Hybrid neural network control design for uncertain switched nonlinear systems with external disturbances: Application to ship maneuvering system

Abstract

In this work, the problem of hybrid neural network control for a class of switched uncertain nonlinear systems in strict-feedback form is considered. To approximate unknown nonlinear functions, an improved whale optimization algorithm (IWOA)-based hybrid neural network controller is introduced. Based on hybrid neural network approximation ability, a new hybrid neural network controller is constructed via the backstepping technique and the common Lyapunov function (CLF) is used for the stability analysis of the proposed model. The suggested method ensures that all signals in the closed-loop system are semi-global uniform ultimate bounded (SGUUB), and the tracking error converges to a small neighborhood of zero. Finally, the proposed scheme is applied to a ship maneuvering system and a numerical example to verify its effectiveness.

Keywords

Hybrid neural network common Lyapunov function improved whale optimization algorithm switched nonlinear systems

Introduction

Switched systems are a type of hybrid system that consists of several subsystems and a switching rule that selects a subsystem to be active for a period of time. In recent years, the control challenges of switched systems have widely attracted more attention because of their important roles in engineering applications like power systems, mobile robots, ducted rocket systems, and so on (Bao et al., 2010; Lee and Jiang, 2008; Liu et al., 2016). Basic problems related to stability of control systems and synthesis of switched linear and nonlinear systems has been discussed by Zhai et al. (2006), Wu and Liu (2017), and Liberzon and Morse (1999).

Stability analysis is well recognized to be the most important factor for nonlinear systems. The stability analysis and control design of switched nonlinear systems have been widely explored using common Lyapunov functions (CLFs) (Vu and Liberzon, 2005) and multiple Lyapunov functions (MLFs) (She and Xue, 2014). It is well accepted that having a CLF for all subsystems can ensure the stability of switched systems with arbitrary switching (Zhu et al., 2020). As a result, the CLF technique has been widely used in the development of controllers and in the stability analysis of switched systems (Aleksandrov et al., 2011; Moulay et al., 2007; Peng and Xu, 2022). Finally, several outstanding results for switched systems with arbitrary switching have been reported (Bali et al., 2022; Fainshil et al., 2009; Zhai et al., 2017).

In the last few years, adaptive stabilization of nonlinear systems has received a lot of attention, and many achievements in the areas of analysis and control of switching systems (Han et al., 2009a; Li and Xiang, 2019; Yang and Yue, 2018). Particularly, researchers have focused their attention on a neural network (NN)-based adaptive control strategy for uncertain nonlinear systems in strict-feedback mode (Chu et al., 2021; Zhou et al., 2018). Zhou et al. (2022a, 2022b, 2022c) focused on the delay intermittent control problem for stochastic system with time-varying multi-weights network and stabilization issue for stochastic complex networks and stochastic coupled systems. An adaptive control technique using the backstepping method and NN is suggested by Lian et al. (2010) for a class of uncertain strict-feedback nonlinear systems. A class of unknown nonlinear dynamical systems were examined by Liu and Tong (2015) using adaptive fuzzy control. Dwell time and average dwell time switching approach for adaptive neural control for switched nonlinear systems was investigated by Han et al. (2009b), Ghazisaeedi and Tavazoei (2021), and Yin et al. (2018).

In real-world problems, switched nonlinear systems inevitably have uncertainties, and there have been some research for dealing with such systems, but most of the controlled nonlinear systems require the uncertainties to satisfy the matching conditions (Kanellakopoulos et al., 1991; Lei and Lin, 2007) for the analysis of global stability. In many situations, however, we are unable to obtain a priori knowledge of system uncertainty, which can only be described by completely unknown functions (Sakhre et al., 2017). The approximation ability of NNs or fuzzy logic systems has been investigated in the literature (Li et al., 2011; Singh and Jain, 2016; Yu et al., 2020; Zhu et al., 2022) to solve the control challenges for switched systems. As a result, a number of major results have been presented for uncertainties existing in nonlinear systems (Qi et al., 2018; Roy and Kar, 2016; Wang et al., 2018, 2019).

The training procedure is one of the difficult aspects in designing NNs. The complex nature of NNs and unknown set of primary parameters like initial weights and learning rate, affects the convergence speed and accuracy of NNs (Singh et al., 2018). To overcome this problem, different bio-inspired and evolutionary methods have been developed in recent years (Aljarah et al., 2018; Angeline et al., 1994; Singh and Jain, 2018). However, in these methods, there are numerous challenges such as local minima, saddle points, and vanishing gradients due to which these methods suffer and hence needs to be improved. To handle these issues, we proposed an improved whale optimization algorithm (IWOA) to assign initial weights to the NN. It is verified that the IWOA-based NN has better training performance, faster convergence rate, and greater prediction ability than standard radial basis function NN (Huang and Xiang, 2016; Li and Wang, 2018; Yoo, 2016).

Inspired by the above discussions, this paper considers the hybrid neural network (HNN) based adaptive tracking control for a class of switched uncertain nonlinear systems in strict-feedback form with external disturbances. The following is a list of the paper’s major aspects:

A HNN control problem for uncertain switched nonlinear systems with bounded external disturbances is proposed.

To approximate the unknown functions, an IWOA-based HNN is developed. The complexity analysis of the proposed controller has also been discussed.

By choosing design parameters suitably, the stability analysis of the proposed controller is determined by the CLF method. It has been shown that all signals in the closed-loop systems remain bounded in the sense of semi-global uniform ultimate boundedness.

Finally, the effectiveness of the proposed controller is verified by a real-world application in ship maneuvering system and a numerical example.

The remaining paper is written out as follows. In section 2, problem formulation and some preliminary work are presented. In Section 3, a systematic design process for HNN controller and the stability analysis for the proposed model are described. Section 4 provides two simulation examples, one of them is the application to the ship maneuvering system to verify the effectiveness of the proposed model. Section 5 concludes the work.

Notations: In this article, we have used standard notations. $ℜ^{n}$ stands for $n$ -dimensional Euclidean space, ${\bar{ξ}}_{i} (.) = [ξ_{1}, . . ., ξ_{i}]^{T}$ represents the state vector with T being transpose of the vector/matrix. $e_{i}$ is the tracking error, $δ (Z)$ is the estimation error, $y_{d}$ is the desired trajectory, $y$ is the output trajectory, $q_{i, p}$ is the external disturbances, $g_{i, p}$ are positive constants, and $f_{i, p} (ξ_{i})$ are unknown smooth functions.

Problem formulation and preliminaries

Consider the following uncertain switched nonlinear system with bounded disturbance

{\begin{matrix} {\overset{\cdot}{ξ}}_{i} = g_{i, ϱ (t)} ξ_{i + 1} + f_{i, ϱ (t)} ({\bar{ξ}}_{i}) + q_{i, ϱ (t)}, 1 \leq i \leq n - 1 \\ {\overset{\cdot}{ξ}}_{n} = g_{n, ϱ (t)} v + f_{n, ϱ (t)} ({\bar{ξ}}_{n}) + q_{n, ϱ (t)}, \\ y = ξ_{1} \end{matrix}

(1)

where ${\bar{ξ}}_{i} = [ξ_{1}, \dots, ξ_{i}]^{T} \in ℜ^{i}$ , $i = 1, \dots, n$ is the system state. The system output is denoted by $y \in ℜ$ and $ϱ (t) : [0, \infty) \to$ $S = {1, 2, \dots, s}$ is a switching signal, which takes the values from the compact set $S$ . If $ϱ (t) = p$ , then the $p th$ subsystem is active while the other subsystems are inactive. $v$ denote the controller input of the $p th$ subsystem. $q_{i, p}$ represents the external disturbance satisfying $| q_{i, p} | \leq {\bar{q}}_{i, p}$ . $f_{i, ρ} (ξ)$ are unknown smooth functions with $f_{i, 0} (0) = 0$ and $g_{i, p}$ is a positive constant.

Remark 1: System (1) has a strict-feedback form of structure which can also be used to describe a wide range of real-world systems, including continuously stirred tank reactors with two feed stream modes (Li et al., 2020) and mass-spring-damper systems (Wang and Long, 2020) with controller switching.

Controller objectives and assumptions

The control objective of this paper is to design state feedback controller such that:

All signals in the system under consideration are bounded under arbitrary switching;

The system output $y$ tracks the reference signal $y_{d} (t)$ .

IWOA

The whale optimization algorithm (WOA) is a bio-inspired meta-heuristic algorithm based on the swarm foraging behavior of humpback whales proposed by Mirjalili and Lewis (2016) in 2016. The algorithm has a basic concept, good performance, and only a few adjustment parameters (Chen et al., 2020). Like other heuristic algorithms, WOA also suffers from low accuracy and slow convergence speed of local optimum solutions. To handle this problem, an IWOA (Mafarja and Mirjalili, 2017) has been used with new convergence factor $\vec{a}$ and inertia weight vector $α$ . The main steps of IWOA are explained below:

Encircling operation

\vec{X} (t + 1) = α . {\vec{X}}^{*} (t) - \vec{A} . \vec{D}, if p < 0.5

(2)

where $\vec{D} = | \vec{C} . {\vec{X}}^{*} (t) - \vec{X} (t) |$ , ${\vec{X}}^{*}$ is the position of best solution obtained. In equation (2), the term $α$ is the inertia weight coefficient, and it is defined as follews

α = \frac{1}{2} + \frac{1}{2} \times {(\frac{T}{max T})}^{2}

(3)

where $T$ and $max T$ are the current iteration number and maximum iteration numbers set, respectively. As the iteration time increases, the value of $α$ increases from 0.5 to 1 which improves the convergence rate and accuracy of the algorithm.

Shrinking operation

\vec{X} (t + 1) = {\vec{D}}^{'} . e^{bk} . \cos (2 π k) + α . {\vec{X}}^{*} (t), if p \geq 0.5

(4)

where ${\vec{D}}^{'} = | {\vec{X}}^{*} (t) - \vec{X} (t) |$ and $k$ is the random number used within the limit $[- 1, 1]$ .

