Adaptive neural tracking control for near-space vehicles with stochastic disturbances

Abstract

In this article, an adaptive neural tracking controller is designed for near-space vehicles with stochastic disturbances and unknown parametric uncertainties. Based on the great nonlinear function approximation capability of neural networks, the unknown system uncertainties are tackled using the radial basis function neural networks. Furthermore, on the basis of stochastic Lyapunov stability theory, an adaptive tracking control scheme is developed for near-space vehicle which can guarantee the closed-loop system stability. Under the developed adaptive neural control scheme, all closed-loop system signals are bounded in the sense of probability, and the tracking error converges to a small neighborhood of the origin. Finally, simulation results are provided to illustrate the proposed adaptive neural control scheme that can guarantee the satisfactory tracking performance for the attitude motion of the near-space vehicle with stochastic disturbances.

Keywords

Near-space vehicles stochastic disturbances neural network adaptive control backstepping

Introduction

Near-space vehicles (NSVs) are novel aerospace vehicles that have attracted more and more attention. With the advantages of traditional aircrafts and spacecrafts, such as high mobility and wide flight envelope,^1

–4 it has raised concerns on military and civilian application. As a kind of flight control systems, NSV has the complicated characteristics, such as strong nonlinearity, strong coupling, system uncertainties, special working environment, and multiple mission modes. The robust flight control of NSV becomes a challenging and valuable work, and some control approaches have been proposed in the literature.^{5

–15} Meanwhile, the control design as one of the most important research topics in NSV, the tracking control problem has been extensively studied. The novel global neural dynamic surface tracking control scheme is presented for hypersonic flight vehicle in the study by Xu et al.¹⁶ The guaranteed transient performance-based attitude control scheme was proposed for NSV with input saturation by Chen et al.¹⁷ In the study by Chen et al.,¹⁸ a robust sliding mode control strategy was presented for attitude dynamics of NSV. In the study by Cheng et al.,¹⁹ an online support vector regression compensated nonlinear-generalized predictive control (NGPC) system was presented for hypersonic vehicles.

As is well known, the NSV is a multiple-input multiple-output (MIMO) nonlinear system subject to external deterministic disturbances which will further increase the design difficulty of the reliable and accurate flight control scheme. Over the past few decades, the disturbance observer–based control schemes have been well studied in the literature.^{20

–30} To efficiently handle the unknown disturbance based on the dynamic information of disturbance, the adaptive dynamic surface control scheme based on disturbance observer was proposed for the NSV with input saturation by Chen and Yu.³¹ In the study by Chen et al.,³² on the basis of terminal sliding mode technique and disturbance observer method, an anti-disturbance control scheme was developed for the hypersonic flight vehicles with input saturation. In the study by Cheng et al.,³³ an NGPC strategy based on sliding mode disturbance observer (SMDO) was designed for the hypersonic NSV attitude control system. However, stochastic disturbances always exist in practical systems which need to be further considered. Especially, the flight attitude control of NSV will become more complicated if above mentioned aspects are simultaneously considered.

Recently, some significant results on backstepping-based control scheme^{34

–41} for deterministic nonlinear systems have been generalized to stochastic nonlinear systems. For instance, by introducing a quartic Lyapunov function, a backstepping design scheme was proposed for a class of strict-feedback stochastic nonlinear systems by Deng et al.⁴² Based on the great nonlinear function approximation capability of neural networks (NNs) or fuzzy logic systems, by combining adaptive fuzzy control or adaptive neural control with backstepping technique, many adaptive control schemes can be used to efficiently handle the uncertainties of stochastic nonlinear systems. Furthermore, some meaningful results were presented for stochastic nonlinear systems on the basis of the quartic Lyapunov function in the literature.^{43

–55} It is well known that mechanical systems are often subjected to stochastic disturbances, which significantly affect the performance. By reasonably introducing random noise, a method to construct stochastic Lagrangian control systems was given and an adaptive tracking controller was designed by Cui et al.⁵⁶ Using geometric stochastic feedback control, a robust asymptotic stabilization of rigid body attitude dynamics with stochastic input torque was studied by Samiei et al.⁵⁷ In earlier study,⁵⁸ a new robust stochastic control methodology was developed for unmanned aerial vehicles. In a realistic environment, according to the characteristics of the NSV, stochastic components and random noises are included in the models which will lead to the considerable difficulty in the design stage of the flight control system for the NSV with stochastic disturbances.

Motivated by above analysis, in this article, using backstepping technology, an adaptive neural tracking control scheme will be developed for the NSV with stochastic disturbances, unknown parametric uncertainties, and external disturbances. This article is organized as follows: The stochastic nonlinear control system of NSV with stochastic disturbance and mathematical preliminaries is given in “Problem statements and preliminaries” section. The design and stability analysis of an adaptive neural tracking controller is proposed in “Adaptive neural tracking control design” section. In “Simulation study” section, simulation results are given to demonstrate the effectiveness of our developed robust attitude control approaches. Conclusions are addressed in the last section.

Throughout this article, ∥ ⋅ ∥ stands for the Euclidean norm (or Frobenius norm for a matrix); λ_min(⋅) and λ_max(⋅) denote the minimum and maximum eigenvalues of a matrix, respectively. Let E[⋅] and P{⋅} represent the expectation operator and the probability of a random variable, respectively. $F$ is the σ-algebra of the subsets of sample space Ξ, and P is the probability measure.

