Prescribed performance dynamic neural network control for a flexible hypersonic vehicle with unknown control directions

Abstract

In this article, we examine the system control design of a flexible hypersonic vehicle with an unknown direction control. Prescribed performance controls, backstepping controls, and radial basis function neural network are used to design the controllers. Different from a traditional radial basis function neural network, the fully tuned dynamic radial basis function neural network has a better approximation ability; and the weight vector, respective centers, and width of the Gaussian function of neural network are regulated by adaptive laws designed in the controller. The proof and analysis of stability are taken in this article for the fully tuned dynamic neural network introduced to control the system. Furthermore, prescribed performance control can guarantee the tracking errors satisfy the specified conditions. The unknown control direction is solved with the Nussbaum function in the controller. Finally, the simulations demonstrate the effectiveness and corrective measures of the control strategy.

Keywords

Hypersonic vehicle flexible prescribed performance neural network

Introduction

An aircraft that has speeds greater than Mach Five is considered to be a hypersonic vehicle. Right now, around the world, there are many scholars who are involved in studying hypersonic vehicles. Designing the control system is considered to be a paramount objective in the technology of the hypersonic vehicle.

The control system design methods of hypersonic vehicles have been researched by a large number of scholars. For example, the $H_{\infty}$ method is being used to design the controller of a hypersonic vehicle¹ and a linear variable parameter controller has been designed based on structural dynamic methods.² The aforementioned works are based on linear models that could not comprehensively portray the dynamic behaviors of a hypersonic vehicle. An adaptive controller is designed in Yu et al.;³ the feedback controller is designed based on observer technique in Sigthorsson et al.;⁴ and the model following control methods is researched in Groves et al.,⁵ but the uncertainties of the models are not fully considered in these three methods.

Nonlinear robust control method is used in Wang and Stengel,⁶ Parker et al.,⁷ and Fiorentini et al.,⁸ but the flexibility issues are not researched in the papers. The adaptive neural network controller is designed in Xu et al.⁹ and Wallner and Well,¹⁰ but the approximate errors of the neural network are not solved to impact the tracking effect. The fuzzy logic method is used for the altitude control of X-38 in Wu,¹¹ while the uncertainties and external disturbance of a hypersonic vehicle are solved to fulfill the altitude control by the robust controller in Marrison and Stengel.¹² In the study of Song et al.,¹³ the adaptive sliding mode controller is designed based on backstepping to solve the uncertainty in the models, but the external disturbance of the aircraft is not considered. Stabilization control for a hypersonic vehicle with a wave body shape is researched based on the adaptive quadratic form method in Huo et al.¹⁴ Nevertheless, the Taylor linearization model near equilibrium point could not accurately describe the overall dynamic characteristics of the hypersonic vehicle. Although the hypersonic vehicle in Xu et al.¹⁵ with an axisymmetric body shape can be stabilized through the adaptive sliding controller, it cannot solve the mismatch uncertainty. The flexibility issues are considered to be an important factor influencing the stability of aircraft, but are not studied in the above works. Robust adaptive controller in Sigthorsson et al.⁴ and an adaptive neural network controller in Wallner and Well¹⁰ and Cheng et al.¹⁶ are designed for a flexible hypersonic vehicle. The controller in Kuipers et al.¹⁷ is designed based on the linear variable parameter control method. Adami and Fiorentini^17–21 have researched the flexibility issues of hypersonic vehicle with the adaptive control and the nonlinear robust adaptive control methods. While these papers have considered the flexible modes and increased the size of the system, but while doing so, the controller has been more difficult to operate. In the above works, the stability has been researched without considering the transient performance. In Li et al.²² and Bu et al.,²³ the control system is designed to assure stability and transient performance with the prescribed performance control method, but the unknown control direction issues are not discussed in the works. The direction of motion could not be ascertained when the control coefficient sign and direction are unknown. Drastic changes in the aerodynamic and external conditions are the reasons for the perturbation of the model, and this results in the control direction being difficult to obtain accurately. To implement effective control, the problem of the unknown control direction must be solved. Consequently, the study of the control system, with regard to the control direction, for a flexible hypersonic vehicle is very much necessary, which at this time is rare.

Motivated by the above discussions for a longitudinal dynamic system of a hypersonic vehicle, the inner-loop control system and the outer-loop control system are designed, respectively, in this article. Also, in this article, the unknown control direction issue is solved with the Nussbaum function, and the flexible modes are considered. The uncertainties of the hypersonic vehicle system are approximated online with fully tuned dynamic radial basis function (RBF) neural network being introduced in the control system, specifically. The prescribed performance controllers are devised via backstepping.

Problem formulation and preliminaries

Model description

In this article, the dynamic model taken into consideration is developed by Bolender and Doman in Parker et al.⁷ for a flexible hypersonic vehicle. The equations are formulated as follows

\overset{\cdot}{h} = V \sin γ

(1)

\overset{\cdot}{V} = \frac{1}{m} (T \cos α - D) - g \sin γ

(2)

\overset{\cdot}{α} = \frac{1}{mV} (- T \sin α - L) + ω_{z} + \frac{g}{V} \cos γ

(3)

J_{z} {\overset{\cdot}{ω}}_{z} = M + {\tilde{ψ}}_{1} {\overset{\cdot\cdot}{η}}_{1} + {\tilde{ψ}}_{2} {\overset{\cdot\cdot}{η}}_{2}

(4)

k_{1} {\overset{\cdot\cdot}{η}}_{1} = - 2 ζ_{1} ω_{1} {\overset{\cdot}{η}}_{1} - ω_{1}^{2} η_{1} + N_{1} - {\tilde{ψ}}_{1} \frac{M}{J_{z}} - \frac{{\tilde{ψ}}_{1} {\tilde{ψ}}_{2} {\overset{\cdot\cdot}{η}}_{2}}{J_{z}}

(5)

k_{2} {\overset{\cdot\cdot}{η}}_{2} = - 2 ζ_{2} ω_{2} {\overset{\cdot}{η}}_{2} - ω_{2}^{2} η_{2} + N_{2} - {\tilde{ψ}}_{2} \frac{M}{J_{z}} - \frac{{\tilde{ψ}}_{1} {\tilde{ψ}}_{2} {\overset{\cdot\cdot}{η}}_{2}}{J_{z}}

(6)

where $L$ , $D$ , $M$ , $T$ , $N_{1}$ , and $N_{2}$ are defined as

L \approx \frac{1}{2} ρ V^{2} S C_{L} (α, δ_{e}), D \approx \frac{1}{2} ρ V^{2} S C_{D} (α, δ_{e})

(7)

M \approx z_{T} T + \frac{1}{2} ρ V^{2} S \bar{c} [C_{M, α} (α) + C_{M, δ_{e}} (δ_{e})]

(8)

T \approx C_{T}^{α^{3}} α^{3} + C_{T}^{α^{2}} α^{2} + C_{T}^{α} α + C_{T}^{0}

(9)

\begin{matrix} N_{1} & \approx N_{1}^{α^{2}} α^{2} + N_{1}^{α} α + N_{1}^{0}, N_{2} \approx N_{2}^{α^{2}} α^{2} \\ + N_{2}^{α} α + N_{2}^{δ_{e}} δ_{e} + N_{2}^{0} \end{matrix}

(10)

with

C_{L} = C_{L}^{α} α + C_{L}^{δ_{e}} δ_{e} + C_{L}^{0}

(11)

C_{D} = C_{D}^{α^{2}} α^{2} + C_{D}^{α} α + C_{D}^{δ_{e}^{2}} δ_{e}^{2} + C^{δ_{e}} δ_{e} + C_{D}^{0}

(12)

C_{M, α} (α) = C_{M, α}^{α^{2}} α^{2} + C_{M, α}^{α} α + C_{M, α}^{0}, C_{M, δ_{e}} (δ_{e}) = c_{e} δ_{e}

(13)

C_{T}^{α^{3}} = β_{1} (h, \bar{q}) Φ + β_{2} (h, \bar{q}), C_{T}^{α^{2}} = β_{3} (h, \bar{q}) Φ + β_{4} (h, \bar{q})

(14)

C_{T}^{α} = β_{5} (h, \bar{q}) Φ + β_{6} (h, \bar{q}), C_{T}^{0} = β_{7} (h, \bar{q}) Φ + β_{8} (h, \bar{q})

(15)

Assumption 1

$T \cdot \sin α$ is far less than $L$ in equation (3); the effect on $α$ from $T \sin α$ can be ignored, so $T \cdot \sin α \approx 0$ can be gained.

Assumption 2

In this article, it is assumed that the hypersonic vehicle keeps cruising, and the thrust only comes from a scramjet engine, and thrust vector control is not in the research contents. The effect on $M$ from thrust can be ignored in equation (4), so $z_{T} T \approx 0$ can be gained in equation (8).

Assumption 3

The effect on aerodynamic force from the rudder is ignored.

Remark 1

The above assumptions are logical in the case of approximation through the analysis of Groves et al.,⁵ Parker et al.,⁷ Cheng et al.,¹⁶ and Ge and Huang.²⁴

The model of hypersonic vehicle can also be described as

\overset{\cdot}{V} = f_{V} + a_{Φ} Φ

(16)

\overset{\cdot}{α} = f_{α} + ω_{z}

(17)

{\overset{\cdot}{ω}}_{z} = f_{ω} + a_{δ} δ_{e}

(18)

k_{1} {\overset{\cdot\cdot}{η}}_{1} = - 2 ζ_{1} ω_{1} {\overset{\cdot}{η}}_{1} - ω_{1}^{2} η_{1} + N_{1} - {\tilde{ψ}}_{1} \frac{M}{J_{z}} - \frac{{\tilde{ψ}}_{1} {\tilde{ψ}}_{2} {\overset{\cdot\cdot}{η}}_{2}}{J_{z}}

(19)

k_{2} {\overset{\cdot\cdot}{η}}_{2} = - 2 ζ_{2} ω_{2} {\overset{\cdot}{η}}_{2} - ω_{2}^{2} η_{2} + N_{2} - {\tilde{ψ}}_{2} \frac{M}{J_{z}} - \frac{{\tilde{ψ}}_{1} {\tilde{ψ}}_{2} {\overset{\cdot\cdot}{η}}_{2}}{J_{z}}

(20)

where $f_{V}$ , $f_{α}$ , $f_{ω}$ , $a_{δ}$ , and $a_{Φ}$ include the uncertainties form of the drastic changes of the aerodynamic parameters, external disturbances, and the flexible deformation. The term $f_{ω}$ include the terms ${\overset{\cdot\cdot}{η}}_{1}$ and ${\overset{\cdot\cdot}{η}}_{2}$ . Needing to be emphasized is that $f_{V}$ , $f_{α}$ , $f_{ω}$ , $a_{δ}$ , and $a_{Φ}$ are considered to be continuous smooth functions.

Prescribed performance control

Control objectives are described as follows:

Actual states $α$ , $ω_{z}$ can track the desired command signals $α_{d}$ , $ω_{zd}$ , and all the signals of the closed-loop system are bounded.

The steady and transient performance of the output error $e (t) = α - α_{d}$ can be satisfied.

Assumption 4

All order derivatives of the desired command signals are known and are continuously bounded.

Assumption 5

Initial error $| e_{1} (0) | = | α (0) - α_{d} (0) |$ is bounded, and the upper bound is known as a constant.

Definition 1

A smooth function $λ (t)$ : $R_{+} \to R_{+} 7$ will be called a performance function if it satisfies the following:

$λ (t)$ is positive and decreasing.

$lim_{t \to \infty} λ (t) = λ_{\infty} > 0$ .

To achieve the second control target, the error $e (t)$ evolves strictly within the prescribed performance boundary (PPB) as follows

- κ λ (t) < e (t) < λ (t), e (0) > 0

(21)

- λ (t) < e (t) < κ λ (t), e (0) < 0

(22)

with $t \in [0, \infty), κ \in [0, 1]$ .

According to Definition 1, a feasible performance function that satisfies all the previous properties is formulated as

λ (t) = (λ_{0} - λ_{\infty}) e^{- ξ t} + λ_{\infty}

(23)

where $λ_{0}$ , $λ_{\infty}$ , and $ξ > 0$ are appropriately chosen positive constants, and $λ_{\infty}$ is prescribed to be the upper range value of steady error. The maximum overshoot of error $e (t)$ should be no greater than $κ λ_{0}$ . The lower bound of convergence rate of $e (t)$ is consistent with the decay rate of $λ (t)$ . According to above description, feasible choices for performance $λ (t)$ and constant $κ$ can guarantee that the steady and transient performance of the track error evolves strictly within PPB. The behavioral bound imposed by equation (16), on the error $e (t)$ , is shown in Figure 1.

Figure 1.

The PPB on the error $e (t)$ : (a) the situation of equation (21) and (b) the situation of equation (22).

Realizing the fact that it is a difficult task to design the controller with equations (21) and (22), we will introduce the error transformation function to transform the original unequal “constrained” systems (21) and (22) into an equivalent one.

The error transformation function is defined as follows

e (t) = λ (t) G (ε)

(24)

where $ε$ is the transformed error and $G (ε)$ denotes the error transformation function and it satisfies the following:

Error transformation function $G (ε)$ is smooth and strictly increasing.