Hunting operation

\vec{X} (t + 1) = {\vec{X}}_{rand} - \vec{A} . \vec{D}

(5)

where ${\vec{X}}_{rand}$ is the random position vector and $\vec{D} = | \vec{C} . {\vec{X}}_{rand} - \vec{X} (t) |$ . The vectors $\vec{A}$ and $\vec{C}$ are determined as

\vec{A} = 2 \vec{a} . \vec{r} - \vec{a}

(6)

\vec{C} = 2 . \vec{r}

(7)

where $\vec{r}$ represents the random vector in $[0, 1]$ . Equation (6) represents the calculation of the convergence factor $\vec{a}$ which is defined as

\vec{a} = 2 + 2 . \cos (\frac{π}{2} \times (1 + \frac{T}{max T}))

(8)

where $T$ and $max T$ are the current iteration number and maximum iteration number set, respectively.

Let $n$ be the size of the population in $d$ -dimensional space. The time complexity of each step of IWOA is as follows:

Total $n$ agents distributed in $n$ -dimensional search space is requires $n . d$ times.

Calculate the fitness of each agent and select the best one needs $n (n + 1) / 2$ times.

Updation of parameters requires constant time.

Update the position of $n$ agents in $d$ -dimensional space requires $n . d$ times.

Hence, the total running time comes under whole loop (for which maxiteration is $T$ ). Therefore, the total complexity of IWOA is $O (T . [n d + (n (n + 1) / 2) + c]) ≅ O (T . [n d + (n (n + 1) / 2)])$ . Now, when $d \approx n$ , then the total complexity of IWOA is $O (n^{2} . T)$ .

HNN

In this work, we use an IWOA for training an NN. It provides optimized weights to the NN in order to improve its efficiency. The resulting IWOA-based NN called HNN defined as $F_{nn} = W^{T} S (Z)$ , and it is used to approximate any continuous function $F (Z) : ℜ^{n} \to ℜ$ over a compact set $Ω_{Z} \subset R^{q}$ such that for any $ϵ > 0$

F (Z) = {W^{*}}^{T} S (Z) + δ (Z)

(9)

where $δ (Z)$ is the approximation error satisfy the condition $| δ (Z) | \leq ϵ$ , and $W^{*}$ represents the ideal weight vector and it is defined as

W^{*} = \underset{W \in ℜ^{N}}{\arg min} {sup_{Z \in Ω_{Z}} | F (Z) - W^{T} S (Z) |}

(10)

where $S (Z) = [s_{1} (Z), \dots, s_{n} (Z)]$ is the basis function vector. The number of nodes of HNN is $n$ and $W = [w_{1}, \dots, w_{n}] \in ℜ^{N}$ is the weight vector of HNN. $s_{i} (Z)$ is chosen as Gaussian function and it is defined as

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

(11)

where $μ_{i} = [μ_{i 1}, \dots, μ_{in}]^{T}$ is the center of the receptive field and $ν > 0$ is the width of the Gaussian function. The basic diagram of HNN is shown in Figure 1 where $H$ is the weight vector from hidden to output layer. Consider $n$ , $l$ , and $m$ number of neurons in the input, hidden, and output layer, respectively, of NN, the input layer computation given as

F (Z) = W^{T} S (Z)

(12)

which requires $O (l * n)$ computational steps for hidden layer (11). The output layer (13) requires $O (l * m)$ computational steps

O_{j} = \sum_{k = 1}^{m} s_{k} (Z) H_{k, j}

(13)

where $H_{k, j}$ is the weight vector from $k th$ hidden neuron to $j th$ output neuron. Therefore, the total complexity of NN is $O (l * (n + m))$ , if $m = 1$ and $l = n$ , then its complexity is $O (n^{2})$ .

Hence, the total computational complexity of HNN is $O (n^{2}) + O (n^{2} . T) \approx O (n^{2})$ when $T << n$ and $O (n^{2}) + O (n^{2} . T) \approx O (n^{3})$ when $T = n$ .

In order to fulfill the control goal, we need to give the following conventional assumptions.

Assumption 1. The reference trajectory $y_{d} (t)$ and its $n$ th order derivatives with respect to time not only continuous but also bounded, that is, there exists constant $d > 0$ such that $| y_{d} | \leq d$ .

Assumption 2. There exist strictly increasing smooth function $ϕ_{i, p}$ with $ϕ_{i, p} (0) = 0$ for functions $f_{i, p}$ in (1) such that for $i = 1, \dots, n$

| f_{i, p} | \leq ϕ_{i, p} (∥ ξ ∥)

Assumption 3. For $i = 1, \dots, n$ , the unknown external disturbance functions $q_{i, p}$ satisfies

| q_{i, p} | \leq {\bar{q}}_{i, p}

where ${\bar{q}}_{i, p}$ are known constants.

Lemma 1 (Chen et al., 2012). From the transformation of variable $e_{i} = ξ_{i} - ζ_{i - 1}$ for $i = 1, \dots, n$ , we have the following inequality

∥ ξ ∥ \leq \sum_{i = 1}^{n} | e_{i} | ψ_{i, p} ({\hat{η}}_{i}) + d^{*}

(14)

where $ψ_{i, p} ({\hat{η}}_{i}) = 1 / {\underline{g}}_{i} (- k_{i} - l_{i}^{2} - β_{i}^{2} {\hat{η}}_{i} S_{i}^{T} (Z_{i}) S_{i} (Z_{i})) + 1, i =$ $1, \dots, n - 1$ $ψ_{n, p} ({\hat{η}}_{n}) = 1$ and $d^{*}$ is positive constant.

Lemma 2 (Chen et al., 2010). For any $γ \in ℜ$ , there exist a constant $ς > 0$ such that

| γ | \leq γ \tanh (\frac{γ}{ς}) + δ ς, δ = 0.2785

(15)

Lemma 3 (Dragomir, 2017). (Young’s inequality) For any vectors $x, y \in ℜ^{n}$ , we have

xy \leq \frac{1}{p} | x |^{p} + \frac{1}{q} | y |^{q}

(16)

where $p$ and $q$ are positive constants with $(p - 1) (q - 1) = 1$ .

HNN controller design and stability analysis

This section presents an HNN-based adaptive control scheme for system (1) via backstepping method. The systematic backstepping method has $n$ steps and is based on the following coordinate change

e_{1} = ξ_{1} - y_{d}

(17)

e_{i} = ξ_{i} - ζ_{i - 1}, 1 \leq i \leq n

(18)

where $ζ_{0} = y_{d}$ and $ζ_{i}$ is the virtual control law and it is defined as

ζ_{i} = - \frac{e_{i}}{{\underline{g}}_{i}} (k_{i} + l_{i}^{2} + β_{i}^{2} {\hat{η}}_{i} S_{i}^{T} (Z_{i}) S_{i} (Z_{i}))

(19)

where $k_{i}$ , $l_{i}$ and $β_{i}$ are positive design parameters, ${\underline{g}}_{i} = min {g_{i, p}, p \in M}$ , ${\hat{η}}_{i}$ is the estimation of $η_{i}$ and it is defined as ${\tilde{η}}_{i} = η_{i} - {\hat{η}}_{i}$ , $Z_{i} = ({\bar{ξ}}_{i}^{T}, y_{d}, {\overset{\cdot}{y}}_{d}, \dots, y_{d}^{(i)}, {\hat{η}}_{i})^{T}$ . The real control law is selected as

v = - \frac{e_{n}}{{\underline{g}}_{n}} (k_{n} + l_{n}^{2} + β_{n}^{2} {\hat{η}}_{n} S_{n}^{T} (Z_{n}) S_{n} (Z_{n}))

(20)

where $k_{n}$ , $l_{n}$ , and $β_{n}$ are positive design parameters, ${\underline{g}}_{n} = min {g_{n, p}, p \in M}$ , ${\hat{η}}_{n}$ is the estimation of $η_{n}$ , and it is defined as ${\tilde{η}}_{n} = η_{n} - {\hat{η}}_{n}$ , $Z_{n} = ({\bar{ξ}}_{n}^{T}, y_{d}, {\overset{\cdot}{y}}_{d}, \dots, y_{d}^{(n)}, {\hat{η}}_{n})^{T}$ .

The adaptive law is chosen as

{\overset{\cdot}{\hat{η}}}_{i} = r_{i} β_{i}^{2} e_{i}^{2} S_{i}^{T} (Z_{i}) S_{i} (Z_{i}) - ϖ_{i} {\hat{η}}_{i}

(21)

where $r_{i}$ and $ϖ_{i}$ are positive design parameters.

Remark 2: It has been demonstrated that global asymptotic stability for switched systems with arbitrary switching is equivalent to the presence of a CLF. As a result, the CLF design method is used in this study to try to solve the control problem of the switched systems (1) under arbitrary switching.

