Adaptive Control of Switched Nonlinear Systems

Abstract

It has been shown in Ye, Neurocomputing 71:3373–3378 (2008), Yu, et al., Neurocomputing 156:245–251 (2015), Peng, et al., Neurocomputing 149:132–141 (2015), Miao, Neurocomputing 111:184–189 (2013), Xu, Neurocomputing 154:337–346 (2015), [1–5] that the adaptive backstepping technique is a powerful tool which has been widely used to solve some complex optimization problems and applied in the fields of industry and engineering.

4.1 Background and Motivation

It has been shown in [1–5] that the adaptive backstepping technique is a powerful tool which has been widely used to solve some complex optimization problems and applied in the fields of industry and engineering. Recently, many adaptive backstepping-based control methods have been used in switched nonlinear systems; see, for example, [6–10] and the references therein. The authors in [11] solved the problem of adaptive stabilization for a class of uncertain switched nonlinear systems whose non-switching part consists of feedback linearizable dynamics. In [12], the authors investigated the problem of adaptive stabilization for a class of switched nonlinearly parameterized systems where the solvability of the adaptive stabilization problem for subsystems is unnecessary.

It is well known that the stability of a switched system under arbitrary switching can be guaranteed if a CLF exists for all subsystems [13]. Therefore, CLF has been extensively used for control synthesis of switched linear systems [14–17]. Recently, there have been some results on the global stabilization problem for switched nonlinear systems in strict-feedback form under arbitrary switchings by using the backstepping technique [9, 18]. Meanwhile, [19] investigated the global stabilization problem for a class of switched nonlinear systems in p-normal form by the so-called power integrator backstepping design method.

In practice, uncertainties inevitably exist in many practical systems. In recent years, some attentions has been paid to both general nonlinear systems and switched nonlinear systems with uncertainties, but most of the obtained results require that the uncertainties should satisfy some additional conditions. However, in many cases, we cannot get the knowledge of system uncertainty a priori, which can only be described by completely unknown functions. In this case, the excellent approximation capability of neural networks (or fuzzy logic systems) has been explored in the literature to tackle the corresponding control problems for either switched systems or non-switched systems. Thus, many significant results have been proposed. To list a few, the authors in [20] investigated the control problem of nonlinear pure-feedback systems with unknown nonlinear functions by using the implicit function theorem and NN approximation. The adaptive tracking control problem for a class of uncertain nonlinear strict-feedback systems is solved by [21] using fuzzy logic system approximation. A practical design method is developed by [22] for cooperative tracking control of higher-order nonlinear systems with a dynamic leader. For a class of switched uncertain nonlinear systems without the measurements of the system states, the problem of adaptive neural tracking control via output-feedback was solved in [23] by using a novel switched filter.

However, few results on adaptive tracking control have been developed for lower triangular switched nonlinear systems with completely unknown uncertainties. On the other hand, most system models of the above-mentioned results about adaptive control for switched nonlinear systems are in the strict-feedback form that limits applications of the results to more general switched nonlinear systems. Therefore, considering the adaptive tracking control for switched nonstrict-feedback nonlinear systems with completely unknown uncertainties is more reasonable. In this chapter, the adaptive tracking control problem is investigated for both strict-feedback and nonstrict-feedback switched nonlinear systems with completely unknown uncertainties.

Notations: In this chapter, the notations are standard. $\mathbb {R}^{n}$ denotes the n-dimensional Euclidean space, the notation $\left\| \cdot \right\|$ refers to the Euclidean vector norm. $\mathbb {R}^{+}$ is the set of all nonnegative real numbers. For positive integers $1\le i\le n,\,1\le j\le m,\,$ we also denote $\varXi _{i,\max }=\max \{\varXi _{i,j}:\,1\le j\le m\},\,\varXi _{i,\min }=\min \{\varXi _{i,j}:1\le j\le m\}.$ $\mathscr {C}^{i}$ stands for a set of functions with continuous $i^{th}$ partial derivatives. For a given matrix A (or vector v), $A^{T}$ (or $v^{T}$) denotes its transpose, and $Tr\{A\}$ denotes its trace when A is a square. $\mathscr {K}$ represents the set of functions: $\mathbb {R}^{+}\rightarrow \mathbb {R}^{+}$, which are continuous, strictly increasing and vanishing at zero; $\mathscr {K}_{\infty }$ denotes a set of functions which is of class $\mathscr {K}$ and unbounded.

4.2 Adaptive Control of Switched Strict-Feedback Nonlinear Systems

4.2.1 Problem Formulation and Preliminaries

Consider a class of switched nonlinear systems in the following form,

$$\begin{aligned} \dot{x}_{i}&=g_{i,\sigma (t)}x_{i+1}+f_{i,\sigma (t)}(\bar{x}_{i}),\text { }i=1,2,\ldots ,n-1,\nonumber \\ \dot{x}_{n}&=g_{n,\sigma (t)}u_{\sigma (t)}+f_{n,\sigma (t)}(\bar{x} _{n}),\nonumber \\ y&=x_{1}, \end{aligned}$$

(4.1)

where $\bar{x}_{i}:=(x_{1},x_{2},\ldots ,x_{i})^{T}\in \mathbb {R} ^{i},\,i=1,2,\ldots ,n\$is the system state, y is the system output; $\sigma (t):[0,+\infty )\rightarrow M=\{1,2,\ldots ,m\}$ is the switching signal; $u_{k}\in R$ is the control input of the $k^{th}$ subsystem, For any $i=1,2,\ldots ,n$ and $k=1,2,\ldots ,m$, $f_{i,k}(\bar{x}_{i})$ is an unknown smooth nonlinear function representing the system uncertainty, and $g_{i,k}$ is a positive constant.

Our control objective is to design state-feedback controllers such that the output of system (4.1) tracks a given time-varying signal $y_{d}(t)$ within a bounded error and all the signals of the closed-loop systems remain bounded under arbitrary switchings.

Assumption 4.1

The tracking target $y_{d}(t)$ and its time derivatives up to the $n^{th}$ order are continuous and bounded.

In the controller design and stability analysis procedure, fuzzy logic systems will be used to approximate the unknown functions. Therefore, the following useful concept and lemma are first recalled.

Fuzzy logic systems include some IF-THEN rules, and the $i^{th}$ IF-THEN rule is written as

$$ \mathbb {R}_{i}:\text { If }x_{1}\text { is }F_{1}^{i}\text { and ... and }x_{n}\text { is }F_{n}^{i}\text { then }y\text { is }B^{i}, $$

where $\mathbf {x}=[x_{1},x_{2},...,x_{n}]^{T}\in \mathbb {R} ^{n}$, and $y\in \mathbb {R}$ are the input and output of the fuzzy logic systems, respectively. $F_{1} ^{i},F_{2}^{i},...,F_{n}^{i}$ and $B^{i}$ are fuzzy sets in $\mathbb {R}$. By using the strategy of singleton fuzzification, the product inference and the center-average defuzzification, the fuzzy logic system can be formulated as

$$ y(\mathbf {x})=\frac{\sum \limits _{i=1}^{N}w_{i} {\displaystyle \prod \limits _{j=1}^{n}} \mu _{F_{j}^{l}}(x_{j})}{\sum \limits _{i=1}^{N}\left[ {\displaystyle \prod \limits _{j=1}^{n}} \mu _{F_{j}^{l}}(x_{j})\right] }, $$

where N is the number of IF-THEN rules; $w_{i}$ is the point at which fuzzy membership function $\mu _{B^{i}}(w_{i})=1.$ Let

$$\begin{aligned} s_{i}(\mathbf {x})= {\displaystyle \prod \limits _{j=1}^{n}} \mu _{F_{j}^{l}}(x_{j})/\sum \limits _{i=1}^{N}\left[ {\displaystyle \prod \limits _{j=1}^{n}} \mu _{F_{j}^{l}}(x_{j})\right] , S(\mathbf {x})=[s_{1}(\mathbf {x}),\ldots ,s_{N}(\mathbf {x})]^{T} \end{aligned}$$

and $W=[w_{1},w_{2},\ldots ,w_{N}]^{T}.$ Then the fuzzy logic system can be rewritten as

$$\begin{aligned} y=W^{T}S(\mathbf {x}), \end{aligned}$$

(4.2)

If all memberships are chosen as Gaussian functions, the following lemma holds.

Lemma 4.1

[24] Let $f(\mathbf {x})$ be a continuous function defined on a compact set $\varOmega .\,$Then, for a given desired level of accuracy $\varepsilon >0,\,$there exists a fuzzy logic system (4.2) such that

$$ \sup _{x\in \varOmega }\left| f(\mathbf {x})-W^{T}S(\mathbf {x})\right| \le \varepsilon . $$

Remark 4.1

Lemma 4.1 plays a key role in the following design procedure and it indicates that any given real continuous function f(x) can be represented by the linear combination of the basis function vector $S(\mathbf {x})$ within a bounded error $\varepsilon$. That is, $f(\mathbf {x})=W^{T}S(\mathbf {x})+\delta (\varepsilon ),$ $\left| \delta (\varepsilon )\right| \le \varepsilon .$ It is noted that $0<S^{T}S\le 1.$

4.2.2 Main Results

In this section, we present an adaptive fuzzy control scheme for system (4.1) via the backstepping technique. In Sect. 3.1, a detailed design procedure was given. In each step, a common virtual control function $\alpha _{i}$ should be designed by using an appropriate common Lyapunov function $V_{i},$ and the control law $u_{k}$ is finally designed.

4.2.2.1 Adaptive Control Design Under Multiple Adaptive Laws

In this subsection, a systemic control design procedure under multiple adaptive laws is presented. Design the control laws as

$$\begin{aligned} u_{k}=-\frac{1}{g_{n,k}}\left( \frac{\hat{\theta }_{n}}{2\zeta _{n,\min }^{2}} z_{n}+\lambda _{n}z_{n}+\frac{z_{n}}{2}\right) , \end{aligned}$$

(4.3)

where $\zeta _{n,k}$ and $\lambda _{n}$ are positive design parameters, $\zeta _{n,\min }=\min \{\zeta _{n,k}:k\in M\}$, $\hat{\theta }_{n}$ is the estimation of $\theta _{n}=\left\| W_{n,\max }\right\| ^{2}$, $W_{n,\max }=\max \{W_{n,k}:k\in M\}$ and $W_{n,k}$ is used in fuzzy logic system $W_{n,k}^{T}S_{n,k}(\mathbf {x})$ to approximate the unknown function $\hat{f}_{n,k} (\mathbf {x})$. $\hat{f}_{n,k}(\mathbf {x})$ is specified in the proof of Theorem 4.1.

