Research ArticleRobust Finite-Time119867

infinControl for Linear Time-Varying

Descriptor Systems with Jumps

Xiaoming Su and Adiya Bao

School of Science Shenyang University of Technology Shenyang 110870 China

Correspondence should be addressed to Adiya Bao syeaadygmailcom

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

Academic Editor Valter J S Leite

Copyright copy 2015 X Su and A Bao This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The finite-time119867infincontrol 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 isobtainedThen the result is extended to finite-time119867

infinfor the system Based on the condition state feedback controller is designed

such that the closed-loop system is finite-time boundedness and satisfies 1198712gain 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 feasibilityproblem involving DLMIs and LMIs which can be solved by using existing linear algorithms Finally a numerical example is givento illustrate the effectiveness of the method

1 Introduction

In practice a system could be stable but completely uselessbecause it possesses undesirable transient performancesThus it is useful to consider the stability of such systemsonly over a finite time The concept of finite-time stability(FTS) was first introduced in the Russian literature [1ndash3]Later during the 1960s this concept appeared in the literature[4 5] Until now there are many valuable results for thistype of stability In [6] the FTS problem for continuous-time linear time-varying system with finite jumps is dealtwith and the finite-time analysis and designed problems forcontinuous-time time-varying linear system are dealt with in[7] Sufficient conditions for FTS and finite-time stabilizationhave been provided in the control literature see [8ndash10] Theproblem of FTS of linear system via impulsive control at fixedtimes and variable times was considered in [11] FTS in thepresence of exogenous inputs leads to the concept of finite-time boundedness (FTB) In other words a system is saidto be FTB if given a bound on the initial condition anda characterization of the set of admissible inputs the statevariables remain below the prescribed limit for all inputs inthe set Necessary and sufficient conditions for FTS and FTBare presented and both the state feedback and the outputfeedback problems are considered in [12]

On the other hand in the past three decades descriptorsystem theory has been well studied since it often betterdescribe physical systems than regular ones In time-varyingcases many problems based on descriptor system havebeen extensively studied and many interesting results havebeen extended The controllability observability impulsivecontrollability and impulsive observability problem of time-varying singular system has been discussed in [13 14] Thefinite-time stability (FTS) problems of time-varying linearsingular system have been studied [15ndash17] The strict LMIsolving problem for descriptor system has been discussed in[18ndash20]

The systemwe consider in this paper is a class of uncertainlinear time-varying descriptor system with finite state jumpsand time-varying norm-bounded disturbance This class ofsystem exists in the real world which displays a certain kindof dynamics with impulse effect at time instant that is thestate jumps which cannot be described by pure continuous orpure discrete models Recently some results on time-varyingdescriptor system with jumps have been reported Stabilityrobust stabilization and 119867

infincontrol of singular impulsive

system were studied in [21] The problem of stability andstabilization of singular Markovian jump system with exter-nal discontinuities and saturating inputs were addressed in[22] Paper [23] formulated and studied a model for singular

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 950685 9 pageshttpdxdoiorg1011552015950685

2 Mathematical Problems in Engineering

impulsive delayed systems with uncertainty from nonlinearperturbations In [24 25] sufficient conditions for FTS oflinear time-varying singular system with impulses at fixedtimes were given in terms of matrix inequalities

We tackle the problemof the FTB finite-time119867infincontrol

and the controller design Our results are different fromthose previous results In Section 2 the problem we dealwith is precisely stated and some preliminary definitions andnotations are provided In Section 3 first the result of FTBand finite-time 119867

infincontrol problem analysis is given and

then based on the result the controller design problem isconsidered In this section we also discuss the numericalalgorithms to solve the DDLMIs In Section 4 simpleexamples are presented to illustrate the applicability of ourresults Finally some concluding remarks are provided inSection 5

2 Problem Statement

We consider a time-varying descriptor system with jumps atfixed time described by

119864 (119905) = 119860 (119905) 119909 (119905) + 119861 (119905) 119906 (119905) + 119866 (119905) 120596 (119905) (119905 = 120591119896)

Δ119909 (120591119896) = 119860

119896119909 (120591minus

119896) (119905 = 120591

119896)

119910 (119905) = 119862 (119905) 119909 (119905) + 119863 (119905) 119906 (119905) + 119865 (119905) 120596 (119905)

(1)

where 119909(119905) isin 119877119899 is the state vector 119906(119905) isin 119877

119897 is the input120596(119905) isin 119877

119898 is the disturbance input 119910(119905) isin 119877119902 is the output

119864119909(1199050) = 119909

0is the initial value of the system state 119860(119905) isin

119877119899times119899 119861(119905) isin 119877

119899times119897 119866(119905) isin 119877119899times119898 119862(119905) isin 119877

119902times119899 119863(119905) isin 119877119902times119897

119865(119905) isin 119877119902times119898 and 119864 isin 119877

119899times119899 are continuous matrix functionswhile 119864 is singular

119860 (119905) = 119860 (119905) + Δ119860 (119905)

119861 (119905) = 119861 (119905) + Δ119861 (119905)

(2)

whereΔ119860(119905)Δ119861(119905) are unknownmatrices representing time-varying parameter uncertainties that can be described as

[Δ119860 (119905) Δ119861 (119905)] = 119872Δ (119905) [119873119886119873119887] (3)

where119872119873119886 119873119887are known real constant matrix and 119865(119905) isin

119877119902times119904 is unknown matrix function with Lebesgue-measurable

elements and satisfies Δ119879(119905)Δ(119905) le 119868 when Δ119860(119905) and Δ119861(119905)are zero matrix system (1) is called norminal system

119860119896isin 119877119899times119899 119896 = 1 2 119873 is time-invariant matrix which

reflects the discontinuity of the state trajectory of (1) minus1 lt

120582max119860119896 lt 0 System (1) exhibits a finite jump from 119909(120591119896) to

119909(120591+

119896) at fixed time sequence 120591

119896 (1199050lt 1205911lt sdot sdot sdot lt 120591

119898le 119879 lt

120591119898+1

lt sdot sdot sdot ) and Δ119909(120591119896) = 119909(120591

+

119896) minus 119909(120591

minus

119896) we assume that the

evolution of the state vector 119909(119905) is left continuous at each 120591119896

namely

119909 (120591119896) = 119909 (120591

minus

119896) = limℎrarr0

+

119909 (120591119896minus ℎ)

119909 (120591+

119896) = limℎrarr0

+

119909 (120591119896+ ℎ)

(4)

Moreover 120596(119905) isin 1198712[0 119879] is the disturbance input and sat-

isfies

int

119879

0

120596119879

(119905) 120596 (119905) 119889119905 le 119889 119889 ge 0 (5)

In this paper we investigate the behavior of system (1)within a finite time interval [119905

0 119879] Firstly we propose the

following definitions and lemmas

Definition 1 (regular and impulse-free) (1) The time-varyingdescriptor system 119864(119905) = 119860(119905)119909(119905) + 119866(119905)120596(119905) is said to beuniformly regular in time interval [119905

0 119879] if for any 119905 isin [119905

0 119879]

there exists a scalar 119904 such that det(119904119864 minus 119860(119905)) = 0(2) The time-varying descriptor system 119864(119905) =

119860(119905)119909(119905) + 119866(119905)120596(119905) is said to be impulse-free in time interval[1199050 119879] if for any 119905 isin [119905

0 119879] there exists a scalar 119904 such that

deg(det(119904119864 minus 119860(119905))) = rank(119864(119905))

From [26] we know that the continuity of matrix func-tions119860(119905)119861(119905) and119866(119905) and the uniform regularity of system(1) assure the existence and uniqueness of the solution to(1199050 119879] So in this paper we suppose (1) is uniformly regular

Definition 2 ([8] (finite-time boundedness (FTB) for timendashvarying descriptor system with jumps)) Given a positivescalar 119879 and a positive definite matrix-valued function 119877(sdot)system (1) is said to be FTB with respect to (119879 119877(sdot) 119889) if

119909119879

0119877 (1199050) 1199090le 1 997904rArr 119909

119879

(119905) 119877 (119905) 119909 (119905) le 1 119905 isin [1199050 1199050+ 119879]

(6)

Lemma 3 Given matrices Ω Γ and Ξ with appropriate di-mensions and with Ω symmetrical then

Ω + ΓΔ (119905) Ξ + Ξ119879

Δ119879

(119905) Γ119879

lt 0 (7)

for any Δ(119905) satisfying Δ119879(119905)Δ(119905) le 119868 if and only if there existsa scalar 120576 gt 0 such that

Ω + 120576ΓΓ119879

+ 120576minus1

Ξ119879

Ξ lt 0 (8)

The problem of finite-time 119867infin

to be addressed in thepaper can be formulated as finding a state feedback controller119906 = 119870(119905)119909 such that the following conclusions are held forthe close-loop system below

119864 (119905) = 119860119888(119905) 119909 (119905) + 119866 (119905) 120596 (119905) (119905 = 120591

119896)

Δ119909 (120591119896) = 119860

119896119909 (120591minus

119896) (119905 = 120591

119896)

119910 (119905) = 119862119888(119905) 119909 (119905) + 119865 (119905) 120596 (119905)

(9)

(1) The closed-loop system (9) is FTB with respect to (119879119877(sdot) 119889)

(2) The controlled output 119910(119905) satisfies int1198791199050

(119910119879

(119905)119910(119905) minus

1205742

120596119879

(119905)120596(119905))119889119905 lt 119872(1199090 1199050) for any nonzero 120596(119905)

where 120574 gt 0 is a prescribed scalar

A system is called finite-time 119867infin

if the two conditionsabove are satisfied

Mathematical Problems in Engineering 3

3 Main Result

The system can be decomposed (1) into two subsystems asfollows

1(119905) = 119860

11(119905) 1199091(119905) + 119860

22(119905) 1199092(119905) + 119861

1(119905) 119906 (119905)

+ 1198661(119905) 120596 (119905)

0 = 11986021(119905) 1199091(119905) + 119860

22(119905) 1199092(119905) + 119861

2(119905) 119906 (119905) + 119866

2(119905) 120596 (119905)

119910 (119905) = 1198621(119905) 1199091(119905) + 119862

2(119905) 1199092(119905) + 119863 (119905) 119906 (119905) + 119865 (119905) 120596 (119905)

(10)

where

119875119864119876 = [

119868 0

0 0

] 119875119860 (119905) 119876 = [

11986011(119905) 119860

12(119905)

11986021(119905) 119860

22(119905)

]

119875119866 (119905) = [

1198661(119905)

1198662(119905)

]

119876minus1

119909 (119905) = [1199091(119905) 1199092(119905)] 119862 (119905) 119876 = [119862

1(119905) 119862

2(119905)]

(11)

119875119876 are nonsingular matrix and it is easy to find that system(1) is impulse-free in time interval [119905

0 119879] for any inital value

1199090 if matrix function 119860

22(119905) is invertible

The following theorem gives a sufficient condition forFTB of system (1)

Theorem 4 Uncertain time-varying descriptor system withjumps (1) is said to be FTB with respect to (119879 119877(sdot) 119889) if thereexists a nonsingular and piecewise continuously differentialmatrix-valued function Γ(sdot) such that

119864119879

Γ (119905) = Γ119879

(119905) 119864 ge 119877 (119905) ge 0 (12a)

[

[

[

[

[

Ω (119905) Γ119879

(119905) 119866 (119905) Γ119879

(119905)119872

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0

119872119879

Γ (119905) 0 120576119868

]

]

]

]

]

lt 0 (12b)

119864119879

Γ (1199050) = Γ119879

(1199050) 119864 le

1

1 + 119889

119877 (1199050) (12c)

are fulfilled over [1199050 119879] where

Ω (119905) = 119860

119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) + 119864119879

Γ (119905) + 120576119873119879

119886119873119886

(13)

Proof By Schur complement (12b) is equivalent to

[

[

Ω0(119905) Γ

119879

(119905) 119866 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868

]

]

lt 0 (14)

where Ω0(119905) = 119860

119879

(119905)Γ(119905) + Γ119879

(119905)119860(119905) + 119864119879

Γ(119905) + 120576119873119879

119886119873119886+

(1120576)Γ119879

(119905)119872119872119879

Γ(119905)

Decompose (14) as

[

[

[

119860

119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) + 119864119879

Γ (119905) Γ119879

(119905) 119866 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868

]

]

]

+ 120576[

119873119879

1198861198731198860

0 0

] +

1

120576

[

Γ119879

(119905)119872119872119879

Γ (119905) 0

0 0

]

(15)

From Lemma 3 it is easy to derive

[

[

[

119860

119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) + 119864119879

Γ (119905) Γ119879

(119905) 119866 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868

]

]

]

+ [

Γ119879

(119905)119872 0

0 0

][

Δ (119905) 0

0 0

] [

1198731198860

0 0

]

+ ([

Γ119879

(119905)119872 0

0 0

][

Δ (119905) 0

0 0

] [

1198731198860

0 0

])

119879

(16)

so (14) is equivalent to the following LMI

[

[

Ω1(119905) Γ

119879

(119905) 119866 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868

]

]

lt 0 (17)

where Ω1(119905) = 119860

119879

(119905)Γ(119905) + Γ119879

(119905)119860(119905) + 119864119879

Γ(119905) From (12a)(12b) (12c) and

119876minus119879

Γ (119905) 119875 = [

Γ1(119905) Γ2(119905)

Γ3(119905) Γ4(119905)

] (18)

we have Γ2(119905) = 0 and Γ

1(119905) is symmetric from (17) we have

Ω1(119905) lt 0

Ω1(119905) = 119876

119879

119860119879

(119905) 119875119879

119875minus119879

Γ (119905) 119876

+ 119876119879

Γ119879

(119905) 119875minus1

119875119860 (119905) 119876 + 119876119879

119864119879

119875119879

119875minus119879

Γ (119905) 119876

= [

119860119879

11(119905) 119860

119879

21(119905)

119860119879

12(119905) 119860

119879

22(119905)

] [

Γ1(119905) Γ2(119905)

Γ3(119905) Γ4(119905)

]

+ [

Γ119879

1(119905) Γ119879

3(119905)

Γ119879

2(119905) Γ119879

4(119905)

] [

11986011(119905) 119860

12(119905)

11986021(119905) 119860

22(119905)

]

+ [

119868 0

0 0

][

Γ1(119905) Γ2(119905)

Γ3(119905) Γ4(119905)

]

= [

⋆ ⋆

⋆ 119860119879

22(119905) Γ4(119905) + Γ

119879

4(119905) 11986022(119905)

] lt 0

(19)

Obviously 11986011987922(119905)Γ4(119905) + Γ

119879

4(119905)11986022(119905) lt 0 So 119860

22(119905) is invert-

ible and system is impulse-free in time interval [1199050 119879] for any

initial value

4 Mathematical Problems in Engineering

Consider the Lyapunov function

119881 (119905 119909 (119905)) = 119909119879

(119905) 119864119879

Γ (119905) 119909 (119905) (20)

Then differentiating 119881(119905 119909(119905)) with respect to time 119905 on thetime interval 119905 isin (119905

0 1205911) we obtain

(119905 119909 (119905)) = 119909119879

(119905) (119860119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) + 119864119879

Γ (119905)) 119909 (119905)

+ 120596119879

(119905) 119866119879

(119905) Γ (119905) 119909 (119905)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905)

= 119909119879

(119905) Ω1(119905) 119909 (119905) + 120596

119879

(119905) 119866119879

(119905) Γ (119905) 119909 (119905)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905)

(21)

Construct a new vector as follows

119911 (119905) = [

119909 (119905)

120596 (119905)

] (22)

By (17) it is easy to see that

119911119879

(119905)[

[

Ω1(119905) Γ

119879

(119905) 119866 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868

]

]

119911 (119905)

= 119909119879

(119905) Ω1(119905) 119909 (119905) + 120596

119879

(119905) 119866119879

(119905) Γ (119905) 119909 (119905)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905) minus

1

1 + 119889

120596119879

120596 (119905)

= (119905 119909 (119905)) minus

1

1 + 119889

120596119879

(119905) 120596 (119905) lt 0

(23)

It follows that

(119905 119909 (119905)) lt

1

1 + 119889

120596119879

(119905) 120596 (119905) (24)

Integrating both sides of (24) from 1199050to 119905 in which 119905 isin (119905

0 1205911)

and noting that 119909119879(1199050)119877(1199050)119909(1199050) lt 1 we obtain

119881 (119905 119909 (119905)) lt 119881 (1199050 119909 (1199050)) +

1

1 + 119889

int

119905

1199050

120596119879

(120591) 120596 (120591) 119889120591

lt

1

1 + 119889

119909119879

0119877 (1199050) 1199090+

1

1 + 119889

int

119905

1199050

120596119879

(120591) 120596 (120591) 119889120591

lt

1

1 + 119889

+

1

1 + 119889

int

119905

1199050

120596119879

(120591) 120596 (120591) 119889120591 119905 isin (1199050 1205911)

(25)

When 119905 = 1205911 since 119909(120591minus

1) = 119909(120591

1) and 120591minus

1isin (1199050 1205911) by (25) we

also have

119881 (1205911 119909 (1205911)) lt

1

1 + 119889

+

1

1 + 119889

int

1205911

1199050

120596119879

(120591) 120596 (120591) 119889120591 (26)

Now we consider the situation 119905 isin (1205911 1205912) Integrating both

sides of (24) from 120591+

1to 119905 it is obvious to have

119881 (119905 119909 (119905)) lt

1

1 + 119889

int

119905

1205911

120596119879

(120591) 120596 (120591) 119889120591 + 119881 (120591+

1 119909 (120591+

1))

119905 isin (1205911 1205912)

(27)

and according to minus1 lt 1198601198962lt 0

119881 (120591+

1 119909 (120591+

1)) = 119909

119879

(120591+

1) 119864119879

Γ (120591+

1) 119909 (120591+

1)

= [(119868 + 1198601) 119909 (1205911)]119879

119864119879

Γ (120591+

1)

sdot [(119868 + 1198601) 119909 (1205911)]

lt 119909119879

(1205911) 119864119879

Γ (1205911) 119909 (1205911)

le

1

1 + 119889

+

1

1 + 119889

int

1205911

1199050

120596119879

(120591) 120596 (120591) 119889120591

(28)

so when 119905 isin (1205911 1205912] we have

119881 (119905 119909 (119905)) lt

1

1 + 119889

+

1

1 + 119889

int

119905

1199050

120596119879

(120591) 120596 (120591) 119889120591 (29)

By the same progress it is easy to have

119909119879

(119905) 119877 (119905) 119909 (119905) le 119881 (119905 119909 (119905)) le 1 (30)

For norminal system it is easy to obtain the followingresult

Corollary 5 Norminal time-varying descriptor system withjumps is said to be FTBwith respect to (119879 119877(sdot) 119889) if there existsa nonsingular and piecewise continuously differential matrix-valued function Γ(sdot) such that (12a) (12b) (12c) (17) and (24)are fulfilled over [119905

0 119879]

Theorem 6 Given a scalar 120574 gt 0 uncertain time-varyingdescriptor system with jumps (1) is said to be FTB with respectto (119879 119877(sdot) 119889) and satisfies int119879

1199050

(119910119879

(119905)119910(119905) minus 1205742

120596119879

(119905)120596(119905))119889119905 lt

0 There exists a nonsingular and piecewise continuouslydifferential matrix-valued function Γ(sdot) such that

119864119879

Γ (119905) = Γ119879

(119905) 119864 ge 119877 (119905) ge 0 (31a)

[

[

[

[

[

[

[

[

Ξ (119905) Γ119879

(119905) 119866 (119905) 119862119879

(119905) 119865 (119905) Γ119879

(119905)119872

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0 0

119865119879

(119905) 119862 (119905) 0 119865119879

(119905) 119865 (119905) minus 1205742

119868 0

119872119879

Γ (119905) 0 0 120576119868

]

]

]

]

]

]

]

]

lt 0

(31b)

119864119879

Γ (1199050) = Γ119879

(1199050) 119864 le

1

1 + 119889

119877 (1199050) (31c)

Mathematical Problems in Engineering 5

are fulfilled over [1199050 119879] where

Ξ (119905) = 119864119879

Γ (119905) + 119860

119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) + 119862119879

(119905) 119862 (119905)

+ 120576119873119879

119886119873119886

(32)

