Adaptive Control of Switched Stochastic Nonlinear Systems

Abstract

The last chapter has discussed adaptive control design methods for switched nonlinear systems with uncertainties.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

5.1 Background and Motivation

The last chapter discussed adaptive control design methods for switched nonlinear systems with uncertainties. However, the system structures considered in the last chapter are somewhat simple, which greatly limits the applications of the results in practice.

It is well known that stochastic disturbance is inevitably encountered in practical systems. Therefore, control of stochastic systems with or without switching has become an active research field and received much attention recently, see, e.g., [1–5] and the references therein. The authors in [6] considered global stabilization for high-order stochastic nonlinear systems with stochastic integral input-to-state stability inverse dynamics. The moment stability and sample path stability of switched stochastic linear systems were investigated in [7]. In [8] dissipativity-based sliding mode control for switched stochastic linear systems was adopted. Stabilization problems for stochastic nonlinear systems with Markovian switching were studied in [9]. The $p^{th}$ moment exponential stability and global asymptotic stability in probability for a class of switched stochastic nonlinear retarded systems with asynchronous switching were solved in [10].

Moreover, dead-zone characteristics are encountered in many physical components of control systems. They are particularly common in actuators, such as hydraulic servovalves and electric servomotors. They also appear in biomedical systems. The system model is more realistic and reliable when the dead-zone nonlinearities are taken into consideration.

On the other hand, since the input-to-state stability (ISS) property was proposed in [11], it has rapidly become an important tool to investigate the stability problem of nonlinear systems. In view of the crucial importance of ISS, it is natural to introduce this concept to switched nonlinear systems. In this chapter, we consider some control problems of switched high-order nonlinear systems. Some complex dynamics such as stochastic disturbances, uncertainties, dead-zone nonlinearities and input-to-state stability inverse dynamics are considered in the systems under investigations. The considered mathematical models can provide a good description of a large number of practical switched nonlinear systems.

Notation $\mathbb {R}$ denotes the n-dimensional space, $\mathbb {R}^{n}$ is the set of all nonnegative real numbers. $\mathscr {C}^{i}$ 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 that is of class $\mathscr {K}$ and unbounded. In addition, $\left\| \cdot \right\|$ refers to the Euclidean vector norm. $\mathbb {R}$ denotes the n-dimensional space, $\mathbb {R}^{+}$ denotes the set of all nonnegative real numbers, and $\mathbb {R}^{*}=\{q\in \mathbb {R}^{+} :q\ge 1$ is an odd integer$\}$. $\mathscr {C}^{i}$ denotes a set of all 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. $\mathscr {K}$ denotes the set of all functions: $\mathbb {R}^{+}\rightarrow \mathbb {R}^{+}$, which are continuous, strictly increasing and vanishing at zero; $\mathscr {K}_{\infty }$ denotes a set of functions that are of class $\mathscr {K}$ and unbounded. In addition, $\left\| \cdot \right\|$ refers to the Euclidean vector norm.

5.2 Adaptive Tracking Control for Switched Stochastic Nonlinear Systems with Unknown Actuator Dead-Zone

5.2.1 Problem Formulation and Preliminaries

Consider the following switched stochastic nonlinear system in nonstrict-feedback form.

$$\begin{aligned} dx_{i}&= (g_{i,\sigma (t)}x_{i+1}+f_{i,\sigma (t)}(x))dt+\psi _{i,\sigma (t)}^{T}(x)dw,\nonumber \\&1\le i\le n-1,\nonumber \\ dx_{n}&= (g_{n,\sigma (t)}v_{\sigma (t)}+f_{n,\sigma (t)}(x))dt+\psi _{n,\sigma (t)}^{T}(x)dw,\nonumber \\ v_{\sigma (t)}&= D_{\sigma (t)}(u_{\sigma (t)}),\nonumber \\ y&= x_{1}, \end{aligned}$$

(5.1)

where $x=(x_{1},\,x_{2},\ldots ,x_{n})^{T}\in \mathbb {R}^{n}\$ is the system state, w is an r-dimensional independent standard Brownian motion defined on the complete probability space $\left( \varOmega , \mathscr {F} ,\left\{ \mathscr {F} _{t}\right\} _{t\ge 0},P\right)$ with $\varOmega$ being a sample space, $\mathscr {F}$ being a $\sigma$-field, $\left\{ \mathscr {F} _{t}\right\} _{t\ge 0}$ being a filtration, and P being a probability measure, and y is the system output; $\sigma (t):[0,\infty )\rightarrow M=\{1,2,\ldots ,m\}$ represents the switching signal; $v_{\sigma (t)} ,\,u_{\sigma (t)}\in \mathbb {R}$ are the actuator output and input. For any $i=1,2,\ldots ,n$ and $k\in M,\,\,f_{i,k}(x):$ $\mathbb {R}^{n}\rightarrow \mathbb {R},\,\psi _{i,k}:\mathbb {R}^{n}\rightarrow \mathbb {R}^{r}\$ are locally Lipschitz unknown nonlinear functions and $\,g_{i,k}$ are positive known constants.

The nonsymmetric dead-zone nonlinearity is considered in the chapter, which is defined as the form in [12]:

$$\begin{aligned} v_{k}=D_{k}(u_{k})=\left\{ \begin{array} [c]{ll} m_{r_{k}}(u_{k}-b_{r_{k}}), &{} u_{k}\ge b_{r_{k}}\\ 0, &{} -b_{l_{k}}<u_{k}<b_{r_{k}}\\ m_{l_{k}}(u_{k}+b_{l_{k}}), &{} u_{k}\le -b_{l_{k}} \end{array} \right. \end{aligned}$$

(5.2)

Here, $m_{r_{k}}>0$ and $m_{l_{k}}>0$ represent the right and the left slopes of the dead-zone characteristic. $b_{r_{k}}>0$ and $b_{l_{k}}>0$ stand for the breakpoints of the input nonlinearity.

It is assumed that the nonsymmetric dead-zone nonlinearity can be reformulated as:

$$\begin{aligned} v_{k}=D_{k}^{^{\prime }}(u_{k})+\iota _{k}, \end{aligned}$$

(5.3)

where $D_{k}^{^{\prime }}(u_{k})$ is a smooth function, $\iota _{k}\,$is the error between $D_{k}(u_{k})\,$and $D_{k}^{^{\prime }}(u_{k})$ with $\left| \iota _{k}\right| \le \bar{\iota }_{k}$.

Moreover, we have

$$\begin{aligned} v_{k}&=u_{k}+(D_{k}^{^{\prime }}(u_{k})-u_{k}+\iota _{k})\nonumber \\&=u_{k}+\eta _{k}^{^{\prime }}(u_{k})+\iota _{k}, \end{aligned}$$

(5.4)

where $\eta _{k}^{^{\prime }}(u_{k})=D_{k}^{^{\prime }}(u_{k})-u_{k}$ is an unknown function.

The controller can be designed as

$$\begin{aligned} u_{k}=u_{c_{k}}-u_{\phi _{k}}. \end{aligned}$$

(5.5)

Then (5.4) can be rewritten as

$$\begin{aligned} v_{k}=u_{c_{k}}+\eta _{k}^{^{\prime }}(u_{k})-u_{\phi _{k}}+\iota _{k}. \end{aligned}$$

(5.6)

where $u_{\phi _{k}}$ is the compensator of dead-zone nonlinearity and $u_{c_{k}}$ is a main controller of system (5.1).

Our control objective is to design a state-feedback controller such that the output of system (5.1) can track a given time-varying signal $y_{d}(t)$, and the problem of the actuator dead-zone can be solved. The following assumptions are supposed to be true.

Assumption 5.1

The tracking target $y_{d}(t)$ and its time derivatives up to $n^{th}$ order $y_{d}^{(n)}(t)$ are continuous and bounded; it is further assumed that $|y_{d}(t)|\le d$.

Assumption 5.2

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

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

(5.7)

$$\begin{aligned} \left| \psi _{i,k}(x)\right|&\le \rho _{i,k}(\left\| x\right\| ). \end{aligned}$$

(5.8)

Remark 5.1

The increasing properties of $\phi _{i,k}(\cdot ),\,\rho _{i,k}(\cdot )$ imply that if $a_{i},b_{i}\ge 0,\,$for $i=1,2,\ldots ,n,$ then $\phi _{i,k}(\sum _{i=1}^{n}a_{i})\le \sum _{i=1}^{n}\phi _{i,k}(na_{i})$, $\rho _{i,k}(\sum _{i=1}^{n}b_{i})\le \sum _{i=1}^{n}\rho _{i,k}(nb_{i}).$ Notice that $\phi _{i,k}(s),\,\rho _{i,k}(s)$ are smooth functions, and $\phi _{i,k}(0)=\rho _{i,k}(0)=0.$ Therefore, there exist smooth functions $h_{i,k}(s),\,\eta _{i,k}(s)$ such that $\phi _{i,k}(s)=sh_{i,k}(s),\,\rho _{i,k}(s)=s\eta _{i,k}(s)$ which results in

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

(5.9)

$$\begin{aligned} \rho _{i,k}\left( \sum _{j=1}^{n}b_{j}\right)&\le \sum _{j=1}^{n}nb_{j}\eta _{i,k} (nb_{j}). \end{aligned}$$

(5.10)

We use the radial basis function (RBF) neural networks to approximate any a real function f(Z) over a compact set $\varOmega _{Z}\subset \mathbb {R}^{q}.$ For arbitrary $\bar{\varepsilon }>0,$ there exists a neural network $W^{T}S(Z)$ such that

$$\begin{aligned} f(Z)=W^{T}S(Z)+\varepsilon (Z),\,\varepsilon (Z)\le \bar{\varepsilon }, \end{aligned}$$

(5.11)

where $Z\in \varOmega _{Z}\subset R^{q},$ $W=[w_{1},w_{2},\ldots ,w_{l}]^{T}$ is the ideal constant weight vector, and $S(Z)=[s_{1}(Z)$, $s_{2}(Z)$,$\,\ldots$ ,$\,s_{l}(Z)]^{T}$ is the basis function vector, with $l>1$ being the number of the neural network nodes and $s_{i}(Z)$ being chosen as Gaussian functions; i.e., for $i=1,2,\ldots ,l$,

$$\begin{aligned} s_{i}(Z)=\exp \left[ \frac{-(Z-\mu _{i})^{T}(Z-\mu _{i})}{\zeta _{i}^{2}}\right] , \end{aligned}$$

(5.12)

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

Definition 5.1

For any given $V(x_{i},t)\in \mathscr {C}^{2,1}$ associated with system (5.1), define the differential operator $\mathscr {L}$ as follows;

$$\begin{aligned} \mathscr {L}V=\frac{\partial V}{\partial t}+\frac{\partial V}{\partial x_{i} }F_{i,k}+\frac{1}{2}Tr\left\{ \psi _{i,k}^{T}\frac{\partial ^{2}V}{\partial x_{i}^{2}}\psi _{i,k}\right\} , \end{aligned}$$

(5.13)

where $F_{i,k}=g_{i,k}x_{i+1}+f_{i,k}(x)$.

Definition 5.2

The trajectory $\{x(t),t\ge 0\}$ of switched stochastic system (5.1) is said to be semi-globally uniformly ultimately bounded (SGUUB) in the $p^{th}$ moment, if for some compact set $\varOmega \in \mathbb {R}^{n}$ and any initial state $x_{0}=x(t_{0})$, there exist a constant $\varepsilon >0$, and a time constant $T=T(\varepsilon ,x_{0})$ such that $E(|x(t)|^{p})<\varepsilon$, for all $t>t_{0}+T$. Especially, when $p=2$, it is usually called SGUUB in mean square.

Lemma 5.1

([13]) Suppose that there exist a $C^{2,1}$ function V(x, t) $:\mathbb {R}^{n}\times \mathbb {R}^{+}\rightarrow \mathbb {R}^{+}$, two constants $c_{1}>0$ and $c_{2}>0$, class $\mathscr {K}_{\infty }$ functions $\bar{\alpha }_{1}$ and $\bar{\alpha }_{2}$ such that

$$ \left\{ \begin{array} [c]{l} \bar{\alpha }_{1}(|x|)\le V(x,t)\le \bar{\alpha }_{2}(|x|)\\ \mathscr {L}V\le -c_{1}V(x,t)+c_{2} \end{array} \right. $$

for all $x\in \mathbb {R}^{n}$ and $t>t_{0}$. Then, there is an unique strong solution of system (5.1) for each $x_{0}\in \mathbb {R}^{n}$, that satisfies

$$ E[V(x,t)]\le V(x_{0})e^{-c_{1}t}+\frac{c_{2}}{c_{1}},\,\forall t>t_{0} $$

Lemma 5.2

([14]) For any $\xi \in \mathbb {R}$ and $\varpi >0,$ the following inequality holds:

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

(5.14)

with $\delta =0.2785.$

5.2.2 Main Results

Based on the backstepping technique, a control design and stability analysis procedure is presented in this section. For $i=1,2,\ldots ,n-1,$ define a common virtual control function $\alpha _{i}$ as

$$\begin{aligned} \alpha _{i}=\frac{1}{g_{i,\min }}\left[ -\left( \lambda _{i}+\frac{3}{4}\right) z_{i}-\frac{1}{2a_{i}^{2}}z_{i}^{3}\hat{\theta }S_{i}^{T}S_{i}\right] , \end{aligned}$$

(5.15)

where $\lambda _{i}$, $a_{i}>0$ are design parameters,   $g_{i,\min } =\min \{g_{i,k}:k\in M\}$, $z_{i}$ is the new state variable after the coordinate transformation: $z_{i}=x_{i}-\alpha _{i-1}$, $\alpha _{0}=y_{d}$. $\hat{\theta }$ is an unknown constant that is specified later. $S_{i} =S_{i}(X_{i})$ is the basis function vector. $X_{i}=[\bar{x}_{i}^{T},\bar{\hat{\theta }}_{i},\bar{y}_{d}^{(i)}]^{T}$ with $\bar{x}_{i}=[x_{1} ,x_{2},\ldots ,x_{i}]^{T}$, $\,\bar{\hat{\theta }}_{i}=[\hat{\theta }_{1} ,\hat{\theta }_{2},\ldots ,\hat{\theta }_{i}]^{T}$, $\,\bar{y}_{d}^{(i)}=[y_{d} ,\dot{y}_{d},\ldots ,y_{d}^{(i)}]^{T}$. The z-system is obtained as

$$\begin{aligned} dz_{i}&= (g_{i,k}x_{i+1}+f_{i,k}-\mathscr {L}\alpha _{i-1})dt+\left( \psi _{i,k}-\sum _{j=0}^{i-1}\frac{\partial \alpha _{i-1}}{\partial x_{j}}\psi _{j,k}\right) ^{T}dw,\ 1\le i\le n-1\nonumber \\ dz_{n}&= (g_{n,k}v_{k}+f_{n,k}-\mathscr {L}\alpha _{n-1})dt+\left( \psi _{n,k}-\sum _{j=0}^{n-1}\frac{\partial \alpha _{n-1}}{\partial x_{j}}\psi _{j,k}\right) ^{T}dw, \end{aligned}$$

(5.16)

where the differential operator $\mathscr {L}$ is defined in Definition 5.1; $\mathscr {L}\alpha _{i-1}$ is given by:

$$\begin{aligned} \mathscr {L}\alpha _{i-1}&= \frac{\partial \alpha _{i-1}}{\partial \hat{\theta } }\dot{\hat{\theta }}+\sum _{s=1}^{i-1}\frac{\partial \alpha _{i-1}}{\partial x_{s}}(f_{s,k}+g_{s,k}x_{s+1})\nonumber \\&\quad +\sum _{s=0}^{i-1}\frac{\partial \alpha _{i-1}}{\partial y_{d}^{(s)}} y_{d}^{(s+1)}+\frac{1}{2}\sum _{p,q=1}^{i-1}\frac{\partial ^{2}\alpha _{i-1} }{\partial x_{p}\partial x_{q}}\psi _{p,k}^{T}\psi _{q,k}. \end{aligned}$$

(5.17)

Consider the following common stochastic Lyapunov function candidate

$$\begin{aligned} V=\sum _{i=1}^{n}\frac{1}{4}z_{i}^{4}+\frac{1}{2r_{1}}\tilde{\theta }^{2} +\frac{1}{2r_{2}}\tilde{\vartheta }^{2}, \end{aligned}$$

(5.18)

where $r_{1},\,r_{2}>0$ are design parameters; $\theta$ and $\vartheta$ are specified later. $\hat{\theta }$ and $\hat{\vartheta }$ stand for the estimations of $\theta$ and $\vartheta$, respectively; $\tilde{\theta } =\theta -\hat{\theta }$, $\tilde{\vartheta }=\vartheta -\hat{\vartheta }$.