Step 1: Let $e_{1} = ξ_{1} - y_{d}$ , the time derivative of $e_{1}$ is

\begin{matrix} {\overset{\cdot}{e}}_{1} = {\overset{\cdot}{ξ}}_{1} - {\overset{\cdot}{y}}_{d} \\ = g_{1, p} ξ_{2} + f_{1, p} + q_{1, p} - {\overset{\cdot}{y}}_{d} \end{matrix}

(22)

The Lyapunov function candidate is chosen as

V_{1} = \frac{1}{2} e_{1}^{2} + \frac{1}{2 r_{1}} {\tilde{η}}_{1}^{2}

(23)

where $r_{1} > 0$ is a constant. Differentiating $V_{1}$ with respect to time $t$ , we have

\begin{matrix} {\overset{\cdot}{V}}_{1} = e_{1} {\overset{\cdot}{e}}_{2} - \frac{1}{r_{1}} {\tilde{η}}_{1} {\overset{\cdot}{\hat{η}}}_{1} \\ = e_{1} (g_{1, p} ξ_{2} + f_{1, p} (ξ) + q_{1, p} - {\overset{\cdot}{y}}_{d}) - \frac{1}{r_{1}} {\tilde{η}}_{1} {\overset{\cdot}{\hat{η}}}_{1} \\ = e_{1} g_{1, p} ξ_{2} + e_{1} f_{1, p} + e_{1} q_{1, p} - e_{1} {\overset{\cdot}{y}}_{d} - \frac{1}{r_{1}} {\tilde{η}}_{1} {\overset{\cdot}{\hat{η}}}_{1} \end{matrix}

(24)

Let us take the following error variable

e_{2} = ξ_{2} - ζ_{1}

(25)

where

ζ_{1} = - \frac{e_{1}}{{\underline{g}}_{1}} (k_{1} + l_{1}^{2} + β_{1}^{2} {\hat{η}}_{1} S_{1}^{T} S_{1})

(26)

By using equation (25) in equation (24), we have

\begin{matrix} {\overset{\cdot}{V}}_{1} = e_{1} g_{1, p} (e_{2} + ζ_{1}) + e_{1} f_{1, p} + e_{1} q_{1, p} - e_{1} {\overset{\cdot}{y}}_{d} - \frac{1}{r_{1}} {\tilde{η}}_{1} {\overset{\cdot}{\hat{η}}}_{1} \\ = e_{1} g_{1, p} e_{2} + e_{1} g_{1, p} ζ_{1} + e_{1} f_{1, p} + e_{1} q_{1, p} - e_{1} {\overset{\cdot}{y}}_{d} - \frac{1}{r_{1}} {\tilde{η}}_{1} {\overset{\cdot}{\hat{η}}}_{1} \\ = e_{1} g_{1, p} e_{2} + e_{1} g_{1, p} (- \frac{e_{1}}{{\underline{g}}_{1}} (k_{1} + l_{1}^{2} + β_{1}^{2} {\hat{η}}_{1} S_{1}^{T} S_{1})) \\ + e_{1} f_{1, p} + e_{1} q_{1, p} - e_{1} {\overset{\cdot}{y}}_{d} - \frac{1}{r_{1}} {\tilde{η}}_{1} {\overset{\cdot}{\hat{η}}}_{1} \\ = e_{1} g_{1, p} e_{2} - k_{1} e_{1}^{2} - l_{1}^{2} e_{1}^{2} - e_{1}^{2} β_{1}^{2} {\hat{η}}_{1} S_{1}^{T} S_{1} + e_{1} f_{1, p} - e_{1} {\overset{\cdot}{y}}_{d} \\ + e_{1} q_{1, p} - \frac{1}{r_{1}} {\tilde{η}}_{1} {\overset{\cdot}{\hat{η}}}_{1} \end{matrix}

(27)

Now, by using the completion of square, we have

e_{1} q_{1, p} \leq \frac{e_{1}^{2}}{2} + \frac{{\bar{q}}_{1, p}^{2}}{2}

(28)

Also,

\begin{matrix} e_{1} f_{1, p} (ξ) \leq | e_{1} | ϕ_{1, p} (∥ ξ ∥) \\ \leq | e_{1} | ϕ_{1, p} (\sum_{j = 1}^{n} | e_{j} | ψ_{j, p} ({\hat{η}}_{j}) + d^{*}) \\ \leq | e_{1} | \sum_{j = 1}^{n} ϕ_{1, p} ((n + 1) | e_{j} | ψ_{j, ρ} ({\hat{η}}_{j})) + | e_{1} | ϕ_{j, p} ((n + 1) d^{*}) \\ \leq \frac{1}{2} {ne}_{1}^{2} + \sum_{j = 1}^{n} \frac{1}{2} ϕ_{1}^{2} ((n + 1) | e_{j} | ψ_{j} ({\hat{η}}_{j})) \\ + | e_{1} | ϕ_{j, p} ((n + 1) d^{*}) \\ \leq \frac{1}{2} {ne}_{1}^{2} + \sum_{j = 1}^{n} e_{j}^{2} {\bar{ϕ}}_{1, p}^{2} ({\hat{η}}_{j}) + | e_{1} | ϕ_{j, p} ((n + 1) d^{*}) \end{matrix}

(29)

where ${\bar{ϕ}}_{1, p}^{2} ({\hat{η}}_{j}) = 1 / 2 (n + 1)^{2} ψ_{j, p}^{2} (e_{j}, {\hat{η}}_{j}) κ_{1, p}^{2} ((n + 1) | e_{j} | ψ_{j, p}^{2} ({\hat{η}}_{j})$ . Next by using Lemma 3 on the last term of equation (29), we have the following inequality

\begin{matrix} | e_{1} | ϕ_{1, ρ} ((n + 1) d^{*}) \leq | e_{1} | ϕ_{1, p} ((n + 1) d^{*}) \tanh \\ (\frac{| e_{1} | ϕ_{1, p} ((n + 1) d^{*})}{ς_{1}}) + δ ς_{1} \end{matrix}

(30)

By using equations (28)–(30) in equation (27), we have

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq e_{1} g_{1, p} e_{2} - k_{1} e_{1}^{2} - l_{1}^{2} e_{1}^{2} - e_{1}^{2} β_{1}^{2} {\hat{η}}_{1} S_{1}^{T} S_{1} + e_{1} {\bar{F}}_{1, p} (Z_{1}) \\ + \sum_{j = 2}^{n} e_{j}^{2} {\bar{ϕ}}_{1, p}^{2} ({\hat{η}}_{j}) \\ + \frac{{\bar{q}}_{1, p}^{2}}{2} + δ ς_{1} - e_{1}^{2} \sum_{l = 2}^{n - 1} \sum_{k = 1}^{l} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{1}) - \frac{1}{r_{1}} {\tilde{η}}_{1} {\overset{\cdot}{\hat{η}}}_{1} \end{matrix}

(31)

where

\begin{matrix} {\bar{F}}_{1, p} (Z_{1}) = \frac{1}{2} e_{1} (n + 1) + ϕ_{j, p} ((n + 1) d^{*}) \tanh \\ (\frac{| e_{1} | ϕ_{j, p} ((n + 1) d^{*})}{ς_{1}}) \\ + e_{1} \sum_{l = 1}^{n - 1} \sum_{k = 1}^{l} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{1}) - {\overset{\cdot}{y}}_{d} . \end{matrix}

(32)

As ${\bar{F}}_{1, p} (Z_{1})$ contains the unknown smooth function ${\bar{ϕ}}_{k, p}$ , an HNN $W_{1}^{T} S_{1} (Z_{1})$ is used to approximate ${\bar{F}}_{1, p} (Z_{1})$ such that for any $ϵ > 0$ , we have

{\bar{F}}_{1, p} (Z_{1}) = W_{1}^{T} S_{1} (Z_{1}) + δ_{1} (Z_{1}), | δ_{1} (Z_{1}) | \leq ϵ_{1}

(33)

where $δ_{1} (Z_{1})$ denotes the approximation error.

Now, by using Young’s inequality, we have

\begin{matrix} e_{1} {\bar{F}}_{1, p} (Z_{1}) = e_{1} (W_{1}^{T} S_{1} (Z_{1}) + δ_{1} (Z_{1})) \\ \leq | e_{1} | (∥ W_{1}^{T} ∥ ∥ S_{1} (Z_{1}) ∥ + ϵ_{1}) \\ \leq \frac{β_{1}^{2}}{2} e_{1}^{2} ∥ W_{1} ∥^{2} S_{1}^{T} S_{1} + \frac{l_{1}^{2}}{2} e_{1}^{2} + \frac{ϵ_{1}^{2}}{2 l_{1}^{2}} + \frac{1}{2 β_{1}^{2}} \\ \leq β_{1}^{2} e_{1}^{2} η_{1} S_{1}^{T} S_{1} + l_{1}^{2} e_{1}^{2} + \frac{ϵ_{1}^{2}}{l_{1}^{2}} + \frac{1}{β_{1}^{2}} \end{matrix}

(34)

where $η_{1} = ∥ W_{1} ∥^{2}$ . Substituting equation (34) into equation (31), we have

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq e_{1} g_{1, p} e_{2} - k_{1} e_{1}^{2} - l_{1}^{2} e_{1}^{2} - e_{1}^{2} β_{1}^{2} {\hat{η}}_{1} S_{1}^{T} S_{1} + β_{1}^{2} e_{1}^{2} η_{1} S_{1}^{T} S_{1} \\ + l_{1}^{2} e_{1}^{2} + \frac{ϵ_{1}^{2}}{l_{1}^{2}} + \frac{1}{β_{1}^{2}} + \sum_{j = 2}^{n} e_{j}^{2} {\bar{ϕ}}_{1, p}^{2} ({\hat{η}}_{j}) \\ + \frac{{\bar{q}}_{1, p}^{2}}{2} + δ ς_{1} - e_{1}^{2} \sum_{l = 2}^{n - 1} \sum_{k = 1}^{l} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{1}) - \frac{1}{r_{1}} {\tilde{η}}_{1} {\overset{\cdot}{\hat{η}}}_{1} + e_{1} g_{1, p} e_{2} \\ \leq - k_{1} e_{1}^{2} + e_{1}^{2} β_{1}^{2} S_{1}^{T} S_{1} (η_{1} - {\hat{η}}_{1}) + \frac{ϵ_{1}^{2}}{l_{1}^{2}} + \frac{1}{β_{1}^{2}} \\ + \sum_{j = 2}^{n} e_{j}^{2} {\bar{ϕ}}_{1, p}^{2} ({\hat{η}}_{j}) + \frac{{\bar{q}}_{1, p}^{2}}{2} + δ ς_{1} - e_{1}^{2} \sum_{l = 2}^{n - 1} \sum_{k = 1}^{l} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{1}) \\ - \frac{1}{r_{1}} {\tilde{η}}_{1} {\overset{\cdot}{\hat{η}}}_{1} + e_{1} g_{1, p} e_{2} \end{matrix}

(35)

Since ${\tilde{η}}_{1} = η_{1} - {\hat{η}}_{1}$ . Therefore, equation (35) becomes