The adaptive laws are defined as the solution to the following differential equations,

$$\begin{aligned} \dot{\hat{\theta }}_{i}=\frac{r_{i}}{2\zeta _{i,\min }^{2}}z_{i}^{2}-\beta _{i}\hat{\theta }_{i}, \end{aligned}$$

(4.4)

where $r_{i},$ $\zeta _{n,k}$ and $\beta _{i}$ are positive design parameters, $\zeta _{n,\min }=\min \{\zeta _{n,k}:k\in M\}$, and the choice of $\hat{\theta }_{j}(0)$, $j=1,2,\ldots ,n$ are required to satisfy $\hat{\theta }_{j}(0)\ge 0$ such that $\hat{\theta }_{j}\ge 0.$ Now, we state one of our main results as follows.

Theorem 4.1

Consider the closed-loop system (4.1) with the controllers (4.3) and the adaptive laws (4.4). For $1\le i\le n,\,k\in M,$ there exists $W_{i,k}^{T}S_{i,k}(\mathbf {x})$ such that $\sup _{\mathbf {x}\in \varOmega }\left| \hat{f}_{i,k}(\mathbf {x})-W_{i,k}^{T}S_{i,k}(\mathbf {x})\right| \le \varepsilon _{i,k}$ in the sense that the approximation error $\varepsilon _{i,k}$ is bounded, and all the initial values of $\hat{\theta }_{i}$ satisfy $\hat{\theta }_{i}(0)\ge 0$. Then, the tracking error and closed-loop signals are bounded.

Proof

For $1\le i\le n-1$, we define the common virtual control functions as $\alpha _{i}$ which are required to be in the form:

$$\begin{aligned} \alpha _{i}(X_{i})=-\frac{1}{g_{i,\min }}\left( \frac{\hat{\theta }_{i}}{2\zeta _{i,\min }^{2}}+\lambda _{i}+\frac{1}{2}\right) z_{i}, \end{aligned}$$

(4.5)

where $\zeta _{i,k}$ is a positive design parameter, $\zeta _{i,\min } =\min \{\zeta _{i,k}:k\in M\}$, $g_{i,\min }=\min \{g_{i,k}:k\in M\}$, $\lambda _{i}=g_{i,\max }+c_{i}$, $g_{i,\max }=\max \{g_{i,k}:k\in M\}$ and $c_{i}$ is a positive constant. $\hat{\theta }_{i}$ is the estimation of $\theta _{i}=\left\| W_{i,\max }\right\| ^{2}$ where $W_{i,\max } =\max \{W_{i,k}:k\in M\}$ and $W_{i,k}$ is used in fuzzy logic system $W_{i,k}^{T}S_{i,k}(\mathbf {x})$ to approximate the unknown function $\hat{f}_{i,k} (\mathbf {x})$. $X_{i}=[\bar{x}_{i}^{T},\bar{\hat{\theta }}_{i},\bar{y}_{d}^{(i)}]^{T}$ with $\bar{x}_{i}^{T}=[x_{1},x_{2},\ldots ,x_{i}]^{T}$, $\,\bar{\hat{\theta }} _{i}=[\hat{\theta }_{1},\hat{\theta }_{2},\ldots ,\hat{\theta }_{i}]^{T}$, $\,\bar{y}_{d}^{(i)}=[y_{d},\dot{y}_{d},\ldots ,y_{d}^{(i)}]^{T}$ and $\bar{y}_{d}^{(i)}$ being the $i^{th}$ derivative of $y_{d}$.

Step 1. Denote $z_{1}=x_{1}-y_{d},\,z_{2}=x_{2}-\alpha _{1}.$ Consider a Lyapunov function candidate as

$$\begin{aligned} V_{1}=\frac{1}{2}z_{1}^{2}. \end{aligned}$$

(4.6)

For any $k\in M,\,$the derivative of $V_{1}$ is given by

$$\begin{aligned} \dot{V}_{1}&=z_{1}(g_{1,k}\alpha _{1}+g_{1,k}z_{2}+f_{1,k}-\dot{y} _{d})\nonumber \\&=z_{1}(g_{1,k}\alpha _{1}+g_{1,k}z_{2}+\hat{f}_{1,k}), \end{aligned}$$

(4.7)

where $\hat{f}_{1,k}=f_{1,k}-\dot{y}_{d}.$ By Lemma 4.1, the following equation can be obtained,

$$\begin{aligned} {\hat{f}}_{1,k}=W_{1,k}^{T}S_{1,k}(X_{1})+\delta _{1,k}(X_{1}),\,\left| \delta _{1,k}(X_{1})\right| \le \varepsilon _{1,k}. \end{aligned}$$

(4.8)

Remark 4.2

It should be pointed out that the fuzzy logic system is used to approximate the redefined unknown nonlinear function ${\hat{f}}_{1,k}$ that includes the unknown function $f_{1,k}$ and the derivative of the desired output rather than the unknown function $f_{1,k}$ only.

Substituting (4.8) into (4.7), one gets that

$$\begin{aligned} \dot{V}_{1}&=g_{1,k}z_{1}\alpha _{1}+g_{1,k}z_{1}z_{2}+z_{1}W_{1,k} ^{T}S_{1,k}(z_{1})+z_{1}\delta (z_{1})\nonumber \\&\le g_{1,k}z_{1}\alpha _{1}+g_{1,k}z_{1}z_{2}+\frac{1}{2\zeta _{1,k}^{2} }z_{1}^{2}\left\| W_{1,k}\right\| ^{2}\nonumber \\&\quad +\frac{\zeta _{1,k}^{2}+\varepsilon _{1,k}^{2}}{2}+\frac{1}{2}z_{1}^{2}, \end{aligned}$$

(4.9)

where $\zeta _{1,k}$ is a positive design parameter.

A feasible virtual control function can be constructed as

$$\begin{aligned} \alpha _{1}=-\frac{1}{g_{1,\min }}\left( \frac{\hat{\theta }_{1}}{2\zeta _{1,\min }^{2} }+\lambda _{1}+\frac{1}{2}\right) z_{1}, \end{aligned}$$

(4.10)

where $\lambda _{1}=$ $g_{1,\max }+c_{1}$ with $c_{1}$ being a positive constant.

By substituting (4.1) into (4.9), one has

$$\begin{aligned} \dot{V}_{1} \le -\lambda _{1}z_{1}^{2}+\left( \frac{\left\| W_{1,k}\right\| ^{2}}{2\zeta _{1,k}^{2}}-\frac{g_{1,k}\hat{\theta }_{1}}{2g_{1,\min } \zeta _{1,\min }^{2}}\right) z_{1}^{2} +\frac{\zeta _{1,k}^{2}+\varepsilon _{1,k}^{2}}{2}+g_{1,k}z_{1}z_{2}. \end{aligned}$$

(4.11)

Step 2. Let $z_{3}=x_{3}-\alpha _{2},$ and choose

$$\begin{aligned} V_{2}=V_{1}+\frac{1}{2}z_{2}^{2}. \end{aligned}$$

(4.12)

For any $k\in M,\,$the time derivative of $V_{2}$ is given by

$$\begin{aligned} \dot{V}_{2}&=\dot{V}_{1}+z_{2}(g_{2,k}\alpha _{2}+g_{2,k}z_{3}+f_{2,k} -\dot{\alpha }_{1})\nonumber \\&=\dot{V}_{1}+z_{2}(g_{2,k}\alpha _{2}+g_{2,k}z_{3}+\hat{f}_{2,k}), \end{aligned}$$

(4.13)

where $\hat{f}_{2,k}=f_{2,k}-\dot{\alpha }_{1},\,\dot{\alpha }_{1} =\frac{\partial \alpha _{1}}{\partial x_{1}}\dot{x}_{1}+\frac{\partial \alpha _{1}}{\partial \hat{\theta }_{1}}\dot{\hat{\theta }}_{1}+\sum _{i=0}^{1} \frac{\partial \alpha _{1}}{\partial y_{d}^{(i)}}y_{d}^{(i+1)}.$

By Lemma 4.1, the following equation can be obtained,

$$\begin{aligned} \hat{f}_{2,k}=W_{2,k}^{T}S_{2,k}(X_{2})+\delta _{2,k}(X_{2}),\,\left| \delta _{2,k}(X_{2})\right| \le \varepsilon _{2,k}. \end{aligned}$$

(4.14)

Substituting (4.14) into (4.13), yields that

$$\begin{aligned} \dot{V}_{2}&=\dot{V}_{1}+g_{2,k}z_{2}\alpha _{2}+g_{2,k}z_{2} z_{3}+z_{2}(W_{2,k}^{T}S(z_{2})+\delta _{2,k}(z_{2}))\nonumber \\&\le \dot{V}_{1}+g_{2,k}z_{2}\alpha _{2}+g_{2,k}z_{2}z_{3} +\frac{1}{2\zeta _{2,k}^{2}}z_{2}^{2}\left\| W_{2,k}\right\| ^{2}+\frac{\zeta _{2,k}^{2}+\varepsilon _{2,k}^{2}}{2}+\frac{1}{2}z_{2}^{2}, \end{aligned}$$

(4.15)

where $\zeta _{2,k}$ is a positive design parameter.

Design the virtual control function $\alpha _{2}$ as

$$\begin{aligned} \alpha _{2}=-\frac{1}{g_{2,\min }}\left( \frac{\hat{\theta }_{2}}{2\zeta _{2,\min }^{2} }+\lambda _{2}+\frac{1}{2}\right) z_{2}, \end{aligned}$$

(4.16)

where $\lambda _{2}=$ $g_{2,\max }+c_{2}$ with $c_{2}$ being a positive constant.

