Robust Stabilization for Uncertain Non-Minimum Phase Switched Nonlinear System under Arbitrary Switchings

Abstract

This paper addresses the problem of stabilization of non-minimum phase switched nonlinear systems where the internal dynamics with symmetries or non-symmetries of each mode may be unstable. The authors initially build a stabilizing Lyapunov controller for each mode in order to stabilize its own unstable internal dynamics. The proposed approach is based on the exact input-output feedback linearization technique and the Lyapunov stability theory. The stability results for non-minimum phase switched nonlinear systems with arbitrary switching rules are then obtained using generalized Gronwall–Bellman inequalities. Finally, numerical examples are provided to demonstrate the efficacy of the achieved results.

1. Introduction

Due to the importance of switched nonlinear systems (SNLSs) in both theory and practice, they have attracted a lot of attention recently [1,2,3,4,5,6,7,8,9]. They are composed of several modes, each with its own subsystem and a switching signal that governs the switching behaviors. Switched systems correctly depict the structural hybrid aspects of today’s practical systems. Many physical systems have been employed as typical applications in practice, including electrical circuits [10,11], power electronics [12], PT symmetric devices [13,14], referring to Parity-Time symmetry, the car industry [15], network control [16,17], chemical processes [18,19], etc. Switched systems, on the other hand, have sparked a great deal of interest in the field of control due to its practical and theoretical applications.

In the last two decades, switching system stability has been an important basic research challenge. This task is exceedingly complicated to resolve because of the hybrid nature of how switched systems operate. Several methods, including the multiple Lyapunov function technique [20,21,22,23,24], the common Lyapunov function method, the switched Lyapunov function method, the mean residence time approach, and variants thereof, have been proposed to address stability and robust stabilization issues in SNLSs [25,26]. However, the problem of unstable subsystems remains difficult and requires further investigation. In this case, there are two aspects to consider: i. create a suitable switching law to stabilize a switched system in the absence of control input [25] or ii. create a controller as well as a switching law to stabilize a switched control system [25].

Despite the various results reported by previous studies, little effort has been made to stabilize nonlinear systems where there are unstable internal dynamics with symmetries or non-symmetries. Nonlinear systems with non-minimum phase switching nonlinear modes, in which each nonlinear mode might have non-minimum phase, have also garnered considerable attention. The study of the stability of non-minimum phase SNLSs applied to multi-agent systems was recommended in [27]. In previous work [28], we have investigated a control approach based on multi-diffeomorphism for the stabilization of a class of non-minimum phase SNLSs. Using an approximated non-minimum phase model, Oishi et al. [29] explored the output tracking of non-minimum phase switched nonlinear systems. In reference [30], H∞ control has been introduced for a class of non-minimum phase cascade SNLSs. As a result, [31] looks at the challenge of resilient stabilization in a type of SNLS with unknown dynamics, where every subsystem illustrates a non-minimum phase.

We are motivated to investigate the stability of a nonlinear system with uncertainty and non-minimum phase switching when switching is arbitrary. This paper’s significant contributions are as follows: (1) the creation of a special controller in each mode to make up for its own unstable internal dynamics, resulting in the autonomous stabilization of each mode. Consequently, each subsystem’s controller has excellent performance and is resistant to unpredictability. (2) Using modified Gronwall–Bellman inequalities, we provide stability results for the non-minimum phase switched nonlinear system with uncertainty under arbitrary switching procedures [32]. The remainder of the work is arranged as follow. Section 2 includes some preliminary background and problem formulation. Section 3 includes the essential results of our proposed work. Some necessary requirements to guarantee the stability of a non-minimum phase switched nonlinear system with uncertainty under arbitrary switching are proposed in Section 4. Simulation results concerning a non-minimum phase SNLS example is provided to show the effectiveness of the proposed technique are found in Section 5.

2. Background and Problem Formulation

Let’s look at the following SNLS with uncertain control dynamics:

$$\{\begin{array}{l}\dot{x}\hspace{0.17em}=\hspace{0.17em}{f}_{\sigma }\left(x\left(t\right),\hspace{0.17em}\theta \right)\hspace{0.17em}+\hspace{0.17em}{g}_{\sigma }\left(x\left(t\right),\hspace{0.17em}\theta \right)\hspace{0.17em}{u}_{\sigma }\\ y\hspace{0.17em}=\hspace{0.17em}h\left(x\right)\end{array}$$

(1)

where $x\hspace{0.17em}\left(t\right)\in \hspace{0.17em}{\Re }^n$ are accessible states. Let M = {1, 2, …, m}, with m standing for the number modes, and σ(t): [0,∞) → M denotes the switching signal. We suppose that σ(t) is a continuous piecewise constant function. $\forall \hspace{0.17em}i\hspace{0.17em}\in \hspace{0.17em}\mathsf{{\rm M}},\hspace{0.17em}\hspace{0.17em}{u}_i\hspace{0.17em}\in \hspace{0.17em}\Re$ is the input; $f_i\left(\hspace{0.17em}.\hspace{0.17em}\right)$ and $g_i\left(\hspace{0.17em}.\hspace{0.17em}\right)$ are smooth functions; the function $h\left(x\right)$ represents the output signal, and $\theta$ denotes uncertainties of the parameters of plant.

Assume that for any mode $i\hspace{0.17em}\in \mathsf{{\rm M}}=\left\{1,\hspace{0.17em}\hspace{0.17em}2,\hspace{0.17em}\hspace{0.17em}\dots ,\hspace{0.17em}m\right\}$, we could discover a function y_i and a partition $x\hspace{0.17em}=\hspace{0.17em}\left[\begin{array}{cc}{\xi }_i^T& {\eta }_i^T\end{array}\right]{\hspace{0.17em}}^T$ where ${\xi }_i\hspace{0.17em}\in \hspace{0.17em}{\Re }^r\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}{\eta }_i\hspace{0.17em}\in \hspace{0.17em}\hspace{0.17em}{\Re }^{\left(n-r\right)}$. Consequently, there exists a set of diffeomorphisms: $T_i\left(x\right)=\hspace{0.17em}\hspace{0.17em}{\left[\begin{array}{ccccccc}h\left(x\right)& L_{f}h\left(x\right)& \dots & L_f^{r}h\left(x\right)& {\chi }_{1,\hspace{0.17em}i}\left(x\right)& \dots & {\chi }_{\left(n-r\right),\hspace{0.17em}i}\left(x\right)\end{array}\right]}^T,$ where $L_f^{j}h\left(x\right)\ne 0\hspace{0.17em},\hspace{0.17em}j\hspace{0.17em}\ge \hspace{0.17em}0$, denotes the Lie derivative of $h\left(x\right)$ with respect $f\left(x\right)$, and $L_f^{j}h\left(x\right)=\hspace{0.17em}\frac{\partial \hspace{0.17em}f\left(x\right)}{\partial \hspace{0.17em}x}\hspace{0.17em}h\left(x\right)$ and ${\chi }_{1,\hspace{0.17em}i}(x),\hspace{0.17em}\hspace{0.17em}{\chi }_{2,\hspace{0.17em}i}(x),\dots ,\hspace{0.17em}\hspace{0.17em}{\chi }_{\left(n-r\right),\hspace{0.17em}i}(x)$ are nonlinear scalar functions of x which can be explicit by the following relation:

$$\{\begin{array}{l}{L}_g{\chi }_{j,i}\left(x\right)=0\\ j=1,\hspace{0.17em}2,\dots ,\hspace{0.17em}n-r\\ i=1,\hspace{0.17em}2\hspace{0.17em},\dots ,\hspace{0.17em}m\end{array}$$

(2)

As a result, the System (1) has the following normal form [31,32,33].

$$\{\begin{array}{l}\{\begin{array}{l}{\dot{\xi }}_1\hspace{0.17em}=\hspace{0.17em}{\xi }_2\\ ⋮\\ {\dot{\xi }}_r\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{a}_i\left(\xi ,\hspace{0.17em}\eta \right)\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}{\varphi }_i\left(\xi ,\hspace{0.17em}\eta ,\hspace{0.17em}\theta \right)+\hspace{0.17em}{b}_i\left(\xi ,\hspace{0.17em}\eta \right)\hspace{0.17em}{u}_i\\ {\dot{\eta }}_1\hspace{0.17em}=\hspace{0.17em}\mathsf{\Psi }{\hspace{0.17em}}_{1,\hspace{0.17em}i}\left(\xi ,\hspace{0.17em}\eta \right)\\ ⋮\\ {\dot{\eta }}_{\left(n-r\right)}\hspace{0.17em}=\hspace{0.17em}\mathsf{\Psi }{\hspace{0.17em}}_{\left(n-r\right),\hspace{0.17em}i}\left(\xi ,\hspace{0.17em}\eta \right)\end{array}\\ y\hspace{0.17em}=\hspace{0.17em}{\xi }_1\end{array}$$

(3)

where $a_i\left(\xi ,\hspace{0.17em}\eta ,\hspace{0.17em}\hspace{0.17em}\theta \right)\ne \hspace{0.17em}0$,$\hspace{0.17em}{b}_i\left(\xi ,\hspace{0.17em}\eta \right)\hspace{0.17em}\ne 0\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\xi \hspace{0.17em}\in \hspace{0.17em}{\Re }^r$ is the vector of input-output signals; $\eta \hspace{0.17em}\in \hspace{0.17em}{\Re }^{n-r}$ denotes the vector of internal states; $\theta$ denotes uncertainties of the parameters of system; and ${\varphi }_i\left(\xi ,\hspace{0.17em}\eta ,\hspace{0.17em}\theta \right)$ stands for the nonlinear perturbation of the system.