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq - k_{1} e_{1}^{2} + e_{1}^{2} β_{1}^{2} S_{1}^{T} S_{1} {\tilde{η}}_{1} + \frac{ϵ_{1}^{2}}{l_{1}^{2}} + \frac{1}{β_{1}^{2}} + \sum_{j = 2}^{n} e_{j}^{2} {\bar{ϕ}}_{1, p}^{2} ({\hat{η}}_{j}) \\ + \frac{{\bar{q}}_{1, p}^{2}}{2} + δ ς_{1} - e_{1}^{2} \sum_{l = 2}^{n - 1} \sum_{k = 1}^{l} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{1}) \\ - \frac{1}{r_{1}} {\tilde{η}}_{1} {\overset{\cdot}{\hat{η}}}_{1} + e_{1} g_{1, p} e_{2} \\ \leq - k_{1} e_{1}^{2} + \frac{1}{r_{1}} {\tilde{η}}_{1} (r_{1} e_{1}^{2} β_{1}^{2} S_{1}^{T} S_{1} - {\overset{\cdot}{\hat{η}}}_{1}) + \frac{ϵ_{1}^{2}}{l_{1}^{2}} + \frac{1}{β_{1}^{2}} \\ + \sum_{j = 2}^{n} e_{j}^{2} {\bar{ϕ}}_{1, p}^{2} ({\hat{η}}_{j}) + \frac{{\bar{q}}_{1, p}^{2}}{2} + δ ς_{1} \\ - e_{1}^{2} \sum_{l = 2}^{n - 1} \sum_{k = 1}^{l} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{1}) + e_{1} g_{1, p} e_{2} \end{matrix}

(36)

Now, by using the adaptive law (21) with $i = 1$ in equation (35), we have

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq - k_{1} e_{1}^{2} + \frac{1}{r_{1}} {\tilde{η}}_{1} ϖ_{1} {\hat{η}}_{1} + \frac{ϵ_{1}^{2}}{l_{1}^{2}} + \frac{1}{β_{1}^{2}} + \sum_{j = 2}^{n} e_{j}^{2} {\bar{ϕ}}_{1, p}^{2} ({\hat{η}}_{j}) \\ + \frac{{\bar{q}}_{1, p}^{2}}{2} + δ ς_{1} - e_{1}^{2} \sum_{l = 2}^{n - 1} \sum_{k = 1}^{l} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{1}) + e_{1} g_{1, p} e_{2} \\ \leq - k_{1} e_{1}^{2} + \frac{1}{r_{1}} {\tilde{η}}_{1} {\hat{η}}_{1} ϖ_{1} + \sum_{j = 2}^{n} e_{j}^{2} {\bar{ϕ}}_{1, p}^{2} ({\hat{η}}_{j}) \\ - e_{1}^{2} \sum_{l = 2}^{n - 1} \sum_{k = 1}^{l} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{1}) + e_{1} g_{1, p} e_{2} + D_{1} \end{matrix}

(37)

where $D_{1}$ = $\frac{ϵ_{1}^{2}}{l_{1}^{2}} + \frac{1}{β_{1}^{2}} + \frac{{\bar{q}}_{1, p}^{2}}{2} + δ ς_{1}$ .

Step i $(2 \leq i \leq n - 1) :$ For $e_{i} = ξ_{i} - ζ_{i - 1}$ , the differentiation of $e_{i}$ with respect to time $t$ is given as

\begin{matrix} {\overset{\cdot}{e}}_{i} = {\overset{\cdot}{ξ}}_{i} - {\overset{\cdot}{ζ}}_{i - 1} \\ = g_{i, p} ξ_{i + 1} + f_{i, p} + q_{i, p} - {\overset{\cdot}{ζ}}_{i - 1} \end{matrix}

(38)

where

\begin{matrix} {\overset{\cdot}{ζ}}_{i - 1} = \sum_{k = 1}^{i - 1} \frac{\partial ζ_{i - 1}}{\partial ξ_{k}} (f_{k, p} + q_{k, p}) + \sum_{k = 1}^{i - 1} \frac{\partial ζ_{i - 1}}{\partial ξ_{k}} (g_{k, p} ξ_{k + 1}) \\ + \sum_{k = 0}^{i - 1} \frac{\partial ζ_{i - 1}}{\partial y_{d}^{(k)}} y_{d}^{(k + 1)} + \sum_{k = 1}^{i - 1} \frac{\partial ζ_{i - 1}}{\partial {\hat{η}}_{k}} {\overset{\cdot}{\hat{η}}}_{k} \end{matrix}

(39)

The CLF candidate is chosen as

V_{i} = V_{i - 1} + \frac{1}{2} e_{i}^{2} + \frac{1}{2 r_{i}} {\tilde{η}}_{i}^{2}

(40)

where $r_{i}$ is a positive design constant. Take the time derivative of equation (40), we have

{\overset{\cdot}{V}}_{i} = {\overset{\cdot}{V}}_{i - 1} + e_{i} {\overset{\cdot}{e}}_{i} - \frac{1}{r_{i}} {\tilde{η}}_{i} {\overset{\cdot}{\hat{η}}}_{i}

(41)

By using equation (38) in equation (41), we have

\begin{matrix} {\overset{\cdot}{V}}_{i} = {\overset{\cdot}{V}}_{i - 1} + e_{i} g_{i, p} ξ_{i + 1} + e_{i} f_{i, p} + e_{i} q_{i, p} - e_{i} {\overset{\cdot}{ζ}}_{i - 1} - \frac{1}{r_{i}} {\tilde{η}}_{i} {\overset{\cdot}{\hat{η}}}_{i} \\ \leq {\overset{\cdot}{V}}_{i - 1} + e_{i} g_{i, p} ξ_{i + 1} + e_{i} f_{i, p} + e_{i} q_{i, p} \\ - e_{i} \sum_{k = 1}^{i - 1} \frac{\partial ζ_{i - 1}}{\partial ξ_{k}} (f_{k, p} + q_{k, p}) - e_{i} \sum_{k = 1}^{i - 1} \frac{\partial ζ_{i - 1}}{\partial ξ_{k}} (g_{k, p} ξ_{k + 1}) \\ - e_{i} \sum_{k = 0}^{i - 1} \frac{\partial ζ_{i - 1}}{\partial y_{d}^{(k)}} y_{d}^{(k + 1)} \\ - e_{i} \sum_{k = 1}^{i - 1} \frac{\partial ζ_{i - 1}}{\partial {\hat{η}}_{k}} {\overset{\cdot}{\hat{η}}}_{k} - \frac{1}{r_{i}} {\tilde{η}}_{i} {\overset{\cdot}{\hat{η}}}_{i} \end{matrix}

(42)

where the term ${\overset{\cdot}{V}}_{i - 1}$ in equation (42) can be generated in the following form by using the step 1 technique

\begin{matrix} {\overset{\cdot}{V}}_{i - 1} \leq - \sum_{j = 1}^{i - 1} k_{j} e_{j}^{2} + \sum_{j = 1}^{i - 1} \frac{1}{r_{j}} {\tilde{η}}_{j} {\hat{η}}_{j} ϖ_{j} + \sum_{s = 1}^{i - 1} \sum_{k = 1}^{s} \sum_{j = i}^{n} e_{j}^{2} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{j}) \\ - \sum_{s = 1}^{i - 1} \sum_{l = i}^{n - 1} \sum_{k = 1}^{l} e_{s}^{2} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{s}) + \sum_{j = 1}^{i - 1} e_{j} g_{j, p} e_{j + 1} + \sum_{j = 1}^{i - 1} D_{j} \end{matrix}

(43)

where $D_{j} = (ϵ_{j}^{2} / l_{j}^{2}) + (1 / β_{j}^{2}) + ({\bar{q}}_{j, p}^{2} / 2) + δ ς_{j}$ .

The following two inequalities can be obtained by following the approach described in step 1

\begin{matrix} - e_{i} \sum_{k = 1}^{i - 1} \frac{\partial ζ_{i - 1}}{\partial ξ_{k}} f_{k, p} (ξ) \leq \frac{1}{2} {ne}_{i}^{2} \sum_{k = 1}^{i - 1} {(\frac{\partial ζ_{i - 1}}{\partial ξ_{k}})}^{2} + \sum_{k = 1}^{i - 1} \sum_{j = 1}^{n} \\ e_{j}^{2} {\bar{ϕ}}_{k, ρ}^{2} ({\hat{η}}_{j}) + | e_{i} | \sum_{k = 1}^{i - 1} | \frac{\partial ζ_{i - 1}}{\partial ξ_{k}} | {\bar{ϕ}}_{k, p} ((n + 1) d^{*}) \end{matrix}

(44)

e_{i} f_{i, p} (ξ) \leq \frac{1}{2} {ne}_{i}^{2} + \sum_{j = 1}^{n} e_{j}^{2} {\bar{ϕ}}_{i, p}^{2} ({\hat{η}}_{j}) + | e_{i} | ϕ_{i, p} ((n + 1) d^{*})

(45)

In addition, as with equation (30), the following statement is correct

| e_{i} | Y_{i, p} \leq e_{i} Y_{i, p} \tanh (\frac{e_{i} Y_{i, p}}{ς_{i}}) + δ ς_{i}

(46)

where $Y_{i, p} = \sum_{k = 1}^{i - 1} | \partial ζ_{i - 1} / \partial ξ_{k} | ϕ_{k, p} ((n + 1) d^{*}) + ϕ_{i, p} ((n + 1) d^{*}$ .