Then, one can get from (4.11), (4.15) and (4.16) that

$$\begin{aligned} \dot{V}_{2} \le \sum _{j=1}^{2}\left\{ -\lambda _{j}z_{j}^{2}+\frac{\zeta _{j,k}^{2}+\varepsilon _{j,k}^{2}}{2}+g_{j,k}z_{j}z_{j+1} +\left( \frac{\left\| W_{j,k}\right\| ^{2}}{2\zeta _{j,k}^{2}} -\frac{g_{j,k}\hat{\theta }_{j}}{2g_{j,\min }\zeta _{j,\min }^{2}}\right) z_{j}^{2}\right\} . \end{aligned}$$

(4.17)

Step i. Let $z_{i+1}=x_{i+1}-\alpha _{i},$ and assume that we have finished the first $i-1\,(2\le i\le n)$ steps. That is, for the following collection of auxiliary $(z_{1},\ldots ,z_{i-1})$-equations

$$\begin{aligned} \dot{z}_{j}=g_{j,k}x_{j+1}+\phi _{j,k}(X_{j}),\text { }j=1,\ldots ,i-1, \end{aligned}$$

(4.18)

where

$$\begin{aligned} \phi _{j,k}(X_{j})&=f_{j,k}(\bar{x}_{j})-\sum _{l=1}^{j-1}\frac{\partial \alpha _{j-1}}{\partial x_{l}}\dot{x}_{l}-\sum _{l=1}^{j-1}\frac{\partial \alpha _{j-1}}{\partial \hat{\theta }_{l} }\dot{\hat{\theta }}_{l}-\sum _{l=0}^{j-1}\frac{\partial \alpha _{j-1}}{\partial y_{d}^{(l)}}y_{d}^{(l+1)}. \end{aligned}$$

(4.19)

We have a set of common virtual control functions as (4.5). A common Lyapunov function can be designed as

$$\begin{aligned} V_{i-1}=\frac{1}{2}\sum _{j=1}^{i-1}z_{j}^{2}. \end{aligned}$$

(4.20)

For any $k\in M,$ the time derivative of $V_{i-1}$ satisfies

$$\begin{aligned} \dot{V}_{i-1} \le&\sum _{j=1}^{i-1}\left\{ -\lambda _{j}z_{j}^{2}+\frac{\zeta _{j,k}^{2}+\varepsilon _{j,k}^{2}}{2}+g_{j,k}z_{j}z_{j+1}\right. \nonumber \\&+\left. \left( \frac{\left\| W_{j,k}\right\| ^{2}}{2\zeta _{j,k}^{2}} -\frac{g_{j,k}\hat{\theta }_{j}}{2g_{j,\min }\zeta _{j,\min }^{2}}\right) z_{j}^{2}\right\} , \end{aligned}$$

(4.21)

where $\zeta _{j,k}$ is a positive design parameter.

Choose

$$\begin{aligned} V_{i}=V_{i-1}+\frac{1}{2}z_{i}^{2}. \end{aligned}$$

(4.22)

Analogous to the procedures above, the following inequality can be obtained

$$\begin{aligned} \dot{V}_{i}&\le \sum _{j=1}^{i}\left\{ -\lambda _{j}z_{j}^{2}+\frac{\zeta _{j,k}^{2}+\varepsilon _{j,k}^{2}}{2}+g_{j,k}z_{j}z_{j+1}\right. \nonumber \\&\quad +\left. \left( \frac{\left\| W_{j,k}\right\| ^{2}}{2\zeta _{j,k}^{2}} -\frac{g_{j,k}\hat{\theta }_{j}}{2g_{j,\min }\zeta _{j,\min }^{2}}\right) z_{j}^{2}\right\} . \end{aligned}$$

(4.23)

Step n. By repeatedly using the inductive argument above, a common Lyapunov function, a common virtual control function and state-feedback controllers are chosen, respectively, as

$$\begin{aligned} V_{n}&=\sum _{j=1}^{n}\left\{ \frac{1}{2}z_{l}^{2}+\frac{1}{2r_{j}}\tilde{\theta }_{j}^{2}\right\} ,\end{aligned}$$

(4.24)

$$\begin{aligned} \alpha _{n-1}&=-\frac{1}{g_{n-1,\min }}\left( \frac{\hat{\theta }_{n-1}}{2\zeta _{n-1,\min }^{2}}z_{n-1}+\lambda _{n-1}z_{n-1}+\frac{z_{n-1}}{2}\right) ,\end{aligned}$$

(4.25)

$$\begin{aligned} u_{k}&=-\frac{1}{g_{n,k}}\left( \frac{\hat{\theta }_{n}}{2\zeta _{n,\min }^{2} }z_{n}+\lambda _{n}z_{n}+\frac{z_{n}}{2}\right) , \end{aligned}$$

(4.26)

where $\theta _{j}=\left\| W_{j,\max }\right\| ^{2},\ \tilde{\theta } _{j}=\theta _{j}-\hat{\theta }_{j}$ $(j=1,2,\ldots ,n)$ are the error between $\theta _{j}$ and its estimation $\hat{\theta }_{j}$.

For any $k\in M,$ the time derivative of $V_{n}$ satisfies

$$\begin{aligned} \dot{V}_{n}&\le \sum _{j=1}^{n-1}\left\{ -\lambda _{j}z_{j}^{2}+g_{j,k} z_{j}z_{j+1}+\frac{\zeta _{j,k}^{2}+\varepsilon _{j,k}^{2}}{2}-\frac{1}{r_{j} }\tilde{\theta }_{j}\dot{\hat{\theta }}_{j}\right. \nonumber \\&\quad +\left. \left( \frac{\left\| W_{j,k}\right\| ^{2}}{2\zeta _{j,k}^{2}} -\frac{g_{j,k}\hat{\theta }_{j}}{2g_{j,\min }\zeta _{j,\min }^{2}}\right) z_{j} ^{2}\right\} -\lambda _{n}z_{n}^{2}\nonumber \\&\quad +\frac{\zeta _{n,k}^{2}+\varepsilon _{n,k}^{2}}{2}-\frac{1}{r_{n}} \tilde{\theta }_{n}\dot{\hat{\theta }}_{n}+\left( \frac{\left\| W_{n,k}\right\| ^{2}}{2\zeta _{n,k}^{2}} -\frac{\hat{\theta }_{n}}{2\zeta _{n,\min }^{2}}\right) z_{n}^{2}, \end{aligned}$$

(4.27)

where $\lambda _{j}=g_{j,\max }+c_{j},$ and $c_{j}$ is a positive constant.

Substituting (4.4) into (4.27) gives that

$$\begin{aligned} \dot{V}_{n}&\le \sum _{j=1}^{n-1}\left\{ -\lambda _{j}z_{j}^{2}+g_{j,k} z_{j}z_{j+1}+\frac{\zeta _{j,k}^{2}+\varepsilon _{j,k}^{2}}{2}+\frac{1}{r_{j} }\beta _{j}\tilde{\theta }_{j}\hat{\theta }_{j}\right. \nonumber \\&\ \quad +\left. \left( \frac{\left\| W_{j,k}\right\| ^{2}}{2\zeta _{j,k}^{2}} -\frac{g_{j,k}\hat{\theta }_{j}}{2g_{j,\min }\zeta _{j,\min }^{2}}-\frac{\tilde{\theta }_{j}}{2\zeta _{j,\min }^{2}}\right) z_{j}^{2}\right\} \nonumber \\&\ \quad -\lambda _{n}z_{n}^{2}+\frac{\zeta _{n,k}^{2}+\varepsilon _{n,k}^{2} }{2}+\frac{1}{r_{n}}\beta _{n}\tilde{\theta }_{n}\hat{\theta }_{n}+\left( \frac{\left\| W_{n,k}\right\| ^{2}}{2\zeta _{n,k}^{2}} -\frac{\theta _{n}}{2\zeta _{n,\min }^{2}}\right) z_{n}^{2}\nonumber \\&\le \sum _{j=1}^{n}\left\{ -\lambda _{j}z_{j}^{2}+\frac{\zeta _{j,k}^{2} +\varepsilon _{j,k}^{2}}{2}+\frac{1}{r_{j}}\beta _{j}\tilde{\theta }_{j} \hat{\theta }_{j}\right\} +\sum _{j=1}^{n-1}g_{j,k}z_{j}z_{j+1}. \end{aligned}$$

(4.28)

It is not difficult to see that

$$\begin{aligned} \sum _{j=1}^{n-1}g_{j,k}z_{j}z_{j+1}\le g_{j,\max }\sum _{j=1}^{n}z_{j}^{2}, \end{aligned}$$

(4.29)

and

$$\begin{aligned} \tilde{\theta }_{j}\hat{\theta }_{j}=\tilde{\theta }_{j}(\theta _{j}-\tilde{\theta }_{j})\le -\frac{1}{2}\tilde{\theta }_{j}^{2}+\frac{1}{2}\theta _{j}^{2}. \end{aligned}$$

(4.30)

One can get from (4.28), (4.29) and (4.30) that

$$\begin{aligned} \dot{V}_{n}&\le \sum _{j=1}^{n}\left\{ -c_{j}z_{j}^{2}-\frac{1}{2r_{j}}\beta _{j}\tilde{\theta }_{j}^{2}\right\} +\sum _{j=1}^{n}\left\{ \frac{\zeta _{j,\max }^{2}+\varepsilon _{j,\max }^{2}}{2}+\frac{1}{2r_{j}}\beta _{j}\theta _{j}^{2}\right\} . \end{aligned}$$

(4.31)

Let $a_{0}=\min \{2c_{j},\beta _{j}:1\le j\le n\},\,b_{0}=\sum _{j=1} ^{n}\{\frac{1}{2r_{j}}\beta _{j}\theta _{j}^{2}+\frac{\zeta _{j,\max } ^{2}+\varepsilon _{j,\max }^{2}}{2}\}$. One has

$$\begin{aligned} \dot{V}_{n}\le -a_{0}V_{n}+b_{0}. \end{aligned}$$

(4.32)

According to the comparison principle, one gets

$$\begin{aligned} V_{n}(t)\le \left( V_{n}(0)-\frac{b_{0}}{a_{0}}\right) e^{-a_{0}t}+\frac{b_{0}}{a_{0} },\,t\ge 0. \end{aligned}$$

(4.33)

Inequality (4.33) indicates that all the signals in the closed-loop system are bounded. In particular, we have

$$\begin{aligned} \lim _{t\rightarrow \infty }\left| z_{1}\right| \le \sqrt{\frac{2b_{0} }{a_{0}}.} \end{aligned}$$

(4.34)

The proof is completed here. $\square$

4.2.2.2 Adaptive Control Design Under One Adaptive Law

In this subsection, a controller design approach with one adaptive law is presented.

The control laws are chosen as

$$\begin{aligned} u_{k}=-\frac{1}{g_{n,k}}\left( \frac{\hat{\theta }}{2\zeta _{n,\min }^{2}}z_{n} +\lambda _{n}z_{n}+\frac{z_{n}}{2}\right) , \end{aligned}$$

(4.35)

where $\zeta _{n,k}$ and $\lambda _{n}$ are positive design parameters, $\zeta _{n,\min }=\min \{\zeta _{n,k}:k\in M\}$, $\hat{\theta }$ is the estimation of $\theta =\sum _{i=1}^{n}\left\| W_{i,\max }\right\| ^{2}$, $W_{i,\max }=\max \{W_{i,k}:k\in M\}$ and $W_{i,k}$ is used in fuzzy logic system $W_{i,k}^{T}S_{i,k}(x)$ to approximate the unknown function $\hat{f}_{i,k}(x)$.

