Mathematical Problems in Engineering

Volume 2015, Article ID 950685, 9 pages

http://dx.doi.org/10.1155/2015/950685

## Robust Finite-Time Control for Linear Time-Varying Descriptor Systems with Jumps

School of Science, Shenyang University of Technology, Shenyang 110870, China

Received 30 December 2014; Revised 9 March 2015; Accepted 10 March 2015

Academic Editor: Valter J. S. Leite

Copyright © 2015 Xiaoming Su and Adiya Bao. 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

The finite-time control problem is addressed for uncertain time-varying descriptor system with finite jumps and time-varying norm-bounded disturbance. Firstly, a sufficient condition of finite-time boundedness for the abovementioned class of system is obtained. Then the result is extended to finite-time for the system. Based on the condition, state feedback controller is designed such that the closed-loop system is finite-time boundedness and satisfies gain. The conditions are given in terms of differential linear matrix inequalities (DLMIs) and linear matrix inequalities (LMIs), and such conditions require the solution of a feasibility problem involving DLMIs and LMIs, which can be solved by using existing linear algorithms. Finally, a numerical example is given to illustrate the effectiveness of the method.

#### 1. Introduction

In practice a system could be stable but completely useless because it possesses undesirable transient performances. Thus it is useful to consider the stability of such systems only over a finite time. The concept of finite-time stability (FTS) was first introduced in the Russian literature [1–3]. Later during the 1960s this concept appeared in the literature [4, 5]. Until now, there are many valuable results for this type of stability. In [6] the FTS problem for continuous-time linear time-varying system with finite jumps is dealt with and the finite-time analysis and designed problems for continuous-time time-varying linear system are dealt with in [7]. Sufficient conditions for FTS and finite-time stabilization have been provided in the control literature; see [8–10]. The problem of FTS of linear system via impulsive control at fixed times and variable times was considered in [11]. FTS in the presence of exogenous inputs leads to the concept of finite-time boundedness (FTB). In other words a system is said to be FTB if, given a bound on the initial condition and a characterization of the set of admissible inputs, the state variables remain below the prescribed limit for all inputs in the set. Necessary and sufficient conditions for FTS and FTB are presented, and both the state feedback and the output feedback problems are considered in [12].

On the other hand, in the past three decades, descriptor system theory has been well studied since it often better describe physical systems than regular ones. In time-varying cases, many problems based on descriptor system have been extensively studied and many interesting results have been extended. The controllability, observability, impulsive controllability, and impulsive observability problem of time-varying singular system has been discussed in [13, 14]. The finite-time stability (FTS) problems of time-varying linear singular system have been studied [15–17]. The strict LMI solving problem for descriptor system has been discussed in [18–20].

The system we consider in this paper is a class of uncertain linear time-varying descriptor system with finite state jumps and time-varying norm-bounded disturbance. This class of system exists in the real world which displays a certain kind of dynamics with impulse effect at time instant, that is, the state jumps, which cannot be described by pure continuous or pure discrete models. Recently, some results on time-varying descriptor system with jumps have been reported. Stability, robust stabilization, and control of singular impulsive system were studied in [21]. The problem of stability and stabilization of singular Markovian jump system with external discontinuities and saturating inputs were addressed in [22]. Paper [23] formulated and studied a model for singular impulsive delayed systems with uncertainty from nonlinear perturbations. In [24, 25], sufficient conditions for FTS of linear time-varying singular system with impulses at fixed times were given in terms of matrix inequalities.

We tackle the problem of the FTB, finite-time control, and the controller design. Our results are different from those previous results. In Section 2, the problem we deal with is precisely stated and some preliminary definitions and notations are provided. In Section 3, first, the result of FTB and finite-time control problem analysis is given, and then based on the result, the controller design problem is considered. In this section, we also discuss the numerical algorithms to solve the D/DLMIs. In Section 4, simple examples are presented to illustrate the applicability of our results. Finally, some concluding remarks are provided in Section 5.

#### 2. Problem Statement

We consider a time-varying descriptor system with jumps at fixed time described bywhere is the state vector; is the input; is the disturbance input; is the output. is the initial value of the system state. , , , , , , and are continuous matrix functions while is singular: where , are unknown matrices representing time-varying parameter uncertainties that can be described aswhere are known real constant matrix and is unknown matrix function with Lebesgue-measurable elements and satisfies ; when and are zero matrix, system (1) is called norminal system.

, is time-invariant matrix which reflects the discontinuity of the state trajectory of (1), . System (1) exhibits a finite jump from to at fixed time sequence and ; we assume that the evolution of the state vector is left continuous at each , namely,Moreover, is the disturbance input and satisfies

In this paper we investigate the behavior of system (1) within a finite time interval . Firstly, we propose the following definitions and lemmas.

*Definition 1 (regular and impulse-free). *(1) The time-varying descriptor system is said to be uniformly regular in time interval , if for any there exists a scalar , such that .

(2) The time-varying descriptor system is said to be impulse-free in time interval , if for any there exists a scalar , such that .

From [26], we know that the continuity of matrix functions , , and and the uniform regularity of system (1) assure the existence and uniqueness of the solution to . So in this paper we suppose (1) is uniformly regular.

