Uniformly asymptotic stability of second-order linear time-varying systems

Abstract

This paper is concerned with the stability of second-order linear time-varying systems. By utilizing the Lyapunov approach, a generally uniformly asymptotic stability criterion is obtained by adding the system matrices into the quadratic Lyapunov candidate function. In the case of the derivative of the Lyapunov candidate function is semi-positive definite, the stability criterion is also efficient. Based on the proposed results, the systems with uncertain disturbances such as structured independent and structured dependent perturbations are considered. Using the matrix measure and the singular value theory, the bounds of the uncertainties are obtained that guarantee the system uniformly asymptotically stable, while the bounds of state feedback control input are also derived to stabilize the second-order linear time-varying systems. Finally, several numerical examples are given to prove the validity and correctness of the proposed criteria with existing ones.

Introduction

Second-order systems can represent many practical processes in nature, such as RLC circuit networks, armature-controlled DC motors, and spring-mass-damper systems, etc.^1–3 For example, one of the spring-mass-damper systems is the buffer. Buffer devices are the main components of energy generation in the process of absorption and dissipation. Buffer can be seen everywhere in life, such as shock absorbers for automobiles and buffers for energy-consuming collisions. Therefore, it is necessary to study the stability of the second-order linear time-varying (SLTV) systems. Different from linear time-invariant systems, there are several different definitions for linear time-varying (LTV) systems, including stability, asymptotic stability, and exponential stability, which can distinguish uniform stability from non-uniform stability according to the relationship with the initial time of the system. For first-order LTV systems, the efficient tool for testing the stability is the Lyapunov approach. The main idea is to find a positive definite energy function and derive the conditions by solving the Linear Matrix Inequalities (LMIs). For time-invariant systems, researchers generally select the quadratic matrix of the derivative of Lyapunov function as an identity matrix, then obtain the condition by employing LMIs.^4–8 However, the above method is not suitable for LTV systems, that is, it is very difficult to acquire the integral of the time-varying matrix. Therefore, the general stability criterion of the LTV systems is still an open and unsolvable problem in control theory field.

Early researchers are devoted to first-order time-invariant systems. Anderson and Moore considered the first-order scalar system. ⁹ Due to the simplicity of the scalar system, then the trajectory of the state variable can be obtained directly. For the stability of the time-invariant systems can be determined directly in terms of the locations of all the eigenvalues, while the stability of the LTV systems cannot be linked to the parameters of the system which can deicide the eigenvalues. Further, most of the scholars usually select the Lyapunov approach to deal with it. Some of them use the eigenvalues of the matrix to determine the positive or negative definiteness. ¹⁰ If all eigenvalues of the quadratic matrix are smaller or larger than zero, then the negative or positive definiteness of it can be determined. On the base of the eigenvalues, later developed a singular values method directly related to eigenvalues in the actual field. Xi gave the robust stable condition for the uncertain LTV systems by the singular values theory. ¹¹ Chen and Chou paid their attention to stability analysis of LTV systems with both time-varying structured and unstructured perturbations. ¹² Recently, a lot of researches have appeared for different LTV systems.^13–20

Second-order systems capture the practical process of many mechanical systems, such as spacecraft orbital maneuver and spacecraft rendezvous, etc. It is challenging to obtain analytical solutions for these systems. The general method of the stability analysis of second-order systems is to transform the original system into a first-order form. Lu et al. utilized the matrix measure and Kronecker product to analyze the stability. ²¹ The matrix measure is related to the eigenvalue and the matrix norm. Using the matrix measure theory, we can transform the SLTV system into its corresponding comparison systems and the exponential stability of the system can be obtained. ²² However, the conditions are hard to be satisfied, and the scope of application is limited. Besides, Tadi directly selected the quadratic form as the Lyapunov candidate function without transforming the system equation. ²³ Getting the stability result of the eigenvalue and matrix measure, Cao and Shu focused on the robust stability bounds for multi-degree of freedom linear systems with structured perturbations, concluding the asymptotic stability bounds for the second-order time-invariant systems. ²⁴ Cao et al. aimed at the second-order damped systems with nonlinear uncertainties, proposing asymptotic stability criterion based on the eigenvalue. ²⁵ Diwekar et al. considered the robust controller design according to the structured uncertainties. ²⁶ With the arranged eigenvalue sequence, they proposed a simple procedure to select control gains which can guarantee the stability of the system. Adegas et al. paid their attention to linear second-order systems, by using LMIs to obtain the asymptotic stability results. ²⁷ From the above discussion, the basic stability problem of second-order time-invariant systems is solved. Recently, Sun et al. combined the matrix measure of the SLTV system with a special form, through the linear transformation, they deduced the exponentially stable criterion. ²⁸ Tung et al. considered the singular SLTV system by improving the approach in Sun et al. ²⁸ Due to the system cannot be transformed into first-order from, Tung used the linear transformation and Volterra operator to obtaining the exponentially stable result, nevertheless, the calculating process is intricate instead. ²⁹ Roup and Bernstein mainly investigated the adaptive stabilization of the SLTV system where the coefficient matrices have time-varying bounds and obtained the equilibrium set which leads the system to uniform stability. ³⁰

However, few results are available for the uniformly asymptotic stability of SLTV systems, and also the robust stability. The existing results can only be applied to first-order systems. This augmentation destroys some advantages of the originally second-order models, such as controllability, full-actuated characteristic, and so on. Therefore, it is very necessary and meaningful to derive the results based on the coefficient matrices of the SLTV system. Our research team has achieved a series of results for SLTV systems. Gu et al. proposed the robust stable criterion of uncertain SLTV systems based on the original system coefficients matrices, then gave the stability conditions which could be utilized to design controller for the extending space structures system of the spacecraft. ³¹ Further, Gu et al. considered the parametric design approach to LTV systems by using dynamic compensator and multi-objective optimization.^32,33 In this paper, we focus on the uniformly asymptotic stability and robust stability of SLTV systems, further, the feedback stabilization control law is designed to verify the proposed approach.

The major contributions of this research are introduced as follows. Firstly, we propose a constructive method for the Lyapunov function based on the originally coefficient matrices, as well as the uniformly asymptotic stability conditions. Secondly, considering structured independent and structured dependent perturbations, the conditions of robust stability are also given. Thirdly, based on the proposed criterion, we design a simple symmetrical state feedback controller that stabilizes the closed-loop system.

The remainder of this article is organized as follows. The problem formulation and preliminaries are given in Section 2. In Sections 3 and 4, the conditions of uniformly asymptotic stability for SLTV systems are proposed, also the state feedback control law. Further, uncertain disturbances such as structured independent and structured dependent perturbations are considered in Sections 5 and 6. Finally, several numerical examples are given to verify the proposed criterion in Section 7.

Problem formulation and preliminaries

Consider a class of SLTV systems as follow:

$$M(t)\stackrel{··}{x}(t)+N(t)\stackrel{·}{x}(t)+K(t)x(t)=0,t\in J,$$

(1)

where $M(t)=M^{\mathrm{T}}(t)\in {\mathbb{R}}^{n\times n}$ , $N(t)\in {\mathbb{R}}^{n\times n}$ and $K(t)\in {\mathbb{R}}^{n\times n}$ are all piece-wisely differentiable with respect to $t$ and uniformly bounded time-varying symmetric coefficient matrices, $x(t)\in {\mathbb{R}}^n$ is the state variable, and $J=[t^{*},\infty )$ , $t^{*}$ is a finite number.

