Global Output Feedback Stabilization for a Class of Nonlinear Cascade Systems

Abstract

This paper focuses on the problem of global output feedback stabilization for a class of nonlinear cascade systems with time-varying output function. By using double-domination approach, an output feedback controller is developed to guarantee the global asymptotic stability of closed-loop system. The novel control strategy successfully constructs a unified Lyapunov function, which is suitable for both upper-triangular and lower-triangular systems. Finally, two numerical examples are provided to illustrate the effectiveness of a control strategy.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

It is well known that global output feedback stabilization is viewed as one of the most challenging fields of nonlinear control. Researchers have not yet found any unified way to handle the problem of global output feedback stabilization because the measure of states is difficult. Fortunately, with the help of nonseparation principle [1], homogeneous domination approach [2], and backstepping method, many interesting results such as [3–11] have been achieved.

It is worth pointing out that the structures of system output and nonlinear functions determine the possible forms of observer and controller. More specifically, the uncertainty of nonlinearities has led to the emergence of many kinds of observers, including high-gain observer, homogeneous observer, and time-varying observer. For example, [12] solved the problem of global output feedback stabilization based on linear high-gain observer for a class of uncertain nonlinear systems, where controller is independent of higher-order nonlinearities. Under uncertain linear growth condition in [13], a dynamic high-gain observer is proposed without requiring precise information of output function. References [14, 15] achieved system global stabilization by using time-varying observer, which uses the appropriate functions of time, rather than the dynamic compensator. Since some nonlinear functions satisfy neither the linear growth nor Lipschitz condition in practice, the existing approaches are not suitable. Therefore, [16–19] proposed homogeneous domination method to overcome this obstacle. Based on the existing results, some special observers are proposed, such as dual-observer [20] and reduced-observer [21]. In practice, complex systems are usually composed of simple subsystems. Therefore, cascade systems have become one of the most interesting topics of nonlinear systems. A great deal of research has been devoted to this subject over the last decades, as evidenced by the comprehensive books of [22, 23]. However, when zero-dynamics exist and obey mild conditions, the tracking problem cannot be solved by trivially extending the corresponding results without zero-dynamics; that is, there do not exist appropriate observers to tracking states of cascade systems. As further investigation, researchers now consider cascade connections in which the nonlinear systems are globally stable, but the input subsystem is more complex than just an integrator; for instance, [24–26] successfully investigated output feedback stabilization for uncertain cascade systems under growth condition. Regrettably, their approaches are only suitable for lower-triangular cascade systems. On the other hand, some literatures [27] achieved global output feedback stabilization when output function depends only on a state. References [28, 29] required that output function be continuous differentiability and initial value equals zero when output is unknown. The above conditions are restrictive; researchers turned to study time-varying output function. For instance, [30] further investigated the problem of global output feedback stabilization for a class of nonlinear systems with unknown measurement sensitivity. Meanwhile, a new method, namely, dual-domination approach, is proposed in [30].

In view of the above argument, an interesting question is proposed simultaneously: Is it possible to find a new approach to solve the problem of global output feedback stabilization for nonlinear cascade systems with unknown time-varying output function, which is suitable for both upper-triangular and lower-triangular systems? Based on above analysis and references, we will solve aforementioned question and provide satisfactory answer. There are three troublesome difficulties throughout the paper. The first is to find the appropriate Lyapunov function that is independent of the derivative of output function, since output function is unknown and does not satisfy differentiability. The second is to choose allowable sensitivity error, since it appears in the construction of controller. The third is to design rational observers to successfully track states, since nonlinearities and output function are unknown. A novel observer is proposed, which is different from the existing results [24, 25].

The main contributions of this paper are divided into three aspects: (i) double-domination approach is provided to handle time-varying sensitivity and uncertain nonlinearities, which is suitable for both upper-triangular and lower-triangular systems; (ii) linear observer does not rely on precise information of nonlinearities and output function; (iii) the construction of Lyapunov function avoids the use of the differentiability of output function.

2. Preliminaries and Problem Formulation

2.1. Preliminaries

We will adopt the following notations throughout this paper. $\mathbb{R}$ denotes the set of real numbers, ${\mathbb{R}}^{+}$ denotes the set of all nonnegative real numbers, and ${\mathbb{R}}^n$ denotes Euclidean space with dimension $n$. For any real vector $x=[x_{\mathrm{1}},\dots ,x_n{]}^T\in {\mathbb{R}}^n$, the norm ${‖x‖}_p$ is defined by ${‖x‖}_p={\left({\sum }_{i=\mathrm{1}}^n{x}_i^p\right)}^{\mathrm{1}/p}$, $\mathrm{1}\le p<\infty$. For matrix $A={\left(a_{ij}\right)}^{n\times n}$, $i=\mathrm{1},\dots ,n$, $j=\mathrm{1},\dots ,n$, $‖A‖$ denotes the norm $‖A‖=\sqrt{{\lambda }_{max}(A^{T}A)}$, where ${\lambda }_{max}(A^{T}A)$ denotes maximum eigenvalue of square matrix $A^{T}A$. ${\mathcal{C}}^i$ denotes the set of all functions with continuous $i$th partial derivatives. A continuous function $\alpha :[\mathrm{0},a)\to [\mathrm{0},\infty )$ is said to belong to class $\mathcal{K}$ if it is strictly increasing and $\alpha (\mathrm{0})=\mathrm{0}$. It is said to belong to class ${\mathcal{K}}_{\infty }$ if $a=\infty$ and $\alpha (r)=\infty$ as $r\to \infty$. For a continuously differentiable function $V:{\mathbb{R}}^n\to {\mathbb{R}}^{+}$, it is positive definite if $V(x)\ge \mathrm{0}$ and $V(x)=\mathrm{0}$ if and only if $x=\mathrm{0}$; it is radially unbounded if $V(x)\to \infty$ as $‖x‖\to \infty$. The arguments of functions are sometimes simplified; for instance, a function $f(x(t))$ is denoted by $f(x)$ or $f$ and ${‖x‖}_{\mathrm{2}}$ is denoted by $‖x‖$.

In the following, we list three lemmas that play an important role in proving the main results, and their proofs can be found in [31–33].

Lemma 1 (see [31]).

Let $c$ and $d$ be positive constants; given any positive real-valued function $\gamma (x,y)$, the following inequality holds:$$\begin{array}{}\text{(1)}& {\left|x\right|}^c{\left|y\right|}^d\le \frac{c}{c+d}\gamma \left(x,y\right){\left|x\right|}^{c+d}+\frac{d}{c+d}\gamma {\left(x,y\right)}^{-c/d}{\left|y\right|}^{c+d}.\end{array}$$

Lemma 2 (see [32]).

For any $x={\left[x_{\mathrm{1}},\dots ,x_n\right]}^T\in {\mathbb{R}}^n$, the following inequality holds: $$\begin{array}{}\text{(2)}& {‖x‖}_{\mathrm{2}}\le {‖x‖}_{\mathrm{1}}\le \sqrt{n}{‖x‖}_{\mathrm{2}}.\end{array}$$

Lemma 3 (see [33]).

For $x\in \mathbb{R}$ and $y\in \mathbb{R}$, $p\ge \mathrm{1}$ is an integer, and the following inequality holds:$$\begin{array}{}\text{(3)}& {\left|x+y\right|}^p\le {\mathrm{2}}^{p-\mathrm{1}}\left|x^p+y^p\right|.\end{array}$$

2.2. Problem Formulation

This paper investigates the nonlinear cascade system described by$$\begin{array}{c}\text{(4)}& \dot{z}=f_{\mathrm{0}}\left(t,z,x,u\right),{\dot{x}}_i=x_{i+\mathrm{1}}+f_i\left(t,z,x,u\right),\hspace{1em}i=\mathrm{1},\dots ,n-\mathrm{1},\\ {\dot{x}}_n=u+f_n\left(t,z,x,u\right),\\ y\left(t\right)=\theta \left(t\right)x_{\mathrm{1}}\left(t\right),\end{array}$$where $x(t)={\left[x_{\mathrm{1}}\left(t\right),\dots ,x_n\left(t\right)\right]}^T\in {\mathbb{R}}^n$ and $z(t)\in {\mathbb{R}}^m$ are system states with the initial values $x(\mathrm{0})=x_{\mathrm{0}}$, $z(\mathrm{0})=z_{\mathrm{0}}$, $u(t)\in \mathbb{R}$, $y(t)\in \mathbb{R}$ being control input and output, respectively. $f_{\mathrm{0}}:{\mathbb{R}}^{+}\times {\mathbb{R}}^m\times {\mathbb{R}}^n\times \mathbb{R}\to {\mathbb{R}}^m$ is continuous function with $f_{\mathrm{0}}(t,\mathrm{0,0},\mathrm{0})=\mathrm{0}$ and globally Lipschitz with respect to $x$; $f_i:{\mathbb{R}}^{+}\times {\mathbb{R}}^m\times {\mathbb{R}}^n\times \mathbb{R}\to \mathbb{R}$, $i=\mathrm{1},\dots ,n$, are continuous functions with $f_i(t,\mathrm{0,0},\mathrm{0})=\mathrm{0}$. $\theta (t):{\mathbb{R}}^{+}\to \mathbb{R}$ is a continuous function that represents time-varying sensitivity.

The following assumptions are needed.

Assumption 4.

For the continuous function $\theta (t)$, there is a known positive parameter $\overline{\theta }$ satisfying $\left|\mathrm{1}-\theta \left(t\right)\right|\le \overline{\theta }<\tilde{\theta }<\mathrm{1}$, where $\overline{\theta }$ is an allowable sensitivity error and $\tilde{\theta }$ is the upper bound of the allowable sensitivity error.

Assumption 5.

There exists a positive-definite and radially unbounded function $U(z)\in {\mathcal{C}}^{\mathrm{2}}$ such that$$\begin{array}{c}\text{(5)}& \frac{\partial U}{\partial z}{f}_{\mathrm{0}}\left(t,z,\mathrm{0},u\right)\le -c_{\mathrm{1}}{‖z‖}^{\mathrm{2}},‖\frac{\partial U}{\partial z}‖\le c_{\mathrm{2}}‖z‖,\end{array}$$where $c_{\mathrm{1}}\in (c_{\mathrm{1}}^{\ast },\infty )$, $c_{\mathrm{2}}\in \left[\mathrm{0},c_{\mathrm{2}}^{\ast }\right)$, and $c_{\mathrm{1}}^{\ast }$ and $c_{\mathrm{2}}^{\ast }$ are positive constants.

Assumption 6.

There exists a constant $c\ge \mathrm{0}$ such that$$\begin{array}{c}\text{(6)}& \left|f_i\left(·\right)\right|\le c\left(‖z‖+\left|x_{i+\mathrm{2}}\right|+\cdots +\left|x_n\right|\right),\hspace{1em}i=\mathrm{1},\dots ,n-\mathrm{2},\left|f_{n-\mathrm{1}}\left(·\right)\right|\le c‖z‖,\\ \left|f_n\left(·\right)\right|\le c‖z‖.\end{array}$$

Remark 7.

Since output function contains unknown parameter, it implies that the scope of this paper is more general than [1, 2, 12, 16] whose output function is equal to $x_{\mathrm{1}}$.