The system equation is obtained by rewriting the normal form (3) into System (4):

$$\{\begin{array}{l}\dot{\xi }\hspace{0.17em}={\overline{A}}_r\hspace{0.17em}\xi +\hspace{0.17em}{\overline{\varphi }}_i\hspace{0.17em}\left(\xi ,\hspace{0.17em}\eta ,\hspace{0.17em}\theta \right)\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}{\overline{b}}_i\left(\xi ,\hspace{0.17em}\eta \right)\hspace{0.17em}{u}_i\\ \dot{\eta }\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\mathsf{\Psi }}_{j,\hspace{0.17em}i}\left(\xi \hspace{0.17em},\hspace{0.17em}\eta \right)\hspace{0.17em}\hspace{0.17em}\\ \hspace{0.17em}i\hspace{0.17em}\in \hspace{0.17em}\mathsf{{\rm M}}\hspace{0.17em}=\hspace{0.17em}\left\{1,\hspace{0.17em}2,\hspace{0.17em}\dots ,m\right\}\end{array}$$

(4)

where

$${\overline{A}}_r\hspace{0.17em}\in \hspace{0.17em}{\Re }^{\left(r\times r\right)}=\left[\begin{array}{ccccc}0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & 1\\ 0& 0& \cdots & \cdots & 0\end{array}\right],\hspace{0.17em}\hspace{0.17em}$$

$${\overline{\varphi }}_i\hspace{0.17em}\left(\xi ,\hspace{0.17em}\eta ,\hspace{0.17em}\theta \right)\in \hspace{0.17em}{\Re }^{\left(r\times 1\right)}=\hspace{0.17em}{\left[\begin{array}{cccc}0& \cdots & 0& {\varphi }_i\hspace{0.17em}\left(\xi ,\hspace{0.17em}\eta ,\hspace{0.17em}\theta \right)\end{array}\right]}^T$$

and

$${\overline{b}}_i\left(\xi ,\hspace{0.17em}\eta \right)\in \hspace{0.17em}{\Re }^{\left(r\times 1\right)}=\hspace{0.17em}{\left[\begin{array}{cccc}0& \cdots & 0& b_i\left(\xi ,\hspace{0.17em}\eta \right)\hspace{0.17em}\end{array}\right]}^T$$

where m is the number of subsystems. $\forall \hspace{0.17em}i\hspace{0.17em}\in \hspace{0.17em}M,\hspace{0.17em}\hspace{0.17em}{u}_{i\hspace{0.17em}}\in \hspace{0.17em}\hspace{0.17em}\Re$ is the input; $\dot{\eta }\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\mathsf{\Psi }}_{j,\hspace{0.17em}i}\left(0\hspace{0.17em},\hspace{0.17em}\eta \right)$ is the unstable zero dynamics.

The mode i will be non-minimum phase if the zero dynamics $\dot{\eta }\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{\mathsf{\Psi }}_{j,\hspace{0.17em}i}\left(0\hspace{0.17em},\hspace{0.17em}\eta \right)$ are unstable. The major purpose of this project is to develop robust controllers capable of global system stabilization (1).

3. Main Results

The purpose of this research is to develop robust controllers capable of globally stabilizing the non-minimum phase switched nonlinear system (1), which has unstable internal dynamics in each mode. Our objective at this stage is to develop a control law u_i that stabilizes the input–output dynamics of various modes. To do this, we must first recast the internal dynamics by decoupling their linear and nonlinear components:

$$\dot{\eta }=\hspace{0.17em}{\mathsf{\Psi }}_{j,\hspace{0.17em}i}\left(\xi \hspace{0.17em},\hspace{0.17em}\eta \right)=\hspace{0.17em}{\overline{A}}_{\xi }^i\xi +{\overline{A}}_{\eta }^i\hspace{0.17em}\eta +{\vartheta }^i\left(\xi ,\eta \right)$$

(5)

The closed-loop system (4) is thus as follows:

$$\left[\begin{array}{l}\dot{\xi }\\ \dot{\eta }\end{array}\right]={\tilde{A}}^i\hspace{0.17em}\left[\begin{array}{l}\xi \\ \eta \end{array}\right]+B\hspace{0.17em}{v}^i+\hspace{0.17em}\hspace{0.17em}\left[\begin{array}{l}{0}_r\\ {\vartheta }^i\left(\xi ,\hspace{0.17em}\eta \right)\end{array}\right]$$

(6)

with

$${\tilde{A}}^i=\hspace{0.17em}\left[\begin{array}{cc}{\overline{A}}_r& 0_r\\ {\overline{A}}_{\xi }^i& {\overline{A}}_{\eta }^i\end{array}\right]\hspace{0.17em}$$

and

$$\hspace{0.17em}\hspace{0.17em}B=\hspace{0.17em}\left[\begin{array}{c}{0}_{\left(r-1\right)}\\ 1\\ 0_{\left(n-r\right)}\end{array}\right]$$

The dynamic feedback synthesis fulfilling the precise input–output feedback linearization control aim is provided by:

$$u_i=\frac{-{\varphi }_i\hspace{0.17em}\left(\xi ,\hspace{0.17em}\eta ,\hspace{0.17em}\theta \right)}{b_i\hspace{0.17em}\left(\xi ,\hspace{0.17em}\eta \right)}+\frac{v_i}{b_i\hspace{0.17em}\left(\xi ,\hspace{0.17em}\eta \right)}\hspace{0.17em}\hspace{0.17em}$$

(7)

For each mode i ∈ M, we propose the following control technique for the auxiliary input:

$$v^i=-K^i\hspace{0.17em}\left[\begin{array}{c}\xi \\ \eta \end{array}\right]+v_{NL}^i\left(\xi ,\hspace{0.17em}\eta ,\hspace{0.17em}\theta \right)$$

(8)

where $K^i=\hspace{0.17em}\hspace{0.17em}\left[\begin{array}{cccc}{k}_1^i& k_2^i& \dots & k_n^i\end{array}\right]$ is a row vector of constant gains, and $v_{NL}^i$ is a nonlinear control law.

By combining (6) and (8), we obtain

$$\left[\begin{array}{c}\dot{\xi }\\ \dot{\eta }\end{array}\right]=\left({\tilde{A}}^i\hspace{0.17em}-\hspace{0.17em}B\hspace{0.17em}{K}^i\right)\left[\begin{array}{c}\xi \\ \eta \end{array}\right]+\left[\begin{array}{c}{0}_{r-1}\\ v_{NL}^i\left(\xi ,\hspace{0.17em}\eta ,\hspace{0.17em}\theta \right)\\ {\vartheta }^i\left(\xi ,\hspace{0.17em}\eta \right)\end{array}\right]$$

(9)

Given the equation’s state space description of the system (9). If $K^i$ is selected in such a way that $\left({\tilde{A}}^i\hspace{0.17em}-\hspace{0.17em}B\hspace{0.17em}{K}^i\right)$ is a Hurwitz polynomial, then $\forall \hspace{0.17em}{Q}^i>0,\hspace{0.17em}\exists \hspace{0.17em}{P}^i>0$ such that:

$${\left({\tilde{A}}^i\hspace{0.17em}-\hspace{0.17em}B\hspace{0.17em}{K}^i\right)}^T{P}^i+P^i\left({\tilde{A}}^i\hspace{0.17em}-\hspace{0.17em}B\hspace{0.17em}{K}^i\right)=-Q^i$$

(10)

Let $V^i\left(\xi ,\hspace{0.17em}\eta \right)$ be a Lyapunov candidate function for each mode i in closed-loop

$$V^i\left(\xi ,\hspace{0.17em}\eta \right)=\hspace{0.17em}{\left[\begin{array}{c}{\xi }_1\\ {\xi }_2\\ ⋮\\ {\xi }_r\\ {\eta }_{1,i}\\ ⋮\\ {\eta }_{\left(n-r\right),\hspace{0.17em}i}\end{array}\right]}^T\hspace{0.17em}\hspace{0.17em}{P}^i\hspace{0.17em}\hspace{0.17em}\left[\begin{array}{c}{\xi }_1\\ {\xi }_2\\ ⋮\\ {\xi }_r\\ {\eta }_{1,i}\\ ⋮\\ {\eta }_{\left(n-r\right),\hspace{0.17em}i}\end{array}\right]\hspace{0.17em}$$

(11)

$V^i\left(\xi ,\hspace{0.17em}\eta \right)$ is positive definite on ${\Re }^n$. Its derivative is given by

$${\dot{V}}^i\left(\xi ,\hspace{0.17em}\eta \right)=\hspace{0.17em}-{\left[\begin{array}{c}\xi \\ \eta \end{array}\right]}^T\hspace{0.17em}{Q}^i\hspace{0.17em}\hspace{0.17em}\left[\begin{array}{c}\xi \\ \eta \end{array}\right]\hspace{0.17em}+\hspace{0.17em}2\hspace{0.17em}{\left[\begin{array}{ccc}{0}_{\left(r-1\right)}& v_{NL}^i& {\vartheta }^i\left(\xi ,\hspace{0.17em}\eta \right)\end{array}\right]}^T\hspace{0.17em}{P}^i\hspace{0.17em}\left[\begin{array}{c}\xi \\ \eta \end{array}\right]\hspace{0.17em}$$

(12)

We propose to choose $v_{NL}^i$ which satisfies

$${\left[\begin{array}{ccc}{0}_{\left(r-1\right)}& v_{NL}^i& {\vartheta }^i\left(\xi ,\hspace{0.17em}\eta \right)\end{array}\right]}^T\hspace{0.17em}{P}^i\hspace{0.17em}\hspace{0.17em}\left[\begin{array}{c}\xi \\ \eta \end{array}\right]\hspace{0.17em}\hspace{0.17em}=0$$

(13)

i.e.

$$v_{NL}^i\left(\xi ,\hspace{0.17em}\eta ,\hspace{0.17em}\theta \right)=\frac{-{\displaystyle \sum _{k=r+1}^n{\vartheta }_k^i\left(\xi ,\hspace{0.17em}\eta \right)P_k^l\hspace{0.17em}\left[\begin{array}{c}\xi \\ \eta \end{array}\right]}}{{\mathrm{P}}_r^l\left[\begin{array}{c}\xi \\ \eta \end{array}\right]}$$

(14)

We can show that $v_{NL}^i\hspace{0.17em}\to \hspace{0.17em}0$ when $\left(\xi ,\hspace{0.17em}\hspace{0.17em}\eta \right)\hspace{0.17em}\to \hspace{0.17em}0$.

Note that $\begin{array}{cccc}{P}_1^l,\hspace{0.17em}& \hspace{0.17em}{P}_2^l& ,\hspace{0.17em}\dots ,& P_n^l\end{array}$ represent the row vectors of the matrix $P^l$, and

$${\vartheta }_{}^i\left(\xi ,\hspace{0.17em}\eta \right)\hspace{0.17em}=\hspace{0.17em}{\left[\begin{array}{ccc}{\vartheta }_1^i\left(\xi ,\hspace{0.17em}\eta \right)& \dots & {\vartheta }_{\left(n-r\right)}^i\left(\xi ,\hspace{0.17em}\eta \right)\end{array}\right]}^T$$

(15)

we have then

$${\dot{V}}^i\left(\xi ,\hspace{0.17em}\eta \right)=\hspace{0.17em}-{\left[\begin{array}{c}\xi \\ \eta \end{array}\right]}^T\hspace{0.17em}{Q}^i\hspace{0.17em}\hspace{0.17em}\left[\begin{array}{c}\xi \\ \eta \end{array}\right]\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}<\hspace{0.17em}\hspace{0.17em}0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall \hspace{0.17em}\left(\xi ,\hspace{0.17em}\eta \right)\hspace{0.17em}\in \hspace{0.17em}{\Re }^n$$

(16)

The system (4) has the same normal form as the equation system (17):

$$\dot{x}=\hspace{0.17em}{A}^i\hspace{0.17em}x\left(t\right)\hspace{0.17em}+\hspace{0.17em}{F}_i\left(x\left(t\right),\hspace{0.17em}\theta \right)$$

(17)

where $A^i=\left({\tilde{A}}^i\hspace{0.17em}-\hspace{0.17em}B\hspace{0.17em}{K}^i\right)$ and $F_i\left(x\left(t\right),\hspace{0.17em}\theta \right)=\left[\begin{array}{c}{0}_{r-1}\\ v_{NL}^i\left(\xi ,\hspace{0.17em}\eta ,\hspace{0.17em}\theta \right)\\ {\vartheta }^i\left(\xi ,\hspace{0.17em}\eta \right)\end{array}\right]$.