Problem statements and preliminaries

Problem formulation and some assumptions

The uncertain attitude dynamics with external disturbance of the NSV can be described as the following MIMO nonlinear system³:

\begin{matrix} \dot{Ω} = F_{1} (Ω) + Δ F_{1} (Ω) + G_{1} (Ω) ω + d_{1} \\ \dot{ω} = F_{2} (Ω, ω) + Δ F_{2} (Ω, ω) + G_{2} (Ω, ω) M_{c} + d_{2} \end{matrix}

where $Ω = [α, β, μ]^{T} \in ℝ^{3}$ is the attitude angle vector representing attack angle, sideslip angle, and roll angle, respectively. $ω = [p, q, r]^{T} \in ℝ^{3}$ is the attitude angular rate vector representing roll angular rate, pitch angular rate, and yaw angular rate, respectively. $M_{c} = [l_{c}, m_{c}, n_{c}]^{T} \in ℝ^{3}$ is the control moment vector representing rolling moment, pitching moment, and yaw moment, respectively. $F_{1} \in ℝ^{3}$ and $F_{2} \in ℝ^{3}$ are the known state function vectors; $G_{1} \in ℝ^{3 \times 3}$ and $G_{2} \in ℝ^{3 \times 3}$ are the known system control matrices which are invertible. $Δ F_{1}, Δ F_{2} \in ℝ^{3}$ are the unknown modeling errors which are smooth functions, and $d_{1}, d_{2} \in ℝ^{3}$ are the unknown external disturbances.

By considering the influence of the stochastic disturbance torque, as shown in the study by Zhu,⁵⁹ the attitude dynamics in the body frame is obtained as follows

\begin{array}{l} \dot{Ω} = F_{1} (Ω) + Δ F_{1} (Ω) + G_{1} (Ω) ω + d_{1}, \\ G_{2}^{- 1} (Ω, ω) \dot{ω} = G_{2}^{- 1} (Ω, ω) (F_{2} (Ω, ω) + Δ F_{2} (Ω, ω)) + M_{c} + G_{2}^{- 1} (Ω, ω) d_{2} + Λ (Ω, ω) ξ \end{array}

where $Λ (Ω, ω) ξ$ is caused by white noise (i.e. stochastic disturbance signal) ξ, and $Λ (Ω, ω) \in ℝ^{3 \times m}$ is a Borel measurable function matrix which is locally bounded and locally Lipschitz continuous, and $Λ (0, 0) = 0$ .

According to Section 5.5 of the reference,⁶⁰ with $\frac{d W}{d t}$ instead of ξ, we obtain the following stochastic differential equation in the Stratonovich sense as

\begin{array}{l} d Ω = (F_{1} (Ω) + Δ F_{1} (Ω) + G_{1} (Ω) ω + d_{1}) d t \\ d ω = (F_{2} (Ω, ω) + Δ F_{2} (Ω, ω) + G_{2} (Ω, ω) M_{c} + d_{2}) d t + G_{2} (Ω, ω) Λ (Ω, ω) \circ d W \end{array}

where W is a m-dimensional independent Wiener process.

To facilitate the controller design for attitude dynamics with stochastic disturbances, the system (3) needs to be transformed into the equivalent $I t \hat{o}$ type stochastic differential equation as follows

\begin{matrix} {[G_{2}^{- 1} (Ω, ω) d ω]}_{i} = {[G_{2}^{- 1} (Ω, ω) (F_{2} (Ω, ω) + Δ F_{2} (Ω, ω)) + M_{c} + G_{2}^{- 1} (Ω, ω) d_{2}]}_{i} d t \\ + \frac{1}{2} \sum_{k = 1}^{3} \sum_{j = 1}^{m} Λ_{k j} \frac{\partial Λ_{i j}}{\partial ω_{k}} d t + \sum_{j = 1}^{m} Λ_{i j} d W_{j} (t) \end{matrix}

where ${[\cdot]}_{i}, i = 1, 2, 3$ denotes the ith element of a vector, and $\frac{1}{2} \sum_{k = 1}^{3} \sum_{j = 1}^{m} Λ_{k j} \frac{\partial Λ_{i j}}{\partial ω_{k}}$ is the Wong–Zakai correction term.⁵¹ Suppose that the power spectral density (PSD) of white noise ξ equals $\frac{1}{2 π} Π$ , which is equivalent to the fact $d W = Π d B$ , where $Π \in ℝ^{m \times m}$ is an unknown positive matrix and B is a m-dimensional independent standard Wiener process. It is a fact that $E {d W d W^{T}} = Π E {d B d B^{T}} Π^{T} = Π Π^{T} d t$ .⁴² For the ease of control design, equation (4) can be represented using the matrix form of an $I t \hat{o}$ type nonlinear stochastic differential equation as

{\begin{array}{l} d Ω = (F_{1} (Ω) + Δ F_{1} (Ω) + G_{1} (Ω) ω + d_{1}) d t \\ d ω = (F_{2} (Ω, ω) + Δ F_{2} (Ω, ω) + G_{2} (Ω, ω) M_{c} + d_{2} + \bar{G} (Ω, ω)) d t + G_{2} (Ω, ω) Λ (Ω, ω) Π d B (t) \\ y = Ω \end{array}

where ${\bar{g}}_{i} = \frac{1}{2} \sum_{k = 1}^{3} \sum_{j = 1}^{m} Λ_{k j} \frac{\partial Λ_{i j}}{\partial ω_{k}}$ , and $\bar{G} (Ω, ω) = G_{2} (Ω, ω) [{\bar{g}}_{1}, {\bar{g}}_{2}, {\bar{g}}_{3}]^{T} \in ℝ^{3}$ is the Wong–Zakai coefficient vector.

