A Note on Practical Stability of Nonlinear Vibration Systems with Impulsive Effects

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.

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 state 𝑥=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 𝜆_min(𝐴) and 𝜆_max(𝐴) mean the minimal and maximal eigenvalues of matrix 𝐴, respectively.

2. Problem Formulation

Consider the following nonlinear system 𝑥̇(𝑡)=𝑓(𝑥(𝑡)), (2.1) where 𝑥(𝑡)∈𝑅^𝑛 is the state variable, 𝑓∈𝐶[𝑅^𝑛,𝑅^𝑛]. We can construct the fuzzy model for (2.1) as follows: rule (𝑖): if 𝑧₁(𝑡) is 𝑀_𝑖1,…, and 𝑧_𝑝(𝑡) is 𝑀_𝑖𝑝, THEN 𝑥(𝑡)=𝐴_𝑖𝑥(𝑡), 𝑖=1,2,…,𝑟, in which 𝑧_𝑖(𝑡),…,𝑧_𝑝(𝑡) are the premise variables, each 𝑀_𝑖𝑗(𝑗=1,2,…,𝑝) is a fuzzy set, and 𝐴_𝑖∈𝑅^𝑛×𝑛 is a constant matrix. With a center-average defuzzifier, the overall fuzzy system is represented as 𝑥̇(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝑡)𝐴_𝑖𝑥(𝑡), (2.2) where 𝑟 is the number of fuzzy implications, ℎ_𝑖(𝑡)=𝑤_𝑖(𝑡)/(∑^𝑟_𝑖=1𝑤_𝑖(𝑡)), 𝑤_𝑖(𝑡)=∏^𝑝_𝑙=1𝑀_𝑖𝑙(𝑧_𝑙(𝑡)), and 𝑀_𝑖𝑙(𝑧_𝑙(𝑡)) is the grade of membership of 𝑧_𝑙(𝑡) in 𝑀_𝑖𝑙. Of course, ℎ_𝑖(𝑡)≥0 and ∑^𝑟_𝑖=1ℎ_𝑖(𝑡)=1. Note that system (2.2) can locally represent system (2.1).

Disturbances, acting on system (2.1), are given by a sequence {𝑡_𝑘,𝐼_𝑘(𝑥(𝑡_𝑘))}, where 0<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯, 𝑡_𝑘→∞ as 𝑘→∞, and 𝐼_𝑘∈𝐶[𝑅^𝑛,𝑅^𝑛] denotes the incremental change of the state at time 𝑡_𝑘. Thus, the following impulsive differential equation is obtained 𝑥̇(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝑡)𝐴_𝑖𝑥(𝑡),∀𝑡≥0,𝑡≠𝑡_𝑘,Δ𝑥(𝑡)=𝐼_𝑘(𝑥(𝑡)),𝑡=𝑡_𝑘,𝑘∈𝑁, (2.3) in which Δ𝑥(𝑡)=𝑥(𝑡⁺_𝑘)−𝑥(𝑡_𝑘),𝑥(𝑡⁺_𝑘)=lim_{𝑡→𝑡+𝑘}𝑥(𝑡_𝑘),𝑘∈𝑁.

Assume that 𝐼_𝑘(0)≡0 for all 𝑘 so that the trivial solution of (2.3) exists. Denote 𝑆(𝜌)={𝑥∈𝑅^𝑛∶‖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}. 𝐶_𝐾={𝑎∈𝐶[𝑅²₊,𝑅₊]∶𝑎(𝑡,𝑢)∈𝐾 for each 𝑡∈𝑅₊}.

Definition 2.1. Let 𝑉∶𝑅₊×𝑅^𝑛→𝑅₊, then 𝑉 is said to belong to class 𝜐₀ if:(1)𝑉is continuous in (𝑡_𝑘−1,𝑡_𝑘]×𝑅^𝑛 and for each 𝑥∈𝑅^𝑛,𝑘=1,2,…, lim_{→𝑡𝑡,𝑦+𝑘,𝑥}𝑉𝑡,𝑦lim=𝑉𝑡⁺_𝑘,𝑥 (2.4) exits;(2)𝑉 is locally Lipschitzian in 𝑥.