Remark 8.

In terms of the appearance of $\theta (t)$, two obstacles will be encountered. The first is to find an appropriate observer, which does not use the information of output function. The second is to find the feasible range of $\theta (t)$, because the information of $\theta (t)$ will be used in the design of controller.

3. Main Results

3.1. Output Feedback Controller Design for Upper-Triangular Case

We now summarise main results of this paper.

Theorem 9.

For system (4) under Assumptions 4–6, there exists an output feedback controller such that states of the closed-loop system are uniformly bounded over $[\mathrm{0},+\infty )$ and ${\mathrm{l}\mathrm{i}\mathrm{m}}_{t\to +\infty }x(t)=\mathrm{0}$.

Proof.

The proof is in four parts. At first, a linear observer with a domination gain is introduced to reconstruct all the states. Secondly, an output feedback controller composed of another domination gain is constructed to counteract the destabilized terms. Finally, a delicate selection of gains is provided and strict analysis is performed to guarantee that the closed-loop systems are globally asymptotically stable.

Part I: Design of an Observer. We construct the following linear observer:$$\begin{array}{c}\text{(7)}& {\dot{\widehat{x}}}_i={\widehat{x}}_{i+\mathrm{1}}-\frac{a_i}{L^i}{\widehat{x}}_{\mathrm{1}},\hspace{1em}i=\mathrm{1},\dots ,n-\mathrm{1},{\dot{\widehat{x}}}_n=u-\frac{a_n}{L^n}{\widehat{x}}_{\mathrm{1}},\end{array}$$where $a_i>\mathrm{0}$ are coefficients of Hurwitz polynomial $p_{\mathrm{1}}(s)=s^n+a_{\mathrm{1}}{s}^{n-\mathrm{1}}+\cdots +a_{n-\mathrm{1}}s+a_n$ and the domination gain $L\ge \mathrm{1}$ will be determined later. Define the estimation error as follows:$$\begin{array}{}\text{(8)}& {\epsilon }_i=L^{i-\mathrm{1}}\left(x_i-{\widehat{x}}_i\right),\hspace{1em}i=\mathrm{1},\dots ,n.\end{array}$$Then, the error equation can be rewritten as$$\begin{array}{}\text{(9)}& \dot{\epsilon }=\frac{\mathrm{1}}{L}{A}_{\epsilon }\epsilon +\frac{\mathrm{1}}{L}{D}_{\epsilon }{x}_{\mathrm{1}}+F,\end{array}$$where$$\begin{array}{c}\text{(10)}& \epsilon =\left[\begin{array}{c}{\epsilon }_{\mathrm{1}}\\ {\epsilon }_{\mathrm{2}}\\ ⋮\\ {\epsilon }_n\end{array}\right],A_{\epsilon }=\left[\begin{array}{cccc}-a_{\mathrm{1}}& \mathrm{1}& \cdots & \mathrm{0}\\ ⋮& ⋮& \ddots & ⋮\\ -a_{n-\mathrm{1}}& \mathrm{0}& \cdots & \mathrm{1}\\ -a_n& \mathrm{0}& \cdots & \mathrm{0}\end{array}\right],\\ D_{\epsilon }=\left[\begin{array}{c}{a}_{\mathrm{1}}\\ a_{\mathrm{2}}\\ ⋮\\ a_n\end{array}\right],\\ F=\left[\begin{array}{c}{f}_{\mathrm{1}}\left(·\right)\\ L{f}_{\mathrm{2}}\left(·\right)\\ ⋮\\ L^{n-\mathrm{1}}{f}_n\left(·\right)\end{array}\right].\end{array}$$Since $A_{\epsilon }$ is Hurwitz, there exists a symmetric positive-definite matrix $P_{\mathrm{1}}\in {\mathbb{R}}^{n\times n}$ such that $A_{\epsilon }^T{P}_{\mathrm{1}}+P_{\mathrm{1}}{A}_{\epsilon }=-I$, where $I$ is an identity matrix of ${\mathbb{R}}^{n\times n}$. Consider positive-definite and radially unbounded function $V_{\epsilon }(\epsilon )={\epsilon }^T{P}_{\mathrm{1}}\epsilon$; a direct calculation gives$$\begin{array}{}\text{(11)}& {\dot{V}}_{\epsilon }=-\frac{\mathrm{1}}{L}{\epsilon }^T\epsilon +\mathrm{2}\frac{\mathrm{1}}{L}{\epsilon }^T{P}_{\mathrm{1}}{D}_{\epsilon }{x}_{\mathrm{1}}+\mathrm{2}{\epsilon }^T{P}_{\mathrm{1}}F\le -\frac{\mathrm{1}}{L}{‖\epsilon ‖}^{\mathrm{2}}+\mathrm{2}\frac{\mathrm{1}}{L}‖\epsilon ‖·‖D_{\epsilon }‖·‖P_{\mathrm{1}}‖·\left|x_{\mathrm{1}}\right|+\mathrm{2}‖\epsilon ‖·‖P_{\mathrm{1}}‖·‖F‖.\end{array}$$According to Assumption 6 and Lemma 2, we obtain$$\begin{array}{}\text{(12)}& ‖F‖=\sqrt{{\left|f_{\mathrm{1}}\right|}^{\mathrm{2}}+L^{\mathrm{2}}{\left|f_{\mathrm{2}}\right|}^{\mathrm{2}}+\cdots +L^{\mathrm{2}\left(n-\mathrm{1}\right)}{\left|f_n\right|}^{\mathrm{2}}}\le \left|f_{\mathrm{1}}\right|+L\left|f_{\mathrm{2}}\right|+\cdots +L^{n-\mathrm{1}}\left|f_n\right|\le c{L}^{n-\mathrm{1}}n‖z‖+{\displaystyle \sum }_{i=\mathrm{3}}^{n}c{L}^{i-\mathrm{3}}\left(i-\mathrm{2}\right)\left|x_i\right|.\end{array}$$Substituting (12) into (11), one has$$\begin{array}{}\text{(13)}& {\dot{V}}_{\epsilon }\le -\frac{\mathrm{1}}{L}{‖\epsilon ‖}^{\mathrm{2}}+\mathrm{2}c‖P_{\mathrm{1}}‖·‖\epsilon ‖\left(L^{n-\mathrm{1}}n‖z‖+{\displaystyle \sum }_{i=\mathrm{3}}^n{L}^{i-\mathrm{3}}\left(i-\mathrm{2}\right)\left|x_i\right|\right)+\frac{\mathrm{2}}{L}‖P_{\mathrm{1}}‖·‖\epsilon ‖·‖D_{\epsilon }‖·\left|x_{\mathrm{1}}\right|.\end{array}$$In addition, Lemma 1 shows that$$\begin{array}{}\text{(14)}& \frac{\mathrm{2}}{L}‖P_{\mathrm{1}}‖·‖\epsilon ‖·‖D_{\epsilon }‖·\left|x_{\mathrm{1}}\right|\le \frac{\mathrm{1}}{\mathrm{2}L}{‖\epsilon ‖}^{\mathrm{2}}+\frac{\mathrm{1}}{L}{K}_{\mathrm{2}}{\left|x_{\mathrm{1}}\right|}^{\mathrm{2}},\end{array}$$and$$\begin{array}{}\text{(15)}& \mathrm{2}c‖P_{\mathrm{1}}‖·‖\epsilon ‖\left(L^{n-\mathrm{1}}n‖z‖+{\displaystyle \sum }_{i=\mathrm{3}}^n{L}^{i-\mathrm{3}}\left(i-\mathrm{2}\right)\left|x_i\right|\right)\le {\displaystyle \sum }_{i=\mathrm{3}}^n{K}_i{L}^{\mathrm{2}\left(i-\mathrm{2}\right)}{x}_i^{\mathrm{2}}+\frac{K_{\mathrm{1}}}{L^{\mathrm{2}}}{‖\epsilon ‖}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}cn{L}^{\mathrm{2}n}‖P_{\mathrm{1}}‖·{‖z‖}^{\mathrm{2}},\end{array}$$where $K_{\mathrm{1}}=c(\mathrm{2}n+(n-\mathrm{1})(n-\mathrm{2}))‖P_{\mathrm{1}}‖\ge \mathrm{0}$, $K_{\mathrm{2}}=\mathrm{2}{‖D_{\epsilon }‖}^{\mathrm{2}}{‖P_{\mathrm{1}}‖}^{\mathrm{2}}>\mathrm{0}$, and $K_i=\left(\mathrm{1}/\mathrm{2}\right)c(i-\mathrm{2})‖P_{\mathrm{1}}‖\ge \mathrm{0},\hspace{0.17em}\hspace{0.17em}i=\mathrm{3},\dots ,n$, are independent of a domination gain $L$. Therefore, substituting (14) and (15) into (13) yields$$\begin{array}{}\text{(16)}& {\dot{V}}_{\epsilon }\le -\frac{\mathrm{1}}{\mathrm{2}L}{‖\epsilon ‖}^{\mathrm{2}}+\frac{K_{\mathrm{1}}}{L^{\mathrm{2}}}{‖\epsilon ‖}^{\mathrm{2}}+\frac{\mathrm{1}}{L}{K}_{\mathrm{2}}{\left|x_{\mathrm{1}}\right|}^{\mathrm{2}}+{\displaystyle \sum }_{i=\mathrm{3}}^n{K}_i{L}^{\mathrm{2}\left(i-\mathrm{2}\right)}{x}_i^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}cn{L}^{\mathrm{2}n}‖P_{\mathrm{1}}‖·{‖z‖}^{\mathrm{2}}.\end{array}$$