The adaptive law is defined as the solution to the following differential equation:

$$\begin{aligned} \dot{\hat{\theta }}=\sum _{j=1}^{n}\frac{r}{2\zeta _{j,\min }^{2}}z_{j}^{2} -\beta \hat{\theta }, \end{aligned}$$

(4.36)

where r, $\zeta _{j,k}$ and $\beta$ are positive design parameters, $\zeta _{j,\min }=\min \{\zeta _{j,k}:k\in M\}$ and the choice of $\hat{\theta }(0)$ is required to satisfy $\hat{\theta }(0)\ge 0$ such that $\hat{\theta }\ge 0.$

Next, we give another main result of the chapter.

Theorem 4.2

Consider the closed-loop system (4.1) with the controllers (4.35) and the adaptive laws (4.36). For $1\le i\le n,\,k\in M,$ there exists $W_{i,k}^{T}S_{i,k}(\mathbf {x})$ such that $\sup _{x\in \varOmega }\left| \hat{f}_{i,k}(\mathbf {x})-W_{i,k}^{T}S_{i,k}(\mathbf {x})\right| \le \varepsilon _{i,k}$ in the sense that the approximation error $\varepsilon _{i,k}$ is bounded, and the initial value of $\hat{\theta }$ satisfies $\hat{\theta }(0)\ge 0$. Then, the tracking error and closed-loop signals are bounded.

Proof

For $1\le i\le n-1$, define the common virtual control functions $\alpha _{i}$ as:

$$\begin{aligned} \alpha _{i}(X_{i})=-\frac{1}{g_{i,\min }}\left( \frac{\hat{\theta }}{2\zeta _{i,\min }^{2}}+\lambda _{i}+\frac{1}{2}\right) z_{i}, \end{aligned}$$

(4.37)

where $\zeta _{i,k}$ is a positive design parameter, $\zeta _{i,\min } =\min \{\zeta _{i,k}:k\in M\}$, $g_{i,\min }=\min \{g_{i,k}:k\in M\}$, $\lambda _{i}=g_{i,\max }+c_{i}$, $g_{i,\max }=\max \{g_{i,k}:k\in M\}$ and $c_{i}$ is a positive constant. $\hat{\theta }$ is the estimation of $\theta =\sum _{i=1}^{n}\left\| W_{i,\max }\right\| ^{2},\,$ $X_{i} =[\bar{x}_{i}^{T},\hat{\theta },\bar{y}_{d}^{(i)}]^{T}$ where $\bar{x}_{i} ^{T}=[x_{1},x_{2},\ldots ,x_{i}]^{T}$, $\,\bar{y}_{d}^{(i)}=[y_{d},\dot{y} _{d},\ldots ,y_{d}^{(i)}]^{T}$ and $\bar{y}_{d}^{(i)}$ being the $i^{th}$ derivative of $y_{d}$.

Consider a common Lyapunov function

$$\begin{aligned} V=\sum _{j=1}^{n}\frac{1}{2}z_{j}^{2}+\frac{1}{2r}\tilde{\theta }^{2}, \end{aligned}$$

(4.38)

where$\ \tilde{\theta }=\theta -\hat{\theta }$ is the error between $\theta$ and its estimation $\hat{\theta }$.

For any $k\in M,$ the time derivative of V satisfies

$$\begin{aligned} \dot{V}&=\sum _{i=1}^{n-1}z_{i}(g_{i,k}\alpha _{i}+g_{i,k}z_{i+1} +f_{i,k}-\dot{\alpha }_{i-1})\nonumber \\&\quad +z_{n}(g_{n,k}u_{k}+f_{n,k}-\dot{\alpha }_{n-1})-\frac{1}{r} \tilde{\theta }\dot{\hat{\theta }}\nonumber \\&=\sum _{i=1}^{n-1}z_{i}(g_{i,k}\alpha _{i}+g_{i,k}z_{i+1}+\hat{f} _{i,k})\nonumber \\&\quad +z_{n}(g_{n,k}u_{k}+\hat{f}_{n,k})-\frac{1}{r}\tilde{\theta }\dot{\hat{\theta }} \end{aligned}$$

(4.39)

where $\hat{f}_{i,k}=f_{i,k}-\dot{\alpha }_{i-1},\,\dot{\alpha }_{i-1} =\sum _{l=1}^{j-1}\frac{\partial \alpha _{j-1}}{\partial x_{l}}\dot{x}_{l} \quad +\frac{\partial \alpha _{j-1}}{\partial \hat{\theta }}\dot{\hat{\theta }} +\sum _{l=0}^{j-1}\frac{\partial \alpha _{j-1}}{\partial y_{d}^{(l)}} y_{d}^{(l+1)}.$

For $1\le i\le n$, the following equation can be obtained by using Lemma 4.1.

$$\begin{aligned} \hat{f}_{i,k}=W_{i,k}^{T}S_{i,k}(X_{i})+\delta _{i,k}(X_{i}),\,\left| \delta _{2,k}(X_{i})\right| \le \varepsilon _{i,k}. \end{aligned}$$

(4.40)

Substituting (4.36) and (4.37) into (4.39), one has

$$\begin{aligned} \dot{V}&\le \frac{\beta }{r}\tilde{\theta }\hat{\theta }+\sum _{j=1} ^{n-1}\left\{ -\lambda _{j}z_{j}^{2}+g_{j,k}z_{j}z_{j+1}+\frac{\zeta _{j,k} ^{2}+\varepsilon _{j,k}^{2}}{2}\right. \nonumber \\&\ \quad +\left. \left( \frac{\left\| W_{j,k}\right\| ^{2}}{2\zeta _{j,k}^{2}} -\frac{g_{j,k}\hat{\theta }}{2g_{j,\min }\zeta _{j,\min }^{2}}-\frac{\tilde{\theta }}{2\zeta _{j,\min }^{2}}\right) z_{j}^{2}\right\} \nonumber \\&\ \quad -\lambda _{n}z_{n}^{2}+\frac{\zeta _{n,k}^{2}+\varepsilon _{n,k}^{2} }{2}+\left( \frac{\left\| W_{n,k}\right\| ^{2}}{2\zeta _{n,k}^{2}}-\frac{\theta }{2\zeta _{n,\min }^{2}}\right) z_{n}^{2}\nonumber \\&\le \frac{\beta }{r}\tilde{\theta }\hat{\theta }+\sum _{j=1}^{n}\left\{ -\lambda _{j}z_{j}^{2}+\frac{\zeta _{j,k}^{2}+\varepsilon _{j,k}^{2}}{2}\right\} +\sum _{j=1}^{n-1}g_{j,k}z_{j}z_{j+1}. \end{aligned}$$

(4.41)

The rest of proof is omitted here as it is similar to (4.29)–(4.34). $\square$

4.2.3 Simulation Results

In this section, an example is provided to demonstrate the effectiveness of our main results.

Consider the following switched nonlinear system

$$\begin{aligned} \dot{x}_{1}&=g_{1,\sigma (t)}x_{2}+f_{1,\sigma (t)},\nonumber \\ \dot{x}_{2}&=g_{2,\sigma (t)}u_{\sigma (t)}+f_{2,\sigma (t)},\nonumber \\ y&=x_{1},\nonumber \\ y_{d}&=\sin t, \end{aligned}$$

(4.42)

where $g_{1,1}=2,\,g_{1,2}=1,\,f_{1,1}=x_{1},\,f_{1,2}=\sin x_{1} ,\,g_{2,1}=2,\,g_{2,2}=1,\,f_{2,1}=x_{1}x_{2},\,f_{2,2}=x_{1}x_{2}^{2}.$ First, the controllers under multiple adaptive laws are designed by Theorem 4.1. The initial conditions are $x_{1}(0)=0.05,\,x_{2}(0)=0.05,\,$ and $\hat{\theta }_{1}(0)=\hat{\theta }_{2}(0)=0.$ We choose $c_{1}=2,\,c_{2}=1,\,r_{1} =10,\,r_{2}=3,\,\beta _{1}=\beta _{2}=0.02,\,\varsigma _{1,1}=0.25,\,\varsigma _{1,2}=3,\,\varsigma _{2,1}=0.5,\,\varsigma _{2,2}=1.8.\,$ Second, the controllers under one adaptive law is designed by Theorem 2, and the initial conditions are $x_{1}(0)=0.05,\,x_{2}(0)=0.05,\, \hat{\theta }(0)=0.$ We choose $c_{1}=2,\,c_{2}=1,\,r=12,\,\beta =0.025,\,\varsigma _{1,1} =0.25,\,\varsigma _{1,2}=3,\,\varsigma _{2,1}=1.5,\,\varsigma _{2,2}=1.8.\,$ The objective is to design the controllers $u_{k}$ such that y can track a desired trajectory $y_{d}$ under arbitrary switchings.

According to Theorem 4.1, the adaptive laws $\hat{\theta }_{1},\,\hat{\theta } _{2}$ and the control law $u_{k}$ are chosen, respectively, as

$$\begin{aligned} \dot{\hat{\theta }}_{1}&=\frac{r_{1}}{2\zeta _{1,1}^{2}}z_{1}^{2}-\beta _{1}\hat{\theta }_{1},\,\dot{\hat{\theta }}_{2}=\frac{r_{2}}{2\zeta _{2,1}^{2} }z_{2}^{2}-\beta _{2}\hat{\theta }_{2},\\ u_{1}&=-\frac{1}{g_{2,1}}\left( \frac{\hat{\theta }_{2}}{2\zeta _{2,1}^{2}} z_{2}+\lambda _{2}z_{2}+\frac{z_{2}}{2}\right) ,\\ u_{2}&=-\frac{1}{g_{2,2}}\left( \frac{\hat{\theta }_{2}}{2\zeta _{2,1}^{2}} z_{2}+\lambda _{2}z_{2}+\frac{z_{2}}{2}\right) , \end{aligned}$$

where $z_{1}=x_{1}-y_{d},\,z_{2}=x_{2}-\alpha _{1},\,\lambda _{2}=c_{2}+g_{2,1}$. The virtual control function $\alpha _{1}$ is given by

$$ \alpha _{1}=-\frac{1}{g_{1,2}}\left( \frac{\hat{\theta }_{1}}{2\zeta _{1,1}^{2}} z_{1}+\lambda _{1}z_{1}+\frac{z_{1}}{2}\right) , $$

where $\lambda _{1}=c_{1}+g_{1,1}$. The controller design based on Theorem 4.1 is completed here. In the next, another design according to Theorem 2 is presented.