• Defintion 1. ³⁴ The LTV system $\stackrel{·}{x}(t)=A(t)x(t)$ is said to be

• (1) stable if for any $\epsilon >0$ and for any $t_0\in J$ , there exists a $\delta =\delta (t_0,\epsilon )>0$ such that $|x(t_0)|\in [0,\delta ]\Rightarrow |x(t)|\in [0,\epsilon ],\forall t,t_0\in J,t\ge t_0$ ;

• (2) uniformly stable if $\delta$ in Item 1 is independent of $t_0$ ;

• (3) asymptotically stable (AS) if it is stable and, for any $t_0\in J$ and any $\epsilon >0$ , there exist two positive scalars $\eta =\eta (t_0)$ and $T=T(t_0,\epsilon )$ such that $|x(t_0)|\in [0,\eta ]\Rightarrow |x(t)|\in [0,\epsilon ],\forall t\ge T+t_0$ ;

• (4) uniformly asymptotically stable (UAS) if it is uniformly stable and $\eta$ and $T$ in Item 3 are independent of $t_0$ .

• Defintion 2. ³⁵ The matrix measure of ${\mu }_p\left[A\right]$ or directional derivative is defined as the derivative induced matrix norm at point $I$ in the direction of $A$

$$\begin{array}{c}{\mu }_p\left[A\right]=\underset{\delta \to 0^{+}}{lim}\frac{\parallel I+\delta A{\parallel }_p-\parallel I{\parallel }_p}{\delta }\hfill \\ =\underset{\delta \to 0^{+}}{lim}\frac{\parallel I+\delta A{\parallel }_p-1}{\delta },\hfill \end{array}$$

(2)

where $I$ is the identity matrix of the same dimension as $A$ , the last part of the above equation requires that $p=1,\infty$ or $s$ , and

$${\mu }_{\infty }\left[A\right]=\underset{i}{max}[A_{\mathrm{ii}}+\sum _{j\ne i}|A_{\mathrm{ij}}|],$$

(3)

$${\mu }_1\left[A\right]=\underset{j}{max}[A_{\mathrm{jj}}+\sum _{i\ne j}|A_{\mathrm{ij}}|],$$

(4)

$${\mu }_s[A]=\frac{1}{2}{\lambda }_{max}[A^{\mathrm{T}}+A],$$

(5)

$$-{\mu }_s[-A]=\frac{1}{2}{\lambda }_{max}[A^{\mathrm{T}}+A].$$

(6)

For matrices $A,B\in {\mathbb{R}}^{n\times n}$ , we have the following matrix measure properties

1. $-{\mu }_p[-A]\le \mathrm{Re}(\lambda (A))\le {\mu }_p\left[A\right]\le \parallel A{\parallel }_p$ ,

2. ${\mu }_p[A+B]\le {\mu }_p\left[A\right]+{\mu }_p\left[B\right]$ ,

3. ${\mu }_p\left[\gamma A\right]=\gamma {\mu }_p\left[A\right],\gamma \ge 0$ .

Lemma 1. ⁵ For the LTV system

$$\stackrel{·}{x}(t)=A(t)x(t),$$

(7)

where $A(t)\in {\mathbb{R}}^{n\times n}$ is piece-wisely continuous and uniformly bounded. Then the LTV system is uniformly asymptotically stable if and only if for an arbitrary uniformly bounded and uniformly symmetric positive definite matrix $Q(t)\in {\mathbb{R}}^{n\times n}$ , there exists a uniformly bounded and uniformly symmetric positive definite matrix $P(t)$ satisfying the following Lyapunov differential equation

$$\stackrel{·}{P}(t)+A^{\mathrm{T}}(t)P(t)+P(t)A(t)+Q(t)=0,t\in J.$$

(8)

• Lemma 2. ³⁶ The LTV system (7) is uniformly asymptotically stable if and only if for an arbitrary uniformly bounded and uniformly symmetric semi-positive definite matrix

$$Q(t)=C^{\mathrm{T}}(t)C(t),$$

(9)

where the matrix pair $\{A(t),C(t)\}$ is uniformly completely observable, and there exists a uniformly bounded and uniformly symmetric positive-definite matrix $P(t)$ satisfying Lyapunov differential equation (8).

Main results

Let

$$X(t)=\left[\begin{array}{c}x(t)\hfill \\ \multicolumn{1}{c}{\stackrel{·}{x}(t)}\end{array}\right],$$

(10)

then the SLTV System (1) can be rewritten as the following first-order form

$$\stackrel{·}{X}(t)=A(t)X(t),$$

(11)

where

$$A(t)=\left[\begin{array}{cc}0\hfill & I\hfill \\ \multicolumn{1}{c}{-M^{-1}(t)K(t)}& -M^{-1}(t)N(t)\hfill \end{array}\right].$$

(12)

Selecting a Lyapunov function candidate as the following quadratic form.

$$V(X,t)=X^T(t)P(t)X(t).$$

(13)

For intuition, explicit time dependence is dropped in following derivation, we can calculate the derivative of $V(x,t)$ along the solution of the system (11).

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}(X,t)={\stackrel{·}{X}}^{\mathrm{T}}\mathrm{PX}+X^{\mathrm{T}}\stackrel{·}{P}X+X^{\mathrm{T}}P\stackrel{·}{X}}\\ \multicolumn{1}{c}{=X^{\mathrm{T}}[A^{\mathrm{T}}P+\mathrm{PA}+\stackrel{·}{P}]X}\\ \multicolumn{1}{c}{=X^{\mathrm{T}}[-Q]X.}\end{array}$$

(14)

According to the Lyapunov stability theory and matrix theory, the system (11) is uniformly asymptotically stable if the following condition hold:

$$|P|\le \alpha I,|-Q|\le \beta I,P\ge \mathrm{cI},-Q\le -\mathrm{dI},t\in J,$$

(15)

where $\alpha ,\beta ,c,d$ are finite positive constants, $I$ is the identity matrix, and the condition (15) means $P$ and $-Q$ are uniformly bounded and uniformly positive and negative definite and

$$V(X,t)>0,\stackrel{·}{V}(X,t)<0,t\in J.$$

(16)

We partition the matrix $P$ as

$$P=\left[\begin{array}{cc}{P}_1\hfill & P_2\hfill \\ \multicolumn{1}{c}{P_2}& P_3\hfill \end{array}\right],$$