$G (ε)$ should be within the inequality “constrained” described as follows

{\begin{matrix} - κ < G (ε) < 1, e (0) > 0 \\ - 1 < G (ε) < κ, e (0) < 0 \end{matrix}

$G (ε)$ should be within the equality “constrained” described as follows

{\begin{matrix} lim_{ε \to - \infty} G (ε) = - κ, \\ lim_{ε \to + \infty} G (ε) = 1, \end{matrix} e (0) > 0

(25)

{\begin{matrix} lim_{ε \to - \infty} G (ε) = - 1, \\ lim_{ε \to + \infty} G (ε) = κ, \end{matrix} e (0) < 0

(26)

According to above definition, we know that $λ (t) > 0$ . If $e (0) > 0$ , then $- κ λ (t) < λ (t) G (ε) < λ (t)$ is tenable. It is known that $- κ λ (t) < e (t) < λ (t)$ can be gained by equation (24). In the same way, if $e (0) < 0$ , then we can get $- λ (t) < e (t) < κ λ (t)$ . If we can get $ε = P [e (t) / λ (t)]$ by means of the inverse transformation of the error transformation function, where $P = G^{- 1}$ . If $ε (t) \in ℓ_{\infty}$ with $t \in [0, \infty)$ , we can get equations (21) and (22). According to the properties of performance function, we know that the track error can be limited to the assemblage $Ω$ described as follows

Ω = {e \in R : | e (t) | \leq λ_{\infty}}

(27)

Remark 2

If $e (0) = 0$ , $κ$ cannot be set as zero, so that $ε (0)$ avoids to converge toward infinity.

Remark 3

According to Assumption 5, we can devise $λ (0) > | e (0) |$ .

Controller design

Inner-loop controller design

Assumption 6

All the states of the system are located in the compact set $Ω \in R^{n}$ .

Assumption 7

If the control coefficient $a_{δ}$ is unknown, ${\bar{a}}_{δ} > 0$ and ${\underline{a}}_{δ} > 0$ are known constants, and ${\overset{⌢}{a}}_{δ}$ is a smooth function, we can get $0 < {\underline{a}}_{δ} \leq | a_{δ} | \leq {\overset{⌢}{a}}_{δ} \leq {\bar{a}}_{δ}$ .

Remark 4

According to Assumption 7, control coefficient $a_{δ}$ is strictly positive or negative.

Assumption 8

The derivative of control coefficient $a_{δ}$ is bounded, and we can get $| {\overset{\cdot}{a}}_{δ} | \leq {\hat{a}}_{δ}$ with ${\hat{a}}_{δ} > 0$ .

Definition 2

If an arbitrary continuous function $N (ς) : R \to R$ possesses the following properties

lim_{ς \to \infty} sup \frac{1}{ς} \int_{0}^{ς} N (ς) d ς = \infty

(28)

lim_{ς \to \infty} inf \frac{1}{ς} \int_{0}^{ς} N (ς) d ς = - \infty

(29)

then it can be called the Nussbaum function, formulated as $N (ς) = ς^{2} \cos (ς)$ in this article.

Lemma 1

$V (\cdot)$ and $ς (\cdot)$ are smooth functions located in the interval $[0, t_{f})$ , and we know $V (t) \geq 0$ , and $N (\cdot)$ is the Nussbaum function if $\forall t \in [0, t_{f})$ .²⁵ Furthermore, $\int_{0}^{t} [θ_{0} N (ς (τ)) + 1] \overset{\cdot}{ς} (τ) d τ$ , $V (t)$ and $ς (t)$ are bounded and located in the interval $[0, t_{f})$ , if we devise appropriate constants $θ_{0} \neq 0$ and $c_{0}$ to gain

V (t) \leq \int_{0}^{t} [θ_{0} N (ς (τ)) + 1] \overset{\cdot}{ς} (τ) d τ + c_{0}, \forall t \in [0, t_{f})

(30)

Lemma 2

$V (\cdot)$ and $ς (\cdot)$ are smooth functions located in the interval $[0, t_{f})$ , and we know $V (t) \geq 0$ , and $N (\cdot)$ is the Nussbaum function if $\forall t \in [0, t_{f})$ .²⁶ Furthermore, $\int_{0}^{t} [θ_{0} N (ς (τ)) + 1] \overset{\cdot}{ς} (τ) d τ$ , $V (t)$ and $ς (t)$ are bounded and located in the interval $[0, t_{f})$ , if we devise appropriate constants $θ_{0} \neq 0$ , $c_{1} > 0$ , and $c_{0}$ to gain

V (t) \leq e^{- c_{1} t} \int_{0}^{t} [θ_{0} N (ς (τ)) + 1] \overset{\cdot}{ς} (τ) e^{c_{1} τ} d τ + c_{0}, \forall t \in [0, t_{f})

(31)

Step 1

Define tracking error as

e_{1} = α - α_{d}

(32)

where $α_{d}$ is the desired commanding signal of attack angle $α$ , and define the error transformation function as

e_{1} = λ_{1} G_{1} (ε_{1})

(33)

with $ε_{1}$ as the transformed error, then the time derivatives of $e_{1}$ and $ε_{1}$ are expressed as

{\overset{\cdot}{e}}_{1} = f_{α} + ω_{z} - {\overset{\cdot}{α}}_{d}

(34)

{\overset{\cdot}{ε}}_{1} = a_{1} (- \frac{{\overset{\cdot}{λ}}_{1}}{λ_{1}} e_{1} + {\overset{\cdot}{e}}_{1}) = a_{1} (- \frac{{\overset{\cdot}{λ}}_{1}}{λ_{1}} e_{1} + f_{α} + ω_{z} - {\overset{\cdot}{α}}_{d})

(35)

with $a_{1} = \frac{1}{λ_{1}} \frac{\partial P_{1}}{\partial (e_{1} / λ_{1})}$ .

Define the second tracking error as

e_{2} = ω_{z} - ω_{zd}

(36)

where $ω_{zd}$ is the desired commanding signal of rate of pitch $ω_{z}$ . The fully tuned dynamic RBF neural network described as follows shows better ability to approximation unknown functions than traditional neural network.

Lemma 3

$Ω \in R^{n}$ is a compact subset; there is an unknown arbitrary continuous function $f : Ω \mapsto R^{r}$ which can be approached by RBF neural network to arbitrary $χ$ , and the relationship between input vector $X$ and output $f$ of RBF neural network can be defined as

f = W^{*} T ϕ^{*} (X) + χ

where $W^{*} \in R^{l \times r}$ is the optimal weight vector, $ϕ^{*} : R^{n} \mapsto R^{l}$ is RBF chosen as the following Gaussian function

ϕ^{*} = {[\exp [\frac{- {‖ X - ξ_{1}^{*} ‖}^{2}}{σ_{1}^{*^{2}}}] \dots \exp [\frac{- {‖ X - ξ_{l}^{*} ‖}^{2}}{σ_{l}^{*^{2}}}]]}^{T}

where $ξ_{i}^{*} (i = 1, 2, \dots, l)$ is the optimal respective centers of RBF function, $σ_{i}^{*} (i = 1, 2, \dots, l)$ is the optimal width of the Gaussian function. $χ$ is called the reconstruction error of the neural network, and we can get $| χ | \leq χ_{α}$ , where $χ_{α}$ is an unknown positive constant called upper bound of $χ$ .

According to the above analysis of Lemma 3, it is easy to get that there exist optimal $W_{1}^{*}$ , $ϕ_{1}^{*}$ such that

f_{α} = W_{1}^{*} T ϕ_{1}^{*} (X_{1}) + χ_{1}

(37)

with

{ϕ_{1}}^{*} = {[\exp [\frac{- {‖ X_{1} - ξ_{11}^{*} ‖}^{2}}{σ_{11}^{*^{2}}}] \dots \exp [\frac{- {‖ X_{1} - ξ_{1 l}^{*} ‖}^{2}}{σ_{1 l}^{*^{2}}}]]}^{T}

(38)

In engineering practice, we cannot get the optimal $W_{1}^{*}$ , $ξ_{1 i}^{*}$ , and $σ_{1 i}^{*}$ , so $W_{1}^{*}$ , $ξ_{1 i}^{*}$ , and $σ_{1 i}^{*}$ can be estimated by ${\hat{W}}_{1}$ , ${\hat{ξ}}_{1 i}$ , and ${\hat{σ}}_{1 i}$ , respectively. The estimating errors can be defined as ${\tilde{W}}_{1} = {\hat{W}}_{1} - W_{1}^{*}$ , ${\tilde{ξ}}_{1 i} = {\hat{ξ}}_{1 i} - ξ_{1 i}^{*}$ , and ${\tilde{σ}}_{1 i} = {\hat{σ}}_{1 i} - σ_{1 i}^{*}$ , respectively, to get the approximation error of RBF neural network, and the approximation error can derived as

\begin{matrix} {\hat{W}}_{1}^{T} {\hat{ϕ}}_{1} - W_{1}^{* T} ϕ_{1}^{*} = {\tilde{W}}_{1}^{T} ({\hat{ϕ}}_{1} - {\hat{ϕ}}_{1}^{ξ'} * {\hat{ξ}}_{1} - {\hat{ϕ}}_{1}^{σ'} * {\hat{σ}}_{1}) \\ + {\hat{W}}_{1}^{T} ({\hat{ϕ}}_{1}^{ξ'} * {\tilde{ξ}}_{1} + {\hat{ϕ}}_{1}^{σ'} * {\tilde{σ}}_{1}) + d_{1 α} \end{matrix}

(39)

with

\begin{matrix} {\hat{ϕ}}_{1}^{ξ'} * {\hat{ξ}}_{1} ≜ {[\begin{matrix} \frac{\partial ({\hat{ϕ}}_{11})}{\partial {\hat{ξ}}_{11}^{T}} {\hat{ξ}}_{11} & \frac{\partial ({\hat{ϕ}}_{12})}{\partial {\hat{ξ}}_{12}^{T}} {\hat{ξ}}_{12} & . . . & \frac{\partial ({\hat{ϕ}}_{1 l})}{\partial {\hat{ξ}}_{1 l}^{T}} {\hat{ξ}}_{1 l} \end{matrix}]}^{T} \\ {\hat{ϕ}}_{1}^{σ'} * {\hat{σ}}_{1} ≜ {[\begin{matrix} \frac{\partial ({\hat{ϕ}}_{11})}{\partial {\hat{σ}}_{11}} {\hat{σ}}_{11} & \frac{\partial ({\hat{ϕ}}_{12})}{\partial {\hat{σ}}_{12}} {\hat{σ}}_{12} & . . . & \frac{\partial ({\hat{ϕ}}_{1 l})}{\partial {\hat{σ}}_{1 l}} {\hat{σ}}_{1 l} \end{matrix}]}^{T} \\ {\hat{ϕ}}_{1}^{ξ'} * {\tilde{ξ}}_{1} ≜ {[\begin{matrix} \frac{\partial ({\hat{ϕ}}_{11})}{\partial {\hat{ξ}}_{11}^{T}} {\tilde{ξ}}_{11} & \frac{\partial ({\hat{ϕ}}_{12})}{\partial {\hat{ξ}}_{12}^{T}} {\tilde{ξ}}_{12} & . . . & \frac{\partial ({\hat{ϕ}}_{1 l})}{\partial {\hat{ξ}}_{1 l}^{T}} {\tilde{ξ}}_{1 l} \end{matrix}]}^{T} \\ {\hat{ϕ}}_{1}^{σ'} * {\tilde{σ}}_{1} ≜ {[\begin{matrix} \frac{\partial ({\hat{ϕ}}_{11})}{\partial {\hat{σ}}_{11}} {\tilde{σ}}_{11} & \frac{\partial ({\hat{ϕ}}_{12})}{\partial {\hat{σ}}_{12}} {\tilde{σ}}_{12} & . . . & \frac{\partial ({\hat{ϕ}}_{1 l})}{\partial {\hat{σ}}_{1 l}} {\tilde{σ}}_{1 l} \end{matrix}]}^{T} \end{matrix}

The $d_{1 α}$ from equation (39) is the residual term of the approximation error, so we have

\begin{matrix} | d_{1 α} | & \leq ‖ {ξ_{1}}^{*} ‖ ‖ {\hat{W}}_{1}^{T} {\hat{ϕ}}_{1}^{ξ} ‖ + ‖ {σ_{1}}^{*} ‖ ‖ {\hat{W}}_{1}^{T} {\hat{ϕ}}_{1}^{σ} ‖ \\ + ‖ {W_{1}}^{* T} ‖ ‖ {\hat{ϕ}}_{1}^{ξ'} * {\hat{ξ}}_{1} ‖ + ‖ {W_{1}}^{* T} ‖ ‖ {\hat{ϕ}}_{1}^{σ'} * {\hat{σ}}_{1} ‖ \\ + \sqrt{l_{α}} ‖ {W_{1}}^{* T} ‖ \end{matrix}

(40)

The virtual control law $ω_{zd}$ is designed as

ω_{zd} = - k_{1} a_{1} ε_{1} - {\hat{W}}_{1}^{T} {\hat{ϕ}}_{1} + {\overset{\cdot}{α}}_{d} + \frac{{\overset{\cdot}{λ}}_{1}}{λ_{1}} e_{1} - υ_{1}

(41)

where $k_{1} > 0$ is a design parameter, and $υ_{1}$ is the robust term designed by equation (59)

The Lyapunov function is devised as

\begin{matrix} V_{1} & = \frac{1}{2} ε_{1}^{2} + \frac{1}{2} tr [{\tilde{W}}_{1}^{T} Γ_{W_{1}}^{- 1} {\tilde{W}}_{1}] \\ + \frac{1}{2} \sum_{i = 1}^{l} ({\tilde{ξ}}_{1 i}^{T} Γ_{1 ξ}^{- 1} {\tilde{ξ}}_{1 i}) + \frac{1}{2} \sum_{i = 1}^{l} Γ_{1 σ}^{- 1} {\tilde{σ}}_{1 i}^{2} \end{matrix}

(42)

where $Γ_{W_{1}} = Γ_{W_{1}}^{T} > 0$ , $Γ_{1 ξ} = Γ_{1 ξ}^{T} > 0$ , and $Γ_{1 σ} > 0$ are design parameters.

