Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 754268, 5 pages

http://dx.doi.org/10.1155/2013/754268

## Finite-Time Boundedness Control of Time-Varying Descriptor Systems

^{1}School of Science, Shenyang University of Technology, Shenyang 110870, China^{2}School of Science, Northeastern University, Shenyang 110004, China

Received 29 August 2013; Accepted 2 December 2013

Academic Editor: Yijing Wang

Copyright © 2013 Xiaoming Su 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.

#### Abstract

This paper mainly studies a control problem of finite-time boundedness of time-varying descriptor systems. Firstly, a sufficient and necessary condition of finite-time stability is given, then a sufficient condition of finite-time boundedness for time-varying descriptor systems is given. Secondly, we analyze the finite-time boundedness control problem and design the finite-time state feedback controller; the controller is given based on LMIs for time-varying descriptor systems and time-varying uncertain descriptor systems, respectively. Finally, a numerical example is given to prove the effectiveness of the method.

#### 1. Introduction

Recently, descriptor systems (which are also known as *singular systems, semistate systems, systems of differential-algebraic equations, or generalized state space systems.*) theory has been well studied since they are very important from the engineering point of view. Let the time-varying descriptor systems be described as . They arise naturally in many physical applications such as electrical networks, aircraft and robot dynamics, neutral delay and large-scale systems, economics and optimization problem, biology, constrained mechanics, as result of partial discretization of partial differential equation, and so forth. The time-varying descriptor system has the common properties of general descriptor system and some special properties. In control theory, the time-varying descriptor system stability is mainly Lyapunov stability; however, Lyapunov stability deals with the whole state performance of the system, but it does not reflect a transient performance of the system. The transient performance refers to the system stability in a short period, but it is different from Lyapunov stability. In engineering, a whole stability of the system may have a very bad transient performance, which would cause a bad effect in the engineering. Therefore, people tend to be more concerned about the transient performance of the system [1–5] than the overall steady state performance of the system.

In order to study the transient performance of the system, scholars have given the concept of finite-time stability. Finite-time stability of time-varying descriptor system is a new field; scholars have mainly studied the finite-time stability of linear system for years and also made some corresponding results. For example, [6–10] proposed the definition of finite-time stability and finite-time boundedness; [5–8] proposed the sufficient conditions for the finite-time stability of general linear system; [1, 11] studied the input and output finite-time stability for time-varying descriptor system.

In addition, a finite-time stability problem with external disturbance is called finite-time boundedness; the concept of finite-time boundedness came from finite-time stability. We have some preliminary research results about the finite-time boundedness. For example, [12–14] studied the finite-time bounded problem of the linear time-varying system with impulse; [15–18] proposed the design methods of dynamic compensators; [19–21] discussed the finite-time control problems of uncertain system with disturbance. Scholars introduced the definition of finite-time stability and have given the necessary and sufficient conditions for finite-time stability of the descriptor system, but the finite-time stability problems of time-varying descriptor system have no available research results, especially when is time-variant. So, it is necessary to study finite-time stability of the time-varying uncertain descriptor system.

The paper is divided into two parts: firstly, we deals with the problem of finite-time stability of time-varying descriptor system. We give the necessary and sufficient condition of finite-time stability and finite-time boundedness of system with time-varying function matrix ; the unstable system can be controlled by the state feedback controllers. We also make the corresponding research for time-varying uncertain descriptor system. Secondly, we design a finite-time boundedness state feedback controller for time-varying uncertain descriptor system to make the close-loop system finite-time boundedness.

The paper is organized as follows. In Section 2, we give some results of finite-time stability and boundedness for time-varying descriptor system. In Section 3, some results of time-varying uncertain descriptor system are provided. In Section 4, a numerical example is presented to illustrate the efficiency of the proposed result.

#### 2. Finite-Time Control of Time-Varying Descriptor Systems

##### 2.1. Finite-Time Stable of Time-Varying Descriptor Systems

*Definition 1. *Time-varying matrix is singular on time interval , if there exists a such that rank .

Consider the following time-varying descriptor systems
where is the state. is given continuous matrix-valued function. is singular on time interval .

*Definition 2. *The time-varying descriptor system (1) is said to be finite-time stable with respect to with positive definite matrix , and given three positive scalars , with , if , then , for all .