Part II: Construction of a Controller. Consider the following system:$$\begin{array}{c}\text{(17)}& {\dot{x}}_{\mathrm{1}}=x_{\mathrm{2}}+f_{\mathrm{1}}\left(t,z,x,u\right),{\dot{\widehat{x}}}_i={\widehat{x}}_{i+\mathrm{1}}+\frac{a_i}{L^i}\left({\epsilon }_{\mathrm{1}}-x_{\mathrm{1}}\right),\hspace{1em}i=\mathrm{2},\dots ,n-\mathrm{1},\\ {\dot{\widehat{x}}}_n=u+\frac{a_n}{L^n}\left({\epsilon }_{\mathrm{1}}-x_{\mathrm{1}}\right).\end{array}$$Define the following change of coordinates:$$\begin{array}{c}\text{(18)}& {\xi }_{\mathrm{1}}=x_{\mathrm{1}},{\xi }_i=\frac{L^{i-\mathrm{1}}}{H^{i-\mathrm{1}}}{\widehat{x}}_i,\\ \upsilon =\frac{L^n}{H^n}u,\\ \hspace{1em}i=\mathrm{2},\dots ,n,\end{array}$$where $H\ge \mathrm{1}$ is a domination gain that will be determined later. Using above coordinate transform, (17) can be rewritten as$$\begin{array}{c}\text{(19)}& {\dot{\xi }}_{\mathrm{1}}=\frac{H}{L}{\xi }_{\mathrm{2}}+\frac{\mathrm{1}}{L}{\epsilon }_{\mathrm{2}}+f_{\mathrm{1}}\left(t,z,x,u\right),{\dot{\xi }}_i=\frac{H}{L}{\xi }_{i+\mathrm{1}}+\frac{a_i}{L{H}^{i-\mathrm{1}}}\left({\epsilon }_{\mathrm{1}}-{\xi }_{\mathrm{1}}\right),\hspace{1em}i=\mathrm{2},\dots ,n-\mathrm{1},\\ {\dot{\xi }}_n=\frac{H}{L}\upsilon +\frac{a_n}{L{H}^{n-\mathrm{1}}}\left({\epsilon }_{\mathrm{1}}-{\xi }_{\mathrm{1}}\right).\end{array}$$Design the output feedback control law as follows:$$\begin{array}{}\text{(20)}& \upsilon =-b_{n}y-b_{n-\mathrm{1}}{\xi }_{\mathrm{2}}-\cdots -b_{\mathrm{2}}{\xi }_{n-\mathrm{1}}-b_{\mathrm{1}}{\xi }_n,\end{array}$$where $b_i>\mathrm{0}$, $i=\mathrm{1},\dots ,n$, are coefficients of Hurwitz polynomial $p_{\mathrm{2}}(s)=s^n+b_{\mathrm{1}}{s}^{n-\mathrm{1}}+\cdots +b_{n-\mathrm{1}}s+b_n$. Substituting (20) into (19), we have$$\begin{array}{}\text{(21)}& \dot{\xi }=\frac{H}{L}{A}_{\xi }\xi +\frac{H}{L}{D}_{\xi }{b}_n\left(\mathrm{1}-\theta \left(t\right)\right){\xi }_{\mathrm{1}}+\frac{B_{\mathrm{2}}}{L}{\epsilon }_{\mathrm{2}}+\frac{B_{\mathrm{1}}}{LH}\left({\epsilon }_{\mathrm{1}}-{\xi }_{\mathrm{1}}\right)+\overline{F},\end{array}$$where$$\begin{array}{c}\text{(22)}& A_{\xi }=\left[\begin{array}{cccc}\mathrm{0}& \mathrm{1}& \cdots & \mathrm{0}\\ ⋮& ⋮& \ddots & ⋮\\ \mathrm{0}& \mathrm{0}& \cdots & \mathrm{1}\\ -b_n& -b_{n-\mathrm{1}}& \cdots & -b_{\mathrm{1}}\end{array}\right],B_{\mathrm{1}}=\left[\begin{array}{c}\mathrm{0}\\ a_{\mathrm{2}}\\ \frac{a_{\mathrm{3}}}{H}\\ ⋮\\ \frac{a_n}{H^{n-\mathrm{2}}}\end{array}\right],\\ B_{\mathrm{2}}=\left[\begin{array}{c}\mathrm{1}\\ \mathrm{0}\\ ⋮\\ \mathrm{0}\end{array}\right],\\ D_{\xi }=\left[\begin{array}{c}\mathrm{0}\\ ⋮\\ \mathrm{0}\\ \mathrm{1}\end{array}\right],\\ \overline{F}=\left[\begin{array}{c}{f}_{\mathrm{1}}\left(·\right)\\ \mathrm{0}\\ ⋮\\ \mathrm{0}\end{array}\right].\end{array}$$$A_{\xi }$ is a Hurwitz matrix that shows that there exists a symmetric positive-definite matrix $P_{\mathrm{2}}$ such that $A_{\xi }^T{P}_{\mathrm{2}}+P_{\mathrm{2}}{A}_{\xi }=-I$. Choose scalar function $V_{\xi }(\xi )={\xi }^T{P}_{\mathrm{2}}\xi$, which is positive-definite and radially bounded. Noting that $‖B_{\mathrm{2}}‖=‖D_{\xi }‖=\mathrm{1}$, the time derivative of $V_{\xi }(\xi )$ along the trajectories of (21) is$$\begin{array}{}\text{(23)}& {\dot{V}}_{\xi }=-\frac{H}{L}{\xi }^T\xi +\frac{\mathrm{2}H}{L}{\xi }^T{P}_{\mathrm{2}}{D}_{\xi }{b}_n\left(\mathrm{1}-\theta \left(t\right)\right){\xi }_{\mathrm{1}}+\mathrm{2}{\xi }^T{P}_{\mathrm{2}}\overline{F}+\mathrm{2}{\xi }^T{P}_{\mathrm{2}}\left(\frac{B_{\mathrm{1}}}{LH}{\epsilon }_{\mathrm{1}}+\frac{B_{\mathrm{2}}}{L}{\epsilon }_{\mathrm{2}}-\frac{B_{\mathrm{1}}}{LH}{\xi }_{\mathrm{1}}\right)\le -\frac{H}{L}{‖\xi ‖}^{\mathrm{2}}+\mathrm{2}{b}_n\left|1-\theta \left(t\right)\right|\frac{H}{L}‖P_{\mathrm{2}}‖·{‖\xi ‖}^{\mathrm{2}}+\mathrm{2}‖\xi ‖·‖P_{\mathrm{2}}‖·\left|f_1\right|+\mathrm{2}‖\xi ‖·‖P_{\mathrm{2}}‖\left(\frac{\mathrm{1}}{L}‖\epsilon ‖+\frac{‖B_{\mathrm{1}}‖}{LH}‖\epsilon ‖\right)+\frac{\mathrm{2}}{LH}‖P_{\mathrm{2}}‖·‖B_{\mathrm{1}}‖·{‖\xi ‖}^{\mathrm{2}}.\end{array}$$Firstly, by virtue of Assumption 6 and Lemma 1, it holds that$$\begin{array}{}\text{(24)}& \mathrm{2}‖\xi ‖·‖P_{\mathrm{2}}‖·\left|f_{\mathrm{1}}\right|\le \mathrm{2}c‖\xi ‖·‖P_{\mathrm{2}}‖\left(‖z‖+\left|x_{\mathrm{3}}\right|+\cdots +\left|x_n\right|\right)\le \frac{c\left(n-\mathrm{1}\right)}{L^{\mathrm{2}}}‖P_{\mathrm{2}}‖·{‖\xi ‖}^{\mathrm{2}}+c‖P_{\mathrm{2}}‖L^{\mathrm{2}}\left({‖z‖}^{\mathrm{2}}+x_{\mathrm{3}}^{\mathrm{2}}+\cdots +x_n^{\mathrm{2}}\right)\triangleq \frac{{\overline{K}}_{\mathrm{1}}}{L^{\mathrm{2}}}{‖\xi ‖}^{\mathrm{2}}+{\overline{K}}_{\mathrm{2}}{L}^{\mathrm{2}}{\displaystyle \sum }_{i=\mathrm{3}}^n{x}_i^{\mathrm{2}}+{\overline{K}}_{\mathrm{2}}{L}^{\mathrm{2}}{‖z‖}^{\mathrm{2}},\end{array}$$where ${\overline{K}}_{\mathrm{1}}=c(n-\mathrm{1})‖P_{\mathrm{2}}‖\ge \mathrm{0}$ and ${\overline{K}}_{\mathrm{2}}=c‖P_{\mathrm{2}}‖\ge \mathrm{0}$. Furthermore, it is deduced from Lemma 1 that$$\begin{array}{c}\text{(25)}& \frac{\mathrm{2}}{L}‖\xi ‖·‖P_{\mathrm{2}}‖·‖\epsilon ‖\le \frac{\mathrm{1}}{\mathrm{8}L}{‖\epsilon ‖}^{\mathrm{2}}+\frac{\mathrm{8}}{L}{‖P_{\mathrm{2}}‖}^{\mathrm{2}}·{‖\xi ‖}^{\mathrm{2}},\frac{\mathrm{2}\gamma }{LH}‖\xi ‖·‖P_{\mathrm{2}}‖·‖\epsilon ‖\le \frac{\mathrm{1}}{\mathrm{8}L}{‖\epsilon ‖}^{\mathrm{2}}+\frac{\mathrm{8}{\gamma }^{\mathrm{2}}}{L{H}^{\mathrm{2}}}{‖P_{\mathrm{2}}‖}^{\mathrm{2}}·{‖\xi ‖}^{\mathrm{2}},\end{array}$$and because of $H>\mathrm{1}$, one can conclude that$$\begin{array}{}\text{(26)}& ‖B_{\mathrm{1}}‖={\left(\sum _{i=\mathrm{2}}^n\frac{\mathrm{1}}{H^{\mathrm{2}\left(i-\mathrm{2}\right)}}{a}_i^{\mathrm{2}}\right)}^{\mathrm{1}/\mathrm{2}}\le {\left(\sum _{i=\mathrm{2}}^n{a}_i^{\mathrm{2}}\right)}^{\mathrm{1}/\mathrm{2}}\stackrel{\Delta }{=}\gamma .\end{array}$$Consequently, one obtains$$\begin{array}{}\text{(27)}& {\dot{V}}_{\xi }\le -\frac{H}{L}\left(\mathrm{1}-\mathrm{2}{b}_n\left|\mathrm{1}-\theta \left(t\right)\right|·‖P_{\mathrm{2}}‖\right){‖\xi ‖}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{4}L}{‖\epsilon ‖}^{\mathrm{2}}+\frac{\mathrm{8}{\gamma }^{\mathrm{2}}}{L{H}^{\mathrm{2}}}{‖P_{\mathrm{2}}‖}^{\mathrm{2}}·{‖\xi ‖}^{\mathrm{2}}+\frac{\mathrm{8}}{L}{‖P_{\mathrm{2}}‖}^{\mathrm{2}}·{‖\xi ‖}^{\mathrm{2}}+\frac{\mathrm{2}\gamma }{LH}‖P_{\mathrm{2}}‖·{‖\xi ‖}^{\mathrm{2}}+\frac{{\overline{K}}_{\mathrm{1}}}{L^{\mathrm{2}}}{‖\xi ‖}^{\mathrm{2}}+{\overline{K}}_{\mathrm{2}}{L}^{\mathrm{2}}{\displaystyle \sum }_{i=\mathrm{3}}^n{x}_i^{\mathrm{2}}+{\overline{K}}_{\mathrm{2}}{L}^{\mathrm{2}}{‖z‖}^{\mathrm{2}}\le -\frac{H}{L}\left(\mathrm{1}-\mathrm{2}{b}_n\left|\mathrm{1}-\theta \left(t\right)\right|·‖P_{\mathrm{2}}‖\right){‖\xi ‖}^{\mathrm{2}}+{\overline{K}}_{\mathrm{2}}{L}^{\mathrm{2}}{\displaystyle \sum }_{i=\mathrm{3}}^n{x}_i^{\mathrm{2}}+{\overline{K}}_{\mathrm{2}}{L}^{\mathrm{2}}{‖z‖}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{4}L}{‖\epsilon ‖}^{\mathrm{2}}+\frac{H}{L}\left(\frac{\mathrm{8}{\gamma }^{\mathrm{2}}}{H}{‖P_{\mathrm{2}}‖}^{\mathrm{2}}+\frac{\mathrm{8}}{H}{‖P_{\mathrm{2}}‖}^{\mathrm{2}}+\frac{\mathrm{2}\gamma }{H}‖P_{\mathrm{2}}‖+\frac{{\overline{K}}_{\mathrm{1}}}{L}\right){‖\xi ‖}^{\mathrm{2}}\triangleq -\frac{H}{L}\left(\mathrm{1}-\mathrm{2}{b}_n\left|\mathrm{1}-\theta \left(t\right)\right|·‖P_{\mathrm{2}}‖\right){‖\xi ‖}^{\mathrm{2}}+{\overline{K}}_{\mathrm{2}}{L}^{\mathrm{2}}{\displaystyle \sum }_{i=\mathrm{3}}^n{x}_i^{\mathrm{2}}+{\overline{K}}_{\mathrm{2}}{L}^{\mathrm{2}}{‖z‖}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{4}L}{‖\epsilon ‖}^{\mathrm{2}}+\frac{H}{L}\left(\frac{\overline{K_{\mathrm{3}}}}{H}+\frac{{\overline{K}}_{\mathrm{1}}}{L}\right){‖\xi ‖}^{\mathrm{2}},\end{array}$$where ${\overline{K}}_{\mathrm{3}}=\mathrm{2}\gamma \left(\mathrm{1}+\mathrm{4}\gamma ‖P_{\mathrm{2}}‖\right)‖P_{\mathrm{2}}‖+\mathrm{8}{‖P_{\mathrm{2}}‖}^{\mathrm{2}}>\mathrm{0}$ is independent of the domination gains $L$ and $H$.