The time derivatives of equation (42) are expressed as

\begin{matrix} {\overset{\cdot}{V}}_{1} = ε_{1} {\overset{\cdot}{ε}}_{1} & + tr [{\tilde{W}}_{1}^{T} Γ_{W_{1}}^{- 1} {\overset{\cdot}{\tilde{W}}}_{1}] + \sum_{i = 1}^{l} ({\tilde{ξ}}_{1 i}^{T} Γ_{1 ξ}^{- 1} {\overset{\cdot}{\tilde{ξ}}}_{1 i}) \\ + \sum_{i = 1}^{l} Γ_{1 σ}^{- 1} {\tilde{σ}}_{1 i} {\overset{\cdot}{\tilde{σ}}}_{1 i} \end{matrix}

(43)

Substituting equation (35) into equation (43) leads to

\begin{matrix} {\overset{\cdot}{V}}_{1} & = ε_{1} a_{1} (- \frac{{\overset{\cdot}{λ}}_{1}}{λ_{1}} e_{1} + f_{α} + ω_{z} - {\overset{\cdot}{α}}_{d}) \\ + tr [{\tilde{W}}_{1}^{T} Γ_{W_{1}}^{- 1} {\overset{\cdot}{\tilde{W}}}_{1}] + \sum_{i = 1}^{l} ({\tilde{ξ}}_{1 i}^{T} Γ_{1 ξ}^{- 1} {\overset{\cdot}{\tilde{ξ}}}_{1 i}) \\ + \sum_{i = 1}^{l} Γ_{1 σ}^{- 1} {\tilde{σ}}_{1 i} {\overset{\cdot}{\tilde{σ}}}_{1 i} \end{matrix}

(44)

Assuming $ω_{z} = ω_{zd}$ , and substituting virtual control law in equations (41) and (37) into equation (44) leads to

\begin{matrix} {\overset{\cdot}{V}}_{1} = & - k_{1} (ε_{1} a_{1})^{2} - ε_{1} a_{1} ({\hat{W}}_{1}^{T} {\hat{ϕ}}_{1} - {W_{1}^{*}}^{T} ϕ_{1}^{*}) \\ + ε_{1} a_{1} χ_{1} - ε_{1} a_{1} υ_{1} \\ + tr [{\tilde{W}}_{1}^{T} Γ_{W_{1}}^{- 1} {\overset{\cdot}{\tilde{W}}}_{1}] + \sum_{i = 1}^{l} ({\tilde{ξ}}_{1 i}^{T} Γ_{1 ξ}^{- 1} {\overset{\cdot}{\tilde{ξ}}}_{1 i}) \\ + \sum_{i = 1}^{l} Γ_{1 σ}^{- 1} {\tilde{σ}}_{1 i} {\overset{\cdot}{\tilde{σ}}}_{1 i} \end{matrix}

(45)

According to ${\overset{\cdot}{\tilde{W}}}_{1} = {\overset{\cdot}{\hat{W}}}_{1}$ , ${\overset{\cdot}{\tilde{ξ}}}_{1 i} = {\overset{\cdot}{\hat{ξ}}}_{1 i}$ , and ${\overset{\cdot}{\tilde{σ}}}_{1 i} = {\overset{\cdot}{\hat{σ}}}_{1 i}$ , we obtain

\begin{matrix} {\overset{\cdot}{V}}_{1} = & - k_{1} (a_{1} ε_{1})^{2} - a_{1} ε_{1} ({\hat{W}}_{1}^{T} {\hat{ϕ}}_{1} - W_{1}^{* T} ϕ_{1}^{*}) \\ + a_{1} ε_{1} χ_{1} - a_{1} ε_{1} υ_{1} \\ + tr [{\tilde{W}}_{1}^{T} Γ_{W_{1}}^{- 1} {\overset{\cdot}{\hat{W}}}_{1}] + \sum_{i = 1}^{l} ({\tilde{ξ}}_{1 i}^{T} Γ_{1 ξ}^{- 1} {\overset{\cdot}{\hat{ξ}}}_{1 i}) \\ + \sum_{i = 1}^{l} Γ_{1 σ}^{- 1} {\tilde{σ}}_{1 i} {\overset{\cdot}{\hat{σ}}}_{1 i} \end{matrix}

(46)

Substituting equation (39) into equation (46) leads to

\begin{matrix} {\overset{\cdot}{V}}_{1} = & - k_{1} (a_{1} ε_{1})^{2} - a_{1} ε_{1} {\tilde{W}}_{1}^{T} ({\hat{ϕ}}_{1} - {\hat{ϕ}}_{1}^{ξ'} * {\hat{ξ}}_{1} - {\hat{ϕ}}_{1}^{σ'} * {\hat{σ}}_{1}) - a_{1} ε_{1} d_{1 α} \\ - a_{1} ε_{1} {\hat{W}}_{1}^{T} ({\hat{ϕ}}_{1}^{ξ'} * {\tilde{ξ}}_{1} + {\hat{ϕ}}_{1}^{σ'} * {\tilde{σ}}_{1}) + a_{1} ε_{1} χ_{1} - a_{1} ε_{1} υ_{1} \\ + tr [{\tilde{W}}_{1}^{T} Γ_{W_{1}}^{- 1} {\overset{\cdot}{\hat{W}}}_{1}] + \sum_{i = 1}^{l} ({\tilde{ξ}}_{1 i}^{T} Γ_{1 ξ}^{- 1} {\overset{\cdot}{\hat{ξ}}}_{1 i}) + \sum_{i = 1}^{l} Γ_{1 σ}^{- 1} {\tilde{σ}}_{1 i} {\overset{\cdot}{\hat{σ}}}_{1 i} \end{matrix}

(47)

According to the above analysis, the adaptive laws of neural network can be designed as

{\overset{\cdot}{\hat{W}}}_{1} = Γ_{W_{1}} ({\hat{ϕ}}_{1} - {\hat{ϕ}}_{1}^{ξ'} * {\hat{ξ}}_{1} - {\hat{ϕ}}_{1}^{σ'} * {\hat{σ}}_{1}) a_{1} ε_{1} - Γ_{W_{1}} δ_{W_{1}} {\hat{W}}_{1}

(48)

{\overset{\cdot}{\hat{ξ}}}_{1 i} = Γ_{1 ξ} a_{1} ε_{1} {\hat{ϕ}}_{1 i}^{ξ'} {\hat{W}}_{1 i} - Γ_{1 ξ} δ_{1 ξ} {\hat{ξ}}_{1 i}

(49)

{\overset{\cdot}{\hat{σ}}}_{1 i} = Γ_{1 σ} a_{1} ε_{1} {\hat{W}}_{1 i}^{T} {\hat{ϕ}}_{1 i}^{σ'} - Γ_{1 σ} δ_{1 σ} {\hat{σ}}_{1 i}

(50)

where $δ_{W_{1}} > 0$ , $δ_{1 ξ} > 0$ , and $δ_{1 σ} > 0$ are the design parameters; and ${\hat{W}}_{1 i}$ , ${\hat{ϕ}}_{1 i}^{ξ'}$ , and ${\hat{ϕ}}_{1 i}^{σ'}$ are the ith line vectors.

Substituting equations (48)–(50) into equation (47) leads to

\begin{matrix} {\overset{\cdot}{V}}_{1} = & - k_{1} (a_{1} ε_{1})^{2} - a_{1} ε_{1} d_{1 α} + a_{1} ε_{1} χ_{1} - a_{1} ε_{1} υ_{1} \\ - tr [{\tilde{W}}_{1}^{T} δ_{W_{1}} {\hat{W}}_{1}] - \sum_{i = 1}^{l} ({\tilde{ξ}}_{1 i}^{T} δ_{1 ξ} {\hat{ξ}}_{1 i}) - \sum_{i = 1}^{l} δ_{1 σ} {\tilde{σ}}_{1 i} {\hat{σ}}_{1 i} \end{matrix}

(51)

Substituting equation (40) into equation (51) leads to

\begin{matrix} {\overset{\cdot}{V}}_{1} & \leq - k_{1} (a_{1} ε_{1})^{2} - a_{1} ε_{1} υ_{1} + | a_{1} ε_{1} | ‖ {ξ_{1}}^{*} ‖ ‖ {\hat{W}}_{1}^{T} {\hat{ϕ}}_{1}^{ξ} ‖ \\ + | a_{1} ε_{1} | ‖ {σ_{1}}^{*} ‖ ‖ {\hat{W}}_{1}^{T} {\hat{ϕ}}_{1}^{σ} ‖ + | a_{1} ε_{1} | χ_{1 α} \\ + | a_{1} ε_{1} | ‖ W_{1}^{* T} ‖ ‖ {\hat{ϕ}}_{1}^{ξ'} * {\hat{ξ}}_{1} ‖ + | a_{1} ε_{1} | ‖ W_{1}^{* T} ‖ ‖ {\hat{ϕ}}_{1}^{σ'} * {\hat{σ}}_{1} ‖ \\ + | a_{1} ε_{1} | \sqrt{l_{α}} ‖ W_{1}^{* T} ‖ \\ - tr [{\tilde{W}}_{1}^{T} δ_{W_{1}} {\hat{W}}_{1}] - \sum_{i = 1}^{l} ({\tilde{ξ}}_{1 i}^{T} δ_{1 ξ} {\hat{ξ}}_{1 i}) - \sum_{i = 1}^{l} δ_{1 σ} {\tilde{σ}}_{1 i} {\hat{σ}}_{1 i} \end{matrix}

(52)

According to the Cauchy–Schwarz inequalities, we get

| a_{1} ε_{1} | ‖ ξ_{1}^{*} ‖ ‖ {\hat{W}}_{1}^{T} {\hat{ϕ}}_{1}^{ξ} ‖ \leq {(a_{1} ε_{1})}^{2} {‖ {\hat{W}}_{1}^{T} {\hat{ϕ}}_{1}^{ξ} ‖}^{2} / ϖ_{1} + \frac{ϖ_{1}}{4} {‖ ξ_{1}^{*} ‖}^{2}

(53)

| a_{1} ε_{1} | ‖ σ_{1}^{*} ‖ ‖ {\hat{W}}_{1}^{T} {\hat{ϕ}}_{1}^{σ} ‖ \leq {(a_{1} ε_{1})}^{2} {‖ {\hat{W}}_{1}^{T} {\hat{ϕ}}_{1}^{σ} ‖}^{2} / ϖ_{1} + \frac{ϖ_{1}}{4} {‖ σ_{1}^{*} ‖}^{2}

(54)

| a_{1} ε_{1} | ‖ W_{1}^{* T} ‖ ‖ {\hat{ϕ}}_{1}^{ξ'} * {\hat{ξ}}_{1} ‖ \leq {(a_{1} ε_{1})}^{2} {‖ {\hat{ϕ}}_{1}^{ξ'} * {\hat{ξ}}_{1} ‖}^{2} / ϖ_{1} + \frac{ϖ_{1}}{4} {‖ W_{1}^{* T} ‖}^{2}

(55)

| a_{1} ε_{1} | ‖ W_{1}^{* T} ‖ ‖ {\hat{ϕ}}_{1}^{σ'} * {\hat{σ}}_{1} ‖ \leq {(a_{1} ε_{1})}^{2} {‖ {\hat{ϕ}}_{1}^{σ'} * {\hat{σ}}_{1} ‖}^{2} / ϖ_{1} + \frac{ϖ_{1}}{4} {‖ W_{1}^{* T} ‖}^{2}

(56)

| a_{1} ε_{1} | (\sqrt{l_{α}} ‖ W_{1}^{* T} ‖ + χ_{1 α}) \leq 2 {(a_{1} ε_{1})}^{2} / ϖ_{1} + \frac{l_{α} ϖ_{1}}{4} {‖ W_{1}^{* T} ‖}^{2} + \frac{ϖ_{1}}{4} χ_{1 α}^{2}

(57)

where $ϖ_{1} > 0$ is the design parameter; substituting equations (53)–(57) into equation (52) leads to

\begin{matrix} {\overset{\cdot}{V}}_{1} & \leq - k_{1} {(a_{1} ε_{1})}^{2} - a_{1} ε_{1} υ_{1} + {(a_{1} ε_{1})}^{2} {‖ {\hat{W}}_{1}^{T} {\hat{ϕ}}_{1}^{ξ} ‖}^{2} / ϖ_{1} + \frac{ϖ_{1}}{4} {‖ ξ_{1}^{*} ‖}^{2} + \frac{ϖ_{1}}{4} χ_{1 α}^{2} \\ + {(a_{1} ε_{1})}^{2} {‖ {\hat{W}}_{1}^{T} {\hat{ϕ}}_{1}^{σ} ‖}^{2} / ϖ_{1} + \frac{ϖ_{1}}{4} {‖ σ_{1}^{*} ‖}^{2} + {(a_{1} ε_{1})}^{2} {‖ {\hat{ϕ}}_{1}^{ξ'} * {\hat{ξ}}_{1} ‖}^{2} / ϖ_{1} + \frac{ϖ_{1}}{4} {‖ W_{1}^{* T} ‖}^{2} \\ + {(a_{1} ε_{1})}^{2} {‖ {\hat{ϕ}}_{1}^{σ'} * {\hat{σ}}_{1} ‖}^{2} / ϖ_{1} + \frac{ϖ_{1}}{4} {‖ W_{1}^{* T} ‖}^{2} + 2 {(a_{1} ε_{1})}^{2} / ϖ_{1} + \frac{l_{α} ϖ_{1}}{4} {‖ W_{1}^{* T} ‖}^{2} \\ - tr [{\tilde{W}}_{1}^{T} δ_{W_{1}} {\hat{W}}_{1}] - \sum_{i = 1}^{l} ({\tilde{ξ}}_{1 i}^{T} δ_{1 ξ} {\hat{ξ}}_{1 i}) - \sum_{i = 1}^{l} δ_{1 σ} {\tilde{σ}}_{1 i} {\hat{σ}}_{1 i} \end{matrix}

(58)