Lemma 5.3

From the coordinate transformations $z_{i}=x_{i}-\alpha _{i-1}$, $i=1,2,\ldots ,n,$ $\alpha _{0}=y_{d}$, the following results hold,

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

(5.19)

where $\varphi _{i}(z_{i},\hat{\theta })=\frac{1}{g_{i,\min }}[(\lambda _{i} +\frac{3}{4})+\frac{1}{2a_{i}^{2}}z_{i}^{2}\hat{\theta }S_{i}^{T}S_{i}]+1,$ for $i=1,2,\ldots ,n-1,$ and $\varphi _{n}=1$.

Proof

From Assumption 5.1 and (5.15), one can get that

$$\begin{aligned} \left\| x\right\|&\le \sum _{i=1}^{n}\left| x_{i}\right| \\&\le \sum _{i=1}^{n}\left( \left| z_{i}\right| +\left| \alpha _{i-1}\right| \right) \\&\le \sum _{i=1}^{n}\left| z_{i}\right| +y_{d}+\sum _{i=1}^{n-1}\left( \frac{1}{g_{i,\min }}[(\lambda _{i}+\frac{3}{4})+\frac{1}{2a_{i}^{2}}z_{i}^{2}\hat{\theta }S_{i}^{T}S_{i}]\right) \left| z_{i}\right| \\&\le \sum _{i=1}^{n}\left| z_{i}\right| \varphi _{i}(z_{i} ,\hat{\theta })+d. \end{aligned}$$

The proof of Lemma 5.3 is completed here.    $\square$

The $\mathscr {L}V$ can be given by

$$\begin{aligned} \mathscr {L}V=&\sum _{i=1}^{n-1}\left\{ z_{i}^{3}\left( f_{i,k} +g_{i,k}x_{i+1}-\sum _{s=0}^{i-1}\frac{\partial \alpha _{i-1}}{\partial y_{d}^{(s)}}y_{d}^{(s+1)}\right. \right. -\frac{\partial \alpha _{i-1}}{\partial \hat{\theta }}\dot{\hat{\theta }}\\&-\sum _{s=1}^{i-1}\frac{\partial \alpha _{i-1}}{\partial x_{s}}(f_{s,k} +g_{s,k}x_{s+1})-\left. \frac{1}{2}\sum _{p,q=1}^{i-1}\frac{\partial ^{2}\alpha _{i-1} }{\partial x_{p}\partial x_{q}}\psi _{p,k}^{T}\psi _{q,k}\right) \\&+\left. \frac{3}{2}z_{i}^{2}\left\| \psi _{i,k}-\sum _{j=0}^{i-1} \frac{\partial \alpha _{i-1}}{\partial x_{j}}\psi _{j,k}\right\| ^{2}\right\} +z_{n}^{3}\left( f_{n,k}+g_{n,k}v_{k}-\sum _{s=0}^{n-1}\frac{\partial \alpha _{n-1}}{\partial y_{d}^{(s)}}y_{d}^{(s+1)}\right. \\&-\sum _{s=1}^{n-1}\frac{\partial \alpha _{n-1}}{\partial x_{s}}(f_{s,k} +g_{s,k}x_{s+1})-\frac{\partial \alpha _{n-1}}{\partial \hat{\theta }}\dot{\hat{\theta }}-\left. \frac{1}{2}\sum _{p,q=1}^{n-1}\frac{\partial ^{2}\alpha _{n-1} }{\partial x_{p}\partial x_{q}}\psi _{p,k}^{T}\psi _{q,k}\right) \\&-\frac{1}{r_{1}}\tilde{\theta }\dot{\hat{\theta }} +\frac{3}{2}z_{n}^{2}\left\| \psi _{n,k}-\sum _{j=0}^{n-1}\frac{\partial \alpha _{n-1}}{\partial x_{j}}\psi _{n,k}\right\| ^{2}-\frac{1}{r_{2}}\tilde{\vartheta }\dot{\hat{\vartheta }}\\ \end{aligned}$$

$$\begin{aligned} =&\sum _{i=1}^{n}\left\{ z_{i}^{3}\left( f_{i,k}-\sum _{s=1}^{i-1} \frac{\partial \alpha _{i-1}}{\partial x_{s}}f_{s,k}\right. \right. -\sum _{s=0}^{i-1}\frac{\partial \alpha _{i-1}}{\partial y_{d}^{(s)}} y_{d}^{(s+1)}-\sum _{s=1}^{i-1}\frac{\partial \alpha _{i-1}}{\partial x_{s} }g_{s,k}x_{s+1}\nonumber \\&-\left. \frac{\partial \alpha _{i-1}}{\partial \hat{\theta }}\dot{\hat{\theta }}-\frac{1}{2}\sum _{p,q=1}^{i-1}\frac{\partial ^{2}\alpha _{i-1}}{\partial x_{p}\partial x_{q}}\psi _{p,k}^{T}\psi _{q,k}\right) +\left. \frac{3}{2}z_{i}^{2}\left\| \psi _{i,k}-\sum _{j=0}^{i-1} \frac{\partial \alpha _{i-1}}{\partial x_{j}}\psi _{j,k}\right\| ^{2}\right\} \nonumber \\&-\frac{1}{r_{1}}\tilde{\theta }\dot{\hat{\theta }} -\frac{1}{r_{2}}\tilde{\vartheta }\dot{\hat{\vartheta }}+\sum _{i=1} ^{n-1}z_{i}^{3}g_{i,k}x_{i+1}+z_{n}^{3}g_{n,k}v_{k}. \end{aligned}$$

(5.20)

By resorting to Assumption 5.2 and Lemma 5.3, one has that

$$\begin{aligned}&z_{i}^{3}(f_{i,k}-\sum _{s=1}^{i-1}\frac{\partial \alpha _{i-1}}{\partial x_{s}}f_{s,k}(x))\nonumber \\ =&-z_{i}^{3}\sum _{s=1}^{i}\frac{\partial \alpha _{i-1}}{\partial x_{s}}f_{s,k}(x)\nonumber \\ \le&\frac{3}{4}nz_{i}^{4}\sum _{s=1}^{i}\left( \frac{\partial \alpha _{i-1} }{\partial x_{s}}\right) ^{\frac{4}{3}}+\sum _{s=1}^{i}\sum _{l=1}^{n}z_{l}^{4} \bar{\phi }_{s,k}^{4}(z_{l},\hat{\theta })+\left| z_{i}^{3}\right| \sum _{s=1}^{i}\left| \frac{\partial \alpha _{i-1}}{\partial x_{s}}\right| \phi _{s,k}((n+1)d), \end{aligned}$$

(5.21)

where $\bar{\phi }_{s,k}^{4}(z_{l}$, $\hat{\theta })=\frac{1}{4}(n+1)^{4} \varphi _{l}^{4}(z_{l}$, $\hat{\theta })h_{s,k}^{4}((n+1)|z_{l}|\varphi _{l}(z_{l}$, $\hat{\theta }))$, $\frac{\partial \alpha _{0}}{\partial x_{i}}=0$ and $\frac{\partial \alpha _{i-1}}{\partial x_{i}}=-1.$

Then, the following inequality can be obtained,

$$\begin{aligned} \frac{3}{2}&z_{i}^{2}\left\| \psi _{i,k}-\sum _{j=0}^{i-1}\frac{\partial \alpha _{i-1}}{\partial x_{j}}\psi _{j,k}\right\| ^{2}\nonumber \\&\le \frac{9}{8}i^{2}(n+1)^{2}nz_{i}^{4} +\sum _{l=1}^{n}z_{l}^{4}\bar{\rho }_{i,k}^{4}(z_{l},\hat{\theta }) +\sum _{j=1}^{i-1}\sum _{l=1}^{n}z_{l}^{4}\bar{\rho }_{j,k}^{4}(z_{l} ,\hat{\theta })\nonumber \\&\quad +\frac{9}{8}i^{2}(n+1)^{2}nz_{i}^{4}\sum _{j=1}^{i-1}(\frac{\partial \alpha _{i-1}}{\partial x_{j}})^{4} +\frac{9}{8}i^{2}(n+1)^{2}z_{i}^{4}l_{ii}^{-2}\rho _{i,k}^{4} ((n+1)d)\nonumber \\&\quad +\sum _{j=1}^{i}l_{ij}^{2} +\frac{9}{8}i^{2}(n+1)^{2}z_{i}^{4}\sum _{j=1}^{i-1}(\frac{\partial \alpha _{i-1}}{\partial x_{j}})^{4}l_{ij}^{-2}\rho _{j,k}^{4}((n+1)d), \end{aligned}$$

(5.22)

where $l_{ij}$ is a positive constant, and $\frac{\partial \alpha _{0}}{\partial x_{j}}=0$ because $\alpha _{0}=y_{d}$, and

$$\begin{aligned}&- \frac{1}{2}z_{i}^{3}\sum _{p,q=1}^{i-1}\frac{\partial ^{2}\alpha _{i-1} }{\partial x_{p}\partial x_{q}}\psi _{p,k}^{T}\psi _{q,k}\nonumber \\&\le (i-1)\sum _{s=1}^{i-1}\sum _{l=1}^{n}z_{l}^{4}\bar{\rho }_{s,k}^{4} (z_{l},\hat{\theta })\nonumber \\&\quad +\frac{1}{8}(n+1)^{2}nz_{i}^{6}\sum _{s=1}^{i-1}\sum _{j=1}^{i-1}\left( \frac{\partial ^{2}\alpha _{i-1}}{\partial x_{s}\partial x_{j}}\right) ^{2}\nonumber \\&\quad +\frac{1}{2}(n+1)\left| z_{i}^{3}\right| \sum _{s=1}^{i-1}\sum _{j=1}^{i-1}\left| \frac{\partial ^{2}\alpha _{i-1}}{\partial x_{s}\partial x_{j}}\right| \rho _{s,k}^{2}((n+1)d), \end{aligned}$$

(5.23)

where $\bar{\rho }_{s,k}^{4}(z_{l},\hat{\theta })=\frac{1}{2}(n+1)^{4} \varphi _{l}^{4}(z_{l},\hat{\theta })\eta _{s,k}^{4}((n+1)|z_{l}|\varphi _{l}(z_{l}$, $\hat{\theta }))$, $s=1,2,\ldots ,i-1.$

Substituting (5.21), (5.22) and (5.23) into (5.20) gives that

$$\begin{aligned}&\mathscr {L}V\le \sum _{i=1}^{n}\frac{3}{4}nz_{i}^{4}\sum _{s=1}^{i} \left( \frac{\partial \alpha _{i-1}}{\partial x_{s}}\right) ^{\frac{4}{3}} +\sum _{i=1}^{n}\sum _{s=1}^{i}\sum _{l=1}^{n}z_{l}^{4}\bar{\phi }_{s,k} ^{4}(z_{l},\hat{\theta })\nonumber \\&+\sum _{i=1}^{n}\left| z_{i}^{3}\right| \sum _{s=1}^{i}\left| \frac{\partial \alpha _{i-1}}{\partial x_{s}}\right| \phi _{s,k} ((n+1)d) +\sum _{i=1}^{n}\sum _{s=1}^{i-1}\sum _{l=1}^{n}(i-1)z_{l}^{4}\bar{\rho } _{s,k}^{4}(z_{l},\hat{\theta })\nonumber \\&+\sum _{i=1}^{n}\sum _{s=1}^{i-1}\sum _{j=1}^{i-1}\frac{1}{8}(n+1)^{2} nz_{i}^{6}\left( \frac{\partial ^{2}\alpha _{i-1}}{\partial x_{s}\partial x_{j}}\right) ^{2}\nonumber \\&+\sum _{i=1}^{n}\sum _{s=1}^{i-1}\sum _{j=1}^{i-1}\frac{1}{2}(n+1)\left| z_{i}^{3}\right| \rho _{s,k}^{2}((n+1)d)\left| \frac{\partial ^{2} \alpha _{i-1}}{\partial x_{s}\partial x_{j}}\right| \nonumber \\&+\sum _{i=1}^{n}\left\{ \frac{9}{8}i^{2}(n+1)^{2}nz_{i}^{4}+\sum _{l=1} ^{n}z_{l}^{4}\bar{\rho }_{i,k}^{4}(z_{l},\hat{\theta })\right. +\sum _{j=1}^{i-1}\sum _{l=1}^{n}z_{l}^{4}\bar{\rho }_{j,k}^{4}(z_{l} ,\hat{\theta })\nonumber \\&+\frac{9}{8}i^{2}(n+1)^{2}nz_{i}^{4}\sum _{j=1}^{i-1}\left( \frac{\partial \alpha _{i-1}}{\partial x_{j}}\right) ^{4} +\frac{9}{8}i^{2}(n+1)^{2}z_{i}^{4}l_{ii}^{-2}\rho _{i,k}^{4}((n+1)d)+\sum _{j=1}^{i}l_{ij}^{2}\nonumber \\&+\left. \frac{9}{8}i^{2}(n+1)^{2}z_{i}^{4}\sum _{j=1}^{i-1}\left( \frac{\partial \alpha _{i-1}}{\partial x_{j}}\right) ^{4}l_{ij}^{-2}\rho _{j,k} ^{4}((n+1)d)\right\} \nonumber \\&+\sum _{i=1}^{n}z_{i}^{3}\left( -\sum _{s=0}^{i-1}\frac{\partial \alpha _{i-1}}{\partial y_{d}^{(s)}}y_{d}^{(s+1)}-\frac{\partial \alpha _{i-1} }{\partial \hat{\theta }}\dot{\hat{\theta }}\right. -\left. \sum _{s=1}^{i-1}\frac{\partial \alpha _{i-1}}{\partial x_{s}} g_{s,k}x_{s+1}\right) \nonumber \\&+\sum _{i=1}^{n-1}z_{i}^{3}g_{i,k}x_{i+1}+z_{n}^{3}g_{n,k}v_{k}-\frac{1}{r_{1}}\tilde{\theta }\dot{\hat{\theta }}-\frac{1}{r_{2}}\tilde{\vartheta }\dot{\hat{\vartheta }}. \end{aligned}$$