Also

e_{i} q_{i, p} \leq (e_{i}^{2} / 2) + ({\bar{q}}_{i, p}^{2} / 2)

(47)

- e_{i} \sum_{k = 1}^{i - 1} \frac{\partial ζ_{i - 1}}{\partial ξ_{k}} q_{k, p} \leq e_{i}^{2} \sum_{k = 1}^{i - 1} {(\frac{\partial ζ_{i - 1}}{\partial ξ_{k}})}^{2} + \sum_{k = 1}^{i - 1} \frac{{\bar{q}}_{k, p}^{2}}{2}

(48)

By using equations (43)–(48) in equation (42), we have

\begin{matrix} {\overset{\cdot}{V}}_{i} \leq - \sum_{j = 1}^{i - 1} k_{j} e_{j}^{2} + \sum_{j = 1}^{i - 1} \frac{1}{r_{j}} {\tilde{η}}_{j} {\hat{η}}_{j} ϖ_{j} + \sum_{s = 1}^{i} \sum_{k = 1}^{s} \sum_{j = i + 1}^{n} e_{j}^{2} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{j}) \\ - \sum_{s = 1}^{i} \sum_{l = i + 1}^{n - 1} \sum_{k = 1}^{l} e_{s}^{2} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{s}) + \sum_{j = 1}^{i - 1} e_{j} g_{j, p} e_{j + 1} + \sum_{j = 1}^{i - 1} D_{j} \\ + e_{i} g_{i, p} (e_{i + 1} + ζ_{i}) + \frac{1}{2} {ne}_{i}^{2} + \frac{e_{i}^{2}}{2} + \frac{{\bar{q}}_{i, p}^{2}}{2} \\ + e_{i} \sum_{l = 1}^{n - 1} \sum_{k = 1}^{l} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{i}) + \frac{1}{2} {ne}_{i}^{2} \sum_{k = 1}^{i - 1} {(\frac{\partial ζ_{i - 1}}{\partial ξ_{k}})}^{2} \\ + e_{i} Y_{i, p} \tanh (\frac{e_{i} Y_{i, p}}{ς_{i}}) + δ ς_{i} \\ + e_{i}^{2} \sum_{k = 1}^{i - 1} {(\frac{\partial ζ_{i - 1}}{\partial ξ_{k}})}^{2} + \sum_{k = 1}^{i - 1} \frac{{\bar{q}}_{k, p}^{2}}{2} - e_{i} \sum_{k = 1}^{i - 1} \frac{\partial ζ_{i - 1}}{\partial ξ_{k}} (g_{k, p} ξ_{k + 1}) \\ - e_{i} \sum_{k = 0}^{i - 1} \frac{\partial ζ_{i - 1}}{\partial y_{d}^{(k)}} y_{d}^{(k + 1)} - e_{i} \sum_{k = 1}^{i - 1} \frac{\partial ζ_{i - 1}}{\partial {\hat{η}}_{k}} {\overset{\cdot}{\hat{η}}}_{k} - \frac{1}{r_{i}} {\tilde{η}}_{i} {\overset{\cdot}{\hat{η}}}_{i} \end{matrix}

(49)

with $e_{i + 1} = ξ_{i + 1} - ζ_{i}$ .

Now by using the virtual control law (19), we have the following inequality

e_{i} g_{i, p} ζ_{i} \leq - k_{i} e_{i}^{2} - l_{i}^{2} e_{i}^{2} - e_{i}^{2} β_{i}^{2} {\hat{η}}_{i} S_{i}^{T} S_{i}

(50)

Therefore, by using equation (50) in equation (49), we have

\begin{matrix} {\overset{\cdot}{V}}_{i} \leq - \sum_{j = 1}^{i} k_{j} e_{j}^{2} + \sum_{j = 1}^{i - 1} \frac{1}{r_{j}} {\tilde{η}}_{j} {\hat{η}}_{j} ϖ_{j} + \sum_{s = 1}^{i} \sum_{k = 1}^{s} \sum_{j = i + 1}^{n} e_{j}^{2} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{j}) \\ - \sum_{s = 1}^{i} \sum_{l = i + 1}^{n - 1} \sum_{k = 1}^{l} e_{s}^{2} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{s}) + \sum_{j = 1}^{i} e_{j} g_{j, p} e_{j + 1} + \sum_{j = 1}^{i - 1} D_{j} \\ - l_{i}^{2} e_{i}^{2} - e_{i}^{2} β_{i}^{2} {\hat{η}}_{i} S_{i}^{T} S_{i} + e_{i} {\bar{F}}_{i, p} + δ ς_{i} + \sum_{k = 1}^{i} \frac{{\bar{q}}_{k, p}^{2}}{2} - \frac{1}{r_{i}} {\tilde{η}}_{i} {\overset{\cdot}{\hat{η}}}_{i} \end{matrix}

(51)

where

\begin{matrix} {\bar{F}}_{i, p} = \frac{1}{2} n e_{i} + \frac{e_{i}}{2} + \sum_{l = 1}^{n - 1} \sum_{k = 1}^{l} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{i}) + \frac{1}{2} n e_{i} \sum_{k = 1}^{i - 1} {(\frac{\partial ζ_{i - 1}}{\partial ξ_{k}})}^{2} \\ + Y_{i, p} \tanh (\frac{e_{i} Y_{i, p}}{ς_{i}}) + e_{i} \sum_{k = 1}^{i - 1} {(\frac{\partial ζ_{i - 1}}{\partial ξ_{k}})}^{2} \\ - e_{i} \sum_{k = 1}^{i - 1} \frac{\partial ζ_{i - 1}}{\partial ξ_{k}} (g_{k, p} ξ_{k + 1}) - e_{i} \sum_{k = 0}^{i - 1} \frac{\partial ζ_{i - 1}}{\partial y_{d}^{(k)}} y_{d}^{(k + 1)} \\ - e_{i} \sum_{k = 1}^{i - 1} \frac{\partial ζ_{i - 1}}{\partial {\hat{η}}_{k}} {\overset{\cdot}{\hat{η}}}_{k} \end{matrix}

(52)

To approximate ${\bar{F}}_{i, p} (Z_{i})$ , an HNN $W_{i}^{T} S_{i} (Z_{i})$ is used, such that for any $ϵ > 0$ , we have

{\bar{F}}_{i, p} (Z_{i}) = W_{i}^{T} S_{i} (Z_{i}) + δ_{i} (Z_{i}), | δ_{i} (Z_{i}) | \leq ϵ_{i}

(53)

where $δ_{i} (Z_{i})$ denotes the approximation error. We may observe that, similarly to equation (34)

e_{i} {\bar{F}}_{i, p} (Z_{i}) \leq β_{i}^{2} e_{i}^{2} η_{i} S_{i}^{T} S_{i} + l_{i}^{2} e_{i}^{2} + \frac{ϵ_{i}^{2}}{l_{i}^{2}} + \frac{1}{β_{i}^{2}}

(54)

Now, by using equation (54) in equation (51), we have

\begin{matrix} {\overset{\cdot}{V}}_{i} \leq - \sum_{j = 1}^{i} k_{j} e_{j}^{2} + \sum_{j = 1}^{i - 1} \frac{1}{r_{j}} {\tilde{η}}_{j} {\hat{η}}_{j} ϖ_{j} + \sum_{s = 1}^{i} \sum_{k = 1}^{s} \sum_{j = i + 1}^{n} e_{j}^{2} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{j}) \\ - \sum_{s = 1}^{i} \sum_{l = i + 1}^{n - 1} \sum_{k = 1}^{l} e_{s}^{2} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{s}) + \sum_{j = 1}^{i} e_{j} g_{j, p} e_{j + 1} + \sum_{j = 1}^{i - 1} D_{j} \\ - l_{i}^{2} e_{i}^{2} - e_{i}^{2} β_{i}^{2} {\hat{η}}_{i} S_{i}^{T} S_{i} + β_{i}^{2} e_{i}^{2} η_{i} S_{i}^{T} S_{i} + l_{i}^{2} e_{i}^{2} \\ + \frac{ϵ_{i}^{2}}{l_{i}^{2}} + \frac{1}{β_{i}^{2}} + δ ς_{i} + \sum_{k = 1}^{i} \frac{{\bar{q}}_{k, p}^{2}}{2} - \frac{1}{r_{i}} {\tilde{η}}_{i} {\overset{\cdot}{\hat{η}}}_{i} \end{matrix}

(55)

After rearranging equation (55), we have

\begin{matrix} {\overset{\cdot}{V}}_{i} \leq - \sum_{j = 1}^{i} k_{j} e_{j}^{2} + \sum_{j = 1}^{i - 1} \frac{1}{r_{j}} {\tilde{η}}_{j} {\hat{η}}_{j} ϖ_{j} + \sum_{s = 1}^{i} \sum_{k = 1}^{s} \sum_{j = i + 1}^{n} e_{j}^{2} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{j}) \\ - \sum_{s = 1}^{i} \sum_{l = i + 1}^{n - 1} \sum_{k = 1}^{l} e_{s}^{2} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{s}) + \sum_{j = 1}^{i} e_{j} g_{j, p} e_{j + 1} + \sum_{j = 1}^{i - 1} D_{j} \\ + β_{i}^{2} e_{i}^{2} S_{i}^{T} S_{i} (η_{i} - {\hat{η}}_{i}) + \frac{ϵ_{i}^{2}}{l_{i}^{2}} + \frac{1}{β_{i}^{2}} + δ ς_{i} \\ + \sum_{k = 1}^{i} \frac{{\bar{q}}_{k, p}^{2}}{2} - \frac{1}{r_{i}} {\tilde{η}}_{i} {\overset{\cdot}{\hat{η}}}_{i} \end{matrix}

(56)

Since ${\tilde{η}}_{i} = η_{i} - {\hat{η}}_{i}$ , therefore equation (56) becomes

\begin{matrix} {\overset{\cdot}{V}}_{i} \leq - \sum_{j = 1}^{i} k_{j} e_{j}^{2} + \sum_{j = 1}^{i - 1} \frac{1}{r_{j}} {\tilde{η}}_{j} {\hat{η}}_{j} ϖ_{j} + \sum_{s = 1}^{i} \sum_{k = 1}^{s} \sum_{j = i + 1}^{n} e_{j}^{2} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{j}) \\ - \sum_{s = 1}^{i} \sum_{l = i + 1}^{n - 1} \sum_{k = 1}^{l} e_{s}^{2} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{s}) + \sum_{j = 1}^{i} e_{j} g_{j, p} e_{j + 1} + \sum_{j = 1}^{i - 1} D_{j} \\ + \frac{1}{r_{i}} {\tilde{η}}_{i} (r_{i} β_{i}^{2} e_{i}^{2} S_{i}^{T} S_{i} - {\overset{\cdot}{\hat{η}}}_{i}) + \frac{ϵ_{i}^{2}}{l_{i}^{2}} + \frac{1}{β_{i}^{2}} + δ ς_{i} + \sum_{k = 1}^{i} \frac{{\bar{q}}_{k, p}^{2}}{2} \end{matrix}