According to Theorem 4.2, an adaptive law $\hat{\theta }$ and the control law $u_{1},\,u_{2}$ are chosen, respectively, as

$$\begin{aligned} \dot{\hat{\theta }}&=\frac{r}{2\zeta _{1,1}^{2}}z_{1}^{2}+\frac{r}{2\zeta _{2,1}^{2}}z_{2}^{2}-\beta \hat{\theta },\\ u_{1}&=-\frac{1}{g_{2,1}}\left( \frac{\hat{\theta }}{2\zeta _{2,1}^{2}} z_{2}+\lambda _{2}z_{2}+\frac{z_{2}}{2}\right) ,\\ u_{2}&=-\frac{1}{g_{2,2}}\left( \frac{\hat{\theta }}{2\zeta _{2,1}^{2}} z_{2}+\lambda _{2}z_{2}+\frac{z_{2}}{2}\right) , \end{aligned}$$

where $z_{1}=x_{1}-y_{d},\,z_{2}=x_{2}-\alpha _{1},\,\lambda _{2}=c_{1} +g_{2,1}.$

The virtual control function $\alpha _{1}$ is given as

$$ \alpha _{1}=-\frac{1}{g_{1,2}}\left( \frac{\hat{\theta }}{2\zeta _{1,1}^{2}} z_{1}+\lambda _{1}z_{1}+\frac{z_{1}}{2}\right) , $$

where $\lambda _{1}=c_{1}+g_{1,1}.$

Fig. 4.1

Tracking performances

Fig. 4.2

Responses of the tracking error $y-y_{d}$

Fig. 4.3

Responses of the adaptive laws

Fig. 4.4

Switching signal

The simulation results are shown in Figs. 4.1, 4.2, 4.3 and 4.4, respectively. Figure 4.1 shows the system output y and reference signal $y_{d}$. Figure 4.2 depicts the response of the tracking error $y-y_{d}$. Figure 4.3 illustrates the trajectory of the adaptive law. Figure 4.4 demonstrates the evolution of the switching signal. From Figs. 4.1, 4.2 and 4.3, it can be seen that the output y of both controllers can track the target signal $y_{d}$ well, and all the closed-loop signals remain bounded.

4.2.4 Conclusions

The tracking control problem for switched strict-feedback nonlinear systems with completely unknown nonlinear functions is given. The application of the adaptive backstepping technique is extended to a class of switched nonlinear systems with unknown uncertainties. The stability analysis shows that the designed controllers can ensure all the closed-loop signals remain bounded, and the system output converges to a small neighborhood of the reference signal.

4.3 Adaptive Control of Switched Nonstrict-Feedback Nonlinear Systems

4.3.1 Problem Formulation and Preliminaries

In this section, the following nonlinear switched system in nonstrict-feedback form is considered:

$$\begin{aligned} \dot{x}_{i}&=g_{i,\sigma (t)}x_{i+1}+f_{i,\sigma (t)}(x)+w_{i,\sigma (t)}, 1\le i\le n-1\nonumber \\ \dot{x}_{n}&=g_{n,\sigma (t)}u_{\sigma (t)}+f_{n,\sigma (t)}(x)+w_{n,\sigma (t)}\nonumber \\ y&=x_{1} \end{aligned}$$

(4.43)

where $x=(x_{1},\,x_{2},\ldots ,x_{n})^{T}\in \mathbb {R}^{n}\$is the system state, y is the system output; $\sigma (t):[0,\infty )\rightarrow M=\{1,2,\ldots ,m\}$ is the switching signal; $u_{k}\in \mathbb {R}$ is the control input of the k-th subsystem. For any $i=1,2,\ldots ,n$ and $k\in M,\,\,f_{i,k}(x)$ are unknown smooth nonlinear functions satisfying locally Lipschitz conditions, $g_{i,k}$ are positive constants, and $w_{i,k}$ is the bounded external disturbance of the system.

Our control objective is to design state-feedback controllers such that the output of system (4.43) tracks a given time-varying signal $y_{d}(t)$ and all the signals of the closed-loop systems remain bounded under arbitrary switchings.

Assumption 4.2

The tracking target $y_{d}(t)$ and its time derivatives up to the $n^{th}$ order are continuous and bounded. It is further assumed that there exists a positive constant d such that $\left| y_{d}\right| \le d.$

Assumption 4.3

There exist strictly increasing smooth functions $\phi _{i,k}(.)s:\mathbb {R}^{+}\rightarrow \mathbb {R}^{+}$ with $\phi _{i,k}(0)=0$ such that for $i=1,2,\ldots ,n-1,\,k\in M,$

$$ |f_{i,k}(x)|\le \phi _{i,k}(\left\| x\right\| ). $$

Remark 4.3

The increasing property of $\phi _{i,k}(.)$ means that if $a_{i}\ge 0,$ $i=1,2,\ldots ,n,$ then $\phi _{i,k}(\sum _{i=1}^{n}a_{i})\le \sum _{i=1}^{n}\phi _{i,k}(na_{i}).$ Note that $\phi _{i,k}(s)$ is a smooth function, and $\phi _{i,k}(0)=0.$ Therefore, there exists a smooth function $p_{i,k}(s)$ such that $\phi _{i,k}(s)=sp_{i,k}(s),$ which gives that

$$\begin{aligned} \phi _{i,k}\left( \sum _{i=1}^{n}a_{i}\right) \le \sum _{i=1}^{n}na_{i}p_{i,k}(na_{i}). \end{aligned}$$

(4.44)

In the control design procedure, radial basis function (RBF) neural networks are used to approximate a continuous function f(X) on a compact set $\varOmega \in R^{q}.$ For any $\varepsilon >0,$ there exists a neural network $\varPhi ^{T}P(X)$ such that

$$\begin{aligned} \sup _{x\in \varOmega }\left| f(X)-\varPhi ^{T}P(X)\right| \le \varepsilon , \end{aligned}$$

(4.45)

where $P(X)=[p_{1}(X),p_{2}(X),\ldots ,p_{l}(X)]^{T}$ is the basis function vector, $\varPhi =[\phi _{1},\phi _{2},\ldots , \phi _{l}]^{T}$ is the ideal constant weight vector with $l>1$ being the number of the neural network nodes and $p_{i}(X)$ are chosen as the form:

$$\begin{aligned} p_{i}(X)=\exp \left( \frac{-(X-\mu _{i})^{T}(X-\mu _{i})}{\zeta _{i}^{2}}\right) , \end{aligned}$$

(4.46)

where $\zeta _{i}$ is the width of the Gaussian function, and $\mu _{i} =[\mu _{i1},\mu _{i2},\ldots ,\mu _{iq}]^{T}$ is the center vector.

Remark 4.4

The readers may refer to [25] for more details about neural networks. Inequality (4.45) indicates that any given real continuous function f(X) can be represented by the linear combination of the basis function vector P(X) within a bounded error $\varepsilon$.

Lemma 4.2

For any $\xi \in R$ and $\varpi >0,$ the following inequality holds,

$$\begin{aligned} 0\le \left| \xi \right| -\xi \tanh \left( \frac{\xi }{\varpi }\right) \le \delta \varpi \end{aligned}$$

(4.47)

where $\delta =0.2785.$

4.3.2 Adaptive Control Design Based on Neural Networks

In this section, a backstepping-based adaptive control design procedure is presented. For the $i^{th}$ subsystem, define a common virtual control function $\alpha _{i}$ as

$$\begin{aligned} \alpha _{i}(X_{i})=-\frac{z_{i}}{\underline{g}_{i}}\left( \lambda _{i} +l_{i}^{2}+\eta _{i}^{2}\hat{\theta }P_{i}^{T}(X_{i})P_{i}(X_{i})\right) . \end{aligned}$$

(4.48)

where $\lambda _{i},$ $l_{i}$ and $\eta _{i}$ are positive design parameters; $\underline{g}_{i}=\min \{g_{i,k},\,k\in M\}$; $\hat{\theta }$ is the estimation of $\theta$ which is an unknown constant and is specified later; $X_{i}=(\bar{x}_{i}^{T},y_{d},\dot{y}_{d},\ldots ,y_{d} ^{(i)},\hat{\theta })^{T}$, $\bar{x}_{i}=$ $(x_{1},x_{2},\ldots ,x_{i})^{T}$; $P_{i}(X_{i})$ represents the basis function of the ith neural network system. Subsequently, a set of the variable change of coordinates is defined as $z_{i}=x_{i}-\alpha _{i-1}.$ Then, the z-system after coordinate transform is that

$$\begin{aligned} \dot{z}_{i}&=g_{i,k}x_{i+1}+f_{i,k}(x)+w_{i,k}-\dot{\alpha }_{i-1},1\le i\le n-1\nonumber \\ \dot{z}_{n}&=g_{n,k}u_{k}+f_{n,k}(x)+w_{n,k}-\dot{\alpha }_{n-1} \end{aligned}$$

(4.49)

where $\alpha _{0}=y_{d}.$

For $i=1,2,\ldots ,n-1,\,$the time derivative of $\alpha _{i-1}$ is given by

$$\begin{aligned} \dot{\alpha }_{i-1}=\sum _{s=1}^{i-1}\frac{\partial \alpha _{i-1}}{\partial x_{s} }(f_{s,k}+g_{s,k}x_{s+1}+w_{s,k})+\sum _{s=1}^{i-1}\frac{\partial \alpha _{i-1} }{\partial \hat{\theta }}\dot{\hat{\theta }}+\sum _{s=0}^{i-1}\frac{\partial \alpha _{i-1}}{\partial y_{d}^{(s)}}y_{d}^{(s+1)}\qquad \end{aligned}$$

(4.50)

where $\sum _{s=1}^{0}\frac{\partial \alpha _{i-1}}{\partial x_{s}} (f_{s,k}+g_{s,k}x_{s+1}+w_{s,k})=0,$ and $\sum _{s=1}^{0}\frac{\partial \alpha _{i-1}}{\partial \hat{\theta }}\dot{\hat{\theta }}=0.$

The controller can be chosen as

$$\begin{aligned} u=-\frac{z_{n}}{\underline{g}_{n}}\left( \lambda _{n}+l_{n}^{2}+\eta _{n} ^{2}\hat{\theta }P_{n}^{T}(X_{n})P_{n}(X_{n})\right) , \end{aligned}$$

(4.51)

where $\lambda _{n},$ $l_{n}$ and $\eta _{n}$ are positive design parameters; $\underline{g}_{n}=\min \{g_{n,k},\,k\in M\}$; $X_{n} =(\bar{x}_{i}^{T},y_{d},\dot{y}_{d},\ldots ,y_{d}^{(n)},\hat{\theta })^{T}$; $P_{n}(X_{n})$ represents the basis function vector of the $n^{th}$ neural network system.

The adaptive law is designed as

$$\begin{aligned} \dot{\hat{\theta }}=\sum _{i=1}^{n}r\eta _{i}^{2}z_{i}^{2}P_{i}^{T}P_{i} -\beta \hat{\theta } \end{aligned}$$

(4.52)

where r and $\beta$ are positive design parameters.