(5.24)

Define $U_{i,k}$ as

$$\begin{aligned} U_{i,k}=&\sum _{s=1}^{i}|\frac{\partial \alpha _{i-1}}{\partial x_{s}} |\phi _{s,k}((n+1)d)\nonumber \\&+\frac{1}{2}(n+1)\sum _{s=1}^{i-1}\sum _{j=1}^{i-1}\left| \frac{\partial ^{2}\alpha _{i-1}}{\partial x_{s}\partial x_{j}}\right| \rho _{s,k}^{2}((n+1)d). \end{aligned}$$

(5.25)

By using Lemma 5.2 one has

$$\begin{aligned} \left| z_{i}^{3}\right| U_{i,k}\le z_{i}^{3}U_{i,k}\tanh (\frac{z_{i}^{3}U_{i,k}}{\varpi _{i,k}})+\delta \varpi _{i,k}. \end{aligned}$$

(5.26)

Note that

$$\begin{aligned} \sum _{i=1}^{n-1}z_{i}^{3}g_{i,k}x_{i+1}=\sum _{i=1}^{n-1}z_{i}^{3} g_{i,k}z_{i+1}+\sum _{i=1}^{n-1}g_{i,k}z_{i}^{3}\alpha _{i}, \end{aligned}$$

(5.27)

Therefore, one has

$$\begin{aligned}&\sum _{i=1}^{n}\sum _{s=1}^{i}\sum _{l=1}^{n}z_{l}^{4}\bar{\phi }_{s,k} ^{4}(z_{l},\hat{\theta }) =\sum _{i=1}^{n}z_{i}^{4}\sum _{s=1}^{n}(n-s+1)\bar{\phi }_{s,k}^{4} (z_{i},\hat{\theta }),\\&\sum _{i=1}^{n}(i-1)\sum _{s=1}^{i-1}\sum _{l=1}^{n}z_{l}^{4}\bar{\rho } _{s,k}^{4}(z_{l},\hat{\theta }) =\sum _{i=1}^{n}z_{i}^{4}\sum _{s=1}^{n-1}(n-s)(i-1)\bar{\rho }_{s,k} ^{4}(z_{i},\hat{\theta }),\\&\sum _{i=1}^{n}\sum _{j=1}^{i}\sum _{l=1}^{n}z_{l}^{4}\bar{\rho }_{j,k} ^{4}(z_{l},\hat{\theta }) =\sum _{i=1}^{n}z_{i}^{4}\sum _{j=1}^{n}(n-j+1)\bar{\rho }_{j,k}^{4} (z_{i},\hat{\theta }). \end{aligned}$$

For any $i=1,2,\ldots ,n$ and $k\in M$, define $\bar{f}_{i,k}$ as

$$\begin{aligned} \bar{f}_{i,k}&= \frac{3}{4}nz_{i}\sum _{s=1}^{i}\left( \frac{\partial \alpha _{i-1} }{\partial x_{s}}\right) ^{\frac{4}{3}} +z_{i}\sum _{s=1}^{n}(n-s+1)\bar{\phi }_{s,k}^{4}(z_{i},\hat{\theta })\\&\quad +z_{i}\sum _{s=1}^{n-1}(n-s)(i-1)\bar{\rho }_{s,k}^{4}(z_{i},\hat{\theta }) +\sum _{s=1}^{i-1}\sum _{j=1}^{i-1}\frac{1}{8}(n+1)^{2}nz_{i}^{3}\left( \frac{\partial ^{2}\alpha _{i-1}}{\partial x_{s}\partial x_{j}}\right) ^{2}\\&\quad +\frac{9}{8}i^{2}(n+1)^{2}z_{i}\sum _{j=1}^{i-1}\left( \frac{\partial \alpha _{i-1} }{\partial x_{j}}\right) ^{4}l_{ij}^{-2}\rho _{j,k}^{4}((n+1)d)\\&\quad +z_{i}\sum _{j=1}^{n}(n-j+1)\bar{\rho }_{j,k}^{4}(z_{i},\hat{\theta }) +\frac{9}{8}i^{2}(n+1)^{2}nz_{i}\\&\quad +\frac{9}{8}i^{2}(n+1)^{2}z_{i}l_{ii}^{-2}\rho _{i,k}^{4}((n+1)d) +\frac{9}{8}i^{2}(n+1)^{2}nz_{i}\sum _{j=1}^{i-1}\left( \frac{\partial \alpha _{i-1}}{\partial x_{j}}\right) ^{4}\\&\quad -\sum _{s=0}^{i-1}\frac{\partial \alpha _{i-1}}{\partial y_{d}^{(s)}} y_{d}^{(s+1)}-\frac{\partial \alpha _{i-1}}{\partial \hat{\theta }}\dot{\hat{\theta }}-\sum _{s=1}^{i-1}\frac{\partial \alpha _{i-1}}{\partial x_{s} }g_{s,k}x_{s+1} \end{aligned}$$

$$\begin{aligned}&+U_{i,k}\tanh (\frac{z_{i}^{3}U_{i,k}}{\varpi _{i,k}})+g_{i,k}z_{i+1}, \end{aligned}$$

(5.28)

with $z_{n+1}=0.$

Substituting (5.6) and (5.26)–(5.28) into (5.24) yields that

$$\begin{aligned} \mathscr {L}V\le&\sum _{i=1}^{n-1}z_{i}^{3}(\bar{f}_{i,k}+g_{i,k}\alpha _{i})+z_{n}^{3}\bar{f}_{n,k} +z_{n}^{3}g_{n,k}(u_{c_{k}}+\eta _{k}^{^{\prime }}-u_{\phi _{k}}+\iota _{k})\nonumber \\&-\frac{1}{r_{1}}\tilde{\theta }\dot{\hat{\theta }}-\frac{1}{r_{2}} \tilde{\vartheta }\dot{\hat{\vartheta }}+\sum _{i=1}^{n}\left( \delta \varpi _{i,k}+\sum _{j=1}^{i}l_{ij}^{2}\right) . \end{aligned}$$

(5.29)

By exploring the neural networks’ approximation capability and Young’s inequality, one can get the following inequalities.

$$\begin{aligned} z_{i}^{3}\bar{f}_{i,k}&= z_{i}^{3}W_{i,k}^{T}S_{i,k}+z_{i}^{3} \varepsilon _{i,k}\nonumber \\&\le \frac{1}{2a_{i}^{2}}z_{i}^{6}\left\| W_{i,k}\right\| ^{2} S_{i,k}^{T}S_{i,k}+\frac{a_{i}^{2}}{2}+\frac{3}{4}z_{n}^{4} +\frac{\bar{\varepsilon }_{i,k}^{4}}{4},\nonumber \\&\le \frac{1}{2a_{i}^{2}}z_{i}^{6}\theta _{i}S_{i}^{T}S_{i}+\frac{a_{i}^{2} }{2}+\frac{3}{4}z_{i}^{4}+\frac{\bar{\varepsilon }_{i}^{4}}{4},\end{aligned}$$

(5.30)

$$\begin{aligned} z_{n}^{3}(\eta _{k}^{^{\prime }}+\iota _{k})&= z_{n}^{3}W_{\eta ,k}^{T}S_{\eta ,k}+z_{n}^{3}(\varepsilon _{\eta ,k} +\iota _{k})\nonumber \\&\le \frac{1}{2a_{\eta }^{2}}z_{n}^{6}\vartheta _{\eta }S_{\eta }^{T}S_{\eta }+\frac{a_{\eta }^{2}}{2}+\frac{3z_{n}^{4}+\bar{\varepsilon }_{\eta }^{4}}{4}, \end{aligned}$$

(5.31)

where $\theta _{i,k}=\left\| W_{i,k}\right\| ^{2}$, $\vartheta _{\eta ,k}=\left\| W_{\eta ,k}\right\| ^{2}$, $\theta _{i}=\max \{\theta _{i,k}:k\in M\}$, $\vartheta _{\eta }=\max \{\vartheta _{\eta ,k}:k\in M\}$, $\left| \varepsilon _{i,k}\right| \le \bar{\varepsilon }_{i}$, $\left| \varepsilon _{\eta ,k}+\iota _{k}\right| \le \bar{\varepsilon }_{\eta }$.

Substituting (5.30) and (5.31) into (5.29) gives

$$\begin{aligned} \mathscr {L}V&\le \sum _{i=1}^{n-1}z_{i}^{3}\left( \frac{z_{i}^{3}\theta _{i} }{2a_{i}^{2}}S_{i}^{T}S_{i}+g_{i,k}\alpha _{i}\right) +z_{n}^{3}\left( \frac{z_{n}^{3}\theta _{n}}{2a_{n}^{2}}S_{n}^{T}S_{n} +g_{n,k}u_{c_{k}}\right) \nonumber \\&+z_{n}^{3}g_{n,k}\left( \frac{1}{2a_{\eta }^{2}}z_{n}^{3}\theta _{\eta }S_{\eta } ^{T}S_{\eta }-u_{\phi _{k}}\right) +g_{n,k}\left( \frac{a_{\eta }^{2}}{2}+\frac{3}{4}z_{n}^{4}+\frac{\bar{\varepsilon }_{\eta }^{4}}{4}\right) \nonumber \\&+\sum _{i=1}^{n}\left( \frac{2a_{i}^{2}+3z_{i}^{4}+\bar{\varepsilon }_{i}^{4}}{4}\right) -\frac{1}{r_{1}}\tilde{\theta }\dot{\hat{\theta }} -\frac{1}{r_{2}}\tilde{\vartheta }\dot{\hat{\vartheta }}+\sum _{i=1} ^{n}\left( \delta \varpi _{i}+\sum _{j=1}^{i}l_{ij}^{2}\right) , \end{aligned}$$

(5.32)

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

Design the virtual control function as

$$\begin{aligned} \alpha _{i}=\frac{1}{g_{i,\min }}\left[ -\left( \lambda _{i}+\frac{3}{4}\right) z_{i}-\frac{1}{2a_{i}^{2}}z_{i}^{3}\hat{\theta }S_{i}^{T}S_{i}\right] , \end{aligned}$$

(5.33)

where $\hat{\theta }=\sum _{i=1}^{n}\hat{\theta }_{i}$ is the estimation of $\theta$; $\,\lambda _{i}>0$ is a design parameter.

The actual actuator input is given as

$$\begin{aligned} u_{k}=u_{c_{k}}-u_{\phi _{k}}, \end{aligned}$$

(5.34)

where

$$\begin{aligned} u_{c_{k}}&=\frac{1}{g_{n,k}}\left[ -\left( \lambda _{n}+\frac{3}{4}\right) z_{n}-\frac{1}{2a_{n}^{2}}z_{n}^{3}\hat{\theta }S_{n}^{T}S_{n}\right] ,\end{aligned}$$

(5.35)

$$\begin{aligned} u_{\phi _{k}}&=\left( \lambda _{\eta }+\frac{3}{4}\right) z_{n}+\frac{g_{n,\max }}{2a_{\eta }^{2}g_{n,k}}z_{n}^{3}\hat{\vartheta }S_{\eta }^{T}S_{\eta }, \end{aligned}$$

(5.36)

$\lambda _{n}$, $\lambda _{\eta }$, $a_{n}$, $a_{\eta }>0$ are design parameters, $g_{n,\max }=\max \{g_{n,k}$, $k\in M\}$, $g_{n,\min } =\min \{g_{n,k}$, $k\in M\}$, $\hat{\vartheta }$ is the estimation of $\vartheta$.

The adaptive laws can be designed as

$$\begin{aligned} \dot{\hat{\theta }}&=\sum _{i=1}^{n}\frac{r_{1}}{2a_{i,\min }^{2}}z_{i} ^{6}S_{i}^{T}S_{i}-\beta _{1}\hat{\theta },\end{aligned}$$

(5.37)

$$\begin{aligned} \dot{\hat{\vartheta }}&=\frac{g_{n,\max }r_{2}}{2a_{\eta ,\min }^{2}}z_{n} ^{6}S_{\eta }^{T}S_{\eta }-\beta _{2}\hat{\vartheta }. \end{aligned}$$

(5.38)

Then, one can get from (5.32)–(5.38) that

$$\begin{aligned} \mathscr {L}V\le&-\sum _{i=1}^{n}\lambda _{i}z_{i}^{4}-\lambda _{\eta } z_{n}^{4}+g_{n,k}\left( \frac{a_{\eta }^{2}}{2}+\frac{\bar{\varepsilon }_{\eta }^{4} }{4}\right) +\sum _{i=1}^{n}\left( \frac{a_{i}^{2}}{2}+\frac{\bar{\varepsilon }_{i}^{4}}{4}\right) +\frac{\beta _{1}}{r_{1}}\tilde{\theta }\hat{\theta }\nonumber \\&+\frac{\beta _{2}}{r_{2}}\tilde{\vartheta }\hat{\vartheta }+\sum _{i=1} ^{n}\left( \delta \varpi _{i}+\sum _{j=1}^{i}l_{ij}^{2}\right) . \end{aligned}$$

(5.39)

It is clear that

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

(5.40)

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

(5.41)

Combining (5.39) with (5.40) and (5.41) yields that

$$\begin{aligned} \mathscr {L}V\le&-\sum _{i=1}^{n}\lambda _{i}z_{i}^{4}-\frac{\beta _{1} }{2r_{1}}\tilde{\theta }^{2}-\frac{\beta _{2}}{2r_{2}}\tilde{\vartheta } ^{2} +g_{n,k}\left( \frac{a_{\eta }^{2}}{2}+\frac{\bar{\varepsilon }_{\eta }^{4}}{4}\right) +\sum _{i=1}^{n}\left( \frac{a_{i}^{2}}{2}+\frac{\bar{\varepsilon }_{i}^{4}}{4}\right) \nonumber \\&+\sum _{i=1}^{n}\left( \delta \varpi _{i}+\sum _{j=1}^{i}l_{ij}^{2}\right) +\frac{\beta _{1}\theta ^{2}}{2r_{1}}+\frac{\beta _{2}\vartheta ^{2}}{2r_{2} }\nonumber \\ \le&-p_{0}V+q_{0}, \end{aligned}$$

(5.42)

where $\lambda _{n}:=\lambda _{n}+\lambda _{\eta },\,p_{0}=\min \{4\lambda _{i},\beta _{1},\beta _{2}:1\le i\le n\},\,q_{0}=\sum _{i=1}^{n}(\frac{a_{i}^{2}}{2}+\frac{\bar{\varepsilon }_{i}^{4}}{4})+\sum _{i=1}^{n}\left( \delta \varpi _{i}+\sum _{j=1}^{i}l_{ij}^{2}\right) +\frac{\beta _{1}\theta ^{2} }{2r_{1}}+\frac{\beta _{2}\vartheta ^{2}}{2r_{2}}+g_{n,k}(\frac{a_{\eta }^{2}}{2}+\frac{\bar{\varepsilon }_{\eta }^{4}}{4}).$

By using Lemma 5.1, we have

$$\begin{aligned} \frac{dE[V(t)]}{dt}\le -p_{0}E[V(t)]+q_{0}; \end{aligned}$$

(5.43)

Then, the following inequality holds

$$\begin{aligned} 0\le E[V(t)]\le V(0)e^{-p_{0}t}+\frac{q_{0}}{p_{0}}, \end{aligned}$$

(5.44)

where $V(0)=\sum _{j=1}^{n}\frac{z_{j}^{2}(0)}{4}+\frac{1}{2r_{1}}\tilde{\theta }(0)^{2}+\frac{1}{2r_{2}}\tilde{\vartheta }(0)^{2}$. Equation (5.44) implies that all the signals in the closed-loop system are bounded in probability. In particular, we have

$$\begin{aligned} E[|z_{i}|^{4}]\le \frac{4q_{0}}{p_{0}},\quad t\rightarrow \infty . \end{aligned}$$

(5.45)

Now, we are ready to provide our main result in the following theorem.