The control object of this article is that the output of system (5) can track the desired bounded signal $Ω_{d} = [α_{d}, β_{d}, μ_{d}]^{T}$ under the proposed adaptive neural controller which can ensure all the closed-loop system signals bounded in the sense of probability, and the tracking errors converge to a small neighborhood of the origin. For simplicity, in the following, the state vectors Ω, ω will be omitted from the corresponding functions. To facilitate the adaptive neural controller design for the NSV described by equation (5), the following assumptions are given as follows.

Assumption 1

For the nonlinear attitude dynamics system (5) of NSV with stochastic disturbances, the gain matrices G₁ and G₂ are invertible.³ In addition, there exist unknown positive constants ${\bar{g}}_{i}$ and $g_{i}^{*}$ such that $λ_{\max} (G_{i}^{T} G_{i}) \leq {\bar{g}}_{i}^{2}$ and $∥ G_{i} ∥ \leq g_{i}^{*}, i = 1, 2$ .

Assumption 2

For all t > 0, there exist the unknown positive constants ϖ_i such that $∥ Ω_{d}^{(i)} ∥ \leq ϖ_{i}, i = 1, 2$ .³

Assumption 3

For the time-varying unknown external disturbances d_i(t), there exist unknown positive constants ${\bar{d}}_{i}$ such that $∥ d_{i} ∥ \leq {\bar{d}}_{i}, i = 1, 2$ .

Assumption 4

There exist unknown positive constants $ρ_{1}, ρ_{2}$ and known smooth nonnegative functions $ψ_{1} (Ω, ω)$ , $ψ_{2} (Ω, ω)$ such that $∥ Λ (Ω, ω) ∥ \leq ρ_{1} ψ_{1} (Ω, ω)$ and $∥ \bar{G} (Ω, ω) ∥ \leq g_{2}^{*} ρ_{2} ψ_{2} (Ω, ω)$ .⁵⁶

Mathematical preliminaries

Consider the following stochastic nonlinear system⁶¹

d x = f (x) d t + g (x) d B, \forall x \in R^{n}

where $x \in R^{n}$ is the state of the system, B is a m-dimensional independent standard Wiener process defined on the complete probability space $(Ξ, F, P)$ , and $f (\cdot) : R^{n} \to R^{n}$ and $g (\cdot) : R^{n} \to R^{n \times m}$ are locally Lipschitz functions in x and satisfy f(0) = 0 and g(0) = 0, respectively.

Definition 1

For any given $V (x) \in C^{2}$ , associated with the stochastic differential equation (6), the differential operator L is defined as follows⁶¹

L V = \frac{\partial V}{\partial x} f (x) + \frac{1}{2} Tr {g^{T} (x) \frac{\partial^{2} V}{\partial x^{2}} g (x)}

where $Tr (\cdot)$ is the trace of a matrix, and $\frac{1}{2} Tr {g^{T} (x) \frac{\partial^{2} V}{\partial x^{2}} g (x)}$ is called $I t \hat{o}$ correction term.

Definition 2

The stochastic process $x (t), t \geq 0$ of stochastic system (6) is said to be bounded in probability, if⁶¹

lim_{c \to \infty} sup_{0 \leq t < \infty} P {| | x (t) | | > c} = 0

Definition 3

The stochastic process⁶¹ x(t) of equation (6) is said to be semi-globally uniformly ultimately bounded (SGUUB) in pth moment, if for a given compact set $Ω^{*} \in R^{n}$ and any initial condition $x_{0} = x (t_{0})$ , there exist a constant ε > 0 and a time constant $T = T (ε, x_{0})$ such that $E [| | x (t {) | |}^{p}] < ε$ for all $t > t_{0} + T .$ Especially, when p = 2, it is usually called SGUUB in mean square.

Lemma 1

Consider the stochastic nonlinear system (6).⁴² If there exists a positive definite, radially unbounded, twice continuously differentiable Lyapunov function $V : R^{n} \to R$ and constants $a_{0} > 0, b_{0} \geq 0$ such that

L V (x) \leq - a_{0} V (x) + b_{0}

then, (i) the system has an unique solution almost surely and (ii) the system is bounded in probability.

Lemma 2 (Young’s inequality)

For $\forall (x^{*}, y^{*}) \in R^{2}$ , the following inequality holds⁴⁵

x^{*} y^{*} \leq \frac{ϑ^{s}}{s} | x^{*} |^{s} + \frac{1}{ι ϑ^{ι}} | y^{*} |^{ι}

where $ϑ > 0, s > 1, ι > 1$ and $(s - 1) (ι - 1) = 1.$

Neural networks

The radial basis function NNs (RBFNNs) are used to approximate an unknown continuous function $f (Z) : R^{r} \to R$ , which can be expressed as¹⁷