Proof By the proof of Theorem 4 (31b) is equivalent to thefollowing LMI

[

[

[

[

[

Ξ0(119905) Γ

119879

(119905) 119866 (119905) 119862119879

(119905) 119865 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0

119865119879

(119905) 119862 (119905) 0 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

lt 0 (33)

where Ξ0(119905) = 119860

119879

(119905)Γ(119905) + Γ119879

(119905)119860(119905) + 119864119879

Γ(119905) + 120576119873119879

119886119873119886+

(1120576)Γ119879

(119905)119872119872119879

Γ(119905) + 119862119879

(119905)119862(119905)By Lemma 3 it is easy to have

Θ (119905)

=

[

[

[

[

[

Ξ1(119905) + 119862

119879

(119905) 119862 (119905) Γ119879

(119905) 119866 (119905) 119862119879

(119905) 119865 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0

119865119879

(119905) 119862 (119905) 0 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

lt 0

(34)

where

Ξ1(119905) = 119864

119879

Γ (119905) + 119860119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) lt 0 (35)

Since 119862119879(119905)119862(119905) gt 0 we have

[

[

Ξ1(119905) Γ

119879

(119905) 119866 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868

]

]

lt 0 (36)

By (31a) (31b) and (36) and the proof of Theorem 4 system(1) is FTB

Now we will show that the following performance func-tion is bounded

119869 = int

119879

1199050

(119910119879

(119905) 119910 (119905) minus 1205742

120596119879

(119905) 120596 (119905)) lt 119881 (1199090 1199050) (37)

Construct a new vector as follows

119911 (119905) =[

[

[

119909 (119905)

120596 (119905)

120596 (119905)

]

]

]

(38)

By (34) we have

119911119879

(119905) Θ (119905) 119911 (119905)

= 119909119879

(119905) Ξ1(119905) 119909 (119905) + 120596

119879

(119905) 119866119879

(119905) Γ (119905) 119909 (t)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905) + 119909119879

(119905) 119862119879

(119905) 119862 (119905) 119909 (119905)

+ 120596119879

(119905) 119865119879

(119905) 119862 (119905) 119909 (119905) + 119909119879

(119905) 119862119879

(119905) 119865 (119905) 120596 (119905)

+ 120596119879

(119905) 119865119879

(119905) 119865 (119905) 120596 (119905) minus (

1

1 + 119889

+ 1205742

)120596119879

(119905) 120596 (119905)

= 119909119879

(119905) Ξ1(119905) 119909 (119905) + 120596

119879

(119905) 119866119879

(119905) Γ (119905) 119909 (119905)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905) minus

1

1 + 119889

120596119879

(119905) 120596 (119905)

+ 119910119879

(119905) 119910 (119905) minus 1205742

120596119879

(119905) 120596 (119905)

= (119905) minus

1

1 + 119889

120596119879

(119905) 120596 (119905) + 119910119879

(119905) 119910 (119905)

minus 1205742

120596119879

(119905) 120596 (119905) lt 0

(39)

Therefore for 119905 isin (120591119896 120591119896+1

) 119896 = 0 1 2 119898 we have

119869120591119896+1

= int

120591119896+1

120591119896

119911119879

(119905) Θ (119905) 119911 (119905) 119889119905

minus (int

120591119896+1

120591119896

(119905) minus

1

1 + 119889

120596119879

(119905) 120596 (119905)) 119889119905

(40)

with minus119881(119879) minus int

119879

1199050

(1(1 + 119889))120596119879

(119905)120596(119905) lt 0 and for Θ(119905) lt 0

we have

119869 =

119898+1

sum

119894=0

119869120591119894+1

= int

119879

1199050

119911119879

(119905) Θ (119905) 119911 (119905) 119889119905 + 119881 (1199050) minus 119881 (119879)

minus int

119879

1199050

1

1 + 119889

120596119879

(119905) 120596 (119905) 119889119905 lt 119881 (1199050)

(1205910= 1199050 120591119898+1

= 119879)

(41)

This completes the proof of the theorem

For norminal system it is easy to obtain the followingresult

Corollary 7 Given a scalar 120574 gt 0 norminal time-varyingdescriptor system with jumps (1) is said to be FTB with respectto (119879 119877(sdot) 119889) and satisfies int119879

1199050

(119910119879

(119905)119910(119905) minus 1205742

120596119879

(119905)120596(119905))119889119905 lt

0 There exists a nonsingular and piecewise continuouslydifferential matrix-valued function Γ(sdot) such that (34) (42a)(42b) (42c) and (45) are fulfilled over [119905

0 119879]

6 Mathematical Problems in Engineering

Theorem 8 Given a scalar 120574 gt 0 and a state feedback con-troller 119906 = 119870(119905)119909 the close-loop system (9) is said to be FTBwith respect to (119879 119877(sdot) 119889) and satisfies 119869 = int

119879

1199050

(119910119879

(119905)119910(119905) minus

1205742

120596119879

(119905)120596(119905))119889119905 lt 119881(1199090 1199050) If there exists a nonsingular and

piecewise continuously differential matrix-valued function Γ(sdot)and a continuously differential matrix-valued function 119871(sdot)

such that

Γ

119879

(119905) 119864119879

= 119864Γ (119905) ge 119877 (119905) ge 0 (42a)

[

[

[

[

[

[

[

[

[

[

[

Π (119905) 119866 (119905) 0 Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905) Γ

119879

(119905)119873119879

119886+ 119871119879

(119905)119873119879

119887

119866119879

(119905) minus

1

1 + 119889

119868 0 0 0

0 0 minus1205742

119868 119865119879

(119905) 0

119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905) 0 119865 (119905) minus119868 0

119873119886Γ (119905) + 119873

119887119871 (119905) 0 0 0 minus120576119868

]

]

]

]

]

]

]

]

]

]

]

lt 0 (42b)

Γ

119879

(1199050) 119864119879

= 119864Γ (1199050) le

1

1 + 119889

119877 (1199050) (42c)

are fulfilled over [1199050 119879] the state feedback gain is obtained with

119870(119905) = 119871(119905)Γ

minus1

(119905) where

Π (119905) = minus 119864Γ (119905) + Γ

119879

(119905) 119860

119879

(119905) + 119860 (119905) Γ (119905) + 119871119879

(119905) 119861

119879

(119905)

+ 119861 (119905) 119871 (119905) + 120576119872119872119879

(43)

Proof By Schur complement and (42b) we have

[

[

[

[

[

[

[

[

Π (119905) 119866 (119905) 0 Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)

119866119879

(119905) minus

1

1 + 119889

119868 0 0

0 0 minus1205742

119868 119865119879

(119905)

119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905) 0 119865 (119905) minus119868

]

]

]

]

]

]

]

]

+

1

120576

[

[

[

[

[

[

Γ119879

(119905)119873119879

119886+ 119871119879

(119905)119873119879

119887

0

0

0

]

]

]

]

]

]

[119873119886Γ (119905) + 119873

119887119871 (119905) 0 0 0] lt 0

(44)

Continue using Schur complement and then (44) is equiva-lent to

[

[

[

Π0(119905) ⋆ ⋆

⋆ ⋆ ⋆

⋆ ⋆ ⋆

]

]

]

+

[

[

[

[

Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)

0

119865119879

(119905)

]

]

]

]

[119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905) 0 119865 (119905)]

=

[

[

[

[

Π0(119905) ⋆ (Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)) 119865 (119905)

⋆ ⋆ ⋆

119865119879

(119905) (119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905)) ⋆ 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

lt 0

(45)

Mathematical Problems in Engineering 7

where Π0(119905) = Π(119905) + (1120576)(Γ

119879

(119905)119873119879

119886+ 119871119879

(119905)119873119879

119887)(119873119886Γ(119905) +

119873119887119871(119905))

We rewrite (45) as

[

[

[

[

[

[

[

Π1(119905) 119866 (119905) (Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)) 119865 (119905)

119866119879

(119905) minus

1

1 + 119889

119868 0

119865119879

(119905) (119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905)) 0 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

]

]

lt 0 (46)

since119870(119905) = 119871(119905)Γ

minus1

(119905) and

Π1(119905) = minus119864

Γ (119905) + Γ

119879

(119905) (119860

119879

(119905) + 119870119879

(119905) 119861

119879

(119905))

+ (119860 (119905) + 119861 (119905)119870 (119905)) Γ (119905) + 120576119872119872119879

+

1

120576

Γ

119879

(119905) (119873119886+ 119873119887119870 (119905))

119879

(119873119886+ 119873119887119870 (119905)) Γ (119905)

+ Γ

119879

(119905) (119862119879

(119905) + 119870119879

(119905) 119863119879

(119905))

sdot (119862 (119905) + 119863 (119905)119870 (119905)) Γ (119905)

(47)

By Lemma 3 we have

Π1(119905) = minus119864

Γ (119905) + Γ

119879

(119905) 119862119879

119888(119905) 119862119888(119905) Γ (119905)

+ Γ

119879

(119905) (119860

119879

(119905) + 119870119879

(119905) 119861

119879

(119905))

+ (119860 (119905) + 119861 (119905)119870 (119905)) Γ (119905)

+ Γ

119879

(119905) (119872Δ (119905)119873119886+119872Δ (119905)119873

119887119870 (119905))

119879

+ (119872Δ (119905)119873119886+119872Δ (119905)119873

119887119870 (119905)) Γ

= minus119864Γ (119905) + Γ

119879

(119905) 119862119879

119888(119905) 119862119888(119905) Γ (119905) + Γ

119879

(119905) 119860119879

119888(119905)

+ 119860119888(119905) Γ (119905)

(48)

Taking Γ(119905) = Γ

minus1

(119905) since Γ119879(119905)119864 = 119864119879

Γ(119905) and 119868 =

Γ(119905)Γ(119905) 0 = Γ(119905)Γ(119905) + Γ(119905)Γ(119905) it is easy to have

minusΓ

minus119879

(119905) 119864Γ (119905) Γ

minus1

(119905) = minus119864119879

Γ (119905)Γ (119905) Γ (119905) = 119864

119879

Γ (119905)

(49)

Pre- and postmultiplying (46) by diag(Γminus119879(119905) 119868 119868) anddiag(Γminus1(119905) 119868 119868) we have

[

[

[

[

[

Π2(119905) Γ

119879

(119905) 119866 (119905) 119862119879

119888(119905) 119865 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0

119865119879

(119905) 119862119888(119905) 0 119865

119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

lt 0 (50)

where Π2(119905) = 119864Γ(119905) + 119862

119879

119888(119905)119862119888(119905) + Γ

119879

(119905)119860119888(119905) + 119860

119879

119888(119905)Γ(119905)

By Theorem 6 close-loop system is FTB and satisfies 119869 lt119881(1199090 1199050)

For norminal system it is easy to obtain the followingresult

Corollary 9 Given a scalar 120574 gt 0 and a state feedback con-troller 119906 = 119870(119905)119909 the norminal close-loop system is said to beFTBwith respect to (119879 119877(sdot) 119889) and satisfies 119869 = int

119879

1199050

(119910119879

(119905)119910(119905)minus

1205742

120596119879

(119905)120596(119905))119889119905 lt 119881(1199090 1199050) If there exists a nonsingular and

piecewise continuously differential matrix-valued function Γ(sdot)and a continuously differential matrix-valued function 119871(sdot)

such that (50) (42a) and (42c) are fulfilled over [1199050 119879] the

state feedback gain is obtained with 119870(119905) = 119871(119905)Γ

minus1

(119905)

All the conditions in this paper are expressed in terms oftime-varyingDDLMIs However with an appropriate choiceof the structure of the unknown matrix function 119875(119905) it canbe turned into a ldquostandardrdquo LMI problem The unknownmatrix function119875(119905) has been assumed to be piecewise affinethat is

119875 (0) (or (119875 (119905))) = Π0

1

119875 (119905) (or (119875 (119905))) = Π0

119896+ Π119904

119896(119905 minus (119896 minus 1) 119879

119904)

(119905) (or (119875 (119905))) = Π119904

119896

119896 isin 119873 119896 lt 119896 119905 isin [(119896 minus 1) 119879119904 119896119879119904]

119875 (119905) (or (119875 (119905))) = Π0

119896+1

+ Π0

119896+1

(119905 minus 119896119879119904)

(119905) (or (119875 (119905))) = Π0

119896+1

119905 isin [119896119879119904 119879]

(51)

where 119896 = max 119896 isin 119873+ 119896 lt 119879119879119904

Hence the conditions are reduced to a set of LMIs Notethat the conditions are not strict LMI conditions due to119864119879

Γ(119905) = Γ119879

(119905)119864 ge 0 and this may cause a big troublein checking the conditions numerically In order to translatethe nonstrict LMI into strict LMI we can use the followinglemma

8 Mathematical Problems in Engineering

Lemma 10 (see [27 28]) Let 119883 isin 119877119899times119899 be symmetric such

that 119864119879119871119883119864119871gt 0 and let 119879 isin 119877

(119899minus119903)times(119899minus119903) be nonsingular Then119883119864 + 119867

119879

119879119878119879 is nonsingular and its inverse is expressed as

(119883119864 + 119867119879

119879119878119879

)

minus1

= X119864119879

+ 119878T119867 (52)

whereX is symmetric and T is a nonsingular matrix with

119864119879

119877X119864119877= (119864119879

119871119883119864119871)

minus1

T = (119878119879

119878)

minus1

119879minus1

(119867119867119879

)

minus1

(53)

where 119867 and 119878 are any matrix with full row rank and satisfy119872119864 = 0 and 119864119878 = 0 respectively 119864 is decomposed as 119864 =

119864119871119864119879

119877with 119864

119871isin 119877119899times119903 and 119864

119877isin 119877119899times119903 are of full column rank

LetΠ0119896= 1198830

119896119864+119867

119879

1198790

119896119878119879 andΠ119904

119896= 119883119904

119896119864+119867

119879

119879119904

119896119878119879 Using

Lemma 10 we can get (1198830119896119864+119867119879

1198790

119896119878119879

)minus1

= X0119896119864119879

+119878T0119896119867 and

(119883119904

119896119864+119867

119879

119879119904

119896119878119879

)minus1

= X119904119896119864119879

+119878T0119896119867 In this way the condition

119864119879

Γ(119905) = Γ119879

(119905)119864 ge 0 is satisfied so the nonstrict LMIs havebeen translated into strict LMIs Exploiting the MATLABLMI toolbox it is possible to find matrices 119883119904

119896 1198790

119896 119879119904

119896 1198830

119896(or

X0119896X119904119896 T0119896 T 119904119896) such that Γ(119905) (or Γ(119905)) and 119871(119905) verify the

conditions

4 Numerical Example

Consider the linear time-varying descriptor system withjumps defined by

119864 = [

1 0

0 0

] 119860 = [

0 1

2 (119905 + 1) 3 (119905 + 1)2]

119861 = [

1

1

] 119865 = [05 1] 119860119896= [

001 0

0 0

]

119866 = [

1 0

0 1

] 119863 = 1 119862 = [1 1]

119872 = [

01

01

] 119873119886= [02 01] 119873

119887= [01 01]

(54)

Given 120596(119905) = [sin(2120587119905 + 1) cos(2120587119905 minus 2)]119879 Ω = [1525] 119877 = diag(12 1) 120574 = 15 and 120576 = 1654 It is easy tosee

119864119877= 119864119871= [

1

0

] (55)

We choose119867119879 = 119878 = [0 1]119879 Solving the LMIs by the numer-

ical algorithm in the previous section using MATLAB LMItoolbox it is possible to find two piecewise affinematrix func-tions Γ(119905) and 119871(119905) which verify the conditions ofTheorem 8Therefore the following state feedback control law can beobtained (see Figure 1)

15 2 25

0

05

1

15

2

25

3

minus05

minus15

minus1

Figure 1 1198701(119905) (solid line) and 119870

2(119905) (dashed line)

5 Conclusions

In this paper we have formulated and studied the finite-time119867infin

control problem of uncertain time-varying descriptorsystem with jumps at fixed times A sufficient condition forFTB and a sufficient condition for finite-time 119867

infinof the

system have been presented in terms of DLMIs and LMIsThen the state feedback controller has been designed toguarantee the closed-loop systemrsquos FTB and satisfy the 119871

2

gain At last two numerical examples have been used toillustrate the main result

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by the National Nature ScienceFoundation of China (no 61074005) and the Talent Project oftheHighEducation of Liaoning province China underGrantno LR2012005

References

[1] G V Kamenkov ldquoOn stability of motion over a finite intervalof timerdquo Journal of Applied Mathematics and Mechanics vol 17pp 529ndash540 1953

[2] A A Lebedev ldquoOn stability of motion during a given intervalof timerdquo Journal of Applied Mathematics and Mechanics vol 18pp 139ndash148 1954

[3] A A Lebedev ldquoOn the problem of stability of motion overa finite interval of timerdquo Journal of Applied Mathematics andMechanics vol 18 pp 75ndash94 1954

[4] P Dorato ldquoShort time stability in linear time-varying systemsrdquoin Proceedings of the IRE International Convention Record Part4 pp 83ndash87 1961

[5] LWeiss andE F Infante ldquoFinite time stability under perturbingforces and on product spacesrdquo IEEE Transactions on AutomaticControl vol 12 no 1 pp 54ndash59 1967

Mathematical Problems in Engineering 9

[6] F Amato R Ambrosino M Ariola and C Cosentino ldquoFinite-time stability of linear time-varying systems with jumpsrdquoAutomatica vol 45 no 5 pp 1354ndash1358 2009

[7] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoIEEETransactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[8] F Amato M Ariola and C Cosentino ldquoFinite-time stabiliza-tion via dynamic output feedbackrdquo Automatica vol 42 no 2pp 337ndash342 2006

[9] G Garcia S Tarbouriech and J Bernussou ldquoFinite-timestabilization of linear time-varying continuous systemsrdquo IEEETransactions on Automatic Control vol 54 no 2 pp 364ndash3692009

[10] Y Shen ldquoFinite-time control of linear parameter-varying sys-tems with norm-bounded exogenous disturbancerdquo Journal ofControlTheory and Applications vol 6 no 2 pp 184ndash188 2008

[11] L Liu and J T Sun ldquoFinite-time stabilization of linear systemsvia impulsive controlrdquo International Journal of Control vol 81no 6 pp 905ndash909 2008

[12] F Amato M Ariola and C Cosentino ldquoFinite-time control oflinear time-varying systems via output feedbackrdquo in Proceedingsof the American Control Conference (ACC rsquo05) pp 4722ndash4726Portland OR USA June 2005

[13] C-J Wang ldquoControllability and observability of linear time-varying singular systemsrdquo IEEE Transactions on AutomaticControl vol 44 no 10 pp 1901ndash1905 1999

[14] C-J Wang and H-E Liao ldquoImpulse observability and impulsecontrollability of linear time-varying singular systemsrdquo Auto-matica vol 37 no 11 pp 1867ndash1872 2001

[15] N A Kablar ldquoFinite-time stability of time-varying linearsingular systemsrdquo in Proceedings of the 37th IEEE Conference onDecision and Control vol 4 pp 3831ndash3836 Tampa Fla USADecember 1998

[16] NAKablar andD LDebeljkovic ldquoFinite-time stability robust-ness of time-varying linear singular systemsrdquo in Proceedings ofthe 3rd Asian Control Conference Shanghai China July 2000

[17] N A Kablar and D L Debeljkovic ldquoFinite-time instabilityof time-varying linear singular systemsrdquo in Proceedings of theAmerican Control Conference vol 3 pp 1796ndash1800 San DiegoCalif USA June 1999

[18] G Zhang Y Xia and P Shi ldquoNew bounded real lemma fordiscrete-time singular systemsrdquo Automatica vol 44 no 3 pp886ndash890 2008

[19] M Chadli H R Karimi and P Shi ldquoOn stability and stabi-lization of singular uncertain Takagi-Sugeno fuzzy systemsrdquoJournal of the Franklin Institute vol 351 no 3 pp 1453ndash14632014

[20] M Chadli A Abdo and S X Ding ldquoH minus Hinfin fault detectionfilter design for discrete-time Takagi-Sugeno fuzzy systemrdquoAutomatica vol 49 no 7 pp 1996ndash2005 2013

[21] J Yao Z-H Guan G R Chen and D W C Ho ldquoStabilityrobust stabilization and 119867

infincontrol of singular-impulsive

systems via switching controlrdquo Systems amp Control Letters vol55 no 11 pp 879ndash886 2006

[22] J Raouf and E K Boukas ldquoStabilisation of singular Markovianjump systems with discontinuities and saturating inputsrdquo IETControl Theory amp Applications vol 3 no 7 pp 971ndash982 2009