4. Stability Analysis

In this section, we explain our primary discoveries on the stabilization of nonlinear systems with unstable internal dynamics that are non-minimum phase switched. The purpose of this study is to look at the asymptotic behavior of a global switched nonlinear system based on the Gronwall–Bellman inequality.

The developed in this context is based on the use of the assumptions, lemmas, and theorems for non-minimum phase switched nonlinear systems listed below (17).

Assumption 1.

The matrices $\left(A^i,\hspace{0.17em}\hspace{0.17em}i\hspace{0.17em}\in \hspace{0.17em}\mathsf{\Lambda }\right)$, for the non-minimum phase switched nonlinear system (17), are Hurwitz. Therefore, there exist two positive constants $\mathsf{{\rm M}}$and$\lambda$such that:

$$\Vert e^{\hspace{0.17em}{A}^{i\hspace{0.17em}t}}\Vert \hspace{0.17em}\le \hspace{0.17em}\hspace{0.17em}\mathsf{{\rm M}}\hspace{0.17em}{e}^{-\lambda \hspace{0.17em}t} \mathrm{for} i\hspace{0.17em}\in \hspace{0.17em}\mathsf{\Lambda }$$

(18)

Assumption 2.

It is required for the non-minimum phase switched nonlinear system (17) that:

$$A_i\hspace{0.17em}{A}_j\hspace{0.17em}=\hspace{0.17em}{A}_j\hspace{0.17em}{A}_i\hspace{0.17em} \mathrm{for} i,\hspace{0.17em}j\in \hspace{0.17em}\mathsf{\Lambda }$$

(19)

Assumption 3.

The function $F_i\left(x\left(t\right),\hspace{0.17em}\theta \right)$ fulfills the following conditions for the non-minimum phase switched nonlinear system (17):

$$F_i\left(x\left(t\right),\hspace{0.17em}\theta \right)\hspace{0.17em}\le \hspace{0.17em}{\epsilon }_1\Vert x\Vert \hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\epsilon }_2{\Vert x\Vert }^p\hspace{0.17em}+\hspace{0.17em}\alpha \left(t\right)$$

(20)

where $p\hspace{0.17em}\ge \hspace{0.17em}0,\hspace{0.17em}{\epsilon }_i\hspace{0.17em}i=1,\hspace{0.17em}2$, and the function $\alpha \left(t\right)$ satisfies

$${\displaystyle {\int }_0^{+\infty }{e}^{\lambda \hspace{0.17em}t}\alpha \left(t\right)}\hspace{0.17em}dt\hspace{0.17em}<\hspace{0.17em}+\infty \hspace{0.17em}$$

(21)

The following lemmas [32] are quite important in proving our key conclusions.

Lemma 1.

Let $z\left(t\right),\hspace{0.17em}F\left(t\right)$ and $\ell \left(t\right)$ be non-negative functions satisfying the inequality

$$z\hspace{0.17em}\left(t\right)\hspace{0.17em}\hspace{0.17em}\le \hspace{0.17em}\hspace{0.17em}k+\hspace{0.17em}{\displaystyle {\int }_a^t\left[F\left(s\right)\hspace{0.17em}z\left(s\right)\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\ell \left(s\right)\hspace{0.17em}{z}^n\left(s\right)\hspace{0.17em}\right]\hspace{0.17em}ds\hspace{0.17em}}\hspace{0.17em} \mathrm{for} t\hspace{0.17em}\in \hspace{0.17em}I=\hspace{0.17em}[a\hspace{0.17em}\hspace{0.17em}b]$$

(22)

where $k\hspace{0.17em}\ge \hspace{0.17em}0$ and $n\hspace{0.17em}\hspace{0.17em}>\hspace{0.17em}1$ are constants. If the inequality below holds,

$$\hspace{0.17em}{k}^{\left(1-n\right)}-\hspace{0.17em}\left(n-1\right)+\hspace{0.17em}{\displaystyle {\int }_a^t\ell \left(s\right)\hspace{0.17em}{e}^{{\displaystyle \underset{a}{\overset{s}{\int }}\left(n-1\right)\hspace{0.17em}F\left(\tau \right)\hspace{0.17em}d\tau \hspace{0.17em}}}ds\hspace{0.17em}}\hspace{0.17em}>\hspace{0.17em}0$$

(23)

then, for $t\hspace{0.17em}\in \hspace{0.17em}I$

$$z\hspace{0.17em}\left(t\right)\hspace{0.17em}\hspace{0.17em}\le \hspace{0.17em}\frac{e^{{\displaystyle \underset{a}{\overset{t}{\int }}\hspace{0.17em}F\left(s\right)\hspace{0.17em}ds\hspace{0.17em}}}}{{\left(k^{1-n}\hspace{0.17em}-\hspace{0.17em}\left(n-1\right)\hspace{0.17em}{\displaystyle \underset{a}{\overset{t}{\int }}\ell \left(s\right)\hspace{0.17em}{e}^{{\displaystyle \underset{a}{\overset{s}{\int }}\hspace{0.17em}\left(n-1\right)F\left(\tau \right)\hspace{0.17em}d\tau \hspace{0.17em}}}ds}\right)}^{\frac{1}{n-1}}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$$

(24)

Lemma 2.

Suppose that $z\left(t\right),\hspace{0.17em}f\left(t\right)$ and $\ell \left(t\right)$ are nonnegative functions accomplishing the inequality

$$z\hspace{0.17em}\left(t\right)\hspace{0.17em}\hspace{0.17em}\le \hspace{0.17em}\hspace{0.17em}k+\hspace{0.17em}{\displaystyle {\int }_a^t\left[F\left(s\right)\hspace{0.17em}z\left(s\right)\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\ell \left(s\right)\hspace{0.17em}{z}^n\left(s\right)\hspace{0.17em}\right]\hspace{0.17em}ds\hspace{0.17em}}\hspace{0.17em} \mathrm{for} t\hspace{0.17em}\in \hspace{0.17em}I=\hspace{0.17em}[a\hspace{0.17em}\hspace{0.17em}b]$$

(25)

where$k\hspace{0.17em}\ge \hspace{0.17em}0$and$0\hspace{0.17em}\le \hspace{0.17em}\hspace{0.17em}n\hspace{0.17em}\hspace{0.17em}<\hspace{0.17em}1$are constants. Then for$t\hspace{0.17em}\in \hspace{0.17em}I$

$$z\hspace{0.17em}\left(t\right)\hspace{0.17em}\hspace{0.17em}\le \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{e}^{{\displaystyle \underset{a}{\overset{t}{\int }}\hspace{0.17em}F\left(s\right)\hspace{0.17em}ds\hspace{0.17em}}}{\left(k^{1-n}\hspace{0.17em}-\hspace{0.17em}\left(1-n\right)\hspace{0.17em}{\displaystyle \underset{a}{\overset{t}{\int }}\ell \left(s\right)\hspace{0.17em}e{-}^{{\displaystyle \underset{a}{\overset{s}{\int }}\hspace{0.17em}\left(1-n\right)F\left(\tau \right)\hspace{0.17em}d\tau \hspace{0.17em}}}ds}\right)}^{\frac{1}{n-1}}$$

(26)

Each mode i’s stability analysis is presented here. Based on the value of p, we may now provide the primary findings. There are two possibilities.

For case 1: p > 1, we have the following theorem:

Theorem 1.

Assume that assumptions 1, 2, and 3 hold for $p>\hspace{0.17em}0$ and $\lambda >\hspace{0.17em}{\mathsf{{\rm M}}}^N{\epsilon }_1$ and that the following inequality is verified

$$\Vert x\left(0\right)\Vert \hspace{0.17em}\hspace{0.17em}<\hspace{0.17em}\hspace{0.17em}{\left(\frac{\lambda -\hspace{0.17em}{\mathsf{{\rm M}}}^N{\epsilon }_1}{{\mathsf{{\rm M}}}^{Np}{\epsilon }_2}\right)}^{\frac{1}{1-\hspace{0.17em}p}}-\hspace{0.17em}{\displaystyle {\int }_0^{\infty }{e}^{\lambda t}\alpha \left(t\right)\hspace{0.17em}dt\hspace{0.17em}}$$

(27)

As a result, the non-minimum phase switched nonlinear system (17) with uncertainty, where any mode may be non-minimum phase, is exponentially stable.

Moreover

$$\Vert x\left(t\right)\Vert \hspace{0.17em}\hspace{0.17em}\le \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{e^{-\left(\lambda -\hspace{0.17em}{\mathsf{{\rm M}}}^N{\epsilon }_1\right)\hspace{0.17em}t}}{{\left({\mathsf{{\rm M}}}^{N\left(1-p\right)}\hspace{0.17em}{\left(\Vert x\left(0\right)\Vert \hspace{0.17em}+\hspace{0.17em}{\displaystyle {\int }_0^{\infty }{e}^{\lambda s}\alpha \left(s\right)\hspace{0.17em}ds\hspace{0.17em}}\right)}^{1-p}\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\left(\frac{\hspace{0.17em}{\mathsf{{\rm M}}}^N{\epsilon }_2}{\lambda \hspace{0.17em}-{\mathsf{{\rm M}}}^N{\epsilon }_1}\right)\right)}^{\left(\frac{1}{p\hspace{0.17em}-1}\right)}}\hspace{0.17em}$$

(28)

Proof (Theorem 1).