The robust term can be designed as

υ_{1} = a_{1} ε_{1} ({‖ {\hat{W}}_{1}^{T} {\hat{ϕ}}_{1}^{ξ} ‖}^{2} + {‖ {\hat{W}}_{1}^{T} {\hat{ϕ}}_{1}^{σ} ‖}^{2} + {‖ {\hat{ϕ}}_{1}^{ξ'} * {\hat{ξ}}_{1} ‖}^{2} + {‖ {\hat{ϕ}}_{1}^{σ'} * {\hat{σ}}_{1} ‖}^{2} + 2) / ϖ_{1}

(59)

Substituting equation (59) into equation (58) leads to

\begin{matrix} {\overset{\cdot}{V}}_{1} & \leq - k_{1} (a_{1} ε_{1})^{2} + \frac{ϖ_{1}}{4} ‖ ξ_{1}^{*} ‖^{2} + \frac{ϖ_{1}}{4} ‖ σ_{1}^{*} ‖^{2} + \frac{2 ϖ_{1} + l_{α} ϖ_{1}}{4} ‖ W_{1}^{* T} ‖^{2} \\ + \frac{ϖ_{1}}{4} χ_{1 α}^{2} - tr [{\tilde{W}}_{1}^{T} δ_{W_{1}} {\hat{W}}_{1}] - \sum_{i = 1}^{l} ({\tilde{ξ}}_{1 i}^{T} δ_{1 ξ} {\hat{ξ}}_{1 i}) - \sum_{i = 1}^{l} δ_{1 σ} {\tilde{σ}}_{1 i} {\hat{σ}}_{1 i} \end{matrix}

(60)

According to the matrix property $tr {y x^{T}} = x^{T} y$ , we can get $2 tr {{\tilde{a}}^{T} \hat{a}} = ‖ \tilde{a} ‖^{2} + ‖ \hat{a} ‖^{2} - ‖ a^{*} ‖^{2} \geq ‖ \tilde{a} ‖^{2} - ‖ a^{*} ‖^{2}$ ; furthermore, we obtain the following inequalities

- tr ({\tilde{W}}_{1}^{T} δ_{W_{1}} {\hat{W}}_{1}) \leq - \frac{δ_{W_{1}}}{2} ({‖ {\tilde{W}}_{1} ‖}^{2} - {‖ W_{1}^{*} ‖}^{2})

(61)

- \sum_{i = 1}^{l} ({\tilde{ξ}}_{1 i}^{T} δ_{1 ξ} {\hat{ξ}}_{1 i}) \leq - \frac{δ_{1 ξ}}{2} ({‖ {\tilde{ξ}}_{1} ‖}^{2} - {‖ ξ_{1}^{*} ‖}^{2})

(62)

- \sum_{i = 1}^{l} δ_{1 σ} {\tilde{σ}}_{1 i} {\hat{σ}}_{1 i} \leq - \frac{δ_{1 σ}}{2} ({‖ {\tilde{σ}}_{1} ‖}^{2} - {‖ σ_{1}^{*} ‖}^{2})

(63)

Substituting equation (63) into equation (60) leads to

\begin{matrix} {\overset{\cdot}{V}}_{1} & \leq - k_{1} (a_{1} ε_{1})^{2} - \frac{δ_{W_{1}}}{2} ‖ {\tilde{W}}_{1} ‖^{2} - \frac{δ_{1 ξ}}{2} ‖ {\tilde{ξ}}_{1} ‖^{2} \\ - \frac{δ_{1 σ}}{2} ‖ {\tilde{σ}}_{1} ‖^{2} + \frac{ϖ_{1} + 2 δ_{1 ξ}}{4} ‖ ξ_{1}^{*} ‖^{2} \\ + \frac{ϖ_{1} + 2 δ_{1 σ}}{4} ‖ σ_{1}^{*} ‖^{2} + \frac{2 ϖ_{1} + l_{α} ϖ_{1} + 2 δ_{W_{1}}}{4} ‖ W_{1}^{* T} ‖^{2} \\ + \frac{ϖ_{1}}{4} χ_{1 α}^{2} \end{matrix}

(64)

Equation (64) can be written as

\begin{matrix} {\overset{\cdot}{V}}_{1} & \leq - k_{1} (a_{1} ε_{1})^{2} - \frac{δ_{W_{1}}}{2} ‖ {\tilde{W}}_{1} ‖^{2} - \frac{δ_{1 ξ}}{2} ‖ {\tilde{ξ}}_{1} ‖^{2} \\ - \frac{δ_{1 σ}}{2} ‖ {\tilde{σ}}_{1} ‖^{2} + Ψ_{1} \end{matrix}

(65)

with

\begin{matrix} Ψ_{1} & = \frac{ϖ_{1} + 2 δ_{1 ξ}}{4} ‖ ξ_{1}^{*} ‖^{2} + \frac{ϖ_{1} + 2 δ_{1 σ}}{4} ‖ σ_{1}^{*} ‖^{2} \\ + \frac{2 ϖ_{1} + l_{α} ϖ_{1} + 2 δ_{W_{1}}}{4} ‖ W_{1}^{* T} ‖^{2} + \frac{ϖ_{1}}{4} χ_{1 α}^{2} \end{matrix}

(66)

where $Ψ_{1}$ is a bounded constant, and according to equation (65), we get

{\overset{\cdot}{V}}_{1} \leq - γ_{1} V_{1} + Ψ_{1}

(67)

with

γ_{1} = min {2 k_{1} \frac{δ_{W_{1}}}{λ_{max} (Γ_{W_{1}}^{- 1})} \frac{δ_{1 ξ}}{λ_{max} (Γ_{1 ξ}^{- 1})} \frac{δ_{1 σ}}{Γ_{1 σ}^{- 1}}}

(68)

Then, we obtain

V_{1} (t) \leq V_{1} (0) e^{- γ_{1} t} + \frac{Ψ_{1}}{γ_{1}} \leq V_{1} (0) + \frac{Ψ_{1}}{γ_{1}}, \forall t \geq 0

(69)

According to the above analysis, the state error $ε_{1}$ of the system and the estimation errors of parameters are bounded. But there is an error $e_{2}$ between $ω_{z}$ and $ω_{zd}$ in the system, so equation (67) should be written as

{\overset{\cdot}{V}}_{1} \leq - γ_{1} V_{1} + a_{1} ε_{1} e_{2} + Ψ_{1}

(70)

The Nussbaum function must be used to solve the issue that the symbol of control coefficient $a_{δ}$ is unknown in Step 2, leading to $a_{1} ε_{1} e_{2}$ not able to offset by the design of the control law with the backstepping method. A feasible method should be introduced to solve this issue. According to equation (65), we obtain

\begin{matrix} {\overset{\cdot}{V}}_{1} & \leq - k_{1} (a_{1} ε_{1})^{2} - \frac{δ_{W_{1}}}{2} ‖ {\tilde{W}}_{1} ‖^{2} - \frac{δ_{1 ξ}}{2} ‖ {\tilde{ξ}}_{1} ‖^{2} \\ - \frac{δ_{1 σ}}{2} ‖ {\tilde{σ}}_{1} ‖^{2} + Ψ_{1} + a_{1}^{2} e_{2}^{2} - {(\frac{1}{2} ε_{1} - a_{1} e_{2})}^{2} \\ \leq - k_{1} (a_{1} ε_{1})^{2} - \frac{δ_{W_{1}}}{2} ‖ {\tilde{W}}_{1} ‖^{2} - \frac{δ_{1 ξ}}{2} ‖ {\tilde{ξ}}_{1} ‖^{2} \\ - \frac{δ_{1 σ}}{2} ‖ {\tilde{σ}}_{1} ‖^{2} + Ψ_{1} + a_{1}^{2} e_{2}^{2} \end{matrix}

(71)

According to equation (71), it is obvious that errors $ε_{1}$ , ${\tilde{W}}_{1}$ , ${\tilde{ξ}}_{1}$ , and ${\tilde{σ}}_{1}$ can be regulated to be bound if error $e_{2}$ in equation (71) is bounded; therefore, we can design the actual control law to achieve the control targets with $e_{2}$ bounded.

Step 2

Taking the time derivative of $e_{2}$ , we get

{\overset{\cdot}{e}}_{2} = f_{ω} + a_{δ} δ_{e} - {\overset{\cdot}{ω}}_{zd}

(72)

where ${\overset{\cdot}{ω}}_{zd}$ can be expressed as

\frac{d ω_{zd}}{d t} = \frac{d}{d t} (- k_{1} a_{1} ε_{1} - {\hat{W}}_{1}^{T} {\hat{ϕ}}_{1} + {\overset{\cdot}{α}}_{d} + \frac{\overset{\cdot}{λ}}{λ} e_{1} - υ_{1})

(73)

The “computational expansion” issue from the backstepping design is present in equation (73) to bring huge difficulties for the controller design. By the analysis of equation (73), we know that $\frac{d ψ}{d t} = \frac{d}{d t} (- {\hat{W}}_{1}^{T} {\hat{ϕ}}_{1} + {\overset{\cdot}{α}}_{d} - υ_{1})$ can be computed and considered as a part of input vector of neural network. Then equation (72) can also be rewritten as

{\overset{\cdot}{e}}_{2} = {\bar{f}}_{ω} + a_{δ} δ_{e}

(74)

where ${\bar{f}}_{ω} = f_{ω} - {\overset{\cdot}{ω}}_{zd}$ is an unknown smooth function.

Finally, the actual controller $δ_{e}$ is designed as

δ_{e} = μ_{1} N (ς_{1}) [e_{2} + {\hat{W}}_{2}^{T} ϕ_{2}]

(75)

where $μ_{1} > 0$ is the design parameter and $N (ς)$ is the Nussbaum function chosen. ${\hat{W}}_{2}$ and $ϕ_{2}$ are defined in the rest of the content.

Choose the following Lyapunov function candidate

V_{2} = \frac{1}{2 Θ} e_{2}^{2} + \frac{1}{2} {\tilde{W}}_{2}^{T} Γ_{W_{2}}^{- 1} {\tilde{W}}_{2}

(76)

where $Γ_{W_{2}} = Γ_{W_{2}}^{T} > 0$ is the design parameter and $Θ = | a_{δ} |$ can be studied and analyzed. The actual value of $Θ$ is not needed; ${\tilde{W}}_{2}$ is defined in the rest of the content.

Taking the time derivative of equation (76), we have

\begin{matrix} {\overset{\cdot}{V}}_{2} = \frac{1}{Θ} e_{2} {\overset{\cdot}{e}}_{2} + {\tilde{W}}_{2}^{T} Γ_{W_{2}}^{- 1} {\overset{\cdot}{\tilde{W}}}_{2} - \frac{\overset{\cdot}{Θ}}{2 Θ^{2}} e_{2}^{2} \\ = \frac{1}{Θ} e_{2} ({\bar{f}}_{ω} + a_{δ} δ_{e}) + {\tilde{W}}_{2}^{T} Γ_{W_{2}}^{- 1} {\overset{\cdot}{\tilde{W}}}_{2} - \frac{\overset{\cdot}{Θ}}{2 Θ^{2}} e_{2}^{2} \\ = e_{2} (\frac{{\bar{f}}_{ω}}{Θ} + \frac{a_{δ}}{Θ} δ_{e}) + {\tilde{W}}_{2}^{T} Γ_{W_{2}}^{- 1} {\overset{\cdot}{\tilde{W}}}_{2} - \frac{\overset{\cdot}{Θ}}{2 Θ^{2}} e_{2}^{2} \\ = e_{2} (\frac{{\bar{f}}_{ω}}{Θ} + θ_{1} δ_{e}) + {\tilde{W}}_{2}^{T} Γ_{W_{2}}^{- 1} {\overset{\cdot}{\tilde{W}}}_{2} - \frac{\overset{\cdot}{Θ}}{2 Θ^{2}} e_{2}^{2} \end{matrix}

(77)

with $θ_{1} = 1$ or $θ_{1} = - 1$ .

Remark 5

Noting the fact that the symbol of a continuous smooth function $a_{δ}$ is positive or negative, we can get that $Θ$ is smooth, continuous, and differentiable.

The unknown continuous function $\frac{{\bar{f}}_{ω}}{Θ}$ can be approximated online by RBF neural network²⁷ as follows

\frac{{\bar{f}}_{ω}}{Θ} = μ_{1} W_{2}^{* T} ϕ_{2} + χ_{2}

(78)

with

ϕ_{2} = {[\exp [\frac{- {‖ X_{2} - ξ_{21} ‖}^{2}}{σ_{21}^{2}}] \dots \exp [\frac{- {‖ X_{2} - ξ_{2 l} ‖}^{2}}{σ_{2 l}^{2}}]]}^{T}

(79)

where $ξ_{2 i} (i = 1, 2, \dots, l)$ is the optimal respective centers of RBF function, $l$ is the number of hidden layer nodes, and $σ_{2 i} (i = 1, 2, \dots, l)$ is the width of the Gaussian function. The input vector of neural network can be devised as $X_{2} = [α ω_{z} ψ]^{T}$ , $χ_{2}$ is the reconstruction error of the neural network, and the upper bound of $χ_{2}$ is an unknown positive constant $χ_{2 ω}$ ; therefore, we get $| χ_{2} | \leq χ_{2 ω}$ . The estimation value of $W_{2}^{*}$ can be defined as ${\hat{W}}_{2}$ , and the estimated error is defined as ${\tilde{W}}_{2} = {\hat{W}}_{2} - W_{2}^{*}$ .