\hat{f} (Z) = Φ^{T} S (Z)

where $\hat{f} (Z)$ is the approximation of f(Z), and $Z \in Ω_{Z} \subset R^{r}$ is the input vector with r being the NNs input dimension. $Φ = [w_{1}, w_{2}, \dots, w_{l}]^{T} \in R^{l}$ is the weight vector. l > 1 denotes the NN node number. $S (Z) = [s_{1} (Z), s_{2} (Z), \dots, s_{l} (Z {)]}^{T} \in R^{l}$ is the basis function vector with s_i(Z) being chosen as the commonly used Gaussian function of the form

s_{i} (Z) = \exp [- \frac{{(Z - μ_{i})}^{T} (Z - μ_{i})}{η^{2}}]

where $μ_{i} = [μ_{i 1}, \dots, μ_{i r}]^{T}, i = 1, 2, \dots, l$ is the center of the receptive field and η is the width of the Gaussian function. The RBFNNs (equation (11)) can approximate any continuous function f(Z) defined over a compact set $Ω_{Z} \subset R^{r}$ to arbitrary any accuracy δ. Then, there exists an NN $Φ^{*}^{T} S (Z)$ such that

f (Z) = Φ^{*}^{T} S (Z) + δ, \forall Z \in Ω_{Z} \subset R^{q}

where δ is the approximation error and satisfies |δ| ≤ δ* with δ* > 0. Φ* is the optimal weight vector and it is expressed in the following form

Φ^{*} : = \arg min_{Φ \in R^{l}} {sup_{Z \in Ω_{Z}} | f (Z) - Φ^{T} S (Z) |}

Adaptive neural tracking control design

In this section, the backstepping design is based on the following coordinate transformations

z_{1} = Ω - Ω_{d}, z_{2} = ω - υ_{1}

where z₁ and z₂ are called the error variables, and υ₁ is the virtual control law which will be given later.

To begin with the backstepping design procedure, for i = 1, 2, denote $θ_{i} = ∥ Φ_{i}^{*} ∥$ with $Φ_{i}^{*}$ being the optimal weight vector, and $ρ = \max {| | Π {| |}_{F}^{4} ρ_{1}^{4}, g_{2}^{*}^{\frac{4}{3}} ρ_{2}^{\frac{4}{3}}}$ . The estimations of the unknown parameters θ_i and ρ are ${\hat{θ}}_{i}$ and $\hat{ρ}$ , respectively. And the parameter estimation errors are ${\tilde{θ}}_{i} = θ_{i} - {\hat{θ}}_{i}$ and $\tilde{ρ} = ρ - \hat{ρ}$ , respectively.

Step 1

Noting $z_{1} = Ω - Ω_{d}$ , the error dynamic is

d z_{1} = (F_{1} + Δ F_{1} + G_{1} ω + d_{1} - {\dot{Ω}}_{d}) d t

As shown in “Neural networks” section, the unknown uncertainty ΔF₁ will be approximated using the RBFNNs. Thus, there exists an NN approximation $Φ_{1}^{*}^{T} S_{1} (Ω)$ such that

Δ F_{1} = Φ_{1}^{* T} S_{1} (Ω) + δ_{1}

where the approximation error vector δ₁ satisfies $∥ δ_{1} ∥ \leq δ_{1}^{*}$ with $δ_{1}^{*} > 0$ .

Substituting equation (17) into equation (16) yields

\begin{matrix} d z_{1} = (F_{1} + Φ_{1}^{*^{T}} S_{1} + δ_{1} + G_{1} ω + d_{1} - {\dot{Ω}}_{d}) d t \\ = (F_{1} + Φ_{1}^{*^{T}} S_{1} + G_{1} ω + D_{1} - {\dot{Ω}}_{d}) d t \end{matrix}

According to assumption 3, the compounded disturbance $D_{1} = δ_{1} + d_{1}$ is bounded, that is, there exists a positive constant $ζ_{1} > 0$ such that $∥ D_{1} ∥ \leq ζ_{1} .$ The estimation of the unknown parameter ζ₁ is ${\hat{ζ}}_{1} .$ And the parameter estimation error is ${\tilde{ζ}}_{1} = ζ_{1} - {\hat{ζ}}_{1}$ .

The virtual control law υ₁ is designed as

υ_{1} = - G_{1}^{- 1} (K_{1} z_{1} + \frac{3 | | G_{1} {| |}^{\frac{4}{3}}}{4 ς^{\frac{4}{3}}} z_{1} + F_{1} - {\dot{y}}_{d} + {\hat{θ}}_{1} \frac{3 ∥ S_{1} ∥^{\frac{4}{3}}}{4 τ_{1}^{\frac{4}{3}}} z_{1} + {\hat{ζ}}_{1} \frac{3}{4 ς_{1}^{\frac{4}{3}}} z_{1})

where $K_{1} = K_{1}^{T} > 0$ is a design matrix, and $ς, τ_{1}, ς_{1}$ are design positive constants.

The adaptive laws are designed as follows

{\dot{\hat{θ}}}_{1} = γ_{1} (\frac{3 ∥ S_{1} ∥^{\frac{4}{3}}}{4 τ_{1}^{\frac{4}{3}}} ∥ z_{1} ∥^{4} - {\hat{θ}}_{1})

{\dot{\hat{ζ}}}_{1} = ξ_{1} (\frac{3}{4 ς_{1}^{\frac{4}{3}}} ∥ z_{1} ∥^{4} - {\hat{ζ}}_{1})

where $γ_{1} and ξ_{1}$ are positive design parameters.