Theorem 3. *The following statements are equivalent:*(i)*system (1) is FTS respect to .*(ii)*For all , , where is the state transition matrix and is positive definite matrix.*(iii)*For all , the differential Lyapunov inequality, with terminal and initial conditions(a) , where
(b), is positive definite matrix, admits a piecewise continuously differentiable symmetric solution .*

*Proof. * Let ; assume , have
Therefore, system (1) is FTS.

by contradiction. Let us assume
Let , , such that ; then (4) implies that
therefore
Obviously, it contradicts the initial assumption that system (1) is FTS.

Let , .

Then (a) implies that is negative definite along the trajectories of system (1).

Now , then for a generic , such that
By contradiction. Because system (1) is FTS. Let for a small , for all , such that
Let be the solution of
And assume that , such that
Now let , , such that .

Then (11) implies
from (9) we obtain that
Integrating (13) from to we have
Therefore
which contradicts (8).

##### 2.2. Finite-Time Boundedness of Time-Varying Descriptor Systems

Consider the following time-varying descriptor systems where is the state and is exogenous input. , are given constant matrices. is a singular function matrix, and .

The exogenous disturbance is time varying and satisfies the constraint

*Definition 4. *The system (16) subject to an exogenous disturbance satisfies (17) and is said to be finite-time bounded with respect to with positive definite matrices , and given three positive scalars , with , if , then .

Theorem 5. *The time-varying descriptor system (16) is finite-time bounded, if there exist two positive definite nonsingular matrices , , such that
**
hold, where denote the maximum eigenvalue of the argument,
*

*Proof. *Let .

We have
Therefore,
integrating from to , with , we have
From (19), we have
Therefore, system (16) is FTB. The proof is completed.

##### 2.3. Finite-Time Bounded Control of Time-Varying Descriptor Systems

Consider the following time-varying descriptor systems where is the state; is control input; is exogenous input. are given continuous matrix-valued functions. is a singular function matrix, and .

Find a state feedback control law where . Let . Then the closed-loop system is given by

Theorem 6. *The time-varying descriptor system (19) is finite-time bounded, if there exist a symmetric positive definite matrix , and a nonsingular matrix , such that inequality (19), (20) and the following condition hold:
**
Moreover, the state feedback control law is given by
**
where denote the maximum eigenvalue of the argument,
*

*Proof. *From Theorem 5, the conditions for FTB is that there exist a nonsingular matrix such that (19), (20) and the following matrix inequality hold:
where
According to Schur’s theorem, it is easy to see that (32) can be rewritten as
Using (30), (34), we have
Since is symmetric, then
By Schur’s theorem, (36) is equivalent to (29). The proof is completed.

##### 2.4. Finite-Time Bounded Control of Time-Varying Uncertain Descriptor Systems

Consider the following time-varying descriptor systems where is the state; is control input; is exogenous input. , , are given continuous matrix-valued functions. is a singular function matrix, and . where is unknown and satisfies

Theorem 7. *The time-varying descriptor system (37) is finite-time bounded, if there exist a symmetric positive definite matrix , and a nonsingular matrix , such that inequality (19), (20), and (32) hold, and the state feedback control law is given by
**
where
*

*Proof. *Applying (40) to (34), and is symmetric, then the proof is completed.

#### 3. Numerical Example

Consider a time-varying descriptor system (26) with Let , , . Exist , such that So, condition (29), (19), (20) hold. the system (32) is finite-time bounded and the state feedback controller with

#### 4. Conclusions

In this paper, we have studied the finite-time stability and given a sufficient and necessary condition of the finite-time stability; the state feedback controller was designed, then we studied the finite-time boundedness of time-varying descriptor systems, and a sufficient and necessary condition of the finite-time boundedness is given and a state feedback controller was designed. In the end, a numerical example is given to prove the effectiveness of the method.

#### Acknowledgment

This project was supported by the National Nature Science Foundation of China (no. 61074005), the Talent Project of the High Education of Liaoning province, China, under Grant no. LR2012005.