[23] Z-H Guan C W Chan A Y T Leung and G ChenldquoRobust stabilization of singular-impulsive-delayed systemswith nonlinear perturbationsrdquo IEEE Transactions on Circuitsand Systems vol 48 no 8 pp 1011ndash1019 2001

[24] S W Zhao J T Sun and L Liu ldquoFinite-time stability oflinear time-varying singular systems with impulsive effectsrdquoInternational Journal of Control vol 81 no 11 pp 1824ndash18292008

[25] J Xu and J Sun ldquoFinite-time stability of linear time-varyingsingular impulsive systemsrdquo IET ControlTheory amp Applicationsvol 4 no 10 pp 2239ndash2244 2010

[26] L G Wu and D W C Ho ldquoSliding mode control of singularstochastic hybrid systemsrdquo Automatica vol 46 no 4 pp 779ndash783 2010

[27] E Uezato and M Ikeda ldquoStrict LMI conditions for stabilityrobust stabilization and 119867

infincontrol of descriptor systemsrdquo in

Proceedings of the 38th IEEE Conference on Decision amp Control(CDC rsquo99) vol 4 pp 4092ndash4097 IEEE Phoenix Ariz USADecember 1999

[28] G P Lu and D W C Ho ldquoContinuous stabilization controllersfor singular bilinear systems the state feedback caserdquo Automat-ica vol 42 no 2 pp 309ndash314 2006

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

impulsive delayed systems with uncertainty from nonlinearperturbations In [24 25] sufficient conditions for FTS oflinear time-varying singular system with impulses at fixedtimes were given in terms of matrix inequalities

We tackle the problemof the FTB finite-time119867infincontrol

and the controller design Our results are different fromthose previous results In Section 2 the problem we dealwith is precisely stated and some preliminary definitions andnotations are provided In Section 3 first the result of FTBand finite-time 119867

infincontrol problem analysis is given and

then based on the result the controller design problem isconsidered In this section we also discuss the numericalalgorithms to solve the DDLMIs In Section 4 simpleexamples are presented to illustrate the applicability of ourresults Finally some concluding remarks are provided inSection 5

2 Problem Statement

We consider a time-varying descriptor system with jumps atfixed time described by

119864 (119905) = 119860 (119905) 119909 (119905) + 119861 (119905) 119906 (119905) + 119866 (119905) 120596 (119905) (119905 = 120591119896)

Δ119909 (120591119896) = 119860

119896119909 (120591minus

119896) (119905 = 120591

119896)

119910 (119905) = 119862 (119905) 119909 (119905) + 119863 (119905) 119906 (119905) + 119865 (119905) 120596 (119905)

(1)

where 119909(119905) isin 119877119899 is the state vector 119906(119905) isin 119877

119897 is the input120596(119905) isin 119877

119898 is the disturbance input 119910(119905) isin 119877119902 is the output

119864119909(1199050) = 119909

0is the initial value of the system state 119860(119905) isin

119877119899times119899 119861(119905) isin 119877

119899times119897 119866(119905) isin 119877119899times119898 119862(119905) isin 119877

119902times119899 119863(119905) isin 119877119902times119897

119865(119905) isin 119877119902times119898 and 119864 isin 119877

119899times119899 are continuous matrix functionswhile 119864 is singular

119860 (119905) = 119860 (119905) + Δ119860 (119905)

119861 (119905) = 119861 (119905) + Δ119861 (119905)

(2)

whereΔ119860(119905)Δ119861(119905) are unknownmatrices representing time-varying parameter uncertainties that can be described as

[Δ119860 (119905) Δ119861 (119905)] = 119872Δ (119905) [119873119886119873119887] (3)

where119872119873119886 119873119887are known real constant matrix and 119865(119905) isin

119877119902times119904 is unknown matrix function with Lebesgue-measurable

elements and satisfies Δ119879(119905)Δ(119905) le 119868 when Δ119860(119905) and Δ119861(119905)are zero matrix system (1) is called norminal system

119860119896isin 119877119899times119899 119896 = 1 2 119873 is time-invariant matrix which

reflects the discontinuity of the state trajectory of (1) minus1 lt

120582max119860119896 lt 0 System (1) exhibits a finite jump from 119909(120591119896) to

119909(120591+

119896) at fixed time sequence 120591

119896 (1199050lt 1205911lt sdot sdot sdot lt 120591

119898le 119879 lt

120591119898+1

lt sdot sdot sdot ) and Δ119909(120591119896) = 119909(120591

+

119896) minus 119909(120591

minus

119896) we assume that the

evolution of the state vector 119909(119905) is left continuous at each 120591119896

namely

119909 (120591119896) = 119909 (120591

minus

119896) = limℎrarr0

+

119909 (120591119896minus ℎ)

119909 (120591+

119896) = limℎrarr0

+

119909 (120591119896+ ℎ)

(4)

Moreover 120596(119905) isin 1198712[0 119879] is the disturbance input and sat-

isfies

int

119879

0

120596119879

(119905) 120596 (119905) 119889119905 le 119889 119889 ge 0 (5)

In this paper we investigate the behavior of system (1)within a finite time interval [119905

0 119879] Firstly we propose the

following definitions and lemmas

Definition 1 (regular and impulse-free) (1) The time-varyingdescriptor system 119864(119905) = 119860(119905)119909(119905) + 119866(119905)120596(119905) is said to beuniformly regular in time interval [119905

0 119879] if for any 119905 isin [119905

0 119879]

there exists a scalar 119904 such that det(119904119864 minus 119860(119905)) = 0(2) The time-varying descriptor system 119864(119905) =

119860(119905)119909(119905) + 119866(119905)120596(119905) is said to be impulse-free in time interval[1199050 119879] if for any 119905 isin [119905

0 119879] there exists a scalar 119904 such that

deg(det(119904119864 minus 119860(119905))) = rank(119864(119905))

From [26] we know that the continuity of matrix func-tions119860(119905)119861(119905) and119866(119905) and the uniform regularity of system(1) assure the existence and uniqueness of the solution to(1199050 119879] So in this paper we suppose (1) is uniformly regular

Definition 2 ([8] (finite-time boundedness (FTB) for timendashvarying descriptor system with jumps)) Given a positivescalar 119879 and a positive definite matrix-valued function 119877(sdot)system (1) is said to be FTB with respect to (119879 119877(sdot) 119889) if

119909119879

0119877 (1199050) 1199090le 1 997904rArr 119909

119879

(119905) 119877 (119905) 119909 (119905) le 1 119905 isin [1199050 1199050+ 119879]

(6)

Lemma 3 Given matrices Ω Γ and Ξ with appropriate di-mensions and with Ω symmetrical then

Ω + ΓΔ (119905) Ξ + Ξ119879

Δ119879

(119905) Γ119879

lt 0 (7)

for any Δ(119905) satisfying Δ119879(119905)Δ(119905) le 119868 if and only if there existsa scalar 120576 gt 0 such that

Ω + 120576ΓΓ119879

+ 120576minus1

Ξ119879

Ξ lt 0 (8)

The problem of finite-time 119867infin

to be addressed in thepaper can be formulated as finding a state feedback controller119906 = 119870(119905)119909 such that the following conclusions are held forthe close-loop system below

119864 (119905) = 119860119888(119905) 119909 (119905) + 119866 (119905) 120596 (119905) (119905 = 120591

119896)

Δ119909 (120591119896) = 119860

119896119909 (120591minus

119896) (119905 = 120591

119896)

119910 (119905) = 119862119888(119905) 119909 (119905) + 119865 (119905) 120596 (119905)

(9)

(1) The closed-loop system (9) is FTB with respect to (119879119877(sdot) 119889)

(2) The controlled output 119910(119905) satisfies int1198791199050

(119910119879

(119905)119910(119905) minus

1205742

120596119879

(119905)120596(119905))119889119905 lt 119872(1199090 1199050) for any nonzero 120596(119905)

where 120574 gt 0 is a prescribed scalar

A system is called finite-time 119867infin

if the two conditionsabove are satisfied

Mathematical Problems in Engineering 3

3 Main Result

The system can be decomposed (1) into two subsystems asfollows

1(119905) = 119860

11(119905) 1199091(119905) + 119860

22(119905) 1199092(119905) + 119861

1(119905) 119906 (119905)

+ 1198661(119905) 120596 (119905)

0 = 11986021(119905) 1199091(119905) + 119860

22(119905) 1199092(119905) + 119861

2(119905) 119906 (119905) + 119866

2(119905) 120596 (119905)

119910 (119905) = 1198621(119905) 1199091(119905) + 119862

2(119905) 1199092(119905) + 119863 (119905) 119906 (119905) + 119865 (119905) 120596 (119905)

(10)

where

119875119864119876 = [

119868 0

0 0

] 119875119860 (119905) 119876 = [

11986011(119905) 119860

12(119905)

11986021(119905) 119860

22(119905)

]

119875119866 (119905) = [

1198661(119905)

1198662(119905)

]

119876minus1

119909 (119905) = [1199091(119905) 1199092(119905)] 119862 (119905) 119876 = [119862

1(119905) 119862

2(119905)]

(11)

119875119876 are nonsingular matrix and it is easy to find that system(1) is impulse-free in time interval [119905

0 119879] for any inital value

1199090 if matrix function 119860

22(119905) is invertible

The following theorem gives a sufficient condition forFTB of system (1)

Theorem 4 Uncertain time-varying descriptor system withjumps (1) is said to be FTB with respect to (119879 119877(sdot) 119889) if thereexists a nonsingular and piecewise continuously differentialmatrix-valued function Γ(sdot) such that

119864119879

Γ (119905) = Γ119879

(119905) 119864 ge 119877 (119905) ge 0 (12a)

[

[

[

[

[

Ω (119905) Γ119879

(119905) 119866 (119905) Γ119879

(119905)119872

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0

119872119879

Γ (119905) 0 120576119868

]

]

]

]

]

lt 0 (12b)

119864119879

Γ (1199050) = Γ119879

(1199050) 119864 le

1

1 + 119889

119877 (1199050) (12c)

are fulfilled over [1199050 119879] where

Ω (119905) = 119860

119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) + 119864119879

Γ (119905) + 120576119873119879

119886119873119886

(13)

Proof By Schur complement (12b) is equivalent to

[

[

Ω0(119905) Γ

119879

(119905) 119866 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868

]

]

lt 0 (14)

where Ω0(119905) = 119860

119879

(119905)Γ(119905) + Γ119879

(119905)119860(119905) + 119864119879

Γ(119905) + 120576119873119879

119886119873119886+

(1120576)Γ119879

(119905)119872119872119879

Γ(119905)

Decompose (14) as

[

[

[

119860

119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) + 119864119879

Γ (119905) Γ119879

(119905) 119866 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868

]

]

]

+ 120576[

119873119879

1198861198731198860

0 0

] +

1

120576

[

Γ119879

(119905)119872119872119879

Γ (119905) 0

0 0

]

(15)

From Lemma 3 it is easy to derive

[

[

[

119860

119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) + 119864119879

Γ (119905) Γ119879

(119905) 119866 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868

]

]

]

+ [

Γ119879

(119905)119872 0

0 0

][

Δ (119905) 0

0 0

] [

1198731198860

0 0

]

+ ([

Γ119879

(119905)119872 0

0 0

][

Δ (119905) 0

0 0

] [

1198731198860

0 0

])

119879

(16)

so (14) is equivalent to the following LMI

[

[

Ω1(119905) Γ

119879

(119905) 119866 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868

]

]

lt 0 (17)

where Ω1(119905) = 119860

119879

(119905)Γ(119905) + Γ119879

(119905)119860(119905) + 119864119879

Γ(119905) From (12a)(12b) (12c) and

119876minus119879

Γ (119905) 119875 = [

Γ1(119905) Γ2(119905)

Γ3(119905) Γ4(119905)

] (18)

we have Γ2(119905) = 0 and Γ

1(119905) is symmetric from (17) we have

Ω1(119905) lt 0

Ω1(119905) = 119876

119879

119860119879

(119905) 119875119879

119875minus119879

Γ (119905) 119876

+ 119876119879

Γ119879

(119905) 119875minus1

119875119860 (119905) 119876 + 119876119879

119864119879

119875119879

119875minus119879

Γ (119905) 119876

= [

119860119879

11(119905) 119860

119879

21(119905)

119860119879

12(119905) 119860

119879

22(119905)

] [

Γ1(119905) Γ2(119905)

Γ3(119905) Γ4(119905)

]

+ [

Γ119879

1(119905) Γ119879

3(119905)

Γ119879

2(119905) Γ119879

4(119905)

] [

11986011(119905) 119860

12(119905)

11986021(119905) 119860

22(119905)

]

+ [

119868 0

0 0

][

Γ1(119905) Γ2(119905)

Γ3(119905) Γ4(119905)

]

= [

⋆ ⋆

⋆ 119860119879

22(119905) Γ4(119905) + Γ

119879

4(119905) 11986022(119905)

] lt 0

(19)

Obviously 11986011987922(119905)Γ4(119905) + Γ

119879

4(119905)11986022(119905) lt 0 So 119860

22(119905) is invert-

ible and system is impulse-free in time interval [1199050 119879] for any

initial value

4 Mathematical Problems in Engineering

Consider the Lyapunov function

119881 (119905 119909 (119905)) = 119909119879

(119905) 119864119879

Γ (119905) 119909 (119905) (20)

Then differentiating 119881(119905 119909(119905)) with respect to time 119905 on thetime interval 119905 isin (119905

0 1205911) we obtain

(119905 119909 (119905)) = 119909119879

(119905) (119860119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) + 119864119879

Γ (119905)) 119909 (119905)

+ 120596119879

(119905) 119866119879

(119905) Γ (119905) 119909 (119905)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905)

= 119909119879

(119905) Ω1(119905) 119909 (119905) + 120596

119879

(119905) 119866119879

(119905) Γ (119905) 119909 (119905)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905)

(21)

Construct a new vector as follows

119911 (119905) = [

119909 (119905)

120596 (119905)

] (22)

By (17) it is easy to see that

119911119879

(119905)[

[

Ω1(119905) Γ

119879

(119905) 119866 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868

]

]

119911 (119905)

= 119909119879

(119905) Ω1(119905) 119909 (119905) + 120596

119879

(119905) 119866119879

(119905) Γ (119905) 119909 (119905)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905) minus

1

1 + 119889

120596119879

120596 (119905)

= (119905 119909 (119905)) minus

1

1 + 119889

120596119879

(119905) 120596 (119905) lt 0

(23)

It follows that

(119905 119909 (119905)) lt

1

1 + 119889

120596119879

(119905) 120596 (119905) (24)

Integrating both sides of (24) from 1199050to 119905 in which 119905 isin (119905

0 1205911)

and noting that 119909119879(1199050)119877(1199050)119909(1199050) lt 1 we obtain

119881 (119905 119909 (119905)) lt 119881 (1199050 119909 (1199050)) +

1

1 + 119889

int

119905

1199050

120596119879

(120591) 120596 (120591) 119889120591

lt

1

1 + 119889

119909119879

0119877 (1199050) 1199090+

1

1 + 119889

int

119905

1199050

120596119879

(120591) 120596 (120591) 119889120591

lt

1

1 + 119889

+

1

1 + 119889

int

119905

1199050

120596119879

(120591) 120596 (120591) 119889120591 119905 isin (1199050 1205911)

(25)

When 119905 = 1205911 since 119909(120591minus

1) = 119909(120591

1) and 120591minus

1isin (1199050 1205911) by (25) we

also have

119881 (1205911 119909 (1205911)) lt

1

1 + 119889

+

1

1 + 119889

int

1205911

1199050

120596119879

(120591) 120596 (120591) 119889120591 (26)

Now we consider the situation 119905 isin (1205911 1205912) Integrating both

sides of (24) from 120591+

1to 119905 it is obvious to have

119881 (119905 119909 (119905)) lt

1

1 + 119889

int

119905

1205911

120596119879

(120591) 120596 (120591) 119889120591 + 119881 (120591+

1 119909 (120591+

1))

119905 isin (1205911 1205912)

(27)

and according to minus1 lt 1198601198962lt 0

119881 (120591+

1 119909 (120591+

1)) = 119909

119879

(120591+

1) 119864119879

Γ (120591+

1) 119909 (120591+

1)

= [(119868 + 1198601) 119909 (1205911)]119879

119864119879

Γ (120591+

1)

sdot [(119868 + 1198601) 119909 (1205911)]

lt 119909119879

(1205911) 119864119879

Γ (1205911) 119909 (1205911)

le

1

1 + 119889

+

1

1 + 119889

int

1205911

1199050

120596119879

(120591) 120596 (120591) 119889120591

(28)

so when 119905 isin (1205911 1205912] we have

119881 (119905 119909 (119905)) lt

1

1 + 119889

+

1

1 + 119889

int

119905

1199050

120596119879

(120591) 120596 (120591) 119889120591 (29)

By the same progress it is easy to have

119909119879

(119905) 119877 (119905) 119909 (119905) le 119881 (119905 119909 (119905)) le 1 (30)

For norminal system it is easy to obtain the followingresult

Corollary 5 Norminal time-varying descriptor system withjumps is said to be FTBwith respect to (119879 119877(sdot) 119889) if there existsa nonsingular and piecewise continuously differential matrix-valued function Γ(sdot) such that (12a) (12b) (12c) (17) and (24)are fulfilled over [119905

0 119879]

Theorem 6 Given a scalar 120574 gt 0 uncertain time-varyingdescriptor system with jumps (1) is said to be FTB with respectto (119879 119877(sdot) 119889) and satisfies int119879

1199050

(119910119879

(119905)119910(119905) minus 1205742

120596119879

(119905)120596(119905))119889119905 lt

0 There exists a nonsingular and piecewise continuouslydifferential matrix-valued function Γ(sdot) such that

119864119879

Γ (119905) = Γ119879

(119905) 119864 ge 119877 (119905) ge 0 (31a)

[

[

[

[

[

[

[

[

Ξ (119905) Γ119879

(119905) 119866 (119905) 119862119879

(119905) 119865 (119905) Γ119879

(119905)119872

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0 0

119865119879

(119905) 119862 (119905) 0 119865119879

(119905) 119865 (119905) minus 1205742

119868 0

119872119879

Γ (119905) 0 0 120576119868

]

]

]

]

]

]

]

]

lt 0

(31b)

119864119879

Γ (1199050) = Γ119879

(1199050) 119864 le

1

1 + 119889

119877 (1199050) (31c)

Mathematical Problems in Engineering 5

are fulfilled over [1199050 119879] where

Ξ (119905) = 119864119879

Γ (119905) + 119860

119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) + 119862119879

(119905) 119862 (119905)

+ 120576119873119879

119886119873119886

(32)

Proof By the proof of Theorem 4 (31b) is equivalent to thefollowing LMI

[

[

[

[

[

Ξ0(119905) Γ

119879

(119905) 119866 (119905) 119862119879

(119905) 119865 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0

119865119879

(119905) 119862 (119905) 0 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

lt 0 (33)

where Ξ0(119905) = 119860

119879

(119905)Γ(119905) + Γ119879

(119905)119860(119905) + 119864119879

Γ(119905) + 120576119873119879

119886119873119886+

(1120576)Γ119879

(119905)119872119872119879

Γ(119905) + 119862119879

(119905)119862(119905)By Lemma 3 it is easy to have

Θ (119905)

=

[

[

[

[

[

Ξ1(119905) + 119862

119879

(119905) 119862 (119905) Γ119879

(119905) 119866 (119905) 119862119879

(119905) 119865 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0

119865119879

(119905) 119862 (119905) 0 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

lt 0

(34)

where

Ξ1(119905) = 119864

119879

Γ (119905) + 119860119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) lt 0 (35)

Since 119862119879(119905)119862(119905) gt 0 we have

[

[

Ξ1(119905) Γ

119879

(119905) 119866 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868

]

]

lt 0 (36)

By (31a) (31b) and (36) and the proof of Theorem 4 system(1) is FTB

Now we will show that the following performance func-tion is bounded

119869 = int

119879

1199050

(119910119879

(119905) 119910 (119905) minus 1205742

120596119879

(119905) 120596 (119905)) lt 119881 (1199090 1199050) (37)

Construct a new vector as follows

119911 (119905) =[

[

[

119909 (119905)

120596 (119905)

120596 (119905)

]

]

]

(38)

By (34) we have

119911119879

(119905) Θ (119905) 119911 (119905)

= 119909119879

(119905) Ξ1(119905) 119909 (119905) + 120596

119879

(119905) 119866119879

(119905) Γ (119905) 119909 (t)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905) + 119909119879

(119905) 119862119879

(119905) 119862 (119905) 119909 (119905)

