Adaptive finite-time neural backstepping control for multiple-input–multiple-output uncertain nonlinear systems with full state constraints

Abstract

This paper investigates the issue of finite-time tracking control for multiple-input–multiple-output nonlinear systems subject to uncertainties and full state constraints. To deal with full state constraints directly, integral barrier Lyapunov functionals (iBLF) are introduced. By using finite-time stability theory, an iBLF-based adaptive finite-time neural control scheme is presented. To solve the problem of “explosion of complexity” in the design of traditional backstepping control, a new finite-time convergent differentiator is presented. Through stability analysis, all closed-loop signals are proved to be semi-globally uniformly ultimately bounded, the finite time convergence can be guaranteed, and the state constraints are never violated. Finally, the attitude tracking simulations for an autonomous airship are conducted to verify the effectiveness of the proposed scheme.

Keywords

Adaptive control finite-time barrier Lyapunov functionals state constraints neural networks

Introduction

Recently, tremendous attention has focused on the tracking control problem associated with multiple-input–multiple-output (MIMO) uncertain nonlinear systems. Many meaningful control technologies, such as backstepping control, sliding-mode control, and H $\infty$ control, have been studied and applied in practice (see Aouaouda et al., 2015; Wang et al., 2015; Zhou et al., 2020).To compensate for the uncertainties, fuzzy logic systems (FLSs)/neural networks (NNs)-based backstepping methods have been widely examined. However, the issue of “explosion of complexity” in backstepping control has always been a difficult one, restricting its practical application. To handle this problem, dynamic surface control (DSC) was presented by Swaroop et al. (2000). Command-filtered backstepping was studied by using a high-order low-pass filter (Dong et al., 2012). Meanwhile, sliding-mode differentiator provides another effective way to deal with this problem (Yi et al., 2017).

In practice, the output/state of nonlinear systems are often subject to various constraints, such as aerial vehicles (Wei et al., 2019; Zheng and Xie, 2019), robot manipulation systems (He et al., 2018; Tang et al., 2016), and marine vehicles (Zhang and Wu, 2020). The transgression of constraints may reduce system performance or even lead to dangerous. In dealing with output constraints, a barrier Lyapunov function (BLF)-based control method was proposed by Tee et al. (2009), which is an effective solution to the problem of state-constrained control. Many researchers have solved the issue of practical systems subject constraints using BLF (Peng et al., 2019; Wei et al., 2020b; Xu et al., 2019). On this basis, the problem is extended to asymmetric time-varying constraints using asymmetric time-varying BLF (ATBLF) (Tee et al., 2011). Meanwhile, tan-type BLF-based approaches were presented to handle the systems with or without state constraints (Gao et al., 2019; Jin and Xu, 2014). Nevertheless, when using BLF-based control approaches, the state constraints must be transformed into error constraints. Moreover, the bounds of the virtual control signals need to be known exactly. To handle the state constraints directly, integral BLFs (iBLF)-based control methods were introduced by Tee and Ge (2012) and Wei et al. (2020a), which extend the initial feasible solution to the entire constraint space.

It is noteworthy that the studies above are based on asymptotic stability theory. However, the finite-time converges of the system output can not be guaranteed. In recent years, the finite-time convergence problem has been extensively investigated. In Yu et al. (2005), a finite-time control method based on the terminal sliding mode was presented. Meanwhile, many fruitful studies on finite-time control methods for uncertain nonlinear systems have been investigated (Guan et al., 2019; Li et al., 2019b; Sun et al., 2020; Wang et al., 2018b). Unfortunately, none of the finite-time control approaches mentioned above are applicable to state-constrained systems. To apply finite-time control method to state-constrained systems, log-type BLF-based finite-time control methods have been studied (see Li et al., 2018, 2019a). Meanwhile, tan-type BLF-based finite-time constrained control approaches have been investigated (Wang and Wu, 2019; Xia et al., 2019). Nevertheless, the above methods can not deal with the state constraint problem directly. The problem of finite-time control scheme using iBLF has also not been investigated.

The finite-time tracking control problem for a class of MIMO uncertain nonlinear state-constrained systems is studied in this paper. The main novelties of this paper are listed as follows:

Different from the existing BLF-based finite-time control methods (Li et al., 2019a; Xia et al., 2019), the finite-time control scheme proposed in this paper is an iBLF approach. By using iBLF, the state constraints can be handled directly. Thus, the initial feasible solutions of the system states can be relaxed.

Considering the existing adaptive finite-time control methods (Li et al., 2019; Wang et al., 2018a; 2018b; Zhao et al., 2018), the range of the key parameter $l$ is $l \in Ω_{l} : = {l | l = \frac{2 n - 1}{2 n + 1}, n \in N^{+}, n \geq 2}$ . In this paper, the range of $l$ is enlarged to $l \in Ω_{l} : = {l | \frac{1}{2} < l < 1}$ .

This paper is organized as follows. Some basic assumptions and preliminaries are presented in Section 2. The controller design process is presented in Section 3. The stability analysis is provided in Section 4. The simulations are provided in Section 5. Finally, the conclusion is presented in Section 6.

Problem formulation and preliminaries

System description and basic assumptions

Consider the MIMO strict-feedback nonlinear system as follows

{\begin{matrix} {\overset{\cdot}{x}}_{s, i_{s}} = f_{s, i_{s}} (ξ) + g_{s, i_{s}} (ξ) x_{s, i_{s} + 1}, \\ s = 1, 2, \dots, n, i_{s} = 1, \dots, m_{j} - 1 \\ {\overset{\cdot}{x}}_{s, m_{s}} = f_{s, m_{s}} (ξ) + g_{s, m_{s}} (ξ) u_{s} (t) \\ y_{s} = x_{s, 1}, \end{matrix}

(1)

where $x_{s, i_{s}}$ , $y_{s}$ , and $u_{s} (t)$ denote the states, output, and input of system (1), respectively. $ξ = [x_{1}^{T}, \dots, x_{n}^{T}]$ with $x_{s} = [x_{s, 1}, \dots, x_{s, m_{s}}]^{T}$ . $f_{s, i_{s}} (ξ)$ and $f_{s, m_{s}} (ξ)$ are unknown smooth nonlinear functions. $g_{s, i_{s}} (ξ)$ and $g_{s, m_{s}} (ξ)$ are known nonzero nonlinear functions.

The objective of this article is to construct an adaptive finite-time control approach using iBLF such that the system output is driven to track the reference trajectory in finite time, and the full state constraints are not transgressed.

Notations: To simplify the notations, without special explanation, $f_{s, i_{s}}$ , $g_{s, i_{s}}$ are utilized to denote $f_{s, i_{s}} (ξ)$ , $g_{s, i_{s}} (ξ)$ , respectively. $ƛ_{max} (Γ)$ and $ƛ_{min} (Γ)$ are utilized to represent the largest and smallest eigenvalues of $Γ$ , respectively.

Assumption 1: The control gains $g_{s, i_{s}}$ are known, and there exist unknown constants ${\bar{g}}_{s, i_{s}} > 0$ and ${\underline{g}}_{s, i_{s}} > 0$ that make the inequality $0 < {\underline{g}}_{s, i_{s}} < | g_{s, i_{s}} | < {\bar{g}}_{s, i_{s}}$ holds.

Assumption 2: The reference trajectory $y_{r} (t) \in R^{m}$ is continuously differentiable, and for any $b_{s, 1} > 0$ , there exist positive constants $Q_{0 s}$ and $Y_{0 s}$ , $s = 1, \dots, n$ , such that the signal $y_{rs} (t)$ and its time derivatives meet the following inequalities

| y_{rs} (t) | \leq Q_{0 s} < b_{s, 1} and | y_{rs}^{i} (t) | < Y_{0 j},

(2)

for $t \geq 0$ and $i = 1, 2, \dots, n - 1$ .

Remark 1: Assumption 1 and Assumption 2 are standard assumptions in trajectory-tracking problems (Chen et al., 2015; Liu et al., 2019). Assumption 1 is a sufficient condition to guarantee the design of the controller and the virtual control effectively. Without loss of generality, we assume that $g_{s, i_{s}} > 0$ .

Finite time

Definition 1: (Wang et al., 2018b) The equilibrium $η = 0$ of the nonlinear system $\overset{\cdot}{η} = f (η, u)$ is semi-global practical finite-time stable (SGPFS) if for all $η (t_{0}) = η_{0}$ , if there exist a positive scalar $ε$ and a setting time $T (ε, η_{0}) < \infty$ such that the inequality $∥ η ∥ < ε$ hold for all $t \geq t_{0} + T$ .

