# A Note on Practical Stability of Nonlinear Vibration Systems with

Authors: Qishui Zhong; Hongcai Li; Hui Liu; Juebang Yu
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper addresses the issue of vibration characteristics of nonlinear systems with impulsive
effects. By utilizing a T-S fuzzy model to represent a nonlinear system, a general strict practical
stability criterion is derived for nonlinear impulsive systems.


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## Body

## 1. Introduction

In recent years, nonlinear vibration and its control have been widely studied due to undesirable or harmful behaviors under many circumstances [1–6]. Furthermore, during working condition, plates usually work with disturbances which can be described with impulsive effects, and impulsive differential equations usually used to dealt with this kind of problem. A number of papers have deal with the theory of impulsive differential equations [7–10] and its applications to many kind of systems [11, 12]. Impulsive control has been demonstrated to be an effective and attractive control method to stabilize linear and nonlinear systems [11, 13], especially to stabilize various chaotic vibration systems (e.g., see [14–16]).Furthermore, a highly nonlinear system can usually be represented by the T-S fuzzy model [17]. The T-S fuzzy model is described by fuzzy IF-THEN rules where the consequent parts represent local linear models for nonlinear systems. Then we can use linear theory to analyze chaotic systems by means of T-S fuzzy models at a certain domain [18, 19].In the study of the Lyapunov stability, an interesting set of problems deals with bringing sets close to a certain state, rather than the statex=0 [20]. The desired state of system may be mathematically unstable and yet the system may oscillate sufficiently near this state that its performance is acceptable. Many problems fall into this category including the travel of a space vehicle between two points, an aircraft or a missile which may oscillate around a mathematically unstable course yet its performance may be acceptable, and the problem in a chemical process of keeping the temperature within certain bounds. Such considerations led to the notion of practical stability which is neither weaker nor stronger than the Lyapunov stability [21, 22]. In a practical control problem, one aims at controlling a system into a certain region of interest instead of an exact point. If one wants to control a system to an exact point, the expense may be prohibitively high in some cases.In this paper, we study the practical stability for the nonlinear impulsive system based on T-S fuzzy model. After a general strictly practical stability  criterion is derived, some simple and easily verified sufficient conditions are given to stabilize the nonlinear vibration system.Notation:
A
>
0
(
<
0
) means A is a symmetrical positive (negative) definite matrix. R+, R+, and N stand for, respectively, the set of all positive real numbers, nonnegative real numbers, and the set of natural numbers. ∥x∥ denotes the Euclidian norm of vector x, and λmin(A) and λmax(A) mean the minimal and maximal eigenvalues of matrix A, respectively.


## 2. Problem Formulation

Consider the following nonlinear system(2.1)x˙(t)=f(x(t)),
where x(t)∈Rn is the state variable, f∈C[Rn,Rn]. We can construct the fuzzy model for (2.1) as follows: rule (i): if z1(t) is Mi1,…, and zp(t) is Mip, THEN x(t)=Aix(t), i=1,2,…,r, in which zi(t),…,zp(t) are the premise variables, each Mij(j=1,2,…,p) is a fuzzy set, and Ai∈Rn×n is a constant matrix. With a center-average defuzzifier, the overall fuzzy system is represented as
(2.2)x˙(t)=∑i=1rhi(t)Aix(t),
where r is the number of fuzzy implications, hi(t)=wi(t)/(∑i=1rwi(t)), wi(t)=∏l=1p‍Mil(zl(t)), and Mil(zl(t)) is the grade of membership of zl(t) in Mil. Of course, hi(t)≥0 and ∑i=1rhi(t)=1. Note that system (2.2) can locally represent system (2.1).Disturbances, acting on system (2.1), are given by a sequence {tk,Ik(x(tk))}, where 0<t1<t2<⋯<tk<⋯, tk→∞ as k→∞, and Ik∈C[Rn,Rn] denotes the incremental change of the state at time tk. Thus, the following impulsive differential equation is obtained
(2.3)x˙(t)=∑i=1rhi(t)Aix(t),∀t≥0,t≠tk,Δx(t)=Ik(x(t)),t=tk,k∈N,
in which Δx(t)=x(tk+)-x(tk),x(tk+)=limt→tk+x(tk),k∈N.Assume thatIk(0)≡0 for all k so that the trivial solution of (2.3) exists. Denote S(ρ)={x∈Rn:∥x∥<ρ}.We will introduce the following classes of function spaces, definitions, and theorems for future use:K={a∈C[R+,R+]:
a is strictly increasing and a(0)=0}.CK={a∈C[R+2,R+]:a(t,u)∈K for each t∈R+}.Definition 2.1.
LetV:R+×Rn→R+, then V is said to belong to class υ0 if:(1)
Vis continuous in (tk-1,tk]×Rn and for each x∈Rn,k=1,2,…,
(2.4)lim⁡(t,y)→(tk+,x)⁡V(t,y)lim⁡⁡=V(tk+,x)
exits;(2)
V is locally Lipschitzian in x.Definition 2.2.
For(t,x)∈(tk-1,tk]×Rn, we define the derivatives of V(t,x) as
(2.5)D+V(t,x)=lim⁡s→0+⁡sup⁡1s[V(t+s,x+sf(t,x))-V(t,x)],(2.6)D-V(t,x)=lim⁡s→0-⁡inf⁡1s[V(t+s,x+sf(t,x))-V(t,x)].Definition 2.3.
The trivial solution of the system (2.1) is said to be (1)
practically stable, if given(λ,A) with 0<λ<A, one has ∥x0∥<λ implies ∥x(t)∥<A,t≥t0 for some t0∈R+;(2)
strictly stable, if(1) holds and for every μ≤λ there exists B<μ such that ∥x0∥>μ implies ∥x(t)∥>B,t≥t0.Theorem 2.4 (see [23]).
Suppose that(1)
0<λ<A<ρ;(2)
there existsV1:R+×S(ρ)→R+,V1∈v0,a1∈CK,b1∈K such that a1(t0,λ)≤b1(A) for some t0∈R+, and for (t,x)∈R+×S(ρ),
(2.7)b1(‖x‖)≤V1(t,x)≤a1(t0,‖x‖),D+V1(t,x)≤0,t≠tk,V1(t+,x+Ik(x))≤V1(t,x),t=tk,(3)
there existsV2:R+×S(ρ)→R+,V2∈v0,a2,b2∈K such that for (t,x)∈R+×S(ρ),
(2.8)b2(‖x‖)≤V2(t,x)≤a2(‖x‖)D-V2(t,x)≥0,t≠tk,V2(t+,x+Ik(x))≥V2(t,x),t=tk.Then, the trivial solution of (2.1) is strictly practically stable.