(57)

Now, with the adaptation law (21) in consideration, it is easy to achieve

\begin{matrix} {\overset{\cdot}{V}}_{i} \leq - \sum_{j = 1}^{i} k_{j} e_{j}^{2} + \sum_{j = 1}^{i} \frac{1}{r_{j}} {\tilde{η}}_{j} {\hat{η}}_{j} ϖ_{j} + \sum_{s = 1}^{i} \sum_{k = 1}^{s} \sum_{j = i + 1}^{n} e_{j}^{2} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{j}) \\ - \sum_{s = 1}^{i} \sum_{l = i + 1}^{n - 1} \sum_{k = 1}^{l} e_{s}^{2} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{s}) + \sum_{j = 1}^{i} e_{j} g_{j, p} e_{j + 1} + \sum_{j = 1}^{i} D_{j} \end{matrix}

(58)

where $D_{j} = (ϵ_{j}^{2} / l_{j}^{2}) + (1 / β_{j}^{2}) + \sum_{k = 1}^{j} ({\bar{q}}_{k, p}^{2} / 2) + δ ς_{j}$

Step n: For $e_{n} = ξ_{n} - ζ_{n - 1}$ , we have

{\overset{\cdot}{e}}_{n} = g_{n, p} v + f_{n, p} + q_{n, p} - {\overset{\cdot}{ζ}}_{n - 1}

(59)

where ${\overset{\cdot}{ζ}}_{n - 1}$ defined in equation (39) with $i = n$ .

The CLF candidate is chosen as

V_{n} = V_{n - 1} + \frac{1}{2} e_{n}^{2} + \frac{1}{2 r_{n}} {\tilde{η}}_{n}^{2}

(60)

Differentiating $V_{n}$ with respect to the time $t$ , we have

\begin{matrix} {\overset{\cdot}{V}}_{n} = {\overset{\cdot}{V}}_{n - 1} + e_{n} (g_{n, p} v + f_{n, p} + q_{n, p} - {\overset{\cdot}{ζ}}_{n - 1}) \\ = {\overset{\cdot}{V}}_{n - 1} + e_{n} g_{n, p} v + e_{n} f_{n, p} + e_{n} q_{n, p} - e_{n} {\overset{\cdot}{ζ}}_{n - 1} \end{matrix}

(61)

Now, design the real control law $v$ as in equation (20), we have the following inequality

e_{n} g_{n, p} v \leq - k_{n} e_{n}^{2} - l_{n}^{2} e_{n}^{2} - e_{n}^{2} β_{n}^{2} {\hat{η}}_{n} S_{n}^{T} S_{n}

(62)

Similar to equation (47) with $i = n$ , we have the following inequality

e_{n} q_{n, p} \leq \frac{e_{n}^{2}}{2} + \frac{{\bar{q}}_{n, p}^{2}}{2}

(63)

As a result, by proceeding in the same manner and employing inequalities (62) and (63) in equation (61), we have

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq - \sum_{j = 1}^{n - 1} k_{j} e_{j}^{2} + \sum_{j = 1}^{n - 1} \frac{1}{r_{j}} {\tilde{η}}_{j} {\hat{η}}_{j} ϖ_{j} + \sum_{j = 1}^{n - 1} e_{j} g_{j, p} e_{j + 1} \\ + \sum_{j = 1}^{n - 1} D_{j} + \frac{e_{n}^{2}}{2} + \frac{{\bar{q}}_{n, p}^{2}}{2} - k_{n} e_{n}^{2} - l_{n}^{2} e_{n}^{2} - e_{n}^{2} β_{n}^{2} {\hat{η}}_{n} S_{n}^{T} S_{n} \\ + e_{n} f_{n, p} - e_{n} \sum_{k = 1}^{n - 1} \frac{\partial ζ_{n - 1}}{\partial ξ_{k}} f_{k, p} (ξ) - e_{n} \sum_{k = 1}^{n - 1} \frac{\partial ζ_{n - 1}}{\partial ξ_{k}} (g_{k, p} ξ_{k + 1}) \\ - e_{n} \sum_{k = 0}^{n - 1} \frac{\partial ζ_{n - 1}}{\partial y_{d}^{(k)}} y_{d}^{(k + 1)} - e_{n} \sum_{k = 1}^{n - 1} \frac{\partial ζ_{n - 1}}{\partial {\hat{η}}_{k}} {\overset{\cdot}{\hat{η}}}_{k} - \frac{1}{r_{n}} {\tilde{η}}_{n} {\overset{\cdot}{\hat{η}}}_{n} \\ + e_{n}^{2} \sum_{l = 1}^{n - 1} \sum_{k = 1}^{l} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{n}) \end{matrix}

(64)

Now after rearranging the terms, we have

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq - \sum_{j = 1}^{n - 1} k_{j} e_{j}^{2} + \sum_{j = 1}^{n - 1} \frac{1}{r_{j}} {\tilde{η}}_{j} {\hat{η}}_{j} ϖ_{j} + \sum_{j = 1}^{n - 1} e_{j} g_{j, p} e_{j + 1} \\ + \sum_{j = 1}^{n - 1} D_{j} + \frac{{\bar{q}}_{n, p}^{2}}{2} \\ - k_{n} e_{n}^{2} - l_{n}^{2} e_{n}^{2} - e_{n}^{2} β_{n}^{2} {\hat{η}}_{n} S_{n}^{T} S_{n} + e_{m} {\bar{F}}_{n, p} - \frac{1}{r_{n}} {\tilde{η}}_{n} {\overset{\cdot}{\hat{η}}}_{n} \end{matrix}

(65)

where

\begin{matrix} {\bar{F}}_{n, p} = \frac{e_{n}}{2} + f_{n, p} - \sum_{k = 1}^{n - 1} \frac{\partial ζ_{n - 1}}{\partial ξ_{k}} f_{k, p} (ξ) - \sum_{k = 1}^{n - 1} \frac{\partial ζ_{n - 1}}{\partial ξ_{k}} (g_{k, p} ξ_{k + 1}) \\ - \sum_{k = 0}^{n - 1} \frac{\partial ζ_{n - 1}}{\partial y_{d}^{(k)}} y_{d}^{(k + 1)} - \sum_{k = 1}^{n - 1} \frac{\partial ζ_{n - 1}}{\partial {\hat{η}}_{k}} {\overset{\cdot}{\hat{η}}}_{k} + e_{n} \sum_{l = 1}^{n - 1} \sum_{k = 1}^{l} {\bar{ϕ}}_{k, p}^{2} ({\hat{η}}_{n}) . \end{matrix}

(66)

Now, an HNN $W_{n}^{T} S_{n} (Z_{n})$ is used to approximate ${\bar{F}}_{n, p}$ such that for any $ϵ > 0$ , we have

{\bar{F}}_{n, p} = W_{n}^{T} S_{n} (Z_{n}) + δ_{n} (Z_{n}); | δ_{n} (Z_{n}) | \leq ϵ_{n}

(67)

Similarly, we have

e_{n} {\bar{F}}_{n, p} (Z_{n}) \leq β_{n}^{2} e_{n}^{2} η_{n} S_{n}^{T} S_{n} + l_{n}^{2} e_{n}^{2} + \frac{ϵ_{n}^{2}}{l_{n}^{2}} + \frac{1}{β_{n}^{2}}

(68)

By using equation (68) in equation (65), we have

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq - \sum_{j = 1}^{n - 1} k_{j} e_{j}^{2} + \sum_{j = 1}^{n - 1} \frac{1}{r_{j}} {\tilde{η}}_{j} {\hat{η}}_{j} ϖ_{j} + \sum_{j = 1}^{n - 1} e_{j} g_{j, p} e_{j + 1} \\ + \sum_{j = 1}^{n - 1} D_{j} + \frac{{\bar{q}}_{n, p}^{2}}{2} - k_{n} e_{n}^{2} \\ - l_{n}^{2} e_{n}^{2} - e_{n}^{2} β_{n}^{2} {\hat{η}}_{n} S_{n}^{T} S_{n} + β_{n}^{2} e_{n}^{2} η_{1} S_{n}^{T} S_{n} + l_{n}^{2} e_{n}^{2} \\ + \frac{ϵ_{n}^{2}}{l_{n}^{2}} + \frac{1}{β_{n}^{2}} - \frac{1}{r_{n}} {\tilde{η}}_{n} {\overset{\cdot}{\hat{η}}}_{n} \end{matrix}

(69)

After rearranging the terms, we have

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq - \sum_{j = 1}^{n} k_{j} e_{j}^{2} + \sum_{j = 1}^{n - 1} \frac{1}{r_{j}} {\tilde{η}}_{j} {\hat{η}}_{j} ϖ_{j} + \sum_{j = 1}^{n - 1} e_{j} g_{j, p} e_{j + 1} \\ + \sum_{j = 1}^{n - 1} D_{j} + \frac{{\bar{q}}_{n, p}^{2}}{2} \\ + β_{n}^{2} e_{n}^{2} S_{n}^{T} S_{n} (η_{n} - {\hat{η}}_{n}) + \frac{ϵ_{n}^{2}}{l_{n}^{2}} + \frac{1}{β_{n}^{2}} - \frac{1}{r_{n}} {\tilde{η}}_{n} {\overset{\cdot}{\hat{η}}}_{n} \end{matrix}

(70)

Since ${\tilde{η}}_{n} = η_{n} - {\hat{η}}_{n}$ ; therefore, equation (70) becomes

