[{\mathit{h}}_1({\mathit{x}}_1)S_1+{\sigma }_1{\hat{\mathit{W}}}_1]$$

(22)

where $\mathit{\Gamma }1=\mathit{\Gamma }{1}^T>0$ is an adaptation gain matrix and ${\sigma }_1$ is a positive constant.

Substituting equations (20) and (21) into equation (19) leads to

$${\stackrel{·}{S}}_1=g_V(-k_V{S}_1+{\tilde{\mathit{W}}}_1^T{\mathit{h}}_1({\mathit{x}}_1)-{\mu }_1)$$

(23)

where ${\tilde{\mathit{W}}}_1={\hat{\mathit{W}}}_1-{\mathit{W}}_1^{*}$ .

Design the following Lyapunov function candidate

$$W_V=\frac{1}{2g_V}{S}_1^2+\frac{{{\tilde{\mathit{W}}}_1}^T\mathit{\Gamma }{1}^{-1}{\tilde{\mathit{W}}}_1}{2}$$

(24)

Differentiating equation (24) along equations (22) and (23) yields

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{W}}_V=\frac{1}{g_V}{S}_1{\stackrel{·}{S}}_1-\frac{{\stackrel{·}{g}}_V}{2g_V^2}{S}_1^2+{{\tilde{\mathit{W}}}_1}^T\mathit{\Gamma }{1}^{-1}{\stackrel{·}{\hat{\mathit{W}}}}_1}\\ =-(k_V+\frac{{\stackrel{·}{g}}_V}{2g_V^2})S_1^2-{\mu }_1{S}_1-{\sigma }_1{{\tilde{\mathit{W}}}_1}^T\hat{\mathit{W}}\hfill \end{array}$$

(25)

Considering that $2{{\tilde{\mathit{W}}}_1}^T{\hat{\mathit{W}}}_1\ge ||{\tilde{\mathit{W}}}_1|{|}^2-||{{\mathit{W}}_1}^{*}|{|}^2$ , ${\mu }_1{S}_1\le |{\mu }_{1M}{S}_1|\le ({\mu }_{1M}{S}_1{)}^2/2+1/2$ , the following inequality holds

$${\stackrel{·}{W}}_V\le -(k_V+\frac{{\stackrel{·}{g}}_V}{2g_V^2}-\frac{{\mu }_{1M}^2}{2})S_1^2+\frac{1}{2}{\sigma }_1\Vert {{\mathit{W}}_1}^{*}{\Vert }^2+\frac{1}{2}$$

(26)

Let $k_V\ge g_{\mathrm{VM}}/(2{\underset{¯}{g}}_V^2)+{\mu }_{1M}^2/2$ . Thus, the tracking error function S₁ can be invariant to the following set

$${\mathrm{\Omega }}_{S_1}=\left\{S_1|\left|S_1\right|\le \sqrt{\frac{(\frac{1}{2}{\sigma }_1{\Vert {{\mathit{W}}_1}^{*}\Vert }^2+\frac{1}{2})}{(k_V+\frac{{\stackrel{·}{g}}_V}{2g_V^2}-\frac{{\mu }_{1M}^2}{2})}}\right\}$$

(27)

The radius of ${\mathrm{\Omega }}_{S_1}$ can be made arbitrarily small by choosing sufficiently large $k_V$ and sufficiently small ${\sigma }_1$ . Considering that the polynomial s + λ₁ is Hurwitz since λ₁ > 0, also the tracking error e₁ is bounded. That is, the neural controller can make V→V_ref.

Adaptive neural controller design for the altitude subsystem

In what follows, the design of adaptive neural controller for altitude subsystem includes two steps. In Step 1, an adaptive neural controller is developed for the h-γ subsystem. Step 2 represents the neural controller design for the θ-Q subsystem.

• Step 1. Define the flight-path angle tracking error e₂ and error function S₂

$$e_2=\gamma -{\gamma }_d$$

(28)

$$S_2=(\frac{d}{\mathrm{dt}}+{\lambda }_2)\underset{0}{\overset{t}{\int }}{e}_{2}d\tau$$

(29)

where ${\lambda }_2>0$ is a design parameter.