(17)

where the element $P_2$ is a $n\times n$ symmetric matrix. Calculating the items of equation (14)

$$\begin{array}{c}\multicolumn{1}{c}{A^{\mathrm{T}}P=\left[\begin{array}{cc}0\hfill & -K^{\mathrm{T}}{M}^{-1}\hfill \\ \multicolumn{1}{c}{I}& -N^{\mathrm{T}}{M}^{-1}\hfill \end{array}\right]\left[\begin{array}{cc}{P}_1\hfill & P_2\hfill \\ \multicolumn{1}{c}{P_2}& P_3\hfill \end{array}\right]}\\ =\left[\begin{array}{cc}-K^{\mathrm{T}}{M}^{-1}{P}_2\hfill & -K^{\mathrm{T}}{M}^{-1}{P}_3\hfill \\ \multicolumn{1}{c}{P_1-N^{\mathrm{T}}{M}^{-1}{P}_2}& P_2-N^{\mathrm{T}}{M}^{-1}{P}_3\hfill \end{array}\right],\hfill \\ \multicolumn{1}{c}{\mathrm{PA}=\left[\begin{array}{cc}{P}_1\hfill & P_2\hfill \\ \multicolumn{1}{c}{P_2}& P_3\hfill \end{array}\right]\left[\begin{array}{cc}0\hfill & I\hfill \\ \multicolumn{1}{c}{-M^{-1}K}& -M^{-1}N\hfill \end{array}\right]}\\ =\left[\begin{array}{cc}-P_2{M}^{-1}K\hfill & P_1-P_2{M}^{-1}N\hfill \\ \multicolumn{1}{c}{-P_3{M}^{-1}K}& P_2-P_3{M}^{-1}N\hfill \end{array}\right]\hfill \\ \stackrel{·}{P}=\left[\begin{array}{cc}{\stackrel{·}{P}}_1\hfill & {\stackrel{·}{P}}_2\hfill \\ \multicolumn{1}{c}{{\stackrel{·}{P}}_2}& {\stackrel{·}{P}}_3\hfill \end{array}\right],\hfill \end{array}$$

then we have

$$-Q=A^{\mathrm{T}}P+\mathrm{PA}+\stackrel{·}{P}=\left[\begin{array}{cc}{Q}_{11}\hfill & Q_{12}\hfill \\ \multicolumn{1}{c}{Q_{21}}& Q_{22}\hfill \end{array}\right],$$

(18)

where

$$\{\begin{array}{cc}\multicolumn{1}{c}{Q_{11}}& =-K^{\mathrm{T}}{M}^{-1}{P}_2-P_2{M}^{-1}K+{\stackrel{·}{P}}_1,\hfill \\ \multicolumn{1}{c}{Q_{12}}& =-K^{\mathrm{T}}{M}^{-1}{P}_3+P_1-P_2{M}^{-1}N+{\stackrel{·}{P}}_2,\hfill \\ \multicolumn{1}{c}{Q_{21}}& =P_1-N^{\mathrm{T}}{M}^{-1}{P}_2-P_3{M}^{-1}K+{\stackrel{·}{P}}_2,\hfill \\ \multicolumn{1}{c}{Q_{22}}& =2P_2-N^{\mathrm{T}}{M}^{-1}{P}_3-P_3{M}^{-1}N+{\stackrel{·}{P}}_3.\hfill \end{array}$$

• Theorem 1. The SLTV system (1) is uniformly asymptotically stable if there exist scalar constants $\delta$ , $\epsilon$ , $\eta$ and positive constants $c$ , $d$ satisfying the following conditions

• 1.

$$\overline{P}=\frac{P+P^{\mathrm{T}}}{2}\ge \mathrm{cI},t\in J,$$

(19)

where

$$P=\left[\begin{array}{cc}K+\delta N+\eta K\hfill & \epsilon M\hfill \\ \multicolumn{1}{c}{\epsilon M}& M\hfill \end{array}\right].$$

(20)

• 2.

$$-\overline{Q}=\frac{-Q-Q^{\mathrm{T}}}{2}\le -\mathrm{dI},t\in J,$$

(21)

where

$$-Q=\left[\begin{array}{cc}{Q}_{11}\hfill & Q_{12}\hfill \\ \multicolumn{1}{c}{Q_{21}}& Q_{22}\hfill \end{array}\right],$$

(22)

and

$$\{\begin{array}{cc}\multicolumn{1}{c}{Q_{11}}& =-\epsilon K^{\mathrm{T}}-\epsilon K+\stackrel{·}{K}+\delta \stackrel{·}{N}+\eta \stackrel{·}{K},\hfill \\ \multicolumn{1}{c}{Q_{12}}& =-K^{\mathrm{T}}+K+\delta N+\eta K-\epsilon N+\epsilon \stackrel{·}{M},\hfill \\ \multicolumn{1}{c}{Q_{21}}& =\delta N+\eta K-\epsilon N^{\mathrm{T}}+\epsilon \stackrel{·}{M},\hfill \\ \multicolumn{1}{c}{Q_{22}}& =2\epsilon M-N^{\mathrm{T}}-N+\stackrel{·}{M}.\hfill \end{array}$$

Proof. Define a Lyapunov function candidate as

$$V(x,t)=x^{\mathrm{T}}\mathrm{Px}=x^{\mathrm{T}}\left[\begin{array}{cc}K+\delta N+\eta K\hfill & \epsilon M\hfill \\ \multicolumn{1}{c}{\epsilon M}& M\hfill \end{array}\right]x.$$

Substitute the above equation into (18), we have the $-Q$ in the form of equation (22). According to the Lyapunov stability theory, the system (1) is uniformly asymptotically stable if $-Q<0,t\in J$ . Due to that matrices $P$ and $Q$ are not symmetric, we introduce the following matrices

$$\overline{P}=\frac{P+P^{\mathrm{T}}}{2},-\overline{Q}=\frac{-Q-Q^{\mathrm{T}}}{2},$$

(23)

it is easy to prove that the matrix $P$ is positive definite if $\overline{P}>0$ , also the matrix $\overline{Q}$ . This implies that the system (1) is uniformly asymptotically stable if inequalities (19) and (21) hold. The proof is finished.

In applications, we can choose $\delta =\epsilon =\eta =0$ for convenience, the following corollary can be easily obtained from Theorem 1.

• Remark 1. Theorem 1 is a general form. For convenience of application, we choose $\delta =\epsilon =\eta =0$ . The uniformly asymptotically stable conditions are simplified as

$$\overline{P}\ge \mathrm{cI},t\in J,$$

(24)

where

$$P=\left[\begin{array}{cc}K\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& M\hfill \end{array}\right]$$

and

$$-\overline{Q}\le -\mathrm{dI},t\in J,$$

(25)

where

$$-Q=\left[\begin{array}{cc}\stackrel{·}{K}\hfill & -K^{\mathrm{T}}+K\hfill \\ \multicolumn{1}{c}{0}& -N^{\mathrm{T}}-N+\stackrel{·}{M}\hfill \end{array}\right].$$

If the above conditions are not met, that is, the matrices $\overline{P}$ and $-\overline{Q}$ are not uniformly bounded or uniformly positive and negative definite. We only can obtain the positive and negative definiteness of $\overline{P}$ and $-\overline{Q}$ , then the asymptotic stability of the system (1) can be obtained as

$$\overline{P}=\frac{P+P^{\mathrm{T}}}{2}>0,$$

$$P=\left[\begin{array}{cc}K+\delta N+\eta K\hfill & \epsilon M\hfill \\ \multicolumn{1}{c}{\epsilon M}& M\hfill \end{array}\right]$$

(26)

and

$$-\overline{Q}=\frac{-Q-Q^{\mathrm{T}}}{2}<0,$$

(27)

where the matrix Q is given as (22).