Consider the Lyapunov function candidate as

V_{1} = \frac{1}{4} {(z_{1}^{T} z_{1})}^{2} + \frac{1}{2 γ_{1}} {\tilde{θ}}_{1}^{2} + \frac{1}{2 ξ_{1}} {\tilde{ζ}}_{1}^{2}

From equation (7), and combining with equations (19) to (21), we have

\begin{array}{l} L V_{1} = z_{1}^{T} z_{1} z_{1}^{T} (F_{1} + Φ_{1}^{*^{T}} S_{1} + G_{1} ω + D_{1} - {\dot{y}}_{d}) - \frac{1}{γ_{1}} {\tilde{θ}}_{1} {\dot{\hat{θ}}}_{1} - \frac{1}{ξ_{1}} {\tilde{ζ}}_{1} {\dot{\hat{ζ}}}_{1} \\ = - z_{1}^{T} z_{1} z_{1}^{T} K_{1} z_{1} - z_{1}^{T} z_{1} z_{1}^{T} \frac{3 | | G_{1} {| |}^{\frac{4}{3}}}{4 ς^{\frac{4}{3}}} z_{1} + z_{1}^{T} z_{1} z_{1}^{T} G_{1} z_{2} + z_{1}^{T} z_{1} z_{1}^{T} Φ_{1}^{*^{T}} S_{1} + z_{1}^{T} z_{1} z_{1}^{T} D_{1} - {\hat{θ}}_{1} \frac{3 ∥ S_{1} ∥^{\frac{4}{3}}}{4 τ_{1}^{\frac{4}{3}}} {(z_{1}^{T} z_{1})}^{2} \\ - {\hat{ζ}}_{1} \frac{3}{4 ς_{1}^{\frac{4}{3}}} {(z_{1}^{T} z_{1})}^{2} - \frac{1}{γ_{1}} {\tilde{θ}}_{1} {\dot{\hat{θ}}}_{1} - \frac{1}{ξ_{1}} {\tilde{ζ}}_{1} {\dot{\hat{ζ}}}_{1} \\ \leq - λ_{\min} (K_{1}) {(z_{1}^{T} z_{1})}^{2} - \frac{3 | | G_{1} {| |}^{\frac{4}{3}}}{4 ς^{\frac{4}{3}}} {(z_{1}^{T} z_{1})}^{2} + z_{1}^{T} z_{1} z_{1}^{T} G_{1} z_{2} + z_{1}^{T} z_{1} z_{1}^{T} Φ_{1}^{*^{T}} S_{1} + z_{1}^{T} z_{1} z_{1}^{T} D_{1} - θ_{1} \frac{3 ∥ S_{1} ∥^{\frac{4}{3}}}{4 τ_{1}^{\frac{4}{3}}} {(z_{1}^{T} z_{1})}^{2} \\ - ζ_{1} \frac{3}{4 ς_{1}^{\frac{4}{3}}} {(z_{1}^{T} z_{1})}^{2} + {\tilde{θ}}_{1} {\hat{θ}}_{1} + {\tilde{ζ}}_{1} {\hat{ζ}}_{1} \end{array}

According to Lemma 2, we have

z_{1}^{T} z_{1} z_{1}^{T} Φ_{1}^{*^{T}} S_{1} (Ω) \leq ∥ z_{1} ∥^{3} ∥ Φ_{1}^{*} ∥ ∥ S_{1} (Ω) ∥ \leq θ_{1} \frac{3 ∥ S_{1} ∥^{\frac{4}{3}}}{4 τ_{1}^{\frac{4}{3}}} {(z_{1}^{T} z_{1})}^{2} + \frac{θ_{1} τ_{1}^{4}}{4}

z_{1}^{T} z_{1} z_{1}^{T} G_{1} z_{2} \leq \frac{3 | | G_{1} {| |}^{\frac{4}{3}}}{4 ς^{\frac{4}{3}}} (z_{1}^{T} z_{1}) + \frac{ς_{_{}}^{4}}{4} {(z_{2}^{T} z_{2})}^{2}

z_{1}^{T} z_{1} z_{1}^{T} D_{1} \leq ζ_{1} \frac{3}{4 ς_{1}^{\frac{4}{3}}} {(z_{1}^{T} z_{1})}^{2} + \frac{ζ_{1} ς_{1}^{4}}{4}

{\tilde{θ}}_{1} {\hat{θ}}_{1} = \frac{1}{2} (θ_{1}^{2} - {\tilde{θ}}_{1}^{2} - {\hat{θ}}_{1}^{2}) \leq \frac{1}{2} (θ_{1}^{2} - {\tilde{θ}}_{1}^{2})

{\tilde{ζ}}_{1} {\hat{ζ}}_{1} = \frac{1}{2} (ζ_{1}^{2} - {\tilde{ζ}}_{1}^{2} - {\hat{ζ}}_{1}^{2}) \leq \frac{1}{2} (ζ_{1}^{2} - {\tilde{ζ}}_{1}^{2})

Substituting equations (24) to (28) into equation (23) yields

L V_{1} \leq - λ_{min} (K_{1}) {(z_{1}^{T} z_{1})}^{2} + \frac{ς^{4}}{4} {(z_{2}^{T} z_{2})}^{2} + \frac{θ_{1} τ_{1}^{4}}{4} + \frac{ζ_{1} ς_{1}^{4}}{4} + \frac{1}{2} (θ_{1}^{2} - {\tilde{θ}}_{1}^{2}) + \frac{1}{2} (ζ_{1}^{2} - {\tilde{ζ}}_{1}^{2})

Step 2

According to $z_{2} = ω - α_{1}$ and the $I t \hat{o}$ formula, we have

d z_{2} = (F_{2} + Δ F_{2} + G_{2} M_{c} + d_{2} + \bar{G} - {\dot{α}}_{1}) d t + G_{2} Λ Π d B

Similar to step 1, the RBFNNs will be utilized to tackle the unknown uncertainty ΔF₂, and the optimal approximation can be written as

Δ F_{2} = Φ_{2}^{*^{T}} S_{2} (Ω, ω) + δ_{2}

where the approximation error vector δ₂ satisfies $∥ δ_{2} ∥ \leq δ_{2}^{*}$ with $δ_{2}^{*} > 0$ .