Differentiating equation (29) along equations (12) and (28), we obtain

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{S}}_2={\stackrel{·}{e}}_2+{\lambda }_2{e}_2=g_{\gamma }\theta +f_{\gamma }+d_{\gamma }}\\ +{\lambda }_2{e}_2-{\stackrel{·}{\gamma }}_d=g_{\gamma }(\theta -F_{\gamma })\hfill \end{array}$$

(30)

where $F_{\gamma }=-(g_{\gamma }^{-1}{f}_{\gamma }+g_{\gamma }^{-1}{\lambda }_2{e}_2-g_{\gamma }^{-1}{\stackrel{·}{\gamma }}_d+g_{\gamma }^{-1}{d}_{\gamma })$ is also completely unknown. We utilize the RBF NN equation (14) to estimate it

$$F_{\gamma }={\mathit{W}}_2^{*T}{\mathit{h}}_2({\mathit{x}}_2)+{\mu }_2,\left|{\mu }_2\right|\le {\mu }_{2M}$$

(31)

Let ${\hat{\mathit{W}}}_2$ be the estimation of ${\mathit{W}}_2^{*}$ and take ${\theta }_d$ as the virtual controller based on back-stepping. We design the following adaptive neural controller

$${\theta }_d=-k_{\gamma }{S}_2+F_{\gamma }=-k_{\gamma }{S}_2+{\hat{\mathit{W}}}_2^T{\mathit{h}}_2({\mathit{x}}_2)$$

(32)

where $k_{\gamma }>0$ is a design parameter. ${\hat{\mathit{W}}}_2$ is updated by the following adaptive law

$${\stackrel{·}{\hat{\mathit{W}}}}_2=-\mathit{\Gamma }2[{\mathit{h}}_2({\mathit{x}}_2)S_2+{\sigma }_2{\hat{\mathit{W}}}_2]$$

(33)

where $\mathit{\Gamma }2=\mathit{\Gamma }{2}^T>0$ is an adaptation gain matrix and ${\sigma }_2$ is a positive constant.

• Step 2. Define pitch angle tracking error e₃ and error function S₃

$$e_3=\theta -{\theta }_d$$

(34)

$$S_3={(\frac{d}{\mathrm{dt}}+{\lambda }_3)}^2\underset{0}{\overset{t}{\int }}{e}_{3}d\tau$$

(35)

where ${\lambda }_3>0$ is a design parameter.

Differentiating equation (35) along equations (13) and (34) yields

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{S}}_3={\lambda }_3^2{e}_3+2{\lambda }_3{\stackrel{·}{e}}_3+{\stackrel{··}{e}}_3=g_Q{\delta }_e+f_Q+d_Q}\\ +{\lambda }_3^2{e}_3+2{\lambda }_3{\stackrel{·}{e}}_3-{\stackrel{··}{\theta }}_d=g_Q({\delta }_e-F_Q)\hfill \end{array}$$

(36)

where $F_Q=-(g_Q^{-1}{f}_Q+g_Q^{-1}{\lambda }_3^2{e}_3+g_Q^{-1}2{\lambda }_3{\stackrel{·}{e}}_3-g_Q^{-1}{\stackrel{··}{\theta }}_d+g_Q^{-1}{d}_Q)$ is unknown and needs to be approached by the RBF NN equation (14).

$$F_Q={\mathit{W}}_3^{*T}{\mathit{h}}_3({\mathit{x}}_3)+{\mu }_3,\left|{\mu }_3\right|\le {\mu }_{3M}$$

(37)

Take ${\hat{\mathit{W}}}_3$ as the estimation of ${\mathit{W}}_3^{*}$ . We design the following adaptive neural controller

$${\delta }_e=k_Q{S}_3+F_Q=k_Q{S}_3+{\hat{\mathit{W}}}_3^T{\mathit{h}}_3({\mathit{x}}_3)$$

(38)

where $k_Q>0$ is a design parameter. ${\hat{\mathit{W}}}_3$ is updated by the following adaptive law

$${\stackrel{·}{\hat{\mathit{W}}}}_3=\mathit{\Gamma }3[{\mathit{h}}_3({\mathit{x}}_3)S_3-{\sigma }_3{\hat{\mathit{W}}}_3]$$

(39)

where $\mathit{\Gamma }3=\mathit{\Gamma }{3}^T>0$ is an adaptation gain matrix and ${\sigma }_3$ is a positive constant.

Substituting equations (32) and (33) into equation (30), and equations (37) and (38) into equation (36), we have

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{S}}_2=g_{\gamma }(e_3+{\theta }_d-F_{\gamma })=g_{\gamma }{e}_3}\\ +g_{\gamma }[-k_{\gamma }{S}_2+{\tilde{\mathit{W}}}_2^T{\mathit{h}}_2({\mathit{x}}_2)-{\mu }_2]\hfill \end{array}$$

(40)

$${\stackrel{·}{S}}_3=g_Q[k_Q{S}_3+{\tilde{\mathit{W}}}_3^T{\mathit{h}}_3({\mathit{x}}_3)-{\mu }_3]$$

(41)

where ${\tilde{\mathit{W}}}_2={\hat{\mathit{W}}}_2-{\mathit{W}}_2^{*}$ , ${\tilde{\mathit{W}}}_3={\hat{\mathit{W}}}_3-{\mathit{W}}_3^{*}$ .