• Corollary 1. Assume that the matrices $M(t)$ , $N(t)$ and $K(t)\in {\mathbb{R}}^{n\times n}$ of the SLTV system (1) are piece-wise continuous and uniformly bounded diagonal. The SLTV system (1) is stable if the following conditions are satisfied

• 1.

$$P=\left[\begin{array}{cc}{M}^{-1}K\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& I\hfill \end{array}\right]>0,$$

(28)

• 2.

$$-Q\le 0,$$

(29)

where

$$\begin{array}{c}Q=\left[\begin{array}{cc}0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 2M^{-1}N\hfill \end{array}\right],\hfill \\ \mathrm{or}\hfill \\ Q=\left[\begin{array}{cc}-{\stackrel{·}{M}}^{-1}K-M^{-1}\stackrel{·}{K}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right],\hfill \end{array}$$

(30)

• 3.

$$\mathrm{rank}\left[\begin{array}{c}\mathrm{sI}-A\hfill \\ \multicolumn{1}{c}{Q}\end{array}\right]=2n.$$

(31)

Proof. Choose $P$ as

$$P=\left[\begin{array}{cc}{M}^{-1}K\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& I\hfill \end{array}\right],$$

and substitute $P$ into equation (18), we get

$$Q=\left[\begin{array}{cc}-{\stackrel{·}{M}}^{-1}K-M^{-1}\stackrel{·}{K}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 2M^{-1}N\hfill \end{array}\right].$$

In some cases, the matrix $-Q$ can be guaranteed the negative definiteness, that is, $-Q$ is semi-negative definite, we cannot deduce the stability of the system (1) based on Theorem 1. For this situation, we take $-Q$ as equation (30). When the condition (31) holds, the matrix pair $\{Q,A\}$ is observable, according to the Lyapunov stability theory, the system (1) is stable. The proof is completed.

• Remark 2. Corollary 1 is to tackle the case that the matrix $Q$ is semi-positive definite, in other words, the conditions in Theorem 1 are invalid. Different from the Corollary 1, there is another particular stability criterion based on the Lemma 2. For the SLTV system (1), assume that the system matrix $A$ is uniformly bounded, if there exists a bounded and uniformly symmetric positive definite matrix $P$ such that the matrix $Q$ is bounded and uniformly symmetric semi-positive definite and can be written to the following form

$$\begin{array}{c}Q=-\left[\begin{array}{cc}-{\stackrel{·}{M}}^{-1}K-M^{-1}\stackrel{·}{K}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -2M^{-1}N\hfill \end{array}\right],\hfill \\ \mathrm{if}{\stackrel{·}{M}}^{-1}K+M^{-1}\stackrel{·}{K}=0,\mathrm{then}\hfill \\ Q=\left[\begin{array}{cc}0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 2M^{-1}N\hfill \end{array}\right]=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{C^{\mathrm{T}}}\end{array}\right]\left[\begin{array}{cc}0\hfill & C\hfill \end{array}\right]\ge 0,\hfill \\ \mathrm{otherwise}2M^{-1}N=0,\mathrm{then}\hfill \\ \multicolumn{1}{c}{Q=-\left[\begin{array}{cc}{\stackrel{·}{M}}^{-1}K+M^{-1}\stackrel{·}{K}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]}\\ =\left[\begin{array}{c}{C}^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{0}\end{array}\right]\left[\begin{array}{cc}C\hfill & 0\hfill \end{array}\right]\ge 0.\hfill \end{array}$$

(32)

From the above two forms of the matrix $Q$ , if the matrix pair $\{A,C\}$ is uniformly completely observable, the system (1) is uniformly asymptotically stable.

In general, the completely observable condition require calculation of the state transition matrix. ⁹ For a time-varying system, the state transition matrix is difficult to solve.

State feedback control

In this section, we propose a method to design a state feedback control law for SLTV systems. Consider the SLTV system (1) with $n=2$ . We choose the following state feedback control

$$u=C_0(t)x+C_1(t)\stackrel{·}{x},$$

(33)

where

$$C_0(t)=\left[\begin{array}{cc}{k}_{01}(t)\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& k_{02}(t)\hfill \end{array}\right],C_1(t)=\left[\begin{array}{cc}{k}_{11}(t)\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& k_{12}(t)\hfill \end{array}\right],$$

and matrices $C_0(t)$ and $C_1(t)$ are feedback gain matrices, $k_{\mathrm{ij}}(t),i=0,1,j=1,2$ are elements of feedback gain matrices. Using the Lyapunov method to obtain the bounds of the feedback control gain matrices such that the SLTV system (1) is stable.

• Theorem 2. The system (1) is uniformly asymptotically stable, if there exists a scalar $\epsilon$ satisfying the following conditions

• 1.

$$\overline{P}\ge \mathrm{cI},t\in J,$$

(34)

where

$$P=\left[\begin{array}{cc}K\hfill & \epsilon M\hfill \\ \multicolumn{1}{c}{\epsilon M}& M\hfill \end{array}\right],$$

• 2.

$$\overline{\mu }[-\overline{Q}]\le -\mathrm{dI},t\in J,$$

(35)

where

$$-Q=\left[\begin{array}{cc}{Q}_{11}\hfill & Q_{12}\hfill \\ \multicolumn{1}{c}{Q_{21}}& Q_{22}\hfill \end{array}\right],$$

(36)

with

$$\{\begin{array}{c}{Q}_{11}=(K-B{C}_0{)}^{\mathrm{T}}-\epsilon (K-B{C}_0)+\stackrel{·}{K},\hfill \\ \multicolumn{1}{c}{Q_{12}=-(K-B{C}_0)+K-\epsilon (N-B{C}_1)+\epsilon \stackrel{·}{M},}\\ \multicolumn{1}{c}{Q_{21}=K-\epsilon (N-B{C}_1{)}^{\mathrm{T}}-(K-B{C}_0)+\epsilon \stackrel{·}{M},}\\ \multicolumn{1}{c}{Q_{22}=2\epsilon M-(N-B{C}_1{)}^{\mathrm{T}}-(N-B{C}_1)+\stackrel{·}{M}.}\end{array}$$

Proof. Substituting the control law (33) into the SLTV system (1), yields

$$\begin{array}{c}M(t)\stackrel{··}{x}+(N(t)-B(t)C_1(t))\stackrel{·}{x}\\ +(K(t)-B(t)C_0(t))x=0.\end{array}$$

Selecting the Lyapunov candidate function $P$ as the following form

$$P=\left[\begin{array}{cc}K\hfill & \epsilon M\hfill \\ \multicolumn{1}{c}{\epsilon M}& M\hfill \end{array}\right],t\in J,$$

then, the matrix $\overline{P}$ is uniformly positive definite. Deducing the derivative of Lyapunov candidate function and calculating the matrix measure of $-\overline{Q}$

$$\overline{\mu }[-\overline{Q}]=\overline{\mu }[\frac{-Q-Q^{\mathrm{T}}}{2}].$$

According to Lemma 1, we have

$$\begin{array}{c}-\overline{Q}<-\mathrm{dI},{\lambda }_{max}[-\overline{Q}]\le -\mathrm{dI},\hfill \\ \overline{\mu }[-\overline{Q}]\le -\mathrm{dI},t\in J,\hfill \end{array}$$

(37)

where ${\lambda }_{max}[·]$ denotes the maximum eigenvalue of the related matrix, $\overline{\mu }[-\overline{Q}]$ represents the upper bound of the matrix measure of $-\overline{Q}$ . The uniformly asymptotically stable bound of the control input can be obtained. The proof is finished