+ 120596119879

(119905) 119865119879

(119905) 119862 (119905) 119909 (119905) + 119909119879

(119905) 119862119879

(119905) 119865 (119905) 120596 (119905)

+ 120596119879

(119905) 119865119879

(119905) 119865 (119905) 120596 (119905) minus (

1

1 + 119889

+ 1205742

)120596119879

(119905) 120596 (119905)

= 119909119879

(119905) Ξ1(119905) 119909 (119905) + 120596

119879

(119905) 119866119879

(119905) Γ (119905) 119909 (119905)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905) minus

1

1 + 119889

120596119879

(119905) 120596 (119905)

+ 119910119879

(119905) 119910 (119905) minus 1205742

120596119879

(119905) 120596 (119905)

= (119905) minus

1

1 + 119889

120596119879

(119905) 120596 (119905) + 119910119879

(119905) 119910 (119905)

minus 1205742

120596119879

(119905) 120596 (119905) lt 0

(39)

Therefore for 119905 isin (120591119896 120591119896+1

) 119896 = 0 1 2 119898 we have

119869120591119896+1

= int

120591119896+1

120591119896

119911119879

(119905) Θ (119905) 119911 (119905) 119889119905

minus (int

120591119896+1

120591119896

(119905) minus

1

1 + 119889

120596119879

(119905) 120596 (119905)) 119889119905

(40)

with minus119881(119879) minus int

119879

1199050

(1(1 + 119889))120596119879

(119905)120596(119905) lt 0 and for Θ(119905) lt 0

we have

119869 =

119898+1

sum

119894=0

119869120591119894+1

= int

119879

1199050

119911119879

(119905) Θ (119905) 119911 (119905) 119889119905 + 119881 (1199050) minus 119881 (119879)

minus int

119879

1199050

1

1 + 119889

120596119879

(119905) 120596 (119905) 119889119905 lt 119881 (1199050)

(1205910= 1199050 120591119898+1

= 119879)

(41)

This completes the proof of the theorem

For norminal system it is easy to obtain the followingresult

Corollary 7 Given a scalar 120574 gt 0 norminal time-varyingdescriptor system with jumps (1) is said to be FTB with respectto (119879 119877(sdot) 119889) and satisfies int119879

1199050

(119910119879

(119905)119910(119905) minus 1205742

120596119879

(119905)120596(119905))119889119905 lt

0 There exists a nonsingular and piecewise continuouslydifferential matrix-valued function Γ(sdot) such that (34) (42a)(42b) (42c) and (45) are fulfilled over [119905

0 119879]

6 Mathematical Problems in Engineering

Theorem 8 Given a scalar 120574 gt 0 and a state feedback con-troller 119906 = 119870(119905)119909 the close-loop system (9) is said to be FTBwith respect to (119879 119877(sdot) 119889) and satisfies 119869 = int

119879

1199050

(119910119879

(119905)119910(119905) minus

1205742

120596119879

(119905)120596(119905))119889119905 lt 119881(1199090 1199050) If there exists a nonsingular and

piecewise continuously differential matrix-valued function Γ(sdot)and a continuously differential matrix-valued function 119871(sdot)

such that

Γ

119879

(119905) 119864119879

= 119864Γ (119905) ge 119877 (119905) ge 0 (42a)

[

[

[

[

[

[

[

[

[

[

[

Π (119905) 119866 (119905) 0 Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905) Γ

119879

(119905)119873119879

119886+ 119871119879

(119905)119873119879

119887

119866119879

(119905) minus

1

1 + 119889

119868 0 0 0

0 0 minus1205742

119868 119865119879

(119905) 0

119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905) 0 119865 (119905) minus119868 0

119873119886Γ (119905) + 119873

119887119871 (119905) 0 0 0 minus120576119868

]

]

]

]

]

]

]

]

]

]

]

lt 0 (42b)

Γ

119879

(1199050) 119864119879

= 119864Γ (1199050) le

1

1 + 119889

119877 (1199050) (42c)

are fulfilled over [1199050 119879] the state feedback gain is obtained with

119870(119905) = 119871(119905)Γ

minus1

(119905) where

Π (119905) = minus 119864Γ (119905) + Γ

119879

(119905) 119860

119879

(119905) + 119860 (119905) Γ (119905) + 119871119879

(119905) 119861

119879

(119905)

+ 119861 (119905) 119871 (119905) + 120576119872119872119879

(43)

Proof By Schur complement and (42b) we have

[

[

[

[

[

[

[

[

Π (119905) 119866 (119905) 0 Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)

119866119879

(119905) minus

1

1 + 119889

119868 0 0

0 0 minus1205742

119868 119865119879

(119905)

119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905) 0 119865 (119905) minus119868

]

]

]

]

]

]

]

]

+

1

120576

[

[

[

[

[

[

Γ119879

(119905)119873119879

119886+ 119871119879

(119905)119873119879

119887

0

0

0

]

]

]

]

]

]

[119873119886Γ (119905) + 119873

119887119871 (119905) 0 0 0] lt 0

(44)

Continue using Schur complement and then (44) is equiva-lent to

[

[

[

Π0(119905) ⋆ ⋆

⋆ ⋆ ⋆

⋆ ⋆ ⋆

]

]

]

+

[

[

[

[

Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)

0

119865119879

(119905)

]

]

]

]

[119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905) 0 119865 (119905)]

=

[

[

[

[

Π0(119905) ⋆ (Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)) 119865 (119905)

⋆ ⋆ ⋆

119865119879

(119905) (119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905)) ⋆ 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

lt 0

(45)

Mathematical Problems in Engineering 7

where Π0(119905) = Π(119905) + (1120576)(Γ

119879

(119905)119873119879

119886+ 119871119879

(119905)119873119879

119887)(119873119886Γ(119905) +

119873119887119871(119905))

We rewrite (45) as

[

[

[

[

[

[

[

Π1(119905) 119866 (119905) (Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)) 119865 (119905)

119866119879

(119905) minus

1

1 + 119889

119868 0

119865119879

(119905) (119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905)) 0 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

]

]

lt 0 (46)

since119870(119905) = 119871(119905)Γ

minus1

(119905) and

Π1(119905) = minus119864

Γ (119905) + Γ

119879

(119905) (119860

119879

(119905) + 119870119879

(119905) 119861

119879

(119905))

+ (119860 (119905) + 119861 (119905)119870 (119905)) Γ (119905) + 120576119872119872119879

+

1

120576

Γ

119879

(119905) (119873119886+ 119873119887119870 (119905))

119879

(119873119886+ 119873119887119870 (119905)) Γ (119905)

+ Γ

119879

(119905) (119862119879

(119905) + 119870119879

(119905) 119863119879

(119905))

sdot (119862 (119905) + 119863 (119905)119870 (119905)) Γ (119905)

(47)

By Lemma 3 we have

Π1(119905) = minus119864

Γ (119905) + Γ

119879

(119905) 119862119879

119888(119905) 119862119888(119905) Γ (119905)

+ Γ

119879

(119905) (119860

119879

(119905) + 119870119879

(119905) 119861

119879

(119905))

+ (119860 (119905) + 119861 (119905)119870 (119905)) Γ (119905)

+ Γ

119879

(119905) (119872Δ (119905)119873119886+119872Δ (119905)119873

119887119870 (119905))

119879

+ (119872Δ (119905)119873119886+119872Δ (119905)119873

119887119870 (119905)) Γ

= minus119864Γ (119905) + Γ

119879

(119905) 119862119879

119888(119905) 119862119888(119905) Γ (119905) + Γ

119879

(119905) 119860119879

119888(119905)

+ 119860119888(119905) Γ (119905)

(48)

Taking Γ(119905) = Γ

minus1

(119905) since Γ119879(119905)119864 = 119864119879

Γ(119905) and 119868 =

Γ(119905)Γ(119905) 0 = Γ(119905)Γ(119905) + Γ(119905)Γ(119905) it is easy to have

minusΓ

minus119879

(119905) 119864Γ (119905) Γ

minus1

(119905) = minus119864119879

Γ (119905)Γ (119905) Γ (119905) = 119864

119879

Γ (119905)

(49)

Pre- and postmultiplying (46) by diag(Γminus119879(119905) 119868 119868) anddiag(Γminus1(119905) 119868 119868) we have

[

[

[

[

[

Π2(119905) Γ

119879

(119905) 119866 (119905) 119862119879

119888(119905) 119865 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0

119865119879

(119905) 119862119888(119905) 0 119865

119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

lt 0 (50)

where Π2(119905) = 119864Γ(119905) + 119862

119879

119888(119905)119862119888(119905) + Γ

119879

(119905)119860119888(119905) + 119860

119879

119888(119905)Γ(119905)

By Theorem 6 close-loop system is FTB and satisfies 119869 lt119881(1199090 1199050)

For norminal system it is easy to obtain the followingresult

Corollary 9 Given a scalar 120574 gt 0 and a state feedback con-troller 119906 = 119870(119905)119909 the norminal close-loop system is said to beFTBwith respect to (119879 119877(sdot) 119889) and satisfies 119869 = int

119879

1199050

(119910119879

(119905)119910(119905)minus

1205742

120596119879

(119905)120596(119905))119889119905 lt 119881(1199090 1199050) If there exists a nonsingular and

piecewise continuously differential matrix-valued function Γ(sdot)and a continuously differential matrix-valued function 119871(sdot)

such that (50) (42a) and (42c) are fulfilled over [1199050 119879] the

state feedback gain is obtained with 119870(119905) = 119871(119905)Γ

minus1

(119905)

All the conditions in this paper are expressed in terms oftime-varyingDDLMIs However with an appropriate choiceof the structure of the unknown matrix function 119875(119905) it canbe turned into a ldquostandardrdquo LMI problem The unknownmatrix function119875(119905) has been assumed to be piecewise affinethat is

119875 (0) (or (119875 (119905))) = Π0

1

119875 (119905) (or (119875 (119905))) = Π0

119896+ Π119904

119896(119905 minus (119896 minus 1) 119879

119904)

(119905) (or (119875 (119905))) = Π119904

119896

119896 isin 119873 119896 lt 119896 119905 isin [(119896 minus 1) 119879119904 119896119879119904]

119875 (119905) (or (119875 (119905))) = Π0

119896+1

+ Π0

119896+1

(119905 minus 119896119879119904)

(119905) (or (119875 (119905))) = Π0

119896+1

119905 isin [119896119879119904 119879]

(51)

where 119896 = max 119896 isin 119873+ 119896 lt 119879119879119904

Hence the conditions are reduced to a set of LMIs Notethat the conditions are not strict LMI conditions due to119864119879

Γ(119905) = Γ119879

(119905)119864 ge 0 and this may cause a big troublein checking the conditions numerically In order to translatethe nonstrict LMI into strict LMI we can use the followinglemma

8 Mathematical Problems in Engineering

Lemma 10 (see [27 28]) Let 119883 isin 119877119899times119899 be symmetric such

that 119864119879119871119883119864119871gt 0 and let 119879 isin 119877

(119899minus119903)times(119899minus119903) be nonsingular Then119883119864 + 119867

119879

119879119878119879 is nonsingular and its inverse is expressed as

(119883119864 + 119867119879

119879119878119879

)

minus1

= X119864119879

+ 119878T119867 (52)

whereX is symmetric and T is a nonsingular matrix with

119864119879

119877X119864119877= (119864119879

119871119883119864119871)

minus1

T = (119878119879

119878)

minus1

119879minus1

(119867119867119879

)

minus1

(53)

where 119867 and 119878 are any matrix with full row rank and satisfy119872119864 = 0 and 119864119878 = 0 respectively 119864 is decomposed as 119864 =

119864119871119864119879

119877with 119864

119871isin 119877119899times119903 and 119864

119877isin 119877119899times119903 are of full column rank

LetΠ0119896= 1198830

119896119864+119867

119879

1198790

119896119878119879 andΠ119904

119896= 119883119904

119896119864+119867

119879

119879119904

119896119878119879 Using

Lemma 10 we can get (1198830119896119864+119867119879

1198790

119896119878119879

)minus1

= X0119896119864119879

+119878T0119896119867 and

(119883119904

119896119864+119867

119879

119879119904

119896119878119879

)minus1

= X119904119896119864119879

+119878T0119896119867 In this way the condition

119864119879

Γ(119905) = Γ119879

(119905)119864 ge 0 is satisfied so the nonstrict LMIs havebeen translated into strict LMIs Exploiting the MATLABLMI toolbox it is possible to find matrices 119883119904

119896 1198790

119896 119879119904

119896 1198830

119896(or

X0119896X119904119896 T0119896 T 119904119896) such that Γ(119905) (or Γ(119905)) and 119871(119905) verify the

conditions

4 Numerical Example

Consider the linear time-varying descriptor system withjumps defined by

119864 = [

1 0

0 0

] 119860 = [

0 1

2 (119905 + 1) 3 (119905 + 1)2]

119861 = [

1

1

] 119865 = [05 1] 119860119896= [

001 0

0 0

]

119866 = [

1 0

0 1

] 119863 = 1 119862 = [1 1]

119872 = [

01

01

] 119873119886= [02 01] 119873

119887= [01 01]

(54)

Given 120596(119905) = [sin(2120587119905 + 1) cos(2120587119905 minus 2)]119879 Ω = [1525] 119877 = diag(12 1) 120574 = 15 and 120576 = 1654 It is easy tosee

119864119877= 119864119871= [

1

0

] (55)

We choose119867119879 = 119878 = [0 1]119879 Solving the LMIs by the numer-

ical algorithm in the previous section using MATLAB LMItoolbox it is possible to find two piecewise affinematrix func-tions Γ(119905) and 119871(119905) which verify the conditions ofTheorem 8Therefore the following state feedback control law can beobtained (see Figure 1)

15 2 25

0

05

1

15

2

25

3

minus05

minus15

minus1

Figure 1 1198701(119905) (solid line) and 119870

2(119905) (dashed line)

5 Conclusions

In this paper we have formulated and studied the finite-time119867infin

control problem of uncertain time-varying descriptorsystem with jumps at fixed times A sufficient condition forFTB and a sufficient condition for finite-time 119867

infinof the

system have been presented in terms of DLMIs and LMIsThen the state feedback controller has been designed toguarantee the closed-loop systemrsquos FTB and satisfy the 119871

2

gain At last two numerical examples have been used toillustrate the main result

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by the National Nature ScienceFoundation of China (no 61074005) and the Talent Project oftheHighEducation of Liaoning province China underGrantno LR2012005

References

[1] G V Kamenkov ldquoOn stability of motion over a finite intervalof timerdquo Journal of Applied Mathematics and Mechanics vol 17pp 529ndash540 1953

[2] A A Lebedev ldquoOn stability of motion during a given intervalof timerdquo Journal of Applied Mathematics and Mechanics vol 18pp 139ndash148 1954

[3] A A Lebedev ldquoOn the problem of stability of motion overa finite interval of timerdquo Journal of Applied Mathematics andMechanics vol 18 pp 75ndash94 1954

[4] P Dorato ldquoShort time stability in linear time-varying systemsrdquoin Proceedings of the IRE International Convention Record Part4 pp 83ndash87 1961

[5] LWeiss andE F Infante ldquoFinite time stability under perturbingforces and on product spacesrdquo IEEE Transactions on AutomaticControl vol 12 no 1 pp 54ndash59 1967

Mathematical Problems in Engineering 9

[6] F Amato R Ambrosino M Ariola and C Cosentino ldquoFinite-time stability of linear time-varying systems with jumpsrdquoAutomatica vol 45 no 5 pp 1354ndash1358 2009

[7] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoIEEETransactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[8] F Amato M Ariola and C Cosentino ldquoFinite-time stabiliza-tion via dynamic output feedbackrdquo Automatica vol 42 no 2pp 337ndash342 2006

[9] G Garcia S Tarbouriech and J Bernussou ldquoFinite-timestabilization of linear time-varying continuous systemsrdquo IEEETransactions on Automatic Control vol 54 no 2 pp 364ndash3692009

[10] Y Shen ldquoFinite-time control of linear parameter-varying sys-tems with norm-bounded exogenous disturbancerdquo Journal ofControlTheory and Applications vol 6 no 2 pp 184ndash188 2008

[11] L Liu and J T Sun ldquoFinite-time stabilization of linear systemsvia impulsive controlrdquo International Journal of Control vol 81no 6 pp 905ndash909 2008

[12] F Amato M Ariola and C Cosentino ldquoFinite-time control oflinear time-varying systems via output feedbackrdquo in Proceedingsof the American Control Conference (ACC rsquo05) pp 4722ndash4726Portland OR USA June 2005

[13] C-J Wang ldquoControllability and observability of linear time-varying singular systemsrdquo IEEE Transactions on AutomaticControl vol 44 no 10 pp 1901ndash1905 1999

[14] C-J Wang and H-E Liao ldquoImpulse observability and impulsecontrollability of linear time-varying singular systemsrdquo Auto-matica vol 37 no 11 pp 1867ndash1872 2001

[15] N A Kablar ldquoFinite-time stability of time-varying linearsingular systemsrdquo in Proceedings of the 37th IEEE Conference onDecision and Control vol 4 pp 3831ndash3836 Tampa Fla USADecember 1998

[16] NAKablar andD LDebeljkovic ldquoFinite-time stability robust-ness of time-varying linear singular systemsrdquo in Proceedings ofthe 3rd Asian Control Conference Shanghai China July 2000

[17] N A Kablar and D L Debeljkovic ldquoFinite-time instabilityof time-varying linear singular systemsrdquo in Proceedings of theAmerican Control Conference vol 3 pp 1796ndash1800 San DiegoCalif USA June 1999

[18] G Zhang Y Xia and P Shi ldquoNew bounded real lemma fordiscrete-time singular systemsrdquo Automatica vol 44 no 3 pp886ndash890 2008

[19] M Chadli H R Karimi and P Shi ldquoOn stability and stabi-lization of singular uncertain Takagi-Sugeno fuzzy systemsrdquoJournal of the Franklin Institute vol 351 no 3 pp 1453ndash14632014

[20] M Chadli A Abdo and S X Ding ldquoH minus Hinfin fault detectionfilter design for discrete-time Takagi-Sugeno fuzzy systemrdquoAutomatica vol 49 no 7 pp 1996ndash2005 2013

[21] J Yao Z-H Guan G R Chen and D W C Ho ldquoStabilityrobust stabilization and 119867

infincontrol of singular-impulsive

systems via switching controlrdquo Systems amp Control Letters vol55 no 11 pp 879ndash886 2006

[22] J Raouf and E K Boukas ldquoStabilisation of singular Markovianjump systems with discontinuities and saturating inputsrdquo IETControl Theory amp Applications vol 3 no 7 pp 971ndash982 2009

[23] Z-H Guan C W Chan A Y T Leung and G ChenldquoRobust stabilization of singular-impulsive-delayed systemswith nonlinear perturbationsrdquo IEEE Transactions on Circuitsand Systems vol 48 no 8 pp 1011ndash1019 2001

[24] S W Zhao J T Sun and L Liu ldquoFinite-time stability oflinear time-varying singular systems with impulsive effectsrdquoInternational Journal of Control vol 81 no 11 pp 1824ndash18292008

[25] J Xu and J Sun ldquoFinite-time stability of linear time-varyingsingular impulsive systemsrdquo IET ControlTheory amp Applicationsvol 4 no 10 pp 2239ndash2244 2010

[26] L G Wu and D W C Ho ldquoSliding mode control of singularstochastic hybrid systemsrdquo Automatica vol 46 no 4 pp 779ndash783 2010

[27] E Uezato and M Ikeda ldquoStrict LMI conditions for stabilityrobust stabilization and 119867

infincontrol of descriptor systemsrdquo in

Proceedings of the 38th IEEE Conference on Decision amp Control(CDC rsquo99) vol 4 pp 4092ndash4097 IEEE Phoenix Ariz USADecember 1999

[28] G P Lu and D W C Ho ldquoContinuous stabilization controllersfor singular bilinear systems the state feedback caserdquo Automat-ica vol 42 no 2 pp 309ndash314 2006

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

3 Main Result

The system can be decomposed (1) into two subsystems asfollows

1(119905) = 119860

11(119905) 1199091(119905) + 119860

22(119905) 1199092(119905) + 119861

1(119905) 119906 (119905)

+ 1198661(119905) 120596 (119905)

0 = 11986021(119905) 1199091(119905) + 119860

22(119905) 1199092(119905) + 119861

2(119905) 119906 (119905) + 119866

2(119905) 120596 (119905)