Theorem 5.1

Consider the closed-loop system (5.1) with unknown nonsymmetric actuator dead-zone (5.2). Suppose that for $1\le i\le n,$ $k\in M,$ the packaged unknown functions $\bar{f}_{i,k}$ can be approximated by neural networks in the sense that the approximation error $\varepsilon _{i,k}$ are bounded. Under the state feedback controller (5.34) and the adaptive laws (5.37), (5.38), the following statements hold,

(i) All the signals of the closed-loop z-system (5.17) are SGUUB in the fourth moment and

$$ P\left\{ \lim _{t\rightarrow \infty }\sum _{i=1}^{n}E[|z_{i}|^{4}]\le \frac{4q_{0}}{p_{0}}\right\} =1. $$

(ii) The output y of the closed-loop system (5.1) can be almost surely regulated to a small neighborhood of the target signal.

Proof

It is not difficult to complete the proof by using the above developments.    $\square$

5.2.3 Simulation Results

In this section, an example about the control of a ship manoeuvring system are used to illustrate the effectiveness of the obtained results.

The ship maneuvering system can be described by the following Norrbin nonlinear model [15].

$$ T_{\sigma (v_{s})}\dot{h}+h+\alpha _{\sigma (v_{s})}h^{3}=K_{\sigma (v_{s})} \delta +\phi _{\sigma (v_{s})}^{T}(\psi ,h,\delta )w, $$

where $T_{\sigma (v_{s})}$ is the time constant, $h=\dot{\psi }$ denotes the yaw rate, $\psi$ stands for the heading angle, $\alpha _{\sigma (v_{s})}$ is the Norrbin coefficient, $K_{\sigma (v_{s})}$ represents the rudder gain, $\delta$ is the rudder angle and w stands for an r-dimensional independent standard Brownian motion, $\phi _{\sigma (v_{s})}(\psi ,h,\delta ):\mathbb {R}^{3}\rightarrow \mathbb {R}^{3\times r}$ is an unknown function, and $\sigma (v_{s})$ is the switching signal that satisfies:

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

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

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

$$ T_{E,\sigma (v_{s})}\dot{\delta }+\delta =K_{E,\sigma (v_{s})}\delta _{E,\sigma (v_{s})}, $$

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

Let $x_{1}$ $=\psi$, $x_{2}$ $=h$, $x_{3}$ $=\delta$, $v_{\sigma (v_{s})}=\delta _{E,\sigma (v_{s})}$; we can get the following switched nonlinear system model with actuator dead-zone to describe the dynamic behavior of the ship with low speed, middle speed and high speed, respectively.

$$\begin{aligned} dx_{1}&=x_{2}dt,\\ dx_{2}&=(f_{\sigma (v_{s})}+b_{\sigma (v_{s})}x_{3})dt+\phi _{\sigma (v_{s} )}^{T}d\omega ,\\ dx_{3}&=\left( -\frac{1}{T_{E,\sigma (v_{s})}}x_{3}+\frac{K_{E,\sigma (t)} }{T_{E,\sigma (v_{s})}}v_{\sigma (v_{s})}\right) dt,\\ v_{\sigma (v_{s})}&=D(u_{\sigma (v_{s})}) \end{aligned}$$

where $f_{\sigma (v_{s})}=-\frac{1}{T_{\sigma (v_{s})}}x_{2}-\frac{\tau _{\sigma (v_{s})}}{T_{\sigma (v_{s})}}x_{2}^{3},$ $b_{\sigma (v_{s})} =\frac{K_{\sigma (v_{s})}}{T_{\sigma (v_{s})}}$.

The vessel data comes from a ship that has a length overall of 160.9 m. $v_{L}=3.7$ m/s, $K_{1}=32$ s$^{\text {-1}}$, $T_{1}=30$ s, $\tau _{1}=40$ s$^{\text {2}}$, $T_{E,1}=4$ s, $K_{E,1}=2$; $v_{M}=7.5$ m/s, $K_{2}=11.4$ s$^{\text {-1}}$, $T_{2}=63.69$ s, $\tau _{2}=30$ s$^{\text {2}}$, $T_{E,2}=2.5$ s, $K_{E,2}=1$; $v_{T}=15.3$ m/s, $K_{3}=5.1$ s$^{\text {-1}}$, $T_{3}=80.47$ s, $\tau _{3}=25$ s$^{\text {2}}$, $T_{E,3}=1$ s, $K_{E,3}=0.72$. The initial conditions are $x_{1}(0)=2$,$\,x_{2}(0)=-0.05$, $x_{3}(0)=0.03$, $\hat{\theta }(0)=10$, $\hat{\vartheta }(0)=1$. We construct the basis function vectors $S_{1}$, $S_{2}$, $S_{3}$ and $S_{\eta }$ using 11, 15, 21 and 48 nodes, the centers $\mu _{1}$, $\mu _{2}$, $\mu _{3}$, $\mu _{\eta }$ evenly spaced on $[-1.5,$ $4.5]\times [-3,\,4]\times [-10,\,8]$, $[-5,\,4]\times [-30,\,20]\times [-0.5,\,5.5]$, $[-5.5,\,8]\times [-12,\,25]\times [-0.1,\,2]$ and $[-10,\,2]\times [-60,\,2]\times [-0.2,\,10.5]$, and the widths $\zeta _{1}=1.2$, $\zeta _{2}=2.2$, $\zeta _{3}=2$, $\zeta _{\eta }=1.8$. The design parameters are $a_{1}=a_{2}=a_{3}=a_{\eta }=10$, $r_{1}=2,$ $r_{2}=10$, $\beta _{1}=0.5$, $\beta _{2}=0.1$, $\lambda _{1}=\lambda _{2}=\lambda _{3}=5$, and $\lambda _{\eta }=3$. The desired trajectory is $y_{d}=10\sin 0.08t$.

According to Theorem 5.1, the adaptive laws $\dot{\hat{\theta }}$, $\dot{\hat{\vartheta }}$ and the control laws $u_{c_{k}},\,u_{\phi _{k}}$ are chosen, respectively, as

$$\begin{aligned} \dot{\hat{\theta }}&=\sum _{i=1}^{3}0.01z_{i}^{6}S_{i}^{T}S_{i} -0.5\hat{\theta },\\ \,\dot{\hat{\vartheta }}&=0.036z_{3}^{6}S_{\eta }^{T}S_{\eta }-0.1\hat{\vartheta },\\ u_{c_{k}}&=\frac{1}{g_{3,k}}[-5.75z_{3}-0.005z_{3}^{3}\hat{\theta }S_{3} ^{T}S_{3}],\\ u_{\phi _{k}}&=3.75z_{3}+\frac{0.00057}{g_{3,k}}z_{3}^{3}\hat{\vartheta }S_{\eta }^{T}S_{\eta }, \end{aligned}$$

where $u_{k}=u_{c_{k}}-u_{\phi _{k}},\,z_{1}=x_{1}-y_{d},\,z_{2}=x_{2} -\alpha _{1},\,z_{3}=x_{3}-\alpha _{2}$ and $\alpha _{1},$ $\alpha _{2}$ are given by

$$\begin{aligned} \alpha _{1}&=-5.75z_{1}-0.005z_{1}^{3}\hat{\theta }S_{1}^{T}S_{1},\\ \alpha _{2}&=-92z_{2}-0.08z_{2}^{3}\hat{\theta }S_{2}^{T}S_{2}. \end{aligned}$$

In order to give the simulation results, we assume that

$$ v_{k}=D(u_{k})=\left\{ \begin{array} [c]{ll} 10(u_{k}-50), &{} u_{k}\ge 50\\ 0, &{} -60<u_{k}<50\\ 20(u_{k}+60), &{} u_{k}\le -60 \end{array} \right. $$

and $\phi _{1}=0.5x_{1}\sin x_{2}x_{3}$, $\phi _{2}=0.25x_{1}^{2}x_{2}\cos x_{2}$, $\phi _{3}=0.1x_{1}x_{3}$. The simulation results are shown in Figs. 5.1–5.4. Figure 5.1 depicts the responses of system output $\psi$ and target signal $y_{d}$. Figure 5.2 shows the trajectories of adaptive laws. Figure 5.3 demonstrates the responses of $D(u_{c_{k}})$ (without dead-zone compensation controller) and $D(u_{c_{k}}-u_{\phi _{k}})$ (with dead-zone compensation controller) and Fig. 5.4 illustrates the evolution of the switching signal. From Fig. 5.1, it can be seen that the output $\psi$ can track the target signal $y_{d}$ within a small bounded error. On the other hand, Fig. 5.3 proves that the dead-zone nonlinearity can be compensated by $u_{\phi _{k}}$.

Fig. 5.1

Tracking performance

Fig. 5.2

The responses of adaptive laws

Fig. 5.3

The responses of $D(u_{c_{k}}-u_{\phi _{k}})$ and $D(u_{c_{k}})$

Fig. 5.4

The response of switching signal

5.2.4 Conclusions

The tracking control problem for a class of stochastic switched nonlinear systems under arbitrary switchings has been investigated, where the unknown nonsymmetric actuator dead-zone is taken into account. A state feedback controller is designed for the systems under consideration. It is shown that the target signal can be almost surely tracked by the system output within a small bounded error, and the tracking error is SGUUB in $4^{th}$ moment.

5.3 Adaptive Neural Control for Switched Stochastic High-Order Uncertain Nonlinear Systems with SISS Inverse Dynamic

5.3.1 Problem Formulation and Preliminaries

Here, we consider the following stochastic switched high-order nonlinear systems with SISS inverse dynamic,

$$\begin{aligned} d\zeta&=f_{0,\sigma (t)}\left( \zeta ,x_{1}\right) dt+\psi _{0,\sigma (t)}^{T}\left( \zeta ,x_{1}\right) d\omega ,\nonumber \\ dx_{i}&=\left( g_{i,\sigma (t)}(\zeta ,x)x_{i+1}^{p_{i}}+f_{i,\sigma (t)}(\zeta ,x)\right) dt+\psi _{i,\sigma (t)}^{T}\left( \zeta ,x\right) d\omega ,\text { }i=1,2,\ldots ,n-1,\nonumber \\ dx_{n}&=\left( g_{n,\sigma (t)}(\zeta ,x)u_{\sigma (t)}^{p_{n}} +f_{n,\sigma (t)}(\zeta ,x)\right) dt+\psi _{n,\sigma (t)}^{T}\left( \zeta ,x\right) d\omega ,\nonumber \\ y&=x_{1}, \end{aligned}$$

(5.46)

where $\zeta \in \mathbb {R}^{r}$ are immeasurable stochastic inverse dynamics; $x=(x_{1},x_{2},\ldots ,x_{n})^{T}\in \mathbb {R}^{n}$ and $y\in R$ are the system state and output, respectively; $p_{i}$ is a positive odd integer and $\omega$ is an m-dimensional standard Wiener process defined on the complete probability space $\left( \varOmega , \mathscr {F} ,\left\{ \mathscr {F} _{t}\right\} _{t\ge 0},P\right)$ with $\varOmega$ being a sample space, $\mathscr {F}$ being a $\sigma$-field, $\left\{ \mathscr {F} _{t}\right\} _{t\ge 0}$ being a filtration, and P being a probability measure; $\sigma (t):[0,+\infty )\rightarrow M=\{1,2,\ldots ,m\}$ is the switching signal; $u_{k}\in R$ is the control input of the k-th subsystem; $f_{0,k}:\mathbb {R}^{r}\times R\rightarrow \mathbb {R}^{r}$, $\psi _{0,k}:\mathbb {R}^{r}\times \mathbb {R}\rightarrow \mathbb {R}^{m\times r}$; For any $i=1,2\ldots ,n$ and $k=1,2,\ldots ,m$, $f_{i,k}:\mathbb {R}^{r}\times \mathbb {R}^{n}\rightarrow \mathbb {R}$, $\psi _{i,k}:\mathbb {R}^{r}\times \mathbb {R}^{n}\rightarrow \mathbb {R}^{m}$ are unknown nonlinear functions assumed to be locally Lipschitz with $f_{i,k}(0)=0$, $\psi _{i,k}(0)=0$, and $g_{i,k}:R^{r}\times \mathbb {R}^{n}\rightarrow R$ is a strictly either positive or negative known function.

Remark 5.2

System (5.46) reduces to the well-known normal form when $p_{i}=1$, $\zeta =0$ and $m=1$. In the case that $p_{i}>1$, $\zeta =0$ and $m=1$, the Jacobian linearization of the system is neither controllable nor feedback linearizable. This makes the control design very challenging. To solve this problem, Lin and Qian [16] proposed a fruitful deterministic technique: adding a power integrator. Subsequently, many excellent results are proposed based on the adding a power integrator technique, see, e.g., [17–19] and the references therein.

Definition 5.3

For any given $V(x_{i},t)\in \mathscr {C}^{2,1}$ associated with system (5.46), define the differential operator $\mathscr {L}$ as follows,

$$\begin{aligned} \mathscr {L}V=\frac{\partial V}{\partial t}+\frac{\partial V}{\partial x_{i} }F_{i,k}+\frac{1}{2}Tr\left\{ \psi _{i,k}^{T}\frac{\partial ^{2}V}{\partial x_{i}^{2}}\psi _{i,k}\right\} , \end{aligned}$$

(5.47)

where $F_{i,k}=g_{i,\sigma (t)}(\zeta ,x)x_{i+1}^{p_{i}}+f_{i,\sigma (t)} (\zeta ,x)$.

Assumption 5.3

The sign and the upper bound of function $g_{i,k}$ for $1\le i\le n$ and $k\in M,$ are known, and without loss of generality, it is assumed that

$$ 0<\underline{d}_{i}\le g_{i,k}(\zeta ,x)\le \bar{d}_{i}, $$

where $\underline{d}_{i}$ and $\bar{d}_{i}$ stand for the lower and upper bound values of $g_{i,k}(\zeta ,x)$, respectively.