Substituting equation (31) into equation (30) yields

d z_{2} = (F_{2} + Φ_{2}^{*^{T}} S_{2} + G_{2} M_{c} + D_{2} + \bar{G} - {\dot{α}}_{1}) d t + G_{2} Λ Π d B

where $D_{2} = δ_{2} + d_{2}$ is compounded disturbance, and it is bounded, that is, $∥ D_{2} ∥ \leq ζ_{2}$ with $ζ_{2} > 0$ . The estimation of the unknown parameter ζ₂ is ${\hat{ζ}}_{2} .$ And the parameter estimation error is ${\tilde{ζ}}_{2} = ζ_{2} - {\hat{ζ}}_{2}$ .

The control law M_c is designed as follows

M_{c} = - G_{2}^{- 1} (K_{2} z_{2} + \frac{ς^{4}}{4} z_{2} + F_{2} - {\dot{υ}}_{1} + {\hat{θ}}_{2} \frac{3 ∥ S_{2} ∥^{\frac{4}{3}} z_{2}}{4 τ_{2}^{\frac{4}{3}}} + {\hat{ζ}}_{2} \frac{3 z_{2}}{4 ς_{2}^{\frac{4}{3}}} + \hat{ρ} z_{2} (\frac{3}{4 ς_{4}^{2}} | | G_{2} {| |}_{F}^{4} ψ_{1}^{4} + \frac{3}{4 ς_{3}^{\frac{4}{3}}} ψ_{2}^{\frac{4}{3}}))

where $K_{2} = K_{2}^{T} > 0$ is a design matrix, and $τ_{2}, ς_{2}, ς_{3}, ς_{4}$ are design positive constants.

The parameter adaptive laws are designed as follows

{\dot{\hat{θ}}}_{2} = γ_{2} (\frac{3 ∥ S_{2} ∥^{\frac{4}{3}}}{4 τ_{2}^{\frac{4}{3}}} z_{2}^{4} - {\hat{θ}}_{2})

{\dot{\hat{ζ}}}_{2} = ξ_{2} (\frac{3}{4 ς_{2}^{\frac{4}{3}}} z_{2}^{4} - {\hat{ζ}}_{2})

\dot{\hat{ρ}} = ϱ ((\frac{3}{4 ς_{4}^{2}} | | G_{2} {| |}_{F}^{4} ψ_{1}^{4} + \frac{3}{4 ς_{3}^{\frac{4}{3}}} ψ_{2}^{\frac{4}{3}}) z_{2}^{4} - \hat{ρ})

where $γ_{2}, ξ_{2}, and ϱ$ are positive design parameters.

Consider a stochastic Lyapunov function as

V_{2} = V_{1} + \frac{1}{4} {(z_{2}^{T} z_{2})}^{2} + \frac{1}{2 γ_{2}} {\tilde{θ}}_{2}^{2} + \frac{1}{2 ξ_{2}} {\tilde{ζ}}_{2}^{2} + \frac{1}{2 ϱ} {\tilde{ρ}}^{2}

From equation (7), one has

L V_{2} = L V_{1} + z_{2}^{T} z_{2} z_{2}^{T} (F_{2} + Φ_{2}^{*^{T}} S_{2} + G_{2} M_{c} + D_{2} + \bar{G} - {\dot{α}}_{1}) + \frac{1}{2} t r (Π^{T} Λ^{T} G_{2}^{T} (2 z_{2} z_{2}^{T} + z_{2}^{T} z_{2} I) G_{2} Λ Π) - \frac{1}{γ_{2}} {\tilde{θ}}_{2} {\dot{\hat{θ}}}_{2} - \frac{1}{ξ_{2}} {\tilde{ζ}}_{2} {\dot{\hat{ζ}}}_{2} - \frac{1}{ϱ} \tilde{ρ} \dot{\hat{ρ}} = L V_{1} - z_{2}^{T} z_{2} z_{2}^{T} K_{2} z_{2} - z_{2}^{T} z_{2} z_{2}^{T} \frac{ς_{1}^{4}}{4} z_{2} + z_{2}^{T} z_{2} z_{2}^{T} Φ_{2}^{*^{T}} S_{2} + z_{2}^{T} z_{2} z_{2}^{T} \bar{G} + z_{2}^{T} z_{2} z_{2}^{T} D_{2} + \frac{1}{2} t r (Π^{T} Λ^{T} G_{2}^{T} (2 z_{2} z_{2}^{T} + z_{2}^{T} z_{2} I) G_{2} Λ Π) - {\hat{θ}}_{2} \frac{3 ∥ S_{2} ∥^{\frac{4}{3}}}{4 τ_{2}^{\frac{4}{3}}} {(z_{2}^{T} z_{2})}^{2} - {\hat{ζ}}_{2} \frac{3}{4 ς_{2}^{\frac{4}{3}}} {(z_{2}^{T} z_{2})}^{2} - \hat{ρ} (\frac{3}{4 ς_{4}^{2}} | | G_{2} {| |}_{F}^{4} ψ_{1}^{4} + \frac{3}{4 ς_{3}^{\frac{4}{3}}} ψ_{2}^{\frac{4}{3}}) {(z_{2}^{T} z_{2})}^{2} - \frac{1}{γ_{2}} {\tilde{θ}}_{2} {\dot{\hat{θ}}}_{2} - \frac{1}{ξ_{2}} {\tilde{ζ}}_{2} {\dot{\hat{ζ}}}_{2} - \frac{1}{ϱ} \tilde{ρ} \dot{\hat{ρ}}