119910 (119905) = 1198621(119905) 1199091(119905) + 119862

2(119905) 1199092(119905) + 119863 (119905) 119906 (119905) + 119865 (119905) 120596 (119905)

(10)

where

119875119864119876 = [

119868 0

0 0

] 119875119860 (119905) 119876 = [

11986011(119905) 119860

12(119905)

11986021(119905) 119860

22(119905)

]

119875119866 (119905) = [

1198661(119905)

1198662(119905)

]

119876minus1

119909 (119905) = [1199091(119905) 1199092(119905)] 119862 (119905) 119876 = [119862

1(119905) 119862

2(119905)]

(11)

119875119876 are nonsingular matrix and it is easy to find that system(1) is impulse-free in time interval [119905

0 119879] for any inital value

1199090 if matrix function 119860

22(119905) is invertible

The following theorem gives a sufficient condition forFTB of system (1)

Theorem 4 Uncertain time-varying descriptor system withjumps (1) is said to be FTB with respect to (119879 119877(sdot) 119889) if thereexists a nonsingular and piecewise continuously differentialmatrix-valued function Γ(sdot) such that

119864119879

Γ (119905) = Γ119879

(119905) 119864 ge 119877 (119905) ge 0 (12a)

[

[

[

[

[

Ω (119905) Γ119879

(119905) 119866 (119905) Γ119879

(119905)119872

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0

119872119879

Γ (119905) 0 120576119868

]

]

]

]

]

lt 0 (12b)

119864119879

Γ (1199050) = Γ119879

(1199050) 119864 le

1

1 + 119889

119877 (1199050) (12c)

are fulfilled over [1199050 119879] where

Ω (119905) = 119860

119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) + 119864119879

Γ (119905) + 120576119873119879

119886119873119886

(13)

Proof By Schur complement (12b) is equivalent to

[

[

Ω0(119905) Γ

119879

(119905) 119866 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868

]

]

lt 0 (14)

where Ω0(119905) = 119860

119879

(119905)Γ(119905) + Γ119879

(119905)119860(119905) + 119864119879

Γ(119905) + 120576119873119879

119886119873119886+

(1120576)Γ119879

(119905)119872119872119879

Γ(119905)

Decompose (14) as

[

[

[

119860

119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) + 119864119879

Γ (119905) Γ119879

(119905) 119866 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868

]

]

]

+ 120576[

119873119879

1198861198731198860

0 0

] +

1

120576

[

Γ119879

(119905)119872119872119879

Γ (119905) 0

0 0

]

(15)

From Lemma 3 it is easy to derive

[

[

[

119860

119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) + 119864119879

Γ (119905) Γ119879

(119905) 119866 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868

]

]

]

+ [

Γ119879

(119905)119872 0

0 0

][

Δ (119905) 0

0 0

] [

1198731198860

0 0

]

+ ([

Γ119879

(119905)119872 0

0 0

][

Δ (119905) 0

0 0

] [

1198731198860

0 0

])

119879

(16)

so (14) is equivalent to the following LMI

[

[

Ω1(119905) Γ

119879

(119905) 119866 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868

]

]

lt 0 (17)

where Ω1(119905) = 119860

119879

(119905)Γ(119905) + Γ119879

(119905)119860(119905) + 119864119879

Γ(119905) From (12a)(12b) (12c) and

119876minus119879

Γ (119905) 119875 = [

Γ1(119905) Γ2(119905)

Γ3(119905) Γ4(119905)

] (18)

we have Γ2(119905) = 0 and Γ

1(119905) is symmetric from (17) we have

Ω1(119905) lt 0

Ω1(119905) = 119876

119879

119860119879

(119905) 119875119879

119875minus119879

Γ (119905) 119876

+ 119876119879

Γ119879

(119905) 119875minus1

119875119860 (119905) 119876 + 119876119879

119864119879

119875119879

119875minus119879

Γ (119905) 119876

= [

119860119879

11(119905) 119860

119879

21(119905)

119860119879

12(119905) 119860

119879

22(119905)

] [

Γ1(119905) Γ2(119905)

Γ3(119905) Γ4(119905)

]

+ [

Γ119879

1(119905) Γ119879

3(119905)

Γ119879

2(119905) Γ119879

4(119905)

] [

11986011(119905) 119860

12(119905)

11986021(119905) 119860

22(119905)

]

+ [

119868 0

0 0

][

Γ1(119905) Γ2(119905)

Γ3(119905) Γ4(119905)

]

= [

⋆ ⋆

⋆ 119860119879

22(119905) Γ4(119905) + Γ

119879

4(119905) 11986022(119905)

] lt 0

(19)

Obviously 11986011987922(119905)Γ4(119905) + Γ

119879

4(119905)11986022(119905) lt 0 So 119860

22(119905) is invert-

ible and system is impulse-free in time interval [1199050 119879] for any

initial value

4 Mathematical Problems in Engineering

Consider the Lyapunov function

119881 (119905 119909 (119905)) = 119909119879

(119905) 119864119879

Γ (119905) 119909 (119905) (20)

Then differentiating 119881(119905 119909(119905)) with respect to time 119905 on thetime interval 119905 isin (119905

0 1205911) we obtain

(119905 119909 (119905)) = 119909119879

(119905) (119860119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) + 119864119879

Γ (119905)) 119909 (119905)

+ 120596119879

(119905) 119866119879

(119905) Γ (119905) 119909 (119905)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905)

= 119909119879

(119905) Ω1(119905) 119909 (119905) + 120596

119879

(119905) 119866119879

(119905) Γ (119905) 119909 (119905)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905)

(21)

Construct a new vector as follows

119911 (119905) = [

119909 (119905)

120596 (119905)

] (22)

By (17) it is easy to see that

119911119879

(119905)[

[

Ω1(119905) Γ

119879

(119905) 119866 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868

]

]

119911 (119905)

= 119909119879

(119905) Ω1(119905) 119909 (119905) + 120596

119879

(119905) 119866119879

(119905) Γ (119905) 119909 (119905)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905) minus

1

1 + 119889

120596119879

120596 (119905)

= (119905 119909 (119905)) minus

1

1 + 119889

120596119879

(119905) 120596 (119905) lt 0

(23)

It follows that

(119905 119909 (119905)) lt

1

1 + 119889

120596119879

(119905) 120596 (119905) (24)

Integrating both sides of (24) from 1199050to 119905 in which 119905 isin (119905

0 1205911)

and noting that 119909119879(1199050)119877(1199050)119909(1199050) lt 1 we obtain

119881 (119905 119909 (119905)) lt 119881 (1199050 119909 (1199050)) +

1

1 + 119889

int

119905

1199050

120596119879

(120591) 120596 (120591) 119889120591

lt

1

1 + 119889

119909119879

0119877 (1199050) 1199090+

1

1 + 119889

int

119905

1199050

120596119879

(120591) 120596 (120591) 119889120591

lt

1

1 + 119889

+

1

1 + 119889

int

119905

1199050

120596119879

(120591) 120596 (120591) 119889120591 119905 isin (1199050 1205911)

(25)

When 119905 = 1205911 since 119909(120591minus

1) = 119909(120591

1) and 120591minus

1isin (1199050 1205911) by (25) we

also have

119881 (1205911 119909 (1205911)) lt

1

1 + 119889

+

1

1 + 119889

int

1205911

1199050

120596119879

(120591) 120596 (120591) 119889120591 (26)

Now we consider the situation 119905 isin (1205911 1205912) Integrating both

sides of (24) from 120591+

1to 119905 it is obvious to have

119881 (119905 119909 (119905)) lt

1

1 + 119889

int

119905

1205911

120596119879

(120591) 120596 (120591) 119889120591 + 119881 (120591+

1 119909 (120591+

1))

119905 isin (1205911 1205912)

(27)

and according to minus1 lt 1198601198962lt 0

119881 (120591+

1 119909 (120591+

1)) = 119909

119879

(120591+

1) 119864119879

Γ (120591+

1) 119909 (120591+

1)

= [(119868 + 1198601) 119909 (1205911)]119879

119864119879

Γ (120591+

1)

sdot [(119868 + 1198601) 119909 (1205911)]

lt 119909119879

(1205911) 119864119879

Γ (1205911) 119909 (1205911)

le

1

1 + 119889

+

1

1 + 119889

int

1205911

1199050

120596119879

(120591) 120596 (120591) 119889120591

(28)

so when 119905 isin (1205911 1205912] we have

119881 (119905 119909 (119905)) lt

1

1 + 119889

+

1

1 + 119889

int

119905

1199050

120596119879

(120591) 120596 (120591) 119889120591 (29)

By the same progress it is easy to have

119909119879

(119905) 119877 (119905) 119909 (119905) le 119881 (119905 119909 (119905)) le 1 (30)

For norminal system it is easy to obtain the followingresult

Corollary 5 Norminal time-varying descriptor system withjumps is said to be FTBwith respect to (119879 119877(sdot) 119889) if there existsa nonsingular and piecewise continuously differential matrix-valued function Γ(sdot) such that (12a) (12b) (12c) (17) and (24)are fulfilled over [119905

0 119879]

Theorem 6 Given a scalar 120574 gt 0 uncertain time-varyingdescriptor system with jumps (1) is said to be FTB with respectto (119879 119877(sdot) 119889) and satisfies int119879

1199050

(119910119879

(119905)119910(119905) minus 1205742

120596119879

(119905)120596(119905))119889119905 lt

0 There exists a nonsingular and piecewise continuouslydifferential matrix-valued function Γ(sdot) such that

119864119879

Γ (119905) = Γ119879

(119905) 119864 ge 119877 (119905) ge 0 (31a)

[

[

[

[

[

[

[

[

Ξ (119905) Γ119879

(119905) 119866 (119905) 119862119879

(119905) 119865 (119905) Γ119879

(119905)119872

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0 0

119865119879

(119905) 119862 (119905) 0 119865119879

(119905) 119865 (119905) minus 1205742

119868 0

119872119879

Γ (119905) 0 0 120576119868

]

]

]

]

]

]

]

]

lt 0

(31b)

119864119879

Γ (1199050) = Γ119879

(1199050) 119864 le

1

1 + 119889

119877 (1199050) (31c)

Mathematical Problems in Engineering 5

are fulfilled over [1199050 119879] where

Ξ (119905) = 119864119879

Γ (119905) + 119860

119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) + 119862119879

(119905) 119862 (119905)

+ 120576119873119879

119886119873119886

(32)

Proof By the proof of Theorem 4 (31b) is equivalent to thefollowing LMI

[

[

[

[

[

Ξ0(119905) Γ

119879

(119905) 119866 (119905) 119862119879

(119905) 119865 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0

119865119879

(119905) 119862 (119905) 0 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

lt 0 (33)

where Ξ0(119905) = 119860

119879

(119905)Γ(119905) + Γ119879

(119905)119860(119905) + 119864119879

Γ(119905) + 120576119873119879

119886119873119886+

(1120576)Γ119879

(119905)119872119872119879

Γ(119905) + 119862119879

(119905)119862(119905)By Lemma 3 it is easy to have

Θ (119905)

=

[

[

[

[

[

Ξ1(119905) + 119862

119879

(119905) 119862 (119905) Γ119879

(119905) 119866 (119905) 119862119879

(119905) 119865 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0

119865119879

(119905) 119862 (119905) 0 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

lt 0

(34)

where

Ξ1(119905) = 119864

119879

Γ (119905) + 119860119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) lt 0 (35)

Since 119862119879(119905)119862(119905) gt 0 we have

[

[

Ξ1(119905) Γ

119879

(119905) 119866 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868

]

]

lt 0 (36)

By (31a) (31b) and (36) and the proof of Theorem 4 system(1) is FTB

Now we will show that the following performance func-tion is bounded

119869 = int

119879

1199050

(119910119879

(119905) 119910 (119905) minus 1205742

120596119879

(119905) 120596 (119905)) lt 119881 (1199090 1199050) (37)

Construct a new vector as follows

119911 (119905) =[

[

[

119909 (119905)

120596 (119905)

120596 (119905)

]

]

]

(38)

By (34) we have

119911119879

(119905) Θ (119905) 119911 (119905)

= 119909119879

(119905) Ξ1(119905) 119909 (119905) + 120596

119879

(119905) 119866119879

(119905) Γ (119905) 119909 (t)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905) + 119909119879

(119905) 119862119879

(119905) 119862 (119905) 119909 (119905)

+ 120596119879

(119905) 119865119879

(119905) 119862 (119905) 119909 (119905) + 119909119879

(119905) 119862119879

(119905) 119865 (119905) 120596 (119905)

+ 120596119879

(119905) 119865119879

(119905) 119865 (119905) 120596 (119905) minus (

1

1 + 119889

+ 1205742

)120596119879

(119905) 120596 (119905)

= 119909119879

(119905) Ξ1(119905) 119909 (119905) + 120596

119879

(119905) 119866119879

(119905) Γ (119905) 119909 (119905)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905) minus

1

1 + 119889

120596119879

(119905) 120596 (119905)

+ 119910119879

(119905) 119910 (119905) minus 1205742

120596119879

(119905) 120596 (119905)

= (119905) minus

1

1 + 119889

120596119879

(119905) 120596 (119905) + 119910119879

(119905) 119910 (119905)

minus 1205742

120596119879

(119905) 120596 (119905) lt 0

(39)

Therefore for 119905 isin (120591119896 120591119896+1

) 119896 = 0 1 2 119898 we have

119869120591119896+1

= int

120591119896+1

120591119896

119911119879

(119905) Θ (119905) 119911 (119905) 119889119905

minus (int

120591119896+1

120591119896

(119905) minus

1

1 + 119889

120596119879

(119905) 120596 (119905)) 119889119905

(40)

with minus119881(119879) minus int

119879

1199050

(1(1 + 119889))120596119879

(119905)120596(119905) lt 0 and for Θ(119905) lt 0

we have

119869 =

119898+1

sum

119894=0

119869120591119894+1

= int

119879

1199050

119911119879

(119905) Θ (119905) 119911 (119905) 119889119905 + 119881 (1199050) minus 119881 (119879)

minus int

119879

1199050

1

1 + 119889

120596119879

(119905) 120596 (119905) 119889119905 lt 119881 (1199050)

(1205910= 1199050 120591119898+1

= 119879)

(41)

This completes the proof of the theorem

For norminal system it is easy to obtain the followingresult

Corollary 7 Given a scalar 120574 gt 0 norminal time-varyingdescriptor system with jumps (1) is said to be FTB with respectto (119879 119877(sdot) 119889) and satisfies int119879

1199050

(119910119879

(119905)119910(119905) minus 1205742

120596119879

(119905)120596(119905))119889119905 lt

0 There exists a nonsingular and piecewise continuouslydifferential matrix-valued function Γ(sdot) such that (34) (42a)(42b) (42c) and (45) are fulfilled over [119905

0 119879]

6 Mathematical Problems in Engineering

Theorem 8 Given a scalar 120574 gt 0 and a state feedback con-troller 119906 = 119870(119905)119909 the close-loop system (9) is said to be FTBwith respect to (119879 119877(sdot) 119889) and satisfies 119869 = int

119879

1199050

(119910119879

(119905)119910(119905) minus

1205742

120596119879

(119905)120596(119905))119889119905 lt 119881(1199090 1199050) If there exists a nonsingular and

piecewise continuously differential matrix-valued function Γ(sdot)and a continuously differential matrix-valued function 119871(sdot)

such that

Γ

119879

(119905) 119864119879

= 119864Γ (119905) ge 119877 (119905) ge 0 (42a)

[

[

[

[

[

[

[

[

[

[

[

Π (119905) 119866 (119905) 0 Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905) Γ

119879

(119905)119873119879

119886+ 119871119879

(119905)119873119879

119887

119866119879

(119905) minus

1

1 + 119889

119868 0 0 0

0 0 minus1205742

119868 119865119879

(119905) 0

119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905) 0 119865 (119905) minus119868 0

119873119886Γ (119905) + 119873

119887119871 (119905) 0 0 0 minus120576119868

]

]

]

]

]

]

]

]

]

]

]

lt 0 (42b)

Γ

119879

(1199050) 119864119879

= 119864Γ (1199050) le

1

1 + 119889

119877 (1199050) (42c)

are fulfilled over [1199050 119879] the state feedback gain is obtained with

119870(119905) = 119871(119905)Γ

minus1

(119905) where

Π (119905) = minus 119864Γ (119905) + Γ

119879

(119905) 119860

119879

(119905) + 119860 (119905) Γ (119905) + 119871119879

(119905) 119861

119879

(119905)

+ 119861 (119905) 119871 (119905) + 120576119872119872119879

(43)

Proof By Schur complement and (42b) we have

[

[

[

[

[

[

[

[

Π (119905) 119866 (119905) 0 Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)

119866119879

(119905) minus

1

1 + 119889

119868 0 0

0 0 minus1205742

119868 119865119879

(119905)

119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905) 0 119865 (119905) minus119868

]

]

]

]

]

]

]

]

+

1

120576

[

[

[

[

[

[

Γ119879

(119905)119873119879

119886+ 119871119879

(119905)119873119879

119887

0

0

0

]

]

]

]

]

]

[119873119886Γ (119905) + 119873

119887119871 (119905) 0 0 0] lt 0

(44)

Continue using Schur complement and then (44) is equiva-lent to

[

[

[

Π0(119905) ⋆ ⋆

⋆ ⋆ ⋆

⋆ ⋆ ⋆

]

]

]

+

[

[

[

[

Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)

0

119865119879

(119905)

]

]

]

]

[119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905) 0 119865 (119905)]

=

[

[

[

[

Π0(119905) ⋆ (Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)) 119865 (119905)

⋆ ⋆ ⋆

119865119879

(119905) (119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905)) ⋆ 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

lt 0

(45)

Mathematical Problems in Engineering 7

where Π0(119905) = Π(119905) + (1120576)(Γ

119879

(119905)119873119879

119886+ 119871119879

(119905)119873119879

119887)(119873119886Γ(119905) +

119873119887119871(119905))

We rewrite (45) as

[

[

[

[

[

[

[

Π1(119905) 119866 (119905) (Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)) 119865 (119905)

119866119879

(119905) minus

1

1 + 119889

119868 0

119865119879

(119905) (119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905)) 0 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

]

]

lt 0 (46)

since119870(119905) = 119871(119905)Γ

minus1

(119905) and

Π1(119905) = minus119864

Γ (119905) + Γ

119879

(119905) (119860

119879

(119905) + 119870119879

(119905) 119861

119879

(119905))

+ (119860 (119905) + 119861 (119905)119870 (119905)) Γ (119905) + 120576119872119872119879

+

1

120576

Γ

119879

(119905) (119873119886+ 119873119887119870 (119905))

119879

(119873119886+ 119873119887119870 (119905)) Γ (119905)

+ Γ

119879

(119905) (119862119879

(119905) + 119870119879

(119905) 119863119879

(119905))

sdot (119862 (119905) + 119863 (119905)119870 (119905)) Γ (119905)

(47)

By Lemma 3 we have

Π1(119905) = minus119864

Γ (119905) + Γ

119879

(119905) 119862119879

119888(119905) 119862119888(119905) Γ (119905)

+ Γ

119879

(119905) (119860

119879

(119905) + 119870119879

(119905) 119861

119879

(119905))

+ (119860 (119905) + 119861 (119905)119870 (119905)) Γ (119905)

+ Γ

119879

(119905) (119872Δ (119905)119873119886+119872Δ (119905)119873

119887119870 (119905))

119879

+ (119872Δ (119905)119873119886+119872Δ (119905)119873

119887119870 (119905)) Γ

= minus119864Γ (119905) + Γ

119879

(119905) 119862119879

119888(119905) 119862119888(119905) Γ (119905) + Γ

119879

(119905) 119860119879

119888(119905)

+ 119860119888(119905) Γ (119905)

(48)

Taking Γ(119905) = Γ

minus1

(119905) since Γ119879(119905)119864 = 119864119879

Γ(119905) and 119868 =

Γ(119905)Γ(119905) 0 = Γ(119905)Γ(119905) + Γ(119905)Γ(119905) it is easy to have

minusΓ

minus119879

(119905) 119864Γ (119905) Γ

minus1

(119905) = minus119864119879

Γ (119905)Γ (119905) Γ (119905) = 119864

119879

Γ (119905)

(49)

Pre- and postmultiplying (46) by diag(Γminus119879(119905) 119868 119868) anddiag(Γminus1(119905) 119868 119868) we have