Let $\sigma \left(t\right)\hspace{0.17em}=\hspace{0.17em}{i}_k$, for $t\hspace{0.17em}\in \hspace{0.17em}\left[t_k\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}{t}_{k+1}\right)$. From the non-minimum phase switched nonlinear system (17) we have

$$\dot{x}=A_{ik}\hspace{0.17em}x\left(t\right)\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{F}_{ik}\left(x\left(t\right),\hspace{0.17em}\theta \right)\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}t\hspace{0.17em}\in \hspace{0.17em}\left[t_k\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}{t}_{k+1}\right)$$

(29)

For $t\hspace{0.17em}\in \hspace{0.17em}\left[t_k\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}{t}_{k+1}\right)$ which implies that

$$x\left(t\right)\hspace{0.17em}=\hspace{0.17em}e{\hspace{0.17em}}^{A_{ik}\left(t-\hspace{0.17em}{t}_k\right)}\hspace{0.17em}x\left(t_k\right)\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_{t_k}^t\hspace{0.17em}e{\hspace{0.17em}}^{A_{ik}\left(t-\hspace{0.17em}s\right)}\hspace{0.17em}{F}_{ik}\left(s,\hspace{0.17em}x\left(s\right)\right)ds}\hspace{0.17em}$$

(30)

Thus, we have

$$\{\begin{array}{l}x\left(t_1\right)\hspace{0.17em}=\hspace{0.17em}e{\hspace{0.17em}}^{A_{i0}\left(t-\hspace{0.17em}{t}_1\right)}\hspace{0.17em}x\left(t_0\right)\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_0^{t_1}\hspace{0.17em}e{\hspace{0.17em}}^{A_{i0}\left(t_1-\hspace{0.17em}s\right)}\hspace{0.17em}{F}_{i0}\left(s,\hspace{0.17em}x\left(s\right)\right)ds}\\ x\left(t_0\right)\hspace{0.17em}=\hspace{0.17em}x\left(0\right)\hspace{0.17em}\end{array}\hspace{0.17em}$$

(31)

and

$$\begin{array}{l}x\left(t_2\right)\hspace{0.17em}=\hspace{0.17em}e{\hspace{0.17em}}^{A_{i1}\left(t_2-\hspace{0.17em}{t}_1\right)}\hspace{0.17em}x\left(t_1\right)\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_{t_1}^{t_2}e{\hspace{0.17em}}^{A_{i1}\left(t_2-\hspace{0.17em}s\right)}\hspace{0.17em}{F}_{i1}\left(s,\hspace{0.17em}x\left(s\right)\right)ds}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}e{\hspace{0.17em}}^{A_{i1}\left(t_2-\hspace{0.17em}{t}_1\right)}\hspace{0.17em}e{\hspace{0.17em}}^{A_{i0}\hspace{0.17em}{t}_1}\hspace{0.17em}x\left(0\right)\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_0^{t_1}e{\hspace{0.17em}}^{A_{i1}\left(t_2-\hspace{0.17em}{t}_1\right)}\hspace{0.17em}e{\hspace{0.17em}}^{A_{i0}\left(t_1-\hspace{0.17em}s\right)}\hspace{0.17em}{F}_{i0}\left(s,\hspace{0.17em}x\left(s\right)\right)ds\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_{t_1}^{t_2}e{\hspace{0.17em}}^{A_{i1}\left(t_2-\hspace{0.17em}s\right)}\hspace{0.17em}{F}_{i1}\left(s,\hspace{0.17em}x\left(s\right)\right)ds}}\end{array}$$

(32)

Similarly, we can obtain

$$\begin{array}{l}x\left(t_k\right)\hspace{0.17em}=\hspace{0.17em}e{\hspace{0.17em}}^{A_{i_{k-1}}\left(t_k-\hspace{0.17em}{t}_{k-1}\right)}\hspace{0.17em}\dots \hspace{0.17em}\hspace{0.17em}e{\hspace{0.17em}}^{A_{i1}\left(t_2-\hspace{0.17em}{t}_1\right)}\hspace{0.17em}e{\hspace{0.17em}}^{A_{i0}\left(t_1\right)}\hspace{0.17em}x\left(0\right)\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_0^{t_1}e{\hspace{0.17em}}^{A_{i_{k-1}}\left(t_k-\hspace{0.17em}{t}_{k-1}\right)}\hspace{0.17em}\dots \hspace{0.17em}e{\hspace{0.17em}}^{A_{i1}\left(t_2-\hspace{0.17em}{t}_1\right)}\hspace{0.17em}e{\hspace{0.17em}}^{A_{i0}\left(t_1-\hspace{0.17em}s\right)}{F}_{i0}\left(s,\hspace{0.17em}x\left(s\right)\right)ds}\\ +\hspace{0.17em}{\displaystyle {\int }_{t_1}^{t_2}e{\hspace{0.17em}}^{A_{i_{k-1}}\left(t_k-\hspace{0.17em}{t}_{k-1}\right)}\hspace{0.17em}\dots \hspace{0.17em}\hspace{0.17em}e{\hspace{0.17em}}^{A_{i1}\left(t_2-\hspace{0.17em}s\right)}\hspace{0.17em}{F}_{i1}\left(s,\hspace{0.17em}x\left(s\right)\right)ds}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\dots \hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_{t_{k-1}}^{t_k}e{\hspace{0.17em}}^{A_{i_{k-1}}\left(t_k-\hspace{0.17em}s\right)}\hspace{0.17em}{F}_{i_{k-1}}\left(s,\hspace{0.17em}x\left(s\right)\right)ds}\hspace{0.17em}\end{array}$$

(33)

Let’s investigate the behavior of the system (30) for $t\hspace{0.17em}\in \hspace{0.17em}\left[t_k\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}{t}_{k+1}\right),$ we get:

$$\begin{array}{l}x\left(t_k\right)\hspace{0.17em}=\hspace{0.17em}e{\hspace{0.17em}}^{A_{i_k}\left(t-\hspace{0.17em}{t}_k\right)}\hspace{0.17em}e{\hspace{0.17em}}^{A_{i_{k-1}}\left(t_k-\hspace{0.17em}{t}_{k-1}\right)}\dots \hspace{0.17em}\hspace{0.17em}e{\hspace{0.17em}}^{A_{i1}\left(t_2-\hspace{0.17em}{t}_1\right)}\hspace{0.17em}e{\hspace{0.17em}}^{A_{i0}\left(t_1\right)}\hspace{0.17em}x\left(0\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_0^{t_1}\hspace{0.17em}e{\hspace{0.17em}}^{A_{i_k}\left(t-\hspace{0.17em}{t}_k\right)}e{\hspace{0.17em}}^{A_{i_{k-1}}\left(t_k-\hspace{0.17em}{t}_{k-1}\right)}\hspace{0.17em}\dots \hspace{0.17em}e{\hspace{0.17em}}^{A_{i1}\left(t_2-\hspace{0.17em}{t}_1\right)}\hspace{0.17em}e{\hspace{0.17em}}^{A_{i0}\left(t_1-\hspace{0.17em}s\right)}{F}_{i0}\left(s,\hspace{0.17em}x\left(s\right)\right)ds}\\ +\hspace{0.17em}{\displaystyle {\int }_{t_1}^{t_2}\hspace{0.17em}e{\hspace{0.17em}}^{A_{i_k}\left(t-\hspace{0.17em}{t}_k\right)}e{\hspace{0.17em}}^{A_{i_{k-1}}\left(t_k-\hspace{0.17em}{t}_{k-1}\right)}\hspace{0.17em}\dots \hspace{0.17em}\hspace{0.17em}e{\hspace{0.17em}}^{A_{i1}\left(t_2-\hspace{0.17em}s\right)}\hspace{0.17em}{F}_{i1}\left(s,\hspace{0.17em}x\left(s\right)\right)ds}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\dots \hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_{t_{k-1}}^{t_k}\hspace{0.17em}e{\hspace{0.17em}}^{A_{i_k}\left(t-\hspace{0.17em}{t}_k\right)}e{\hspace{0.17em}}^{A_{i_{k-1}}\left(t_k-\hspace{0.17em}s\right)}\hspace{0.17em}{F}_{i_{k-1}}\left(s,\hspace{0.17em}x\left(s\right)\right)ds}\hspace{0.17em}\\ +\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_{t_k}^{t}e{\hspace{0.17em}}^{A_{i_k}\left(t-\hspace{0.17em}s\right)}\hspace{0.17em}{F}_{i_k}\left(s,\hspace{0.17em}x\left(s\right)\right)ds}\hspace{0.17em}\end{array}$$

During the time interval $\hspace{0.17em}\left[0\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}t\right)$, let’s consider ${\delta }_i\left(0,\hspace{0.17em}t\right)$ the time for which the i^th system is activated.

By taking Assumption 2 into account, we have:

$$\begin{array}{l}x\left(t_k\right)\hspace{0.17em}=\hspace{0.17em}e{\hspace{0.17em}}^{A_1{\delta }_1\left(0,\hspace{0.17em}t\right)}\hspace{0.17em}e{\hspace{0.17em}}^{A_{{}_2}{\delta }_2\left(0,\hspace{0.17em}t\right)}\dots \hspace{0.17em}\hspace{0.17em}e{\hspace{0.17em}}^{A_N{\delta }_N\left(0,\hspace{0.17em}t\right)}\hspace{0.17em}e{\hspace{0.17em}}^{A_{i0}\left(t_1\right)}\hspace{0.17em}x\left(0\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_0^{t_1}e{\hspace{0.17em}}^{A_1{\delta }_1\left(s,\hspace{0.17em}t\right)}\hspace{0.17em}e{\hspace{0.17em}}^{A_{{}_2}{\delta }_2\left(s,\hspace{0.17em}t\right)}\hspace{0.17em}\dots e{\hspace{0.17em}}^{A_N{\delta }_N\left(s,\hspace{0.17em}t\right)}{F}_{i0}\left(s,\hspace{0.17em}x\left(s\right)\right)ds}\\ +\hspace{0.17em}{\displaystyle {\int }_{t_1}^{t_2}\hspace{0.17em}e{\hspace{0.17em}}^{A_1{\delta }_1\left(s,\hspace{0.17em}t\right)}\hspace{0.17em}e{\hspace{0.17em}}^{A_{{}_2}{\delta }_2\left(s,\hspace{0.17em}t\right)}\hspace{0.17em}\dots \hspace{0.17em}\hspace{0.17em}e{\hspace{0.17em}}^{A_N{\delta }_N\left(s,\hspace{0.17em}t\right)}\hspace{0.17em}{F}_{i1}\left(s,\hspace{0.17em}x\left(s\right)\right)ds}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\dots \hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_{t_{k-1}}^{t_k}\hspace{0.17em}e{\hspace{0.17em}}^{A_1{\delta }_1\left(s,\hspace{0.17em}t\right)}\hspace{0.17em}e{\hspace{0.17em}}^{A_{{}_2}{\delta }_2\left(s,\hspace{0.17em}t\right)}\hspace{0.17em}{F}_{i_{k-1}}\left(s,\hspace{0.17em}x\left(s\right)\right)ds}\hspace{0.17em}\\ +\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_{t_k}^{t}e{\hspace{0.17em}}^{A_{i_k}\left(t-\hspace{0.17em}s\right)}\hspace{0.17em}{F}_{i_k}\left(s,\hspace{0.17em}x\left(s\right)\right)ds}\hspace{0.17em}\end{array}$$