Definition 2.2. For (𝑡,𝑥)∈(𝑡_𝑘−1,𝑡_𝑘]×𝑅^𝑛, we define the derivatives of 𝑉(𝑡,𝑥) as 𝐷⁺𝑉(𝑡,𝑥)=lim_𝑠→0+1sup𝑠𝑉((,𝐷𝑡+𝑠,𝑥+𝑠𝑓𝑡,𝑥)−𝑉𝑡,𝑥)₋𝑉(𝑡,𝑥)=lim_𝑠→0−1inf𝑠𝑉𝑡+𝑠,𝑥+𝑠𝑓(𝑡,𝑥)−𝑉(𝑡,𝑥). (2.5)(2.6)

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 exists 𝑉₁∶𝑅₊×𝑆(𝜌)→𝑅₊,𝑉₁∈𝑣₀,𝑎₁∈𝐶_𝐾,𝑏₁∈𝐾 such that 𝑎₁(𝑡₀,𝜆)≤𝑏₁(𝐴) for some 𝑡₀∈𝑅₊, and for (𝑡,𝑥)∈𝑅₊×𝑆(𝜌), 𝑏₁(‖𝑥‖)≤𝑉₁(𝑡,𝑥)≤𝑎₁𝑡₀,‖𝑥‖,𝐷⁺𝑉₁(𝑡,𝑥)≤0,𝑡≠𝑡_𝑘,𝑉₁𝑡⁺,𝑥+𝐼_𝑘(𝑥)≤𝑉₁(𝑡,𝑥),𝑡=𝑡_𝑘, (2.7)(3)there exists 𝑉₂∶𝑅₊×𝑆(𝜌)→𝑅₊,𝑉₂∈𝑣₀,𝑎₂,𝑏₂∈𝐾 such that for (𝑡,𝑥)∈𝑅₊×𝑆(𝜌), 𝑏₂(‖𝑥‖)≤𝑉₂(𝑡,𝑥)≤𝑎₂𝑥𝐷(‖‖)₋𝑉₂(𝑡,𝑥)≥0,𝑡≠𝑡_𝑘,𝑉₂𝑡⁺,𝑥+𝐼_𝑘(𝑥)≥𝑉₂(𝑡,𝑥),𝑡=𝑡_𝑘. (2.8)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: 𝑃𝐴_𝑖+𝐴^𝑇_𝑖𝑥𝑡𝑃≤𝛼𝐼,𝑖=1,…,𝑟,_𝑘+𝐼_𝑘𝑥𝑡_𝑘²≤𝜆_min(𝑃)𝜆_max(𝑃)𝑥𝑡_𝑘²,𝑘∈𝑁. (3.1)(3.2)