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq - \sum_{j = 1}^{n} k_{j} e_{j}^{2} + \sum_{j = 1}^{n - 1} \frac{1}{r_{j}} {\tilde{η}}_{j} {\hat{η}}_{j} ϖ_{j} + \sum_{j = 1}^{n - 1} e_{j} g_{j, p} e_{j + 1} \\ + \sum_{j = 1}^{n - 1} D_{j} + \frac{{\bar{q}}_{n, p}^{2}}{2} \\ + \frac{1}{r_{n}} {\tilde{η}}_{n} (r_{n} β_{n}^{2} e_{n}^{2} S_{n}^{T} S_{n} - {\overset{\cdot}{\hat{η}}}_{n}) + \frac{ϵ_{n}^{2}}{l_{n}^{2}} + \frac{1}{β_{n}^{2}} \end{matrix}

(71)

Now, with the adaptation law (21) with $i = n$ in consideration, it is easy to achieve

{\overset{\cdot}{V}}_{n} \leq - \sum_{j = 1}^{n} k_{j} e_{j}^{2} + \sum_{j = 1}^{n} \frac{1}{r_{j}} {\tilde{η}}_{j} {\hat{η}}_{j} ϖ_{j} + \sum_{j = 1}^{n - 1} e_{j} g_{j, p} e_{j + 1} + \sum_{j = 1}^{n} D_{j}

(72)

where $D_{j} = (ϵ_{j}^{2} / l_{j}^{2}) + (1 / β_{j}^{2}) + \sum_{k = 1}^{j} {\bar{q}}_{k, p}^{2} / 2) +$ $(2, \dots, n - 1$ , $D_{n} = ({\bar{q}}_{n, p}^{2} / 2) + (ϵ_{n}^{2} / l_{n}^{2}) + (1 / β_{n}^{2}) .$

Theorem 1: Suppose the switched nonlinear system (1) under Assumptions 1–3. For bounded initial condition, the virtual control law (19), the real control law (20), and the adaptive law (21) guarantee that all the signals in the closed-loop system remain bounded, and when $t \to \infty$ , the following inequality holds for $1 \leq i \leq n$

lim_{t \to \infty} e_{1}^{2} \leq 2 \frac{d_{0}}{a_{0}}

(73)

where $a_{0} = min {2 c_{j}, ϖ_{j} : 1 \leq j \leq n}$ , $d_{0} = \sum_{j = 1}^{n} D_{j} + \sum_{j = 1}^{n} (1 / 2 r_{j}) η_{j}^{2} ϖ_{j} .$

Proof: From equation (72)

{\overset{\cdot}{V}}_{n} \leq - \sum_{j = 1}^{n} k_{j} e_{j}^{2} + \sum_{j = 1}^{n} \frac{1}{r_{j}} {\tilde{η}}_{j} {\hat{η}}_{j} ϖ_{j} + \sum_{j = 1}^{n - 1} e_{j} g_{j, p} e_{j + 1} + \sum_{j = 1}^{n} D_{j} .

(74)

It is easy to see that

\sum_{j = 1}^{n - 1} e_{j} g_{j, p} e_{j + 1} \leq {\bar{g}}_{j, p} \sum_{j = 1}^{n} e_{j}^{2}

(75)

where ${\bar{g}}_{j, p} = max {g_{j, p}, p \in M} .$

Notice that

{\tilde{η}}_{i} {\hat{η}}_{i} = {\tilde{η}}_{i} (η_{i} - {\tilde{η}}_{i}) \leq - \frac{1}{2} {\tilde{η}}_{i}^{2} + \frac{1}{2} η_{i}^{2}

(76)

Therefore, one has

\sum_{j = 1}^{n} \frac{1}{r_{j}} {\tilde{η}}_{j} {\hat{η}}_{j} ϖ_{j} \leq - \sum_{j = 1}^{n} \frac{1}{2 r_{j}} {\tilde{η}}_{j}^{2} ϖ_{j} + \sum_{j = 1}^{n} \frac{1}{2 r_{j}} η_{j}^{2} ϖ_{j}

(77)

Using equations (75) and (77) in equation (74), we have

\begin{matrix} {\overset{\cdot}{V}}_{n} \leq - \sum_{j = 1}^{n} c_{j} e_{j}^{2} - \sum_{j = 1}^{n} \frac{1}{2 r_{j}} {\tilde{η}}_{j}^{2} ϖ_{j} + \sum_{j = 1}^{n} \frac{1}{2 r_{j}} η_{j}^{2} ϖ_{j} + \sum_{j = 1}^{n} D_{j} \\ \leq - a_{0} V_{n} + d_{0} \end{matrix}

(78)

where $a_{0} = min {2 c_{j}, ϖ_{j} : 1 \leq j \leq n}$ , $d_{0} = \sum_{j = 1}^{n} D_{j} + \sum_{j = 1}^{n} (1 / 2 r_{j}) η_{j}^{2} ϖ_{j} .$

Furthermore, one has

V (t) \leq (V (0) - \frac{d_{0}}{a_{0}}) e^{- a_{0} t} + \frac{d_{0}}{a_{0}}

(79)

According to inequality (79), all signals in the closed-loop system are bounded. Furthermore, we have

lim_{t \to \infty} e_{1}^{2} \leq 2 \frac{d_{0}}{a_{0}}

(80)

Hence the proof is completed. The flow diagram of proposed control scheme is shown in Figure 2.

Remark 3: The inequality (80) demonstrates that the state variables will ultimately converge to an area with a radius of $2 (d_{0} / a_{0})$ around the origin. However, since $d_{0}$ depends on the unknown parameters $η_{j}$ , the radius of the neighborhood $2 (d_{0} / a_{0})$ is unknown. Then, despite the fact that it is hard to explicitly estimate $2 (d_{0} / a_{0})$ , it is obvious that lowering $ϖ_{j}$ while raising $r_{j}$ will result in a smaller $2 (d_{0} / a_{0})$ .

Simulation results

To prove the effectiveness of proposed model, two examples are shown in this section. In the first example, we consider the numerical example and the another one is the application to the ship maneuvering system. We choose the radial basis function neural network (RBFNN)-based controller, as a reference for contrast.

Performance criteria

In order to analyze the performance of HNN and comparison with RBFNN, we considered following error parameters. For the given $N$ pair of data point of $(y (t), y_{d} (t))$ , where $y (t)$ is the system output and $y_{d} (t)$ is the reference signal, the different type of errors are defined as follows:

The maximum absolute error (MAE)

MAE = max_{1 \leq t \leq N} | y (t) - y_{d} (t) |

(81)

The sum of squared error (SSE)

SSE = \sum_{t = 1}^{N} (y (t) - y_{d} (t))^{2}

(82)

The mean squared error (MSE)

MSE = \frac{1}{N} \sum_{t = 1}^{N} (y (t) - y_{d} (t))^{2}

(83)

Root mean squared error (RMSE)

RMSE = \sqrt{\frac{1}{N} \sum_{t = 1}^{N} {(y (t) - y_{d} (t))}^{2}}

(84)

Normalized mean squared error (NMSE)

NMSE = \frac{\sum_{t = 1}^{N} (y (t) - y_{d} (t))^{2}}{\sum_{t = 1}^{N} (y (t) - y_{d})^{2}}

(85)

and the best fit rate (BFR)

BFR = (1 - \frac{\sqrt{\sum_{t = 1}^{N} {(y (t) - y_{d} (t))}^{2}}}{\sum_{t = 1}^{N} {(y (t) - y_{d})}^{2}}) 100 %

(86)

Example 1: Consider the second-order switched nonlinear switched system consisting of two subsystems 1 and 2 as shown below.

Case 1: When $ϱ (t) = 1$

{\begin{matrix} {\dot{ξ}}_{1} = g_{1, 1} ξ_{2} + f_{1, 1} (ξ) + q_{1, 1} \\ {\dot{ξ}}_{2} = g_{2, 1} v + f_{2, 1} (ξ) + q_{2, 1} \\ y = ξ_{1} \end{matrix}

(87)

Case 2: When $ϱ (t) = 2$

{\begin{matrix} {\overset{\cdot}{ξ}}_{1} = g_{1, 2} ξ_{2} + f_{1, 2} (ξ) + q_{1, 2} \\ {\overset{\cdot}{ξ}}_{2} = g_{2, 2} v + f_{2, 2} (ξ) + q_{2, 2} \\ y = ξ_{1} \end{matrix}

(88)

where $ξ = [ξ_{1}, ξ_{2}]^{T}$ represents the state variables. The main objective is that the system output $y$ follows the desired trajectory $y_{d} = \cos t$ and all signals in the close-loop system remain bounded. By Theorem 1, the virtual controller, real controller and adaptive laws are selected, respectively, as

ζ_{1} = - \frac{e_{1}}{{\underline{g}}_{1}} (k_{1} + l_{1}^{2} + β_{1}^{2} {\hat{η}}_{1} S_{1}^{T} S_{1})

v = - \frac{e_{2}}{{\underline{g}}_{2}} (k_{2} + l_{2}^{2} + β_{2}^{2} {\hat{η}}_{2} S_{2}^{T} S_{2})

{\overset{\cdot}{\hat{η}}}_{1} = r_{1} β_{1}^{2} e_{1}^{2} S_{1}^{T} S_{1} - ϖ_{1} {\hat{η}}_{1}

{\overset{\cdot}{\hat{η}}}_{2} = r_{2} β_{2}^{2} e_{2}^{2} S_{2}^{T} S_{2} - ϖ_{2} {\hat{η}}_{2}

where $e_{1} = ξ_{1} - y_{d}$ , $e_{2} = ξ_{2} - ζ_{1}$ . The initial condition of the system is chosen as $ξ_{1} (0) = ξ_{2} (0) = 0.02$ and ${\hat{η}}_{1} (0) = {\hat{η}}_{2} (0) = 0$ . The center of the receptive field and the width of the Gaussian function are chosen as $μ_{i} = [- 2, 0] \times [- 1, 0] \times [0, 1] \times [0, 2]$ and $ν_{i} = 2$ , $i = 1, 2$ respectively. The design parameters are chosen as $k_{1} = 10$ , $k_{2} = 10$ , $r_{1} = 0.01$ , $r_{2} = 0.01$ , $β_{1} = 0.1$ , $β_{2} = 0.3$ , $l_{1} = l_{2} = 0.7$ , $ϖ_{1} = ϖ_{2} = 0.025$ , $g_{1, 1} = 2$ , $g_{1, 2} = 1$ , $g_{2, 1} = 8$ , and $g_{2, 2} = 5$ . To obtain the simulation results, we assumed that $f_{1, 1} = ξ_{1}$ , $f_{1, 2} = \cos (ξ_{1})$ , $f_{2, 1} = ξ_{1}^{2} ξ_{2}$ , $f_{2, 2} = ξ_{1} ξ_{2}^{2}$ , $q_{1, 1} = 0.5 \sin (t) \cos (0.2 t)$ , $q_{2, 1} = 0.25 (\sin (t))^{2}$ , $q_{1, 2} = 0.5 \sin (t)$ , and $q_{2, 2} = 0.5 \sin (t) \cos (t)$ . The simulation results are demonstrated in Figures 3 –8. The trajectories of the tracking performance and tracking error under HNN and RBFNN are shown in Figures 3 and 4, respectively. The responses of the adaptive laws are demonstrated in Figure 5. The trajectories of virtual control and the real control law are shown in Figures 6 and 7, respectively. The trajectory of switching signal is shown in Figure 8.