Lemma 4.3

For the variable transformations $z_{i}=x_{i}-\alpha _{i-1},$ $i=1,2,\ldots ,n,$ the following inequality holds,

$$\begin{aligned} \left\| x\right\| \le \sum _{i=1}^{n}\left| z_{i}\right| \varphi _{i}(\hat{\theta })+d \end{aligned}$$

(4.53)

where $\alpha _{0}=y_{d}$, $\varphi _{i}(\hat{\theta })=\frac{1}{\underline{g}_{i}}\left( -(\lambda _{i}+l_{i}^{2})-\eta _{i}^{2}\hat{\theta }P_{i}^{T}(X_{i})P_{i}(X_{i})\right) +1,$ $i=1,2,\ldots ,n-1,$ and $\varphi _{n}=1.$

The main result is given in the following theorem.

Theorem 4.3

Consider the closed-loop system (4.43) with the controller (4.51) and the adaptive law (4.52). For $1\le i\le n, k\in M$, assume that all the unknown nonlinear functions $\bar{f}_{i,k}(x)$ are approximated by neural networks in the sense that the approximation error $\varepsilon _{i,k}$ is bounded. Then, for bounded initial conditions, the target signal can be tracked within a small bounded error and other closed-loop signals remain bounded.

Proof

Consider the common Lyapunov function candidate as

$$\begin{aligned} V=\frac{1}{2}\sum _{i=1}^{n}z_{i}^{2}+\frac{1}{2r}\tilde{\theta }^{2} \end{aligned}$$

(4.54)

where $r>0$ is a design parameter.

The time derivative of V is given by

$$\begin{aligned} \dot{V}=&\sum _{i=1}^{n-1}z_{i}(f_{i,k}+g_{i,k}x_{i+1}+w_{1,k}-\dot{\alpha }_{i-1})\nonumber \\&+z_{n}(f_{n,k}+g_{n,k}u+w_{n,k}-\dot{\alpha }_{n-1})-\frac{1}{r} \tilde{\theta }\dot{\hat{\theta }} \end{aligned}$$

(4.55)

where $\alpha _{0}=y_{d.}$

By using (4.50), the following inequality can be obtained,

$$\begin{aligned} \dot{V}=&\sum _{i=1}^{n-1}z_{i}\left\{ f_{i,k}+g_{i,k}x_{i+1}+w_{i,k} -\sum _{s=0}^{i-1}\frac{\partial \alpha _{i-1}}{\partial y_{d}^{(s)}} y_{d}^{(s+1)}\right. \\&-\left. \sum _{s=1}^{i-1}\frac{\partial \alpha _{i-1}}{\partial x_{s} }(f_{s,k}+g_{s,k}x_{s+1}+w_{s,k})-\frac{\partial \alpha _{i-1}}{\partial \hat{\theta }}\dot{\hat{\theta }}\right\} \\&+z_{n}\left\{ f_{n,k}+g_{n,k}u+w_{n,k}-\sum _{s=0}^{n-1}\frac{\partial \alpha _{n-1}}{\partial y_{d}^{(s)}}y_{d}^{(s+1)}\right. \\&-\left. \sum _{s=1}^{n-1}\frac{\partial \alpha _{n-1}}{\partial x_{s} }(f_{s,k}+g_{s,k}x_{s+1}+w_{s,k})-\frac{\partial \alpha _{n-1}}{\partial \hat{\theta }}\dot{\hat{\theta }}\right\} -\frac{1}{r}\tilde{\theta }\dot{\hat{\theta }}\\ =&\sum _{i=1}^{n}z_{i}\left\{ f_{i,k}-\sum _{s=1}^{i-1}\frac{\partial \alpha _{i-1}}{\partial x_{s}}f_{s,k}\right\} +\sum _{i=1}^{n-1}z_{i}\left\{ g_{i,k}x_{i+1}+w_{i,k}\right. \end{aligned}$$

$$\begin{aligned}&-\left. \sum _{s=0}^{i-1}\frac{\partial \alpha _{i-1}}{\partial y_{d}^{(s)} }y_{d}^{(s+1)}-\sum _{s=1}^{i-1}\frac{\partial \alpha _{i-1}}{\partial x_{s} }(g_{s,k}x_{s+1}+w_{s,k})-\frac{\partial \alpha _{i-1}}{\partial \hat{\theta } }\dot{\hat{\theta }}\right\} \nonumber \\&+z_{n}\left\{ g_{n,k}u+w_{n,k}-\sum _{s=0}^{n-1}\frac{\partial \alpha _{n-1} }{\partial y_{d}^{(s)}}y_{d}^{(s+1)}-\frac{\partial \alpha _{n-1}}{\partial \hat{\theta }}\dot{\hat{\theta }}\right. \nonumber \\&-\left. \sum _{s=1}^{n-1}\frac{\partial \alpha _{n-1}}{\partial x_{s} }(g_{s,k}x_{s+1}+w_{s,k})\right\} -\frac{1}{r}\tilde{\theta }\dot{\hat{\theta }} \end{aligned}$$

(4.56)

By using Assumption 4.3, Lemma 4.3 and Remark 4.3, one has

$$\begin{aligned} z_{i}&\left( f_{i,k} -\sum _{s=1}^{i-1}\frac{\partial \alpha _{i-1}}{\partial x_{s} }f_{s,k}(x)\right) \nonumber \\&\qquad = -z_{i}\sum _{s=1}^{i}\frac{\partial \alpha _{i-1}}{\partial x_{s}} f_{s,k}(x)\le \sum _{s=1}^{i}|z_{i}\frac{\partial \alpha _{i-1}}{\partial x_{s} }|\left| f_{s,k}(x)\right| \nonumber \\&\qquad \le \sum _{s=1}^{i}|z_{i}\frac{\partial \alpha _{i-1}}{\partial x_{s}} |\phi _{s,k}(\left\| x\right\| )\nonumber \\&\qquad \le \sum _{s=1}^{i}\sum _{j=1}^{n}|z_{i}\frac{\partial \alpha _{i-1}}{\partial x_{s}}||z_{j}|\bar{\phi }_{s,k}(|z_{j}|\varphi _{j}(\theta ))+\sum _{s=1} ^{i}|z_{i}\frac{\partial \alpha _{i-1}}{\partial x_{s}}|\phi _{s,k}(d)\nonumber \\&\qquad \le \sum _{s=1}^{i}\sum _{j=1}^{n}\frac{1}{2}z_{i}^{2}\left( \frac{\partial \alpha _{i-1}}{\partial x_{s}}\right) ^{2}+\sum _{s=1}^{i}\sum _{j=1}^{n}\frac{1}{2}z_{j}^{2}\bar{\phi }_{s,k}^{2}(|z_{j}|\varphi _{j}(\theta ))\nonumber \\&\qquad \quad +\sum _{s=1}^{i}|z_{i}\frac{\partial \alpha _{i-1}}{\partial x_{s}}|\phi _{s,k}(d) \end{aligned}$$

(4.57)

where $\bar{\phi }_{s,k}(|z_{j}|\varphi _{j}(\theta ))=\varphi _{j}(\theta )h_{s,k}(|z_{j}|\varphi _{j}(\theta ))$.

Substituting (4.57) into (4.56) yields that

$$\begin{aligned} \dot{V}\le&\sum _{i=1}^{n}\sum _{s=1}^{i}\sum _{j=1}^{n}\frac{1}{2}z_{i} ^{2}\left( \frac{\partial \alpha _{i-1}}{\partial x_{s}}\right) ^{2}\\&+\sum _{i=1}^{n}\sum _{s=1}^{i}\sum _{j=1}^{n}\frac{1}{2}z_{j}^{2}\bar{\phi }_{s,k}^{2}(|z_{j}|\varphi _{j}(\theta ))+\sum _{i=1}^{n}\sum _{s=1}^{i} |z_{i}\frac{\partial \alpha _{i-1}}{\partial x_{s}}|\phi _{s,k}(d)\\&+\sum _{i=1}^{n-1}z_{i}\left\{ g_{i,k}x_{i+1}+w_{i,k}-\sum _{s=0}^{i-1} \frac{\partial \alpha _{i-1}}{\partial y_{d}^{(s)}}y_{d}^{(s+1)}\right. \\ \quad&-\left. \sum _{s=1}^{i-1}\frac{\partial \alpha _{i-1}}{\partial x_{s} }(g_{s,k}x_{s+1}+w_{s,k})-\frac{\partial \alpha _{i-1}}{\partial \hat{\theta } }\dot{\hat{\theta }}\right\} \\&+z_{n}\left\{ g_{n,k}u+w_{n,k}-\sum _{s=0}^{n-1}\frac{\partial \alpha _{n-1} }{\partial y_{d}^{(s)}}y_{d}^{(s+1)}-\frac{\partial \alpha _{n-1}}{\partial \hat{\theta }}\dot{\hat{\theta }}\right. \end{aligned}$$

$$\begin{aligned} \left. -\sum _{s=1}^{n-1}\frac{\partial \alpha _{n-1}}{\partial x_{s}} (g_{s,k}x_{s+1}+w_{s,k})\right\} -\frac{1}{r}\tilde{\theta }\dot{\hat{\theta }} \end{aligned}$$

(4.58)

One can obtain that

$$\begin{aligned} \sum _{i=1}^{n}\sum _{s=1}^{i}\sum _{j=1}^{n}\frac{1}{2}z_{j}^{2}\bar{\phi } _{s,k}^{2}(|z_{j}|\varphi _{j}(\theta ))=\sum _{i=1}^{n}z_{i}^{2}\sum _{s=1} ^{n}q(n,s)\bar{\phi }_{s,k}^{2}(|z_{j}|\varphi _{j}(\theta )) \end{aligned}$$

(4.59)

where $q(n,s)=\frac{(n-(s-1))}{2}.$

By using Lemma 4.2, the following inequality holds for $\varpi _{i,k}>0,$

$$\begin{aligned} \sum _{s=1}^{i}|z_{i}\frac{\partial \alpha _{i-1}}{\partial x_{s}}|\phi _{s,k}(d)\le z_{i}Z_{i}\tanh \left( \frac{z_{i}Z_{i}}{\varpi _{i,k}}\right) +\delta \varpi _{i,k} \end{aligned}$$

(4.60)

where $Z_{i}=\sum _{s=1}^{i}|z_{i}\frac{\partial \alpha _{i-1}}{\partial x_{s} }|\phi _{s,k}(d).$