• Remark 3. In Theorem 2, the state feedback control gain matrices are deduced for the dimension of $2$ , but the result is suitable for the dimension of $n$ . Thus, the generally state feedback control is given as follows:

$$\begin{array}{c}{C}_0(t)=\left[\begin{array}{cccc}{k}_{01}(t)\hfill & \hfill & \hfill & \hfill \\ \hfill & k_{02}(t)\hfill & \hfill & \hfill \\ \hfill & \hfill & \ddots \hfill & \hfill \\ \hfill & \hfill & \hfill & k_{0n}(t)\hfill \end{array}\right],\hfill \\ \multicolumn{1}{c}{C_1(t)=\left[\begin{array}{cccc}{k}_{11}(t)\hfill & \hfill & \hfill & \hfill \\ \hfill & k_{12}(t)\hfill & \hfill & \hfill \\ \hfill & \hfill & \ddots \hfill & \hfill \\ \hfill & \hfill & \hfill & k_{1n}(t)\hfill \end{array}\right].}\end{array}$$

(38)

Structured independent perturbation

In this section, we consider the following SLTV system (1) with uncertainties (perturbations)

$$\begin{array}{c}\multicolumn{1}{c}{M(t)\stackrel{··}{x}+\{N(t)+D(t)\}\stackrel{·}{x}}\\ \multicolumn{1}{c}{+\{K(t)+G(t)\}x=0,t\in J,}\end{array}$$

(39)

where where $M(t)=M^{\mathrm{T}}(t)\in {\mathbb{R}}^{n\times n}$ , $N(t)\in {\mathbb{R}}^{n\times n}$ and $K(t)\in {\mathbb{R}}^{n\times n}$ are all known piece-wisely differentiable with respect to $t$ and uniformly bounded time-varying symmetric coefficient matrices, $D(t)$ and $G(t)$ denote time-varying structured independent perturbation matrices in the following form

$$|D(t)|\le {\epsilon }_1{U}_1,|G(t)|\le {\epsilon }_2{U}_2,t\in J,$$

(40)

where ${\epsilon }_1$ and ${\epsilon }_2$ are the real positive uncertain parameters, $U_1$ and $U_2$ are real known positive constant matrices.

Let

$$X(t)=\left[\begin{array}{c}x(t)\hfill \\ \multicolumn{1}{c}{\stackrel{·}{x}(t)}\end{array}\right],$$

(41)

then, the uncertain SLTV system (39) can be rewritten into the first-order form

$$\stackrel{·}{X}=A(t)X(t)+E(t)X(t),$$

(42)

where

$$\begin{array}{cc}A(t)\hfill & =\left[\begin{array}{cc}0\hfill & I\hfill \\ \multicolumn{1}{c}{-M^{-1}K}& -M^{-1}N\hfill \end{array}\right],\hfill \\ \multicolumn{1}{c}{E(t)}& =\left[\begin{array}{cc}0\hfill & 0\hfill \\ \multicolumn{1}{c}{-M^{-1}G}& -M^{-1}D\hfill \end{array}\right].\hfill \end{array}$$

• Theorem 3. The uncertain SLTV system (39) with structured independent perturbation (40) is uniformly asymptotically stable if there exist scalar constant $\epsilon$ and positive constants $c$ , $d$ satisfying the following conditions

• 1.

$$\overline{P}\ge \mathrm{cI},t\in J,$$

(43)

where

$$P=\left[\begin{array}{cc}K\hfill & \epsilon M\hfill \\ \multicolumn{1}{c}{\epsilon M}& M\hfill \end{array}\right],$$

• 2.

$$\begin{array}{c}\multicolumn{1}{c}{\overline{\mu }(-\overline{Q})\le -{\epsilon }_1\overline{\mu }[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{\epsilon |M^{-1}|U_1|M|}\end{array}}\\ \begin{array}{c}\epsilon U_1\hfill \\ \multicolumn{1}{c}{|M^{-1}|U_1|M|+U_1}\end{array}]\hfill \\ -{\epsilon }_2\overline{\mu }[\begin{array}{c}\epsilon |M^{-1}|U_2|M|+\epsilon U_2\hfill \\ \multicolumn{1}{c}{U_2}\end{array}\hfill \\ \begin{array}{c}|-M^{-1}|U_2|M|\hfill \\ \multicolumn{1}{c}{0}\end{array}]-d,t\in J,\hfill \end{array}$$

(44)

where

$$\begin{array}{c}\multicolumn{1}{c}{-Q=[\begin{array}{c}-\epsilon K^{\mathrm{T}}-\epsilon K+\stackrel{·}{K}\hfill \\ \multicolumn{1}{c}{-\epsilon N^{\mathrm{T}}+\epsilon \stackrel{·}{M}}\end{array}}\\ \begin{array}{c}\epsilon \stackrel{·}{M}-\epsilon N-K^{\mathrm{T}}+K\hfill \\ \multicolumn{1}{c}{2\epsilon M-N^{\mathrm{T}}-N+\stackrel{·}{M}}\end{array}],\hfill \end{array}$$

(45)

and the matrices $P$ and $-Q$ are uniformly bounded for all $t\in J$ .

Proof. Define a Lyapunov candidate function as

$$P=\left[\begin{array}{cc}K\hfill & \epsilon M\hfill \\ \multicolumn{1}{c}{\epsilon M}& M\hfill \end{array}\right].$$

Calculate the derivative of $V$ along the solution of the system (39)

$$\stackrel{·}{V}=X^{\mathrm{T}}(A^{\mathrm{T}}P+\mathrm{PA}+\stackrel{·}{P})X+X^{\mathrm{T}}(E^{\mathrm{T}}P+\mathrm{PE})X.$$

(46)

Using the definition of eigenvalue to deal with the negative definiteness of $\stackrel{·}{V}$

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}\le X^{\mathrm{T}}[{\lambda }_{max}[A^{\mathrm{T}}P+\mathrm{PA}+\stackrel{·}{P}+E^{\mathrm{T}}P+\mathrm{PE}]]X}\\ \le X^{\mathrm{T}}{\lambda }_{max}[A^{\mathrm{T}}P+\mathrm{PA}+\stackrel{·}{P}]X\hfill \\ +X^{\mathrm{T}}{\lambda }_{max}[E^{\mathrm{T}}P+\mathrm{PE}]X.\hfill \end{array}$$