Lemma 1: (Wang et al., 2018a) Consider the system $\overset{\cdot}{η} = f (η, u)$ . If for smooth positive definite function $V (η)$ , there exist scalars $c_{1} > 0$ , $0 < l < 1$ , and $c_{2} > 0$ , the following inequality holds

\overset{\cdot}{V} (η) \leq - c_{1} V^{l} (η) + c_{2}, t \geq 0 .

(3)

Then, the system $\overset{\cdot}{η} = f (η, u)$ is SGPFS.

Lemma 2: (Qian and Lin, 2015) The following inequality holds for any real variables $ν$ , $η$ , and any positive constants $μ$ , $l$ , and $ℏ$

{| ν |}^{μ} {| η |}^{l} \leq \frac{μ}{μ + κ} ℏ {| ν |}^{μ + l} + \frac{κ}{μ + κ} ℏ^{\frac{- μ}{l}} {| η |}^{μ + l} .

(4)

Lemma 3: (Hardy et al., 1952) For $Z_{s} \in R$ , $s = 1, 2, \dots, n$ , $0 < l < 1$ , the following inequality holds

{(\sum_{s = 1}^{n} | Z_{s} |)}^{l} \leq \sum_{s = 1}^{n} {| Z_{s} |}^{l} \leq n^{1 - l} {(\sum_{s = 1}^{n} | Z_{s} |)}^{l} .

(5)

IBLF

Define the tracking error as $e_{s, i_{s}}$ . $α_{s, i_{s}}$ denotes the virtual control law. Consider the following iBLF candidate as follows

V_{s, i_{s}}^{e} = \int_{0}^{e_{s, i_{s}}} \frac{τ b_{s, i_{s}}^{2}}{b_{s, i_{s}}^{2} - {(τ + a_{s, i_{s} - 1})}^{2}} d τ,

(6)

where $e_{s, i_{s}} = x_{s, i_{s}} - α_{s, i_{s} - 1}$ , $α_{s, 0} = y_{rs}$ , and $α_{s, 1}, \dots, α_{s, i_{s}}$ are continuously differentiable functions, $b_{s, i_{s}}$ is the constraint of the system state. There exists a constant $Q_{s, i_{s}}$ that holds the following inequality

| a_{s, i_{s} - 1} | \leq Q_{s, i_{s}} < b_{s, i_{s}} .

(7)

Thus, $V_{s, i_{s}}^{e}$ Under the set $x_{s, i_{s}} < b_{s, i_{s}}$ , the following inequality can be obtained

\frac{e_{s, i_{s}}^{2}}{2} \leq V_{s, r_{s}}^{e} \leq e_{s, i_{s}}^{2} \int_{0}^{1} \frac{τ b_{s, i_{s}}^{2}}{b_{s, i_{s}}^{2} - {(τ e_{s, i_{s}} + sign e_{s, i_{s}} Q_{s, i_{s}})}^{2}} d τ .

(8)

Lemma 4: (Tee and Ge, 2012) Consider the iBLF $V_{s, i_{s}}^{e}$ in (6), the following inequality can be guaranteed

V_{s, i_{s}}^{e} \leq \frac{b_{s, i_{s}}^{2} e_{s, i_{s}}}{b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2}},

(9)

for $| x_{s, i_{s}} | < b_{s, i_{s}}$ .

Radial basis function neural networks

RBF NNs is utilized to approximate $F (ξ)$ . The following expression can be obtained as

F (ξ) = W^{* T} S (ξ) + ε (ξ),

(10)

where $S (ξ)$ denotes the basis function vector, $ξ$ denotes the input vector, $ε (ξ)$ denotes the approximation error, and $W^{*} \in R^{β}$ denotes the ideal weight vector.For the optimal weight vector $W^{*}$ , we can get the following form

W^{*} = \arg min_{\hat{W} \in Ω_{f}} [sup_{Z \in S_{Z}} | \hat{F} (ξ | \hat{W}) - F (ξ) |] .

(11)

Adaptive finite-time controller design

Step 1: To facilitate the controller design, the tracking errors are defined as follows

\begin{matrix} e_{s, 1} = x_{s, 1} - y_{rs} \\ e_{s, i_{s}} = x_{s, i_{s}} - α_{s, i_{s} - 1}, \end{matrix}

(12)

where $e_{s, 1}$ and $e_{s, i_{s}}$ denote the tracking errors, $y_{rs}$ denotes the reference trajectory, $α_{s, i_{s} - 1}$ is the virtual control. Then, the time derivative of $e_{s, 1}$ can be obtained as follows

\begin{matrix} {\overset{\cdot}{e}}_{s, 1} = f_{s, 1} + g_{s, 1} x_{s, 2} - {\overset{\cdot}{y}}_{rs} \\ = W_{s, 1}^{* T} S_{s, 1} + g_{s, 1} x_{s, 2} - {\overset{\cdot}{y}}_{rs} + ε_{s, 1} . \end{matrix}

(13)

Define ${\hat{W}}_{s, 1}$ as the estimation of $W_{s, 1}^{*}$ . Then, we have ${\tilde{W}}_{s, 1} = {\hat{W}}_{s, 1} - W_{s, 1}^{*}$ .

Construct the Lyapunov function candidate (LFC) as below

\begin{matrix} V_{1} = \sum_{s = 1}^{n} (V_{s, 1}^{e} + \frac{1}{2} {\tilde{W}}_{s, 1}^{T} Γ_{s, 1}^{- 1} {\tilde{W}}_{s, 1}), \end{matrix}

(14)

where $Γ_{s, 1} = Γ_{s, 1}^{T} > 0$ is an adaptation gain parameter. Then, the time derivative of $V_{1}$ can be obtained as follows

\begin{matrix} {\overset{\cdot}{V}}_{1} = \sum_{s = 1}^{n} (\frac{b_{s, 1}^{2} e_{s, 1} {\overset{\cdot}{e}}_{s, 1}}{b_{s, 1}^{2} - x_{s, 1}^{2}} + \frac{\partial V_{s, 1}^{e}}{\partial y_{rs}} {\overset{\cdot}{y}}_{rs} + {\tilde{W}}_{s, 1}^{T} Γ_{s, 1}^{- 1} {\overset{\cdot}{\hat{W}}}_{s, 1}) . \end{matrix}

(15)

By using the substitution $τ = β e_{s, 1}$ (Ge et al., 1999), we have

\begin{matrix} \frac{\partial V_{s, 1}^{e}}{\partial y_{rs}} = e_{s, 1} [\frac{b_{s, 1}^{2}}{b_{s, 1}^{2} - x_{s, 1}^{2}} - γ_{s, 1} (e_{s, 1}, y_{rs})], \end{matrix}

(16)

where

\begin{matrix} γ_{s, 1} = \int_{0}^{1} \frac{τ b_{s, 1}^{2}}{b_{s, 1}^{2} - {(τ e_{s, 1} + y_{rs})}^{2}} d τ \\ = 2 \frac{b_{s, 1}}{e_{s, 1}} \ln \frac{(b_{s, 1} + e_{s, 1} + y_{rs}) (b_{s, 1} - y_{rs})}{(b_{s, 1} - e_{s, 1} - y_{rs}) (b_{s, 1} + y_{rs})} . \end{matrix}

(17)

Using L’Hopital’s Rule, we can infer that

\begin{matrix} lim_{e_{s, 1} \to 0} γ_{s, 1} = lim_{e_{s, 1} \to 0} \frac{b_{s, 1}^{2}}{b_{s, 1}^{2} - {(e_{s, 1} + y_{rs})}^{2}} \\ = \frac{b_{s, 1}^{2}}{b_{s, 1}^{2} - y_{rs}^{2}} . \end{matrix}

(18)

From Assumption 2, we have $| y_{rs} | < b_{s, 1}$ . Thus, we conclude that $γ_{s, 1}$ is well defined in a neighborhood of $e_{s, 1} = 0$ .

Based on (13), (16), then (15) becomes

\begin{matrix} {\overset{\cdot}{V}}_{1} = \sum_{s = 1}^{n} [\frac{b_{s, 1}^{2} e_{s, 1}}{b_{s, 1}^{2} - x_{s, 1}^{2}} (W_{s, 1}^{* T} S_{s, 1} + g_{s, 1} e_{s, 2} + g_{s, 1} α_{s, 1} - {\overset{\cdot}{y}}_{rs} \\ + ε_{s, 1}) + \frac{b_{s, 1}^{2} e_{s, 1} {\overset{\cdot}{y}}_{rs}}{b_{s, 1}^{2} - x_{s, 1}^{2}} - γ_{s, 1} {\overset{\cdot}{y}}_{rs} + {\tilde{W}}_{s, 1}^{T} Γ_{s, 1}^{- 1} {\overset{\cdot}{\hat{W}}}_{s, 1}] . \end{matrix}

(19)

Then, the virtual control signal is given as follows

\begin{matrix} α_{s, 1} = g_{s, 1}^{- 1} [\frac{(b_{s, 1}^{2} - x_{s, 1}^{2}) {\overset{\cdot}{y}}_{rs} γ_{s, 1}}{b_{s, 1}^{2}} - \frac{b_{s, 1}^{2} e_{s, 1}}{b_{s, 1}^{2} - x_{s, 1}^{2}} \\ - \frac{k_{s, 1} b_{s, 1}^{2 l - 2} sign (e_{s, 1}) {| e_{s, 1} |}^{2 l - 1}}{{(b_{s, 1}^{2} - x_{s, 1}^{2})}^{l - 1}} - {\hat{W}}_{s, 1}^{T} S_{s, 1}], \end{matrix}

(20)

where $k_{s, 1} > 0$ , $l \in Ω_{l} : = {l \in R : \frac{1}{2} < l < 1}$ are constants.