(34)

By respecting Assumptions 1 and 3, we conclude that:

$$\begin{array}{l}\Vert x\left(t\right)\Vert \hspace{0.17em}\hspace{0.17em}\le \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathsf{{\rm M}}}^N(e^{-\lambda \hspace{0.17em}t}\Vert x\left(0\right)\Vert \hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_0^{t_1}e{\hspace{0.17em}}^{-\lambda \left(t-s\right)}\hspace{0.17em}\Vert F_{i0}\left(s,\hspace{0.17em}x\left(s\right)\right)\Vert ds\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_{t_1}^{t_2}e{\hspace{0.17em}}^{-\lambda \left(t-s\right)}\Vert \hspace{0.17em}{F}_{i1}\left(s,\hspace{0.17em}x\left(s\right)\right)\Vert ds\hspace{0.17em}\hspace{0.17em}}}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\dots \hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_{t_{k-1}}^{t_k}e{\hspace{0.17em}}^{-\lambda \left(t-s\right)}\Vert \hspace{0.17em}{F}_{i_{k-1}}\left(s,\hspace{0.17em}x\left(s\right)\right)\Vert ds\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_{t_k}^{t}e{\hspace{0.17em}}^{-\lambda \left(t-s\right)}\Vert \hspace{0.17em}{F}_{i_k}\left(s,\hspace{0.17em}x\left(s\right)\right)\Vert ds}})\\ \Vert x\left(t\right)\Vert \hspace{0.17em}\hspace{0.17em}\le \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathsf{{\rm M}}}^N(e^{-\lambda \hspace{0.17em}t}\Vert x\left(0\right)\Vert \hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\epsilon }_1{\displaystyle {\int }_0^{t}e{\hspace{0.17em}}^{-\lambda \left(t-s\right)}\hspace{0.17em}\Vert \hspace{0.17em}x\left(s\right)\Vert ds\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\epsilon }_2{\displaystyle {\int }_0^{t}e{\hspace{0.17em}}^{-\lambda \left(t-s\right)}\hspace{0.17em}{\Vert \hspace{0.17em}x\left(s\right)\Vert }^{p}ds\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_0^{t}e{\hspace{0.17em}}^{-\lambda \left(t-s\right)}\alpha \left(s\right)ds\hspace{0.17em}\hspace{0.17em}}}}\end{array}$$

(35)

Letting $z\left(t\right)\hspace{0.17em}=\hspace{0.17em}{e}^{\lambda t}\hspace{0.17em}\Vert x\left(t\right)\Vert$, we get

$$z\left(t\right)\hspace{0.17em}\hspace{0.17em}\le \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathsf{{\rm M}}}^N(\Vert x\left(0\right)\Vert \hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_0^{\infty }e{\hspace{0.17em}}^{\lambda s}\alpha \left(s\right)\hspace{0.17em}ds\hspace{0.17em})\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\mathsf{{\rm M}}}^N{\epsilon }_1{\displaystyle {\int }_0^{t}z\left(s\right)ds\hspace{0.17em}\hspace{0.17em}}}+\hspace{0.17em}\hspace{0.17em}{\mathsf{{\rm M}}}^N{\epsilon }_2{\displaystyle {\int }_0^{t}e{\hspace{0.17em}}^{\lambda \left(1-p\right)s}\hspace{0.17em}{z}^p\left(s\right)ds\hspace{0.17em}\hspace{0.17em}}$$

(36)

By using the condition (27) of Theorem 1, we get the following inequality:

$$\begin{array}{l}{\mathsf{{\rm M}}}^{N\left(1-p\right)}(\Vert x\left(0\right)\Vert \hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_0^{\infty }e{\hspace{0.17em}}^{\lambda s}\alpha \left(s\right)\hspace{0.17em}ds\hspace{0.17em}{)}^{1-p}\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\left(p-1\right){\mathsf{{\rm M}}}^N{\epsilon }_2{\displaystyle {\int }_0^{t}e{\hspace{0.17em}}^{\lambda \left(1-p\right)s\hspace{0.17em}{\displaystyle {\int }_0^{s}e{\hspace{0.17em}}^{\lambda \left(p-1\right)s}\hspace{0.17em}{\mathsf{{\rm M}}}^N{\epsilon }_{1}d\delta }}ds\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\\ =\hspace{0.17em}{\mathsf{{\rm M}}}^{N\left(1-p\right)}(\Vert x\left(0\right)\Vert \hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_0^{\infty }e{\hspace{0.17em}}^{\lambda s}\alpha \left(s\right)\hspace{0.17em}ds\hspace{0.17em}{)}^{1-p}\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}\frac{{\mathsf{{\rm M}}}^N{\epsilon }_2}{\lambda \hspace{0.17em}-\hspace{0.17em}{\mathsf{{\rm M}}}^N{\epsilon }_1}\hspace{0.17em}\left(1-\hspace{0.17em}\hspace{0.17em}{e}^{\left(p-1\right)\left(\lambda -{\mathsf{{\rm M}}}^N{\epsilon }_1\right)\hspace{0.17em}t}\hspace{0.17em}d\delta \hspace{0.17em}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\\ \ge \hspace{0.17em}\hspace{0.17em}{\mathsf{{\rm M}}}^{N\left(1-p\right)}(\Vert x\left(0\right)\Vert \hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_0^{\infty }e{\hspace{0.17em}}^{\lambda s}\alpha \left(s\right)\hspace{0.17em}ds\hspace{0.17em}{)}^{1-p}\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}\frac{{\mathsf{{\rm M}}}^N{\epsilon }_2}{\lambda \hspace{0.17em}-\hspace{0.17em}{\mathsf{{\rm M}}}^N{\epsilon }_1}\hspace{0.17em}\hspace{0.17em}>\hspace{0.17em}0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\end{array}$$

(37)

By Lemma 1, we have

$$z\hspace{0.17em}\left(t\right)\hspace{0.17em}\hspace{0.17em}\le \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{e^{{\mathsf{{\rm M}}}^N{\epsilon }_1\hspace{0.17em}t}}{\hspace{0.17em}\hspace{0.17em}{\left({\mathsf{{\rm M}}}^{N\left(1-p\right)}(\Vert x\left(0\right)\Vert \hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_0^{\infty }e{\hspace{0.17em}}^{\lambda s}\alpha \left(s\right)\hspace{0.17em}ds\hspace{0.17em}{)}^{1-p}\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}\frac{{\mathsf{{\rm M}}}^N{\epsilon }_2}{\lambda \hspace{0.17em}-\hspace{0.17em}{\mathsf{{\rm M}}}^N{\epsilon }_1}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\right)}^{\left(\frac{1}{p-1}\right)}}\hspace{0.17em}$$

(38)

Similarly, that is

$$x\hspace{0.17em}\left(t\right)\hspace{0.17em}\hspace{0.17em}\le \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{e^{-\left(\lambda -{\mathsf{{\rm M}}}^N{\epsilon }_1\right)\hspace{0.17em}t}}{\hspace{0.17em}\hspace{0.17em}{\left({\mathsf{{\rm M}}}^{N\left(1-p\right)}(\Vert x\left(0\right)\Vert \hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_0^{\infty }e{\hspace{0.17em}}^{\lambda s}\alpha \left(s\right)\hspace{0.17em}ds\hspace{0.17em}{)}^{1-p}\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}\frac{{\mathsf{{\rm M}}}^N{\epsilon }_2}{\lambda \hspace{0.17em}-\hspace{0.17em}{\mathsf{{\rm M}}}^N{\epsilon }_1}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\right)}^{\left(\frac{1}{p-1}\right)}}\hspace{0.17em}$$

(39)

The proof of Theorem 1 is complete.□

For case 2: $0\hspace{0.17em}\hspace{0.17em}\le \hspace{0.17em}\hspace{0.17em}p\hspace{0.17em}\hspace{0.17em}<\hspace{0.17em}\hspace{0.17em}1$, we have the following theorem

Theorem 2.

Assume that hypotheses 1, 2 and 3 hold: $0\hspace{0.17em}\hspace{0.17em}\le \hspace{0.17em}\hspace{0.17em}p\hspace{0.17em}\hspace{0.17em}<\hspace{0.17em}\hspace{0.17em}1$and$\lambda >\hspace{0.17em}{\mathsf{{\rm M}}}^N{\epsilon }_1$.

Then, for each$x\left(t\right)$of the non-minimum phase switched nonlinear system (17), we obtain

$$\Vert x\left(t\right)\Vert \hspace{0.17em}\hspace{0.17em}\le \hspace{0.17em}\hspace{0.17em}{\left(\frac{\hspace{0.17em}{\mathsf{{\rm M}}}^N{\epsilon }_2}{\lambda \hspace{0.17em}-{\mathsf{{\rm M}}}^N{\epsilon }_1}\hspace{0.17em}+\hspace{0.17em}{\mathsf{{\rm M}}}^{N\left(1-p\right)}\hspace{0.17em}{\left(\Vert x\left(0\right)\Vert \hspace{0.17em}+\hspace{0.17em}{\displaystyle {\int }_0^{\infty }{e}^{\lambda s}\alpha \left(s\right)\hspace{0.17em}ds\hspace{0.17em}}\right)}^{1-p}\hspace{0.17em}{e}^{-\left(1-p\right)\hspace{0.17em}\left(\lambda -{\mathsf{{\rm M}}}^N{\epsilon }_1\right)\hspace{0.17em}t}\hspace{0.17em}\right)}^{\frac{1}{1-p}}$$

(40)

Proof (Theorem 2).