Design the following Lyapunov function candidates

$$W_{\gamma }=\frac{1}{2g_{\gamma }}{S}_2^2+\frac{{{\tilde{\mathit{W}}}_2}^T\mathit{\Gamma }{2}^{-1}{\tilde{\mathit{W}}}_2}{2}$$

(42)

$$W_Q=-\frac{1}{2g_Q}{S}_3^2+\frac{{{\tilde{\mathit{W}}}_3}^T\mathit{\Gamma }{3}^{-1}{\tilde{\mathit{W}}}_3}{2}$$

(43)

Differentiating equation (42) along equation (33) and (40), and equation (43) along equations (39) and (41) leads to

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{W}}_{\gamma }=\frac{1}{g_{\gamma }}{S}_2{\stackrel{·}{S}}_2-\frac{{\stackrel{·}{g}}_{\gamma }}{2g_{\gamma }^2}{S}_2^2+{{\tilde{\mathit{W}}}_2}^T\mathit{\Gamma }{2}^{-1}{\stackrel{·}{\hat{\mathit{W}}}}_2}\\ =-(k_{\gamma }+\frac{{\stackrel{·}{g}}_{\gamma }}{2g_{\gamma }^2})S_2^2+S_2{e}_3-{\mu }_2{S}_2-{\sigma }_2{{\tilde{\mathit{W}}}_2}^T{\hat{\mathit{W}}}_2\hfill \end{array}$$

(44)

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{W}}_Q=-\frac{1}{g_Q}{S}_3{\stackrel{·}{S}}_3+\frac{{\stackrel{·}{g}}_Q}{2g_Q^2}{S}_3^2+{{\tilde{\mathit{W}}}_3}^T\mathit{\Gamma }{3}^{-1}{\stackrel{·}{\hat{\mathit{W}}}}_3}\\ =-(k_Q-\frac{{\stackrel{·}{g}}_Q}{2g_Q^2})S_3^2+{\mu }_3{S}_3-{\sigma }_3{{\tilde{\mathit{W}}}_3}^T{\hat{\mathit{W}}}_3\hfill \end{array}$$

(45)

Noting that

$$\begin{array}{c}2{{\tilde{\mathit{W}}}_2}^T{\hat{\mathit{W}}}_2\ge \Vert {\tilde{\mathit{W}}}_2{\Vert }^2-\Vert {{\mathit{W}}_2}^{*}{\Vert }^2,\hfill \\ 2{{\tilde{\mathit{W}}}_3}^T{\hat{\mathit{W}}}_3\ge \Vert {\tilde{\mathit{W}}}_3{\Vert }^2-\Vert {{\mathit{W}}_3}^{*}{\Vert }^2\hfill \end{array}$$

(46)

$$\begin{array}{c}{S}_2{e}_3-{\mu }_2{S}_2\le \left|(e_3-{\mu }_2)S_2\right|\le \frac{{({\mu }_{2M}+\left|e_3\right|)}^2{S}_2^2}{2}+\frac{1}{2},\hfill \\ {\mu }_3{S}_3\le \left|{\mu }_{3M}{S}_3\right|\le \frac{{\left({\mu }_{3M}{S}_3\right)}^2}{2}+\frac{1}{2}\hfill \end{array}$$

(47)

the following inequalities hold

$$\begin{array}{c}\multicolumn{1}{c}{{\stackrel{·}{W}}_{\gamma }\le -(k_{\gamma }+\frac{{\stackrel{·}{g}}_{\gamma }}{2g_{\gamma }^2}-\frac{{({\mu }_{2M}+\left|e_3\right|)}^2}{2})S_2^2}\\ +\frac{1}{2}{\sigma }_2\Vert {{\mathit{W}}_2}^{*}{\Vert }^2+\frac{1}{2}\hfill \end{array}$$

(48)

$${\stackrel{·}{W}}_Q\le -(k_Q-\frac{{\stackrel{·}{g}}_Q}{2g_Q^2}-\frac{{\mu }_{3M}^2}{2})S_3^2+\frac{{\sigma }_3}{2}\Vert {{\mathit{W}}_3}^{*}{\Vert }^2+\frac{1}{2}$$

(49)