The adaptive law is designed as follows

\begin{matrix} {\overset{\cdot}{\hat{W}}}_{s, 1} = Γ_{s, 1} (\frac{b_{s, 1}^{2} e_{s, 1}}{b_{s, 1}^{2} - x_{s, 1}^{2}} S_{s, 1} - κ_{s, 1} {\hat{W}}_{s, 1}), \end{matrix}

(21)

where $κ_{s, 1} > 0$ is a parameter.

By substituting (20) and (21) into (19), we have

\begin{matrix} {\overset{\cdot}{V}}_{1} = \sum_{s = 1}^{n} [- k_{s, 1} {(\frac{b_{s, 1}^{2} e_{s, 1}^{2}}{b_{s, 1}^{2} - x_{s, 1}^{2}})}^{l} - {(\frac{b_{s, 1}^{2} e_{s, 1}}{b_{s, 1}^{2} - x_{s, 1}^{2}})}^{2} \\ + \frac{b_{s, 1}^{2} e_{s, 1} ε_{s, 1}}{b_{s, 1}^{2} - x_{s, 1}^{2}} + \frac{b_{s, 1}^{2} e_{s, 1} g_{s, 1} e_{s, 2}}{b_{s, 1}^{2} - x_{s, 1}^{2}} + κ_{s, 1} {\tilde{W}}_{s, 1} {\hat{W}}_{s, 1}] . \end{matrix}

(22)

According to Young’s inequality, we have

\begin{matrix} \frac{b_{s, 1}^{2} e_{s, 1} g_{s, 1} e_{s, 2}}{b_{s, 1}^{2} - x_{s, 1}^{2}} \leq \frac{1}{2} {(\frac{b_{s, 1}^{2} e_{s, 1}}{b_{s, 1}^{2} - x_{s, 1}^{2}})}^{2} + \frac{1}{2} g_{s, 1}^{2} e_{s, 2}^{2} \\ \frac{b_{s, 1}^{2} e_{s, 1} ε_{s, 1}}{b_{s, 1}^{2} - x_{s, 1}^{2}} \leq \frac{1}{2} {(\frac{b_{s, 1}^{2} e_{s, 1}}{b_{s, 1}^{2} - x_{s, 1}^{2}})}^{2} + \frac{1}{2} ε_{s, 1}^{2} . \\ {\tilde{W}}_{s, 1} {\hat{W}}_{s, 1} \leq \frac{1}{2} ({‖ W_{s, 1}^{*} ‖}^{2} - {‖ {\tilde{W}}_{s, 1} ‖}^{2}) . \end{matrix}

(23)

Submitting (23) into (22), we have

\begin{matrix} {\overset{\cdot}{V}}_{1} = \sum_{s = 1}^{n} [- k_{s, 1} {(\frac{b_{s, 1}^{2} e_{s, 1}^{2}}{b_{s, 1}^{2} - x_{s, 1}^{2}})}^{l} - \frac{κ_{s, 1}}{2} ‖ {\tilde{W}}_{s, 1} ‖^{2} \\ \frac{1}{2} g_{s, 1}^{2} e_{s, 2}^{2} + \frac{1}{2} ε_{s, 1}^{2} + \frac{κ_{s, 1}}{2} ∥ W_{s, 1}^{*} ∥^{2}] . \end{matrix}

(24)

According to Lemma 2, define $ν = 1$ , $η = ‖ {\tilde{W}}_{s, 1} ‖^{2}$ , $μ = 1 - l$ , and $ℏ = l^{\frac{l}{1 - l}}$ , we have

\begin{matrix} {({‖ {\tilde{W}}_{s, 1} ‖}^{2})}^{l} \leq (1 - l) ℏ + {‖ {\tilde{W}}_{s, 1} ‖}^{2} . \end{matrix}

(25)

Submitting (25) into (24), we can obtain

\begin{matrix} {\overset{\cdot}{V}}_{1} = \sum_{s = 1}^{n} [- k_{s, 1} {(\frac{b_{s, 1}^{2} e_{s, 1}^{2}}{b_{s, 1}^{2} - x_{s, 1}^{2}})}^{l} - \frac{κ_{s, 1}}{2} ‖ {\tilde{W}}_{s, 1} ‖^{l} \\ + \frac{1}{2} g_{s, 1}^{2} e_{s, 2}^{2}] + Δ_{1}, \end{matrix}

(26)

where $Δ_{1} = \sum_{s = 1}^{n} \frac{1}{2} ε_{s, 1}^{2} + \frac{κ_{s, 1}}{2} (1 - l) ℏ + \frac{κ_{s, 1}}{2} ∥ W_{s, 1}^{*} ∥^{2}$ .

According to Lemma 4, we can get the following inequality

\begin{matrix} {(\frac{b_{s, 1}^{2} e_{s, 1}^{2}}{b_{s, 1}^{2} - x_{s, 1}^{2}})}^{l} \geq {(\int_{0}^{e_{s, 1}} \frac{τ b_{s, 1}^{2}}{b_{s, 1}^{2} - {(τ - y_{rs})}^{2}} d τ)}^{l} . \end{matrix}

(27)

Substituting (27) into (26), we have

\begin{matrix} {\overset{\cdot}{V}}_{1} = \sum_{s = 1}^{n} [- k_{s, 1} {(\int_{0}^{e_{s, 1}} \frac{τ b_{s, 1}^{2}}{b_{s, 1}^{2} - {(τ - y_{rs})}^{2}} d τ)}^{l} \\ - \frac{κ_{s, 1}}{2} ‖ {\tilde{W}}_{s, 1} ‖^{l} + \frac{1}{2} g_{s, 1}^{2} e_{s, 2}^{2}] + Δ_{1} . \end{matrix}

(28)

Step $i_{s}$ ( $i_{s} = 2, \dots, m - 1$ ) We can get the time derivative of $e_{s, i_{s}}$ as follows

\begin{matrix} {\overset{\cdot}{e}}_{s, i_{s}} = f_{s, i_{s}} + g_{s, i_{s}} x_{s, i_{s} + 1} - {\overset{\cdot}{a}}_{s, i_{s} - 1} \\ = W_{s, i_{s}}^{* T} S_{s, i_{s}} + g_{s, i_{s}} x_{s, i_{s} + 1} - {\overset{\cdot}{a}}_{s, i_{s} - 1} + ε_{s, i_{s}} . \end{matrix}

(29)

Consider the LFC as follows

\begin{matrix} V_{i_{s}} = \sum_{s = 1}^{n} (V_{s, i_{s}}^{e} + \frac{1}{2} {\tilde{W}}_{s, i_{s}}^{T} Γ_{s, i_{s}}^{- 1} {\tilde{W}}_{s, i_{s}}), \end{matrix}

(30)

where $Γ_{s, i_{s}} = Γ_{s, i_{s}}^{T} > 0$ is a parameter. The time derivative of $V_{r_{s}}$ can be obtained as follows

\begin{matrix} {\overset{\cdot}{V}}_{i_{s}} = \sum_{s = 1}^{n} (\frac{b_{s, i_{s}}^{2} e_{s, i_{s}} {\overset{\cdot}{e}}_{s, rj}}{b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2}} + \frac{\partial V_{s, i_{s}}^{e}}{\partial a_{s, i_{s} - 1}} {\overset{\cdot}{α}}_{s, i_{s} - 1} \\ + {\tilde{W}}_{s, i_{s}}^{T} Γ_{s, i_{s}}^{- 1} {\hat{W}}_{s, i_{s}}) . \end{matrix}

(31)

By using the substitution $τ = β e_{s, i_{s}}$ , we have

\begin{matrix} \frac{\partial V_{s, i_{s}}^{e}}{\partial α_{s, i_{s}}} = e_{s, i_{s}} [\frac{b_{s, i_{s}}^{2}}{b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2}} - γ_{s, i_{s}} (e_{s, i_{s}}, α_{s, i_{s}})], \end{matrix}