Assumption 5.4

For $1\le i\le n$ and $k\in M,$ there exists a $\mathscr {C}^{2}$ function $V_{0}(\zeta )$, which is positive definite and proper, such that $\mathscr {L}V_{0}\le -\lambda _{0}\zeta ^{4}+\bar{\lambda } _{0}x_{1}^{p+3}$, where $\lambda _{0}$ and $\bar{\lambda }_{0}$ are positive constants.

Lemma 5.4

Let $p\in \mathbb {R}^{*}$ and $x,\,y$ be real-valued functions. There exists a constant $c>0$ such that

$$ \left| x^{p}-y^{p}\right| \le c\left| x-y\right| \left| (x-y)^{p-1}+y^{p-1}\right| . $$

Lemma 5.5

Suppose that there exists a $\mathscr {C}^{2,1}$ function $V(x,t):\mathbb {R}^{n}\times \mathbb {R}^{+}\rightarrow \mathbb {R}^{+}$, two constants $c_{1}>0$, $c_{2}>0$, and $\mathscr {K}_{\infty }$ functions $\bar{c}_{1}$, $\bar{c}_{2}$ such that

$$ \left\{ \begin{array} [c]{c} \bar{c}_{1}(\left| x\right| )\le V(x,t)\le \bar{c}_{2}(\left| x\right| )\\ \mathscr {L}V(x,t)\le -c_{1}V(x,t)+c_{2} \end{array} \right. $$

for all $x\in R^{n}$ and $t>t_{0}$. Then, there is an unique strong solution for each $x_{0}\in \mathbb {R}^{n}$ and it satisfies:

$$ E[V(x,t)]\le V(x_{0})e^{-c_{1}t}+\frac{c_{2}}{c_{1}},\,\forall t>t_{0}. $$

In the following control design procedure, radial basis function (RBF) neural networks are used to approximate a continuous real function f(X). For arbitrary $\varepsilon >0,$ there exists a neural network $W^{T}S(X)$ such that

$$ f(X)=W^{T}S(X)+\delta (X),\,\delta (X)\le \varepsilon , $$

where $X\in \varOmega _{X}\subset \mathbb {R}^{q}$ is the input vector with q dimension, $S(X)=[s_{1}(X),s_{2}(X),\ldots , s_{l}(X)]^{T}$ is the basis function vector, and $W=[w_{1},w_{2},\ldots ,w_{l}]^{T}$ is the ideal constant weight vector with $l>1$ being the number of the neural network nodes, and $s_{i}(X)$ are chosen as Gaussian functions; i.e., for $i=1,2,\ldots ,l$,

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

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.

Lemma 5.6

Consider the Gaussian RBF networks. Let $\rho :=\frac{1}{2}\min _{i\ne j}||\mu _{i}-\mu _{j}||$; then an upper bound of ||S(X)|| is taken as

$$ ||S(X)||\le \sum _{k=0}^{\infty }3q(k+2)^{q-1}e^{-2\rho ^{2}k^{2}/\zeta ^{2}}:=D. $$

It has been proven in [20] that the constant D in Lemma 5.6 is a limited value and is independent of the variable X.

5.3.2 Main Results

In the following, the adaptive tracking control design is carried out by using a standard backstepping procedure. Firstly, define $p=\max _{i=1,\ldots ,n}\{p_{i}\}$. The following lemma is also given.

Lemma 5.7

Suppose that the Lyapunov function

$$ V(\xi _{1},\ldots ,\xi _{n})=\sum _{i=1}^{n}\frac{\xi _{i}^{p-p_{i}+4}}{p-p_{i}+4} $$

is positive-definite and proper, satisfying

$$\begin{aligned} \mathscr {L}V\le -\sum _{i=1}^{n}\xi _{i}^{p+3}+\phi . \end{aligned}$$

(5.48)

Then, the following inequality holds

$$ \mathscr {L}V\le -a_{0}V+b_{0}, $$

where

$$ a_{0}=\min \left( \phi ^{(p_{i}-1)/(p+3)}\right) ,b_{0}=(n+1)\phi . $$

Proof

Let $a=\phi ^{1/(p+3)}$ and $b=\xi _{i}$. Then, by using Young’s inequality

$$\begin{aligned} a^{p_{i}-1}b^{p-p_{i}+4}&\le \frac{p_{i}-1}{p+3}a^{p+3}+\frac{p-p_{i} +4}{p+3}b^{p+3}\\&\le a^{p+3}+b^{p+3}, \end{aligned}$$

which implies that

$$\begin{aligned} -\xi _{i}^{p+3}\le -\phi ^{(p_{i}-1)/(p+3)}\xi _{i}^{p-p_{i}+4}+\phi . \end{aligned}$$

(5.49)

Substituting (5.49) into (5.48) yields that

$$ \mathscr {L}V\le -\sum _{i=1}^{n}\phi ^{(p_{i}-1)/(p+1)}\xi _{i}^{p-p_{i} +4}+(n+1)\phi . $$

The proof of Lemma 5.7 is completed here.    $\square$

Step 1: Define the variable $z_{1}=x_{1}$. Then, consider the following Lyapunov function candidate

$$ V_{1}=\frac{\zeta ^{4}}{4}+\frac{z_{1}^{p-p_{1}+4}}{p-p_{1}+4}. $$

It follows from (5.47) and Assumption 5.4 that

$$\begin{aligned} \mathscr {L}V_{1}=-\lambda _{0}\zeta ^{4}+\bar{\lambda }_{0}z_{1}^{p+3} +z_{1}^{p-p_{1}+3}\left( g_{1,k}x_{2}^{p_{1}}+f_{1,k}\right) +\frac{p-p_{1}+3}{2}\left\| \psi _{1,k}\right\| ^{2}z_{1}^{p-p_{1}+2}, \end{aligned}$$

(5.50)

where $f_{1,k}$ and $\left\| \psi _{1,k}\right\| ^{2}$ are unknown. Then, two neural networks $W_{1,k}S_{1}$ and $\varPhi _{1,k}P_{1,k}$ are used to approximate the unknown function $f_{1,k}$ and the norm $\left\| \psi _{1,k}\right\|$ such that for any given $\varepsilon _{1,k}>0$ and $\tau _{1,k}>0$,

$$\begin{aligned} f_{1,k}&=W_{1,k}^{T}S_{1,k}(X_{1})+\delta _{1,k}(X_{1}),\\ \left\| \psi _{1,k}\right\| ^{2}&=\varPhi _{1,k}^{T}P_{1,k}(X_{1} )+\bar{\delta }_{1,k}(X_{1}), \end{aligned}$$

where $X_{1}:=[\zeta ^{T},x^{T}]^{T}\in R^{r+n}$, $\left| \delta _{1,k}(X)\right| \le \varepsilon _{1,k}$, $\bar{\delta }_{1,k}(X_{1} )\le \tau _{1,k}$.

One can get from the Young’s inequality and Lemma 5.6 that

$$\begin{aligned}&z_{1}^{p-p_{1}+3}f_{1,k}\nonumber \\&=z_{1}^{p-p_{1}+3}(W_{1,k}^{T}S_{1,k}(X_{1})+\delta _{1,k}(X_{1} ))\nonumber \\&\le \frac{p-p_{1}+3}{p+3}l_{1}^{\frac{p+3}{p-p_{1}+3}}z_{1}^{p+3}\left\| W_{1,k}\right\| ^{\frac{p+3}{p-p_{1}+3}}\left\| S_{1,k}\right\| ^{\frac{p+3}{p-p_{1}+3}}+\frac{p_{1}}{p+3}l_{1}^{-\frac{p+3}{p_{1}} }\nonumber \\&\text { }+\frac{p-p_{1}+3}{p+3}\eta _{1}^{\frac{p+3}{p-p_{1}+3}}z_{1} ^{p+3}+\frac{p_{1}}{p+3}\eta _{1}^{-\frac{p+3}{p_{1}}}\varepsilon _{1,k} ^{\frac{p+3}{p_{1}}}\nonumber \\&\le l_{1}^{\frac{p+3}{p-p_{1}+3}}z_{1}^{p+3}\left\| W_{1,k}\right\| ^{\frac{p+3}{p-p_{1}+3}}\left\| S_{1,k}\right\| ^{\frac{p+3}{p-p_{1}+3} }+\eta _{1}^{\frac{p+3}{p-p_{1}+3}}z_{1}^{p+3}+l_{1}^{-\frac{p+3}{p_{1}}} +\eta _{1}^{-\frac{p+3}{p_{1}}}\varepsilon _{1,k}^{\frac{p+3}{p_{1}}}\nonumber \\&\le z_{1}^{p+3}\left( l_{1}^{\frac{p+3}{p-p_{1}+3}}\theta _{1}D_{1} ^{\frac{p+3}{p-p_{1}+3}}+\eta _{1}^{\frac{p+3}{p-p_{1}+3}}\right) +b_{1}, \end{aligned}$$

(5.51)

where $l_{1},\,\eta _{1}>0$ are design parameters; $\left\| S_{1}\right\| \le D_{1}$; $\theta _{1}:=\max \{\left\| W_{1,k}\right\| ^{(p+3)/(p-p_{1} +3)}:k\in M\}$; $b_{1}=l_{1}^{-(p+3)/p_{1}}+\eta _{1}^{-(p+3)/p_{1}} \varepsilon _{1}^{(p+3)/p_{1}}.$

Moreover, one has that

$$\begin{aligned}&\left\| \psi _{1,k}\right\| ^{2}z_{1}^{p-p_{1}+2}\nonumber \\&=z_{1}^{p-p_{1}+2}\left( \varPhi _{1,k}^{T}P_{1,k}(X_{1})+\bar{\delta } _{1,k}(X_{1})\right) \nonumber \\&\le z_{1}^{p-p_{1}+2}\varPhi _{1,k}^{T}P_{1,k}+z_{1}^{p-p_{1}+2}\bar{\delta }_{1,k}\nonumber \\&\le \frac{p-p_{1}+2}{p+3}\xi _{1}^{\frac{p+3}{p-p_{1}+2}}z_{1} ^{p+3}\left\| \varPhi _{1,k}\right\| ^{\frac{p+3}{p-p_{1}+2}}\left\| P_{1,k}\right\| ^{\frac{p+3}{p-p_{1}+2}}+\frac{p_{1}+1}{p+3}\xi _{1} ^{-\frac{p+3}{p_{1}+1}}\nonumber \\&\text { }+\frac{p-p_{1}+2}{p+3}m_{1}^{\frac{p+3}{p-p_{1}+2}}z_{1} ^{p+3}+\frac{p_{1}+1}{p+3}m_{1}^{-\frac{p+3}{p_{1}+1}}\tau _{1,k}^{\frac{p+3}{p_{1}+1}}\nonumber \\&\le z_{1}^{p+3}\left( \xi _{1}^{\frac{p+3}{p-p_{1}+2}}\varphi _{1} Q_{1}^{\frac{p+3}{p-p_{1}+2}}+m_{1}^{\frac{p+3}{p-p_{1}+2}}\right) +\bar{b}_{1}, \end{aligned}$$

(5.52)

where $\xi _{1}$, $\ m_{1}>0$ are design parameters; $\left\| P_{1}\right\| \le Q_{1}$; $\bar{b}_{1}=\xi _{1}^{-(p+3)/(p_{1}+1)}$ $+$ $m_{1}^{-(p+3)/(p_{1}+1)}\tau _{1}^{(p+3)/(p_{1}+1)}$; $\varphi _{1} =\max \{\left\| \varPhi _{1,k}\right\| ^{(p+3)/(p-p_{1}+2)}:k\in M\}$.

Substituting (5.51) and (5.52) into (5.50), yields that

$$\begin{aligned} \mathscr {L}V_{1}\le&-\lambda _{0}\zeta ^{4}+\bar{\lambda }_{0}z_{1} ^{p+3}+g_{1,k}z_{1}^{p-p_{1}+3}x_{2}^{p_{1}}+z_{1}^{p+3}\left( l_{1} ^{\frac{p+3}{p-p_{1}+3}}\theta _{1}D_{1}^{\frac{p+3}{p-p_{1}+3}}+\eta _{1}^{\frac{p+3}{p-p_{1}+3}}\right) \nonumber \\&+z_{1}^{p+3}\left( \frac{1}{2}(p-p_{1}+3)\xi _{1}^{\frac{p+3}{p-p_{1}+2} }\varphi _{1}Q_{1}^{\frac{p+3}{p-p_{1}+2}}+\frac{1}{2}(p-p_{1}+3)m_{1} ^{\frac{p+3}{p-p_{1}+2}}\right) +\tilde{b}_{1}, \end{aligned}$$

(5.53)

where $\tilde{b}_{1}:=b_{1}+0.5(p-p_{1}+3)\bar{b}_{1}.$

Then, the common virtual control function can be designed as

$$\begin{aligned} \alpha _{1}=&-z_{1}\left\{ \frac{1}{\underline{d}_{1}}\left( \bar{\lambda }_{1}+l_{1}^{\frac{p+3}{p-p_{1}+3}}\hat{\theta }_{1}D_{1}^{\frac{p+3}{p-p_{1}+3}}+\frac{1}{2}(p-p_{1}+3)\xi _{1}^{\frac{p+3}{p-p_{1}+2}}\hat{\varphi }_{1}Q_{1}^{\frac{p+3}{p-p_{1}+2}}\right. \right. \nonumber \\&+\left. \left. \eta _{1}^{\frac{p+3}{p-p_{1}+3}}+\frac{1}{2} (p-p_{1}+3)m_{1}^{\frac{p+3}{p-p_{1}+2}}\right) \right\} ^{\frac{1}{p_{1}} }\nonumber \\ =&-z_{1}\beta _{1}, \end{aligned}$$

(5.54)

where $\hat{\theta }_{1}$, $\hat{\varphi }_{1}$ are the estimations of $\theta _{1}$, $\varphi _{1}$ respectively; $\bar{\lambda }_{1}>1+\bar{\lambda }_{0}$ is a positive design parameter; $\underline{d}_{1}$ is defined in Assumption 5.3.

It follows from (5.53) and (5.54) that

$$\begin{aligned} \mathscr {L}V_{1}\le&-\lambda _{0}\zeta ^{4}-(\bar{\lambda }_{1}-\bar{\lambda }_{0})z_{1}^{p+3}+g_{1,k}z_{1}^{p-p_{1}+3}\left( x_{2}^{p_{1}}-\alpha _{1}^{p_{1}}\right) \nonumber \\&+l_{1}^{\frac{p+3}{p-p_{1}+3}}\tilde{\theta }_{1}D_{1}^{\frac{p+3}{p-p_{1}+3}}z_{1}^{p+3}+0.5(p-p_{1}+3)\xi _{1}^{\frac{p+3}{p-p_{1}+2}} \tilde{\varphi }_{1}Q_{1}^{\frac{p+3}{p-p_{1}+2}}z_{1}^{p+3}+\tilde{b} _{1}, \end{aligned}$$

(5.55)

where $\tilde{\theta }_{1}=\theta _{1}-\hat{\theta }_{1}$, $\tilde{\varphi } _{1}=\varphi _{1}-\hat{\varphi }_{1}.$

Step 2: Denote $z_{2}=x_{2}-\alpha _{1}$, and define $d\alpha _{1}$ as

$$\begin{aligned} d\alpha _{1}&=\left( \sum _{j=1}^{n}\frac{\partial \alpha _{1}}{\partial x_{j}}\left( g_{j,k}x_{j+1}^{p_{2}}+f_{1,k}\right) +\frac{\partial \alpha _{1}}{\partial \hat{\theta }_{1}}\dot{\hat{\theta }}_{1}+\frac{\partial \alpha _{1}}{\partial \hat{\varphi }_{1}}\dot{\hat{\varphi }}_{1}\right) dt+\frac{\partial \alpha _{1}}{\partial x_{1}}\psi _{1,k}^{T}d\omega \nonumber \\&=\bar{a}_{1}dt+\frac{\partial \alpha _{1}}{\partial x_{1}}\psi _{1}^{T} d\omega , \end{aligned}$$

(5.56)

where $\dot{\hat{\theta }}_{1}$ and$\ \dot{\hat{\varphi }}_{1}$ will be specified later, $x_{n+1}:=u$ will given at final step.