## 3. Main Results

In this section, we will give some strict practical stability criteria of nonlinear vibration system (2.3).Theorem 3.1.
The trivial solution of system (2.3) is strictly practically stable if there exists a matrix P>0, α≤0 and the following conditions hold:
(3.1)PAi+AiTP≤αI,i=1,…,r,(3.2)‖x(tk)+Ik(x(tk))‖2≤λmin⁡(P)λmax⁡(P)‖x(tk)‖2,k∈N.Proof.
Consider the Lyapunov functionV1(x(t))=(1/2)xT(t)Px(t) and V2(x(t))=exp(-xT(t)Px(t)). It yields that
(3.3)12λmin⁡(P)‖x(t)‖2≤V1(x(t))≤12λmax⁡(P)‖x(t)‖2,exp⁡(-λmin⁡(P)‖x(t)‖2)≥V2(x(t))≥exp⁡(-λmax⁡(P)‖x(t)‖2).
When t≠tk, since PAi+AiTP≤αI,i=1,2,…,r and α≤0, the derivatives of them are
(3.4)D+V1(x(t))=12x˙T(t)Px(t)+xT(t)Px˙(t)=12∑i=1rhi(t)xT(t){PAi+AiTP}x(t)≤α2‖x(t)‖2≤0,t∈(tk-1,tk),k∈N,x∈S(ρ),(3.5)D-V2(x(t))=-exp⁡(-xT(t)Px(t))(x˙T(t)Px(t)+xT(t)Px˙(t))=-exp⁡(-xT(t)Px(t))(∑i=1rhi(t)xT(t){PAi+AiTP}x(t))≥-αexp⁡(-xT(t)Px(t))‖x(t)‖2≥0,t∈(tk-1,tk),k∈N,x∈S(ρ).
Let a1(x)=xT(t)Px(t), b1(x)=(1/4)V1(x(t)), a2(x)=2exp(-xT(t)Px(t)), and b2(x)=exp(-xT(t)Px(t)), we can easily obtain
(3.6)b1(‖x‖)≤V1(x(t))≤a1(t0,‖x‖),b2(‖x‖)≤V2(x(t))≤a2(‖x‖).
When t=tk, according to (3.2) and (3.6), we have
(3.7)V1(x(t)+Ik(x(t)))=12(x(t)+Ik(x(t)))TP(x(t)+Ik(x(t)))≤12λmax⁡(P)‖(x(t)+Ik(x(t)))‖2≤12λmin⁡(P)‖(x(t))‖2≤V1(x(t)),(3.8)V2(x(t)+Ik(x(t)))=exp⁡((-(x(t)+Ik(x(t)))TP(x(t)+Ik(x(t))))≥exp⁡(-λmax⁡(P)‖(x(t)+Ik(x(t)))‖)2≥exp⁡(-λmin⁡(P)‖(x(t))‖2)≥V2(x(t)).
Then, with Theorem2.4, the trivial solution of system (2.3) is strictly practical stable.Remark 3.2.
If system (2.3) is practically stabilizable, we can design a general linear or nonlinear impulsive control law {tk,Ik(x(tk))} for system (2.1), which can make the system oscillate sufficiently near the aimed state and the performance is considered acceptable, that is, the vibration of system (2.1) is depressed in the sense of practical stability under the impulsive control.
IfIk(x(tk))=Ckx(tk),k∈N, where each Ck∈Rn×n is constant matrix, then system (2.3) is rewritten by
(3.9)x˙(t)=∑i=1rhi(t)Aix(t),∀t≥0,t≠tk,Δx(t)=Ckx(t),t=tk,k∈N.
From (3.2), it is easily obtained that
(3.10)‖I+Ck‖2≤λmin⁡(P)λmax⁡(P),k∈N.
Then, we have the following corollary.Corollary 3.3.
The trivial solution of system (3.9) is strictly practically stable if there exist a matrix P>0, α≤0 and the conditions (3.1) and (3.10) hold.


## 4. Conclusions

In this paper, some strict practical stability criteria have been put forward for nonlinear impulsive systems based on their T-S models. The reported results are helpful to consider the vibration characteristics of nonlinear systems with impulsive disturbances and also to control the vibration of nonlinear system via the impulsive control law.



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*Source: 340450-2012-08-16.xml*