(32)

where

\begin{matrix} γ_{s, i_{s}} = \int_{0}^{1} \frac{τ b_{s, i_{s}}^{2}}{b_{s, i_{s}}^{2} - {(τ e_{s, i_{s}} + a_{s, i_{s} - 1})}^{2}} d τ \\ = 2 \frac{b_{s, i_{s}}}{e_{s, i_{s}}} \ln \frac{(b_{s, i_{s}} + e_{s, i_{s}} + a_{s, i_{s} - 1}) (b_{s, i_{s}} - a_{s, i_{s} - 1})}{(b_{s, i_{s}} - e_{s, i_{s}} - a_{s, i_{s} - 1}) (b_{s, i_{s}} + a_{s, i_{s} - 1})} . \end{matrix}

(33)

Using L’Hopital’s Rule, we know that

\begin{matrix} lim_{e_{s, i_{s}} \to 0} γ_{s, i_{s}} = lim_{e_{s, i_{s}} \to 0} \frac{b_{s, i_{s}}^{2}}{b_{s, i_{s}}^{2} - {(e_{s, i_{s}} + a_{s, i_{s} - 1})}^{2}} \\ = \frac{b_{s, i_{s}}^{2}}{b_{s, i_{s}}^{2} - a_{s, i_{s} - 1}^{2}} . \end{matrix}

(34)

Similarly, we conclude that $γ_{s, i_{s}}$ is well-defined in a neighborhood of $e_{s, i_{s}}$ . Substituting (29) and (32) into (31), we have

\begin{array}{l} {\dot{V}}_{i_{s}} = \sum_{s = 1}^{n} [\frac{b_{s, i_{s}}^{2} e_{s, i_{s}}}{b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2}} (W_{s, i_{s}}^{* T} S_{s, i_{s}} + g_{s, i_{s}} e_{s, i_{s} + 1} \\ + g_{s, i_{s}} a_{s, i_{s}} - {\dot{α}}_{s, i_{s} - 1} + ε_{s, i_{s}}) + \frac{b_{s, i_{s}}^{2} e_{s, i_{s}} {\dot{α}}_{s, i_{s} - 1}}{b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2}}] . \\ - γ_{s, i_{s}} {\dot{α}}_{s, i_{s} - 1} + {\tilde{W}}_{s, i_{s}}^{T} Γ_{s, i_{s}}^{- 1} {\dot{\hat{W}}}_{s, i_{s}} \end{array}

(35)

To solve the problem of “explosion of complexity” caused by the repeated derivation of ${\overset{\cdot}{a}}_{s, i_{s} - 1}$ , a new FTCD is introduced as follows

{\begin{matrix} {\overset{\cdot}{a}}_{s, i_{s} - 1}^{c} = {\overset{\cdot}{a}}_{s, i_{s} - 1}^{d} - L_{1} h_{1} \\ {\overset{\cdot}{a}}_{s, i_{s} - 1}^{d} = - L_{2} h_{2} \end{matrix}

(36)

{\begin{matrix} h_{1} = {| z_{s, i_{s}} |}^{1 / 2} sign (z_{s, i_{s}}) + β z_{s, i_{s}} \\ h_{2} = sign (z_{s, i_{s}}) - β {| z_{s, i_{s}} |}^{1 / 2} sign (z_{s, i_{s}}) + β^{2} z_{s, i_{s}}, \end{matrix}

(37)

where $L_{1}$ , $L_{2}$ , and $β$ are positive constants to be designed; $z_{s, i_{s}} = a_{s, i_{s} - 1}^{c} - a_{s, i_{s} - 1}$ represents the estimation error; $a_{s, i_{s} - 1}^{c}$ and ${\overset{\cdot}{a}}_{s, i_{s} - 1}^{c}$ are the estimation of $a_{s, i_{s} - 1}$ and ${\overset{\cdot}{a}}_{s, i_{s} - 1}$ , respectively. From the analysis by Castillo et al. (2018) that the estimation error of the FTCD will converge to zero in finite time. Therefore, we can get the following expression

{\overset{\cdot}{a}}_{s, i_{s} - 1} = {\overset{\cdot}{a}}_{s, i_{s} - 1}^{c} + δ_{s, i_{s} - 1},

(38)

where $δ_{s, i_{s} - 1}$ is a bound constant.

The virtual control law is designed as

\begin{matrix} α_{s, i_{s}} = g_{s, i_{s}}^{- 1} [\frac{b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2}}{b_{s, i_{s}}^{2}} ({\overset{\cdot}{a}}_{s, i_{s} - 1}^{c} γ_{s, i_{s}} - \frac{e_{s, i_{s}} γ_{s, i_{s}}^{2}}{2}) \\ - \frac{b_{s, i_{s}}^{2} e_{s, i_{s}}}{b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2}} - \frac{k_{s, i_{j}} b_{s, i_{s}}^{2 l - 2} sign (e_{s, i_{s}}) {| e_{s, i_{s}} |}^{2 l - 1}}{{(b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2})}^{l - 1}} \\ - {\hat{W}}_{s, i_{s}}^{T} S_{s, i_{s}} - \frac{(b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2}) g_{s, i_{s} - 1}^{2} e_{s, i_{s}}}{2 b_{s, i_{s}}^{2}}], \end{matrix}

(39)

where $k_{s, i_{s}} > 0$ is a design parameter.

The adaptive weight is designed as

\begin{matrix} {\overset{\cdot}{\hat{W}}}_{s, i_{s}} = Γ_{s, i_{s}} (\frac{b_{s, i_{s}}^{2} e_{s, i_{s}}}{b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2}} S_{s, i_{s}} - κ_{s, i_{s}} {\hat{W}}_{s, i_{s}}), \end{matrix}

(40)

where $κ_{s, i_{s}}$ is a positive constant.

Substituting (38), (39), and (40) into (35), we can obtain

\begin{matrix} {\overset{\cdot}{V}}_{i_{s}} = \sum_{s = 1}^{n} [- k_{s, i_{s}} {(\frac{b_{s, i_{s}}^{2} e_{s, i_{s}}}{b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2}})}^{l} - {(\frac{b_{s, i_{s}}^{2} e_{s, i_{s}}}{b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2}})}^{2} \\ + \frac{b_{s, i_{s}}^{2} e_{s, i_{s}} ε_{s, i_{s}}}{b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2}} - \frac{e_{s, i_{s}}^{2} γ_{s, i_{s}}^{2}}{2} - \frac{1}{2} g_{s, i_{s}}^{2} e_{s, i_{s}}^{2} \\ - δ_{s, i_{s} - 1} e_{s, i_{s}} γ_{s, i_{s}} + \frac{b_{s, i_{s}}^{2} e_{s, i_{s}} g_{s, i_{s}} e_{s, i_{s} + 1}}{b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2}} \\ + κ_{s, i_{s}} {\tilde{W}}_{s, i_{s}}^{T} {\hat{W}}_{s, i_{s}}] . \end{matrix}

(41)

By utilizing Young’s inequality, we have

\begin{matrix} \frac{b_{s, i_{s}}^{2} g_{s, i_{s}} e_{s, i_{s}} e_{s, i_{s} + 1}}{b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2}} \leq \frac{1}{2} {(\frac{b_{s, i_{s}}^{2} e_{s, i_{s}}}{b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2}})}^{2} + \frac{1}{2} g_{s, i_{s}}^{2} e_{s, i_{s} + 1}^{2}, \\ \frac{b_{s, i_{s}}^{2} e_{s, i_{s}} ε_{s, i_{s}}}{b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2}} \leq \frac{1}{2} {(\frac{b_{s, i_{s}}^{2} e_{s, i_{s}}}{b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2}})}^{2} + \frac{ε_{s, i_{s}}^{2}}{2}, \\ - δ_{s, i_{s} - 1} e_{s, i_{s}} γ_{s, i_{s}} \leq \frac{e_{s, i_{s}}^{2} γ_{s, i_{s}}^{2}}{2} + \frac{δ_{s, i_{s} - 1}^{2}}{2}, \\ {\tilde{W}}_{s, i_{s}}^{T} {\hat{W}}_{s, i_{s}} \leq \frac{1}{2} ({‖ W_{s, i_{s}}^{*} ‖}^{2} - {‖ {\tilde{W}}_{s, i_{s}} ‖}^{2}) . \end{matrix}

(42)

Submitting (42) into (41), we can obtain

\begin{matrix} {\overset{\cdot}{V}}_{i_{s}} \leq \sum_{s = 1}^{n} [- k_{s, i_{s}} {(\frac{b_{s, i_{s}}^{2} e_{s, i_{s}}}{b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2}})}^{l} - \frac{κ_{s, i_{s}}}{2} ‖ {\tilde{W}}_{s, i_{s}} ‖^{2} \\ - \frac{1}{2} g_{s, i_{s} - 1}^{2} e_{s, i_{s}}^{2} + \frac{1}{2} g_{s, i_{s}}^{2} e_{s, i_{s} + 1}^{2} \\ + \frac{κ_{s, i_{s}}}{2} ‖ W_{s, i_{s}}^{*} ‖^{2} + \frac{ε_{s, i_{s}}^{2}}{2} + \frac{δ_{s, i_{s} - 1}^{2}}{2}] . \end{matrix}