Table 1 shows the comparison of HNN and RBFNN based on error parameters defined in equations (81)–(86).

Table 1.

Comparison of proposed method based on different assessment error criteria for example 1.

Model	Different error calculations
Model	MAE	SSE	MSE	RMSE	NMSE	BFR
HNN	0.9800	36.7169	0.0094	0.0969	0.0175	99.7104
RBFNN	0.9800	49.2196	0.0216	0.1470	0.0404	99.4242

Example 2. To demonstrate the application of the proposed model presented in this paper, consider the ship maneuvering system (Yuan and Wu, 2011) shown in Figure 9.

where $l$ is the length of ship, $δ$ represent the ruddle angle, $β$ denotes the drift angle, $N$ is moment component on body relative to z-axis, $κ$ is the speed of ship, $κ_{x}, κ_{y}$ are the forward velocity in x-axis and drift velocity along y-axis, respectively. $r$ is the yaw rate, $x, y$ are the components of force on body, and $φ$ is the yaw angle.

The following Norrbin nonlinear model can be used to explain the ship maneuvering system

T \overset{\cdot}{h} + h + τ h^{3} = L δ + q

(89)

where $T$ denotes the time constant, $h = \overset{\cdot}{φ}$ is the yaw rate, $φ$ is the heading angle, $δ$ is the rudder angle, and $q$ represents the external disturbance. We use spiral test for finding the value of $τ$ . The size and shape of a ship determine its dynamic parameters, which can change depending on operational factors including ship speed, draft, trim, and water depth. The rudder system can be described mathematically in the following way

T_{E} \overset{\cdot}{δ} + δ = L_{E} δ_{E}

(90)

where $T_{E}$ denotes the rudder time, $L_{E}$ represents the rudder control gain, and $δ_{E}$ denotes the rudder order.

Let $ξ_{1} = φ$ , $ξ_{2} = h$ , and $ξ_{3} = δ$ , one can get

\begin{matrix} {\overset{\cdot}{ξ}}_{1} = ξ_{2} \\ {\overset{\cdot}{ξ}}_{2} = f_{ϱ (κ)} (ξ_{2}) + b_{ϱ (κ)} ξ_{3} + q_{ϱ (κ)} \\ {\overset{\cdot}{ξ}}_{3} = - \frac{1}{T_{E, ϱ (κ)}} ξ_{3} + \frac{L_{E, ϱ (κ)}}{T_{E, ϱ (κ)}} δ_{E, ϱ (κ)} \end{matrix}

(91)

where $f_{ϱ (κ)} (ξ_{2})$ = $- (1 / T_{E, ϱ (κ)}) ξ_{2} - (τ_{ϱ (κ)} / T_{E, ϱ (κ)}) ξ_{2}^{3}$ , $b_{ϱ (κ)} = L_{E, ϱ (κ)} / T_{E, ϱ (κ)}$ where $ϱ (κ)$ is the switching signal and it is defined as

ϱ (κ) = {\begin{matrix} 1 & 0 < κ \leq κ_{L} \\ 2 & κ_{L} < κ \leq κ_{M} \\ 3 & κ_{M} < κ \leq κ_{T} \end{matrix}

where $κ_{L} = 3$ , $κ_{M} = 8$ , and $κ_{T} = 16$ denotes the value of the low speed, medium speed, and top speed, respectively. Model parameters of the ship maneuvering are given as follows

when $κ = 3.72 m / s$ $(l = 160.9 m, K_{1} = 0.32 s^{- 1}, T_{1} = 30 s,$ $τ_{1} = 40 s^{2}, T_{E, 1} = 4 s, L_{E, 1} = 2)$ .

when $κ = 7.5 m / s$ $(l = 160.9 m, K_{2} = 0.114 s^{- 1}, T_{2} = 63.69 s,$ $τ_{2} = 30 s^{2}, T_{E, 2} = 2.5 s, L_{E, 2} = 1)$ .

when $κ = 15.3 m / s$ $(l = 160.9 m, K_{3} = 0.051 s^{- 1}, T_{3} = 80.47 s,$ $τ_{3} = 25 s^{2}, T_{E, 3} = 1 s, L_{E, 3} = 0.72)$ .

Over-parameterization can be a problem if there are too many adaptive parameters. To deal with this problem, let $\hat{η} = \sum_{i = 1}^{3} {\hat{η}}_{i}$ , $ϖ = \sum_{i = 1}^{3} ϖ_{i}$ and $r = r_{1} = r_{2} = r_{3}$ . Therefore, adaptive law (21) is chosen as

\overset{\cdot}{\hat{η}} = \sum_{i = 1}^{3} r_{i} β_{i}^{2} e_{i}^{2} S_{i}^{T} S_{i} - ϖ_{i} {\hat{η}}_{i}

The virtual control function $ζ_{1}$ and $ζ_{2}$ are chosen as

ζ_{1} = - \frac{e_{1}}{{\underline{g}}_{1}} (k_{1} + l_{1}^{2} + β_{1}^{2} {\hat{η}}_{1} S_{1}^{T} S_{1})

ζ_{2} = - \frac{e_{2}}{{\underline{g}}_{2}} (k_{2} + l_{2}^{2} + β_{2}^{2} {\hat{η}}_{2} S_{2}^{T} S_{2})

The control law $v$ is chosen as

v = - \frac{e_{3}}{{\underline{g}}_{3}} (k_{3} + l_{3}^{2} + β_{3}^{2} {\hat{η}}_{3} S_{3}^{T} S_{3})

The external disturbances are taken as $q_{1} = 0.01 \sin t$ , $q_{2} = 0.015 \cos t$ , and $q_{3} = 0.013 \sin t$ . The initial conditions are given as $[ξ_{1} (00, ξ_{2} (0), ξ_{3} (0)]^{T} = [0.02, 0.02, 0.02]^{T}$ , $\hat{η} (0) = 1$ . The reference signal is $y_{d} = 10 \sin 0.05 t$ . The center of the receptive field and the width of the Gaussian function are chosen as $μ_{i} = [- 2, 0] \times [- 1, 0] \times [0, 1] \times [0, 2]$ and $ν_{i} = 2$ , $i = 1, 2$ , respectively.

The controller parameters are chosen as follows: $k_{1} = 5, k_{2} = 6, k_{3} = 6$ , $l_{1} = 12, l_{2} = 14, l_{3} = 10$ , $r_{1}, r_{2}, r_{3} = 0.02$ , $β_{1} = 8, β_{2} = 10, β_{3} = 12$ , ${\underline{g}}_{1} = 1, {\underline{g}}_{2} = 6.3 \times 10^{- 4}, {\underline{g}}_{3} = 0.4$ and $ϖ_{1}, ϖ_{2}, ϖ_{3} = 0.1$ .

Figures 10 –15 depict the simulation results for Example 2.

The tracking performance under HNN and RBFNN is shown in Figure 10. The tracking error under HNN and RBFNN is depicted in Figure 11. The trajectory of adaptive law is shown in Figure 12. The trajectory of virtual law $ζ_{1}$ is depicted in Figure 13. The trajectory of virtual law $ζ_{2}$ is depicted in Figure 14. The trajectory of switching signal is depicted in Figure 15.

Table 2 shows the comparison of HNN and RBFNN based on error parameters defined in equations (81)–(86).

Table 2.

Comparison of proposed method based on different assessment error criteria for example 2.

Model	Different error calculations
Model	MAE	SSE	MSE	RMSE	NMSE	BFR
HNN	0.200	14.2024	0.000021334	0.00461	0.00000044502	99.00
RBFNN	0.200	16.3604	0.000023554	0.00485	0.00000045602	98.85

From the simulations results, it can be easily seen that both HNN and RBFNN achieved good tracking performances, but the tracking error obtained from HNN is slightly better than RBFNN, which shows the effectiveness of our proposed method. Also, by using the proposed controller, all the closed-loop system signals remain bounded by tuning the parameters suitably. Hence, the simulation results prove the validity of the proposed controller.

Conclusion

In this paper, an HNN controller is presented for a class of switched uncertain nonlinear systems with bounded external disturbance. To approximate unknown functions, an IWOA-based HNN is used. For stability analysis, a CLF is used. The proposed model guarantees that the tracking error converges to the small neighborhood of origin, and all signals in the close-loop system remain bounded under arbitrary switching. In future work, we will focus on the switched system with state delays and interconnected nonlinear systems with time-varying state constraints.

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 MATRICS project grant no. MTR/2021/000478 from the Science and Engineering Research Board (SERB), India.

ORCID iD

Uday Pratap Singh

Data availability statement

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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