It follows from (4.58)–(4.60) that

$$\begin{aligned} \dot{V}\le&\sum _{i=1}^{n}z_{i}^{2}\sum _{s=1}^{i}\frac{n}{2}\left( \frac{\partial \alpha _{i-1}}{\partial x_{s}}\right) ^{2}+\sum _{i=1}^{n}z_{i}^{2}\sum _{s=1}^{n}q(n,s)\bar{\phi }_{s,k}^{2}(|z_{j}|\varphi _{j}(\theta ))\nonumber \\&+\sum _{i=1}^{n}z_{i}Z_{i}\tanh \left( \frac{z_{i}Z_{i}}{\varpi _{i,k}}\right) +\sum _{i=1}^{n}\delta \varpi _{i,k}+\sum _{i=1}^{n-1}z_{i}\Bigg \{g_{i,k} x_{i+1}+w_{i,k} \nonumber \\&-\left. \sum _{s=0}^{i-1}\frac{\partial \alpha _{i-1}}{\partial y_{d}^{(s)} }y_{d}^{(s+1)}-\sum _{s=1}^{i-1}\frac{\partial \alpha _{i-1}}{\partial x_{s} }(g_{s,k}x_{s+1}+w_{s,k})-\frac{\partial \alpha _{i-1}}{\partial \hat{\theta } }\dot{\hat{\theta }}\right\} \nonumber \\&+z_{n}\left\{ g_{n,k}u+w_{n,k}-\sum _{s=0}^{n-1}\frac{\partial \alpha _{n-1} }{\partial y_{d}^{(s)}}y_{d}^{(s+1)}-\frac{\partial \alpha _{n-1}}{\partial \hat{\theta }}\dot{\hat{\theta }}\right. \nonumber \\&-\left. \sum _{s=1}^{n-1}\frac{\partial \alpha _{n-1}}{\partial x_{s} }(g_{s,k}x_{s+1}+w_{s,k})\right\} -\frac{1}{r}\tilde{\theta }\dot{\hat{\theta }}\nonumber \\ =&\sum _{i=1}^{n}z_{i}\left\{ z_{i}\sum _{s=1}^{i}\frac{n}{2}\left( \frac{\partial \alpha _{i-1}}{\partial x_{s}}\right) ^{2}+z_{i}\sum _{s=1}^{n}q(n,s)\bar{\phi }_{s,k}^{2}(|z_{j}|\varphi _{j}(\theta ))\right. \nonumber \\&+Z_{i}\tanh \left( \frac{z_{i}Z_{i}}{\varpi _{i,k}}\right) -\sum _{s=1}^{n-1}\frac{\partial \alpha _{n-1}}{\partial x_{s}}(g_{s,k}x_{s+1}+w_{s,k})\nonumber \\&-\left. \frac{\partial \alpha _{i-1}}{\partial \hat{\theta }}\dot{\hat{\theta }}-\sum _{s=0}^{i-1}\frac{\partial \alpha _{i-1}}{\partial y_{d}^{(s)}} y_{d}^{(s+1)}+w_{i,k}\right\} \nonumber \\&+\sum _{i=1}^{n-1}z_{i}g_{i,k}x_{i+1}+z_{n}g_{n,k}u-\frac{1}{r}\tilde{\theta }\dot{\hat{\theta }}+\sum _{i=1}^{n}\delta \varpi _{i,k} \end{aligned}$$

(4.61)

Note that

$$\begin{aligned} \sum _{i=1}^{n-1}z_{i}g_{i,k}x_{i+1}=\sum _{i=1}^{n-1}z_{i}g_{i,k}z_{i+1} +\sum _{i=1}^{n-1}z_{i}g_{i,k}\alpha _{i} \end{aligned}$$

(4.62)

and define

$$\begin{aligned} \bar{f}_{i,k}=&z_{i}\sum _{s=1}^{i}\frac{n}{2}\left( \frac{\partial \alpha _{i-1} }{\partial x_{s}}\right) ^{2}+z_{i}\sum _{s=1}^{n}q(n,s)\bar{\phi }_{s,k}^{2} (|z_{j}|\varphi _{j}(\theta ))\nonumber \\&+Z_{i}\tanh \left( \frac{z_{i}Z_{i}}{\varpi _{i,k}}\right) -\sum _{s=1}^{n-1}\frac{\partial \alpha _{n-1}}{\partial x_{s}}(g_{s,k}x_{s+1}+w_{s,k})\nonumber \\&-\frac{\partial \alpha _{i-1}}{\partial \hat{\theta }}\dot{\hat{\theta }} -\sum _{s=0}^{i-1}\frac{\partial \alpha _{i-1}}{\partial y_{d}^{(s)}} y_{d}^{(s+1)}+w_{i,k}+g_{i-1,k}z_{i-1} \end{aligned}$$

(4.63)

where $g_{0}=0$ and $z_{0}=0.$

Substituting (4.62) and (4.63) into (4.61) gives that

$$\begin{aligned} \quad \dot{V}\le \sum _{i=1}^{n-1}z_{i}(\bar{f}_{i,k}+g_{i,k}\alpha _{i} )+z_{n}(\bar{f}_{n,k}+g_{n,k}u_{k})+\sum _{i=1}^{n}\delta \varpi _{i,k}-\frac{1}{r}\tilde{\theta }\dot{\hat{\theta }} \end{aligned}$$

(4.64)

The neural network $\varPhi _{i,k}^{T}P_{i,k}$ is utilized to approximate the unknown function $\bar{f}_{i,k}$ such that for any given $\bar{\varepsilon }_{i,k}>0,$

$$\begin{aligned} \bar{f}_{i,k}=\varPhi _{i,k}^{T}P_{i,k}(X_{i})+\varepsilon _{i,k}(X_{i}) \end{aligned}$$

(4.65)

where $X_{i}=(\bar{x}_{i}^{T},y_{d},\dot{y}_{d},\ldots ,y_{d}^{(i)},\hat{\theta }_{1},\hat{\theta }_{2},\ldots ,\hat{\theta }_{i})^{T}$, $\left| \varepsilon _{i,k}\right| \le \bar{\varepsilon }_{i,k},\,\varepsilon _{i,k}$ denotes the approximation error. Thus, for $i=1,2,\ldots ,n,$

$$\begin{aligned} z_{i}\bar{f}_{i,k}&=z_{i}\varPhi _{i,k}^{T}P_{i,k}(X_{i})+z_{i}\varepsilon _{i,k}(X_{i})\nonumber \\&\le \frac{\eta _{i}^{2}}{2}z_{i}^{2}\left\| \varPhi _{i,k}\right\| ^{2}P_{i,k}^{T}P_{i,k}+\frac{1}{2\eta _{i}^{2}}+\frac{l_{i,k}^{2}}{2}z_{i} ^{2}+\frac{\varepsilon _{i,k}^{2}}{2l_{i,k}^{2}}\nonumber \\&\le \eta _{i}^{2}z_{i}^{2}\theta _{i}P_{i}^{T}P_{i}+l_{i}^{2}z_{i}^{2} +\frac{\bar{\varepsilon }_{i}^{2}}{l_{i}^{2}}+\frac{1}{\eta _{i}^{2}} \end{aligned}$$

(4.66)

where $\eta _{i},l_{i}>0$, $\theta _{i,k}=\left\| \varPhi _{i,k}\right\| ^{2}$, $\theta _{i}=\max \{\theta _{i,k}:k\in M\},$ $P_{i}(X_{i})$ and $\bar{\varepsilon }_{i}(X_{i})$ represent the basis function vector and the estimation error belongs to $\theta _{i}.$

The feasible virtual control functions, adaptive laws and controllers are designed, respectively, as

$$\begin{aligned} \alpha _{i}&=-\frac{z_{i}}{\underline{g}_{i}}\left( \lambda _{i}+l_{i} ^{2}+\eta _{i}^{2}\hat{\theta }_{i}P_{i}^{T}P_{i}\right) \end{aligned}$$

(4.67)

$$\begin{aligned} \dot{\hat{\theta }}_{i}&=r_{i}\eta _{i}^{2}z_{i}^{2}P_{i}^{T}P_{i}-\beta _{i}\hat{\theta }_{i}\end{aligned}$$

(4.68)

$$\begin{aligned} u_{k}&=-\frac{z_{n}}{\underline{g}_{n}}\left( \lambda _{n}+l_{n}^{2} +\eta _{n}^{2}\hat{\theta }_{n}P_{n}^{T}P_{n}\right) \end{aligned}$$

(4.69)

where for $i=1,2,\ldots ,n,\,\lambda _{i},\,r_{i},\,\beta _{i}$ are positive design parameters, and $\hat{\theta }_{i}$ is the estimation of $\theta _{i}.$

Consider that too many adaptive parameters ($\hat{\theta }_{1},\ldots ,\hat{\theta }_{n}$) can cause the problem of over-parameterization. Set $r_{1}=r_{2}=\cdots =r_{n}=r,\ \beta _{1}=\beta _{2}=\cdots =\beta _{n}=\beta ,$ and define $\theta =\sum _{i=1}^{n}\theta _{i},\ \hat{\theta }=\sum _{i=1} ^{n}\hat{\theta }_{i}.$ The adaptive laws (4.68) can be changed as follows

$$\begin{aligned} \dot{\hat{\theta }}&=\sum _{i=1}^{n}\dot{\hat{\theta }}_{i}=\sum _{i=1} ^{n}\left( r\eta _{i}^{2}z_{i}^{2}P_{i}^{T}P_{i}-\beta \hat{\theta }_{i}\right) \nonumber \\&=\sum _{i=1}^{n}r\eta _{i}^{2}z_{i}^{2}P_{i}^{T}P_{i}-\beta \hat{\theta }. \end{aligned}$$

(4.70)

Then, the stabilizing functions, the adaptive law and controllers can be designed as (4.48), (4.52) and (4.51) respectively.