Part III: Determination of Domination Gains. According to above arguments, it follows that$$\begin{array}{c}\text{(28)}& x_{\mathrm{1}}={\xi }_{\mathrm{1}},x_i=\frac{\mathrm{1}}{L^{i-\mathrm{1}}}{\epsilon }_i+\frac{H^{i-\mathrm{1}}}{L^{i-\mathrm{1}}}{\xi }_i,\hspace{1em}i=\mathrm{2},\dots ,n.\end{array}$$Using Lemma 3, there is$$\begin{array}{}\text{(29)}& x_i^{\mathrm{2}}={\left(\frac{\mathrm{1}}{L^{i-\mathrm{1}}}{\epsilon }_i+\frac{H^{i-\mathrm{1}}}{L^{i-\mathrm{1}}}{\xi }_i\right)}^{\mathrm{2}}\le \frac{\mathrm{2}}{L^{\mathrm{2}\left(i-\mathrm{1}\right)}}{‖\epsilon ‖}^{\mathrm{2}}+\frac{\mathrm{2}{H}^{\mathrm{2}\left(i-\mathrm{1}\right)}}{L^{\mathrm{2}\left(i-\mathrm{1}\right)}}{‖\xi ‖}^{\mathrm{2}},\end{array}$$and putting (29) together with (27), we have$$\begin{array}{}\text{(30)}& {\dot{V}}_{\xi }\le -\frac{H}{L}\left(\mathrm{1}-\mathrm{2}{b}_n\left|\mathrm{1}-\theta \left(t\right)\right|·‖P_{\mathrm{2}}‖\right){‖\xi ‖}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{4}L}{‖\epsilon ‖}^{\mathrm{2}}+\frac{H}{L}\left(\frac{\overline{K_{\mathrm{3}}}}{H}+\frac{{\overline{K}}_{\mathrm{1}}}{L}\right){‖\xi ‖}^{\mathrm{2}}+\mathrm{2}{\overline{K}}_{\mathrm{2}}{\displaystyle \sum }_{i=\mathrm{3}}^n\frac{\mathrm{1}}{L^{\mathrm{2}\left(i-\mathrm{2}\right)}}{‖\epsilon ‖}^{\mathrm{2}}+\mathrm{2}{\overline{K}}_{\mathrm{2}}{\displaystyle \sum }_{i=\mathrm{3}}^n\frac{H^{\mathrm{2}\left(i-\mathrm{1}\right)}}{L^{\mathrm{2}\left(i-\mathrm{2}\right)}}{‖\xi ‖}^{\mathrm{2}}+{\overline{K}}_{\mathrm{2}}{L}^{\mathrm{2}}{‖z‖}^{\mathrm{2}}\le -\frac{H}{L}\left(\mathrm{1}-\mathrm{2}{b}_n\left|\mathrm{1}-\theta \left(t\right)\right|·‖P_{\mathrm{2}}‖\right){‖\xi ‖}^{\mathrm{2}}+\frac{H}{L}\left(\frac{{\widehat{K}}_{\mathrm{3}}}{H}+\frac{{\widehat{K}}_{\mathrm{1}}}{L}\right){‖\xi ‖}^{\mathrm{2}}+\frac{\mathrm{1}}{L^{\mathrm{2}}}{\widehat{K}}_{\mathrm{4}}{‖\epsilon ‖}^{\mathrm{2}}+{\widehat{K}}_{\mathrm{2}}{L}^{\mathrm{2}}{‖z‖}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{4}L}{‖\epsilon ‖}^{\mathrm{2}},\end{array}$$where ${\widehat{K}}_{\mathrm{1}}={\overline{K}}_{\mathrm{1}}+\mathrm{2}{\overline{K}}_{\mathrm{2}}(H^{\mathrm{3}}+\cdots +H^{\mathrm{2}n-\mathrm{3}})\ge \mathrm{0},\hspace{0.17em}\hspace{0.17em}{\widehat{K}}_{\mathrm{2}}={\overline{K}}_{\mathrm{2}}\ge \mathrm{0},\hspace{0.17em}\hspace{0.17em}{\widehat{K}}_{\mathrm{3}}={\overline{K}}_{\mathrm{3}}>\mathrm{0},\hspace{0.17em}\hspace{0.17em}{\widehat{K}}_{\mathrm{4}}=\mathrm{2}(n-\mathrm{2}){\overline{K}}_{\mathrm{2}}\ge \mathrm{0}$. Now, we choose the allowable sensitivity error $\overline{\theta }$ as$$\begin{array}{}\text{(31)}& \overline{\theta }<\tilde{\theta }=\frac{\mathrm{1}}{\mathrm{2}{b}_n‖P_{\mathrm{2}}‖}.\end{array}$$Because of $\left|\mathrm{1}-\theta \right|\le \overline{\theta }$, one has$$\begin{array}{}\text{(32)}& \mathrm{1}-\mathrm{2}{b}_n\left|\mathrm{1}-\theta \right|·‖P_{\mathrm{2}}‖\ge \mathrm{1}-\mathrm{2}{b}_n\overline{\theta }‖P_{\mathrm{2}}‖\triangleq \beta .\end{array}$$Obviously, $\mathrm{0}<\beta <\mathrm{1}$ and (30) can be rewritten as$$\begin{array}{}\text{(33)}& {\dot{V}}_{\xi }\le -\frac{H}{L}\beta {‖\xi ‖}^{\mathrm{2}}+\frac{H}{L}\left(\frac{{\widehat{K}}_{\mathrm{3}}}{H}+\frac{{\widehat{K}}_{\mathrm{1}}}{L}\right){‖\xi ‖}^{\mathrm{2}}+\frac{\mathrm{1}}{L^{\mathrm{2}}}{\widehat{K}}_{\mathrm{4}}{‖\epsilon ‖}^{\mathrm{2}}+{\widehat{K}}_{\mathrm{2}}{L}^{\mathrm{2}}{‖z‖}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{4}L}{‖\epsilon ‖}^{\mathrm{2}}.\end{array}$$Similarly, substituting (29) into (16) yields$$\begin{array}{}\text{(34)}& {\dot{V}}_{\epsilon }\le -\frac{\mathrm{1}}{\mathrm{2}L}{‖\epsilon ‖}^{\mathrm{2}}+\frac{K_{\mathrm{2}}}{L}{‖\xi ‖}^{\mathrm{2}}+\frac{K_{\mathrm{1}}}{L^{\mathrm{2}}}{‖\epsilon ‖}^{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{2}}cn{L}^{\mathrm{2}n}‖P_{\mathrm{1}}‖{‖z‖}^{\mathrm{2}}+\mathrm{2}{\displaystyle \sum }_{i=\mathrm{3}}^n\frac{K_i}{L^{\mathrm{2}}}{‖\epsilon ‖}^{\mathrm{2}}+\mathrm{2}{\displaystyle \sum }_{i=\mathrm{3}}^n\frac{K_i{H}^{\mathrm{2}\left(i-\mathrm{1}\right)}}{L^{\mathrm{2}}}{‖\xi ‖}^{\mathrm{2}}\le -\frac{\mathrm{1}}{\mathrm{2}L}{‖\epsilon ‖}^{\mathrm{2}}+\frac{\mathrm{1}}{L^{\mathrm{2}}}{\widehat{K}}_{\mathrm{5}}{‖\epsilon ‖}^{\mathrm{2}}+\frac{\mathrm{1}}{L}{\widehat{K}}_{\mathrm{6}}{‖\xi ‖}^{\mathrm{2}}+\frac{H{\widehat{K}}_{\mathrm{7}}}{L^{\mathrm{2}}}{‖\xi ‖}^{\mathrm{2}}+{\widehat{K}}_{\mathrm{8}}{L}^{\mathrm{2}n}{‖z‖}^{\mathrm{2}},\end{array}$$where ${\widehat{K}}_{\mathrm{5}}=K_{\mathrm{1}}+\mathrm{2}(K_{\mathrm{3}}+\cdots +K_n)\ge \mathrm{0},\hspace{0.17em}\hspace{0.17em}{\widehat{K}}_{\mathrm{6}}=K_{\mathrm{2}}>\mathrm{0},{\hspace{0.17em}\hspace{0.17em}\widehat{K}}_{\mathrm{7}}=\mathrm{2}(K_{\mathrm{3}}{H}^{\mathrm{3}}+\cdots +K_n{H}^{\mathrm{2}n-\mathrm{3}})\ge \mathrm{0},\hspace{0.17em}\hspace{0.17em}{\widehat{K}}_{\mathrm{8}}=\left(\mathrm{1}/\mathrm{2}\right)cn‖P_{\mathrm{1}}‖\ge \mathrm{0}.$ Applying Assumption 5 and the property of Lipschitz function, one has$$\begin{array}{}\text{(35)}& \frac{\partial U}{\partial z}{f}_{\mathrm{0}}\left(t,z,x,u\right)=\frac{\partial U}{\partial z}{f}_{\mathrm{0}}\left(t,z,x,u\right)-\frac{\partial U}{\partial z}{f}_{\mathrm{0}}\left(t,z,\mathrm{0},u\right)+\frac{\partial U}{\partial z}{f}_{\mathrm{0}}\left(t,z,\mathrm{0},u\right)\le ‖\frac{\partial U}{\partial z}‖·‖f_{\mathrm{0}}\left(t,z,x,u\right)-f_{\mathrm{0}}\left(t,z,\mathrm{0},u\right)‖+\frac{\partial U}{\partial z}{f}_{\mathrm{0}}\left(t,z,\mathrm{0},u\right)\le {\overline{c}}_{\mathrm{2}}‖z‖·‖x‖-c_{\mathrm{1}}{‖z‖}^{\mathrm{2}},\end{array}$$where ${\overline{c}}_{\mathrm{2}}$ is a positive constant. By Lemma 1, it is easy to obtain that $‖x‖\le \left|x_{\mathrm{1}}\right|+\cdots +\left|x_n\right|$, and, according to Lemma 2, the following inequalities hold:$$\begin{array}{c}\text{(36)}& {\overline{c}}_{\mathrm{2}}‖z‖·\left(\left|x_{\mathrm{2}}\right|+\cdots +\left|x_n\right|\right)\le \frac{c_{\mathrm{1}}}{\mathrm{2}}{‖z‖}^{\mathrm{2}}+\frac{{\overline{c}}_{\mathrm{2}}^{\mathrm{2}}}{\mathrm{2}{c}_{\mathrm{1}}}\left(x_{\mathrm{2}}^{\mathrm{2}}+\cdots +x_n^{\mathrm{2}}\right),{\overline{c}}_{\mathrm{2}}‖z‖·\left|x_{\mathrm{1}}\right|\le \frac{{\overline{c}}_{\mathrm{2}}^{\mathrm{2}}}{\mathrm{2}L}{x}_{\mathrm{1}}^{\mathrm{2}}+\frac{L}{\mathrm{2}}{‖z‖}^{\mathrm{2}}.