[

[

[

[

[

Π2(119905) Γ

119879

(119905) 119866 (119905) 119862119879

119888(119905) 119865 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0

119865119879

(119905) 119862119888(119905) 0 119865

119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

lt 0 (50)

where Π2(119905) = 119864Γ(119905) + 119862

119879

119888(119905)119862119888(119905) + Γ

119879

(119905)119860119888(119905) + 119860

119879

119888(119905)Γ(119905)

By Theorem 6 close-loop system is FTB and satisfies 119869 lt119881(1199090 1199050)

For norminal system it is easy to obtain the followingresult

Corollary 9 Given a scalar 120574 gt 0 and a state feedback con-troller 119906 = 119870(119905)119909 the norminal close-loop system is said to beFTBwith respect to (119879 119877(sdot) 119889) and satisfies 119869 = int

119879

1199050

(119910119879

(119905)119910(119905)minus

1205742

120596119879

(119905)120596(119905))119889119905 lt 119881(1199090 1199050) If there exists a nonsingular and

piecewise continuously differential matrix-valued function Γ(sdot)and a continuously differential matrix-valued function 119871(sdot)

such that (50) (42a) and (42c) are fulfilled over [1199050 119879] the

state feedback gain is obtained with 119870(119905) = 119871(119905)Γ

minus1

(119905)

All the conditions in this paper are expressed in terms oftime-varyingDDLMIs However with an appropriate choiceof the structure of the unknown matrix function 119875(119905) it canbe turned into a ldquostandardrdquo LMI problem The unknownmatrix function119875(119905) has been assumed to be piecewise affinethat is

119875 (0) (or (119875 (119905))) = Π0

1

119875 (119905) (or (119875 (119905))) = Π0

119896+ Π119904

119896(119905 minus (119896 minus 1) 119879

119904)

(119905) (or (119875 (119905))) = Π119904

119896

119896 isin 119873 119896 lt 119896 119905 isin [(119896 minus 1) 119879119904 119896119879119904]

119875 (119905) (or (119875 (119905))) = Π0

119896+1

+ Π0

119896+1

(119905 minus 119896119879119904)

(119905) (or (119875 (119905))) = Π0

119896+1

119905 isin [119896119879119904 119879]

(51)

where 119896 = max 119896 isin 119873+ 119896 lt 119879119879119904

Hence the conditions are reduced to a set of LMIs Notethat the conditions are not strict LMI conditions due to119864119879

Γ(119905) = Γ119879

(119905)119864 ge 0 and this may cause a big troublein checking the conditions numerically In order to translatethe nonstrict LMI into strict LMI we can use the followinglemma

8 Mathematical Problems in Engineering

Lemma 10 (see [27 28]) Let 119883 isin 119877119899times119899 be symmetric such

that 119864119879119871119883119864119871gt 0 and let 119879 isin 119877

(119899minus119903)times(119899minus119903) be nonsingular Then119883119864 + 119867

119879

119879119878119879 is nonsingular and its inverse is expressed as

(119883119864 + 119867119879

119879119878119879

)

minus1

= X119864119879

+ 119878T119867 (52)

whereX is symmetric and T is a nonsingular matrix with

119864119879

119877X119864119877= (119864119879

119871119883119864119871)

minus1

T = (119878119879

119878)

minus1

119879minus1

(119867119867119879

)

minus1

(53)

where 119867 and 119878 are any matrix with full row rank and satisfy119872119864 = 0 and 119864119878 = 0 respectively 119864 is decomposed as 119864 =

119864119871119864119879

119877with 119864

119871isin 119877119899times119903 and 119864

119877isin 119877119899times119903 are of full column rank

LetΠ0119896= 1198830

119896119864+119867

119879

1198790

119896119878119879 andΠ119904

119896= 119883119904

119896119864+119867

119879

119879119904

119896119878119879 Using

Lemma 10 we can get (1198830119896119864+119867119879

1198790

119896119878119879

)minus1

= X0119896119864119879

+119878T0119896119867 and

(119883119904

119896119864+119867

119879

119879119904

119896119878119879

)minus1

= X119904119896119864119879

+119878T0119896119867 In this way the condition

119864119879

Γ(119905) = Γ119879

(119905)119864 ge 0 is satisfied so the nonstrict LMIs havebeen translated into strict LMIs Exploiting the MATLABLMI toolbox it is possible to find matrices 119883119904

119896 1198790

119896 119879119904

119896 1198830

119896(or

X0119896X119904119896 T0119896 T 119904119896) such that Γ(119905) (or Γ(119905)) and 119871(119905) verify the

conditions

4 Numerical Example

Consider the linear time-varying descriptor system withjumps defined by

119864 = [

1 0

0 0

] 119860 = [

0 1

2 (119905 + 1) 3 (119905 + 1)2]

119861 = [

1

1

] 119865 = [05 1] 119860119896= [

001 0

0 0

]

119866 = [

1 0

0 1

] 119863 = 1 119862 = [1 1]

119872 = [

01

01

] 119873119886= [02 01] 119873

119887= [01 01]

(54)

Given 120596(119905) = [sin(2120587119905 + 1) cos(2120587119905 minus 2)]119879 Ω = [1525] 119877 = diag(12 1) 120574 = 15 and 120576 = 1654 It is easy tosee

119864119877= 119864119871= [

1

0

] (55)

We choose119867119879 = 119878 = [0 1]119879 Solving the LMIs by the numer-

ical algorithm in the previous section using MATLAB LMItoolbox it is possible to find two piecewise affinematrix func-tions Γ(119905) and 119871(119905) which verify the conditions ofTheorem 8Therefore the following state feedback control law can beobtained (see Figure 1)

15 2 25

0

05

1

15

2

25

3

minus05

minus15

minus1

Figure 1 1198701(119905) (solid line) and 119870

2(119905) (dashed line)

5 Conclusions

In this paper we have formulated and studied the finite-time119867infin

control problem of uncertain time-varying descriptorsystem with jumps at fixed times A sufficient condition forFTB and a sufficient condition for finite-time 119867

infinof the

system have been presented in terms of DLMIs and LMIsThen the state feedback controller has been designed toguarantee the closed-loop systemrsquos FTB and satisfy the 119871

2

gain At last two numerical examples have been used toillustrate the main result

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by the National Nature ScienceFoundation of China (no 61074005) and the Talent Project oftheHighEducation of Liaoning province China underGrantno LR2012005

References

[1] G V Kamenkov ldquoOn stability of motion over a finite intervalof timerdquo Journal of Applied Mathematics and Mechanics vol 17pp 529ndash540 1953

[2] A A Lebedev ldquoOn stability of motion during a given intervalof timerdquo Journal of Applied Mathematics and Mechanics vol 18pp 139ndash148 1954

[3] A A Lebedev ldquoOn the problem of stability of motion overa finite interval of timerdquo Journal of Applied Mathematics andMechanics vol 18 pp 75ndash94 1954

[4] P Dorato ldquoShort time stability in linear time-varying systemsrdquoin Proceedings of the IRE International Convention Record Part4 pp 83ndash87 1961

[5] LWeiss andE F Infante ldquoFinite time stability under perturbingforces and on product spacesrdquo IEEE Transactions on AutomaticControl vol 12 no 1 pp 54ndash59 1967

Mathematical Problems in Engineering 9

[6] F Amato R Ambrosino M Ariola and C Cosentino ldquoFinite-time stability of linear time-varying systems with jumpsrdquoAutomatica vol 45 no 5 pp 1354ndash1358 2009

[7] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoIEEETransactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[8] F Amato M Ariola and C Cosentino ldquoFinite-time stabiliza-tion via dynamic output feedbackrdquo Automatica vol 42 no 2pp 337ndash342 2006

[9] G Garcia S Tarbouriech and J Bernussou ldquoFinite-timestabilization of linear time-varying continuous systemsrdquo IEEETransactions on Automatic Control vol 54 no 2 pp 364ndash3692009

[10] Y Shen ldquoFinite-time control of linear parameter-varying sys-tems with norm-bounded exogenous disturbancerdquo Journal ofControlTheory and Applications vol 6 no 2 pp 184ndash188 2008

[11] L Liu and J T Sun ldquoFinite-time stabilization of linear systemsvia impulsive controlrdquo International Journal of Control vol 81no 6 pp 905ndash909 2008

[12] F Amato M Ariola and C Cosentino ldquoFinite-time control oflinear time-varying systems via output feedbackrdquo in Proceedingsof the American Control Conference (ACC rsquo05) pp 4722ndash4726Portland OR USA June 2005

[13] C-J Wang ldquoControllability and observability of linear time-varying singular systemsrdquo IEEE Transactions on AutomaticControl vol 44 no 10 pp 1901ndash1905 1999

[14] C-J Wang and H-E Liao ldquoImpulse observability and impulsecontrollability of linear time-varying singular systemsrdquo Auto-matica vol 37 no 11 pp 1867ndash1872 2001

[15] N A Kablar ldquoFinite-time stability of time-varying linearsingular systemsrdquo in Proceedings of the 37th IEEE Conference onDecision and Control vol 4 pp 3831ndash3836 Tampa Fla USADecember 1998

[16] NAKablar andD LDebeljkovic ldquoFinite-time stability robust-ness of time-varying linear singular systemsrdquo in Proceedings ofthe 3rd Asian Control Conference Shanghai China July 2000

[17] N A Kablar and D L Debeljkovic ldquoFinite-time instabilityof time-varying linear singular systemsrdquo in Proceedings of theAmerican Control Conference vol 3 pp 1796ndash1800 San DiegoCalif USA June 1999

[18] G Zhang Y Xia and P Shi ldquoNew bounded real lemma fordiscrete-time singular systemsrdquo Automatica vol 44 no 3 pp886ndash890 2008

[19] M Chadli H R Karimi and P Shi ldquoOn stability and stabi-lization of singular uncertain Takagi-Sugeno fuzzy systemsrdquoJournal of the Franklin Institute vol 351 no 3 pp 1453ndash14632014

[20] M Chadli A Abdo and S X Ding ldquoH minus Hinfin fault detectionfilter design for discrete-time Takagi-Sugeno fuzzy systemrdquoAutomatica vol 49 no 7 pp 1996ndash2005 2013

[21] J Yao Z-H Guan G R Chen and D W C Ho ldquoStabilityrobust stabilization and 119867

infincontrol of singular-impulsive

systems via switching controlrdquo Systems amp Control Letters vol55 no 11 pp 879ndash886 2006

[22] J Raouf and E K Boukas ldquoStabilisation of singular Markovianjump systems with discontinuities and saturating inputsrdquo IETControl Theory amp Applications vol 3 no 7 pp 971ndash982 2009

[23] Z-H Guan C W Chan A Y T Leung and G ChenldquoRobust stabilization of singular-impulsive-delayed systemswith nonlinear perturbationsrdquo IEEE Transactions on Circuitsand Systems vol 48 no 8 pp 1011ndash1019 2001

[24] S W Zhao J T Sun and L Liu ldquoFinite-time stability oflinear time-varying singular systems with impulsive effectsrdquoInternational Journal of Control vol 81 no 11 pp 1824ndash18292008

[25] J Xu and J Sun ldquoFinite-time stability of linear time-varyingsingular impulsive systemsrdquo IET ControlTheory amp Applicationsvol 4 no 10 pp 2239ndash2244 2010

[26] L G Wu and D W C Ho ldquoSliding mode control of singularstochastic hybrid systemsrdquo Automatica vol 46 no 4 pp 779ndash783 2010

[27] E Uezato and M Ikeda ldquoStrict LMI conditions for stabilityrobust stabilization and 119867

infincontrol of descriptor systemsrdquo in

Proceedings of the 38th IEEE Conference on Decision amp Control(CDC rsquo99) vol 4 pp 4092ndash4097 IEEE Phoenix Ariz USADecember 1999

[28] G P Lu and D W C Ho ldquoContinuous stabilization controllersfor singular bilinear systems the state feedback caserdquo Automat-ica vol 42 no 2 pp 309ndash314 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Consider the Lyapunov function

119881 (119905 119909 (119905)) = 119909119879

(119905) 119864119879

Γ (119905) 119909 (119905) (20)

Then differentiating 119881(119905 119909(119905)) with respect to time 119905 on thetime interval 119905 isin (119905

0 1205911) we obtain

(119905 119909 (119905)) = 119909119879

(119905) (119860119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) + 119864119879

Γ (119905)) 119909 (119905)

+ 120596119879

(119905) 119866119879

(119905) Γ (119905) 119909 (119905)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905)

= 119909119879

(119905) Ω1(119905) 119909 (119905) + 120596

119879

(119905) 119866119879

(119905) Γ (119905) 119909 (119905)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905)

(21)

Construct a new vector as follows

119911 (119905) = [

119909 (119905)

120596 (119905)

] (22)

By (17) it is easy to see that

119911119879

(119905)[

[

Ω1(119905) Γ

119879

(119905) 119866 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868

]

]

119911 (119905)

= 119909119879

(119905) Ω1(119905) 119909 (119905) + 120596

119879

(119905) 119866119879

(119905) Γ (119905) 119909 (119905)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905) minus

1

1 + 119889

120596119879

120596 (119905)

= (119905 119909 (119905)) minus

1

1 + 119889

120596119879

(119905) 120596 (119905) lt 0

(23)

It follows that

(119905 119909 (119905)) lt

1

1 + 119889

120596119879

(119905) 120596 (119905) (24)

Integrating both sides of (24) from 1199050to 119905 in which 119905 isin (119905

0 1205911)

and noting that 119909119879(1199050)119877(1199050)119909(1199050) lt 1 we obtain

119881 (119905 119909 (119905)) lt 119881 (1199050 119909 (1199050)) +

1

1 + 119889

int

119905

1199050

120596119879

(120591) 120596 (120591) 119889120591

lt

1

1 + 119889

119909119879

0119877 (1199050) 1199090+

1

1 + 119889

int

119905

1199050

120596119879

(120591) 120596 (120591) 119889120591

lt

1

1 + 119889

+

1

1 + 119889

int

119905

1199050

120596119879

(120591) 120596 (120591) 119889120591 119905 isin (1199050 1205911)

(25)

When 119905 = 1205911 since 119909(120591minus

1) = 119909(120591

1) and 120591minus

1isin (1199050 1205911) by (25) we

also have

119881 (1205911 119909 (1205911)) lt

1

1 + 119889

+

1

1 + 119889

int

1205911

1199050

120596119879

(120591) 120596 (120591) 119889120591 (26)

Now we consider the situation 119905 isin (1205911 1205912) Integrating both

sides of (24) from 120591+

1to 119905 it is obvious to have

119881 (119905 119909 (119905)) lt

1

1 + 119889

int

119905

1205911

120596119879

(120591) 120596 (120591) 119889120591 + 119881 (120591+

1 119909 (120591+

1))

119905 isin (1205911 1205912)

(27)

and according to minus1 lt 1198601198962lt 0

119881 (120591+

1 119909 (120591+

1)) = 119909

119879

(120591+

1) 119864119879

Γ (120591+

1) 119909 (120591+

1)

= [(119868 + 1198601) 119909 (1205911)]119879

119864119879

Γ (120591+

1)

sdot [(119868 + 1198601) 119909 (1205911)]

lt 119909119879

(1205911) 119864119879

Γ (1205911) 119909 (1205911)

le

1

1 + 119889

+

1

1 + 119889

int

1205911

1199050

120596119879

(120591) 120596 (120591) 119889120591

(28)

so when 119905 isin (1205911 1205912] we have

119881 (119905 119909 (119905)) lt

1

1 + 119889

+

1

1 + 119889

int

119905

1199050

120596119879

(120591) 120596 (120591) 119889120591 (29)

By the same progress it is easy to have

119909119879

(119905) 119877 (119905) 119909 (119905) le 119881 (119905 119909 (119905)) le 1 (30)

For norminal system it is easy to obtain the followingresult

Corollary 5 Norminal time-varying descriptor system withjumps is said to be FTBwith respect to (119879 119877(sdot) 119889) if there existsa nonsingular and piecewise continuously differential matrix-valued function Γ(sdot) such that (12a) (12b) (12c) (17) and (24)are fulfilled over [119905

0 119879]

Theorem 6 Given a scalar 120574 gt 0 uncertain time-varyingdescriptor system with jumps (1) is said to be FTB with respectto (119879 119877(sdot) 119889) and satisfies int119879

1199050

(119910119879

(119905)119910(119905) minus 1205742

120596119879

(119905)120596(119905))119889119905 lt

0 There exists a nonsingular and piecewise continuouslydifferential matrix-valued function Γ(sdot) such that

119864119879

Γ (119905) = Γ119879

(119905) 119864 ge 119877 (119905) ge 0 (31a)

[

[

[

[

[

[

[

[

Ξ (119905) Γ119879

(119905) 119866 (119905) 119862119879

(119905) 119865 (119905) Γ119879

(119905)119872

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0 0

119865119879

(119905) 119862 (119905) 0 119865119879

(119905) 119865 (119905) minus 1205742

119868 0

119872119879

Γ (119905) 0 0 120576119868

]

]

]

]

]

]

]

]

lt 0

(31b)

119864119879

Γ (1199050) = Γ119879

(1199050) 119864 le

1

1 + 119889

119877 (1199050) (31c)

Mathematical Problems in Engineering 5

are fulfilled over [1199050 119879] where

Ξ (119905) = 119864119879

Γ (119905) + 119860

119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) + 119862119879

(119905) 119862 (119905)

+ 120576119873119879

119886119873119886

(32)

Proof By the proof of Theorem 4 (31b) is equivalent to thefollowing LMI

[

[

[

[

[

Ξ0(119905) Γ

119879

(119905) 119866 (119905) 119862119879

(119905) 119865 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0

119865119879

(119905) 119862 (119905) 0 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

lt 0 (33)

where Ξ0(119905) = 119860

119879

(119905)Γ(119905) + Γ119879

(119905)119860(119905) + 119864119879

Γ(119905) + 120576119873119879

119886119873119886+

(1120576)Γ119879

(119905)119872119872119879

Γ(119905) + 119862119879

(119905)119862(119905)By Lemma 3 it is easy to have

Θ (119905)

=

[

[

[

[

[

Ξ1(119905) + 119862

119879

(119905) 119862 (119905) Γ119879

(119905) 119866 (119905) 119862119879

(119905) 119865 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0

119865119879

(119905) 119862 (119905) 0 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

lt 0

(34)

where

Ξ1(119905) = 119864

119879

Γ (119905) + 119860119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) lt 0 (35)

Since 119862119879(119905)119862(119905) gt 0 we have

[

[

Ξ1(119905) Γ

119879

(119905) 119866 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868

]

]

lt 0 (36)

By (31a) (31b) and (36) and the proof of Theorem 4 system(1) is FTB

Now we will show that the following performance func-tion is bounded

119869 = int

119879

1199050

(119910119879

(119905) 119910 (119905) minus 1205742

120596119879

(119905) 120596 (119905)) lt 119881 (1199090 1199050) (37)

Construct a new vector as follows

119911 (119905) =[

[

[

119909 (119905)

120596 (119905)

120596 (119905)

]

]

]

(38)

By (34) we have

119911119879

(119905) Θ (119905) 119911 (119905)

= 119909119879

(119905) Ξ1(119905) 119909 (119905) + 120596

119879

(119905) 119866119879

(119905) Γ (119905) 119909 (t)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905) + 119909119879

(119905) 119862119879

(119905) 119862 (119905) 119909 (119905)

+ 120596119879

(119905) 119865119879

(119905) 119862 (119905) 119909 (119905) + 119909119879

(119905) 119862119879

(119905) 119865 (119905) 120596 (119905)

+ 120596119879

(119905) 119865119879

(119905) 119865 (119905) 120596 (119905) minus (

1

1 + 119889

+ 1205742

)120596119879

(119905) 120596 (119905)

= 119909119879

(119905) Ξ1(119905) 119909 (119905) + 120596

119879

(119905) 119866119879

(119905) Γ (119905) 119909 (119905)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905) minus

1

1 + 119889

120596119879

(119905) 120596 (119905)

+ 119910119879

(119905) 119910 (119905) minus 1205742

120596119879

(119905) 120596 (119905)

= (119905) minus

1

1 + 119889

120596119879

(119905) 120596 (119905) + 119910119879

(119905) 119910 (119905)

minus 1205742

120596119879

(119905) 120596 (119905) lt 0

(39)