From the properties of matrix measure in Definition 2, we have

$$\begin{array}{c}{\lambda }_{max}[A^{\mathrm{T}}P+\mathrm{PA}+\stackrel{·}{P}]+{\lambda }_{max}[E^{\mathrm{T}}P+\mathrm{PE}]\hfill \\ \multicolumn{1}{c}{\le {\lambda }_{max}[A^{\mathrm{T}}P+\mathrm{PA}+\stackrel{·}{P}]+\mu [E^{\mathrm{T}}P+\mathrm{PE}]}\\ \multicolumn{1}{c}{\le {\lambda }_{max}(-Q)+\mu [|E^{\mathrm{T}}P+\mathrm{PE}|],}\end{array}$$

where

$$\begin{array}{c}|E^{\mathrm{T}}P+\mathrm{PE}|=\hfill \\ \left[\begin{array}{cc}|-\epsilon M^{-1}\mathrm{GM}-\epsilon G|\hfill & |-M^{-1}\mathrm{GM}-\epsilon D|\hfill \\ \multicolumn{1}{c}{|-\epsilon M^{-1}\mathrm{DM}-G|}& |-M^{-1}\mathrm{DM}-D|\hfill \end{array}\right]\hfill \\ \le \left[\begin{array}{cc}|-\epsilon M^{-1}\mathrm{GM}-\epsilon G|\hfill & |-M^{-1}\mathrm{GM}|\hfill \\ \multicolumn{1}{c}{|-G|}& 0\hfill \end{array}\right]\hfill \\ +\left[\begin{array}{cc}0\hfill & |-\epsilon D|\hfill \\ \multicolumn{1}{c}{|-\epsilon M^{-1}\mathrm{DM}|}& |-M^{-1}\mathrm{DM}-D|\hfill \end{array}\right]\hfill \\ \le {\epsilon }_2\left[\begin{array}{cc}\epsilon |M^{-1}|U_2|M|+\epsilon U_2\hfill & |-M^{-1}|U_2|M|\hfill \\ \multicolumn{1}{c}{U_2}& 0\hfill \end{array}\right]\hfill \\ +{\epsilon }_1\left[\begin{array}{cc}0\hfill & \epsilon U_1\hfill \\ \multicolumn{1}{c}{\epsilon |M^{-1}|U_1|M|}& |M^{-1}|U_1|M|+U_1\hfill \end{array}\right]\hfill \end{array}$$

and the matrix $-Q$ is in the form of (45).

As a consequence, for all $t\in J$ , the following inequality holds

$${\overline{\lambda }}_{max}(-\overline{Q})+\overline{\mu }[|E^{\mathrm{T}}P+\mathrm{PE}|]\le -d\mu (I).$$

Further, we can deduce the above inequality

$$\overline{\mu }(-\overline{Q})\le -\overline{\mu }[|E^{\mathrm{T}}P+\mathrm{PE}|]-d,$$

that is, inequality (44) holds, then the system (39) is uniformly asymptotically stable. The proof is finished.

Structured dependent perturbation

In this section, we consider the time-varying structured dependent perturbation matrices $D(t)$ and $G(t)$ in the following form

$$D(t)=\sum _{i=1}^r{q}_i(t)D_i(t),G(t)=\sum _{i=1}^r{q}_i(t)G_i(t),t\in J,$$

(47)

where $D_i(t),G_i(t),i=1,2,\dots ,r$ , which are piece-wise continuous, are also given constant or time-varying matrices and $q_i(t),i=1,2,\dots ,r$ are time-varying uncertain parameters.

• Theorem 4. The uncertain SLTV system (39) with structured dependent perturbation (47) is uniformly asymptotically stable if there exist scalar constant $\epsilon$ and positive constants $c$ , $d$ satisfying the following conditions

• 1.

$$\overline{P}\ge \mathrm{cI},t\in J,$$

(48)

where

$$P=\left[\begin{array}{cc}K\hfill & \epsilon M\hfill \\ \multicolumn{1}{c}{\epsilon M}& M\hfill \end{array}\right],$$

• 2.

$$\sum _{i=1}^r{q}_i{\overline{\varphi }}_i\le \underset{¯}{\mu }(\overline{Q})-d,t\in J,$$

(49)

where

$${\overline{\varphi }}_i=\{\begin{array}{c}\underset{t\in J}{sup}\mu (P_i),q_i\ge 0,\hfill \\ \multicolumn{1}{c}{-\underset{t\in J}{sup}\mu (-P_i),q_i<0,}\end{array}$$

(50)

and

$$\begin{array}{c}{P}_i=[\begin{array}{c}-\epsilon M^{-1}{G}_{i}M-\epsilon G_i\hfill \\ \multicolumn{1}{c}{-\epsilon M^{-1}{D}_{i}M-G_i}\end{array}\hfill \\ \begin{array}{c}-M^{-1}{G}_{i}M-\epsilon D_i\hfill \\ \multicolumn{1}{c}{-M^{-1}{D}_{i}M-D_i}\end{array}],\hfill \\ \multicolumn{1}{c}{i=1,2,\cdots ,r.}\end{array}$$

(51)

Proof. The proof is similar to Theorem 3. Calculate the derivative of $V$ along the solution of the system (39)

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{V}=X^{\mathrm{T}}(A^{\mathrm{T}}P+\mathrm{PA}+\stackrel{·}{P}+E^{\mathrm{T}}P+\mathrm{PE})X}\\ \le X^{\mathrm{T}}{\lambda }_{max}(A^{\mathrm{T}}P+\mathrm{PA}+\stackrel{·}{P}+E^{\mathrm{T}}P+\mathrm{PE})X\hfill \\ \le X^{\mathrm{T}}{\overline{\lambda }}_{max}(A^{\mathrm{T}}P+\mathrm{PA}+\stackrel{·}{P}+E^{\mathrm{T}}P+\mathrm{PE})X.\hfill \end{array}$$

According to the Lyapunov stability theory, $\stackrel{·}{V}<0$ should be guaranteed, that is,

$$\begin{array}{c}{\overline{\lambda }}_{max}(A^{\mathrm{T}}P+\mathrm{PA}+\stackrel{·}{P}+E^{\mathrm{T}}P+\mathrm{PE})\hfill \\ ={\overline{\lambda }}_{max}(-Q+E^{\mathrm{T}}P+\mathrm{PE})\hfill \\ \le {\overline{\lambda }}_{max}(-Q)+{\overline{\lambda }}_{max}(E^{\mathrm{T}}P+\mathrm{PE})\hfill \\ \le {\overline{\sigma }}_{max}(-Q)+\hfill \\ \mu \left(\left[\begin{array}{cc}-\epsilon M^{-1}\mathrm{GM}-\epsilon G\hfill & -M^{-1}\mathrm{GM}-\epsilon D\hfill \\ \multicolumn{1}{c}{-\epsilon M^{-1}\mathrm{DM}-G}& -M^{-1}\mathrm{DM}-D\hfill \end{array}\right]\right)\hfill \\ ={\overline{\sigma }}_{max}(-Q)+\mu \left[\begin{array}{cc}{\mathrm{\Phi }}_{11}\hfill & {\mathrm{\Phi }}_{12}\hfill \\ \multicolumn{1}{c}{{\mathrm{\Phi }}_{21}}& {\mathrm{\Phi }}_{22}\hfill \end{array}\right]\hfill \\ ={\overline{\sigma }}_{max}(-Q)+\sum _{i=1}^r{q}_i\mu \left[\begin{array}{cc}{\mathrm{\Phi }}_{11}\prime \hfill & {\mathrm{\Phi }}_{12}\prime \hfill \\ \multicolumn{1}{c}{{\mathrm{\Phi }}_{21}\prime }& {\mathrm{\Phi }}_{22}\prime \hfill \end{array}\right]\hfill \\ \le {\overline{\sigma }}_{max}(-Q)+\sum _{i=1}^r{q}_i\overline{\mu }(P_i)\hfill \\ \le {\overline{\sigma }}_{max}(-Q)+\sum _{i=1}^r{q}_i{\overline{\varphi }}_i\le -d\mu (I)<0,\hfill \end{array}$$

where

$$\{\begin{array}{cc}\multicolumn{1}{c}{{\mathrm{\Phi }}_{11}}& =\sum _{i=1}^r{q}_i{\mathrm{\Phi }}_{11}\prime =\sum _{i=1}^r{q}_i(-\epsilon M^{-1}{G}_{i}M-\epsilon G_i),\hfill \\ \multicolumn{1}{c}{{\mathrm{\Phi }}_{12}}& =\sum _{i=1}^r{q}_i{\mathrm{\Phi }}_{12}\prime =\sum _{i=1}^r{q}_i(-M^{-1}{G}_{i}M-\epsilon D_i),\hfill \\ \multicolumn{1}{c}{{\mathrm{\Phi }}_{21}}& =\sum _{i=1}^r{q}_i{\mathrm{\Phi }}_{21}\prime =\sum _{i=1}^r{q}_i(-\epsilon M^{-1}{D}_{i}M-G_i),\hfill \\ \multicolumn{1}{c}{{\mathrm{\Phi }}_{22}}& =\sum _{i=1}^r{q}_i{\mathrm{\Phi }}_{22}\prime =\sum _{i=1}^r{q}_i(-M^{-1}{D}_{i}M-D_i).\hfill \end{array}$$