Substituting equations (75) and (78) into equation (77) leads to

\begin{matrix} {\overset{\cdot}{V}}_{2} & = e_{2} μ_{1} W_{2}^{* T} ϕ_{2}^{*} + e_{2} χ_{2} + e_{2}^{2} θ_{1} μ_{1} N (ς_{1}) \\ + e_{2} θ_{1} μ_{1} N (ς_{1}) {\hat{W}}_{2}^{T} ϕ_{2} + {\tilde{W}}_{2}^{T} Γ_{W_{2}}^{- 1} {\overset{\cdot}{\tilde{W}}}_{2} - \frac{\overset{\cdot}{Θ}}{2 Θ^{2}} e_{2}^{2} \end{matrix}

(80)

Equation (80) can be rewritten as

\begin{matrix} {\overset{\cdot}{V}}_{2} = - μ_{1} e_{2}^{2} + μ_{1} e_{2}^{2} [1 + θ_{1} N (ς_{1})] + e_{2} μ_{1} W_{2}^{* T} ϕ_{2} + e_{2} χ_{2} \\ + e_{2} θ_{1} μ_{1} N (ς_{1}) {\hat{W}}_{2}^{T} ϕ_{2} + {\tilde{W}}_{2}^{T} Γ_{W_{2}}^{- 1} {\overset{\cdot}{\tilde{W}}}_{2} - \frac{\overset{\cdot}{Θ}}{2 Θ^{2}} e_{2}^{2} \\ = - μ_{1} e_{2}^{2} + μ_{1} e_{2}^{2} [1 + θ_{1} N (ς_{1})] + e_{2} μ_{1} (W_{2}^{* T} ϕ_{2} - {\hat{W}}_{2}^{T} ϕ_{2}) \\ + e_{2} μ_{1} {\hat{W}}_{2}^{T} ϕ_{2} [1 + θ_{1} N (ς_{1})] + e_{2} χ_{2} + {\tilde{W}}_{2}^{T} Γ_{W_{2}}^{- 1} {\overset{\cdot}{\tilde{W}}}_{2} - \frac{\overset{\cdot}{Θ}}{2 Θ^{2}} e_{2}^{2} \\ = - μ_{1} e_{2}^{2} + (μ_{1} e_{2}^{2} + e_{2} μ_{1} {\hat{W}}_{2}^{T} ϕ_{2}) [1 + θ_{1} N (ς_{1})] + e_{2} χ_{2} \\ - e_{2} μ_{1} ({\hat{W}}_{2}^{T} ϕ_{2} - W_{2}^{* T} ϕ_{2}) + {\tilde{W}}_{2}^{T} Γ_{W_{2}}^{- 1} {\overset{\cdot}{\hat{W}}}_{2} - \frac{\overset{\cdot}{Θ}}{2 Θ^{2}} e_{2}^{2} \end{matrix}

(81)

The adaptive laws can be designed as

{\overset{\cdot}{ς}}_{1} = μ_{1} (e_{2}^{2} + e_{2} {\hat{W}}_{2}^{T} ϕ_{2})

(82)

{\overset{\cdot}{\hat{W}}}_{2} = μ_{1} e_{2} Γ_{W_{2}} ϕ_{2} - δ_{W_{2}} Γ_{W_{2}} {\hat{W}}_{2}

(83)

Substituting equations (82) and (83) into equation (81) leads to

\begin{matrix} {\overset{\cdot}{V}}_{2} & \leq - μ_{1} e_{2}^{2} + {\overset{\cdot}{ς}}_{1} [1 + θ_{1} N (ς_{1})] + | e_{2} | χ_{2 ω} - {\tilde{W}}_{2}^{T} δ_{W_{2}} {\hat{W}}_{2} - \frac{\overset{\cdot}{Θ}}{2 Θ^{2}} e_{2}^{2} \\ \leq - (μ_{1} + \frac{\overset{\cdot}{Θ}}{2 Θ^{2}}) e_{2}^{2} + {\overset{\cdot}{ς}}_{1} [1 + θ_{1} N (ς_{1})] + | e_{2} | χ_{2 ω} - {\tilde{W}}_{2}^{T} δ_{W_{2}} {\hat{W}}_{2} \end{matrix}

(84)

We know

- {\tilde{W}}_{2}^{T} δ_{W_{2}} {\hat{W}}_{2} \leq - \frac{δ_{W_{2}}}{2} ({‖ {\tilde{W}}_{2} ‖}^{2} - {‖ W_{2}^{*} ‖}^{2})

(85)

Substituting equation (85) into equation (84), we get

\begin{matrix} {\overset{\cdot}{V}}_{2} & \leq - (μ_{1} + \frac{\overset{\cdot}{Θ}}{2 Θ^{2}}) e_{2}^{2} - \frac{δ_{W_{2}}}{2} ‖ {\tilde{W}}_{2} ‖^{2} \\ + {\overset{\cdot}{ς}}_{1} [1 + θ_{1} N (ς_{1})] + | e_{2} | χ_{2 ω} + \frac{δ_{W_{2}}}{2} ‖ W_{2}^{*} ‖^{2} \end{matrix}

(86)

Therefore, equation (86) can be rewritten as

\begin{matrix} {\overset{\cdot}{V}}_{2} & \leq - (μ_{1} + \frac{\overset{\cdot}{Θ}}{2 Θ^{2}} - \frac{1}{4}) e_{2}^{2} - \frac{δ_{W_{2}}}{2} ‖ {\tilde{W}}_{2} ‖^{2} \\ + {\overset{\cdot}{ς}}_{1} [1 + θ_{1} N (ς_{1})] - \frac{1}{4} e_{2}^{2} + | e_{2} | χ_{2 ω} + \frac{δ_{W_{2}}}{2} ‖ W_{2}^{*} ‖^{2} \end{matrix}

(87)

According to the following inequality

- \frac{1}{4} e_{2}^{2} + | e_{2} | χ_{2 ω} \leq χ_{2 ω}^{2}

(88)

We obtain

\begin{matrix} {\overset{\cdot}{V}}_{2} & \leq - (μ_{1} + \frac{\overset{\cdot}{Θ}}{2 Θ^{2}} - \frac{1}{4}) e_{2}^{2} - \frac{δ_{W_{2}}}{2} ‖ {\tilde{W}}_{2} ‖^{2} \\ + {\overset{\cdot}{ς}}_{1} [1 + θ_{1} N (ς_{1})] + \frac{δ_{W_{2}}}{2} ‖ W_{2}^{*} ‖^{2} + χ_{2 ω}^{2} \end{matrix}

(89)

If $μ_{1} > | \frac{\overset{\cdot}{Θ}}{2 Θ^{2}} | + \frac{1}{2}$ , we get

{\overset{\cdot}{V}}_{2} \leq - λ_{1} V_{2} + {\overset{\cdot}{ς}}_{1} [1 + θ_{1} N (ς_{1})] + Σ_{1}

(90)

with

Σ_{1} = \frac{δ_{W_{2}}}{2} ‖ W_{2}^{*} ‖^{2} + χ_{2 ω}^{2}

(91)

λ_{1} = min {2 (μ_{1} + \frac{\overset{\cdot}{Θ}}{2 Θ^{2}} - \frac{1}{4}) / {\bar{a}}_{δ}, \frac{δ_{W_{2}}}{λ_{max} (Γ_{W_{2}}^{- 1})}}

(92)

Multiplying both the ends of equation (90) with $e^{λ_{1} t}$ , we get

\frac{d}{d t} (V_{2} e^{λ_{1} t}) \leq {\overset{\cdot}{ς}}_{1} [1 + θ_{1} N (ς_{1})] e^{λ_{1} t} + Σ_{1} e^{λ_{1} t}

(93)

The integration of equation (93) can be described as

\begin{matrix} V_{2} (t) & \leq ϒ_{1} + [V_{2} (0) - ϒ_{1}] e^{- λ_{1} t} \\ + e^{- λ_{1} t} \int_{0}^{t} {\overset{\cdot}{ς}}_{1} [1 + θ_{1} N (ς_{1})] e^{λ_{1} τ} d τ \\ \leq ϒ_{1} + V_{2} (0) e^{- λ_{1} t} + e^{- λ_{1} t} \int_{0}^{t} {\overset{\cdot}{ς}}_{1} [1 + θ_{1} N (ς_{1})] e^{λ_{1} τ} d τ \end{matrix}

(94)

with $ϒ_{1} = Σ_{1} / λ_{1}$ .

According to Lemma 2, it is easy to get errors $e_{2}$ , ${\tilde{W}}_{2}$ that are bounded if $V_{2} (t)$ , $ς_{1}$ are bounded. Furthermore, we can come to the conclusion that $V_{1} (t)$ , $ε_{1}$ , ${\tilde{W}}_{1}$ , ${\tilde{ξ}}_{1}$ , ${\tilde{σ}}_{1}$ , and all the signals of the closed-loop system are bounded.²⁸

The following inequality can be gained if $\int_{0}^{t} | {\overset{\cdot}{ς}}_{1} [1 + θ_{1} N (ς_{1})] e^{λ_{1} τ} | d τ \leq c_{u}$

\begin{matrix} e^{- λ_{1} t} \int_{0}^{t} {\overset{\cdot}{ς}}_{1} [1 + θ_{1} N (ς_{1})] e^{λ_{1} τ} d τ \\ \leq \int_{0}^{t} | {\overset{\cdot}{ς}}_{1} [1 + θ_{1} N (ς_{1})] | e^{λ_{1} (t - τ)} d τ \\ \leq \int_{0}^{t} | {\overset{\cdot}{ς}}_{1} [1 + θ_{1} N (ς_{1})] | d τ \leq c_{u} \end{matrix}

(95)

According to equation (94), we have

V_{2} (t) \leq (ϒ_{1} + c_{u}) + V_{2} (0) e^{- λ_{1} t} \leq (ϒ_{1} + c_{u}) + V_{2} (0)

(96)

Furthermore, we have

e_{2}^{2} (t) \leq 2 Θ V_{2} (t)

(97)

‖ {\tilde{W}}_{2} ‖ \leq 2 V_{2} (t) / λ_{min} (Γ_{W_{2}}^{- 1})

(98)

According to equations (65) and (90), the state errors $ε_{1}$ , $e_{2}$ of the system and the estimation errors of neural network can converge to a neighborhood within the origin.

Theorem 1

Consider the closed-loop system consisting of plants (17) and (18) with control laws (41) and (75), adaptive laws (48)–(50), and robust term (59). Then, all the signals involved are uniformly ultimately bounded. The output state error $e_{1}$ satisfies the demands of prescribed performance.

Outer-loop controller design

For the system (16), velocity tracking error is defined as

e_{V} = V - V_{d}

(99)

where $V_{d}$ is the reference trajectory, and the error transformation function can be defined as

e_{V} = λ' G' (ε_{V})

(100)

where $ε_{V}$ is the transformed error, and the time derivative of $e_{V}$ , $ε_{V}$ are expressed as

{\overset{\cdot}{z}}_{V} = \overset{\cdot}{V} - {\overset{\cdot}{V}}_{d} = f_{V} + a_{Φ} Φ - {\overset{\cdot}{V}}_{d}

(101)

{\overset{\cdot}{ε}}_{V} = a_{V} (- \frac{\overset{\cdot}{λ}'}{λ'} e_{V} + {\overset{\cdot}{e}}_{V}) = a_{V} (- \frac{\overset{\cdot}{λ}'}{λ'} e_{V} + f_{V} + a_{Φ} Φ - {\overset{\cdot}{V}}_{d})

(102)

with $a_{V} = \frac{1}{λ'} \frac{\partial P'}{\partial (e_{V} / λ')}$ .

Choose the following Lyapunov function candidate

\begin{matrix} V & = \frac{1}{2} ε_{V}^{2} + \frac{1}{2} tr [{\tilde{W}}_{V}^{T} Γ_{W_{V}}^{- 1} {\tilde{W}}_{V}] \\ + \frac{1}{2} \sum_{i = 1}^{l} ({\tilde{ξ}}_{Vi}^{T} Γ_{V ξ}^{- 1} {\tilde{ξ}}_{Vi}) + \frac{1}{2} \sum_{i = 1}^{l} Γ_{V σ}^{- 1} {\tilde{σ}}_{Vi}^{2} \end{matrix}

(103)

Some notations of equation (103) will be explained in the rest of the content.

