Research Article | Open Access

Qishui Zhong, Hongcai Li, Hui Liu, Juebang Yu, "A Note on Practical Stability of Nonlinear Vibration Systems with
Impulsive Effects", *Journal of Applied Mathematics*, vol. 2012, Article ID 340450, 7 pages, 2012. https://doi.org/10.1155/2012/340450

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

**Academic Editor:**Junjie Wei

#### 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 [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: * means is a symmetrical positive (negative) definite matrix. , , and stand for, respectively, the set of all positive real numbers, nonnegative real numbers, and the set of natural numbers. denotes the Euclidian norm of vector , and and mean the minimal and maximal eigenvalues of matrix , respectively.

#### 2. Problem Formulation

Consider the following nonlinear system where is the state variable, . We can construct the fuzzy model for (2.1) as follows: rule : if is , and is , THEN , , in which are the premise variables, each is a fuzzy set, and is a constant matrix. With a center-average defuzzifier, the overall fuzzy system is represented as where is the number of fuzzy implications, , , and is the grade of membership of in . Of course, and . Note that system (2.2) can locally represent system (2.1).

Disturbances, acting on system (2.1), are given by a sequence , where , as , and denotes the incremental change of the state at time . Thus, the following impulsive differential equation is obtained in which .

Assume that for all so that the trivial solution of (2.3) exists. Denote .

We will introduce the following classes of function spaces, definitions, and theorems for future use: is strictly increasing and . for each .

*Definition 2.1. *Let , then is said to belong to class if:(1)is continuous in and for each ,
exits;(2) is locally Lipschitzian in .

*Definition 2.2. *For , we define the derivatives of as

*Definition 2.3. *The trivial solution of the system (2.1) is said to be (1)practically stable, if given with , one has implies for some ;(2)strictly stable, if holds and for every there exists such that implies .

Theorem 2.4 (see [23]). *Suppose that*(1)*;*(2)*there exists such that for some , and for ,
*(3)*there exists such that for ,
**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 , and the following conditions hold:
*

* Proof. *Consider the Lyapunov function and . It yields that
When , since and , the derivatives of them are
Let , , , and , we can easily obtain
When , according to (3.2) and (3.6), we have

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 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 , where each is constant matrix, then system (2.3) is rewritten by

From (3.2), it is easily obtained that
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 , 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.

#### References

- B. Lehman, J. Bentsman, S. Verduyn Lunel, and E. I. Verriest, “Vibrational control of nonlinear time lag systems with bounded delay: averaging theory, stabilizability, and transient behavior,”
*IEEE Transactions on Automatic Control*, vol. 39, no. 5, pp. 898–912, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH - A. E. Alshorbagy, M. A. Eltaher, and F. F. Mahmoud, “Free vibration characteristics of a functionally graded beam by finite element method,”
*Applied Mathematical Modelling*, vol. 35, no. 1, pp. 412–425, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - B. Farshi and A. Assadi, “Development of a chaotic nonlinear tuned mass damper for optimal vibration response,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 11, pp. 4514–4523, 2011. View at: Publisher Site | Google Scholar - B. K. Eshmatov, “Nonlinear vibrations and dynamic stability of viscoelastic orthotropic rectangular plates,”
*Journal of Sound and Vibration*, vol. 300, no. 3–5, pp. 709–726, 2007. View at: Publisher Site | Google Scholar - X. Zhou, Y. Wu, Y. Li, and Z. Wei, “Hopf bifurcation analysis of the Liu system,”
*Chaos, Solitons & Fractals*, vol. 36, no. 5, pp. 1385–1391, 2008. View at: Publisher Site | Google Scholar - X. Zhou, Y. Wu, Y. Li, and H. Xue, “Adaptive control and synchronization of a novel hyperchaotic system with uncertain parameters,”
*Applied Mathematics and Computation*, vol. 203, no. 1, pp. 80–85, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov,
*Theory of Impulsive Differential Equations*, Series in Modern Applied Mathematics, World Scientific, Singapore, 1989. - Q. Zhong, J. Bao, Y. Yu, and X. Liao, “Exponential stabilization for discrete Takagi-Sugeno fuzzy systems via impulsive control,”
*Chaos, Solitons & Fractals*, vol. 41, no. 4, pp. 2123–2127, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - X. H. Tang, Z. He, and J. S. Yu, “Stability theorem for delay differential equations with impulses,”
*Applied Mathematics and Computation*, vol. 131, no. 2-3, pp. 373–381, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH - T. Yang, “Impulsive control,”
*IEEE Transactions on Automatic Control*, vol. 44, no. 5, pp. 1081–1083, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Q. Zhong, J. Bao, Y. Yu, and X. Liao, “Impulsive control for T-S fuzzy model-based chaotic systems,”
*Mathematics and Computers in Simulation*, vol. 79, no. 3, pp. 409–415, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. De La Sen and N. Luo, “A note on the stability of linear time-delay systems with impulsive inputs,”
*IEEE Transactions on Circuits and Systems. I*, vol. 50, no. 1, pp. 149–152, 2003. View at: Publisher Site | Google Scholar - X. Liu and G. Ballinger, “Uniform asymptotic stability of impulsive delay differential equations,”
*Computers & Mathematics with Applications*, vol. 41, no. 7-8, pp. 903–915, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH - X. Liu, “Impulsive stabilization and control of chaotic system,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 47, no. 2, pp. 1081–1092, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. Sun, “Impulsive control of a new chaotic system,”
*Mathematics and Computers in Simulation*, vol. 64, no. 6, pp. 669–677, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Q. S. Zhong, J. F. Bao, Y. B. Yu, and X. F. Liao, “Impulsive control for fractional-order chaotic systems,”
*Chinese Physics Letters*, vol. 25, no. 8, pp. 2812–2815, 2008. View at: Publisher Site | Google Scholar - T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,”
*IEEE Transactions on Systems, Man and Cybernetics*, vol. 15, no. 1, pp. 116–132, 1985. View at: Google Scholar - Y. W. Wang, Z. H. Guan, and H. O. Wang, “Impulsive synchronization for Takagi-Sugeno fuzzy model and its application to continuous chaotic system,”
*Physics Letters, Section A*, vol. 339, no. 3–5, pp. 325–332, 2005. View at: Publisher Site | Google Scholar - Y.-W. Wang, Z.-H. Guan, H. O. Wang, and J.-W. Xiao, “Impulsive control for T-S fuzzy system and its application to chaotic systems,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 16, no. 8, pp. 2417–2423, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - I. M. Stamova, “Vector Lyapunov functions for practical stability of nonlinear impulsive functional differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 325, no. 1, pp. 612–623, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - V. Lakshmikantham, S. Leela, and A. A. Martynyuk,
*Practical Stability of Nonlinear Systems*, World Scientific, Teaneck, NJ, USA, 1990. - T. Yang, J. A. K. Suykens, and L. O. Chua, “Impulsive control of nonautonomous chaotic systems using practical stabilization,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 8, no. 7, pp. 1557–1564, 1998. View at: Google Scholar - S. Li, X. Song, and A. Li, “Strict practical stability of nonlinear impulsive systems by employing two Lyapunov-like functions,”
*Nonlinear Analysis: Real World Applications*, vol. 9, no. 5, pp. 2262–2269, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH

#### Copyright

Copyright © 2012 Qishui Zhong et al. 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.