According to Young’s inequality, one has

z_{2}^{T} z_{2} z_{2}^{T} Φ_{2}^{*^{T}} S_{2} \leq ∥ z_{2} ∥^{3} ∥ Φ_{2}^{*} ∥ ∥ S_{2} ∥ \leq θ_{2} \frac{3 ∥ S_{2} ∥^{\frac{4}{3}}}{4 τ_{2}^{\frac{4}{3}}} ∥ z_{2} ∥^{4} + \frac{θ_{2} τ_{2}^{4}}{4}

z_{2}^{T} z_{2} z_{2}^{T} D_{2} \leq ζ_{2} \frac{3}{4 ς_{2}^{\frac{4}{3}}} ∥ z_{2} ∥^{4} + \frac{ζ_{2} ς_{2}^{4}}{4}

z_{2}^{T} z_{2} z_{2}^{T} \bar{G} \leq \frac{3}{4 ς_{3}^{\frac{4}{3}}} ∥ z_{2} ∥^{4} g_{2}^{*}^{\frac{4}{3}} ρ_{2}^{\frac{4}{3}} ψ_{2}^{\frac{4}{3}} + \frac{ς_{3}^{4}}{4}

\begin{array}{l} \frac{1}{2} t r (Π^{T} Λ^{T} G_{2}^{T} (2 z_{2} z_{2}^{T} + z_{2}^{T} z_{2} I) G_{2} Λ Π) \leq \frac{3}{2} ∥ z_{2} ∥^{2} ∥ G_{2} Λ Π ∥_{F}^{2} \\ \leq \frac{3}{4 ς_{4}^{2}} ∥ G_{2} Λ Π ∥_{F}^{4} ∥ z_{2} ∥^{4} + \frac{3 ς_{4}^{2}}{4} \\ \leq \frac{3}{4 ς_{4}^{2}} ∥ z_{2} ∥^{4} ∥ G_{2} ∥_{F}^{4} ψ_{1}^{4} ∥ Π ∥_{F}^{4} ρ_{1}^{4} + \frac{3 ς_{4}^{2}}{4} \end{array}

{\tilde{θ}}_{2} {\hat{θ}}_{2} = \frac{1}{2} (θ_{2}^{2} - {\tilde{θ}}_{2}^{2} - {\hat{θ}}_{2}^{2}) \leq \frac{1}{2} (θ_{2}^{2} - {\tilde{θ}}_{2}^{2})

{\tilde{ζ}}_{2} {\hat{ζ}}_{2} = \frac{1}{2} (ζ_{2}^{2} - {\tilde{ζ}}_{2}^{2} - {\hat{ζ}}_{2}^{2}) \leq \frac{1}{2} (ζ_{2}^{2} - {\tilde{ζ}}_{2}^{2})

\tilde{ρ} \hat{ρ} \leq \frac{1}{2} (ρ^{2} - {\tilde{ρ}}^{2} - {\hat{ρ}}^{2}) \leq \frac{1}{2} (ρ^{2} - {\tilde{ρ}}^{2})

Substituting equations (39) to (45) into equation (38), we have

\begin{matrix} L V_{2} \leq L V_{1} - z_{2}^{T} z_{2} z_{2}^{T} K_{2} z_{2} - z_{2}^{T} z_{2} z_{2}^{T} \frac{ς^{4}}{4} z_{2} \\ + \frac{1}{2} (θ_{2}^{2} - {\tilde{θ}}_{2}^{2}) + \frac{1}{2} (ζ_{2}^{2} - {\tilde{ζ}}_{2}^{2}) \\ + \frac{1}{2} (ρ^{2} - {\tilde{ρ}}^{2}) + \frac{θ_{2} τ_{2}^{4}}{4} + \frac{ζ_{2} ς_{2}^{4}}{4} + \frac{ς_{3}^{4}}{4} + \frac{3 ς_{4}^{2}}{4} \end{matrix}

By combining with equation (29), it yields

\begin{array}{l} L V_{2} \leq - \sum_{j = 1}^{2} λ_{min} (K_{j}) {(z_{j}^{T} z_{j})}^{2} - \frac{1}{2} \sum_{j = 1}^{2} {\tilde{θ}}_{j}^{2} - \frac{1}{2} \sum_{j = 1}^{2} {\tilde{ζ}}_{j}^{2} \\ - \frac{1}{2} {\tilde{ρ}}^{2} + \frac{1}{2} \sum_{j = 1}^{2} θ_{j}^{2} + \frac{1}{2} \sum_{j = 1}^{2} ζ_{j}^{2} + \frac{1}{2} ρ^{2} \\ + \sum_{j = 1}^{2} \frac{θ_{j} τ_{j}^{4}}{4} + \sum_{j = 1}^{2} \frac{ζ_{j} ς_{j}^{4}}{4} + \frac{ς_{3}^{4}}{4} + \frac{3 ς_{4}^{2}}{4} \\ \leq - a_{0} V + b_{0} \end{array}

where

{\begin{array}{l} a_{0} = \min {λ_{\min} (K_{1}), λ_{\min} (K_{2}), γ_{j}, ξ_{j}, ρ, j = 1,2} \\ b_{0} = \frac{1}{2} \sum_{j = 1}^{2} θ_{j}^{2} + \frac{1}{2} \sum_{j = 1}^{2} ζ_{j}^{2} + \frac{1}{2} ρ^{2} + \sum_{j = 1}^{2} \frac{θ_{j} τ_{j}^{4}}{4} + \sum_{j = 1}^{2} \frac{ζ_{j} ς_{j}^{4}}{4} + \frac{ς_{3}^{4}}{4} + \frac{3 ς_{4}^{2}}{4} \end{array}

Thus, from equation (47), and according to the stochastic Lyapunov stability theory, the signals of the closed-loop system are uniformly bounded in the sense of probability. It may directly show that the tracking error z₁ is as small as possible by choosing suitable design parameters, and the estimate errors ${\tilde{θ}}_{i}, {\tilde{ζ}}_{i}, i = 1, 2$ and $\tilde{ρ}$ are all bounded.