The time derivative of $V$ is expressed as

\begin{matrix} \overset{\cdot}{V} & = ε_{V} {\overset{\cdot}{ε}}_{V} + tr [{\tilde{W}}_{V}^{T} Γ_{W_{V}}^{- 1} {\overset{\cdot}{\tilde{W}}}_{V}] \\ + \sum_{i = 1}^{l} ({\tilde{ξ}}_{Vi}^{T} Γ_{V ξ}^{- 1} {\overset{\cdot}{\tilde{ξ}}}_{Vi}) + \sum_{i = 1}^{l} Γ_{V σ}^{- 1} {\tilde{σ}}_{Vi} {\overset{\cdot}{\tilde{σ}}}_{Vi} \end{matrix}

(104)

Substituting equation (102) into equation (104) leads to

\begin{matrix} \overset{\cdot}{V} & = ε_{V} a_{V} (- \frac{\overset{\cdot}{λ}'}{λ'} e_{V} + f_{V} + a_{Φ} Φ - {\overset{\cdot}{V}}_{d}) \\ + tr [{\tilde{W}}_{V}^{T} Γ_{W_{V}}^{- 1} {\overset{\cdot}{\tilde{W}}}_{V}] + \sum_{i = 1}^{l} ({\tilde{ξ}}_{Vi}^{T} Γ_{V ξ}^{- 1} {\overset{\cdot}{\tilde{ξ}}}_{Vi}) \\ + \sum_{i = 1}^{l} Γ_{V σ}^{- 1} {\tilde{σ}}_{Vi} {\overset{\cdot}{\tilde{σ}}}_{Vi} \\ = ε_{V} a_{V} (- \frac{\overset{\cdot}{λ}'}{λ'} e_{V} + f_{V} - {\overset{\cdot}{V}}_{d}) + ε_{V} a_{V} a_{Φ} Φ \\ + tr [{\tilde{W}}_{V}^{T} Γ_{W_{V}}^{- 1} {\overset{\cdot}{\tilde{W}}}_{V}] + \sum_{i = 1}^{l} ({\tilde{ξ}}_{Vi}^{T} Γ_{V ξ}^{- 1} {\overset{\cdot}{\tilde{ξ}}}_{Vi}) + \sum_{i = 1}^{l} Γ_{V σ}^{- 1} {\tilde{σ}}_{Vi} {\overset{\cdot}{\tilde{σ}}}_{Vi} \end{matrix}

(105)

Equation (105) can be rewritten as

\begin{matrix} \overset{\cdot}{V} = ε_{V} a_{V} (- \frac{\overset{\cdot}{λ}'}{λ'} e_{V} + f_{V} - {\overset{\cdot}{V}}_{d}) + ε_{V} a_{V} a_{Φ} Φ \\ + tr [{\tilde{W}}_{V}^{T} Γ_{W_{V}}^{- 1} {\overset{\cdot}{\tilde{W}}}_{V}] + \sum_{i = 1}^{l} ({\tilde{ξ}}_{Vi}^{T} Γ_{V ξ}^{- 1} {\overset{\cdot}{\tilde{ξ}}}_{Vi}) + \sum_{i = 1}^{l} Γ_{V σ}^{- 1} {\tilde{σ}}_{Vi} {\overset{\cdot}{\tilde{σ}}}_{Vi} \end{matrix}

(106)

The control law $Φ$ can be designed as

Φ = N (ς_{2}) ψ_{V}

(107)

where $ψ_{V}$ needs to be designed, and the adaptive law of $ς_{2}$ can be designed as

{\overset{\cdot}{ς}}_{2} = ψ_{V} a_{V} ε_{V}

(108)

Substituting equations (107) and (108) into equation (106) leads to

\begin{matrix} \overset{\cdot}{V} = ε_{V} a_{V} (- \frac{\overset{\cdot}{λ}'}{λ'} e_{V} + f_{V} - {\overset{\cdot}{V}}_{d}) - a_{V} ε_{V} ψ_{V} + {\overset{\cdot}{ς}}_{2} [a_{Φ} N (ς_{2}) + 1] \\ + tr [{\tilde{W}}_{V}^{T} Γ_{W_{V}}^{- 1} {\overset{\cdot}{\tilde{W}}}_{V}] + \sum_{i = 1}^{l} ({\tilde{ξ}}_{Vi}^{T} Γ_{V ξ}^{- 1} {\overset{\cdot}{\tilde{ξ}}}_{Vi}) + \sum_{i = 1}^{l} Γ_{V σ}^{- 1} {\tilde{σ}}_{Vi} {\overset{\cdot}{\tilde{σ}}}_{Vi} \end{matrix}

(109)

with

ψ_{V} = \frac{k_{V}}{a_{V}} ε_{V} - {\overset{\cdot}{V}}_{d} - \frac{\overset{\cdot}{λ}'}{λ'} e_{V} - υ_{V} + {\hat{W}}_{V}^{T} {\hat{ϕ}}_{V}

(110)

The uncertainty term $f_{V}$ can be approximated with fully tuned dynamic neural network expressed as

f_{V} = W_{V}^{* T} ϕ_{V}^{*} + χ_{V}

(111)

with

{ϕ_{V}}^{*} = {[\exp [\frac{- {‖ X_{2} - ξ_{V 1}^{*} ‖}^{2}}{σ_{V 1}^{*^{2}}}] \dots \exp [\frac{- {‖ X_{2} - ξ_{Vl}^{*} ‖}^{2}}{σ_{Vl}^{*^{2}}}]]}^{T}

(112)

where $ξ_{Vi}^{*} (i = 1, 2, \dots, l)$ is the optimal respective centers of RBF function, $l$ is the number of hidden layer nodes, and $σ_{Vi}^{*} (i = 1, 2, \dots, l)$ is the width of the Gaussian function. The input vector of the neural network can be devised as $X_{V} = [V α Φ]^{T}$ , $χ_{V}$ is the reconstruction error of the neural network, and the upper bound of $χ_{V}$ is an unknown positive constant $χ_{2 V}$ ; therefore, we get $| χ_{2} | \leq χ_{2 ω}$ . The estimation values of $W_{V}^{*}$ , $ϕ_{V}^{*}$ , and $σ_{V}^{*}$ can be defined as ${\hat{W}}_{2}$ , ${\hat{ϕ}}_{V}$ , and ${\hat{σ}}_{V}$ , respectively, and the estimated errors are defined as ${\tilde{W}}_{V} = {\hat{W}}_{V} - W_{V}^{*}$ , ${\tilde{ϕ}}_{V} = {\hat{ϕ}}_{V} - ϕ_{V}^{*}$ , and ${\tilde{σ}}_{V} = {\hat{σ}}_{V} - σ_{V}^{*}$ , respectively. Then, the approximation error can be expressed as

\begin{matrix} {\hat{W}}_{V}^{T} {\hat{ϕ}}_{V} & - W_{V}^{* T} ϕ_{V}^{*} = {\tilde{W}}_{V}^{T} ({\hat{ϕ}}_{V} - {\hat{ϕ}}_{V}^{ξ'} * {\hat{ξ}}_{V} - {\hat{ϕ}}_{V}^{σ'} * {\hat{σ}}_{V}) \\ + {\hat{W}}_{V}^{T} ({\hat{ϕ}}_{V}^{ξ'} * {\tilde{ξ}}_{V} + {\hat{ϕ}}_{V}^{σ'} * {\tilde{σ}}_{V}) + d_{2 V} \end{matrix}

(113)

where $d_{2 V}$ is the residual term of the approximation error; we then get the following inequality

\begin{matrix} | d_{2 V} | & \leq ‖ {ξ_{V}}^{*} ‖ ‖ {\hat{W}}_{V}^{T} {\hat{ϕ}}_{V}^{ξ} ‖ + ‖ {σ_{V}}^{*} ‖ ‖ {\hat{W}}_{V}^{T} {\hat{ϕ}}_{V}^{σ} ‖ \\ + ‖ {W_{V}}^{* T} ‖ ‖ {\hat{ϕ}}_{V}^{ξ'} * {\hat{ξ}}_{V} ‖ + ‖ {W_{V}}^{* T} ‖ ‖ {\hat{ϕ}}_{V}^{σ'} * {\hat{σ}}_{V} ‖ \\ + \sqrt{l_{V}} ‖ {W_{V}}^{* T} ‖ \end{matrix}

(114)

Substituting equations (110) and (111) into equation (109) leads to

\begin{matrix} \overset{\cdot}{V} = - k_{V} ε_{V}^{2} + ε_{V} a_{V} ({\hat{W}}_{V}^{T} {\hat{ϕ}}_{V} - W_{V}^{* T} ϕ_{V}^{*}) + ε_{V} a_{V} χ_{V} \\ + a_{V} ε_{V} υ_{V} \\ + {\overset{\cdot}{ς}}_{2} [a_{Φ} N (ς_{2}) + 1] + tr [{\tilde{W}}_{V}^{T} Γ_{W_{V}}^{- 1} {\overset{\cdot}{\tilde{W}}}_{V}] \\ + \sum_{i = 1}^{l} ({\tilde{ξ}}_{Vi}^{T} Γ_{V ξ}^{- 1} {\overset{\cdot}{\tilde{ξ}}}_{Vi}) \\ + \sum_{i = 1}^{l} Γ_{V σ}^{- 1} {\tilde{σ}}_{Vi} {\overset{\cdot}{\tilde{σ}}}_{Vi} \end{matrix}

(115)

Substituting equation (113) into equation (115) leads to

\begin{matrix} \overset{\cdot}{V} = & - k_{V} ε_{V}^{2} + ε_{V} a_{V} {\tilde{W}}_{V}^{T} ({\hat{ϕ}}_{V} - {\hat{ϕ}}_{V}^{ξ'} * {\hat{ξ}}_{V} - {\hat{ϕ}}_{V}^{σ'} * {\hat{σ}}_{V}) \\ + ε_{V} a_{V} {\hat{W}}_{V}^{T} ({\hat{ϕ}}_{V}^{ξ'} * {\tilde{ξ}}_{V} + {\hat{ϕ}}_{V}^{σ'} * {\tilde{σ}}_{V}) \\ + ε_{V} a_{V} d_{2 V} + ε_{V} a_{V} χ_{V} + a_{V} ε_{V} υ_{V} + {\overset{\cdot}{ς}}_{2} [a_{Φ} N (ς_{2}) + 1] \\ + tr [{\tilde{W}}_{V}^{T} Γ_{W_{V}}^{- 1} {\overset{\cdot}{\hat{W}}}_{V}] \\ + \sum_{i = 1}^{l} ({\tilde{ξ}}_{Vi}^{T} Γ_{V ξ}^{- 1} {\overset{\cdot}{\hat{ξ}}}_{Vi}) + \sum_{i = 1}^{l} Γ_{V σ}^{- 1} {\tilde{σ}}_{Vi} {\overset{\cdot}{\hat{σ}}}_{Vi} \end{matrix}

(116)

The adaptive laws of neural network are designed as

\begin{matrix} {\overset{\cdot}{\hat{W}}}_{V} & = Γ_{W_{V}} ({\hat{ϕ}}_{V} - {\hat{ϕ}}_{V}^{ξ'} * {\hat{ξ}}_{V} - {\hat{ϕ}}_{V}^{σ'} * {\hat{σ}}_{V}) a_{V} ε_{V} \\ - Γ_{W_{V}} δ_{W_{V}} {\hat{W}}_{V} \end{matrix}

(117)

{\overset{\cdot}{\hat{ξ}}}_{Vi} = Γ_{V ξ} a_{V} ε_{V} {\hat{ϕ}}_{Vi}^{ξ'} {\hat{W}}_{Vi} - Γ_{V ξ} δ_{V ξ} {\hat{ξ}}_{Vi}

(118)

{\overset{\cdot}{\hat{σ}}}_{Vi} = Γ_{V σ} a_{V} ε_{V} {\hat{W}}_{Vi}^{T} {\hat{ϕ}}_{Vi}^{σ'} - Γ_{V σ} δ_{V σ} {\hat{σ}}_{Vi}

(119)

where $δ_{W_{V}} > 0$ , $δ_{V ξ} > 0$ , and $δ_{V σ} > 0$ are the design parameters, and ${\hat{W}}_{Vi}$ , ${\hat{ϕ}}_{Vi}^{ξ'}$ , and ${\hat{ϕ}}_{Vi}^{σ'}$ are ith line vectors.

Substituting equations (117) and (118) into equation (116) leads to

\begin{matrix} \overset{\cdot}{V} = & - k_{V} ε_{V}^{2} + ε_{V} a_{V} d_{2 V} + ε_{V} a_{V} χ_{V} + a_{V} ε_{V} υ_{V} \\ + {\overset{\cdot}{ς}}_{2} [a_{Φ} N (ς_{2}) + 1] \\ - tr [{\tilde{W}}_{V}^{T} δ_{W_{V}} {\hat{W}}_{V}] - \sum_{i = 1}^{l} ({\tilde{ξ}}_{Vi}^{T} δ_{V ξ} {\hat{ξ}}_{Vi}) \\ - \sum_{i = 1}^{l} {\tilde{σ}}_{Vi} δ_{V σ} {\hat{σ}}_{Vi} \end{matrix}

(120)

According to equations (114), equation (120) can be rewritten as

\begin{matrix} \overset{\cdot}{V} & \leq - k_{V} ε_{V}^{2} + | ε_{V} a_{V} | ‖ {ξ_{V}}^{*} ‖ ‖ {\hat{W}}_{V}^{T} {\hat{ϕ}}_{V}^{ξ} ‖ \\ + | ε_{V} a_{V} | ‖ {σ_{V}}^{*} ‖ ‖ {\hat{W}}_{V}^{T} {\hat{ϕ}}_{V}^{σ} ‖ \\ + | ε_{V} a_{V} | ‖ {W_{V}}^{* T} ‖ ‖ {\hat{ϕ}}_{V}^{ξ'} * {\hat{ξ}}_{V} ‖ \\ + | ε_{V} a_{V} | ‖ {W_{V}}^{* T} ‖ ‖ {\hat{ϕ}}_{V}^{σ'} * {\hat{σ}}_{V} ‖ + a_{V} ε_{V} υ_{V} \\ + | ε_{V} a_{V} | \sqrt{l_{V}} ‖ {W_{V}}^{* T} ‖ + | ε_{V} a_{V} | χ_{2 V} \\ + {\overset{\cdot}{ς}}_{2} [a_{Φ} N (ς_{2}) + 1] \\ - tr [{\tilde{W}}_{V}^{T} δ_{W_{V}} {\hat{W}}_{V}] - \sum_{i = 1}^{l} ({\tilde{ξ}}_{Vi}^{T} δ_{V ξ} {\hat{ξ}}_{Vi}) - \sum_{i = 1}^{l} {\tilde{σ}}_{Vi} δ_{V σ} {\hat{σ}}_{Vi} \end{matrix}

(121)

Owing to Cauchy–Schwarz inequality, we get

| a_{V} ε_{V} | ‖ ξ_{V}^{*} ‖ ‖ {\hat{W}}_{V}^{T} {\hat{ϕ}}_{V}^{ξ} ‖ \leq {(a_{V} ε_{V})}^{2} {‖ {\hat{W}}_{V}^{T} {\hat{ϕ}}_{V}^{ξ} ‖}^{2} / ϖ_{V} + \frac{ϖ_{V}}{4} {‖ ξ_{V}^{*} ‖}^{2}

(122)

| a_{V} ε_{V} | ‖ σ_{V}^{*} ‖ ‖ {\hat{W}}_{V}^{T} {\hat{ϕ}}_{V}^{σ} ‖ \leq {(a_{V} ε_{V})}^{2} {‖ {\hat{W}}_{V}^{T} {\hat{ϕ}}_{V}^{σ} ‖}^{2} / ϖ_{V} + \frac{ϖ_{V}}{4} {‖ σ_{V}^{*} ‖}^{2}