Similarly to the proof of Theorem 1, we may deduce that

$$z\left(t\right)\hspace{0.17em}\hspace{0.17em}\le \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathsf{{\rm M}}}^N(\Vert x\left(0\right)\Vert \hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\displaystyle {\int }_0^{\infty }e{\hspace{0.17em}}^{\lambda s}\alpha \left(s\right)\hspace{0.17em}ds\hspace{0.17em})\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\mathsf{{\rm M}}}^N{\epsilon }_1{\displaystyle {\int }_0^{t}z\left(s\right)ds\hspace{0.17em}\hspace{0.17em}}}+\hspace{0.17em}\hspace{0.17em}{\mathsf{{\rm M}}}^N{\epsilon }_2{\displaystyle {\int }_0^{t}e{\hspace{0.17em}}^{\lambda \left(1-p\right)s}\hspace{0.17em}{z}^p\left(s\right)ds\hspace{0.17em}\hspace{0.17em}}$$

(41)

For all t ≥ 0. Hence, by Lemma 2, we get

$$z\left(t\right)\hspace{0.17em}\le \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{e}^{\hspace{0.17em}\left({\mathsf{{\rm M}}}^N{\epsilon }_1\right)\hspace{0.17em}t}\hspace{0.17em}{\left(\hspace{0.17em}{\mathsf{{\rm M}}}^{N\left(1-p\right)}\hspace{0.17em}{\left(\Vert x\left(0\right)\Vert \hspace{0.17em}+\hspace{0.17em}{\displaystyle {\int }_0^{\infty }{e}^{\lambda s}\alpha \left(s\right)\hspace{0.17em}ds\hspace{0.17em}}\right)}^{1-p}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{\hspace{0.17em}{\mathsf{{\rm M}}}^N{\epsilon }_2}{\lambda \hspace{0.17em}-{\mathsf{{\rm M}}}^N{\epsilon }_1}\left(\hspace{0.17em}{e}^{\left(1-p\right)\hspace{0.17em}\left(\lambda -{\mathsf{{\rm M}}}^N{\epsilon }_1\right)\hspace{0.17em}t\hspace{0.17em}\hspace{0.17em}-1}\right)\hspace{0.17em}\right)}^{\frac{1}{1-p}}$$

(42)

That is,

$$x\left(t\right)\hspace{0.17em}\le \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\left(\hspace{0.17em}{\mathsf{{\rm M}}}^{N\left(1-p\right)}\hspace{0.17em}{\left(\Vert x\left(0\right)\Vert \hspace{0.17em}+\hspace{0.17em}{\displaystyle {\int }_0^{\infty }{e}^{\lambda s}\alpha \left(s\right)\hspace{0.17em}ds\hspace{0.17em}}\right)}^{1-p}\hspace{0.17em}\left(\hspace{0.17em}{e}^{-\left(1-p\right)\hspace{0.17em}\left(\lambda -{\mathsf{{\rm M}}}^N{\epsilon }_1\right)\hspace{0.17em}t\hspace{0.17em}\hspace{0.17em}-1}\right)+\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{\hspace{0.17em}{\mathsf{{\rm M}}}^N{\epsilon }_2}{\lambda \hspace{0.17em}-{\mathsf{{\rm M}}}^N{\epsilon }_1}\left(\hspace{0.17em}1-e^{-\left(1-p\right)\hspace{0.17em}\left(\lambda -{\mathsf{{\rm M}}}^N{\epsilon }_1\right)\hspace{0.17em}t\hspace{0.17em}\hspace{0.17em}}\right)\hspace{0.17em}\right)}^{\frac{1}{1-p}}$$

(43)

The proof of Theorem 2 is complete.□

Remark. 

In this work we can apply Filippov’s theory [34,35] to generate a trajectory that has ensured the best stabilization using the concept of multi-diffeomorphism. This contribution makes it possible to work out a switching strategy in such a way that the recourse to the concept of multi-diffeomorphism makes it possible to guarantee an improvement of the transient state compared to a linearization by feedback based on a single diffeomorphism. The formalism introduced by Filippov in [36] is a powerful tool to define a vector field on the sliding surface and to handle discontinuities. This has been applied in other fields, e.g., in power electronics [37] and energy harvesters [38].

5. Numerical Examples

In this section, two examples are presented to illustrate the effectiveness of the proposed

5.1. Example 1

We provide an illustrated case in this subsection to clarify the applicability and importance of our findings. We investigate the following ambiguous nonlinear system that was proposed by [31]:

$$\{\begin{array}{l}\{\begin{array}{l}{\dot{x}}_1\hspace{0.17em}=\hspace{0.17em}-x_1\hspace{0.17em}+\hspace{0.17em}{x}_2\\ {\dot{x}}_2\hspace{0.17em}=\hspace{0.17em}-3x_2\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{x}_1^3\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\theta \hspace{0.17em}+\hspace{0.17em}\left({\left(sin\left(x_3\right)\right)}^2\hspace{0.17em}+\hspace{0.17em}2\right)\hspace{0.17em}u\\ {\dot{x}}_3\hspace{0.17em}=\hspace{0.17em}{x}_1\hspace{0.17em}-\hspace{0.17em}2x_3\\ {\dot{x}}_4=-\hspace{0.17em}{x}_4\hspace{0.17em}+\hspace{0.17em}{x}_3^2\hspace{0.17em}\end{array}\\ y\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{x}_1\hspace{0.17em}-\hspace{0.17em}3x_3\end{array}$$

(44)

where $\theta =\hspace{0.17em}\hspace{0.17em}0.3\hspace{0.17em}sin\left(3\hspace{0.17em}t\right)$, and $y\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{x}_1\hspace{0.17em}-\hspace{0.17em}3x_3$ represents the output.

Using the method described in [31], we can discover two unique diffeomorphisms:

$$T_1\left(x\right)=\hspace{0.17em}\left[\begin{array}{c}{x}_1\hspace{0.17em}-\hspace{0.17em}3x_3\\ -4x_1\hspace{0.17em}+\hspace{0.17em}6x_3\hspace{0.17em}+\hspace{0.17em}{x}_2\\ x_3\\ x_4\end{array}\right] \mathrm{and} T_2\left(x\right)=\hspace{0.17em}\left[\begin{array}{c}{x}_1\hspace{0.17em}-\hspace{0.17em}3x_3\\ -4x_1\hspace{0.17em}+\hspace{0.17em}6x_3\hspace{0.17em}+\hspace{0.17em}{x}_2\\ x_3\hspace{0.17em}+\hspace{0.17em}{x}_4\\ 2x_3\end{array}\right]$$

As a result, the two modes must be used to modify the uncertain system (44):

$$\mathrm{Mode} 1: \{\begin{array}{l}{\dot{\xi }}_1=\hspace{0.17em}{\xi }_2\\ {\dot{\xi }}_2=\hspace{0.17em}3{\xi }_1\hspace{0.17em}-4{\xi }_2\hspace{0.17em}-9{\eta }_1\hspace{0.17em}+\hspace{0.17em}{\left({\xi }_1\hspace{0.17em}+\hspace{0.17em}3{\eta }_2\right)}^3\hspace{0.17em}+\hspace{0.17em}\theta \left(t\right)+\hspace{0.17em}\left(2+\hspace{0.17em}si{n}^2\left({\eta }_1\right)\right)\hspace{0.17em}{u}_1\\ {\dot{\eta }}_1=\hspace{0.17em}{\eta }_1\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\xi }_1\\ {\dot{\eta }}_2=\hspace{0.17em}{\eta }_1^2\hspace{0.17em}-\hspace{0.17em}{\eta }_2\end{array}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$$

(45)

and

$$\mathrm{Mode} 2: \{\begin{array}{l}{\dot{\xi }}_1=\hspace{0.17em}{\xi }_2\\ {\dot{\xi }}_2=\hspace{0.17em}-3{\xi }_1\hspace{0.17em}-4{\xi }_2\hspace{0.17em}\hspace{0.17em}+{\left(\left(3/2\right)\hspace{0.17em}\hspace{0.17em}{\eta }_2\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\xi }_1\right)}^3\hspace{0.17em}+\theta \left(t\right)\hspace{0.17em}+\hspace{0.17em}\left(2+\hspace{0.17em}si{n}^2\left({\eta }_2/2\right)\right)\hspace{0.17em}{u}_2\\ {\dot{\eta }}_1=\hspace{0.17em}{\xi }_1\hspace{0.17em}-\hspace{0.17em}{\eta }_1\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\frac{{\eta }_2^2}{4}\hspace{0.17em}\\ {\dot{\eta }}_2=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}2{\xi }_1\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\eta }_2\end{array}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$$

(46)

The system (44) is converted by the two modes using the design approach in Section 3:

$$\dot{x}=\hspace{0.17em}{A}^{\sigma \left(t\right)}\hspace{0.17em}x\left(t\right)\hspace{0.17em}+\hspace{0.17em}{F}_{\sigma \left(t\right)}\left(x\left(t\right),\hspace{0.17em}\theta \left(t\right)\right)$$

(47)

where σ(t): [0, ∞) → {1, 2}

For mode 1:

$$\dot{x}=\hspace{0.17em}{A}^1\hspace{0.17em}x\left(t\right)\hspace{0.17em}+\hspace{0.17em}{F}_1\left(x\left(t\right),\hspace{0.17em}\theta \right)$$

(48)

where $A^1=\left({\tilde{A}}^1\hspace{0.17em}-\hspace{0.17em}B\hspace{0.17em}{K}^1\right)$ and $F_1\left(x\left(t\right),\hspace{0.17em}\theta \left(t\right)\right)=\left[\begin{array}{c}0\\ v_{NL}^1\left(x\left(t\right)\right)\\ {\vartheta }^1\left(x\left(t\right)\right)\end{array}\right]$, $\tilde{A}{\hspace{0.17em}}^1\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 1& 0\\ 0& 0& 0& -1\end{array}\right]\hspace{0.17em}$, $B\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\left[\begin{array}{c}0\\ 1\\ 0\\ 0\end{array}\right]$, $K^1\hspace{0.17em}={\left[\begin{array}{cccc}{k}_1^1& k_2^1& k_3^1& k_4^1\end{array}\right]}^T$, ${\vartheta }^1\left(x\left(t\right)\right)\hspace{0.17em}=\hspace{0.17em}\left[\begin{array}{c}0\\ x_3^2\end{array}\right]$ and $v_{NL}^1\left(x\left(t\right)\right)\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{{\left(x_1^{}+3x_4\hspace{0.17em}-3x_3\right)}^3\hspace{0.17em}+\theta \left(t\right)\hspace{0.17em}\hspace{0.17em}}{2+\hspace{0.17em}sin\left(x_3^2\right)}$.