Furthermore, the bounds of uncertainties can be determined by

$$\sum _{i=1}^r{q}_i(t){\overline{\varphi }}_i<{\underset{¯}{\sigma }}_{min}(\overline{Q})-d\Rightarrow \sum _{i=1}^r{q}_i{\overline{\varphi }}_i<\underset{¯}{\mu }(\overline{Q})-d,$$

where ${\underset{¯}{\sigma }}_{min}(\overline{Q})=\underset{t\ge 0}{inf}{\sigma }_{min}(\overline{Q})$ , and $\sigma$ represent the singular values. The proof is completed.

Numerical examples

In this section, we provide several examples to illustrate the obtained results.

Example 1

Consider the following SLTV system

$$\stackrel{··}{x}+\left[\begin{array}{cc}{\sin }^2(t)+1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right]\stackrel{·}{x}+\left[\begin{array}{cc}\frac{1}{2t}\hfill & 3\hfill \\ \multicolumn{1}{c}{-3}& e^{-t}\hfill \end{array}\right]x=0.$$

We choose $\eta =\delta =\epsilon =0$ , then the matrix $P$ can be obtained as

$$P=\left[\begin{array}{cc}K\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& M\hfill \end{array}\right]=\left[\begin{array}{cccc}\frac{1}{2t}\hfill & 3\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-3}& e^{-t}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right],$$

Based on Equation (19), we have $\overline{P}>0$ , $t>0$ . $\overline{P}$ is positive definite but not uniformly positive definite, then we have

$$-Q=\left[\begin{array}{cccc}-\frac{1}{2t^2}\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -e^{-t}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -2{\sin }^2(t)-2\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & -2\hfill \end{array}\right],$$

and

$$-\overline{Q}<0,t>0.$$

Because the matrices $P$ and $-Q$ with the element $\frac{1}{t}$ , this means $P$ and $-Q$ are unbounded. The asymptotic stability of the system can be concluded according to Remark 1. Set the initial time $t_0=0.0001$ , and initial values of the state variable are $x_1(t_0)=0.1,x_2(t_0)=-0.2,x_3(t_0)=1,x_4(t_0)=-1$ . The trajectories of the state variable are shown in Figure 1. From Figure 1, the state variable of the system tends to zero with the increase of time. When the time is large enough, the state variable of the system would remain at zero and without changes. The system is in equilibrium, indicating that zero is the stable point of the system. The system is asymptotically stable.

Figure 1.

Trajectory of state variable of Example 1.

Example 2

Consider the following SLTV system

$$\stackrel{··}{x}+\left[\begin{array}{cc}3\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 4\hfill \end{array}\right]\stackrel{·}{x}+\left[\begin{array}{cc}{\sin }^2(t)+4\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\cos }^2(t)+4\hfill \end{array}\right]x=0,$$

where

$$\begin{array}{cc}M\hfill & =\left[\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right],N=\left[\begin{array}{cc}3\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 4\hfill \end{array}\right],\hfill \\ \multicolumn{1}{c}{K}& =\left[\begin{array}{cc}{\sin }^2(t)+4\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\cos }^2(t)+4\hfill \end{array}\right].\hfill \end{array}$$

According to Theorem 1, we choose the special form of $P$ with $\delta =\eta =0,\epsilon =1$ .

$$\begin{array}{c}P=\left[\begin{array}{c}KM\hfill \\ \multicolumn{1}{c}{MM}\end{array}\right]\hfill \\ =\left[\begin{array}{cccc}{\sin }^2(t)+4\hfill & 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\cos }^2(t)+4\hfill & 0\hfill & 1\hfill \\ \multicolumn{1}{c}{1}& 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & 1\hfill \end{array}\right]>0.5I,\hfill \\ t\ge 0,\hfill \end{array}$$

then we can obtain

$$-Q<-I,t\ge 0.$$

The matrices $\overline{P}$ and $\overline{Q}$ are uniformly bounded and uniformly positive definite, then the system is uniformly asymptotically stable based on Theorem 1. To illustrate the above results, we give the following simulation.

From Figure 2, we can see the system is asymptotically stable with initial values are $x_1(0)=0.1,x_2(0)=0.2,x_3(0)=0.3,x_4(0)=-0.2$ .

Figure 2.

Trajectory of state variable of Example 2.

Example 3

Consider the following scalar system

$$\stackrel{··}{x}+2(6+\frac{1}{2}{\sin }^{2}t)\stackrel{·}{x}+\frac{1}{4t}x=0.$$

(52)

Based on the criterion in, ²⁸ we have

$$a(t)=6+\frac{1}{2}{\sin }^{2}t,b(t)=\frac{1}{4t}.$$

Let $m=\underset{t\ge 0}{inf}a(t)=2$ , then calculate $l$ and $c$ , we get

$$l=\underset{t\ge 0}{sup}max[0,2{\mu }_2(\mathrm{mI}-A(t))]=0,$$

and

$$c=\underset{t\ge 0}{sup}(2\mathrm{ma}(t)-m^{2}I-b(t)),$$

the exact value of $c$ cannot be obtained due to the existence of $\frac{1}{t}$ , that is, the criterion in ²⁸ is invalid. By using the method in this paper, we have

$$P=\left[\begin{array}{cc}K\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& M\hfill \end{array}\right]=\left[\begin{array}{cc}\frac{1}{4t}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right]>0,$$

and

$$-Q=\left[\begin{array}{cc}\stackrel{·}{K}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -2N\hfill \end{array}\right]=\left[\begin{array}{cc}-\frac{1}{4t^2}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -24-2{\sin }^{2}t\hfill \end{array}\right]<0.$$

According to that the matrices $P$ and $-Q$ are symmetric, the system (52) is asymptotically stable without calculating matrices $\overline{P}$ and $-\overline{Q}$ . The simulation result is shown in Figure 3 with the initial condition $x_1(0)=2,x_2(0)=1$ .

Figure 3.

Trajectory of state variable of Example 3.