\end{array}$$Substituting (36) into (35), one has$$\begin{array}{}\text{(37)}& \frac{\partial U}{\partial z}{f}_{\mathrm{0}}\left(t,z,x,u\right)\le -\left(\frac{c_{\mathrm{1}}}{\mathrm{2}}-\frac{L}{\mathrm{2}}\right){‖z‖}^{\mathrm{2}}+\frac{{\overline{c}}_{\mathrm{2}}^{\mathrm{2}}}{\mathrm{2}{c}_{\mathrm{1}}}\left(x_{\mathrm{2}}^{\mathrm{2}}+\cdots +x_n^{\mathrm{2}}\right)+\frac{{\overline{c}}_{\mathrm{2}}^{\mathrm{2}}}{\mathrm{2}L}{x}_{\mathrm{1}}^{\mathrm{2}}.\end{array}$$According to (29) and the definition of norms, it is easy to deduce that$$\begin{array}{c}\text{(38)}& x_{\mathrm{1}}^{\mathrm{2}}\le {‖\xi ‖}^{\mathrm{2}},x_{\mathrm{2}}^{\mathrm{2}}+x_{\mathrm{3}}^{\mathrm{2}}+\cdots +x_n^{\mathrm{2}}\le \frac{\mathrm{2}\left(n-\mathrm{1}\right)}{L^{\mathrm{2}}}{‖\epsilon ‖}^{\mathrm{2}}+\frac{H}{L^{\mathrm{2}}}\mathrm{2}{\widehat{K}}_{\mathrm{9}}{‖\xi ‖}^{\mathrm{2}},\end{array}$$where ${\widehat{K}}_{\mathrm{9}}=H+H^{\mathrm{3}}+\cdots +H^{\mathrm{2}n-\mathrm{3}}$; therefore (37) can be further expressed as$$\begin{array}{}\text{(39)}& \frac{\partial U}{\partial z}{f}_{\mathrm{0}}\left(t,z,x,u\right)\le -\left(\frac{c_{\mathrm{1}}}{\mathrm{2}}-\frac{L}{\mathrm{2}}\right){‖z‖}^{\mathrm{2}}+\frac{{\overline{c}}_{\mathrm{2}}^{\mathrm{2}}\left(n-\mathrm{1}\right)}{c_{\mathrm{1}}{L}^{\mathrm{2}}}{‖\epsilon ‖}^{\mathrm{2}}+\left(\frac{{\overline{c}}_{\mathrm{2}}^{\mathrm{2}}{\widehat{K}}_{\mathrm{9}}H}{c_{\mathrm{1}}{L}^{\mathrm{2}}}+\frac{{\overline{c}}_{\mathrm{2}}^{\mathrm{2}}}{\mathrm{2}L}\right){‖\xi ‖}^{\mathrm{2}}.\end{array}$$Consider positive-definite and radially unbounded function as follows:$$\begin{array}{}\text{(40)}& V=U+V_{\epsilon }+V_{\xi },\end{array}$$and substituting (33), (34), and (39) into (40), there is$$\begin{array}{}\text{(41)}& \dot{V}\le -\left(\frac{c_{\mathrm{1}}}{\mathrm{2}}-\frac{L}{\mathrm{2}}-{\widehat{K}}_{\mathrm{8}}{L}^{\mathrm{2}n}-{\widehat{K}}_{\mathrm{2}}{L}^{\mathrm{2}}\right){‖z‖}^{\mathrm{2}}-\frac{\mathrm{1}}{L}\left(\frac{\mathrm{1}}{\mathrm{4}}-\frac{{\widehat{K}}_{\mathrm{5}}+{\widehat{K}}_{\mathrm{4}}}{L}-\frac{\mathrm{2}{\overline{c}}_{\mathrm{2}}^{\mathrm{2}}\left(n-\mathrm{1}\right)}{L}\right){‖\epsilon ‖}^{\mathrm{2}}-\frac{H}{L}\left(\beta -\frac{{\widehat{K}}_{\mathrm{6}}+{\widehat{K}}_{\mathrm{3}}}{H}-\frac{{\widehat{K}}_{\mathrm{1}}+{\widehat{K}}_{\mathrm{7}}}{L}-\frac{{\overline{c}}_{\mathrm{2}}^{\mathrm{2}}}{\mathrm{2}H}-\frac{{\overline{c}}_{\mathrm{2}}^{\mathrm{2}}{\widehat{K}}_{\mathrm{9}}}{L}\right){‖\xi ‖}^{\mathrm{2}}.\end{array}$$In order to ensure that $\dot{V}\le \mathrm{0}$, we choose the domination gains $L$ and $H$ as follows. Firstly,$$\begin{array}{}\text{(42)}& \beta -\frac{{\widehat{K}}_{\mathrm{6}}+{\widehat{K}}_{\mathrm{3}}}{H}-\frac{{\overline{c}}_{\mathrm{2}}^{\mathrm{2}}}{\mathrm{2}H}\ge \frac{\beta }{\mathrm{2}},\end{array}$$implying that$$\begin{array}{}\text{(43)}& H\ge \frac{\mathrm{2}{\widehat{K}}_{\mathrm{3}}+\mathrm{2}{\widehat{K}}_{\mathrm{6}}+{\overline{c}}_{\mathrm{2}}^{\mathrm{2}}}{\beta },\end{array}$$and, with $H\ge \mathrm{1}$ in hand, one deduces$$\begin{array}{}\text{(44)}& H\ge \max \left\{\mathrm{1},\frac{\mathrm{2}{\widehat{K}}_{\mathrm{3}}+\mathrm{2}{\widehat{K}}_{\mathrm{6}}+{\overline{c}}_{\mathrm{2}}^{\mathrm{2}}}{\beta }\right\}.\end{array}$$As consequence, (41) can be rewritten as$$\begin{array}{}\text{(45)}& \dot{V}\le -\left(\frac{c_{\mathrm{1}}}{\mathrm{2}}-\frac{L}{\mathrm{2}}-{\widehat{K}}_{\mathrm{8}}{L}^{\mathrm{2}n}-{\widehat{K}}_{\mathrm{2}}{L}^{\mathrm{2}}\right){‖z‖}^{\mathrm{2}}-\frac{\mathrm{1}}{L}\left(\frac{\mathrm{1}}{\mathrm{4}}-\frac{{\widehat{K}}_{\mathrm{5}}+{\widehat{K}}_{\mathrm{4}}}{L}-\frac{\mathrm{2}{\overline{c}}_{\mathrm{2}}^{\mathrm{2}}\left(n-\mathrm{1}\right)}{L}\right){‖\epsilon ‖}^{\mathrm{2}}-\frac{H}{L}\left(\frac{\beta }{\mathrm{2}}-\frac{{\widehat{K}}_{\mathrm{1}}+{\widehat{K}}_{\mathrm{7}}}{L}-\frac{{\overline{c}}_{\mathrm{2}}^{\mathrm{2}}{\widehat{K}}_{\mathrm{9}}}{L}\right){‖\xi ‖}^{\mathrm{2}}.\end{array}$$Secondly, we can select L to satisfy the following inequalities:$$\begin{array}{c}\text{(46)}& \frac{\mathrm{1}}{\mathrm{4}}-\frac{{\widehat{K}}_{\mathrm{5}}+{\widehat{K}}_{\mathrm{4}}}{L}-\frac{\mathrm{2}{\overline{c}}_{\mathrm{2}}^{\mathrm{2}}\left(n-\mathrm{1}\right)}{L}\ge \frac{\beta }{\mathrm{4}},\frac{\beta }{\mathrm{2}}-\frac{{\widehat{K}}_{\mathrm{1}}+{\widehat{K}}_{\mathrm{7}}}{L}-\frac{{\overline{c}}_{\mathrm{2}}^{\mathrm{2}}{\widehat{K}}_{\mathrm{9}}}{L}\ge \frac{\beta }{\mathrm{4}},\end{array}$$and, in light of $L\ge \mathrm{1}$, it is straightforward to show that$$\begin{array}{}\text{(47)}& L\ge \max \left\{\mathrm{1},\frac{\mathrm{4}{\widehat{K}}_{\mathrm{1}}+\mathrm{4}{\widehat{K}}_{\mathrm{7}}+\mathrm{4}{\overline{c}}_{\mathrm{2}}^{\mathrm{2}}{\widehat{K}}_{\mathrm{9}}}{\beta },\frac{\mathrm{4}{\widehat{K}}_{\mathrm{4}}+\mathrm{4}{\widehat{K}}_{\mathrm{5}}+\mathrm{8}{\overline{c}}_{\mathrm{2}}^{\mathrm{2}}\left(n-\mathrm{1}\right)}{\mathrm{1}-\beta }\right\},\end{array}$$and $c_{\mathrm{1}}$ satisfies$$\begin{array}{}\text{(48)}& c_{\mathrm{1}}=\frac{\beta }{\mathrm{4}}+c_{\mathrm{1}}^{\ast },\hspace{1em}{c}_{\mathrm{1}}^{\ast }\ge \mathrm{2}\left({\widehat{K}}_{\mathrm{8}}{L}^{\mathrm{2}n}+{\widehat{K}}_{\mathrm{2}}{L}^{\mathrm{2}}\right)+L.\end{array}$$Under above choice of domination gains, it is clear that$$\begin{array}{}\text{(49)}& \dot{V}\le -\frac{\beta }{\mathrm{4}}{‖z‖}^{\mathrm{2}}-\frac{\beta }{\mathrm{4}L}{‖\xi ‖}^{\mathrm{2}}-\frac{\beta }{\mathrm{4}L}{‖\epsilon ‖}^{\mathrm{2}}.\end{array}$$