(43)

According to Lemma 2, we have

\begin{matrix} {({‖ {\tilde{W}}_{s, i_{s}} ‖}^{2})}^{l} \leq (1 - l) ℏ + {‖ {\tilde{W}}_{s, i_{s}} ‖}^{2} . \end{matrix}

(44)

By introducing (44) into (43), we can obtain

\begin{matrix} {\overset{\cdot}{V}}_{i_{s}} \leq \sum_{s = 1}^{n} [- k_{s, i_{s}} {(\frac{b_{s, i_{s}}^{2} e_{s, i_{s}}}{b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2}})}^{l} - \frac{κ_{s, i_{s}}}{2} ‖ {\tilde{W}}_{s, i_{s}} ‖^{2 l} \\ + \frac{1}{2} g_{s, i_{s}}^{2} e_{s, i_{s} + 1}^{2}] + Δ_{i_{s}}, \end{matrix}

(45)

where $Δ_{i_{s}} = \sum_{j = 1}^{m} \frac{κ_{s, i_{s}}}{2} ∥ W_{s, i_{s}}^{*} ∥ + \frac{ε_{s, i_{s}}^{2}}{2} + \frac{δ_{s, i_{s} - 1}^{2}}{2} + \frac{κ_{s, i_{s}}}{2} (1 - l) ℏ$ .

According to Lemma 4, the following inequality can be obtained

{(\frac{b_{s, i_{s}}^{2} e_{s, i_{s}}^{2}}{b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2}})}^{l} \geq {(\int_{0}^{e_{s, i_{s}}} \frac{τ b_{s, i_{s}}^{2}}{b_{s, i_{s}}^{2} - {(τ - α_{s, i_{s} - 1})}^{2}} d τ)}^{l} .

(46)

Substituting (46) into (45), we have

\begin{matrix} {\overset{\cdot}{V}}_{i_{s}} \leq \sum_{s = 1}^{n} [- k_{s, i_{s}} {(\int_{0}^{e_{s, i_{s}}} \frac{τ b_{s, i_{s}}^{2}}{b_{s, i_{s}}^{2} - {(τ - α_{s, i_{s} - 1})}^{2}} d τ)}^{l} \\ - \frac{κ_{s, i_{s}}}{2} ‖ {\tilde{W}}_{s, i_{s}} ‖^{2 l} + \frac{1}{2} g_{s, i_{s}}^{2} e_{s, i_{s} + 1}^{2}] + Δ_{i_{s}} . \end{matrix}

(47)

Step $m_{s}$ Define $e_{s, m_{s}} = x_{s, m_{s}} - α_{s, m_{s} - 1}$ , where $α_{s, m_{s} - 1}$ is the virtual control signal designed in step $m_{s} - 1$ . Then, we can get the time derivative of $e_{s, m_{s}}$ as follows

\begin{matrix} {\overset{\cdot}{e}}_{s, m_{s}} = f_{s, m_{s}} + g_{s, m_{s}} u_{s} - {\overset{\cdot}{a}}_{s, m_{s} - 1} \\ = W_{s, m_{s}}^{* T} S_{s, m_{s}} + g_{s, m_{s}} u_{s} - {\overset{\cdot}{a}}_{s, m_{s} - 1} + ε_{s, m_{s}} . \end{matrix}

(48)

Consider the LFC as follows

\begin{matrix} V_{m_{s}} = \sum_{s = 1}^{n} (V_{s, m_{s}}^{e} + \frac{1}{2} {\tilde{W}}_{s, m_{s}}^{T} Γ_{s, m_{s}}^{- 1} {\hat{W}}_{s, m_{s}}), \end{matrix}

(49)

where $Γ_{s, m_{s}} = Γ_{s, m_{s}}^{T} > 0$ . Then, the time derivative of $V_{m_{s}}$ can be obtained as

\begin{matrix} {\overset{\cdot}{V}}_{m_{s}} = \sum_{s = 1}^{n} (\frac{b_{s, m_{s}}^{2} e_{s, m_{s}} {\overset{\cdot}{e}}_{s, m_{s}}}{b_{s, m_{s}}^{2} - x_{s, m_{s}}^{2}} + \frac{\partial V_{s, m_{s}}^{e}}{\partial α_{s, m_{s} - 1}} {\overset{\cdot}{α}}_{s, m_{s} - 1} + {\tilde{W}}_{s, m_{s}}^{T} Γ_{s, m_{s}}^{- 1} {\overset{\cdot}{\hat{W}}}_{s, m_{s}}) . \end{matrix}

(50)

By using the substitution $τ = β e_{s, m_{s}}$ , we have

\begin{matrix} \frac{\partial V_{s, m_{s}}^{e}}{\partial α_{s, m_{s}}} = e_{s, m_{s}} (\frac{b_{s, m_{s}}^{2}}{b_{s, m_{s}}^{2} - x_{s, m_{s}}^{2}} - γ_{s, m_{s}}), \end{matrix}

(51)

where

\begin{matrix} γ_{s, m_{s}} = \int_{0}^{1} \frac{τ b_{s, m_{s}}^{2}}{b_{s, m_{s}}^{2} - {(τ e_{s, m_{s}} + α_{s, m_{s} - 1})}^{2}} d τ \\ = 2 \frac{b_{s, m_{s}}}{e_{s, m_{s}}} \ln \frac{(b_{s, m_{s}} + e_{s, m_{s}} + α_{s, m_{s} - 1}) (b_{s, m_{s}} - α_{s, m_{s} - 1})}{(b_{s, m_{s}} - e_{s, m_{s}} - α_{s, m_{s} - 1}) (b_{s, m_{s}} + α_{s, m_{s} - 1})} . \end{matrix}

(52)

Using L’Hopital’s Rule, we know that

\begin{matrix} lim_{e_{s, m_{s}} \to 0} γ_{s, m_{s}} = lim_{e_{s, m_{s}} \to 0} \frac{b_{s, m_{s}}^{2}}{b_{s, m_{s}}^{2} - {(e_{s, m_{s}} + α_{s, m_{s} - 1})}^{2}} \\ = \frac{b_{s, m_{s}}^{2}}{b_{s, m_{s}}^{2} - α_{s, m_{s} - 1}^{2}} . \end{matrix}

(53)

Similarly, we conclude that $γ_{s, m_{s}}$ is well-defined in a neighborhood of $e_{s, m_{s}}$ . Substituting (48) and (51) into (50), we have

\begin{matrix} {\overset{\cdot}{V}}_{m_{s}} = \sum_{s = 1}^{n} [\frac{b_{s, m_{s}}^{2} e_{s, m_{s}}}{b_{s, m_{s}}^{2} - x_{s, m_{s}}^{2}} (W_{s, m_{s}}^{* T} S_{s, m_{s}} + g_{s, m_{s}} u_{s} \\ - {\overset{\cdot}{α}}_{s, m_{s} - 1} + ε_{s, m_{s} - 1}) + \frac{b_{s, m_{s}}^{2} e_{s, m_{s}} {\overset{\cdot}{α}}_{s, m_{s} - 1}}{b_{s, m_{s}}^{2} - x_{s, m_{s}}^{2}} \\ - γ_{s, m_{s}} {\overset{\cdot}{α}}_{s, m_{s} - 1} + {\tilde{W}}_{s, m_{s}}^{T} Γ_{s, m_{s}}^{- 1} {\overset{\cdot}{\hat{W}}}_{s, m_{s}}] . \end{matrix}

(54)

To handle the issue of “explosion of complexity”, a FTCD is designed to estimate ${\overset{\cdot}{α}}_{s, m_{s} - 1}$ . Thus, we can reach the following expression

\begin{matrix} {\overset{\cdot}{α}}_{s, m_{s} - 1} = {\overset{\cdot}{α}}_{s, m_{s} - 1}^{c} + δ_{s, m_{s} - 1}, \end{matrix}

(55)

where $δ_{s, m_{s} - 1}$ is bounded.