Simulation study

In this section, simulation results are presented to illustrate the effectiveness of the proposed adaptive neural attitude control scheme for NSV with stochastic disturbances. The attitude dynamics of NSV with stochastic input torque is in the form of system (5).

Suppose that the aerodynamic coefficients and aerodynamic moment coefficients have +20% and −20% uncertainties. On the other hand, suppose that all the external disturbances are assumed to act on the fast-loop system in the form of the moment disturbances d₂ which are given as

d_{2} = [5 \times 10^{5} (sin (5 t) + 0.3),5 \times 10^{5} (cos (5 t) + 0.1), 5 \times 10^{5} sin (6 t)] Nm

The stochastic input disturbance is in the form of the Gaussian white noise $Λ (Ω, ω) ξ \in ℝ^{3}$ where the PSD of the white noise ξ is $Π = (Π_{i j})_{3 \times 3}$ with $Π_{i i} = 0.1$ and $Π_{i j} = 0, (i \neq j)$ , and the coefficient matrix is chosen as⁵⁷

Λ (Ω, ω) = [\begin{matrix} p + q & 0 & p \\ 0 & r & 0 \\ q sin (r) & p & 0 \end{matrix}]

Thus, we obtain

\bar{G} = G_{2} (Ω, ω) {[p + 0.5 q,0.5 p,0.5 q^{2} sin (r) \cos (r)]}^{T}

It is clear that the following inequalities hold

∥ Λ (Ω, ω) ∥_{F} = \sqrt{| p + q |^{2} + q^{2} sin {(r)}^{2} + 2 p^{2} + r^{2}} \leq \sqrt{4 p^{2} + 3 q^{2} + r^{2}} \leq 2 \sqrt{p^{2} + q^{2} + r^{2}} = ρ_{1} ψ_{1}

| | \bar{G} | | \leq 0.5 g_{2}^{*} \sqrt{7 p^{2} + 3 q^{2} + q^{4}} = g_{2}^{*} ρ_{2} ψ_{2}

where $ρ_{1} = 2, ρ_{2} =0.5, ψ_{1} = \sqrt{p^{2} + q^{2} + r^{2}}, and ψ_{2} = \sqrt{7 p^{2} + 3 q^{2} + q^{4}}$ .

The initial values are chosen as H ₀ = 21,000 m, V ₀ = 4000 m/s, α ₀ = 2°, β ₀ = −1°, μ ₀ = 3°, p = q = r = 0°/s. The desired flight attitudes Ω_d are chosen as

Ω_{d} = {\begin{array}{l} α_{d} = {\begin{matrix} - 2^{^{\circ}}, & 6 k \leq t \leq 6 k + 3 \\ 2^{^{\circ}}, & 6 k + 3 \leq t \leq 6 k + 6 \end{matrix} \\ β_{d} {= 0}^{^{\circ}} \\ μ_{d} = 1.7 sin (0.8 π t) + sin (0.3 π t) \end{array}

The designed parameters and matrices of the controller are chosen as $K_{1} = K_{2} = 30 diag {1, 1, 1},$ $γ_{1} = γ_{2} = 1,$ $ξ_{1} = ξ_{2} = 1, ρ = 2,$ ς = 1, $τ_{1} = τ_{2} = ς_{1} = ς_{2} = 5,$ and $ς_{3} = ς_{4} = 10$ . The intermediate control function υ₁, the controller M_c, and the adaptive laws are designed as equations (19), (33), (20), (21), (34), (35), and (36), respectively. From Figures 1 to 3, we observe that the tracking performance has been achieved under the effects of the stochastic disturbances and the system uncertainties. In addition, Figure 4 shows the variation of actual control input signals. From above simulation results, the proposed adaptive neural tracking control scheme is valid for stochastic nonlinear MIMO systems with system uncertainties and stochastic disturbances.

Figure 1.

Response curves of attitude angles under adaptive neural controller.

Figure 2.

Response curves of tracking errors under adaptive neural controller.

Figure 3.

Response curves of attitude angles rate under adaptive neural controller.

Figure 4.

Actual control inputs under adaptive neural controller.

Conclusions

In this article, an adaptive neural controller has been proposed for the NSV with stochastic disturbances and unknown uncertainties. The packaged unknown uncertainties are tackled using the RBFNNs. Furthermore, an effective robust adaptive backstepping-based control scheme has been designed. On the basis of the stochastic Lyapunov stability theory, the stability of the closed-loop system is proved, and the tracking error can converge to a small neighborhood of the origin. Finally, the effectiveness of the proposed robust attitude control scheme was illustrated via the simulation results.

Footnotes

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the National Natural Science Foundation of China under grants 61573184 and 61374212, in part by the Fundamental Research Funds for the Central Universities under grant NE2016101, and in part by the Projects of Anhui Province University Outstanding Youth Talent Support Program under grant gxyq2017068.

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