Part IV: Stability Analysis. Consider transformed systems (9), (20), and (21). By the existence and continuity of solution, the closed-loop system state composed of $Y(t)={\left[z^T,x^T,{\widehat{x}}^T\right]}^T$ can be defined on a time interval $\left[\mathrm{0},t_m\right)$, where $t_m>\mathrm{0}$ may be a finite constant or infinity.

(i)  For the Boundedness of $Y(t)$. Due to the fact that ${\nu }_{\mathrm{1}}(z)\le U(z)\le {\nu }_{\mathrm{2}}(z)$, ${\lambda }_{min}(P){‖x‖}^{\mathrm{2}}\le x^{T}Px\le {\lambda }_{max}(P){‖x‖}^{\mathrm{2}}$, where ${\nu }_{\mathrm{1}}$ and ${\nu }_{\mathrm{2}}$ are ${\mathcal{K}}_{\infty }$ functions; we obtain$$\begin{array}{}\text{(50)}& V=U+V_{\epsilon }+V_{\xi }\ge {\nu }_{\mathrm{1}}\left(z\right)+{\lambda }_{min}\left(P_{\mathrm{1}}\right){‖\epsilon ‖}^{\mathrm{2}}+{\lambda }_{min}\left(P_{\mathrm{2}}\right){‖\xi ‖}^{\mathrm{2}}\ge {\nu }_{\mathrm{1}}\left(z\right)+\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}\left(P_{\mathrm{1}}\right),{\lambda }_{\mathrm{m}\mathrm{i}\mathrm{n}}\left(P_{\mathrm{2}}\right)\right\}{‖\omega \left(t\right)‖}^{\mathrm{2}}\triangleq {\phi }_{\mathrm{1}}\left(‖\varpi \left(t\right)‖\right),\end{array}$$and$$\begin{array}{}\text{(51)}& V=U+V_{\epsilon }+V_{\xi }\le {\nu }_{\mathrm{2}}\left(z\right)+{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(P_{\mathrm{1}}\right){‖\epsilon ‖}^{\mathrm{2}}+{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(P_{\mathrm{2}}\right){‖\xi ‖}^{\mathrm{2}}\le {\nu }_{\mathrm{2}}\left(z\right)+\mathrm{m}\mathrm{a}\mathrm{x}\left\{{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(P_{\mathrm{1}}\right),{\lambda }_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(P_{\mathrm{2}}\right)\right\}{‖\omega \left(t\right)‖}^{\mathrm{2}}\triangleq {\phi }_{\mathrm{2}}\left(‖\varpi \left(t\right)‖\right),\end{array}$$where $\omega (t)=[{\epsilon }^T(t),{\xi }^T(t){]}^T$, $\varpi (t)=[z^T,{\epsilon }^T(t),{\xi }^T(t){]}^T$, and ${\phi }_{\mathrm{1}},{\phi }_{\mathrm{2}}$ are ${\mathcal{K}}_{\infty }$ functions.

For any $ϵ>\mathrm{0}$, noting that ${\mathrm{l}\mathrm{i}\mathrm{m}}_{s\to +\infty }{\phi }_{\mathrm{1}}(s)=+\infty$, one can always find a constant $\beta =\beta (ϵ)>ϵ>\mathrm{0}$, such that ${\phi }_{\mathrm{2}}(ϵ)\le {\phi }_{\mathrm{1}}(\beta )$. From (49), it follows that $V\left(‖\varpi \left(t\right)‖\right)\le V(‖\varpi (0)‖)$, $\forall t\in [\mathrm{0},t_m)$. Then, if $‖\varpi (0)‖<ϵ$, it holds that$$\begin{array}{}\text{(52)}& {\phi }_{\mathrm{1}}\left(‖\varpi \left(t\right)‖\right)\le V\left(‖\varpi \left(t\right)‖\right)\le V\left(‖\varpi \left(\mathrm{0}\right)‖\right)\le {\phi }_{\mathrm{2}}\left(‖\varpi \left(\mathrm{0}\right)‖\right)<{\phi }_{\mathrm{2}}\left(ϵ\right)\le {\phi }_{\mathrm{1}}\left(\beta \right),\hspace{1em}\forall t\in \left[\mathrm{0},t_m\right),\end{array}$$which implies that $‖\varpi (t)‖<\beta$, $\forall t\in \left[\mathrm{0},t_m\right)$; that is, $‖\varpi (t)‖$ is bounded. Therefore, by virtue of estimation errors and coordinates transform, it is easy to get that the state $Y(t)={\left[z^T,x^T,{\widehat{x}}^T\right]}^T$ is bounded on $\left[\mathrm{0},t_m\right)$ as well.

(ii)  $t_m=+\infty$. This claim can be shown by contradiction; if $t_m$ is finite, then $t_m$ would be a finite escape time; that is, the state $Y(t)$ would tend to $\infty$ as $t=t_m$. However, the continuity of solution guarantees the boundedness of $Y(t)$ at $t=t_m$, since $Y(t)$ is bounded on $[\mathrm{0},t_m)$. This is clearly a contradiction. Therefore, the state of the closed-loop system is uniformly bounded over $[\mathrm{0},+\infty )$.

(iii)  ${\mathrm{l}\mathrm{i}\mathrm{m}}_{t\to +\infty }x(t)=\mathrm{0}$ and ${\mathrm{l}\mathrm{i}\mathrm{m}}_{t\to +\infty }z(t)=\mathrm{0}$. Above all, it is clear that$$\begin{array}{}\text{(53)}& \mathrm{0}\le {{\displaystyle \int }}_{\mathrm{0}}^t\left(\frac{\beta }{\mathrm{4}}{‖z‖}^{\mathrm{2}}+\frac{\beta }{\mathrm{4}L}{‖\epsilon ‖}^{\mathrm{2}}+\frac{\beta }{\mathrm{4}L}{‖\xi ‖}^{\mathrm{2}}\right)ds\le -{{\displaystyle \int }}_{\mathrm{0}}^t\dot{V}\left(\varpi \left(s\right)\right)ds<V\left(\varpi \left(\mathrm{0}\right)\right)<+\infty ,\hspace{1em}\forall t\ge \mathrm{0},\end{array}$$which implies that ${\int }_{\mathrm{0}}^{\infty }\left(\left(\beta /\mathrm{4}\right){‖z‖}^{\mathrm{2}}+\left(\beta /\mathrm{4}L\right){‖\epsilon ‖}^{\mathrm{2}}+\left(\beta /\mathrm{4}L\right){‖\xi ‖}^{\mathrm{2}}\right)ds$ exists and is finite. On the other hand, the boundedness of $\varpi (t)$ over $[\mathrm{0},+\infty )$ means that $\dot{\varpi }(t)$ is uniformly bounded in $t$ over $[\mathrm{0},+\infty )$. Thus, $\varpi (t)$ is uniformly continuous in $t$ over $[\mathrm{0},+\infty )$ and so is ${\varpi }^{\mathrm{2}}(t)$. Using well-known Barbalat’s Lemma in [22], one obtains ${\mathrm{l}\mathrm{i}\mathrm{m}}_{t\to +\infty }{z}_i^{\mathrm{2}}(t)=\mathrm{0}$, ${\mathrm{l}\mathrm{i}\mathrm{m}}_{t\to +\infty }{\xi }_i^{\mathrm{2}}(t)=\mathrm{0}$, and ${\mathrm{l}\mathrm{i}\mathrm{m}}_{t\to +\infty }{\epsilon }_i^{\mathrm{2}}(t)=\mathrm{0}$, which shows that ${\mathrm{l}\mathrm{i}\mathrm{m}}_{t\to +\infty }z(t)=\mathrm{0}$, ${\mathrm{l}\mathrm{i}\mathrm{m}}_{t\to +\infty }\xi (t)=\mathrm{0}$, and ${\mathrm{l}\mathrm{i}\mathrm{m}}_{t\to +\infty }\epsilon (t)=\mathrm{0}$. Finally, from (29), it is easy to see that ${\mathrm{l}\mathrm{i}\mathrm{m}}_{t\to +\infty }x(t)=\mathrm{0}$. This completes the proof.

Remark 10.

A double-domination method is proposed to handle the time-varying output function and nonlinearities in the proof of Theorem 9; that is, two domination gains $H$ and $L$ are used to dominate time-varying function $\theta (t)$ and nonlinearities $f_i$, respectively.

Remark 11.

It should be noted from $\tilde{\theta }=\mathrm{1}/\mathrm{2}{b}_n‖P_{\mathrm{2}}‖$ that the upper bound $\tilde{\theta }$ depends on coefficients $b_{\mathrm{1}},\dots ,b_n$ for the Hurwitz polynomial $p_{\mathrm{2}}(s)$. When $b_{\mathrm{1}},\dots ,b_n$ are specified, the corresponding matrix $P_{\mathrm{2}}$ as well as the upper bound $\tilde{\theta }$ can be computed. For example, $\tilde{\theta }=\mathrm{0.4420}$, for $n=\mathrm{2}$; $\tilde{\theta }=\mathrm{0.1501}$, for $n=\mathrm{3}$.

3.2. Extension to Lower-Triangular Case

Some subsystems do not satisfy upper-triangular structure in practical application, so we extend the subsystems to lower-triangular form and impose following assumptions on system (4).

Assumption 12.

For $i=\mathrm{1},\dots ,n$, there exists a constant $c\ge \mathrm{0}$ such that $$\begin{array}{}\text{(54)}& \left|f_i\right|\le c\left(‖z‖+\left|x_{\mathrm{1}}\right|+\cdots +\left|x_i\right|\right).\end{array}$$

Theorem 13.

For a class of nonlinear cascade system (4) under Assumptions 4, 5, and 12, there exists an output feedback controller, such that states of the closed-loop system are uniformly bounded over $[\mathrm{0},+\infty )$ and ${\mathrm{l}\mathrm{i}\mathrm{m}}_{t\to +\infty }x(t)=\mathrm{0}$.

Proof.

The proof is analogous to the proof of Theorem 9 with an obvious modification. To facilitate comparison, we select same notations as Theorem 9 and many similarities will be omitted.