*Definition 2 ([8] (finite-time boundedness (FTB) for time-varying descriptor system with jumps)). *Given a positive scalar and a positive definite matrix-valued function , system (1) is said to be FTB with respect to if

Lemma 3. *Given matrices , , and with appropriate dimensions and with symmetrical, then**for any satisfying if and only if there exists a scalar such that*

*The problem of finite-time to be addressed in the paper can be formulated as finding a state feedback controller, , such that the following conclusions are held for the close-loop system below:(1)The closed-loop system (9) is FTB with respect to .(2)The controlled output satisfies , for any nonzero , where is a prescribed scalar.*

*A system is called finite-time if the two conditions above are satisfied.*

*3. Main Result*

*The system can be decomposed (1) into two subsystems as follows: where are nonsingular matrix, and it is easy to find that system (1) is impulse-free in time interval , for any inital value , if matrix function is invertible.*

*The following theorem gives a sufficient condition for FTB of system (1)*

*Theorem 4. Uncertain time-varying descriptor system with jumps (1) is said to be FTB with respect to , if there exists a nonsingular and piecewise continuously differential matrix-valued function such thatare fulfilled over , where *

* Proof . *By Schur complement (12b) is equivalent towhere .

Decompose (14) asFrom Lemma 3 it is easy to deriveso (14) is equivalent to the following LMI:where . From (12a), (12b), (12c), andwe have and is symmetric; from (17) we have , Obviously . So is invertible, and system is impulse-free in time interval for any initial value.

Consider the Lyapunov functionThen, differentiating with respect to time on the time interval , we obtain Construct a new vector as follows: By (17), it is easy to see that It follows thatIntegrating both sides of (24) from to in which and noting that , we obtain When , since and , by (25) we also haveNow we consider the situation . Integrating both sides of (24) from to it is obvious to haveand according to so when we haveBy the same progress, it is easy to have

*For norminal system it is easy to obtain the following result.*

*Corollary 5. Norminal time-varying descriptor system with jumps is said to be FTB with respect to , if there exists a nonsingular and piecewise continuously differential matrix-valued function such that (12a), (12b), (12c), (17), and (24) are fulfilled over .*

*Theorem 6. Given a scalar , uncertain time-varying descriptor system with jumps (1) is said to be FTB with respect to and satisfies . There exists a nonsingular and piecewise continuously differential matrix-valued function such that are fulfilled over , where *

*Proof. *By the proof of Theorem 4, (31b) is equivalent to the following LMI:where .

By Lemma 3, it is easy to havewhere Since , we haveBy (31a), (31b), and (36) and the proof of Theorem 4, system (1) is FTB.

Now we will show that the following performance function is bounded:Construct a new vector as follows: By (34), we haveTherefore, for , , we havewith and for we haveThis completes the proof of the theorem.

*For norminal system it is easy to obtain the following result.*

*Corollary 7. Given a scalar , norminal time-varying descriptor system with jumps (1) is said to be FTB with respect to and satisfies . There exists a nonsingular and piecewise continuously differential matrix-valued function such that (34), (42a), (42b), (42c), and (45) are fulfilled over .*

*Theorem 8. Given a scalar and a state feedback controller , the close-loop system (9) is said to be FTB with respect to and satisfies . If there exists a nonsingular and piecewise continuously differential matrix-valued function and a continuously differential matrix-valued function such that
are fulfilled over , the state feedback gain is obtained with , where*

*Proof. *By Schur complement and (42b), we haveContinue using Schur complement, and then (44) is equivalent to
where .

We rewrite (45) as
since andBy Lemma 3 we have Taking , since and , , it is easy to havePre- and postmultiplying (46) by and , we havewhere .

By Theorem 6, close-loop system is FTB and satisfies .

*For norminal system it is easy to obtain the following result.*

*Corollary 9. Given a scalar and a state feedback controller , the norminal close-loop system is said to be FTB with respect to and satisfies . If there exists a nonsingular and piecewise continuously differential matrix-valued function and a continuously differential matrix-valued function such that (50), (42a), and (42c) are fulfilled over , the state feedback gain is obtained with .*

*All the conditions in this paper are expressed in terms of time-varying D/DLMIs. However, with an appropriate choice of the structure of the unknown matrix function , it can be turned into a “standard” LMI problem. The unknown matrix function has been assumed to be piecewise affine; that is, where .*

*Hence, the conditions are reduced to a set of LMIs. Note that the conditions are not strict LMI conditions due to , and this may cause a big trouble in checking the conditions numerically. In order to translate the nonstrict LMI into strict LMI, we can use the following lemma.*

*Lemma 10 (see [27, 28]). Let be symmetric such that and let be nonsingular. Then, is nonsingular and its inverse is expressed aswhere is symmetric and is a nonsingular matrix withwhere and are any matrix with full row rank and satisfy and , respectively; is decomposed as with and are of full column rank.*

*Let and . Using Lemma 10, we can get and . In this way the condition is satisfied, so the nonstrict LMIs have been translated into strict LMIs. Exploiting the MATLAB LMI toolbox, it is possible to find matrices (or ), such that (or ) and verify the conditions.*

*4. Numerical Example*

*4. Numerical Example*

*Consider the linear time-varying descriptor system with jumps defined by *

*Given , , , , and . It is easy to seeWe choose . Solving the LMIs by the numerical algorithm in the previous section using MATLAB LMI toolbox, it is possible to find two piecewise affine matrix functions and which verify the conditions of Theorem 8. Therefore, the following state feedback control law can be obtained (see Figure 1).*