For mode 1, we select the quadratic Lyapunov function $V_1\left(x\right)$:

$$V_1\left(x\right)=\frac{1}{2}{x}_3^2\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{1}{2}{x}_4^2\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\frac{{\left(x_1\hspace{0.17em}-\hspace{0.17em}3\hspace{0.17em}{x}_3\right)}^2}{2}\hspace{0.17em}\hspace{0.17em}+\frac{{\left(-4x_1\hspace{0.17em}+x_2+\hspace{0.17em}6x_3\right)}^2}{2}\hspace{0.17em}$$

(49)

The stabilizing controller $u_1$ is:

$$u_1\left(x\right)\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\frac{-x_4{x}_3^2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{v}^1}{1/2\left(-4\hspace{0.17em}{x}_1\hspace{0.17em}+x_2\hspace{0.17em}+\hspace{0.17em}6x_3\right)}$$

(50)

with

$$v^{\left(1\right)}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}-\hspace{0.17em}{k}_1^1\hspace{0.17em}\left(x_1\hspace{0.17em}-\hspace{0.17em}3x_3\right)\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{k}_2^1\hspace{0.17em}\left(-4x_1\hspace{0.17em}+\hspace{0.17em}6x_3\hspace{0.17em}+\hspace{0.17em}{x}_2\right)\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}{k}_3^1\left(x_3\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}{k}_4^1\left(x_4\right)\hspace{0.17em}\hspace{0.17em}+\frac{{\left(x_1^{}+3x_4\hspace{0.17em}-3x_3\right)}^3\hspace{0.17em}+\theta \left(t\right)\hspace{0.17em}\hspace{0.17em}}{2+\hspace{0.17em}sin\left(x_3^2\right)}$$

For mode 2:

$$\dot{x}=\hspace{0.17em}{A}^2\hspace{0.17em}x\left(t\right)\hspace{0.17em}+\hspace{0.17em}{F}_2\left(x\left(t\right),\hspace{0.17em}\theta \right)$$

(51)

where $A^2=\left({\tilde{A}}^2\hspace{0.17em}-\hspace{0.17em}B\hspace{0.17em}{K}^2\right)$ and $F_2\left(x\left(t\right),\hspace{0.17em}\theta \left(t\right)\right)=\left[\begin{array}{c}0\\ v_{NL}^2\left(x\left(t\right)\right)\\ {\vartheta }^2\left(x\left(t\right)\right)\end{array}\right]$, $\tilde{A}{\hspace{0.17em}}^2\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 0& 0\\ 1& 0& -1& 0\\ 2& 0& 0& 1\end{array}\right]\hspace{0.17em}$, $B\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\left[\begin{array}{c}0\\ 1\\ 0\\ 0\end{array}\right]$,$K^2\hspace{0.17em}={\left[\begin{array}{cccc}{k}_1^2& k_2^2& k_3^2& k_4^2\end{array}\right]}^T$, ${\vartheta }^2\left(x\left(t\right)\right)\hspace{0.17em}=\hspace{0.17em}\left[\begin{array}{c}\frac{{\left(2x_3\right)}^2}{4}\\ 0\end{array}\right]$ and $v_{NL}^{\left(2\right)}\left(x\left(t\right)\right)\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\frac{x_1^3\hspace{0.17em}+\theta \left(t\right)\hspace{0.17em}\hspace{0.17em}}{2+\hspace{0.17em}sin{\left(\left(x_3^{}\hspace{0.17em}+x_4\right)/2\right)}^2}$.

For mode 2, we select the quadratic Lyapunov function $V_2\left(x\right)$:

$$V_2\left(x\right)=2x_3^2\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{1}{2}{\left(x_3\hspace{0.17em}+x_4^{}\hspace{0.17em}\right)}^2\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{\left(x_1\hspace{0.17em}-\hspace{0.17em}3\hspace{0.17em}{x}_3\right)}^2\hspace{0.17em}+{\left(-4x_1\hspace{0.17em}+x_2+\hspace{0.17em}6x_3\right)}^2\hspace{0.17em}$$

(52)

The stabilizing controller $u_2$ is:

$$u_2\left(x\right)\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\frac{\left(x_1\hspace{0.17em}-3x_3\right){\left(2x_3\right)}^2/2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}{v}^{\left(2\right)}}{1/2\left(-4\hspace{0.17em}{x}_1\hspace{0.17em}+x_2\hspace{0.17em}+\hspace{0.17em}6x_3\right)}$$

(53)

with

$$v^{\left(2\right)}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}-\hspace{0.17em}{k}_1^2\hspace{0.17em}\left(x_1\hspace{0.17em}-\hspace{0.17em}3x_3\right)\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{k}_2^2\hspace{0.17em}\left(-4x_1\hspace{0.17em}+\hspace{0.17em}6x_3\hspace{0.17em}+\hspace{0.17em}{x}_2\right)\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}{k}_3^2\left(x_3+\hspace{0.17em}{x}_4\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}{k}_4^2\left(2x_3\right)\hspace{0.17em}\hspace{0.17em}+\frac{x_1^3\hspace{0.17em}+\theta \left(t\right)\hspace{0.17em}\hspace{0.17em}}{2+\hspace{0.17em}sin{\left(\left(x_3^{}\hspace{0.17em}+x_4\right)/2\right)}^2}$$

The values of the gain vectors $K^1$ and $K^2$ were chosen so that the matrices $A{\hspace{0.17em}}^1\hspace{0.17em}$ and $A{\hspace{0.17em}}^2\hspace{0.17em}$ are Hurwitz.

$$A{\hspace{0.17em}}^1\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\left[\begin{array}{cccc}-1& 1& 0& 0\\ 3& -4& 0& 0\\ 1& 0& -1& 0\\ 0& 1& 0& -1\end{array}\right]\hspace{0.17em}, F_1\left(x\left(t\right),\hspace{0.17em}\theta \left(t\right)\right)={\left[\begin{array}{cccc}0& \frac{{\left(x_1^{}+3x_4\hspace{0.17em}-3x_3\right)}^3\hspace{0.17em}+0.3\hspace{0.17em}sin\left(3\hspace{0.17em}t\right)\hspace{0.17em}+\hspace{0.17em}{v}^1\hspace{0.17em}}{2+\hspace{0.17em}sin\left(x_3^2\right)}& 0& x_3^2\end{array}\right]}^T$$

and

$$A{\hspace{0.17em}}^2\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\left[\begin{array}{cccc}-1& 1& 0& 0\\ 0& -2& 0& 0\\ 1& 0& -1& 0\\ 2& -3& 0& -1\end{array}\right], F_2\left(x\left(t\right),\hspace{0.17em}\theta \left(t\right)\right)={\left[\begin{array}{cccc}0& \frac{x_1^3\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}0.3\hspace{0.17em}sin\left(3\hspace{0.17em}t\right)\hspace{0.17em}+\hspace{0.17em}{v}^{\left(2\right)}\hspace{0.17em}}{2+\hspace{0.17em}sin{\left(\left(x_3^{}\hspace{0.17em}+x_4\right)/2\right)}^2}& \frac{{\left(2x_3\right)}^2}{4}& 0\end{array}\right]}^T$$

By simple computation, we have $\lambda \left(A{\hspace{0.17em}}^1\right)\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\left\{-1\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}-1,\hspace{0.17em}\hspace{0.17em}-0.2087,\hspace{0.17em}\hspace{0.17em}-4.731\right\}$, $\lambda \left(A{\hspace{0.17em}}^2\right)\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\left\{-1\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}-1,\hspace{0.17em}\hspace{0.17em}-1,\hspace{0.17em}\hspace{0.17em}-2\right\}$

The criteria of Assumptions 1, 2, and 3 are easily met using $\mathsf{{\rm M}}=\hspace{0.17em}1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\lambda =1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\epsilon }_1\hspace{0.17em}=\hspace{0.17em}0.5,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\epsilon }_2\hspace{0.17em}=\hspace{0.17em}0.25,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}p=2$, and $\alpha \left(t\right)\hspace{0.17em}=\hspace{0.17em}{e}^{-2t}$. Since $\lambda >1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathsf{{\rm M}}}^2\hspace{0.17em}{\epsilon }_1=\hspace{0.17em}0.5\hspace{0.17em}$, We can now use Theorem 1 to show that the non-minimum phase switched nonlinear system (47) with uncertainty is exponentially stable if the starting states fulfill $\Vert x\left(0\right)\Vert \hspace{0.17em}\hspace{0.17em}<\hspace{0.17em}\hspace{0.17em}1$.

Moreover, $\Vert x\left(t\right)\Vert \hspace{0.17em}\hspace{0.17em}\le \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{2\left(1+\hspace{0.17em}\Vert x\left(0\right)\Vert \right)}{1\hspace{0.17em}-\hspace{0.17em}\Vert x\left(0\right)\Vert }{e}^{-0.5\hspace{0.17em}t}\hspace{0.17em}$ for all initial states satisfying $\Vert x\left(0\right)\Vert \hspace{0.17em}\hspace{0.17em}<\hspace{0.17em}\hspace{0.17em}1$.

Figure 1 and Figure 2 illustrate the simulation results. The state trajectories with the initial conditions $x\left(0\right)\hspace{0.17em}=\hspace{0.17em}{\left[\begin{array}{cccc}0.31& 0.52& 0.48& 0.12\end{array}\right]}^T$ and the control of the non-minimum phase switched nonlinear system (47) with uncertainty are depicted in these pictures.

The solution of the non-minimum phase switched nonlinear system (47) is asymptotically convergent, as shown in Figure 1 and Figure 2.

It is obvious that the suggested control method has excellent performance and significant resilience in the face of uncertainty.

5.2. Example 2

In this illustrative example, let us consider the following uncertain nonlinear system [28]:

$$\{\begin{array}{l}{\dot{x}}_1\hspace{0.17em}=\hspace{0.17em}-x_3\hspace{0.17em}\\ {\dot{x}}_2\hspace{0.17em}=\hspace{0.17em}sin\left(x_2\right)\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{x}_3\\ {\dot{x}}_3\hspace{0.17em}=\hspace{0.17em}2x_1^2\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}\left(1\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\theta \left(t\right)\right)x_3\hspace{0.17em}+\hspace{0.17em}u\hspace{0.17em}\end{array}$$

(54)

where $\theta =\hspace{0.17em}\hspace{0.17em}0.2\hspace{0.17em}sin\left(4\hspace{0.17em}t\right)$, and $y\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{x}_1$ represents the output.