Therefore for 119905 isin (120591119896 120591119896+1

) 119896 = 0 1 2 119898 we have

119869120591119896+1

= int

120591119896+1

120591119896

119911119879

(119905) Θ (119905) 119911 (119905) 119889119905

minus (int

120591119896+1

120591119896

(119905) minus

1

1 + 119889

120596119879

(119905) 120596 (119905)) 119889119905

(40)

with minus119881(119879) minus int

119879

1199050

(1(1 + 119889))120596119879

(119905)120596(119905) lt 0 and for Θ(119905) lt 0

we have

119869 =

119898+1

sum

119894=0

119869120591119894+1

= int

119879

1199050

119911119879

(119905) Θ (119905) 119911 (119905) 119889119905 + 119881 (1199050) minus 119881 (119879)

minus int

119879

1199050

1

1 + 119889

120596119879

(119905) 120596 (119905) 119889119905 lt 119881 (1199050)

(1205910= 1199050 120591119898+1

= 119879)

(41)

This completes the proof of the theorem

For norminal system it is easy to obtain the followingresult

Corollary 7 Given a scalar 120574 gt 0 norminal time-varyingdescriptor system with jumps (1) is said to be FTB with respectto (119879 119877(sdot) 119889) and satisfies int119879

1199050

(119910119879

(119905)119910(119905) minus 1205742

120596119879

(119905)120596(119905))119889119905 lt

0 There exists a nonsingular and piecewise continuouslydifferential matrix-valued function Γ(sdot) such that (34) (42a)(42b) (42c) and (45) are fulfilled over [119905

0 119879]

6 Mathematical Problems in Engineering

Theorem 8 Given a scalar 120574 gt 0 and a state feedback con-troller 119906 = 119870(119905)119909 the close-loop system (9) is said to be FTBwith respect to (119879 119877(sdot) 119889) and satisfies 119869 = int

119879

1199050

(119910119879

(119905)119910(119905) minus

1205742

120596119879

(119905)120596(119905))119889119905 lt 119881(1199090 1199050) If there exists a nonsingular and

piecewise continuously differential matrix-valued function Γ(sdot)and a continuously differential matrix-valued function 119871(sdot)

such that

Γ

119879

(119905) 119864119879

= 119864Γ (119905) ge 119877 (119905) ge 0 (42a)

[

[

[

[

[

[

[

[

[

[

[

Π (119905) 119866 (119905) 0 Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905) Γ

119879

(119905)119873119879

119886+ 119871119879

(119905)119873119879

119887

119866119879

(119905) minus

1

1 + 119889

119868 0 0 0

0 0 minus1205742

119868 119865119879

(119905) 0

119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905) 0 119865 (119905) minus119868 0

119873119886Γ (119905) + 119873

119887119871 (119905) 0 0 0 minus120576119868

]

]

]

]

]

]

]

]

]

]

]

lt 0 (42b)

Γ

119879

(1199050) 119864119879

= 119864Γ (1199050) le

1

1 + 119889

119877 (1199050) (42c)

are fulfilled over [1199050 119879] the state feedback gain is obtained with

119870(119905) = 119871(119905)Γ

minus1

(119905) where

Π (119905) = minus 119864Γ (119905) + Γ

119879

(119905) 119860

119879

(119905) + 119860 (119905) Γ (119905) + 119871119879

(119905) 119861

119879

(119905)

+ 119861 (119905) 119871 (119905) + 120576119872119872119879

(43)

Proof By Schur complement and (42b) we have

[

[

[

[

[

[

[

[

Π (119905) 119866 (119905) 0 Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)

119866119879

(119905) minus

1

1 + 119889

119868 0 0

0 0 minus1205742

119868 119865119879

(119905)

119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905) 0 119865 (119905) minus119868

]

]

]

]

]

]

]

]

+

1

120576

[

[

[

[

[

[

Γ119879

(119905)119873119879

119886+ 119871119879

(119905)119873119879

119887

0

0

0

]

]

]

]

]

]

[119873119886Γ (119905) + 119873

119887119871 (119905) 0 0 0] lt 0

(44)

Continue using Schur complement and then (44) is equiva-lent to

[

[

[

Π0(119905) ⋆ ⋆

⋆ ⋆ ⋆

⋆ ⋆ ⋆

]

]

]

+

[

[

[

[

Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)

0

119865119879

(119905)

]

]

]

]

[119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905) 0 119865 (119905)]

=

[

[

[

[

Π0(119905) ⋆ (Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)) 119865 (119905)

⋆ ⋆ ⋆

119865119879

(119905) (119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905)) ⋆ 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

lt 0

(45)

Mathematical Problems in Engineering 7

where Π0(119905) = Π(119905) + (1120576)(Γ

119879

(119905)119873119879

119886+ 119871119879

(119905)119873119879

119887)(119873119886Γ(119905) +

119873119887119871(119905))

We rewrite (45) as

[

[

[

[

[

[

[

Π1(119905) 119866 (119905) (Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)) 119865 (119905)

119866119879

(119905) minus

1

1 + 119889

119868 0

119865119879

(119905) (119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905)) 0 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

]

]

lt 0 (46)

since119870(119905) = 119871(119905)Γ

minus1

(119905) and

Π1(119905) = minus119864

Γ (119905) + Γ

119879

(119905) (119860

119879

(119905) + 119870119879

(119905) 119861

119879

(119905))

+ (119860 (119905) + 119861 (119905)119870 (119905)) Γ (119905) + 120576119872119872119879

+

1

120576

Γ

119879

(119905) (119873119886+ 119873119887119870 (119905))

119879

(119873119886+ 119873119887119870 (119905)) Γ (119905)

+ Γ

119879

(119905) (119862119879

(119905) + 119870119879

(119905) 119863119879

(119905))

sdot (119862 (119905) + 119863 (119905)119870 (119905)) Γ (119905)

(47)

By Lemma 3 we have

Π1(119905) = minus119864

Γ (119905) + Γ

119879

(119905) 119862119879

119888(119905) 119862119888(119905) Γ (119905)

+ Γ

119879

(119905) (119860

119879

(119905) + 119870119879

(119905) 119861

119879

(119905))

+ (119860 (119905) + 119861 (119905)119870 (119905)) Γ (119905)

+ Γ

119879

(119905) (119872Δ (119905)119873119886+119872Δ (119905)119873

119887119870 (119905))

119879

+ (119872Δ (119905)119873119886+119872Δ (119905)119873

119887119870 (119905)) Γ

= minus119864Γ (119905) + Γ

119879

(119905) 119862119879

119888(119905) 119862119888(119905) Γ (119905) + Γ

119879

(119905) 119860119879

119888(119905)

+ 119860119888(119905) Γ (119905)

(48)

Taking Γ(119905) = Γ

minus1

(119905) since Γ119879(119905)119864 = 119864119879

Γ(119905) and 119868 =

Γ(119905)Γ(119905) 0 = Γ(119905)Γ(119905) + Γ(119905)Γ(119905) it is easy to have

minusΓ

minus119879

(119905) 119864Γ (119905) Γ

minus1

(119905) = minus119864119879

Γ (119905)Γ (119905) Γ (119905) = 119864

119879

Γ (119905)

(49)

Pre- and postmultiplying (46) by diag(Γminus119879(119905) 119868 119868) anddiag(Γminus1(119905) 119868 119868) we have

[

[

[

[

[

Π2(119905) Γ

119879

(119905) 119866 (119905) 119862119879

119888(119905) 119865 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0

119865119879

(119905) 119862119888(119905) 0 119865

119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

lt 0 (50)

where Π2(119905) = 119864Γ(119905) + 119862

119879

119888(119905)119862119888(119905) + Γ

119879

(119905)119860119888(119905) + 119860

119879

119888(119905)Γ(119905)

By Theorem 6 close-loop system is FTB and satisfies 119869 lt119881(1199090 1199050)

For norminal system it is easy to obtain the followingresult

Corollary 9 Given a scalar 120574 gt 0 and a state feedback con-troller 119906 = 119870(119905)119909 the norminal close-loop system is said to beFTBwith respect to (119879 119877(sdot) 119889) and satisfies 119869 = int

119879

1199050

(119910119879

(119905)119910(119905)minus

1205742

120596119879

(119905)120596(119905))119889119905 lt 119881(1199090 1199050) If there exists a nonsingular and

piecewise continuously differential matrix-valued function Γ(sdot)and a continuously differential matrix-valued function 119871(sdot)

such that (50) (42a) and (42c) are fulfilled over [1199050 119879] the

state feedback gain is obtained with 119870(119905) = 119871(119905)Γ

minus1

(119905)

All the conditions in this paper are expressed in terms oftime-varyingDDLMIs However with an appropriate choiceof the structure of the unknown matrix function 119875(119905) it canbe turned into a ldquostandardrdquo LMI problem The unknownmatrix function119875(119905) has been assumed to be piecewise affinethat is

119875 (0) (or (119875 (119905))) = Π0

1

119875 (119905) (or (119875 (119905))) = Π0

119896+ Π119904

119896(119905 minus (119896 minus 1) 119879

119904)

(119905) (or (119875 (119905))) = Π119904

119896

119896 isin 119873 119896 lt 119896 119905 isin [(119896 minus 1) 119879119904 119896119879119904]

119875 (119905) (or (119875 (119905))) = Π0

119896+1

+ Π0

119896+1

(119905 minus 119896119879119904)

(119905) (or (119875 (119905))) = Π0

119896+1

119905 isin [119896119879119904 119879]

(51)

where 119896 = max 119896 isin 119873+ 119896 lt 119879119879119904

Hence the conditions are reduced to a set of LMIs Notethat the conditions are not strict LMI conditions due to119864119879

Γ(119905) = Γ119879

(119905)119864 ge 0 and this may cause a big troublein checking the conditions numerically In order to translatethe nonstrict LMI into strict LMI we can use the followinglemma

8 Mathematical Problems in Engineering

Lemma 10 (see [27 28]) Let 119883 isin 119877119899times119899 be symmetric such

that 119864119879119871119883119864119871gt 0 and let 119879 isin 119877

(119899minus119903)times(119899minus119903) be nonsingular Then119883119864 + 119867

119879

119879119878119879 is nonsingular and its inverse is expressed as

(119883119864 + 119867119879

119879119878119879

)

minus1

= X119864119879

+ 119878T119867 (52)

whereX is symmetric and T is a nonsingular matrix with

119864119879

119877X119864119877= (119864119879

119871119883119864119871)

minus1

T = (119878119879

119878)

minus1

119879minus1

(119867119867119879

)

minus1

(53)

where 119867 and 119878 are any matrix with full row rank and satisfy119872119864 = 0 and 119864119878 = 0 respectively 119864 is decomposed as 119864 =

119864119871119864119879

119877with 119864

119871isin 119877119899times119903 and 119864

119877isin 119877119899times119903 are of full column rank

LetΠ0119896= 1198830

119896119864+119867

119879

1198790

119896119878119879 andΠ119904

119896= 119883119904

119896119864+119867

119879

119879119904

119896119878119879 Using

Lemma 10 we can get (1198830119896119864+119867119879

1198790

119896119878119879

)minus1

= X0119896119864119879

+119878T0119896119867 and

(119883119904

119896119864+119867

119879

119879119904

119896119878119879

)minus1

= X119904119896119864119879

+119878T0119896119867 In this way the condition

119864119879

Γ(119905) = Γ119879

(119905)119864 ge 0 is satisfied so the nonstrict LMIs havebeen translated into strict LMIs Exploiting the MATLABLMI toolbox it is possible to find matrices 119883119904

119896 1198790

119896 119879119904

119896 1198830

119896(or

X0119896X119904119896 T0119896 T 119904119896) such that Γ(119905) (or Γ(119905)) and 119871(119905) verify the

conditions

4 Numerical Example

Consider the linear time-varying descriptor system withjumps defined by

119864 = [

1 0

0 0

] 119860 = [

0 1

2 (119905 + 1) 3 (119905 + 1)2]

119861 = [

1

1

] 119865 = [05 1] 119860119896= [

001 0

0 0

]

119866 = [

1 0

0 1

] 119863 = 1 119862 = [1 1]

119872 = [

01

01

] 119873119886= [02 01] 119873

119887= [01 01]

(54)

Given 120596(119905) = [sin(2120587119905 + 1) cos(2120587119905 minus 2)]119879 Ω = [1525] 119877 = diag(12 1) 120574 = 15 and 120576 = 1654 It is easy tosee

119864119877= 119864119871= [

1

0

] (55)

We choose119867119879 = 119878 = [0 1]119879 Solving the LMIs by the numer-

ical algorithm in the previous section using MATLAB LMItoolbox it is possible to find two piecewise affinematrix func-tions Γ(119905) and 119871(119905) which verify the conditions ofTheorem 8Therefore the following state feedback control law can beobtained (see Figure 1)

15 2 25

0

05

1

15

2

25

3

minus05

minus15

minus1

Figure 1 1198701(119905) (solid line) and 119870

2(119905) (dashed line)

5 Conclusions

In this paper we have formulated and studied the finite-time119867infin

control problem of uncertain time-varying descriptorsystem with jumps at fixed times A sufficient condition forFTB and a sufficient condition for finite-time 119867

infinof the

system have been presented in terms of DLMIs and LMIsThen the state feedback controller has been designed toguarantee the closed-loop systemrsquos FTB and satisfy the 119871

2

gain At last two numerical examples have been used toillustrate the main result

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by the National Nature ScienceFoundation of China (no 61074005) and the Talent Project oftheHighEducation of Liaoning province China underGrantno LR2012005

References

[1] G V Kamenkov ldquoOn stability of motion over a finite intervalof timerdquo Journal of Applied Mathematics and Mechanics vol 17pp 529ndash540 1953

[2] A A Lebedev ldquoOn stability of motion during a given intervalof timerdquo Journal of Applied Mathematics and Mechanics vol 18pp 139ndash148 1954

[3] A A Lebedev ldquoOn the problem of stability of motion overa finite interval of timerdquo Journal of Applied Mathematics andMechanics vol 18 pp 75ndash94 1954

[4] P Dorato ldquoShort time stability in linear time-varying systemsrdquoin Proceedings of the IRE International Convention Record Part4 pp 83ndash87 1961

[5] LWeiss andE F Infante ldquoFinite time stability under perturbingforces and on product spacesrdquo IEEE Transactions on AutomaticControl vol 12 no 1 pp 54ndash59 1967

Mathematical Problems in Engineering 9

[6] F Amato R Ambrosino M Ariola and C Cosentino ldquoFinite-time stability of linear time-varying systems with jumpsrdquoAutomatica vol 45 no 5 pp 1354ndash1358 2009

[7] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoIEEETransactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[8] F Amato M Ariola and C Cosentino ldquoFinite-time stabiliza-tion via dynamic output feedbackrdquo Automatica vol 42 no 2pp 337ndash342 2006

[9] G Garcia S Tarbouriech and J Bernussou ldquoFinite-timestabilization of linear time-varying continuous systemsrdquo IEEETransactions on Automatic Control vol 54 no 2 pp 364ndash3692009

[10] Y Shen ldquoFinite-time control of linear parameter-varying sys-tems with norm-bounded exogenous disturbancerdquo Journal ofControlTheory and Applications vol 6 no 2 pp 184ndash188 2008

[11] L Liu and J T Sun ldquoFinite-time stabilization of linear systemsvia impulsive controlrdquo International Journal of Control vol 81no 6 pp 905ndash909 2008

[12] F Amato M Ariola and C Cosentino ldquoFinite-time control oflinear time-varying systems via output feedbackrdquo in Proceedingsof the American Control Conference (ACC rsquo05) pp 4722ndash4726Portland OR USA June 2005

[13] C-J Wang ldquoControllability and observability of linear time-varying singular systemsrdquo IEEE Transactions on AutomaticControl vol 44 no 10 pp 1901ndash1905 1999

[14] C-J Wang and H-E Liao ldquoImpulse observability and impulsecontrollability of linear time-varying singular systemsrdquo Auto-matica vol 37 no 11 pp 1867ndash1872 2001

[15] N A Kablar ldquoFinite-time stability of time-varying linearsingular systemsrdquo in Proceedings of the 37th IEEE Conference onDecision and Control vol 4 pp 3831ndash3836 Tampa Fla USADecember 1998

[16] NAKablar andD LDebeljkovic ldquoFinite-time stability robust-ness of time-varying linear singular systemsrdquo in Proceedings ofthe 3rd Asian Control Conference Shanghai China July 2000

[17] N A Kablar and D L Debeljkovic ldquoFinite-time instabilityof time-varying linear singular systemsrdquo in Proceedings of theAmerican Control Conference vol 3 pp 1796ndash1800 San DiegoCalif USA June 1999

[18] G Zhang Y Xia and P Shi ldquoNew bounded real lemma fordiscrete-time singular systemsrdquo Automatica vol 44 no 3 pp886ndash890 2008

[19] M Chadli H R Karimi and P Shi ldquoOn stability and stabi-lization of singular uncertain Takagi-Sugeno fuzzy systemsrdquoJournal of the Franklin Institute vol 351 no 3 pp 1453ndash14632014

[20] M Chadli A Abdo and S X Ding ldquoH minus Hinfin fault detectionfilter design for discrete-time Takagi-Sugeno fuzzy systemrdquoAutomatica vol 49 no 7 pp 1996ndash2005 2013

[21] J Yao Z-H Guan G R Chen and D W C Ho ldquoStabilityrobust stabilization and 119867

infincontrol of singular-impulsive

systems via switching controlrdquo Systems amp Control Letters vol55 no 11 pp 879ndash886 2006

[22] J Raouf and E K Boukas ldquoStabilisation of singular Markovianjump systems with discontinuities and saturating inputsrdquo IETControl Theory amp Applications vol 3 no 7 pp 971ndash982 2009

[23] Z-H Guan C W Chan A Y T Leung and G ChenldquoRobust stabilization of singular-impulsive-delayed systemswith nonlinear perturbationsrdquo IEEE Transactions on Circuitsand Systems vol 48 no 8 pp 1011ndash1019 2001

[24] S W Zhao J T Sun and L Liu ldquoFinite-time stability oflinear time-varying singular systems with impulsive effectsrdquoInternational Journal of Control vol 81 no 11 pp 1824ndash18292008

[25] J Xu and J Sun ldquoFinite-time stability of linear time-varyingsingular impulsive systemsrdquo IET ControlTheory amp Applicationsvol 4 no 10 pp 2239ndash2244 2010

[26] L G Wu and D W C Ho ldquoSliding mode control of singularstochastic hybrid systemsrdquo Automatica vol 46 no 4 pp 779ndash783 2010

[27] E Uezato and M Ikeda ldquoStrict LMI conditions for stabilityrobust stabilization and 119867

infincontrol of descriptor systemsrdquo in

Proceedings of the 38th IEEE Conference on Decision amp Control(CDC rsquo99) vol 4 pp 4092ndash4097 IEEE Phoenix Ariz USADecember 1999

[28] G P Lu and D W C Ho ldquoContinuous stabilization controllersfor singular bilinear systems the state feedback caserdquo Automat-ica vol 42 no 2 pp 309ndash314 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

are fulfilled over [1199050 119879] where

Ξ (119905) = 119864119879

Γ (119905) + 119860

119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) + 119862119879

(119905) 119862 (119905)

+ 120576119873119879

119886119873119886

(32)

Proof By the proof of Theorem 4 (31b) is equivalent to thefollowing LMI

[

[

[

[

[

Ξ0(119905) Γ

119879

(119905) 119866 (119905) 119862119879

(119905) 119865 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0

119865119879

(119905) 119862 (119905) 0 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

lt 0 (33)

where Ξ0(119905) = 119860

119879

(119905)Γ(119905) + Γ119879

(119905)119860(119905) + 119864119879

Γ(119905) + 120576119873119879

119886119873119886+

(1120576)Γ119879

(119905)119872119872119879

Γ(119905) + 119862119879

(119905)119862(119905)By Lemma 3 it is easy to have

Θ (119905)

=

[

[

[

[

[

Ξ1(119905) + 119862

119879

(119905) 119862 (119905) Γ119879

(119905) 119866 (119905) 119862119879

(119905) 119865 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0

119865119879

(119905) 119862 (119905) 0 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

lt 0

(34)

where

Ξ1(119905) = 119864

119879

Γ (119905) + 119860119879

(119905) Γ (119905) + Γ119879

(119905) 119860 (119905) lt 0 (35)

Since 119862119879(119905)119862(119905) gt 0 we have

[

[

Ξ1(119905) Γ

119879

(119905) 119866 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868

]

]

lt 0 (36)

By (31a) (31b) and (36) and the proof of Theorem 4 system(1) is FTB

Now we will show that the following performance func-tion is bounded

119869 = int

119879

1199050

(119910119879

(119905) 119910 (119905) minus 1205742

120596119879

(119905) 120596 (119905)) lt 119881 (1199090 1199050) (37)