Let $k_Q\ge g_{\mathrm{QM}}/(2{\overline{g}}_Q^2)+{\mu }_{3M}^2/2$ . Then, the tracking error function S₃ can be invariant to the following set

$${\mathrm{\Omega }}_{S_3}=\left\{S_3|\left|S_3\right|\le \sqrt{\frac{(\frac{{\sigma }_3}{2}{\Vert {{\mathit{W}}_3}^{*}\Vert }^2+\frac{1}{2})}{(k_Q-\frac{{\stackrel{·}{g}}_Q}{2g_Q^2}-\frac{{\mu }_{3M}^2}{2})}}\right\}$$

(50)

The radius of ${\mathrm{\Omega }}_{S_3}$ can be made arbitrarily small by choosing sufficiently large $k_Q$ and sufficiently small ${\sigma }_3$ . Considering that the polynomial $s^2+2{\lambda }_{3}s+{\lambda }_3^2$ is Hurwitz since λ₃ > 0, the tracking error e₃ is also bounded. Thus, there exists a positive constant ${\overline{e}}_3$ such that ${\overline{e}}_3\ge |e_3|$ .

Let $k_{\gamma }\ge g_{\gamma M}/(2{\underset{¯}{g}}_{\gamma }^2)+({\mu }_{2M}+|{\overline{e}}_3|{)}^2/2$ . Hence, the tracking error function S₂ can be invariant to the following set

$${\mathrm{\Omega }}_{S_2}=\left\{S_2|\left|S_2\right|\le \sqrt{\frac{(\frac{1}{2}{\sigma }_2{\Vert {{\mathit{W}}_2}^{*}\Vert }^2+\frac{1}{2})}{(k_{\gamma }+\frac{{\stackrel{·}{g}}_{\gamma }}{2g_{\gamma }^2}-\frac{{({\mu }_{2M}+\left|e_3\right|)}^2}{2})}}\right\}$$

(51)

By choosing sufficiently large $k_{\gamma }$ and sufficiently small ${\sigma }_2$ , the radius of ${\mathrm{\Omega }}_{S_2}$ can be made arbitrarily small. Noting that the polynomial s + λ₂ is Hurwitz since λ₂ > 0, the tracking error e₂ is also bounded.

According to the analysis results of sections “Adaptive neural controller design for the velocity subsystem” and “Adaptive neural controller design for the altitude subsystem,” it is easy to get the following theorem.

Theorem 1

Consider the closed-loop system consisting of plants (10), (12), and (13) with Assumptions 1 and 2, controllers (21), (32), and (38), and adaptive laws (22), (33), and (39). Then, the tracking errors e₁, e₂, and e₃ are bounded. By choosing sufficiently large $k_V$ , $k_{\gamma }$ , $k_Q$ and sufficiently small ${\sigma }_1$ , ${\sigma }_2$ , ${\sigma }_3$ , then $|e_1|$ , $|e_2|$ , and $|e_3|$ can be arbitrarily small. That is, V→V_ref, θ→θ_d, and γ→γ_d, further, h→h_ref.

Remark 3

According to LaSalle-Yoshizawa Theorem together with the inequalities (26), (48), and (49), it is clear that all the signals in the Lyapunov functions (24), (42), and (43) are uniformly ultimately bounded. Thus, the possible control singularity problem is avoided, since the tracking error functions S _i and NN weights W _i are bounded, where i = 1, 2, 3.

Remark 4

We should select $k_V$ , $k_{\gamma }$ , and $k_Q$ such that $k_V\ge g_{\mathrm{VM}}/(2{\underset{¯}{g}}_V^2)+{\mu }_{1M}^2/2$ , $k_{\gamma }\ge g_{\gamma M}/(2{\underset{¯}{g}}_{\gamma }^2)+({\mu }_{2M}+|{\overline{e}}_3|{)}^2/2$ , $k_Q\ge g_{\mathrm{QM}}/(2{\overline{g}}_Q^2)+{\mu }_{3M}^2/2$ . In the simulation study, we can chose adequately large $k_V$ , $k_{\gamma }$ , and $k_Q$ . For other design parameters, their values should be suitably chosen according to the control performance.

Simulation study

The adaptive neural controllers (21), (32), and (38) with adaptive laws (22), (33), and (39) are tested in MATLAB/Simulink^® in the case of a climbing maneuver from the initial trim conditions, provided in Table 1, to the final trim conditions h = 90,000 ft and V = 8401 ft/s. The reference trajectories of velocity and altitude are generated via the following filters

$$\frac{V_{\mathrm{ref}}\left(s\right)}{V_c\left(s\right)}=\frac{{0.03}^2}{s^2+2\times 0.95\times 0.03s+{0.03}^2}$$

(52)

$$\frac{h_{\mathrm{ref}}\left(s\right)}{h_c\left(s\right)}=\frac{{0.03}^2}{s^2+2\times 0.95\times 0.03\times s+{0.03}^2}$$

(53)

Table 1.