Using the method described in [28], we can discover two unique diffeomorphisms:

$$T_1\left(x\right)\hspace{0.17em}=\hspace{0.17em}\left[\begin{array}{c}{x}_1\hspace{0.17em}\\ -\hspace{0.17em}{x}_3\\ x_2\end{array}\right] \mathrm{and} T_2\left(x\right)\hspace{0.17em}=\hspace{0.17em}\left[\begin{array}{c}{x}_1\hspace{0.17em}\\ -\hspace{0.17em}{x}_3\\ x_1\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{x}_2\end{array}\right]$$

As a result, the two modes must be used to modify the uncertain system (54):

$$\mathrm{Mode} 1: \{\begin{array}{l}{\dot{\xi }}_1=\hspace{0.17em}{\xi }_2\\ {\dot{\xi }}_2=\hspace{0.17em}-2{\xi }_1^2\hspace{0.17em}-\hspace{0.17em}\left(1\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\theta \right)\hspace{0.17em}{\xi }_2\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}{u}_1\\ \dot{\eta }=\hspace{0.17em}Sin\left(\eta \right)\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}{\xi }_2\end{array}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$$

(55)

and

$$\mathrm{Mode} 2: \{\begin{array}{l}{\dot{\xi }}_1=\hspace{0.17em}{\xi }_2\\ {\dot{\xi }}_2=\hspace{0.17em}-2{\xi }_1^2\hspace{0.17em}-\hspace{0.17em}\left(1\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\theta \right)\hspace{0.17em}{\xi }_2\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}{u}_2\\ \dot{\eta }=\hspace{0.17em}Sin\left(\eta \right)\hspace{0.17em}-{\xi }_1\end{array}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$$

(56)

The system (54) is converted by the two modes using the design approach in Section 3:

$$\dot{x}=\hspace{0.17em}{A}^{\sigma \left(t\right)}\hspace{0.17em}x\left(t\right)\hspace{0.17em}+\hspace{0.17em}{F}_{\sigma \left(t\right)}\left(x\left(t\right),\hspace{0.17em}\theta \left(t\right)\right)$$

(57)

where σ(t): [0, ∞) → {1, 2}

For mode 1:

$$\dot{x}=\hspace{0.17em}{A}^1\hspace{0.17em}x\left(t\right)\hspace{0.17em}+\hspace{0.17em}{F}_1\left(x\left(t\right),\hspace{0.17em}\theta \right)$$

(58)

where $A^1=\left({\tilde{A}}^1\hspace{0.17em}-\hspace{0.17em}B\hspace{0.17em}{K}^1\right)$ and $F_1\left(x\left(t\right),\hspace{0.17em}\theta \left(t\right)\right)=\left[\begin{array}{c}0\\ v_{NL}^1\left(x\left(t\right)\right)\\ {\vartheta }^1\left(x\left(t\right)\right)\end{array}\right]$, $\tilde{A}{\hspace{0.17em}}^1\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& -1& 0\end{array}\right]$, $B\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]$,$K^1\hspace{0.17em}={\left[\begin{array}{ccc}{k}_1^1& k_2^1& k_3^1\end{array}\right]}^T$, ${\vartheta }^1\left(x\left(t\right)\right)\hspace{0.17em}=\hspace{0.17em}\left[Sin\left(x_2^{}\right)\right]$ and $v_{NL}^1\left(x\left(t\right)\right)\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{Sin\left(x_2\right)\left(x_1^{}+x_2\hspace{0.17em}-x_3\right)\hspace{0.17em}\hspace{0.17em}}{x_3}$.

For mode 1, we select the quadratic Lyapunov function $V_1\left(x\right)$:

$$V_1\left(x\right)=\frac{1}{2}{x}_1^2\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{1}{2}{x}_3^2\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\frac{1}{2}{x}_2^2\hspace{0.17em}\hspace{0.17em}$$

(59)

The stabilizing controller $u_1$ is:

$$u_1\left(x\right)\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}2x_1^2\hspace{0.17em}+\hspace{0.17em}\left(1\hspace{0.17em}+\theta \right)\hspace{0.17em}{x}_3\hspace{0.17em}-v^1$$

(60)

with $v^{\left(1\right)}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}{k}_1^1\hspace{0.17em}\left(x_1\right)\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{k}_2^1\hspace{0.17em}\left(\hspace{0.17em}{x}_3\right)\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\hspace{0.17em}{k}_3^1\left(x_2\right)\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\frac{Sin\left(x_2\right)\left(x_1^{}+x_2\hspace{0.17em}-x_3\right)\hspace{0.17em}\hspace{0.17em}}{x_3}$.

For mode 2:

$$\dot{x}=\hspace{0.17em}{A}^2\hspace{0.17em}x\left(t\right)\hspace{0.17em}+\hspace{0.17em}{F}_2\left(x\left(t\right),\hspace{0.17em}\theta \right)$$

(61)

where $A^2=\left({\tilde{A}}^2\hspace{0.17em}-\hspace{0.17em}B\hspace{0.17em}{K}^2\right)$ and $F_2\left(x\left(t\right),\hspace{0.17em}\theta \left(t\right)\right)=\left[\begin{array}{c}0\\ v_{NL}^2\left(x\left(t\right)\right)\\ {\vartheta }^2\left(x\left(t\right)\right)\end{array}\right]$, $\tilde{A}{\hspace{0.17em}}^2\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ -1& 0& 0\end{array}\right]$, $B\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]$,$K^2\hspace{0.17em}={\left[\begin{array}{ccc}{k}_1^2& k_2^2& k_3^2\end{array}\right]}^T$, ${\vartheta }^2\left(x\left(t\right)\right)\hspace{0.17em}=\hspace{0.17em}\left[Sin\left(x_1\hspace{0.17em}+\hspace{0.17em}{x}_2\right)\right]$ and $v_{NL}^{\left(2\right)}\left(x\left(t\right)\right)\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\frac{Sin\left(x_1\hspace{0.17em}+\hspace{0.17em}{x}_2\right)\left[2x_1\hspace{0.17em}+\hspace{0.17em}{x}_2\hspace{0.17em}-x_3^{}\hspace{0.17em}\right]\hspace{0.17em}\hspace{0.17em}}{x_3^{}\hspace{0.17em}}$.

For mode 2, we select the quadratic Lyapunov function $V_2\left(x\right)$:

$$V_2\left(x\right)=\hspace{0.17em}\frac{1}{2}{\left(x_2\right)}^2\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}{\left(\hspace{0.17em}{x}_3\right)}^2\hspace{0.17em}+\hspace{0.17em}\frac{1}{2}{\left(x_1\hspace{0.17em}+x_2\right)}^2\hspace{0.17em}$$

(62)

The stabilizing controller $u_2$ is:

$$u_2\left(x\right)\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}2x_1^2\hspace{0.17em}-\hspace{0.17em}\left(1\hspace{0.17em}+\hspace{0.17em}\theta \right)x_3\hspace{0.17em}-v^2$$

(63)

with

$$v^{\left(2\right)}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}-\hspace{0.17em}{k}_1^2\hspace{0.17em}\left(x_1\right)\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{k}_2^2\hspace{0.17em}\left(-x_3\right)\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}{k}_3^2\left(x_1+\hspace{0.17em}{x}_2\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}\frac{Sin\left(x_1\hspace{0.17em}+\hspace{0.17em}{x}_2\right)\left[2x_1\hspace{0.17em}+\hspace{0.17em}{x}_2\hspace{0.17em}-x_3^{}\hspace{0.17em}\right]\hspace{0.17em}\hspace{0.17em}}{x_3^{}\hspace{0.17em}}$$

The values of the gain vectors $K^1$ and $K^2$ were chosen so that the matrices $A{\hspace{0.17em}}^1\hspace{0.17em}$ and $A{\hspace{0.17em}}^2\hspace{0.17em}$ are Hurwitz.

$$A{\hspace{0.17em}}^1\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 1& -1& -1\end{array}\right] , F_1\left(x\left(t\right),\hspace{0.17em}\theta \left(t\right)\right)={\left[\begin{array}{ccc}0& \frac{Sin\left(x_2\right)\left(x_1^{}+x_2\hspace{0.17em}-x_3\right)\hspace{0.17em}\hspace{0.17em}}{x_3}& Sin\left(x_2^{}\right)\end{array}\right]}^T$$

and

$$A{\hspace{0.17em}}^2\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\left[\begin{array}{ccc}-2& 0& 0\\ -1& -2& 0\\ 1& 0& -1\end{array}\right], F_2\left(x\left(t\right),\hspace{0.17em}\theta \left(t\right)\right)={\left[\begin{array}{ccc}0& \frac{Sin\left(x_1\hspace{0.17em}+\hspace{0.17em}{x}_2\right)\left[2x_1\hspace{0.17em}+\hspace{0.17em}{x}_2\hspace{0.17em}-x_3^{}\hspace{0.17em}\right]\hspace{0.17em}\hspace{0.17em}}{x_3^{}\hspace{0.17em}}& Sin\left(x_2^{}+x_1\right)\end{array}\right]}^T$$

By simple computation, we have $\lambda \left(A{\hspace{0.17em}}^1\right)\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\left\{-1\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}-1,\hspace{0.17em}\hspace{0.17em}-1\right\}$, $\lambda \left(A{\hspace{0.17em}}^2\right)\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\left\{-1\hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-2,\hspace{0.17em}\hspace{0.17em}-2\right\}$.

The criteria of Assumptions 1, 2, and 3 are easily met using $\mathsf{{\rm M}}=\hspace{0.17em}1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\lambda =1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\epsilon }_1\hspace{0.17em}=\hspace{0.17em}0.332,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\epsilon }_2\hspace{0.17em}=\hspace{0.17em}0.214,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}p=3$, and $\alpha \left(t\right)\hspace{0.17em}=\hspace{0.17em}{e}^{-3t}$. Since $\lambda >1,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\mathsf{{\rm M}}}^2\hspace{0.17em}{\epsilon }_1=\hspace{0.17em}0.332\hspace{0.17em}$, we can now use Theorem 1 to show that the non-minimum phase switched nonlinear system (57) with uncertainty is exponentially stable if the starting states fulfill $\Vert x\left(0\right)\Vert \hspace{0.17em}\hspace{0.17em}<\hspace{0.17em}\hspace{0.17em}1$.

Moreover, $\Vert x\left(t\right)\Vert \hspace{0.17em}\hspace{0.17em}\le \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{3\left(1+\hspace{0.17em}\Vert x\left(0\right)\Vert \right)}{1\hspace{0.17em}-\hspace{0.17em}\Vert x\left(0\right)\Vert }{e}^{-0.33\hspace{0.17em}t}\hspace{0.17em}$ for all initial states satisfying $\Vert x\left(0\right)\Vert \hspace{0.17em}\hspace{0.17em}<\hspace{0.17em}\hspace{0.17em}1$.

Figure 3 and Figure 4 illustrate the simulation results. The state trajectories with the initial conditions $x\left(0\right)\hspace{0.17em}=\hspace{0.17em}{\left[0.1\hspace{0.17em}\hspace{0.17em}0.5\hspace{0.17em}\hspace{0.17em}0.35\right]}^T$ and the control of the non-minimum phase switched nonlinear system (7) with uncertainty are depicted in these pictures.

The solution of the non-minimum phase switched nonlinear system (57) is asymptotically convergent, as shown in Figure 3 and Figure 4.

It is obvious that the suggested control method has excellent performance and significant resilience in the face of uncertainty.

6. Conclusions

In this paper, we examine at non-minimum phase switched nonlinear systems with uncertainty. Based on the input–output feedback linearization technique and the Lyapunov stability theory, we developed a switching control strategy. Using extended Gronwall–Bellman inequalities, we present stability results for systems with arbitrary switching rules. To explain the major conclusions, an illustrative case is also provided. Future work will involve applying the suggested method to the output tracking trajectory issue for uncertain nonlinear switching non-minimum phase systems.