Firstly, we construct the similar observer as (7)$$\begin{array}{c}\text{(55)}& {\dot{\widehat{x}}}_i={\widehat{x}}_{i+\mathrm{1}}-L^i{a}_i{\widehat{x}}_{\mathrm{1}},\hspace{1em}i=\mathrm{1},\dots ,n-\mathrm{1},{\dot{\widehat{x}}}_n=u-L^n{a}_n{\widehat{x}}_{\mathrm{1}},\end{array}$$where $a_i>\mathrm{0}$ and $L>\mathrm{1}$. With the help of the previous process, define the estimation error$$\begin{array}{}\text{(56)}& {\epsilon }_i=\frac{x_i-{\widehat{x}}_i}{L^{i-\mathrm{1}}},\hspace{1em}i=\mathrm{1},\dots ,n,\end{array}$$and it is straightforward to show that$$\begin{array}{}\text{(57)}& \dot{\epsilon }=L{A}_{\epsilon }\epsilon +L{G}_{\epsilon }{x}_{\mathrm{1}}+F,\end{array}$$and since the definitions of the associated symbol are same as (9), we just give different symbol in the following paper: $F={\left[f_{\mathrm{1}}(·),f_{\mathrm{2}}(·)/L,\dots ,f_n(·)/L^{n-\mathrm{1}}\right]}^T\in {\mathbb{R}}^{n\times \mathrm{1}}$. Consider same scalar function $V_{\epsilon }(\epsilon )={\epsilon }^T{P}_{\mathrm{1}}\epsilon$, which is proper and radially unbounded. Evidently, following (16), we arrive at$$\begin{array}{}\text{(58)}& {\dot{V}}_{\epsilon }\le \left(L{K}_{\mathrm{2}}+K_{\mathrm{3}}\right)x_{\mathrm{1}}^{\mathrm{2}}+\frac{K_{\mathrm{4}}}{L_{\mathrm{1}}^{\mathrm{2}}}{x}_{\mathrm{2}}^{\mathrm{2}}+\cdots +\frac{K_{n+\mathrm{2}}}{L_{\mathrm{1}}^{\mathrm{2}n-\mathrm{2}}}{x}_n^{\mathrm{2}}-\frac{L}{\mathrm{2}}{‖\epsilon ‖}^{\mathrm{2}}+K_{\mathrm{1}}{‖\epsilon ‖}^{\mathrm{2}}+K_{\mathrm{3}}{‖z‖}^{\mathrm{2}},\end{array}$$where $K_{\mathrm{1}}=cn(n+\mathrm{3})‖P_{\mathrm{1}}‖\ge \mathrm{0},\hspace{0.17em}\hspace{0.17em}{K}_{\mathrm{2}}=\mathrm{2}{‖G_{\epsilon }‖}^{\mathrm{2}}{‖P_{\mathrm{1}}‖}^{\mathrm{2}}>\mathrm{0},\hspace{0.17em}\hspace{0.17em}{K}_i=\left(\mathrm{1}/\mathrm{2}\right)(n-i+\mathrm{3})c‖P_{\mathrm{1}}‖\ge \mathrm{0},\hspace{0.17em}\hspace{0.17em}i=\mathrm{3},\dots ,n+\mathrm{2}$, are independent of a domination gain $L$.

Secondly, consider system described by$$\begin{array}{c}\text{(59)}& {\dot{x}}_{\mathrm{1}}=x_{\mathrm{2}}+f_{\mathrm{1}}\left(t,z,x,u\right),{\dot{\widehat{x}}}_i={\widehat{x}}_{i+\mathrm{1}}+L^i{a}_i\left({\epsilon }_{\mathrm{1}}-x_{\mathrm{1}}\right),\hspace{1em}i=\mathrm{2},\dots ,n-\mathrm{1},\\ {\dot{\widehat{x}}}_n=u+L^n{a}_n\left({\epsilon }_{\mathrm{1}}-x_{\mathrm{1}}\right).\end{array}$$Then, introduce the following transformations:$$\begin{array}{c}\text{(60)}& {\xi }_{\mathrm{1}}=x_{\mathrm{1}},{\xi }_i=\frac{{\widehat{x}}_i}{{\left(LH\right)}^{i-\mathrm{1}}},\\ \upsilon =\frac{u}{{\left(LH\right)}^n},\\ \hspace{1em}i=\mathrm{2},\dots ,n,\end{array}$$where $H\ge \mathrm{1}$ is a domination gain to be determined later. With the help of (60), it is easy to see that system (59) can be rewritten as$$\begin{array}{c}\text{(61)}& {\dot{\xi }}_{\mathrm{1}}=LH{\xi }_{\mathrm{2}}+L{\epsilon }_{\mathrm{2}}+f_{\mathrm{1}}\left(t,z,x,u\right),{\dot{\xi }}_i=LH{\xi }_{i+\mathrm{1}}+\frac{L{a}_i}{H^{i-\mathrm{1}}}\left({\epsilon }_{\mathrm{1}}-{\xi }_{\mathrm{1}}\right),\hspace{1em}i=\mathrm{2},\dots ,n-\mathrm{1},\\ {\dot{\xi }}_n=LH\upsilon +\frac{L{a}_n}{H^{n-\mathrm{1}}}\left({\epsilon }_{\mathrm{1}}-{\xi }_{\mathrm{1}}\right).\end{array}$$If the control law is designed as (20), then substituting (20) into (61) leads to$$\begin{array}{}\text{(62)}& \dot{\xi }=LH{A}_{\xi }\xi +LH{G}_{\xi }{b}_n\left(\mathrm{1}-\theta \left(t\right)\right){\xi }_{\mathrm{1}}+L{B}_{\mathrm{2}}{\epsilon }_{\mathrm{2}}+\frac{L}{H}{B}_{\mathrm{1}}\left({\epsilon }_{\mathrm{1}}-{\xi }_{\mathrm{1}}\right)+\overline{F},\end{array}$$where the definitions of notations are the same as (21). Consider the same quadratic function$$\begin{array}{}\text{(63)}& V_{\xi }\left(\xi \right)={\xi }^T{P}_{\mathrm{2}}\xi ,\end{array}$$and, after calculations, we arrive at$$\begin{array}{}\text{(64)}& {\dot{V}}_{\xi }\le -LH\beta {‖\xi ‖}^{\mathrm{2}}+\frac{L}{\mathrm{4}}{‖\epsilon ‖}^{\mathrm{2}}+LH\left(\frac{{\overline{K}}_{\mathrm{1}}}{H}+\frac{{\overline{K}}_{\mathrm{2}}}{L}\right){‖\xi ‖}^{\mathrm{2}}+{\overline{K}}_{\mathrm{3}}{‖z‖}^{\mathrm{2}},\end{array}$$where ${\overline{K}}_{\mathrm{1}}=\mathrm{2}\gamma \left(\mathrm{1}+\mathrm{4}\gamma ‖P_{\mathrm{2}}‖\right)‖P_{\mathrm{2}}‖+\mathrm{8}{‖P_{\mathrm{2}}‖}^{\mathrm{2}}>\mathrm{0},\hspace{0.17em}\hspace{0.17em}{\overline{K}}_{\mathrm{2}}=\mathrm{3}c‖P_{\mathrm{2}}‖\ge \mathrm{0},\hspace{0.17em}\hspace{0.17em}{\overline{K}}_{\mathrm{3}}=c‖P_{\mathrm{2}}‖\ge \mathrm{0}$ are independent of the domination gains $L$ and $H$. It follows from (56) and (60) that$$\begin{array}{c}\text{(65)}& x_{\mathrm{1}}={\xi }_{\mathrm{1}},x_i=L^{i-\mathrm{1}}{\epsilon }_i+{\widehat{x}}_i=L^{i-\mathrm{1}}{\epsilon }_i+{\left(LH\right)}^{i-\mathrm{1}}{\xi }_i,\hspace{1em}i=\mathrm{2},\dots ,n.\end{array}$$Hence, by Lemma 2, it is easy to obtain the following inequality:$$\begin{array}{}\text{(66)}& \frac{\mathrm{1}}{L^{\mathrm{2}\left(i-\mathrm{1}\right)}}{x}_i^{\mathrm{2}}=\frac{\mathrm{1}}{L^{\mathrm{2}\left(i-\mathrm{1}\right)}}{\left(L^{i-\mathrm{1}}{\epsilon }_i+{\left(LH\right)}^{i-\mathrm{1}}{\xi }_i\right)}^{\mathrm{2}}\le \frac{\mathrm{1}}{L^{\mathrm{2}\left(i-\mathrm{1}\right)}}\mathrm{2}\left(L^{\mathrm{2}\left(i-\mathrm{1}\right)}{\epsilon }_i^{\mathrm{2}}+{\left(LH\right)}^{\mathrm{2}\left(i-\mathrm{1}\right)}{\xi }_i^{\mathrm{2}}\right)\le \mathrm{2}{‖\epsilon ‖}^{\mathrm{2}}+\mathrm{2}{H}^{\mathrm{2}\left(i-\mathrm{1}\right)}{‖\xi ‖}^{\mathrm{2}},\hspace{1em}i=\mathrm{2},\dots ,n.\end{array}$$Putting together (66) and (58), there would always hold$$\begin{array}{}\text{(67)}& {\dot{V}}_{\epsilon }\left(\epsilon \right)\le -\frac{L}{\mathrm{2}}{‖\epsilon ‖}^{\mathrm{2}}+{\widehat{K}}_{\mathrm{1}}{‖\epsilon ‖}^{\mathrm{2}}+\left(L{\widehat{K}}_{\mathrm{2}}+H{\widehat{K}}_{\mathrm{4}}\right){‖\xi ‖}^{\mathrm{2}}+{\widehat{K}}_{\mathrm{3}}{‖z‖}^{\mathrm{2}},\end{array}$$where ${\widehat{K}}_{\mathrm{1}}=K_{\mathrm{1}}+\mathrm{2}(K_{\mathrm{4}}+\cdots +K_{n+\mathrm{2}})\ge \mathrm{0},\hspace{0.17em}\hspace{0.17em}{\widehat{K}}_{\mathrm{2}}=K_{\mathrm{2}}+K_{\mathrm{3}}>\mathrm{0},\hspace{0.17em}\hspace{0.17em}{\widehat{K}}_{\mathrm{3}}=K_{\mathrm{3}}\ge \mathrm{0},\hspace{0.17em}\hspace{0.17em}{\widehat{K}}_{\mathrm{4}}=\mathrm{2}(K_{\mathrm{4}}H+\cdots +K_{n+\mathrm{2}}{H}^{\mathrm{2}n-\mathrm{3}})\ge \mathrm{0}$. Furthermore, following (39), we arrive at$$\begin{array}{}\text{(68)}& \frac{\partial U}{\partial z}{f}_{\mathrm{0}}\le -\left(\frac{c_{\mathrm{1}}}{\mathrm{2}}-\frac{L^{\mathrm{2}}+\cdots +L^{\mathrm{2}\left(n-\mathrm{1}\right)}}{\mathrm{2}}\right){‖z‖}^{\mathrm{2}}+H{\widehat{K}}_{\mathrm{5}}{‖\xi ‖}^{\mathrm{2}}+\left(n-\mathrm{1}\right){\overline{c}}_{\mathrm{2}}^{\mathrm{2}}{‖\epsilon ‖}^{\mathrm{2}},\end{array}$$where ${\widehat{K}}_{\mathrm{5}}={\overline{c}}_{\mathrm{2}}^{\mathrm{2}}(H+H^{\mathrm{3}}+\cdots +H^{\mathrm{2}n-\mathrm{3}})$. Finally, let$$\begin{array}{}\text{(69)}& V=U+V_{\epsilon }+V_{\xi },\end{array}$$and a direct calculation yields$$\begin{array}{}\text{(70)}& \dot{V}\le -\left(\frac{c_{\mathrm{1}}}{\mathrm{2}}-\frac{L^{\mathrm{2}}+\cdots +L^{\mathrm{2}\left(n-\mathrm{1}\right)}}{\mathrm{2}}-{\widehat{K}}_{\mathrm{3}}-{\overline{K}}_{\mathrm{3}}\right){‖z‖}^{\mathrm{2}}-L\left(\frac{\mathrm{1}}{\mathrm{4}}-\frac{{\widehat{K}}_{\mathrm{1}}+{\overline{c}}_{\mathrm{2}}^{\mathrm{2}}\left(n-\mathrm{1}\right)}{L}\right){‖\epsilon ‖}^{\mathrm{2}}-LH\left(\beta -\frac{{\overline{K}}_{\mathrm{1}}+{\widehat{K}}_{\mathrm{2}}}{H}-\frac{{\overline{K}}_{\mathrm{2}}+{\widehat{K}}_{\mathrm{4}}+{\widehat{K}}_{\mathrm{5}}}{L}\right){‖\xi ‖}^{\mathrm{2}}.\end{array}$$We choose domination gains $L$ and $H$ as$$\begin{array}{c}\text{(71)}& H\ge \max \left\{\mathrm{1},\frac{\mathrm{2}{\overline{K}}_{\mathrm{1}}+\mathrm{2}{\widehat{K}}_{\mathrm{2}}}{\beta }\right\},L\ge \mathrm{m}\mathrm{a}\mathrm{x}\left\{\mathrm{1},\frac{\mathrm{4}{\widehat{K}}_{\mathrm{1}}+\mathrm{4}{\overline{c}}_{\mathrm{2}}^{\mathrm{2}}\left(n-\mathrm{1}\right)}{\mathrm{1}-\beta },\frac{\mathrm{4}{\overline{K}}_{\mathrm{2}}+\mathrm{4}{\widehat{K}}_{\mathrm{4}}+\mathrm{4}{\widehat{K}}_{\mathrm{5}}}{\beta }\right\},\end{array}$$where $c_{\mathrm{1}}$ satisfies$$\begin{array}{}\text{(72)}& c_{\mathrm{1}}=c_{\mathrm{1}}^{\ast }+\frac{\beta }{\mathrm{4}},\hspace{1em}{c}_{\mathrm{1}}^{\ast }\ge \mathrm{2}{\overline{K}}_{\mathrm{3}}+\mathrm{2}{\widehat{K}}_{\mathrm{3}}+L^{\mathrm{2}}+\cdots +L^{\mathrm{2}n-\mathrm{3}},\end{array}$$and ${\overline{c}}_{\mathrm{2}}$ is a positive constant; there is$$\begin{array}{}\text{(73)}& \dot{V}\le -\frac{\beta }{\mathrm{4}}{‖z‖}^{\mathrm{2}}-\frac{\beta }{\mathrm{4}}{‖\epsilon ‖}^{\mathrm{2}}-\frac{\beta }{\mathrm{4}}{‖\xi ‖}^{\mathrm{2}}.\end{array}$$