Initial trim conditions.

Item	Value	Units
V	7650	ft/s
h	86,000	ft
γ	0	deg
α	1.7840	deg
Q	0	°/s
${\eta }_1$	1.0180	ft
${\stackrel{·}{\eta }}_1$	0	ft/s
${\eta }_2$	0.8349	ft
${\stackrel{·}{\eta }}_2$	0	ft/s

The inputs of RBF NNs are selected as x ₁ = [V, α, S₁, ${\stackrel{·}{V}}_{\mathrm{ref}}$ , q]^T, x ₂ = [V, α, S₂, q]^T, and x ₃ = [V, α, S₃, q]^T with their centers a ₁, a ₂, and a ₃ being evenly spaced in [7650, 8401; 0.0175, 0.0524; −1, 1; 0, 9; 1882, 1887], [7650, 8401; 0.0175, 0.0524; –1 × 10⁻³,2 × 10⁻⁴; 1882, 1887], [7650, 8401; 0.0175, 0.0524; –2 × 10⁻³,1.2×10⁻²; 1882, 1887]. The widths of the Gaussian functions are chosen as b ₁ = [50.64, 0.0023, 0.13, 0.1333, 0.33], b ₂ = [50.64, 0.0023, 0.0001, 0.33], b ₃ = [50.64, 0.0023, 0.0009, 0.33]. The number of NN nodes is chosen as 13. The initial values of RBF NN weights are set to zero. The controller parameters are chosen as follows: λ₁ = 0.1, λ₂ = 0.2, λ₃ = 2, k_V = 1, k_γ = 0.5, k_α = 0.5, $\mathit{\Gamma }1$ =diag{5}, σ₁ = 5 × 10⁻⁴, $\mathit{\Gamma }2$  = diag{50}, σ₂ = 0.15, $\mathit{\Gamma }3$  = diag{2000}, σ₃ = 0.05, k = 1.5, and k_I = 0.5. In the simulation, all the model coefficients in equations (1)–(7) are assumed to be uncertain by defining $C=C_0[1+0.3\sin (0.02\pi t)]$ , where C denotes the value of uncertain coefficient, and C₀ is the normal value of C. Moreover, external disturbances d_V =2.5sin(0.02πt) ft/s², d_Q = 0.5 sin (0.02πt) °/s² are added into the vehicle model in the simulation study.

To show the superiority, the proposed controller is compared with a traditional back-stepping controller presented by Bu et al. ³³ The simulation results are presented in Figures 1–12. Figures 1–4 show that the tracking errors e₁, e₂, e₃, and $\tilde{h}$ are bounded and can be regulated to zero timely. Compared with the control strategy of Bu et al., ³³ the proposed controller can provide higher tracking accuracy. The flight-path angle, angle of attack, pitch angle, pitch rate, the control inputs, and the flexible states achieved by the proposed control method, just as shown in Figures 3–9, are smoother than the ones provided by the control scheme addressed by Bu et al., ³³ Figures 10–12 show the boundness of $||{\hat{\mathit{W}}}_1||$ , $||{\hat{\mathit{W}}}_2||$ , and $||{\hat{\mathit{W}}}_3||$ .

Figure 1.

Velocity tracking performance.

Figure 2.

Altitude tracking performance.

Figure 3.

Flight-path angle tracking performance.

Figure 4.

Pitch angle tracking performance.

Figure 5.

Fuel equivalence ratio.

Figure 6.

Elevator angular deflection.

Figure 7.

Angle of attack.

Figure 8.

Pitch rate.

Figure 9.

${\eta }_1$ and ${\eta }_2$ .

Figure 10.

$\Vert {\hat{\mathit{W}}}_1\Vert$ .

Figure 11.

$\Vert {\hat{\mathit{W}}}_2\Vert$ .

Figure 12.

$\Vert {\hat{\mathit{W}}}_3\Vert$ .

Conclusion

The problem of adaptive neural back-stepping control design for FAHV is studied in this article. Different from the model adopted in the existing studies, a dynamic model including the flexible effects and completely unknown functions is considered in this study. The RBF NN with quite simple adaptive law is employed to approach the lumped system uncertainty, which avoids complex computation. The stability analysis demonstrates the boundness of the tracking errors. Simulation results show that a satisfying velocity and altitude tracking performance can be achieved by the developed controllers. Dealing with the problem of control input constraints is our further work. The future research is focused on prescribed performance-based control.^34,35