#### References

- J. Yao, Z.-H. Guan, G. Chen, and D. W. C. Ho, “Stability, robust stabilization and ${H}_{\infty}$ control of singular-impulsive systems via switching control,”
*Systems & Control Letters*, vol. 55, no. 11, pp. 879–886, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Lin, H. Du, and S. Li, “Finite-time boundedness and ${L}_{2}$-gain analysis for switched delay systems with norm-bounded disturbance,”
*Applied Mathematics and Computation*, vol. 217, no. 12, pp. 5982–5993, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - G. Garcia, S. Tarbouriech, and J. Bernussou, “Finite-time stabilization of linear time-varying continuous systems,”
*IEEE Transactions on Automatic Control*, vol. 54, no. 2, pp. 364–369, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - Z.-H. Guan, C. W. Chan, A. Y. T. Leung, and G. Chen, “Robust stabilization of singular-impulsive-delayed systems with nonlinear perturbations,”
*IEEE Transactions on Circuits and Systems*, vol. 48, no. 8, pp. 1011–1019, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Tong, B. Wu, and F. Huang, “Finite-time bound and L
*L*_{2}-gain analysis for linear time-varying singular impulsive systems,” in*Proceedings of the Chinese Control and Decision Conference (CCDC '12)*, no. 24, 2012. - E. Moulay, M. Dambrine, N. Yeganefar, and W. Perruquetti, “Finite-time stability and stabilization of time-delay systems,”
*Systems & Control Letters*, vol. 57, no. 7, pp. 561–566, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Yao, J.-e. Feng, L. Sun, and Y. Zheng, “Input-output finite-time stability of time-varying linear singular systems,”
*Journal of Control Theory and Applications*, vol. 10, no. 3, pp. 287–291, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Amato, R. Ambrosino, C. Cosentino, and G. DeTommasi, “Input-output finite-time stability of linear systems,” in
*Proceedings of the 17th Mediterranean Conference on Control and Automation*, pp. 342–346, Hessaloniki, Greece, June 2009. - F. Amato, M. Ariola, C. Cosentino, C. T. Abdallah, and P. Dorato, “Necessary and sufficient conditions for finite-time stability of linear systems,” in
*Proceedings of the American Control Conference*, pp. 4452–4456, Denver, Colo, USA, June 2003. View at Scopus - N. A. Kablar and D. L. Debeljkovic, “Finite-time stability of time-varying linear singular systems,” in
*Proceedings of the 37th IEEE Conference on Decision and Control (CDC '98)*, pp. 3831–3836, Tampa, Fla, USA, December 1998. View at Scopus - J. Xu and J. Sun, “Finite-time stability of linear time-varying singular impulsive systems,”
*IET Control Theory & Applications*, vol. 4, no. 10, pp. 2239–2244, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - S. Zhao, J. Sun, and L. Liu, “Finite-time stability of linear time-varying singular systems with impulsive effects,”
*International Journal of Control*, vol. 81, no. 11, pp. 1824–1829, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Amato, M. Ariola, and C. Cosentino, “Finite-time stability of linear time-varying systems: analysis and controller design,”
*IEEE Transactions on Automatic Control*, vol. 55, no. 4, pp. 1003–1008, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - F. Amato, R. Ambrosino, M. Ariola, and C. Cosentino, “Finite-time stability of linear time-varying systems with jumps,”
*Automatica A*, vol. 45, no. 5, pp. 1354–1358, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Amato, M. Ariola, and C. Cosentino, “Finite-time stabilization via dynamic output feedback,”
*Automatica A*, vol. 42, no. 2, pp. 337–342, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Amato, M. Ariola, M. Carbone, and C. Cosentino, “Finite-time output feedback control of linear systems via differential linear matrix conditions,” in
*Proceedings of the 45th IEEE Conference on Decision and Control (CDC '06)*, pp. 5371–5375, San Diego, Calif, USA, December 2006. View at Scopus - P. Dorato, “Short time stability in linear time-varying systems,” in
*Proceedings of the IRE International Convention Record*, pp. 83–87, 1961. - J. Sun and Z. Cheng, “Finite-time control for one kind of uncertain linear singular systems subject to norm bounded uncertainties,” in
*Proceedings of the 5th World Congress on Intelligent Control and Automation*, pp. 980–984, Hangzhou, Chin, June 2004. View at Scopus - F. Amato, M. Ariola, and P. Dorato, “Finite-time control of linear systems subject to parametric uncertainties and disturbances,”
*Automatica*, vol. 37, no. 9, pp. 1459–1463, 2001. View at Publisher · View at Google Scholar · View at Scopus - J.-E. Feng, Z. Wu, and J.-B. Sun, “Finite-time control of linear singular systems with parametric uncertainties and disturbances,”
*Acta Automatica Sinica*, vol. 31, no. 4, pp. 634–637, 2005. View at Google Scholar · View at Scopus - R. Ambrosino, F. Calabrese, C. Cosentino, and G. de Tommasi, “Sufficient conditions for finite-time stability of impulsive dynamical systems,”
*IEEE Transactions on Automatic Control*, vol. 54, no. 4, pp. 861–865, 2009. View at Publisher · View at Google Scholar · View at MathSciNet