Choose the Lyapunov function as

$$ V_{2}=V_{1}+\frac{z_{2}^{p-p_{2}+4}}{p-p_{2}+4}. $$

Then, $\mathscr {L}V_{2}$ is given by

$$\begin{aligned} \mathscr {L}V_{2}\le&-\lambda _{0}\zeta ^{4}-(\bar{\lambda }_{1}-\bar{\lambda }_{0})z_{1}^{p+3}+g_{1,k}z_{1}^{p-p_{1}+3}\left( x_{2}^{p_{1}}-\alpha _{1}^{p_{1}}\right) \nonumber \\&+l_{1}^{\frac{p+3}{p-p_{1}+3}}\tilde{\theta }_{1}D_{1}^{\frac{p+3}{p-p_{1}+3}}z_{1}^{p+3}+0.5(p-p_{1}+3)\xi _{1}^{\frac{p+3}{p-p_{1}+2}} \tilde{\varphi }_{1}Q_{1}^{\frac{p+3}{p-p_{1}+2}}z_{1}^{p+3}+\tilde{b} _{1}\nonumber \\&+z_{2}^{p-p_{2}+3}\left( g_{2,k}x_{3}^{p_{2}}+f_{2,k}-\bar{a}_{1}\right) +\frac{p-p_{2}+3}{2}\left\| \psi _{2,k}-\frac{\partial \alpha _{1}}{\partial x_{1}}\psi _{1,k}\right\| ^{2}z_{2}^{p-p_{2}+2}. \end{aligned}$$

(5.57)

By using Lemma 5.4 and Young’s inequality, one can obtain that

$$\begin{aligned}&\left| g_{1,k}z_{1}^{p-p_{1}+3}(x_{2}^{p_{1}}-\alpha _{1}^{p_{1} })\right| \nonumber \\&\le c_{1}\bar{d}_{1}\left| z_{1}^{p-p_{1}+3}\right| \left| z_{2}\right| \left| z_{2}^{p_{1}-1}+(z_{1}\beta _{1})^{p_{1} -1}\right| \nonumber \\&\le c_{1}\bar{d}_{1}\frac{p-p_{1}+3}{p+3}z_{1}^{p+3}+c_{1}\bar{d}_{1} \frac{p_{1}}{p+3}z_{2}^{p+3}+c_{1}\bar{d}_{1}\frac{p+2}{p+3}z_{1} ^{p+3}\nonumber \\&\quad +c_{1}\bar{d}_{1}\frac{1}{p+3}z_{2}^{p+3}\beta _{1}^{(p_{1} -1)(p+3)}\nonumber \\&\le z_{1}^{p+3}+z_{2}^{p+3}\left( 1+\beta _{1}^{(p_{1}-1)(p+3)}\right) \nonumber \\&=z_{1}^{p+3}+z_{2}^{p+3}\bar{\beta }_{1}, \end{aligned}$$

(5.58)

where $\bar{\beta }_{1}=1+\beta _{1}^{(p_{1}-1)(p+1)}$; $c_{1}$ is chosen as $1/2\bar{d}_{1}$; $\bar{d}_{1}$ is defined in Assumption 5.3.

Substituting (5.58) into (5.57), gives that

$$\begin{aligned} \mathscr {L}V_{2}&\le -\lambda _{0}\zeta ^{4}-(\bar{\lambda }_{1}-\bar{\lambda }_{0}-1)z_{1}^{p+3}+z_{2}^{p+3}\bar{\beta }_{1}+l_{1}^{\frac{p+3}{p-p_{1}+3} }\tilde{\theta }_{1}D_{1}^{\frac{p+3}{p-p_{1}+3}}z_{1}^{p+3}\nonumber \\&\quad +\frac{1}{2}(p-p_{1}+3)\xi _{1}^{\frac{p+3}{p-p_{1}+2}}\tilde{\varphi }_{1}Q_{1}^{\frac{2p+6}{p-p_{1}+2}}z_{1}^{p+3}+\tilde{b}_{1}+z_{2}^{p-p_{2} +3}\left( g_{2,k}x_{3}^{p_{2}}+\bar{f}_{2,k}\right) \nonumber \\&\quad +\frac{1}{2}\left( p-p_{2}+3\right) \bar{\psi }_{2,k}z_{2} ^{p-p_{2}+2}, \end{aligned}$$

(5.59)

where $\bar{f}_{2,k}=f_{2,k}-\bar{\alpha }_{1}$, $\bar{\psi }_{2,k}:=$ $\left\| \psi _{2,k}-\frac{\partial \alpha _{1}}{\partial x_{1}}\psi _{1,k}\right\| ^{2}$. Then, neural networks $W_{2,k}^{T}S_{2,k}(X_{2})$ and $\varPhi _{2,k}^{T}P_{2,k}(X_{2})$ are used to approximate the unknown functions $\bar{f}_{2,k}$ and $\bar{\psi }_{2,k}$ such that for any given $\varepsilon _{2,k}>0$ and $\tau _{2,k}>0$,

$$\begin{aligned} \bar{f}_{2,k}&=W_{2,k}^{T}S_{2,k}(X_{2})+\delta _{2,k}(X_{1}),\\ \bar{\psi }_{2,k}&=\varPhi _{2,k}^{T}P_{2,k}(X_{2})+\bar{\delta }_{2,k}(X_{1}), \end{aligned}$$

where $X_{2}:=[\zeta ^{T},x^{T},\hat{\theta }_{1},\hat{\varphi }_{1}]^{T}\in R^{r+n+2}$, $\left| \delta _{2,k}(X_{2})\right| \le \varepsilon _{2,k}$, $\bar{\delta }_{2,k}(X_{2})\le \tau _{2,k}$.

Similar to the procedure in (5.51), one can obtain that

$$\begin{aligned}&z_{2}^{p-p_{2}+3}f_{2,k}\nonumber \\&=z_{2}^{p-p_{2}+3}(W_{2,k}^{T}S_{2,k}(X_{2})+\delta _{2,k}(X_{2} ))\nonumber \\&\le \frac{p-p_{2}+3}{p+3}l_{2}^{\frac{p+3}{p-p_{2}+3}}z_{2}^{p+3}\left\| W_{2,k}\right\| ^{\frac{p+3}{p-p_{2}+3}}\left\| S_{2,k}\right\| ^{\frac{p+3}{p-p_{2}+3}}+\frac{p_{2}}{p+3}l_{2}^{-\frac{p+3}{p_{2}} }\nonumber \\&\quad +\frac{p-p_{2}+3}{p+3}\eta _{2}^{\frac{p+3}{p-p_{2}+3}}z_{2} ^{p+3}+\frac{p_{2}}{p+3}\eta _{2}^{-\frac{p+3}{p_{2}}}\varepsilon _{2,k} ^{\frac{p+3}{p_{2}}}\nonumber \\&\le l_{2}^{\frac{p+3}{p-p_{2}+3}}z_{2}^{p+3}\left\| W_{2,k}\right\| ^{\frac{p+3}{p-p_{2}+3}}\left\| S_{2,k}\right\| ^{\frac{p+3}{p-p_{2}+3} }+\eta _{2}^{\frac{p+3}{p-p_{2}+3}}z_{2}^{p+3}+l_{2}^{-\frac{p+3}{p_{2}}} +\eta _{2}^{-\frac{p+3}{p_{2}}}\varepsilon _{2,k}^{\frac{p+3}{p_{2}}}\nonumber \\&\le z_{2}^{p+3}\left( l_{2}^{\frac{p+3}{p-p_{2}+3}}\theta _{2}D_{2} ^{\frac{p+3}{p-p_{2}+3}}+\eta _{2}^{\frac{p+3}{p-p_{2}+3}}\right) +b_{2}, \end{aligned}$$

(5.60)

where $l_{2},\,\eta _{2}>0$ are design parameters, $\theta _{2}:=\max \{\left\| W_{2,k}\right\| ^{(p+3)/(p-p_{2}+3)}:k\in M\}$, $b_{2} =l_{2}^{-(p+3)/p_{2}}+\eta _{2}^{-(p+3)/p_{2}}\varepsilon _{2}^{(p+3)/p_{2}}.$

Using a similar way to (5.52), one gets that

$$\begin{aligned}&\bar{\psi }_{2,k}z_{2}^{p-p_{2}+2}\nonumber \\&=z_{2}^{p-p_{2}+2}\left( \varPhi _{2,k}^{T}P_{2,k}(X_{2})+\bar{\delta } _{2,k}(X_{2})\right) \nonumber \\&\le z_{2}^{p-p_{2}+2}\varPhi _{2,k}^{T}P_{2,k}+z_{2}^{p-p_{2}+2}\bar{\delta }_{2,k}\nonumber \\&\le \frac{p-p_{2}+2}{p+3}\xi _{2}^{\frac{p+3}{p-p_{2}+2}}z_{2} ^{p+3}\left\| \varPhi _{2,k}\right\| ^{\frac{p+3}{p-p_{2}+2}}\left\| P_{2,k}\right\| ^{\frac{p+3}{p-p_{2}+2}}+\frac{p_{2}+1}{p+3}\xi _{2} ^{-\frac{p+3}{p_{2}+1}}\nonumber \\&\quad +\frac{p-p_{2}+2}{p+3}m_{2}^{\frac{p+3}{p-p_{2}+2}}z_{2}^{p+3} +\frac{p_{2}+1}{p+3}m_{2}^{-\frac{p+3}{p_{2}+1}}\tau _{2,k}^{\frac{p+3}{p_{2}+1}}\nonumber \\&\le z_{2}^{p+3}\left( \xi _{2}^{\frac{p+3}{p-p_{2}+2}}\varphi _{2} Q_{2}^{\frac{p+3}{p-p_{2}+2}}+m_{2}^{\frac{p+3}{p-p_{2}+2}}\right) +\bar{b}_{2}, \end{aligned}$$

(5.61)

where $\xi _{2}$, $m_{2}>0$ are design parameters; $\bar{b}_{2}=\xi _{2} ^{-\frac{p+3}{p_{2}+1}}+m_{2}^{-\frac{p+3}{p_{2}+1}}\tau _{2}^{\frac{p+3}{p_{2}+1}}$; $\varphi _{2}=\max \{ \left\| \varPhi _{2,k}\right\| ^{\frac{p+3}{p-p_{2}+2}}:k\in M\}$; $P_{2}(X_{2})$ and $\tau _{2}(X_{2})$ represent the basis function vector and the estimation error of $\varphi _{2}$.

Design the common virtual control function as

$$\begin{aligned} \alpha _{2}&=-z_{2}\left\{ \frac{1}{\underline{d}_{2}}\left( \bar{\beta }_{1}+\lambda _{2}+l_{2}^{\frac{p+3}{p-p_{2}+3}}\hat{\theta }_{2}D_{2} ^{\frac{p+3}{p-p_{2}+3}}+\frac{1}{2}(p-p_{2}+3)\xi _{2}^{\frac{p+3}{p-p_{2}+2} }\hat{\varphi }_{2}Q_{2}^{\frac{p+3}{p-p_{2}+2}}\right. \right. \nonumber \\&\left. \left. \quad +\eta _{2}^{\frac{p+3}{p-p_{2}+3}}+\frac{1}{2} (p-p_{2}+3)m_{2}^{\frac{p+3}{p-p_{2}+2}}\right) \right\} ^{\frac{1}{p_{2}} }\nonumber \\&=-z_{2}\beta _{2}, \end{aligned}$$

(5.62)

where $\hat{\theta }_{2}$, $\hat{\varphi }_{2}$ are the estimation of $\theta _{2}$ and $\varphi _{2}$ respectively; $\lambda _{2}>1$ is a design parameter; $\underline{d}_{2}$ is defined in Assumption 5.3.

By substituting (5.60)–(5.62) into (5.59), one has

$$\begin{aligned}&\mathscr {L}V_{2}\\&\le -\lambda _{0}\zeta ^{4}-(\bar{\lambda }_{1}-\bar{\lambda }_{0}-1)z_{1}^{p+3}-\lambda _{2}z_{2}^{p+3}+g_{2,k}z_{2}^{p-p_{2}+3} (x_{3}^{p_{2}}-\alpha _{2}^{p_{2}})\\&\quad +\sum _{j=1}^{2}\left( l_{j}^{\frac{p+3}{p-p_{j}+3}}\tilde{\theta } _{j}D_{j}^{\frac{p+3}{p-p_{j}+3}}z_{j}^{p+3}+0.5(p-p_{j}+3)\xi _{j}^{\frac{p+3}{p-p_{j}+2}}\tilde{\varphi }_{j}Q_{j}^{\frac{p+3}{p-p_{j}+2}}z_{j} ^{p+3}+\tilde{b}_{j}\right) , \end{aligned}$$

where $\tilde{b}_{j}:=b_{j}+0.5(p-p_{j}+3)\bar{b}_{j}$, $\tilde{\theta } _{j}=\theta _{j}-\hat{\theta }_{j}$, $\tilde{\varphi }_{j}=\varphi _{j} -\hat{\varphi }_{j}$, $j=1,2$.