Then, we present the control law as follows

\begin{matrix} u_{s} = g_{s, m_{s}}^{- 1} [\frac{b_{s, m_{s}}^{2} - x_{s, m_{s}}^{2}}{b_{s, m_{s}}^{2}} ({\overset{\cdot}{α}}_{s, m_{s} - 1}^{c} γ_{s, m_{s}} - \frac{e_{s, m_{s}} γ_{s, m_{s}}^{2}}{2}) \\ - {\hat{W}}_{s, m_{s}}^{T} S_{s, m_{s}} - \frac{b_{s, m_{s}}^{2} e_{s, m_{s}}}{2 (b_{s, m_{s}}^{2} - x_{s, m_{s}}^{2})} \\ - \frac{k_{s, m_{s}} b_{s, m_{s}}^{2 l - 2} sign (e_{s, m_{s}}) {| e_{s, m_{s}} |}^{2 l - 1}}{{(b_{s, m_{s}}^{2} - x_{s, m_{s}}^{2})}^{l - 1}} \\ - \frac{(b_{s, m_{s}}^{2} - x_{s, m_{s}}^{2}) g_{s, m_{s} - 1}^{2} e_{s, m_{s}}}{2 b_{s, m_{s}}^{2}}], \end{matrix}

(56)

where $k_{nj} > 0$ denotes the control parameter.

The adaptive weight is presented as follows

\begin{matrix} {\overset{\cdot}{\hat{W}}}_{s, m_{s}} = Γ_{s, m_{s}} (\frac{b_{s, m_{s}}^{2} e_{s, m_{s}}}{b_{s, m_{s}}^{2} - x_{s, m_{s}}^{2}} S_{s, m_{s}} - κ_{s, m_{s}} {\hat{W}}_{s, m_{s}}), \end{matrix}

(57)

where $κ_{s, m_{s}}$ is a positive constant.

Substituting (55), (56), and (57) into (54), we have

\begin{matrix} {\overset{\cdot}{V}}_{m_{s}} = \sum_{s = 1}^{n} [- k_{s, m_{s}} {(\frac{b_{s, m_{s}}^{2} e_{s, m_{s}}}{b_{s, m_{s}}^{2} - x_{s, m_{s}}^{2}})}^{l} - \frac{1}{2} e_{s, m_{s}}^{2} γ_{s, m_{s}}^{2} \\ - \frac{1}{2} {(\frac{b_{s, m_{s}}^{2} e_{s, m_{s}}}{b_{s, m_{s}}^{2} - x_{s, m_{s}}^{2}})}^{2} - δ_{s, m_{s} - 1} e_{s, m_{s}} γ_{s, m_{s}} \\ + \frac{b_{s, m_{s}}^{2} e_{s, m_{s}} ε_{s, m_{s}}}{b_{s, m_{s}}^{2} - x_{s, m_{s}}^{2}} + \frac{1}{2} g_{s, m_{s}}^{2} e_{s, m_{s}}^{2} + κ_{s, m_{s}} {\tilde{W}}_{s, m_{s}}^{T} {\hat{W}}_{s, m_{s}}] . \end{matrix}

(58)

According to Young’s inequality, we can get the following inequalities

\begin{matrix} \frac{b_{s, m_{s}}^{2} e_{s, m_{s}} ε_{s, m_{s}}}{b_{s, m_{s}}^{2} - x_{s, m_{s}}^{2}} \leq \frac{1}{2} {(\frac{b_{s, m_{s}}^{2} e_{s, m_{s}}}{b_{s, m_{s}}^{2} - x_{s, m_{s}}^{2}})}^{2} + \frac{1}{2} ε_{s, m_{s}}^{2}, \\ - δ_{s, m_{s} - 1} e_{s, m_{s}} γ_{s, m_{s}} \leq \frac{1}{2} e_{s, m_{s}}^{2} γ_{s, m_{s}}^{2} + \frac{1}{2} δ_{s, m_{s} - 1}^{2}, \\ {\tilde{W}}_{s, m_{s}}^{T} {\hat{W}}_{s, m_{s}} \leq \frac{1}{2} ({‖ W_{s, m_{s}}^{*} ‖}^{2} - {‖ {\tilde{W}}_{s, m_{s}} ‖}^{2}) . \end{matrix}

(59)

Submitting (59) into (58), we can obtain

\begin{matrix} {\overset{\cdot}{V}}_{m_{s}} = \sum_{s = 1}^{n} [- k_{s, m_{s}} {(\frac{b_{s, m_{s}}^{2} e_{s, m_{s}}}{b_{s, m_{s}}^{2} - x_{s, m_{s}}^{2}})}^{l} - \frac{κ_{s, m_{s}}}{2} ∥ {\tilde{W}}_{s, m_{s}} ∥^{2} \\ + \frac{1}{2} g_{s, m_{s} - 1}^{2} e_{s, m_{s}}^{2} + \frac{1}{2} δ_{s, m_{s} - 1}^{2} + \frac{κ_{s, m_{s}}}{2} ∥ W_{s, m_{s}}^{*} ∥^{2}] . \end{matrix}

(60)

According to Lemma 2, we have

{({‖ {\tilde{W}}_{s, m_{s}} ‖}^{2})}^{l} \leq (1 - l) ℏ + {‖ {\tilde{W}}_{s, m_{s}} ‖}^{2} .

(61)

Submitting (61) into (60), we have

\begin{matrix} {\overset{\cdot}{V}}_{m_{s}} = \sum_{s = 1}^{n} [- k_{s, m_{s}} {(\frac{b_{s, m_{s}}^{2} e_{s, m_{s}}}{b_{s, m_{s}}^{2} - x_{s, m_{s}}^{2}})}^{l} - \frac{1}{2} κ_{s, m_{s}} ‖ {\tilde{W}}_{s, m_{s}} ‖^{2 l} \\ + \frac{1}{2} g_{s, m_{s} - 1}^{2} e_{s, m_{s}}^{2}] + Δ_{m_{s}}, \end{matrix}

(62)

where $Δ_{m_{s}} = \sum_{s = 1}^{n} + \frac{1}{2} κ_{s, m_{s}} (1 - l) ℏ + \frac{1}{2} δ_{s, m_{s} - 1}^{2} + \frac{1}{2} κ_{s, m_{s}} ∥ W_{s, m_{s}}^{*} ∥^{2}$ .

Using to Lemma 4, the following inequality can be obtained

{(\frac{b_{s, m_{s}}^{2} e_{s, m_{s}}^{2}}{b_{s, m_{s}}^{2} - x_{s, m_{s}}^{2}})}^{l} \geq {(\int_{0}^{e_{s, m_{s}}} \frac{τ b_{s, m_{s}}^{2}}{b_{s, m_{s}}^{2} - {(τ - α_{s, m_{s} - 1})}^{2}} d τ)}^{l} .

(63)

Substituting (63) into (62), we have

\begin{matrix} {\overset{\cdot}{V}}_{m_{s}} \leq \sum_{s = 1}^{n} [- k_{s, m_{s}} {(\int_{0}^{e_{s, m_{s}}} \frac{τ b_{s, m_{s}}^{2}}{b_{s, m_{s}}^{2} - {(τ - α_{s, m_{s} - 1})}^{2}} d τ)}^{l} \\ - \frac{κ_{s, m_{s}}}{2} ‖ {\tilde{W}}_{s, m_{s}} ‖^{2 l} + \frac{1}{2} g_{s, m_{s} - 1}^{2} e_{s, m_{s}}^{2}] + Δ_{m_{s}} . \end{matrix}

(64)

Remark 2: As we can see that the state constraints are directly considered by using iBLF, the bounds of the virtual control signals need not to be known exactly. Some similar studies can be found in He et al. (2019) and Liu et al. (2017). However, the problem of finite-time convergence has not been studied. To the best of our knowledge, the adaptive finite-time neural backstepping method using iBLF is proposed for the first time in this paper.

Remark 3: Although there are many fruitful studies in the design of adaptive finite-time control approaches for uncertain nonlinear systems, such as Wang et al. (2018b), the key parameter $l = \frac{2 n - 1}{2 n + 1}, n \geq 2, n \in N^{+}$ is defined in these studies to ensure the controller is meaningful in the real domain. The difference from previous studies is that by introducing $sign (e_{s, 1}) {| e_{s, 1} |}^{2 l - 1}$ in (20), $sign (e_{s, i_{s}}) {| e_{s, i_{s}} |}^{2 l - 1}$ in (39), and $sign (e_{s, m_{s}}) {| e_{s, m_{s}} |}^{2 l - 1}$ in (56), we can expand $l$ to $l \in Ω_{l} : = {l \in R : \frac{1}{2} < l < 1}$ .

Stability analysis

Theorem 1: Consider the system discussed in (1), which satisfies all the assumptions mentioned in this paper. By designing the virtual control signals (20), (39), the adaptive laws (21), (40), and (57), the final controller (56), and the initial value satisfy $x_{s, i_{s}} (0) \in Ω_{x} : = {x_{s, i_{s}} \in R : | x_{s, i_{s}} | < b_{s, i_{s}}, i = 1, \dots, n, i_{s} = 1, \dots, m_{s}}$ . Let $Q_{s, i_{s} - 1} = max | a_{s, i_{s} - 1} ({\bar{x}}_{s, i_{s}}) |, i = 1, \dots, n - 1$ . If the positive constants $k_{s, 1}, \dots, k_{s, i_{s}}$ exist such that $Q_{s, i_{s} - 1} (k_{s, 1}, \dots, k_{s, i_{s}}) < b_{s, i_{s}}$ . Then, the following conclusions can be drawn:

The error signals in the closed-loop system will converge to a small neighbourhood of the origin in finite time.