Substituting (4.48), (4.51) and (4.52) into (4.62) one has

$$\begin{aligned} \dot{V}\le -\sum _{i=1}^{n}\lambda _{i}z_{i}^{2}+\sum _{i=1}^{n}\frac{\beta }{r}\tilde{\theta }\hat{\theta }+\sum _{i=1}^{n}\left( \frac{\bar{\varepsilon }_{i}^{2} }{l_{i}^{2}}+\frac{1}{\eta _{i}^{2}}+\delta \varpi _{i}\right) \end{aligned}$$

(4.71)

where $\varpi _{i}=\max \{\varpi _{i,k},k\in M\}.$

It is true that

$$\begin{aligned} \tilde{\theta }\hat{\theta }=\tilde{\theta }(\theta -\tilde{\theta })\le -\frac{1}{2}\tilde{\theta }^{2}+\frac{1}{2}\theta ^{2} \end{aligned}$$

(4.72)

Then, (4.71) can be rewritten as

$$\begin{aligned} \dot{V}\le&-\frac{1}{2}\sum _{i=1}^{n}\left( 2\lambda _{i}z_{i}^{2}+\frac{\beta }{r}\tilde{\theta }^{2}\right) +\sum _{i=1}^{n}\left( \frac{\bar{\varepsilon }_{i}^{2}}{l_{i}^{2}}+\frac{1}{\eta _{i}^{2}}+\delta \varpi _{i}+\frac{\beta }{2r}\theta ^{2}\right) \nonumber \\ \le&-a_{0}V+b_{0} \end{aligned}$$

(4.73)

where $a_{0}=\min \{2\lambda _{i},\,\beta :1\le i\le n\}$ and $b_{0}=\sum _{i=1}^{n}(\bar{\varepsilon }_{i}^{2}/l_{i}^{2}+1/\eta _{i}^{2}+\delta \varpi _{i}+\frac{\beta }{2r}\theta ^{2}).$

Furthermore

$$\begin{aligned} V(t)\le \left( V(0)-\frac{b_{0}}{a_{0}}\right) e^{-a_{0}t}+\frac{b_{0}}{a_{0}} \end{aligned}$$

(4.74)

which means that all the signals in the closed-loop system are bounded. In particular, we have

$$ \lim _{t\rightarrow \infty }\left| z_{1}\right| \le \sqrt{\frac{2b_{0} }{a_{0}}} $$

The proof is completed here. $\square$

4.3.3 Simulation Results

In this section, the simulation studies for the ship manoeuvring systems shown in Fig. 4.5 are carried out to illustrates the effectiveness of our results.

L :

: length of ship

N :

: moment component on body relative to z-axis

r :

: yaw rate

v :

: speed of ship

$v_{x}$ :

: forward velocity in x-axis

$v_{y}$ :

: drift velocity along y-axis

$\beta$ :

: drift angle

x, y :

: force components on body

$\psi$ :

: yaw angle

$\delta$ :

: rudder angle

Fig. 4.5

Ship manoeuvring system

The ship maneuvering system can be described by the following Norrbin nonlinear model,

$$\begin{aligned} T\dot{h}+h+\tau h^{3}=K\delta +\omega , \end{aligned}$$

(4.75)

where T is the time constant, $h=\dot{\psi }$ denotes the yaw rate, $\psi$ stands for the heading angle, $\tau$ is the Norrbin coefficient, K represents the rudder gain, $\delta$ is the rudder angle and $\omega$ stands for the outside disturbances. The value of $\tau$ can be determined via a spiral test. The ship’s dynamic parameters are basically determined by its size and shape, and may vary with operational conditions such as ship speed, draft, trim, and water depth.

A simplified mathematical model of the rudder system can be described as:

$$\begin{aligned} T_{E}\dot{\delta }+\delta =K_{E}\delta _{E}, \end{aligned}$$

(4.76)

where $T_{E}$ represents the rudder time constant, $\delta$ stands for the actual rudder angle, $K_{E}$ denotes the rudder control gain and $\delta _{E}$ is the rudder order.

Let $x_{1}$ $=\psi$, $x_{2}$ $=h$, $x_{3}$ $=\delta$; one has

$$\begin{aligned} \dot{x}_{1}&=x_{2},\nonumber \\ \dot{x}_{2}&=f+bx_{3}+\omega ,\nonumber \\ \dot{x}_{3}&=-\frac{1}{T_{E}}x_{3}+\frac{K_{E}}{T_{E}}\delta _{E}, \end{aligned}$$

(4.77)

where $f=-\frac{1}{T}x_{2}-\frac{\tau }{T}x_{2}^{3}$ is an unknown nonlinear function, $b=\frac{K}{T}.$

Note that some parameters of the aforementioned system will change when the speed of the ship changes. We adopt the following switched model to depict the dynamic behavior when the ship is at low speed, medium speed and high speed, respectively.

$$\begin{aligned} \dot{x}_{1}&=x_{2},\nonumber \\ \dot{x}_{2}&=f_{\sigma (v)}(x_{2})+b_{\sigma (v)}x_{3}+\omega _{\sigma (v)},\nonumber \\ \dot{x}_{3}&=-\frac{1}{T_{E,\sigma (v)}}x_{3}+\frac{K_{E,\sigma (v)} }{T_{E,\sigma (v)}}\delta _{E,\sigma (v)}, \end{aligned}$$

(4.78)

where $f_{\sigma (v)}(x_{2})=-\frac{1}{T_{\sigma (v)}}x_{2}-\frac{\tau _{\sigma (v)}}{T_{\sigma (v)}}x_{2}^{3},$ $b_{\sigma (v)}=\frac{K_{\sigma (v)} }{T_{\sigma (v)}}$ and $\sigma (v)$ is the switching signal that satisfies:

$$ \sigma (v)=\left\{ \begin{array} [c]{ll} 1, &{} 0<v\le v_{L}\\ 2, &{} v_{L}<v\le v_{M}\\ 3, &{} v_{M}<v\le v_{T} \end{array} \right. $$

where $v_{L},\,v_{M},\,v_{T}$ represent the value of low speed, medium speed and top speed, respectively.

The vessel data comes from a ship that is listed in Table 4.1. The controller parameters are chosen as those in Table 4.2. Furthermore, the outside disturbances are: $w_{1}=0.01\sin t$; $w_{2}=0.015\cos t$; $w_{3}=0.013\sin t$. We construct the basis function vectors $P_{1}$, $P_{2}$ and $P_{3}$ using 7, 15 and 27 nodes, the centers $\mu _{1}$, $\mu _{2}$ and $\mu _{3}$ evenly spaced on $[-3,\,3]\times [-4,\,1]\times [-2,\,2]$, $[-0.5,$ $3.5]\times [-4,\,4]\times [-8,\,8]$ and $[-4,\,4]\times [-30,\,10]\times [-0.5,\,3.5]$, and the widths $\zeta _{1} =2,\,\zeta _{2}=2.5$, and $\,\zeta _{3}=2,$ respectively. The initial conditions are $x_{1}(0)=x_{2}(0)=x_{3}(0)=0.02$, $\hat{\theta }(0)=1$ and the target signal is $y_{d}=10\sin 0.05t.$

Table 4.1 Model parameters of ship maneuvering system

Table 4.2 Controller parameters

To illustrate the effectiveness of the proposed controller, comparison results are presented. The first one uses existing results in [26] and our results, respectively, to control the system when the ship is at a constant speed: low speed. The other one uses existing results [26] and our results respectively to control the system when the ship switches among different speeds.

According to (4.51) and (4.52), the adaptive law $\hat{\theta }$ and the control law $u_{k}$ are chosen, respectively, as

$$\begin{aligned} \dot{\hat{\theta }}&=\sum _{i=1}^{3}r\eta _{i}^{2}z_{i}^{2}P_{i}^{T} P_{i}-\beta \hat{\theta }\\&=0.64z_{1}^{2}P_{1}^{T}P_{1}+z_{2}^{2}P_{2}^{T}P_{2}+1.44z_{3}^{2}P_{3} ^{T}P_{3}-0.1\hat{\theta }\\ u_{k}&=-\frac{z_{3}}{\underline{g}_{3}}\left( \lambda _{3}+l_{3}^{2} +\eta _{3}^{2}\hat{\theta }P_{3}^{T}P_{3}\right) \\&=-2.5z_{3}\left( 105+144P_{3}^{T}P_{3}\right) \end{aligned}$$

where $z_{1}=x_{1}-y_{d},\,z_{2}=x_{2}-\alpha _{1},\,z_{3}=x_{3}-\alpha _{2}.$

The virtual control functions $\alpha _{1}$ and $\alpha _{2}$ are given by

$$\begin{aligned} \alpha _{1}&=-\frac{z_{1}}{\underline{g}_{1}}\left( \lambda _{1}+l_{1} ^{2}+\eta _{1}^{2}\hat{\theta }P_{1}^{T}P_{1}\right) \\&=-z_{1}(146+64\hat{\theta }P_{1}^{T}P_{1})\\ \alpha _{2}&=-\frac{z_{2}}{\underline{g}_{2}}\left( \lambda _{2}+l_{2} ^{2}+\eta _{2}^{2}\hat{\theta }P_{1}^{T}P_{1}\right) \\&=-1.6\times 10^{3}\times z_{2}\left( 199+100\hat{\theta }P_{2}^{T} P_{2}\right) \end{aligned}$$

Figures 4.6, 4.7 and 4.8 show the comparison results by using the existing method in [26] and our approach, respectively. It can be seen that both methods can ensure the target signal is tracked within a small bounded error.

Fig. 4.6

Tracking performances under a constant speed. $y_{d}$ is the target signal; $\psi _{1}$ and $\psi _{2}$ represent the outputs by using existing results in [26] and our results respectively

Fig. 4.7

Responses of tracking errors under a constant speed. $\psi _{1}-y_{d}$ and $\psi _{2}-y_{d}$ stand for the tracking error by using existing results in [26] and our results respectively

Fig. 4.8

Responses of adaptive laws under a constant speed. $\hat{\theta }_{1}$, $\hat{\theta }_{2}$ and $\hat{\theta }_{3}$ denote the adaptive laws by existing results in [26]; $\hat{\theta }$ represents the adaptive law by our results

Fig. 4.9

Tracking performances under switched speeds. $y_{d}$ is the target signal; $\psi _{1}$ and $\psi _{2}$ represent the outputs by using existing results in [26] and our results respectively

Fig. 4.10

Responses of tracking errors under switched speeds. $\psi _{1}-y_{d}$ and $\psi _{2}-y_{d}$ stand for the tracking error by using existing results in [26] and our results respectively

Fig. 4.11

Responses of adaptive laws under switched speeds. $\hat{\theta }_{1}$, $\hat{\theta }_{2}$ and $\hat{\theta }_{3}$ denote the adaptive laws by existing results in [26]; $\hat{\theta }$ represents the adaptive law by our results

Fig. 4.12

Responses of switching signal

Figures 4.6, 4.7, 4.8, 4.9, 4.10 and 4.11 depict the comparison results by using the existing method in [26] and our method under different speeds, and Fig. 4.12 gives the switching evolution among different speeds. From Figs. 4.9 and 4.10, it can be seen that the existing results in [26] cannot guarantee a good tracking performance under switched speeds. However, our method can still ensure the target signal is tracked within a small bounded error. Figure 4.11 indicates that the adaptive law’s number in our results is less than the one in [26].

4.3.4 Conclusions

The problem of adaptive neural tracking control for a class of switched uncertain nonlinear systems in nonstrict-feedback form is investigated. The stability analysis in indicates that the designed controllers can ensure that the target signal can be tracked with a small bounded error and the stability of the system can be kept under arbitrary switchings.