Proof. Consider the Lyapunov function 𝑉₁(𝑥(𝑡))=(1/2)𝑥^𝑇(𝑡)𝑃𝑥(𝑡) and 𝑉₂(𝑥(𝑡))=exp(−𝑥^𝑇(𝑡)𝑃𝑥(𝑡)). It yields that 12𝜆_min(𝑃)‖𝑥(𝑡)‖²≤𝑉₁(𝑥(𝑡≤1))2𝜆_max(𝑃)‖𝑥(𝑡)‖²,exp−𝜆_min(𝑃)‖𝑥(𝑡)‖²≥𝑉₂(𝑥(𝑡))≥exp−𝜆_max(𝑃)‖𝑥(𝑡)‖². (3.3) When 𝑡≠𝑡_𝑘, since 𝑃𝐴_𝑖+𝐴^𝑇_𝑖𝑃≤𝛼𝐼,𝑖=1,2,…,𝑟 and 𝛼≤0, the derivatives of them are 𝐷⁺𝑉₁(𝑥(𝑡=1))2𝑥̇^𝑇(𝑡)𝑃𝑥(𝑡)+𝑥^𝑇(𝑡)(𝑡)=1𝑃𝑥̇2^𝑟_𝑖=1ℎ_𝑖(𝑡)𝑥^𝑇(𝑡)𝑃𝐴_𝑖+𝐴^𝑇_𝑖𝑃𝑥(𝑡)≤𝛼2‖𝑥(𝑡)‖²≤0,𝑡∈𝑡_𝑘−1,𝑡_𝑘𝜌,𝐷,𝑘∈𝑁,𝑥∈𝑆₋𝑉₂(𝑥(𝑡))=−exp−𝑥^𝑇(𝑡)𝑃𝑥(𝑡)𝑥̇^𝑇(𝑡)𝑃𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃𝑥̇(𝑡)=−exp−𝑥^𝑇(𝑡)𝑃𝑥(𝑡)^𝑟_𝑖=1ℎ_𝑖(𝑡)𝑥^𝑇(𝑡)𝑃𝐴_𝑖+𝐴^𝑇_𝑖𝑃𝑥(𝑡)≥−𝛼exp−𝑥^𝑇(𝑡)𝑃𝑥(𝑡)‖𝑥(𝑡)‖²≥0,𝑡∈𝑡_𝑘−1,𝑡_𝑘,𝑘∈𝑁,𝑥∈𝑆𝜌. (3.4)(3.5) Let 𝑎₁(𝑥)=𝑥^𝑇(𝑡)𝑃𝑥(𝑡), 𝑏₁(𝑥)=(1/4)𝑉₁(𝑥(𝑡)), 𝑎₂(𝑥)=2exp(−𝑥^𝑇(𝑡)𝑃𝑥(𝑡)), and 𝑏₂(𝑥)=exp(−𝑥^𝑇(𝑡)𝑃𝑥(𝑡)), we can easily obtain 𝑏₁(‖𝑥‖)≤𝑉₁(𝑥(𝑡))≤𝑎₁𝑡₀,‖𝑥‖,𝑏₂(‖𝑥‖)≤𝑉₂(𝑥(𝑡))≤𝑎₂(‖𝑥‖). (3.6) When 𝑡=𝑡_𝑘, according to (3.2) and (3.6), we have 𝑉₁𝑥(𝑡)+𝐼_𝑘(𝑥(𝑡=1))2𝑥(𝑡)+𝐼_𝑘(𝑥(𝑡))^𝑇𝑃𝑥(𝑡)+𝐼_𝑘(𝑥(𝑡≤1))2𝜆_max(𝑃)𝑥(𝑡)+𝐼_𝑘(𝑥(𝑡))²≤12𝜆_min(𝑃)‖(𝑥(𝑡))‖²≤𝑉₁(𝑥(𝑡,𝑉))₂𝑥(𝑡)+𝐼_𝑘(𝑥(𝑡))=exp−(𝑥(𝑡)+𝐼_𝑘(𝑥(𝑡))^𝑇𝑃𝑥(𝑡)+𝐼_𝑘(𝑥(𝑡))≥exp−𝜆_max(𝑃)𝑥(𝑡)+𝐼_𝑘(𝑥(𝑡))²≥exp−𝜆_min(𝑃)‖(𝑥(𝑡))‖²≥𝑉₂(𝑥(𝑡)). (3.7)(3.8)
Then, with Theorem 2.4, 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 {𝑡_𝑘,𝐼_𝑘(𝑥(𝑡_𝑘))} 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.
If 𝐼_𝑘(𝑥(𝑡_𝑘))=𝐶_𝑘𝑥(𝑡_𝑘),𝑘∈𝑁, where each 𝐶_𝑘∈𝑅^𝑛×𝑛 is constant matrix, then system (2.3) is rewritten by 𝑥̇(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝑡)𝐴_𝑖𝑥(𝑡),∀𝑡≥0,𝑡≠𝑡_𝑘,Δ𝑥(𝑡)=𝐶_𝑘𝑥(𝑡),𝑡=𝑡_𝑘,𝑘∈𝑁. (3.9)
From (3.2), it is easily obtained that 𝐼+𝐶_𝑘²≤𝜆_min(𝑃)𝜆_max(𝑃),𝑘∈𝑁. (3.10) 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.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants no. 50905018 and 51075033, the Foundation of Science and Technology on Vehicle Transmission Laboratory, the Fundamental Research Funds for the Central Universities under Grant no. ZYGX2009J088, and Sichuan Science & Technology Plan under Grant no. 2011JY0001.