$x_{s, i_{s}} (t) \in Ω_{x}$ $\forall t \geq 0$ .

All signals of the closed-loop system are bounded.

Proof: Consider the following LFC

V_{L} = \sum_{i_{s} = 1}^{m_{s}} V_{i_{s}} .

(65)

By taking the derivative of (65) and considering (28), (47), and (62), we have

\begin{matrix} {\overset{\cdot}{V}}_{L} \leq \sum_{i_{s} = 1}^{m_{s}} \sum_{s = 1}^{n} [- b_{s, i_{s}} {(\frac{b_{s, i_{s}}^{2} e_{s, i_{s}}}{b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2}})}^{l} - \frac{κ_{s, i_{s}}}{2} {‖ {\tilde{W}}_{s, i_{s}} ‖}^{2 l}] + \sum_{i_{s} = 1}^{m_{s}} Δ_{i_{s}} . \end{matrix}

(66)

Using Lemma 3, we have

\begin{matrix} {\overset{\cdot}{V}}_{L} \leq - C_{1} V_{L}^{l} + C_{2}, \end{matrix}

(67)

where $C_{1} = \min (k_{s, i_{s}}, κ_{s, i_{s}} Γ_{s, i_{s}})$ , $C_{2} = \sum_{i_{s} = 1}^{m_{s}} Δ_{i_{s}}$ . Let $T_{reach} = \frac{1}{(1 - l) ξ C_{1}} [{V^{1 - l}}_{L} (0) - {(\frac{C_{2}}{(1 - ξ) C_{1}})}^{\frac{1 - l}{l}}]$ , where $V_{L} (0)$ denotes the initial value of $V_{L} (t)$ , $ξ \in (0, 1)$ is an auxiliary parameter. According to the discussion in Wang et al. (2018b), for $\forall t \geq T_{reach}$ , the inequality $V_{L} \leq {[\frac{C_{2}}{(1 - ξ) C_{1}}]}^{1 / l}$ holds. Thus, we have $e_{s, i_{s}}$ and ${\tilde{W}}_{s, i_{s}}$ are bounded. Therefore, we conclude that the error signals will converge to a small neighbourhood of the origin for all $\forall t \geq T_{reach}$ .

To prove (2), we assume that there exists a time $T_{c}$ that makes $x_{s, i_{s}} (T_{c}) = b_{s, i_{s}}$ hold. Thus, we can get the following equation

\begin{matrix} V_{i_{s}} |_{t = T_{c}} = \sum_{s = 1}^{n} (V_{s, i_{s}}^{e} + \frac{1}{2} {\tilde{W}}_{s, i_{s}}^{T} Γ_{s, i_{s}}^{- 1} {\tilde{W}}_{s, i_{s}}) |_{t = T_{c}} \\ = \sum_{s = 1}^{n} \int_{0}^{e_{s, i_{s}} (T_{c})} \frac{τ b_{s, i_{s}}^{2}}{b_{s, i_{s}}^{2} - {(τ + a_{s, i_{s} - 1})}^{2}} d τ \\ \frac{1}{2} {\tilde{W}}_{s, i_{s}}^{T} Γ_{s, i_{s}}^{- 1} {\tilde{W}}_{s, i_{s}} |_{t = T_{c}} \\ \leq \sum_{s = 1}^{n} V_{s, r_{s}}^{e} |_{t = 0} + \frac{1}{2} {\tilde{W}}_{s, i_{s}}^{T} Γ_{s, i_{s}}^{- 1} {\tilde{W}}_{s, i_{s}} |_{t = T_{c}} . \end{matrix}

(68)

Then, $V_{i_{s}} |_{t = T_{c}}$ is bounded. Integrating $V_{s, i_{s}}^{e} |_{T_{c}}$ by parts, we get the following expression

\begin{matrix} V_{s, i_{s}}^{e} |_{t = T_{c}} = b_{s, i_{s}} {[τ \tanh^{- 1} \frac{τ + a_{s, i_{s} - 1}}{b_{s, i_{s}}}]}_{0}^{e_{s, i_{s}}} \\ - b_{s, i_{s}} \int_{0}^{e_{s, i_{s}}} \tanh^{- 1} \frac{τ + a_{s, i_{s} - 1}}{b_{s, i_{s}}} d τ \\ \leq b_{s, i_{s}} a_{s, i_{s} - 1} (T_{c}) \ln \frac{(1 + a_{s, i_{s} - 1} (T_{c})) (1 - x_{s, i_{s}} (T_{c}))}{(1 - a_{s, i_{s} - 1} (T_{c})) (1 + x_{s, i_{s}} (T_{c}))} \\ + \frac{b_{s, i_{s}}^{2}}{2} \ln \frac{b_{s, i_{s}}^{2} - a_{s, i_{s} - 1}^{2} (T_{c})}{b_{s, i_{s}}^{2} - x_{s, i_{s}}^{2} (T_{c})} \\ \leq V_{s, i_{s}}^{e} |_{t = 0}, \end{matrix}

(69)

for all $s = 1, \dots, n, i_{s} = 1, \dots, m$ . If $x_{s, i_{s}} (T_{c}) = b_{s, i_{s}}$ holds, we know from (69) that $V_{s, i_{s}}^{e} |_{t = T_{c}}$ is unbounded. Then, $V_{i_{s}} |_{t = T_{c}}$ is unbounded, which contradicts the results obtained in (68). Therefore, $x_{s, i_{s}} (t) \neq b_{s, i_{s}}$ , for $\forall t > 0$ . Thus, we conclude that for $x_{s, i_{s}} (t) \in Ω_{x}, \forall t > 0$ can be guaranteed.

From the aforementioned discussions, we know that $e_{s, i_{s}}$ and ${\tilde{W}}_{s, i_{s}}$ are bounded. Since $W_{s, i_{s}}^{*}$ is bounded, then ${\hat{W}}_{s, i_{s}}$ is also bounded. Moreover, $| a_{s, i_{s} - 1} (t) | < b_{s, i_{s}}, \forall t > 0$ , and ${\overset{\cdot}{a}}_{s, i_{s} - 1}^{c} (t)$ is bounded. Then, from the expression in (56) we can draw a conclusion that the control input $u (t)$ is bounded. Therefore, all signals are bounded.

Remark 4: From (67), some suggestion on how to select the control parameters can can be obtained. To reduce tracking errors and convergence time, we can increase the constant $C_{1}$ , and reduce the constant $C_{2}$ and $l$ . That is, increasing control parameters $k_{s, i_{s}}$ , $κ_{s, i_{s}} γ_{s, i_{s}}$ , and decreasing control parameters $κ_{s, i_{s}}$ and $l$ might help to reduce the tracking errors and convergence time. Meanwhile, choosing large $k_{s, i_{s}}$ and $κ_{s, i_{s}} γ_{s, i_{s}}$ may lead to increase the cost of energy. Therefore, the selection of related parameters not only meets the inequality conditions presented in this section, but also requires the designer’s simulation test to obtain better control performance.

Simulation studies

To verify the effectiveness of the proposed control method, the attitude tracking simulations for an autonomous airship are considered. Define $x_{s, 1} = {[ϕ, θ, ψ]}^{T}$ , where $ϕ$ , $θ$ , and $ψ$ represent the roll, pitch and yaw angles of the airship, respectively. The angular velocities are described as $x_{s, 2} = {[p, q, r]}^{T}$ , where $p$ , $q$ , and $r$ represent the rolling, pitching and yawing angular velocities. The kinematics and dynamic equations can be constructed as follows

{\begin{matrix} {\overset{\cdot}{x}}_{1, 1} = f_{1, 1} + g_{1, 1} x_{1, 2} \\ {\overset{\cdot}{x}}_{2, 1} = f_{2, 1} + g_{2, 1} x_{2, 2} \\ {\overset{\cdot}{x}}_{3, 1} = f_{3, 1} + g_{3, 1} x_{3, 2} \end{matrix}

(70)

{\begin{matrix} {\overset{\cdot}{x}}_{1, 2} = f_{1, 2} + g_{1, 2} u_{1} \\ {\overset{\cdot}{x}}_{2, 2} = f_{2, 2} + g_{2, 2} u_{2} \\ {\overset{\cdot}{x}}_{3, 2} = f_{3, 2} + g_{3, 2} u_{3} \end{matrix}