Example 4

Revisit the system in Example 2 with the following structured independent perturbation.

$$\stackrel{··}{x}+\{N(t)+D(t)\}\stackrel{·}{x}+\{K(t)+G(t)\}x=0,$$

(53)

where

$$N=\left[\begin{array}{cc}3\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 4\hfill \end{array}\right],K=\left[\begin{array}{cc}{\sin }^2(t)+4\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\cos }^2(t)+4\hfill \end{array}\right],$$

with

$$|D(t)|\le {\epsilon }_1{U}_1,|G(t)|\le {\epsilon }_2{U}_2,U_1=U_2=\left[\begin{array}{cc}1\hfill & 1\hfill \\ \multicolumn{1}{c}{1}& 1\hfill \end{array}\right].$$

Using the analytical method in Theorem 3 to deduce the asymptotically stable bounds. Calculate the Lyapunov function and its derivative with $\epsilon =1$ and

$$\begin{array}{c}P=\left[\begin{array}{cccc}{\sin }^2(t)+4\hfill & 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\cos }^2(t)+4\hfill & 0\hfill & 1\hfill \\ \multicolumn{1}{c}{1}& 0\hfill & 1\hfill & 0\hfill \end{array}\right],\hfill \\ P=\ge \mathrm{cI}=0.5I,t\ge 0.\hfill \end{array}$$

we have $\overline{\mu }[-Q]\le -0.1$ . In this example, we choose $\mu$ as ${\mu }_1$ , $P$ and $-Q$ are uniformly bounded, existing $d=0.1$ . Then, the stable bound is

$$\begin{array}{c}\multicolumn{1}{c}{{\overline{\mu }}_1[-Q]=-1<-{\overline{\mu }}_1[|E^{\mathrm{T}}P+\mathrm{PE}|]-0.1,}\\ \multicolumn{1}{c}{-1<-4{\epsilon }_1-4{\epsilon }_2-0.1,}\\ \multicolumn{1}{c}{{\epsilon }_1+{\epsilon }_2<0.225,\mathrm{for}{\epsilon }_1>0,{\epsilon }_2>0.}\end{array}$$

If the uncertain parameters ${\epsilon }_1$ and ${\epsilon }_2$ are in the above region, the system (53) is uniformly asymptotically stable. For simulation, we choose ${\epsilon }_1=0.1,{\epsilon }_2=0.05$ , the trajectory of state variable is shown in Figure 4.

Figure 4.

Trajectory of state variable of Example 4.

Example 5

Consider the following the feedback control system

$$\stackrel{··}{x}+N(t)\stackrel{·}{x}+K(t)x=B(t)u,$$

where

$$\begin{array}{cc}\multicolumn{1}{c}{N(t)}& =\left[\begin{array}{cc}4\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 6\hfill \end{array}\right],\hfill \\ \multicolumn{1}{c}{K(t)}& =\left[\begin{array}{cc}{\sin }^2(t)+6\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\cos }^2(t)+6\hfill \end{array}\right],\hfill \\ \multicolumn{1}{c}{B(t)}& =\left[\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right],\hfill \\ \multicolumn{1}{c}{u}& =\left[\begin{array}{cc}{k}_{01}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& k_{02}\hfill \end{array}\right]x+\left[\begin{array}{cc}{k}_{11}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& k_{12}\hfill \end{array}\right]\stackrel{·}{x}.\hfill \end{array}$$

Calculate the bounds of the control input based on Theorem 2. Let $\epsilon =1$ , and $\forall t>0$ , we have

$$P=\left[\begin{array}{cccc}{\sin }^2(t)+6\hfill & 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\cos }^2(t)+6\hfill & 0\hfill & 1\hfill \\ \multicolumn{1}{c}{1}& 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & 1\hfill \end{array}\right]\ge 0.5I.$$

Calculate the matrix measure ${\mu }_1$ of $-\overline{Q}$ and $d$ is a positive constant, then we get

$$\{\begin{array}{c}-2{\sin }^2(t)-12+2k_{01}+\sin (2t)\hfill \\ \multicolumn{1}{c}{+|k_{01}+k_{11}-2+\frac{1}{2}{\sin }^2(t)|<-d,}\\ \multicolumn{1}{c}{-2{\cos }^2(t)-12+2k_{02}-\sin (2t)}\\ \multicolumn{1}{c}{+|k_{02}+k_{12}-3+\frac{1}{2}{\cos }^2(t)|<-d,}\\ \multicolumn{1}{c}{-6+2k_{11}+|k_{01}+k_{11}+\frac{1}{2}{\sin }^2(t)-2|<-d,}\\ \multicolumn{1}{c}{-10+2k_{12}+|k_{02}+k_{12}+\frac{1}{2}{\cos }^2(t)-3|<-d.}\end{array}$$

According the boundary property of trigonometric functions, for the situation $t\to \infty$ , we have

$$\{\begin{array}{cc}\multicolumn{1}{c}{2k_{01}}& +|k_{01}+k_{11}|+|\frac{1}{2}{\sin }^2(t)-2|<11.5858-d,\hfill \\ \multicolumn{1}{c}{2k_{02}}& +|k_{02}+k_{12}|+|\frac{1}{2}{\cos }^2(t)-3|<11.3820-d,\hfill \\ \multicolumn{1}{c}{2k_{11}}& +|k_{01}+k_{11}|+|\frac{1}{2}{\sin }^2(t)-2|<6-d,\hfill \\ \multicolumn{1}{c}{2k_{12}}& +|k_{02}+k_{12}|+|\frac{1}{2}{\cos }^2(t)-3|<10-d,\hfill \end{array}$$

and

$$\{\begin{array}{cc}\multicolumn{1}{c}{3k_{01}}& +k_{11}<10.0858-d,\hfill \\ \multicolumn{1}{c}{-k_{01}}& +k_{11}>-10.0858+d,\hfill \\ \multicolumn{1}{c}{3k_{02}}& +k_{12}<8.882-d,\hfill \\ \multicolumn{1}{c}{-k_{02}}& +k_{12}>-8.882+d,\hfill \\ \multicolumn{1}{c}{3k_{11}}& +k_{01}<4.5-d,\hfill \\ \multicolumn{1}{c}{-k_{11}}& +k_{01}>-4.5+d,\hfill \\ \multicolumn{1}{c}{3k_{12}}& +k_{02}<7.5-d,\hfill \\ \multicolumn{1}{c}{-k_{12}}& +k_{02}>-7.5+d,\hfill \end{array}$$

the final result is

$$\{\begin{array}{c}{k}_{11}<0.4268-0.25d,\hfill \\ \multicolumn{1}{c}{k_{02}<2.3933-0.25d,}\\ \multicolumn{1}{c}{-10.0858+d<-k_{01}+k_{11}<4.5-d,}\\ \multicolumn{1}{c}{-8.882+d<-k_{02}+k_{12}<7.5-d.}\end{array}$$

Conclusions

The proposed stability criterion of SLTV systems can be applied for more general situations. By constructing the Lyapunov candidate function with the system parameters, a simple and efficient criterion is proposed to test the uniformly asymptotic stability of SLTV systems. On the base of the proposed stability criterion, the stability of uncertain second-order systems is also discussed. With the help of the matrix measure, some simple stability conditions are deduced, such as robust asymptotic stability, etc. Furthermore, the uniformly asymptotically stable bound is also considered and a simple criterion is obtained. Finally, some numerical examples are introduced to verify the correctness of the stability criteria. The next major work focus on two aspects, one is to extend the results to output feedback control for uncertain SLTV systems, another is to establish a multi-objective optimization for choosing the parameters.