Step i: Suppose at step i $(3\le i\le n-1)$ that, there is a set of virtual control functions $\alpha _{3},\ldots ,\alpha _{n-1}$, defined by

$$\begin{aligned} \alpha _{i}=z_{i}\beta _{i},\,z_{i}=x_{i+1}-\alpha _{i} \end{aligned}$$

(5.63)

and assume that a set of unknown nonlinear functions $\bar{f}_{i,k}$ and $\bar{\psi }_{i,k}$ can be approximated by neural networks $W_{i,k}^{T} S_{i,k}(X_{i})$ and $\varPhi _{i,k}^{T}P_{i,k}(X_{i})$ for any given $\varepsilon _{i,k}>0$, $\tau _{i,k}>0.$

$$\begin{aligned} \bar{f}_{i,k}&=W_{i,k}^{T}S_{i,k}(X_{i})+\delta _{i,k},\,\left| \delta _{i,k} (X_{i})\right| \le \varepsilon _{i,k},\\ \bar{\psi }_{i,k}&=\varPhi _{i,k}^{T}P_{i,k}(X_{i})+\bar{\delta }_{i,k},\,\left| \bar{\delta }_{i,k}(X_{i})\right| \le \tau _{i,k}, \end{aligned}$$

where $X_{i}:=[\zeta ^{T},x^{T},\hat{\theta }_{1},\ldots ,\hat{\theta }_{i} ,\hat{\varphi }_{1},\ldots ,\hat{\varphi }_{i}]^{T}\in R^{r+n+2i}.$

A straightforward calculation gives that

$$\begin{aligned}&\mathscr {L}V_{i}\nonumber \\&\le -\lambda _{0}\zeta ^{4}-(\bar{\lambda }_{1}-\bar{\lambda }_{0}-1)z_{1}^{p+3}-\sum _{j=2}^{i-1}(\lambda _{j}-1)z_{j}^{p+3}\nonumber \\&-\lambda _{i}z_{i}^{p+3}+g_{i,k}z_{i}^{p-p_{i}+3}(x_{i+1}^{p_{i}}-\alpha _{i}^{p_{i} })\nonumber \\&+\sum _{j=1}^{i}\left( l_{j}^{\frac{p+3}{p-p_{j}+3}}\tilde{\theta } _{j}\left\| S_{j}\right\| ^{\frac{p+3}{p-p_{j}+3}}z_{j}^{p+3} +0.5(p-p_{j}+3)\xi _{j}^{\frac{p+3}{p-p_{j}+2}}\tilde{\varphi }_{j}\left\| P_{j}\right\| ^{\frac{p+3}{p-p_{j}+2}}z_{j}^{p+3}+\tilde{b}_{j}\right) , \end{aligned}$$

(5.64)

where $\tilde{\theta }_{j}=\theta _{j}-\hat{\theta }_{j}$, $\tilde{\varphi } _{j}=\varphi _{j}-\hat{\varphi }_{j}$; $[\hat{\theta }_{j}$, $\hat{\varphi }_{j} ]$ is the estimation of

$$\begin{aligned}{}[\theta _{j},\,\varphi _{j}]:&=\max \{\left\| W_{j,k}\right\| ^{\frac{p+3}{p-p_{j}+3}}, \left\| \varPhi _{j,k}\right\| ^{\frac{p+3}{p-p_{j}+2}}:k\in M\}; \\ \tilde{b}_{j}&=b_{j}+0.5(p-p_{j}+3)\bar{b}_{j},\\ b_{j}&=l_{j}^{-(p+3)/p_{j}}+\eta _{j}^{-(p+3)/p_{j}}\varepsilon _{j}^{(p+3)/p_{j}} \end{aligned}$$

and $\bar{b}_{j}=\xi _{j}^{-(p+3)/(p_{j}+1)}+m_{j}^{-(p+3)/(p_{j}+1)}\tau _{j}^{(p+3)/(p_{j}+1)}$.

Step n: Let $z_{n}=x_{n}-\alpha _{n-1},$ define $d\alpha _{n-1}$ as

$$\begin{aligned} d\alpha _{n-1}&=\left( \sum _{j=1}^{n}\frac{\partial \alpha _{n-1}}{\partial x_{j}}\left( g_{j,k}x_{j+1}^{p_{2}}+f_{j,k}\right) +\sum _{j=1}^{n-1} \frac{\partial \alpha _{n-1}}{\partial \hat{\theta }_{j}}\dot{\hat{\theta }} _{j}+\sum _{j=1}^{n-1}\frac{\partial \alpha _{n-1}}{\partial \hat{\varphi }_{j} }\dot{\hat{\varphi }}_{j}\right) dt\\&+\sum _{j=1}^{n-1}\frac{\partial \alpha _{n-1}}{\partial x_{j}}\psi _{j,k}^{T}d\omega \\&=\bar{a}_{n-1}dt+\sum _{j=1}^{n-1}\frac{\partial \alpha _{n-1}}{\partial x_{j}}\psi _{j,k}^{T}d\omega . \end{aligned}$$

where $x_{n+1}:=u$ is provided later.

We construct the Lyapunov function as

$$ V_{n}=V_{n-1}+\frac{z_{n}^{p-p_{n}+4}}{p-p_{n}+4}. $$

By using (5.64), $\mathscr {L}V_{n}$ is given by

$$\begin{aligned}&\mathscr {L}V_{n}\end{aligned}$$

(5.65)

$$\begin{aligned}&\le -\lambda _{0}\zeta ^{4}-(\bar{\lambda }_{1}-\bar{\lambda }_{0}-1)z_{1}^{p+3}-\sum _{j=2}^{n-2}(\lambda _{j}-1)z_{j}^{p+3}\end{aligned}$$

(5.66)

$$\begin{aligned}&-\lambda _{n-1}z_{n-1}^{p+3}+g_{n-1,k}z_{n-1}^{p-p_{n}+3}(x_{n}^{p_{n-1}}-\alpha _{n-1}^{p_{n-1}})\nonumber \\&+\sum _{j=1}^{n-1}\left( l_{j}^{\frac{p+3}{p-p_{j}+3}}\tilde{\theta } _{j}D_{j}^{\frac{p+3}{p-p_{j}+3}}z_{j}^{p+3}+0.5(p-p_{j}+3)\xi _{j}^{\frac{p+3}{p-p_{j}+2}}\tilde{\varphi }_{j}Q_{j}^{\frac{p+3}{p-p_{j}+2}}z_{j} ^{p+3}+\tilde{b}_{j}\right) \nonumber \\&+z_{n}^{p-p_{n}+3}\left( g_{n,k}u^{p_{n}}+\bar{f}_{n,k}\right) +\frac{p-p_{n}+3}{2}\bar{\psi }_{n,k}z_{n}^{p-p_{n}+2}, \end{aligned}$$

(5.67)

where $\bar{f}_{n,k}=f_{n,k}-\bar{\alpha }_{n-1}$, $\bar{\psi }_{n,k}=$ $\left\| \psi _{n,k}-\sum _{i=1}^{n-1}\frac{\partial \alpha _{n-1}}{\partial x_{i}}\psi _{i,k}\right\| ^{2}$. Then, neural networks $W_{n,k}^{T} S_{n,k}(X_{n})$ and $\varPhi _{n,k}^{T}P_{n,k}(X_{n})$ are used to approximate unknown functions $\bar{f}_{n,k}$ and $\bar{\psi }_{n,k}$ such that for any given $\varepsilon _{n,k}>0$ and $\tau _{n,k}>0,$

$$\begin{aligned} \bar{f}_{n,k}&=W_{n,k}^{T}S_{n,k}(X_{n})+\delta _{n,k}(X_{n}),\\ \bar{\psi }_{n,k}&=\varPhi _{n,k}^{T}P_{n,k}(X_{n})+\bar{\delta }_{n,k}(X_{n}), \end{aligned}$$

where $X_{n}:=[\zeta ^{T},x^{T},\hat{\theta }_{1},\ldots ,\hat{\theta }_{n} ,\hat{\varphi }_{1},\ldots ,\hat{\varphi }_{n}]^{T}\in R^{r+3n}$, $\left| \delta _{n,k}(X_{n})\right| \le \varepsilon _{n,k}$, $\bar{\delta } _{n,k}(X_{n})\le \tau _{n,k}$.

Similar to (5.60) and (5.61), one has

$$\begin{aligned} z_{n}^{p-p_{n}+3}\bar{f}_{n,k}&\le z_{n}^{p+3}\left( l_{n}^{\frac{p+3}{p-p_{n}+3}}\theta _{n}D_{n}^{\frac{p+3}{p-p_{n}+3}}+\eta _{n}^{\frac{p+3}{p-p_{n}+3}}\right) +b_{n},\nonumber \\ z_{n}^{p-p_{n}+3}\bar{\psi }_{n,k}&\le z_{n}^{p+3}\left( \xi _{n} ^{\frac{p+3}{p-p_{n}+2}}\varphi _{n}Q_{n}^{\frac{p+3}{p-p_{n}+2}}+m_{n} ^{\frac{p+3}{p-p_{n}+2}}\right) +\bar{b}_{n}, \end{aligned}$$

(5.68)

where $l_{n},\,\eta _{n},\,\xi _{n},\,m_{n}>0$ are design parameters; $\theta _{n}:=\max \{\left\| W_{n,k}\right\| ^{(p+3)/(p-p_{n}+3)}:k\in M\}$; $b_{n}=l_{n}^{-\frac{p+3}{p_{n}}}+\eta _{n}^{-\frac{p+3}{p_{n}} }\varepsilon _{n}^{\frac{p+3}{p_{n}}}$, $\varphi _{n}:=\max \{\left\| \varPhi _{n,k}\right\| ^{(p+3)/(p-p_{n}+2)}:k\in M\}$; $\bar{b}_{n}=\xi _{n}^{-(p+3)/(p_{n}+1)}+m_{n}^{-(p+3)/(p_{n}+1)}\tau _{n}^{(p+3)/(p_{n}+1)}$.

Furthermore, it is not hard to get that

$$\begin{aligned}&\left| g_{n-1,k}z_{n-1}^{p-p_{n}+3}(x_{n}^{p_{n-1}}-\alpha _{n-1}^{p_{n-1}})\right| \nonumber \\&\le c_{n-1}\bar{d}_{n-1}\left| z_{n-1}^{p-p_{n-1}+3}\right| \left| z_{n}\right| \left| z_{n}^{p_{n-1}-1}+(z_{n-1}\beta _{n-1})^{p_{n-1}-1}\right| \nonumber \\&\le c_{n-1}\bar{d}_{n-1}\frac{p-p_{n-1}+3}{p+3}z_{n-1}^{p+3}+c_{n-1} \bar{d}_{n-1}\frac{p_{n-1}}{p+3}z_{n}^{p+3}+c_{n-1}\bar{d}_{n-1}\frac{p+2}{p+3}z_{n-1}^{p+3}\nonumber \\&\quad +c_{n-1}\bar{d}_{n-1}\frac{1}{p+3}z_{n}^{p+3}\beta _{n-1} ^{(p_{n-1}-1)(p+3)}\nonumber \\&\le z_{n-1}^{p+3}+z_{n}^{p+3}\left( 1+\beta _{n-1}^{(p_{n-1} -1)(p+3)}\right) \nonumber \\&=z_{n-1}^{p+3}+z_{n}^{p+3}\bar{\beta }_{n-1}, \end{aligned}$$

(5.69)

where $\bar{\beta }_{n-1}=1+\beta _{n-1}^{(p_{n-1}-1)(p+1)}$, $c_{n-1}$ is chosen as $1/2\bar{d}_{n-1}$, and $\bar{d}_{n-1}$ is defined in Assumption 5.3.

Substituting (5.68) and (5.69) into (5.67), the following inequality can be obtained.

$$\begin{aligned}&\mathscr {L}V_{n}\end{aligned}$$

(5.70)

$$\begin{aligned}&\le -\lambda _{0}\zeta ^{4}-(\bar{\lambda }_{1}-\bar{\lambda }_{0}-1)z_{1}^{p+3}-\sum _{j=2}^{n-1}(\lambda _{j}-1)z_{j}^{p+3}+z_{n}^{p+3} \bar{\beta }_{n-1}\nonumber \\&+\sum _{j=1}^{n-1}\left( l_{j}^{\frac{p+3}{p-p_{j}+3}}\tilde{\theta } _{j}D_{j}^{\frac{p+3}{p-p_{j}+3}}z_{j}^{p+3}+0.5(p-p_{j}+3)\xi _{j}^{\frac{p+3}{p-p_{j}+2}}\tilde{\varphi }_{j}Q_{j}^{\frac{p+3}{p-p_{j}+2}}z_{j} ^{p+3}\right) +\sum _{j=1}^{n}\tilde{b}_{j}\nonumber \\&+z_{n}^{p+3}\left( l_{n}^{\frac{p+3}{p-p_{n}+3}}\theta _{n}D_{n} ^{\frac{p+3}{p-p_{n}+3}}+\eta _{n}^{\frac{p+3}{p-p_{n}+3}}+0.5(p-p_{n} +3)\xi _{n}^{\frac{p+3}{p-p_{n}+2}}\varphi _{n}Q_{n}^{\frac{p+3}{p-p_{n}+2} }\right. \nonumber \\&+\left. 0.5(p-p_{n}+3)m_{n}^{\frac{p+3}{p-p_{n}+2}}\right) +z_{n} ^{p-p_{n}+3}g_{n,k}u^{p_{n}}. \end{aligned}$$

(5.71)

Design the controller u as

$$\begin{aligned} u=&-z_{n}\left\{ \frac{1}{\underline{d}_{n}}\left( \lambda _{n}+\bar{\beta }_{n-1}+l_{n}^{\frac{p+3}{p-p_{n}+3}}\hat{\theta }_{n}D_{n}^{\frac{p+3}{p-p_{n}+3}}+\eta _{n}^{\frac{p+3}{p-p_{n}+3}}\right. \right. \nonumber \\&+\left. \left. 0.5(p-p_{n}+3)\xi _{n}^{\frac{p+3}{p-p_{n}+2}}\hat{\varphi }_{n}Q_{n}^{\frac{p+3}{p-p_{n}+2}}+0.5(p-p_{n}+3)m_{n}^{\frac{p+3}{p-p_{n}+2} }\right) \right\} ^{\frac{1}{p_{n}}}\nonumber \\ =&-z_{n}\beta _{n}, \end{aligned}$$

(5.72)

where $\hat{\theta }_{n}$ is the estimation of $\theta _{n};$ $\lambda _{n}>1$ is a positive design parameter; $d_{n}$ is defined in Assumption 5.3.

It follows from (5.71) and (5.72) that

$$\begin{aligned}&\mathscr {L}V_{n}\le \end{aligned}$$

(5.73)

$$\begin{aligned}&-\lambda _{0}\zeta ^{4}-\sum _{j=1}^{n}(\lambda _{j}-1)z_{j}^{p+3}\nonumber \\&+\sum _{j=1}^{n}\left( l_{j}^{\frac{p+3}{p-p_{j}+3}}\tilde{\theta }_{j} D_{j}^{\frac{p+3}{p-p_{j}+3}}z_{j}^{p+3}+0.5(p-p_{j}+3)\xi _{j}^{\frac{p+3}{p-p_{j}+2}}\tilde{\varphi }_{j}Q_{j}^{\frac{p+3}{p-p_{j}+2}}z_{j} ^{p+3}+\tilde{b}_{j}\right) \end{aligned}$$

(5.74)

where $\lambda _{1}:=\bar{\lambda }_{1}-\bar{\lambda }_{0}$.

Last Step: Choose the final Lyapunov function as

$$\begin{aligned} V=V_{n}+\sum _{j=1}^{n}\left( \frac{1}{2r_{j}}\tilde{\theta }_{j}^{2}+\frac{1}{2\bar{r}_{j}}\tilde{\varphi }_{j}^{2}\right) \end{aligned}$$

(5.75)

where $r_{j}$ is a positive design parameter.