(123)

| a_{V} ε_{V} | ‖ W_{V}^{* T} ‖ ‖ {\hat{ϕ}}_{V}^{ξ'} * {\hat{ξ}}_{V} ‖ \leq {(a_{V} ε_{V})}^{2} {‖ {\hat{ϕ}}_{V}^{ξ'} * {\hat{ξ}}_{V} ‖}^{2} / ϖ_{V} + \frac{ϖ_{V}}{4} {‖ W_{V}^{* T} ‖}^{2}

(124)

| a_{V} ε_{V} | ‖ W_{V}^{* T} ‖ ‖ {\hat{ϕ}}_{V}^{σ'} * {\hat{σ}}_{V} ‖ \leq {(a_{V} ε_{V})}^{2} {‖ {\hat{ϕ}}_{V}^{σ'} * {\hat{σ}}_{V} ‖}^{2} / ϖ_{V} + \frac{ϖ_{V}}{4} {‖ W_{V}^{* T} ‖}^{2}

(125)

| a_{V} ε_{V} | (\sqrt{l_{V}} ‖ W_{V}^{* T} ‖ + χ_{2 V}) \leq 2 {(a_{V} ε_{V})}^{2} / ϖ_{V} + \frac{l_{V} ϖ_{V}}{4} {‖ W_{V}^{* T} ‖}^{2} + \frac{ϖ_{V}}{4} χ_{2 V}^{2}

(126)

where $ϖ_{V} > 0$ is the design parameter, and substituting equations (122)–(126) into equation (121) leads to

\begin{matrix} \overset{\cdot}{V} & \leq - k_{V} ε_{V}^{2} + {(a_{V} ε_{V})}^{2} {‖ {\hat{W}}_{V}^{T} {\hat{ϕ}}_{V}^{ξ} ‖}^{2} / ϖ_{V} + \frac{ϖ_{V}}{4} {‖ ξ_{V}^{*} ‖}^{2} + {(a_{V} ε_{V})}^{2} {‖ {\hat{W}}_{V}^{T} {\hat{ϕ}}_{V}^{σ} ‖}^{2} / ϖ_{V} \\ + \frac{ϖ_{V}}{4} {‖ σ_{V}^{*} ‖}^{2} + {(a_{V} ε_{V})}^{2} {‖ {\hat{ϕ}}_{V}^{ξ'} * {\hat{ξ}}_{V} ‖}^{2} / ϖ_{V} + {(a_{V} ε_{V})}^{2} {‖ {\hat{ϕ}}_{V}^{σ'} * {\hat{σ}}_{V} ‖}^{2} / ϖ_{V} \\ + \frac{ϖ_{V}}{4} {‖ W_{V}^{* T} ‖}^{2} + a_{V} ε_{V} υ_{V} + 2 {(a_{V} ε_{V})}^{2} / ϖ_{V} + \frac{l_{V} ϖ_{V}}{4} {‖ W_{V}^{* T} ‖}^{2} + \frac{ϖ_{V}}{4} {‖ W_{V}^{* T} ‖}^{2} \\ + {\overset{\cdot}{ς}}_{2} [a_{Φ} N (ς_{2}) + 1] + \frac{ϖ_{V}}{4} χ_{2 V}^{2} - tr [{\tilde{W}}_{V}^{T} δ_{W_{V}} {\hat{W}}_{V}] - \sum_{i = 1}^{l} ({\tilde{ξ}}_{Vi}^{T} δ_{V ξ} {\hat{ξ}}_{Vi}) - \sum_{i = 1}^{l} {\tilde{σ}}_{Vi} δ_{V σ} {\hat{σ}}_{Vi} \end{matrix}

(127)

The robust term is designed as

υ_{V} = - a_{V} ε_{V} ({‖ {\hat{W}}_{V}^{T} {\hat{ϕ}}_{V}^{ξ} ‖}^{2} + {‖ {\hat{W}}_{V}^{T} {\hat{ϕ}}_{V}^{σ} ‖}^{2} + {‖ {\hat{ϕ}}_{V}^{ξ^{'}} * {\hat{ξ}}_{V} ‖}^{2} + {‖ {\hat{ϕ}}_{V}^{σ^{'}} * {\hat{σ}}_{V} ‖}^{2} + 2) / ϖ_{V}

(128)

Substituting equation (128) into equation (127) leads to

\begin{matrix} \overset{\cdot}{V} & \leq - k_{V} ε_{V}^{2} + \frac{ϖ_{V}}{4} ‖ ξ_{V}^{*} ‖^{2} + \frac{ϖ_{V}}{4} ‖ σ_{V}^{*} ‖^{2} \\ + \frac{2 ϖ_{V} + l_{V} ϖ_{V}}{4} ‖ W_{V}^{* T} ‖^{2} + {\overset{\cdot}{ς}}_{2} [a_{Φ} N (ς_{2}) + 1] \\ - tr [{\tilde{W}}_{V}^{T} δ_{W_{V}} {\hat{W}}_{V}] - \sum_{i = 1}^{l} ({\tilde{ξ}}_{Vi}^{T} δ_{V ξ} {\hat{ξ}}_{Vi}) \\ - \sum_{i = 1}^{l} {\tilde{σ}}_{Vi} δ_{V σ} {\hat{σ}}_{Vi} + \frac{ϖ_{V}}{4} χ_{2 V}^{2} \end{matrix}

(129)

$2 tr {{\tilde{a}}^{T} \hat{a}} = ‖ \tilde{a} ‖^{2} + ‖ \hat{a} ‖^{2} - ‖ a^{*} ‖^{2} \geq ‖ \tilde{a} ‖^{2} - ‖ a^{*} ‖^{2}$ can be gained according to the properties of $tr {y x^{T}} = x^{T} y$ . Furthermore, we can obtain

- tr ({\tilde{W}}_{V}^{T} δ_{W_{V}} {\hat{W}}_{V}) \leq - \frac{δ_{W_{V}}}{2} ({‖ {\tilde{W}}_{V} ‖}^{2} - {‖ W_{V}^{*} ‖}^{2})

(130)

- \sum_{i = 1}^{l} ({\tilde{ξ}}_{Vi}^{T} δ_{V ξ} {\hat{ξ}}_{Vi}) \leq - \frac{δ_{V ξ}}{2} ({‖ {\tilde{ξ}}_{V} ‖}^{2} - {‖ ξ_{V}^{*} ‖}^{2})

(131)

- \sum_{i = 1}^{l} δ_{V σ} {\tilde{σ}}_{Vi} {\hat{σ}}_{Vi} \leq - \frac{δ_{V σ}}{2} ({‖ {\tilde{σ}}_{V} ‖}^{2} - {‖ σ_{V}^{*} ‖}^{2})

(132)

Substituting equations (130)–(132) into equation (129) leads to

\begin{matrix} \overset{\cdot}{V} & \leq - k_{V} ε_{V}^{2} - \frac{δ_{W_{V}}}{2} ‖ {\tilde{W}}_{V} ‖^{2} - \frac{δ_{V ξ}}{2} ‖ {\tilde{ξ}}_{V} ‖^{2} \\ - \frac{δ_{V σ}}{2} ‖ {\tilde{σ}}_{V} ‖^{2} + \frac{ϖ_{V} + 2 δ_{V ξ}}{4} ‖ ξ_{V}^{*} ‖^{2} \\ + \frac{ϖ_{V} + δ_{V σ}}{4} ‖ σ_{V}^{*} ‖^{2} + \frac{2 ϖ_{V} + l_{V} ϖ_{V} + 2 δ_{W_{V}}}{4} \\ ‖ W_{V}^{* T} ‖^{2} + {\overset{\cdot}{ς}}_{2} [a_{Φ} N (ς_{2}) + 1] + \frac{ϖ_{V}}{4} χ_{2 V}^{2} \end{matrix}

(133)

Equation (133) can be rewritten as

\begin{matrix} \overset{\cdot}{V} & \leq - k_{V} ε_{V}^{2} - \frac{δ_{W_{V}}}{2} ‖ {\tilde{W}}_{V} ‖^{2} - \frac{δ_{V ξ}}{2} ‖ {\tilde{ξ}}_{V} ‖^{2} \\ - \frac{δ_{V σ}}{2} ‖ {\tilde{σ}}_{V} ‖^{2} + {\overset{\cdot}{ς}}_{2} [a_{Φ} N (ς_{2}) + 1] + Ψ_{V} \end{matrix}

(134)

with

\begin{matrix} Ψ_{V} & = \frac{ϖ_{V} + 2 δ_{V ξ}}{4} ‖ ξ_{V}^{*} ‖^{2} + \frac{ϖ_{V} + δ_{V σ}}{4} ‖ σ_{V}^{*} ‖^{2} \\ + \frac{2 ϖ_{V} + l_{V} ϖ_{V} + 2 δ_{W_{V}}}{4} ‖ W_{V}^{* T} ‖^{2} + \frac{ϖ_{V}}{4} χ_{2 V}^{2} \end{matrix}

(135)

According to equation (134), we get

\overset{\cdot}{V} \leq - γ_{V} V + Ψ_{V} + {\overset{\cdot}{ς}}_{2} [a_{Φ} N (ς_{2}) + 1]

(136)

where $Ψ_{V} > 0$ is a constant and

γ_{V} = min {2 k_{V} \frac{δ_{W_{V}}}{λ_{max} (Γ_{W_{V}}^{- 1})} \frac{δ_{V ξ}}{λ_{max} (Γ_{V ξ}^{- 1})} \frac{δ_{V σ}}{Γ_{V σ}^{- 1}}}

(137)

Both the ends of equation (136) being multiplied by $e^{γ_{V} t}$ , we get

\frac{d}{d t} (V e^{γ_{V} t}) \leq Ψ_{V} e^{γ_{V} t} + {\overset{\cdot}{ς}}_{2} [a_{Φ} N (ς_{2}) + 1] e^{γ_{V} t}

(138)

The integration of equation (138) is expressed as

V (t) \leq \frac{Ψ_{V}}{γ_{V}} + V (0) + \int_{0}^{t} {\overset{\cdot}{ς}}_{2} [a_{Φ} N (ς_{2}) + 1] e^{γ_{V} (τ - t)} d τ

(139)

where the control coefficient $a_{Φ}$ is strictly positive or negative. According to Lemma 2, we can obtain the suitable $γ_{V}$ through regulating parameters $k_{V}$ , $δ_{W_{V}}$ , $Γ_{W_{V}}^{- 1}$ , $δ_{V ξ}$ , $Γ_{V ξ}^{- 1}$ , $δ_{V σ}$ , and $Γ_{V σ}^{- 1}$ . Furthermore, we can gain $\int_{0}^{t} {\overset{\cdot}{ς}}_{2} (τ) [a_{Φ} N (ς_{2} (τ)) + 1] d τ$ , where ${\overset{\cdot}{ς}}_{2}$ , $V (t)$ are bounded in the interval $[0, t_{f})$ ; therefore, $V (t)$ , $ε_{V}$ , ${\tilde{W}}_{V}$ , ${\tilde{ξ}}_{V}$ , and ${\tilde{σ}}_{V}$ are bounded.

If $\int_{0}^{t} | {\overset{\cdot}{ς}}_{2} [1 + a_{Φ} N (ς_{1})] e^{γ_{V} τ} | d τ \leq c_{V}$ , we gain

\begin{matrix} e^{- γ_{V} t} \int_{0}^{t} {\overset{\cdot}{ς}}_{2} [1 + a_{Φ} N (ς_{2})] e^{γ_{V} τ} d τ \\ \leq \int_{0}^{t} | {\overset{\cdot}{ς}}_{2} [1 + a_{Φ} N (ς_{2})] | e^{γ_{V} (t - τ)} d τ \leq \int_{0}^{t} | {\overset{\cdot}{ς}}_{2} [1 + a_{Φ} N (ς_{2})] | d τ \leq c_{V} \end{matrix}

(140)

According to equation (139), we obtain

V (t) \leq (\frac{Ψ_{V}}{γ_{V}} + c_{V}) + V (0) e^{- γ_{V} t} \leq (\frac{Ψ_{V}}{γ_{V}} + c_{V}) + V (0)

(141)

According to the Lyapunov function chosen, we get

ε_{V}^{2} (t) \leq 2 V (t)

(142)

‖ {\tilde{W}}_{V} ‖^{2} \leq \frac{2 V (t)}{λ_{min} (Γ_{W_{V}}^{- 1})}

(143)

‖ {\tilde{ξ}}_{V} ‖^{2} \leq \frac{2 V (t)}{λ_{min} (Γ_{ξ_{V}}^{- 1})}

(144)

‖ {\tilde{σ}}_{V} ‖^{2} \leq \frac{2 V (t)}{λ_{min} (Γ_{σ_{V}}^{- 1})}

(145)

where $λ_{min} (\cdot)$ is the minimal eigenvalue of matrix.

According to equations (142)–(145), tracking error $ε_{V}$ and estimation errors ${\tilde{W}}_{V}$ , ${\tilde{ξ}}_{V}$ , and ${\tilde{σ}}_{V}$ of the neural network are all bounded and converge to a neighborhood within the origin.

Theorem 2

Consider the closed-loop system, consisting of plant (16) with control law (107), adaptive laws (108) and (117)–(119), and robust term (128). All the signals involved are then uniformly and ultimately bounded. The output state error satisfies the demands of prescribed performance.