The process of stability analysis is analogous to Theorem 9 and is omitted for the sake of space. This completes the proof.

Remark 14.

The process of Theorems 9 and 13 means that Assumptions 4 and 5 are suitable for two cases of triangular systems. The contribution of these assumptions is that $f_{\mathrm{0}}$ contains all of states of nonlinear cascade systems. Meanwhile, a unified Lyapunov function is successfully constructed and we proposed a novel observer that is different from [24, 26].

4. Simulation Example

As application of the design method, two examples are provided as follows.

Example 1.

Consider nonlinear cascade system:$$\begin{array}{c}\text{(74)}& \dot{z}\left(t\right)=-\mathrm{2}z\left(t\right)-x_{\mathrm{2}}\left(t\right),{\dot{x}}_{\mathrm{1}}\left(t\right)=x_{\mathrm{2}}\left(t\right)+x_{\mathrm{3}}\left(t\right)-\sin \left(x_{\mathrm{1}}\right)z\left(t\right),\\ {\dot{x}}_{\mathrm{2}}\left(t\right)=x_{\mathrm{3}}\left(t\right)+z\left(t\right),\\ {\dot{x}}_{\mathrm{3}}\left(t\right)=u\left(t\right)+z\left(t\right),\\ y\left(t\right)=\theta \left(t\right)x_{\mathrm{1}}\left(t\right),\end{array}$$where $\theta (t)=\mathrm{1}+\mathrm{0.15}\left|\mathrm{s}\mathrm{i}\mathrm{n}\left(\mathrm{6}t\right)\right|$. Obviously, systems satisfy Assumptions 4–6 with $c=\mathrm{1}$, $c_{\mathrm{1}}=\mathrm{204}$, and $c_{\mathrm{2}}=\mathrm{20}$. And we choose $p_{\mathrm{2}}(s)={\left(s+\mathrm{0.4}\right)}^{\mathrm{3}}$; that is, $b_{\mathrm{1}}=\mathrm{1.2}$, $b_{\mathrm{2}}=\mathrm{0.48}$, $b_{\mathrm{3}}=\mathrm{0.064}$, and $\tilde{\theta }=\mathrm{0.1501}$. Besides, for the observer design, we select $\overline{\theta }=\mathrm{0.15}<\tilde{\theta }$, $a_{\mathrm{1}}=\mathrm{4}$, $a_{\mathrm{2}}=\mathrm{3}$, and $a_{\mathrm{3}}=\mathrm{1}$. In consequence, the control law is given as follows:$$\begin{array}{c}\text{(75)}& u\left(t\right)=-\frac{b_{\mathrm{3}}{H}^{\mathrm{3}}}{L^{\mathrm{3}}}y-\frac{b_{\mathrm{2}}{H}^{\mathrm{2}}}{L^{\mathrm{2}}}{\widehat{x}}_{\mathrm{2}}-\frac{b_{\mathrm{1}}H}{L}{\widehat{x}}_{\mathrm{3}},{\dot{\widehat{x}}}_{\mathrm{1}}\left(t\right)={\widehat{x}}_{\mathrm{2}}\left(t\right)-\frac{a_{\mathrm{1}}}{L}{\widehat{x}}_{\mathrm{1}}\left(t\right),\\ {\dot{\widehat{x}}}_{\mathrm{2}}\left(t\right)={\widehat{x}}_{\mathrm{3}}\left(t\right)-\frac{a_{\mathrm{2}}}{L^{\mathrm{2}}}{\widehat{x}}_{\mathrm{1}}\left(t\right),\\ {\dot{\widehat{x}}}_{\mathrm{3}}\left(t\right)=u\left(t\right)-\frac{a_{\mathrm{3}}}{L^{\mathrm{3}}}{\widehat{x}}_{\mathrm{1}}\left(t\right),\end{array}$$where $L=\mathrm{2}$ and $H=\mathrm{7}$. In simulation, initial values are chosen as$$\begin{array}{}\text{(76)}& {\left[z,x_{\mathrm{1}}\left(\mathrm{0}\right),x_{\mathrm{2}}\left(\mathrm{0}\right),x_{\mathrm{3}}\left(\mathrm{0}\right),{\widehat{x}}_{\mathrm{1}}\left(\mathrm{0}\right),{\widehat{x}}_{\mathrm{2}}\left(\mathrm{0}\right),{\widehat{x}}_{\mathrm{3}}\left(\mathrm{0}\right)\right]}^T={\left[-\mathrm{1},-\mathrm{2,1},\mathrm{4,4},-\mathrm{2},-\mathrm{1}\right]}^T,\end{array}$$

and one gets Figures 1–4, which illustrate that the control law (20) is effective.

[figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF]

Example 2.

Consider the following nonlinear cascade system:$$\begin{array}{c}\text{(77)}& \dot{z}\left(t\right)=-z\left(t\right)+x_{\mathrm{1}}\left(t\right),{\dot{x}}_{\mathrm{1}}\left(t\right)=x_{\mathrm{2}}\left(t\right)+\sin \left(x_{\mathrm{1}}\right)z,\\ {\dot{x}}_{\mathrm{2}}\left(t\right)=u\left(t\right)+x_{\mathrm{1}}\left(t\right),\\ y\left(t\right)=\theta \left(t\right)x_{\mathrm{1}}\left(t\right),\end{array}$$where $\theta (t)=\mathrm{1}+\mathrm{0.44}\left|\mathrm{s}\mathrm{i}\mathrm{n}\left(\mathrm{6}t\right)\right|$. Evidently, Assumptions 4, 5, and 12 are satisfied with $c=\mathrm{1}$, $c_1=\mathrm{18}$, and $c_{\mathrm{2}}=\mathrm{1}$. And we choose $p_{\mathrm{2}}(s)={\left(s+\mathrm{0.4}\right)}^{\mathrm{2}}$; that is, $b_{\mathrm{1}}=\mathrm{0.8}$ and $b_{\mathrm{2}}=\mathrm{0.16}$. After sample calculation, $\tilde{\theta }=\mathrm{0.4420}$ and select $\overline{\theta }=\mathrm{0.44}<\tilde{\theta }$, $a_{\mathrm{1}}=\mathrm{5}$, and $a_{\mathrm{2}}=\mathrm{1}$; we construct an output feedback controller as follows:$$\begin{array}{c}\text{(78)}& u\left(t\right)=-b_{\mathrm{2}}{\left(LH\right)}^{\mathrm{2}}y-b_{\mathrm{1}}LH{\widehat{x}}_{\mathrm{2}},{\dot{\widehat{x}}}_{\mathrm{1}}\left(t\right)={\widehat{x}}_{\mathrm{2}}\left(t\right)-a_{\mathrm{1}}L{\widehat{x}}_{\mathrm{1}}\left(t\right),\\ {\dot{\widehat{x}}}_{\mathrm{2}}\left(t\right)=u\left(t\right)-a_{\mathrm{2}}{L}^{\mathrm{2}}{\widehat{x}}_{\mathrm{1}}\left(t\right),\end{array}$$where $L=\mathrm{4}$ and $H=\mathrm{5}$. In the simulation, we choose the initial values as ${\left[z\left(\mathrm{0}\right),x_{\mathrm{1}}\left(\mathrm{0}\right),x_{\mathrm{1}}\left(\mathrm{0}\right),{\widehat{x}}_{\mathrm{1}}\left(\mathrm{0}\right),{\widehat{x}}_{\mathrm{2}}\left(\mathrm{0}\right)\right]}^T={\left[\mathrm{3},-\mathrm{2,4},-\mathrm{1,1.5}\right]}^T$; one can obtain the simulation results as shown in Figures 5–8, which exhibit the effectiveness of the control scheme.

[figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF]

5. Conclusions

This paper solves the problem of global output feedback stabilization for the nonlinear cascade systems with time-varying output function. The construction of the output feedback controller is based on the double-domination method. There still exist some problems to be investigated. For example, (I) [34–36] solved the problem of finite-time stabilization; it is unclear whether scheme can be applied to solve the finite-time stabilization for nonlinear systems with time-varying output function. (II) References [37, 38] proposed the output feedback controller that ensures that the equilibrium is globally asymptotically stable in probability. Then, is it possible to achieve the stabilization of nonlinear stochastic systems by the proposed strategy of this paper? (III) References [39–45] advanced the solution to the stabilization problem of time-delay systems. However, it is unclear whether this method could be used to address the stabilization of time-delay nonlinear systems. (IV) References [46–48] focus on global adaptive state-feedback stabilization for a class of high-order uncertain nonlinear systems. When c in Assumption 6 is unknown, can the approach of this paper be used to solve the problem of adaptive stabilization?

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (61773237, 61374004, and 61473170), China Postdoctoral Science Foundation Funded Project (2017M610414), Shandong Province Quality Core Curriculum of Postgraduate Education (SDYKC17079), Zhejiang Provincial Natural Science Foundation (LY16E050003), and the Ministry of Science and Technology, Taiwan, under Grant (MOST 106-2218-E-006-026-).

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