(71)

where $f_{1} = [f_{1, 1} f_{2, 1} f_{3, 1}]^{T}$ , $g_{1} = [g_{1, 1} g_{2, 1} g_{3, 1}]^{T}$ are expressed as follows

f_{1} = [\begin{matrix} \sin (ϕ) \tan (θ) v + \cos (ϕ) \tan (θ) r \\ \sin (ϕ) r \\ \sin (ϕ) \sec (θ) v \end{matrix}],

(72)

g_{1} = [\begin{matrix} 1 \\ - \cos (ϕ) \\ \cos (ϕ) \sec (θ) \end{matrix}],

(73)

$f_{2} = [f_{1, 2} f_{2, 2} f_{3, 2}]^{T}$ denotes the external forces on the airship, which is expressed as

f_{2} = [\begin{matrix} (ι_{1} r - ι_{2} p) q + ι_{3} z_{G} G \cos θ \sin ϕ \\ ι_{5} rp - ι_{6} (p^{2} - r^{2}) - ι_{7} z_{G} G \sin θ \\ (ι_{2} r + ι_{8} p) q + ι_{9} z_{G} G \cos θ \sin ϕ \end{matrix}],

(74)

and $g_{2} = [g_{1, 2} g_{2, 2} g_{3, 2}]^{T}$ is expressed as follows

g_{2} = [ι_{4} ι_{7} ι_{4}]^{T} .

(75)

We can see that (70) is a strict feedback nonlinear MIMO system. The detailed parameters can be found in Wang et al. (2016). In the simulation, we assume that $f_{1}$ is known while $f_{2}$ is unknown. RBF NNs are designed to compensate for $f_{2}$ .

The desired attitude is expressed as $y_{rs} = {[ϕ_{r}, θ_{r}, ψ_{r}]}^{T} = [1 - e^{- 0.5 t} (\sin (0.5 t) + \cos (0.5 t))] [\frac{π}{6}, \frac{π}{4}, - \frac{π}{3}]^{T}$ . The state constraints are given as $b_{1} = {[\frac{π}{4} \frac{π}{3} \frac{π}{2}]}^{T}$ , $b_{2} = {[1.51.51.5]}^{T}$ . The initial values of the angles and angular velocities are set as $x_{s, 1} = {[\frac{π}{8} - \frac{π}{6} \frac{π}{6}]}^{T}$ rad, $x_{s, 2} = {[- \frac{π}{60} \frac{π}{60} \frac{π}{60}]}^{T}$ rad/s, respectively.

The control parameters are set as $k_{1} = k_{2} = diag (1, 1, 1)$ , $l = 0.6$ . The parameters of FTCD are set as $L_{1} = 10$ , $L_{1} = 0.1$ , and $β = 0.1$ . For the RBF NN approximator, the Gauss function is selected. $ξ_{2, 1} = ξ_{2, 2} = ξ_{2, 3} = {[ϕ, θ, p, q, r]}^{T}$ are set as the input of $S_{s, 2} (ξ)$ . In the hidden layer, a number of 61 nodes are used for each $S_{s, 2} (ξ)$ . The parameters of RBF NNs are selected as $Γ_{1, 2} = Γ_{2, 2} = Γ_{3, 2} = 1 I_{61 \times 61}$ , $κ_{1, 2} = κ_{2, 2} = κ_{3, 2} = 0.1$ , the width of the neural cell $d = 2$ . The initial values of adaptive weights are set as ${\hat{W}}_{s, 2} (0) = 0$

According to the design procedure in Section 3, the control scheme and the adaptive law are presented as

\begin{matrix} α_{s, 1} = g_{s, 1}^{- 1} [\frac{(b_{s, 1}^{2} - x_{s, 1}^{2}) {\overset{\cdot}{y}}_{rs} γ_{s, 1}}{b_{s, 1}^{2}} - \frac{b_{s, 1}^{2} e_{s, 1}}{b_{s, 1}^{2} - x_{s, 1}^{2}} \\ - \frac{k_{s, 1} b_{s, 1}^{2 l - 2} sign e_{s, 1} {| e_{s, 1} |}^{2 l - 1}}{{(b_{s, 1}^{2} - x_{s, 1}^{2})}^{l - 1}} - f_{s, 1}], \end{matrix}

(76)

\begin{matrix} {\overset{\cdot}{\hat{W}}}_{s, 1} = Γ_{s, 1} (\frac{b_{s, 1}^{2} e_{s, 1}}{b_{s, 1}^{2} - x_{s, 1}^{2}} S_{s, 1} - κ_{s, 1} {\hat{W}}_{s, 1}), \end{matrix}

(77)

\begin{matrix} u_{s} = g_{s, 2}^{- 1} [\frac{b_{s, 2}^{2} - x_{s, 2}^{2}}{b_{s, 2}^{2}} ({\overset{\cdot}{α}}_{s, 1}^{c} γ_{s, 2} - \frac{e_{s, 2} γ_{s, 2}^{2}}{2}) \\ - {\hat{W}}_{s, 2}^{T} S_{s, 2} - \frac{k_{s, 2} b_{s, 2}^{2 l - 2} sign e_{s, 2} {| e_{s, 2} |}^{2 l - 1}}{{(b_{s, 2}^{2} - x_{s, 2}^{2})}^{l - 1}} \\ - \frac{b_{s, 2}^{2} e_{s, 2}}{2 (b_{s, 2}^{2} - x_{s, 2}^{2})} - \frac{(b_{s, 2}^{2} - x_{s, 2}^{2}) g_{s, 1}^{2} e_{s, 2}}{2 b_{s, 2}^{2}}] . \end{matrix}

(78)

Furthermore, the iBLF-based traditional adaptive backstepping control approach is selected as comparison. The virtual conreol law, the adaptive law and the control law and are expressed as follows

α_{s, 1} = g_{s, 1}^{- 1} [\frac{(b_{s, 1}^{2} - x_{s, 1}^{2}) {\overset{\cdot}{y}}_{rs} γ_{s, 1}}{b_{s, 1}^{2}} - k_{s, 1} e_{s, 1}],

(79)

\begin{matrix} {\overset{\cdot}{\hat{W}}}_{s, 2} = Γ_{s, 2} (\frac{b_{s, 2}^{2} e_{s, 2}}{b_{s, 2}^{2} - x_{s, 2}^{2}} S_{s, 2} - κ_{s, 2} {\hat{W}}_{s, 2}), \end{matrix}

(80)

\begin{matrix} u_{s} = g_{s, 2}^{- 1} [\frac{b_{s, 2}^{2} - x_{s, 2}^{2}}{b_{s, 2}^{2}} {\overset{\cdot}{α}}_{s, 1}^{c} γ_{s, 2} - {\hat{W}}_{s, 2}^{T} S_{s, 2} \\ - k_{s, 2} e_{s, 2} - \frac{b_{s, 1}^{2} (b_{s, 2}^{2} - x_{s, 2}^{2}) g_{s, 1} e_{s, 1}}{b_{s, 1}^{2} (b_{s, 1}^{2} - x_{s, 1}^{2})}] . \end{matrix}

(81)

To simplify notation, the proposed scheme is labeled as “iBLF-AFTC”, the iBLF-based traditional control method is labeled as “iBLF-TABC”. The values of common control parameters and initial conditions of the two control methods are selected all the same.

Figures 1 –6 demonstrate the corresponding simulation results. The attitude tracking comparisons are shown in Figures 1 –3. As we can see from Figures 1 –3, the tracking performance under iBLF-AFTC method is better than iBLF-TABC. Compared with the iBLF-TABC method, the proposed control scheme has faster convergence speed and smaller tracking error. Figure 4 shows the angular velocities of the autonomous airship. It can be observed from Figures 1 –4 that the full state constraints are never violated. Figures 5 and 6 depict the adaptive weights and approximation performance, respectively. As we can see from Figures 5 and 6, the adaptive RBF NN control can compensate for the uncertainties of the autonomous airship system with full state constraints effectively.

Conclusion

In this paper, the iBLF-based adaptive finite-time neural control method for MIMO uncertain nonlinear systems subject to full state constraints has been designed. IBLFs were used to handle the state constraints. RBF NNs were designed to approximate the uncertainties. To handle the problem of “complex explosion” in backstepping control, a new FTCD was introduced. Under the proposed control method, the output signals of the system can track the reference signals in a finite time, and full state constraints are never violated. Finally, stability analysis and comparison simulations have been conducted to verify the effectiveness of the proposed control approach. In the paper, the input constraint and dead zone problem have not been considered. Thus, when utilizing the designed control method, the actual ability of the actuator should be considered in a practical application. Further research will focus on the issue of adaptive optimal control for nonlinear state-constrained systems.

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 by the National Natural Science Foundation of China [Grant No 62073216, 61973204 and 51906141].

ORCID iD

Yan Wei

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