Remark 6

The Nussbaum function is used to solve the unknown control direction issue in the inner-loop subsystem. Therefore, the robust term could not be introduced to handle the approximation error of a fully tuned dynamic neural network in Step 2 of section “Inner-loop controller design,” but the fully tuned dynamic neural network, having a stronger approximation ability, is introduced to approach uncertainties in Step 1 of section “Inner-loop controller design” and section “Outer-loop controller design.” In section “Outer-loop controller design,” the design method overcomes the influence of the Nussbaum function. The robust term is then introduced to solve the approximation error of the fully tuned dynamic neural network.

Simulation results

We demonstrate the proposed prescribed performance dynamic neural network controllers using the longitudinal dynamic models (1)–(6) of a flexible hypersonic vehicle. All the aerodynamic coefficients and model parameters are the same as Parker et al.’s study.⁷ Let $k_{1} = 2.2$ , $μ_{1} = 30$ , $k_{V} = 0.2$ , $ϖ_{1} = 15$ , $ϖ_{V} = 20$ , $Γ_{W_{1}} = 5.5 I_{15 \times 15}$ , $δ_{W_{1}} = 30.52$ , $Γ_{1 ξ} = 1.2 I_{2 \times 2}$ , $δ_{1 ξ} = 1.5$ , $Γ_{1 σ} = 8.6$ , $δ_{1 σ} = 0.2$ , $Γ_{W_{2}} = 1.08 I_{15 \times 15}$ , $δ_{W_{2}} = 82$ , $Γ_{W_{V}} = 0.025 I_{15 \times 15}$ , $δ_{W_{V}} = 65$ , $Γ_{V ξ} = 2.2 I_{3 \times 3}$ , $δ_{V ξ} = 15.8$ , $Γ_{V σ} = 0.05$ , and $δ_{V σ} = 0.09$ . Prescribed performance functions are designed as $λ_{1} = 3 e^{- 1.0 t} + 0.2$ , $λ' = 7 e^{- 1.0 t} + 4$ , and transformed errors are $ε_{1} = \tan (\frac{π}{2} \frac{e_{1}}{λ_{1}})$ , $ε_{V} = \tan (\frac{z_{V}}{λ'})$ . In this simulation study, the initial trim conditions are listed in Table 1 and the values of force and moment coefficients are listed in Table 2.

Table 1.

Initial trim conditions.

Item	Value	Unit
$h$	$85, 000$	$ft$
$α$	$0.2$	$\deg$
$V$	$3850$	$m / s$
$ω_{z}$	$2$	$\deg / s$
$η_{1}$	$0.3122$	$\deg$
${\overset{\cdot}{η}}_{1}$	$0$	$\deg / s$
$η_{2}$	$0.2114$	$\deg$
${\overset{\cdot}{η}}_{2}$	$0$	$\deg / s$

Table 2.

Values of force and moment coefficients.

Item (unit)	Value	Item (unit)	Value
$C_{L}^{α} (ra d^{- 1})$	$4.6773$	$β_{1} (kg \cdot m^{- 1} \cdot ra d^{- 3})$	$- 5.6093 \times 1 0^{5}$
$C_{L}^{δ_{e}} (ra d^{- 1})$	$0.7622$	$β_{2} (kg \cdot m^{- 1} \cdot ra d^{- 3})$	$- 5.4323 \times 1 0^{5}$
$C_{L}^{0}$	$- 0.018714$	$β_{3} (kg \cdot m^{- 1} \cdot ra d^{- 2})$	$3.913 \times 1 0^{5}$
$C_{M, α}^{α^{2}} (ra d^{- 2})$	$6.2926$	$β_{4} (kg \cdot m^{- 1} \cdot ra d^{- 2})$	$- 2.5213 \times 1 0^{5}$
$C_{M, α}^{α} (ra d^{- 1})$	$2.1335$	$β_{5} (kg \cdot m^{- 1} \cdot ra d^{- 1})$	$5.1867 \times 1 0^{5}$
$C_{M, α}^{0}$	$0.18979$	$β_{6} (kg \cdot m^{- 1} \cdot ra d^{- 1})$	$- 3.5339 \times 1 0^{4}$
$z_{T} (m)$	$2.5481$	$β_{7} (kg \cdot m^{- 1})$	$9.3083 \times 1 0^{4}$
$\bar{c} (m)$	$5.1816$	$β_{8} (kg \cdot m^{- 1})$	$- 1.4725 \times 1 0^{3}$
$c_{e} (m)$	$- 1.2897$	$C_{D}^{α^{2}} (ra d^{- 2})$	$5.8224$
$C_{D}^{α} (ra d^{- 1})$	$- 0.0453$	$C_{D}^{0}$	$0.0101$
$N_{2}^{α^{2}} (k g^{0.5} \cdot m^{- 1} \cdot ra d^{- 2})$	$- 1.9566 \times 10^{3}$	$N_{1}^{α^{2}} (k g^{0.5} \cdot m^{- 1} \cdot ra d^{- 2})$	$5.4590 \times 10^{2}$
$N_{2}^{δ_{e}} (k g^{0.5} \cdot m^{- 1} \cdot ra d^{- 1})$	$4.856 \times 10^{2}$	$N_{1}^{α} (k g^{0.5} \cdot m^{- 1} \cdot ra d^{- 1})$	$1.7817 \times 10^{3}$
$N_{2}^{α} (k g^{0.5} \cdot m^{- 1} \cdot ra d^{- 1})$	$1.1154 \times 10^{3}$	$N_{1}^{0} (k g^{0.5} \cdot m^{- 1})$	$45.78$
$N_{2}^{0} (k g^{0.5} \cdot m^{- 1})$	$- 17.218$

Let $α_{d} = 3.5 °$ , $V_{d} = 3860 m / s$ , $K_{1} = 1.0084$ , $K_{2} = 1.0084$ , $ζ_{1} = 0.02$ , $ζ_{2} = 0.02$ , $ω_{1} = 16.021$ , $ω_{2} = 19.582$ , ${\tilde{ψ}}_{1} = 4223.44$ , and ${\tilde{ψ}}_{2} = 4229.55$ . The number of neural network node is selected as $l = 15$ , and the initial values are selected as

\begin{matrix} {\hat{W}}_{1} = {[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]}^{T}, {\hat{W}}_{2} = {[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]}^{T} \\ {\hat{ξ}}_{1} = [\begin{matrix} 0.15 & 0.015 & 0.13 & 0.025 & 0.116 & 0.05 & 0.115 & 0.03 & 0.15 & 0.06 & 0.08 & 0.25 & 0.04 & 0.25 & 0.056 \\ 0.11 & 0.2 & 0.19 & 0.8 & 0.55 & 0.11 & 0.2 & 0.19 & 0.8 & 0.65 & 0.21 & 0.15 & 0.29 & 0.1 & 0.35 \\ 3849 & 3849.2 & 3850 & 3850.8 & 3851 & 3851.5 & 3852 & 3851.9 & 3851.5 & 3850 & 3849 & 3848 & 3847.5 & 3847.2 & 3847.1 \\ 0.11 & 1.2 & 2.19 & 3.8 & 4.55 & 5.11 & 6.2 & 5.19 & 4.8 & 3.65 & 2.21 & 1.15 & 0.89 & 0.5 & 0.35 \\ 0.11 & 0.2 & 0.19 & 0.8 & 0.55 & 0.11 & 0.2 & 0.19 & 0.8 & 0.65 & 0.21 & 0.15 & 0.29 & 0.1 & 0.35 \end{matrix}] \\ {\hat{σ}}_{1} = {[\begin{matrix} 0.5 & 16 & 0.9 & 15 & 0.1 & 14 & 0.7 & 15 & 9 & 0.2 & 12.6 & 0.6 & 13.8 & 8.8 & 0.5 \end{matrix}]}^{T} \\ ξ_{2} = [\begin{matrix} 0.0032 & 0.0034 & 0.0036 & 0.0033 & 0.0037 & 0.0035 & 0.0038 & 0.0031 & 0.0032 & 0.0036 & 0.0038 & 0.0035 & 0.0034 & 0.0035 & 0.0036 \\ 0.032 & 0.033 & 0.039 & 0.048 & 0.055 & 0.031 & 0.035 & 0.033 & 0.036 & 0.32 & 0.039 & 0.035 & 0.041 & 0.036 & 0.035 \\ 3849 & 3849.2 & 3850 & 3850.8 & 3851 & 3851.5 & 3852 & 3851.9 & 3851.5 & 3850 & 3849 & 3848 & 3847.5 & 3847.2 & 3847.1 \\ 0.26 & 0.21 & 0.22 & 0.39 & 0.34 & 0.33 & 0.29 & 0.35 & 0.28 & 0.39 & 0.3 & 0.31 & 0.33 & 0.32 & 0.37 \\ 0.001 & 0.311 & 0.1 & 0.89 & 0.1011 & 0.9111 & 0.1 & 0.89 & 0.1 & 0.39 & 0.71 & 0.3 & 0.69 & 0.2 & 0.59 \\ 1.26 & 2.21 & 1.22 & 0.59 & 2.34 & 1.33 & 6.29 & 3.35 & 2.28 & 0.29 & 0.13 & 0.21 & 0.13 & 0.12 & 0.17 \end{matrix}] \\ {\hat{ξ}}_{V} = [\begin{matrix} 3848.2 & 3849.1 & 3851.1 & 3850.1 & 3849.2 & 3848.2 & 3849.1 & 3851.1 & 3850.1 & 3847.2 & 3849.2 & 3848.1 & 3847.1 & 3852.1 & 3850.2 \\ 0.002 & 0.003 & 0.005 & 0.004 & 0.006 & 0.006 & 0.004 & 0.005 & 0.004 & 0.006 & 0.006 & 0.004 & 0.005 & 0.004 & 0.006 \\ 0.15 & 0.16 & 0.19 & 0.13 & 0.15 & 0.17 & 0.19 & 0.11 & 1.11 & 0.11 & 0.17 & 0.19 & 0.11 & 1.11 & 0.11 \\ 0.26 & 0.21 & 0.22 & 0.39 & 0.34 & 0.33 & 0.29 & 0.35 & 0.28 & 0.39 & 0.3 & 0.31 & 0.33 & 0.32 & 0.37 \\ 0.011 & 0.311 & 0.1 & 0.89 & 0.11 & 0.11 & 0.1 & 0.89 & 0.1 & 0.39 & 0.71 & 0.3 & 0.9 & 0.2 & 0.59 \\ 0.26 & 1.21 & 1.22 & 0.59 & 2.34 & 1.33 & 6.29 & 3.35 & 2.28 & 0.29 & 0.13 & 0.21 & 0.13 & 0.12 & 0.17 \end{matrix}] \\ {\hat{W}}_{V} = {[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]}^{T}, σ_{2} = {[\begin{matrix} 0.5 & 6 & 0.9 & 5 & 0.1 & 4 & 0.7 & 5 & 9 & 0.2 & 2 & 0.6 & 3 & 8 & 0.5 \end{matrix}]}^{T} \\ {\hat{σ}}_{V} = {[\begin{matrix} 3500 & 4800 & 3900 & 3000 & 3600 & 5500 & 3800 & 6900 & 3060 & 3100 & 3890 & 5670 & 4900 & 2860 & 3950 & 5700 \end{matrix}]}^{T} \end{matrix}

The simulation results are expressed in Figures 2 –19.

Figure 2.

Angle of attack.

Figure 3.

Rate of pitch.

Figure 4.

Elevator deflection.

Figure 5.

Velocity.

Figure 6.

Equivalence ratio.

Figure 7.

Curve of ${\hat{W}}_{1}$ .

Figure 8.

Curve of ${\hat{W}}_{2}$ .

Figure 9.

Curve of ${\hat{W}}_{V}$ .

Figure 10.

Neural network approximation $f_{α}$ .

Figure 11.

Neural network approximation $\frac{{\bar{f}}_{ω}}{Θ}$ .

Figure 12.

Neural network approximation $f_{V}$ .

Figure 13.

Curve of $η_{1}$ .

Figure 14.

Curve of $η_{2}$ .

Figure 15.

Curve of ${\overset{\cdot}{η}}_{1}$ .

Figure 16.

Curve of ${\overset{\cdot}{η}}_{2}$ .

Figure 17.

Curve of $e_{1}$ .

Figure 18.

Curve of $e_{2}$ .

Figure 19.

Curve of $z_{V}$ .

According to the simulation results, it has been shown that altitude tracking errors $e_{1}$ , $e_{2}$ and velocity tracking error $z_{V}$ are successfully limited to the given PPBs shown in Figures 17 –19. The actual values of the attack angle can track the desirable values well in Figure 2. The path rate tracking is shown in Figure 3. The velocity in Figure 5 can track the desirable value well, and the transient and steady performance are satisfactory. The control input, elevator deflection, and equivalence ratio are bounded as shown in Figures 4 and 6. The response of weight matrices of the neural network is bounded as shown in Figures 7 –9. From Figures 10 –12, it is observed that the neural network approximation of uncertainties $f_{α}$ , ${\bar{f}}_{ω} / Θ$ , and $f_{V}$ are bounded and the approximation effects are very good. Also, it is observed from Figures 13 –16 that flexible states and first-order derivatives of the flexible states are bounded and converge to their steady values.

Conclusion

In this article, a control system is designed for a flexible hypersonic vehicle with the prescribed performance control, backstepping control, and RBF neural network method. The prescribed performance control method can guarantee the transient and steady performance of the tracking errors when velocity and altitude are formally specified. In this article, a new RBF neural network (which is called the fully tuned dynamic neural network) is introduced to approximate the following: the uncertainties of the system; the respective centers; the width of the Gaussian function; and the weight vector, which are regulated by adaptive laws. The unknown control direction issue is solved with the Nussbaum function. Finally, the effectiveness and superiority of the proposed control strategy are validated by simulation results. In the future work, the design will be toward the characteristics of flexible hypersonic vehicle dealing with the problem of real-time performance, the learning speed of neural network, and the large flight envelope.

Footnotes

Appendix 1

Handling Editor: Hongwei Wu

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

He-wei Zhao

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