$\mathscr {L}V$ is given by

$$\begin{aligned}&\mathscr {L}V\le \end{aligned}$$

(5.76)

$$\begin{aligned}&-\lambda _{0}\zeta ^{4}-\sum _{j=1}^{n}(\lambda _{j} -1)z_{j}^{p+3}-\sum _{j=1}^{n}\left( \frac{1}{r_{j}}\tilde{\theta }_{j} \dot{\hat{\theta }}_{j}+\frac{1}{\bar{r}_{j}}\tilde{\varphi }_{j}\dot{\hat{\varphi }}_{j}\right) \nonumber \\&+\sum _{j=1}^{n}\left( l_{j}^{\frac{p+3}{p-p_{j}+3}}\tilde{\theta }_{j} D_{j}^{\frac{p+3}{p-p_{j}+3}}z_{j}^{p+3}+0.5(p-p_{j}+3)\xi _{j}^{\frac{p+3}{p-p_{j}+2}}\tilde{\varphi }_{j}Q_{j}^{\frac{p+3}{p-p_{j}+2}}z_{j} ^{p+3}+\tilde{b}_{j}\right) . \end{aligned}$$

(5.77)

The adaptive laws are defined as the solutions to the following differential equations

$$\begin{aligned} \dot{\hat{\theta }}_{j}&=r_{j}l_{j}^{\frac{p+3}{p-p_{j}+3}}D_{j}^{\frac{p+3}{p-p_{j}+3}}z_{j}^{p+3}-B_{j}\hat{\theta }_{j},\nonumber \\ \dot{\hat{\varphi }}_{j}&=\frac{1}{2}(p-p_{j}+3)\bar{r}_{j}\xi _{j} ^{\frac{p+3}{p-p_{j}+2}}Q_{j}^{\frac{p+3}{p-p_{j}+2}}z_{j}^{p+3}-\bar{B} _{j}\hat{\varphi }_{j}, \end{aligned}$$

(5.78)

where $\,j=1,2,\ldots ,n,$ $B_{j},\,\bar{B}_{j}>0$ are design parameters.

This, together with (5.77), means that

$$\begin{aligned} \mathscr {L}V\le -\lambda _{0}\zeta ^{4}-\sum _{j=1}^{n}(\lambda _{j}-1)z_{j} ^{p+3}+\sum _{j=1}^{n}\frac{B_{j}\hat{\theta }_{j}\tilde{\theta }_{j}}{r_{j} }+\sum _{j=1}^{n}\frac{\bar{B}_{j}\hat{\varphi }_{j}\tilde{\varphi }_{j}}{\bar{r}_{j}}+\sum _{j=1}^{n}\tilde{b}_{j}. \end{aligned}$$

(5.79)

Notice that

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

(5.80)

By using (5.79), (5.80) and Lemma 5.7, one has

$$\begin{aligned} \mathscr {L}V\le&-\lambda _{0}\zeta ^{4}-\sum _{j=1}^{n}\left( (\lambda _{j}-1)z_{j}^{p+3}+\frac{B_{j}}{2r_{j}}\tilde{\theta }_{j}^{2}+\frac{\bar{B}_{j}}{2\bar{r}_{j}}\tilde{\varphi }_{j}^{2}\right) \\&+\sum _{j=1}^{n}\left( \frac{B_{j}}{2r_{j}}\theta _{j}^{2}+\frac{\bar{B} _{j}}{2\bar{r}_{j}}\varphi _{j}^{2}+\tilde{b}_{j}\right) \\ \le&-q_{0}V+q_{1}, \end{aligned}$$

where $q_{0}=\min \{(p-p_{i}+4)(\lambda _{j}-1)\phi ^{(p_{i}-1)/(p+3)}$, $B_{j}$, $\bar{B}_{j}$, $2\lambda _{0}:1\le j\le n\}$, $\phi =\sum _{j=1}^{n}(\frac{B_{j}}{2r_{j}}\theta _{j}^{2}+\frac{\bar{B}_{j} }{2\bar{r}_{j}}\varphi _{j}^{2}+\tilde{b}_{j})$, $q_{1}=(\lambda _{j} -1)(n+1)\phi$.

According to Lemma 5.5, we have that

$$\begin{aligned} E[V(x,t)]\le V(x_{0})e^{-q_{0}t}+\frac{q_{1}}{q_{0}},\,\forall t\ge 0, \end{aligned}$$

(5.81)

which indicates that all the signals in the closed-loop system are bounded. The design is completed here. Next, we address our main result.

Theorem 5.2

For $1\le i\le n,k\in M$ assume that all the unknown nonlinear functions $\bar{f}_{i,k}$ and $\bar{\psi }_{i,k}$ can be approximated by neural networks in the sense that the approximation errors are bounded, and all the initial values of $\hat{\theta }_{i}$ and $\hat{\varphi }_{i}$ satisfy $\hat{\theta }_{i}(0)\ge 0$ and $\hat{\varphi }_{i}(0)\ge 0$, respectively. Then, under the state feedback controller (5.72) and the adaptive laws (5.78), the equilibrium at the origin of the closed-loop system is boundedly stable in probability and

$$ P\left\{ \lim _{t\rightarrow \infty }\left( \frac{|\zeta |^{4}}{4}+\sum _{i=1}^{n}\frac{|z_{i}|^{p-p_{i}+4}}{p-p_{i}+4}\right) \le \frac{q_{1}}{q_{0}}\right\} =1. $$

Proof

It is not difficult to complete the proof by the above discussions, and thus we omit the proof here.

In the following, a corollary is given by using only two adaptive laws.

Corollary 5.1

For $1\le i\le n,k\in M$ assume that all the unknown nonlinear functions $\bar{f}_{i,k}$ and $\bar{\psi }_{i,k}$ can be approximated by neural networks in the sense that the approximation errors are bounded, and all the initial values of $\hat{\theta }_{i}$ and $\hat{\varphi }_{i}$ satisfy $\hat{\theta }_{i}(0)\ge 0$ and $\hat{\varphi }_{i}(0)\ge 0$, respectively. Consider the following controller and adaptive laws:

$$\begin{aligned} u=&-z_{n}\left\{ \frac{1}{\underline{d}_{n}}\left( \lambda _{n}+\bar{\beta }_{n-1}+l_{n}^{\frac{p+3}{p-p_{n}+3}}\hat{\theta }D_{n}^{\frac{p+3}{p-p_{n}+3}}+\eta _{n}^{\frac{p+3}{p-p_{n}+3}}\right. \right. \\&+\left. \left. \frac{1}{2}(p-p_{n}+3)\xi _{n}^{\frac{p+3}{p-p_{n}+2}} \hat{\varphi }Q_{n}^{\frac{p+3}{p-p_{n}+2}}+0.5(p-p_{n}+3)m_{n}^{\frac{p+3}{p-p_{n}+2}}\right) \right\} ^{\frac{1}{p_{n}}},\\ \dot{\hat{\theta }}=&\sum _{j=1}^{n}rl_{j}^{\frac{p+3}{p-p_{j}+3}} D_{j}^{\frac{p+3}{p-p_{j}+3}}z_{j}^{p+3}-B\hat{\theta },\\ \dot{\hat{\varphi }}=&\frac{1}{2}\sum _{j=1}^{n}(p-p_{j}+3)\bar{r}\xi _{j}^{\frac{p+3}{p-p_{j}+2}}Q_{j}^{\frac{p+3}{p-p_{j}+2}}z_{j}^{p+3}-\bar{B}\hat{\varphi }, \end{aligned}$$

where $\,\lambda _{n}>1$, $l_{j},\,\xi _{j},\,m_{n},\,\eta _{n},\,r\,,B\,,\bar{r}\,,\bar{B}>0$ are positive design parameters, $\hat{\theta }=\sum _{j=1} ^{n}\hat{\theta }_{j}$, $\hat{\varphi }=\sum _{j=1}^{n}\hat{\varphi }_{j}$. Then, the equilibrium at the origin of the closed-loop system is boundedly stable in probability and

$$ P\left\{ \lim _{t\rightarrow \infty }\left( \frac{|\zeta |^{4}}{4}+\sum _{i=1}^{n}\frac{|z_{i}|^{p-p_{i}+4}}{p-p_{i}+4}\right) \le \frac{q_{1}}{q_{0}}\right\} =1. $$

Proof

It should be pointed out that $\hat{\theta }\ge$ $\hat{\theta }_{j}\ge 0$, $\hat{\varphi }\ge \hat{\varphi }_{j}\ge 0$, $\,j=1,\ldots ,n$. Therefore, we can use $\hat{\theta }$ and $\hat{\varphi }$ instead of $\hat{\theta }_{j}$ and $\hat{\varphi }_{j}$ in (5.54), (5.62), (5.63) and (5.72). In (5.75), the parameters $\tilde{\theta }_{j}$ and $\tilde{\varphi }_{j}$ in Lyapunov function V should be rewritten as $\tilde{\theta }$ and $\tilde{\varphi }.$ The detailed proof is omitted here because it is similar to the one of Theorem 5.2.    $\square$

5.3.3 Simulation Results

An example with two controllers (multiple adaptive laws and two adaptive laws respectively) is presented in the following to demonstrate the effectiveness of our main results.

Consider the following switched stochastic high-order nonlinear systems with SISS inverse dynamics:

$$\begin{aligned}&\sum \nolimits _{1}=\left\{ \begin{array} [c]{l} d\zeta =f_{0,1}\left( \zeta ,x_{1}\right) dt+\psi _{0,1}^{T}\left( \zeta ,x_{1}\right) d\omega ,\\ dx_{1}=\left[ g_{1,1}(\zeta ,x_{1},x_{2})x_{2}^{p_{1}}+f_{1,1}(\zeta ,x_{1},x_{2})\right] dt+\psi _{1,1}^{T}\left( \zeta ,x_{1},x_{2}\right) d\omega ,\\ dx_{2}=\left[ g_{2,1}(\zeta ,x_{1},x_{2})u^{p_{2}}+f_{2,1}(\zeta ,x_{1} ,x_{2})\right] dt+\psi _{2,1}^{T}\left( \zeta ,x_{1},x_{2}\right) d\omega , \end{array} \right. \\&\sum \nolimits _{2}=\left\{ \begin{array} [c]{l} d\zeta =f_{0,2}\left( \zeta ,x_{1}\right) dt+\psi _{0,2}^{T}\left( \zeta ,x_{1}\right) d\omega ,\\ dx_{1}=\left[ g_{1,2}(\zeta ,x_{1},x_{2})x_{2}^{p_{1}}+f_{1,2}(\zeta ,x_{1},x_{2})\right] dt+\psi _{1,2}^{T}\left( \zeta ,x_{1},x_{2}\right) d\omega ,\\ dx_{2}=\left[ g_{2,2}(\zeta ,x_{1},x_{2})u^{p_{2}}+f_{2,2}(\zeta ,x_{1} ,x_{2})\right] dt+\psi _{2,2}^{T}\left( \zeta ,x_{1},x_{2}\right) d\omega , \end{array} \right. \end{aligned}$$

where $g_{1,1}$, $f_{1,1}$, $\psi _{1,1}$, $g_{2,1}$, $f_{2,1}$, $\psi _{2,1}$, $g_{1,2}$, $f_{1,2}$, $\psi _{1,2}$, $g_{2,2}$, $f_{2,2}$, and $\psi _{2,2}$ are all completely unknown functions; $p_{1}=3$, $p_{2}=5$. First, a controller under multiple adaptive laws is designed by Theorem 5.2. The initial conditions are $\zeta (0)=1$ $x_{1}(0)=0.5,\,x_{2}(0)=-0.5$ and $\hat{\theta }_{1}(0)=2$, $\hat{\theta }_{2}(0)=3.5$, $\hat{\varphi }_{1}(0)=3$, $\hat{\varphi }_{2}(0)=4$. The controller parameters are chosen as: $\lambda _{1}=\lambda _{2}=5$, $l_{2}=l_{2}=\eta _{1}=\eta _{2}=\xi _{1}=\xi _{2}=m_{1}=m_{2}=4$, $r_{1} =r_{2}=\bar{r}_{1}=\bar{r}_{2}=1$, $B_{1}=B_{2}=\bar{B}_{1}=\bar{B}_{2}=0.1$. We apply three nodes for each input dimension of $W_{1}^{T}S_{1}$, $W_{2} ^{T}S_{2}$, $\varPhi _{1}^{T}P_{1}$ and $\varPhi _{2}^{T}P_{2}$. Therefore, each of them contains 81 nodes with centers spaced evenly in the interval $[-0.5,$ $0.5]\times [-0.5,\,0.5]\times [-0.5,$ $0.5]\times [-0.5,$ 0.5], and the widths still being equal to 2.5. Second, a controller under two adaptive laws is designed by Corollary 5.1 with same conditions except ${\hat{\theta }}(0)=3$, ${\hat{\varphi }}(0)=4$, $r=1$, $B=0.1$.

Fig. 5.5

Responses of system states by using multiple adaptive laws

Fig. 5.6

Responses of the multiple adaptive laws

Fig. 5.7

Response of switching signal

Fig. 5.8

Responses of system states by using two adaptive laws

Fig. 5.9

Responses of two adaptive laws

Fig. 5.10

Response of switching signal

In order to give the simulation results, it is assumed that $f_{0,1} =-15\zeta +0.1x_{1}^{2}$, $\psi _{0,1}=\left( \zeta ^{2}+0.3x_{1}^{4}\right) ^{1/2}$, $g_{1,1}=\sin (x_{1}x_{2}+\zeta )+2$, $f_{1,1}=x_{1}x_{2}+\zeta$, $\psi _{1,1}=\sin (x_{1}x_{2}+\zeta )$, $g_{2,1}=\cos (x_{1}+x_{2}^{2}+\zeta )+2$, $f_{2,1}=x_{1}x_{2}^{2}+\zeta \sin \zeta$, $\,\psi _{2,1}=x_{1}\cos x_{2} +\zeta ^{2}$; $f_{0,2}=-13\zeta +0.3x_{1}^{2}$, $\psi _{0,2}=\left( 0.17\zeta ^{2}+0.13x_{1}^{4}\right) ^{1/2}$, $g_{1,2}=\sin (x_{1}^{2} +x_{2}+\zeta )+2$, $f_{1,2}=x_{1}^{2}x_{2}+\zeta ^{2}$, $\psi _{1,2}=\sin (x_{1}+x_{2})+\zeta ^{3}$, $g_{2,2}=\cos (x_{1}x_{2}^{2}+\zeta )+2$, $f_{2,2}=x_{1}x_{2}+\zeta \cos \zeta$, $\,\psi _{2,2}=x_{1}\sin (x_{1}x_{2} )+\zeta ^{2}.$

The simulation results based on Theorem 5.2 are shown in Figs. 5.5, 5.6 and 5.7, respectively. Figure 5.5 depicts the responses of system states. The trajectories of adaptive laws are shown in Figs. 5.6, and 5.7 describes the switching signal. From Fig. 5.5, it can be seen that all the system states eventually converge to a small neighborhood of the origin. The simulation results based on Corollary 5.1 are shown in Figs. 5.8, 5.9 and 5.10, respectively. It can be seen that all the system states eventually converge to a small neighborhood of the origin by using only two adaptive laws.

5.3.4 Conclusions

The adaptive neural control for a class of stochastic high-order switched nonlinear systems with SISS inverse dynamic is studied. An adaptive neural control algorithm is proposed. It can be shown that the equilibrium at the origin of the closed-loop system is BIBO stable in probability.