Construct a new vector as follows

119911 (119905) =[

[

[

119909 (119905)

120596 (119905)

120596 (119905)

]

]

]

(38)

By (34) we have

119911119879

(119905) Θ (119905) 119911 (119905)

= 119909119879

(119905) Ξ1(119905) 119909 (119905) + 120596

119879

(119905) 119866119879

(119905) Γ (119905) 119909 (t)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905) + 119909119879

(119905) 119862119879

(119905) 119862 (119905) 119909 (119905)

+ 120596119879

(119905) 119865119879

(119905) 119862 (119905) 119909 (119905) + 119909119879

(119905) 119862119879

(119905) 119865 (119905) 120596 (119905)

+ 120596119879

(119905) 119865119879

(119905) 119865 (119905) 120596 (119905) minus (

1

1 + 119889

+ 1205742

)120596119879

(119905) 120596 (119905)

= 119909119879

(119905) Ξ1(119905) 119909 (119905) + 120596

119879

(119905) 119866119879

(119905) Γ (119905) 119909 (119905)

+ 119909119879

(119905) Γ119879

(119905) 119866 (119905) 120596 (119905) minus

1

1 + 119889

120596119879

(119905) 120596 (119905)

+ 119910119879

(119905) 119910 (119905) minus 1205742

120596119879

(119905) 120596 (119905)

= (119905) minus

1

1 + 119889

120596119879

(119905) 120596 (119905) + 119910119879

(119905) 119910 (119905)

minus 1205742

120596119879

(119905) 120596 (119905) lt 0

(39)

Therefore for 119905 isin (120591119896 120591119896+1

) 119896 = 0 1 2 119898 we have

119869120591119896+1

= int

120591119896+1

120591119896

119911119879

(119905) Θ (119905) 119911 (119905) 119889119905

minus (int

120591119896+1

120591119896

(119905) minus

1

1 + 119889

120596119879

(119905) 120596 (119905)) 119889119905

(40)

with minus119881(119879) minus int

119879

1199050

(1(1 + 119889))120596119879

(119905)120596(119905) lt 0 and for Θ(119905) lt 0

we have

119869 =

119898+1

sum

119894=0

119869120591119894+1

= int

119879

1199050

119911119879

(119905) Θ (119905) 119911 (119905) 119889119905 + 119881 (1199050) minus 119881 (119879)

minus int

119879

1199050

1

1 + 119889

120596119879

(119905) 120596 (119905) 119889119905 lt 119881 (1199050)

(1205910= 1199050 120591119898+1

= 119879)

(41)

This completes the proof of the theorem

For norminal system it is easy to obtain the followingresult

Corollary 7 Given a scalar 120574 gt 0 norminal time-varyingdescriptor system with jumps (1) is said to be FTB with respectto (119879 119877(sdot) 119889) and satisfies int119879

1199050

(119910119879

(119905)119910(119905) minus 1205742

120596119879

(119905)120596(119905))119889119905 lt

0 There exists a nonsingular and piecewise continuouslydifferential matrix-valued function Γ(sdot) such that (34) (42a)(42b) (42c) and (45) are fulfilled over [119905

0 119879]

6 Mathematical Problems in Engineering

Theorem 8 Given a scalar 120574 gt 0 and a state feedback con-troller 119906 = 119870(119905)119909 the close-loop system (9) is said to be FTBwith respect to (119879 119877(sdot) 119889) and satisfies 119869 = int

119879

1199050

(119910119879

(119905)119910(119905) minus

1205742

120596119879

(119905)120596(119905))119889119905 lt 119881(1199090 1199050) If there exists a nonsingular and

piecewise continuously differential matrix-valued function Γ(sdot)and a continuously differential matrix-valued function 119871(sdot)

such that

Γ

119879

(119905) 119864119879

= 119864Γ (119905) ge 119877 (119905) ge 0 (42a)

[

[

[

[

[

[

[

[

[

[

[

Π (119905) 119866 (119905) 0 Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905) Γ

119879

(119905)119873119879

119886+ 119871119879

(119905)119873119879

119887

119866119879

(119905) minus

1

1 + 119889

119868 0 0 0

0 0 minus1205742

119868 119865119879

(119905) 0

119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905) 0 119865 (119905) minus119868 0

119873119886Γ (119905) + 119873

119887119871 (119905) 0 0 0 minus120576119868

]

]

]

]

]

]

]

]

]

]

]

lt 0 (42b)

Γ

119879

(1199050) 119864119879

= 119864Γ (1199050) le

1

1 + 119889

119877 (1199050) (42c)

are fulfilled over [1199050 119879] the state feedback gain is obtained with

119870(119905) = 119871(119905)Γ

minus1

(119905) where

Π (119905) = minus 119864Γ (119905) + Γ

119879

(119905) 119860

119879

(119905) + 119860 (119905) Γ (119905) + 119871119879

(119905) 119861

119879

(119905)

+ 119861 (119905) 119871 (119905) + 120576119872119872119879

(43)

Proof By Schur complement and (42b) we have

[

[

[

[

[

[

[

[

Π (119905) 119866 (119905) 0 Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)

119866119879

(119905) minus

1

1 + 119889

119868 0 0

0 0 minus1205742

119868 119865119879

(119905)

119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905) 0 119865 (119905) minus119868

]

]

]

]

]

]

]

]

+

1

120576

[

[

[

[

[

[

Γ119879

(119905)119873119879

119886+ 119871119879

(119905)119873119879

119887

0

0

0

]

]

]

]

]

]

[119873119886Γ (119905) + 119873

119887119871 (119905) 0 0 0] lt 0

(44)

Continue using Schur complement and then (44) is equiva-lent to

[

[

[

Π0(119905) ⋆ ⋆

⋆ ⋆ ⋆

⋆ ⋆ ⋆

]

]

]

+

[

[

[

[

Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)

0

119865119879

(119905)

]

]

]

]

[119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905) 0 119865 (119905)]

=

[

[

[

[

Π0(119905) ⋆ (Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)) 119865 (119905)

⋆ ⋆ ⋆

119865119879

(119905) (119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905)) ⋆ 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

lt 0

(45)

Mathematical Problems in Engineering 7

where Π0(119905) = Π(119905) + (1120576)(Γ

119879

(119905)119873119879

119886+ 119871119879

(119905)119873119879

119887)(119873119886Γ(119905) +

119873119887119871(119905))

We rewrite (45) as

[

[

[

[

[

[

[

Π1(119905) 119866 (119905) (Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)) 119865 (119905)

119866119879

(119905) minus

1

1 + 119889

119868 0

119865119879

(119905) (119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905)) 0 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

]

]

lt 0 (46)

since119870(119905) = 119871(119905)Γ

minus1

(119905) and

Π1(119905) = minus119864

Γ (119905) + Γ

119879

(119905) (119860

119879

(119905) + 119870119879

(119905) 119861

119879

(119905))

+ (119860 (119905) + 119861 (119905)119870 (119905)) Γ (119905) + 120576119872119872119879

+

1

120576

Γ

119879

(119905) (119873119886+ 119873119887119870 (119905))

119879

(119873119886+ 119873119887119870 (119905)) Γ (119905)

+ Γ

119879

(119905) (119862119879

(119905) + 119870119879

(119905) 119863119879

(119905))

sdot (119862 (119905) + 119863 (119905)119870 (119905)) Γ (119905)

(47)

By Lemma 3 we have

Π1(119905) = minus119864

Γ (119905) + Γ

119879

(119905) 119862119879

119888(119905) 119862119888(119905) Γ (119905)

+ Γ

119879

(119905) (119860

119879

(119905) + 119870119879

(119905) 119861

119879

(119905))

+ (119860 (119905) + 119861 (119905)119870 (119905)) Γ (119905)

+ Γ

119879

(119905) (119872Δ (119905)119873119886+119872Δ (119905)119873

119887119870 (119905))

119879

+ (119872Δ (119905)119873119886+119872Δ (119905)119873

119887119870 (119905)) Γ

= minus119864Γ (119905) + Γ

119879

(119905) 119862119879

119888(119905) 119862119888(119905) Γ (119905) + Γ

119879

(119905) 119860119879

119888(119905)

+ 119860119888(119905) Γ (119905)

(48)

Taking Γ(119905) = Γ

minus1

(119905) since Γ119879(119905)119864 = 119864119879

Γ(119905) and 119868 =

Γ(119905)Γ(119905) 0 = Γ(119905)Γ(119905) + Γ(119905)Γ(119905) it is easy to have

minusΓ

minus119879

(119905) 119864Γ (119905) Γ

minus1

(119905) = minus119864119879

Γ (119905)Γ (119905) Γ (119905) = 119864

119879

Γ (119905)

(49)

Pre- and postmultiplying (46) by diag(Γminus119879(119905) 119868 119868) anddiag(Γminus1(119905) 119868 119868) we have

[

[

[

[

[

Π2(119905) Γ

119879

(119905) 119866 (119905) 119862119879

119888(119905) 119865 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0

119865119879

(119905) 119862119888(119905) 0 119865

119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

lt 0 (50)

where Π2(119905) = 119864Γ(119905) + 119862

119879

119888(119905)119862119888(119905) + Γ

119879

(119905)119860119888(119905) + 119860

119879

119888(119905)Γ(119905)

By Theorem 6 close-loop system is FTB and satisfies 119869 lt119881(1199090 1199050)

For norminal system it is easy to obtain the followingresult

Corollary 9 Given a scalar 120574 gt 0 and a state feedback con-troller 119906 = 119870(119905)119909 the norminal close-loop system is said to beFTBwith respect to (119879 119877(sdot) 119889) and satisfies 119869 = int

119879

1199050

(119910119879

(119905)119910(119905)minus

1205742

120596119879

(119905)120596(119905))119889119905 lt 119881(1199090 1199050) If there exists a nonsingular and

piecewise continuously differential matrix-valued function Γ(sdot)and a continuously differential matrix-valued function 119871(sdot)

such that (50) (42a) and (42c) are fulfilled over [1199050 119879] the

state feedback gain is obtained with 119870(119905) = 119871(119905)Γ

minus1

(119905)

All the conditions in this paper are expressed in terms oftime-varyingDDLMIs However with an appropriate choiceof the structure of the unknown matrix function 119875(119905) it canbe turned into a ldquostandardrdquo LMI problem The unknownmatrix function119875(119905) has been assumed to be piecewise affinethat is

119875 (0) (or (119875 (119905))) = Π0

1

119875 (119905) (or (119875 (119905))) = Π0

119896+ Π119904

119896(119905 minus (119896 minus 1) 119879

119904)

(119905) (or (119875 (119905))) = Π119904

119896

119896 isin 119873 119896 lt 119896 119905 isin [(119896 minus 1) 119879119904 119896119879119904]

119875 (119905) (or (119875 (119905))) = Π0

119896+1

+ Π0

119896+1

(119905 minus 119896119879119904)

(119905) (or (119875 (119905))) = Π0

119896+1

119905 isin [119896119879119904 119879]

(51)

where 119896 = max 119896 isin 119873+ 119896 lt 119879119879119904

Hence the conditions are reduced to a set of LMIs Notethat the conditions are not strict LMI conditions due to119864119879

Γ(119905) = Γ119879

(119905)119864 ge 0 and this may cause a big troublein checking the conditions numerically In order to translatethe nonstrict LMI into strict LMI we can use the followinglemma

8 Mathematical Problems in Engineering

Lemma 10 (see [27 28]) Let 119883 isin 119877119899times119899 be symmetric such

that 119864119879119871119883119864119871gt 0 and let 119879 isin 119877

(119899minus119903)times(119899minus119903) be nonsingular Then119883119864 + 119867

119879

119879119878119879 is nonsingular and its inverse is expressed as

(119883119864 + 119867119879

119879119878119879

)

minus1

= X119864119879

+ 119878T119867 (52)

whereX is symmetric and T is a nonsingular matrix with

119864119879

119877X119864119877= (119864119879

119871119883119864119871)

minus1

T = (119878119879

119878)

minus1

119879minus1

(119867119867119879

)

minus1

(53)

where 119867 and 119878 are any matrix with full row rank and satisfy119872119864 = 0 and 119864119878 = 0 respectively 119864 is decomposed as 119864 =

119864119871119864119879

119877with 119864

119871isin 119877119899times119903 and 119864

119877isin 119877119899times119903 are of full column rank

LetΠ0119896= 1198830

119896119864+119867

119879

1198790

119896119878119879 andΠ119904

119896= 119883119904

119896119864+119867

119879

119879119904

119896119878119879 Using

Lemma 10 we can get (1198830119896119864+119867119879

1198790

119896119878119879

)minus1

= X0119896119864119879

+119878T0119896119867 and

(119883119904

119896119864+119867

119879

119879119904

119896119878119879

)minus1

= X119904119896119864119879

+119878T0119896119867 In this way the condition

119864119879

Γ(119905) = Γ119879

(119905)119864 ge 0 is satisfied so the nonstrict LMIs havebeen translated into strict LMIs Exploiting the MATLABLMI toolbox it is possible to find matrices 119883119904

119896 1198790

119896 119879119904

119896 1198830

119896(or

X0119896X119904119896 T0119896 T 119904119896) such that Γ(119905) (or Γ(119905)) and 119871(119905) verify the

conditions

4 Numerical Example

Consider the linear time-varying descriptor system withjumps defined by

119864 = [

1 0

0 0

] 119860 = [

0 1

2 (119905 + 1) 3 (119905 + 1)2]

119861 = [

1

1

] 119865 = [05 1] 119860119896= [

001 0

0 0

]

119866 = [

1 0

0 1

] 119863 = 1 119862 = [1 1]

119872 = [

01

01

] 119873119886= [02 01] 119873

119887= [01 01]

(54)

Given 120596(119905) = [sin(2120587119905 + 1) cos(2120587119905 minus 2)]119879 Ω = [1525] 119877 = diag(12 1) 120574 = 15 and 120576 = 1654 It is easy tosee

119864119877= 119864119871= [

1

0

] (55)

We choose119867119879 = 119878 = [0 1]119879 Solving the LMIs by the numer-

ical algorithm in the previous section using MATLAB LMItoolbox it is possible to find two piecewise affinematrix func-tions Γ(119905) and 119871(119905) which verify the conditions ofTheorem 8Therefore the following state feedback control law can beobtained (see Figure 1)

15 2 25

0

05

1

15

2

25

3

minus05

minus15

minus1

Figure 1 1198701(119905) (solid line) and 119870

2(119905) (dashed line)

5 Conclusions

In this paper we have formulated and studied the finite-time119867infin

control problem of uncertain time-varying descriptorsystem with jumps at fixed times A sufficient condition forFTB and a sufficient condition for finite-time 119867

infinof the

system have been presented in terms of DLMIs and LMIsThen the state feedback controller has been designed toguarantee the closed-loop systemrsquos FTB and satisfy the 119871

2

gain At last two numerical examples have been used toillustrate the main result

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by the National Nature ScienceFoundation of China (no 61074005) and the Talent Project oftheHighEducation of Liaoning province China underGrantno LR2012005

References

[1] G V Kamenkov ldquoOn stability of motion over a finite intervalof timerdquo Journal of Applied Mathematics and Mechanics vol 17pp 529ndash540 1953

[2] A A Lebedev ldquoOn stability of motion during a given intervalof timerdquo Journal of Applied Mathematics and Mechanics vol 18pp 139ndash148 1954

[3] A A Lebedev ldquoOn the problem of stability of motion overa finite interval of timerdquo Journal of Applied Mathematics andMechanics vol 18 pp 75ndash94 1954

[4] P Dorato ldquoShort time stability in linear time-varying systemsrdquoin Proceedings of the IRE International Convention Record Part4 pp 83ndash87 1961

[5] LWeiss andE F Infante ldquoFinite time stability under perturbingforces and on product spacesrdquo IEEE Transactions on AutomaticControl vol 12 no 1 pp 54ndash59 1967

Mathematical Problems in Engineering 9

[6] F Amato R Ambrosino M Ariola and C Cosentino ldquoFinite-time stability of linear time-varying systems with jumpsrdquoAutomatica vol 45 no 5 pp 1354ndash1358 2009

[7] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoIEEETransactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[8] F Amato M Ariola and C Cosentino ldquoFinite-time stabiliza-tion via dynamic output feedbackrdquo Automatica vol 42 no 2pp 337ndash342 2006

[9] G Garcia S Tarbouriech and J Bernussou ldquoFinite-timestabilization of linear time-varying continuous systemsrdquo IEEETransactions on Automatic Control vol 54 no 2 pp 364ndash3692009

[10] Y Shen ldquoFinite-time control of linear parameter-varying sys-tems with norm-bounded exogenous disturbancerdquo Journal ofControlTheory and Applications vol 6 no 2 pp 184ndash188 2008

[11] L Liu and J T Sun ldquoFinite-time stabilization of linear systemsvia impulsive controlrdquo International Journal of Control vol 81no 6 pp 905ndash909 2008

[12] F Amato M Ariola and C Cosentino ldquoFinite-time control oflinear time-varying systems via output feedbackrdquo in Proceedingsof the American Control Conference (ACC rsquo05) pp 4722ndash4726Portland OR USA June 2005

[13] C-J Wang ldquoControllability and observability of linear time-varying singular systemsrdquo IEEE Transactions on AutomaticControl vol 44 no 10 pp 1901ndash1905 1999

[14] C-J Wang and H-E Liao ldquoImpulse observability and impulsecontrollability of linear time-varying singular systemsrdquo Auto-matica vol 37 no 11 pp 1867ndash1872 2001

[15] N A Kablar ldquoFinite-time stability of time-varying linearsingular systemsrdquo in Proceedings of the 37th IEEE Conference onDecision and Control vol 4 pp 3831ndash3836 Tampa Fla USADecember 1998

[16] NAKablar andD LDebeljkovic ldquoFinite-time stability robust-ness of time-varying linear singular systemsrdquo in Proceedings ofthe 3rd Asian Control Conference Shanghai China July 2000

[17] N A Kablar and D L Debeljkovic ldquoFinite-time instabilityof time-varying linear singular systemsrdquo in Proceedings of theAmerican Control Conference vol 3 pp 1796ndash1800 San DiegoCalif USA June 1999

[18] G Zhang Y Xia and P Shi ldquoNew bounded real lemma fordiscrete-time singular systemsrdquo Automatica vol 44 no 3 pp886ndash890 2008

[19] M Chadli H R Karimi and P Shi ldquoOn stability and stabi-lization of singular uncertain Takagi-Sugeno fuzzy systemsrdquoJournal of the Franklin Institute vol 351 no 3 pp 1453ndash14632014

[20] M Chadli A Abdo and S X Ding ldquoH minus Hinfin fault detectionfilter design for discrete-time Takagi-Sugeno fuzzy systemrdquoAutomatica vol 49 no 7 pp 1996ndash2005 2013

[21] J Yao Z-H Guan G R Chen and D W C Ho ldquoStabilityrobust stabilization and 119867

infincontrol of singular-impulsive

systems via switching controlrdquo Systems amp Control Letters vol55 no 11 pp 879ndash886 2006

[22] J Raouf and E K Boukas ldquoStabilisation of singular Markovianjump systems with discontinuities and saturating inputsrdquo IETControl Theory amp Applications vol 3 no 7 pp 971ndash982 2009

[23] Z-H Guan C W Chan A Y T Leung and G ChenldquoRobust stabilization of singular-impulsive-delayed systemswith nonlinear perturbationsrdquo IEEE Transactions on Circuitsand Systems vol 48 no 8 pp 1011ndash1019 2001

[24] S W Zhao J T Sun and L Liu ldquoFinite-time stability oflinear time-varying singular systems with impulsive effectsrdquoInternational Journal of Control vol 81 no 11 pp 1824ndash18292008

[25] J Xu and J Sun ldquoFinite-time stability of linear time-varyingsingular impulsive systemsrdquo IET ControlTheory amp Applicationsvol 4 no 10 pp 2239ndash2244 2010

[26] L G Wu and D W C Ho ldquoSliding mode control of singularstochastic hybrid systemsrdquo Automatica vol 46 no 4 pp 779ndash783 2010

[27] E Uezato and M Ikeda ldquoStrict LMI conditions for stabilityrobust stabilization and 119867

infincontrol of descriptor systemsrdquo in

Proceedings of the 38th IEEE Conference on Decision amp Control(CDC rsquo99) vol 4 pp 4092ndash4097 IEEE Phoenix Ariz USADecember 1999

[28] G P Lu and D W C Ho ldquoContinuous stabilization controllersfor singular bilinear systems the state feedback caserdquo Automat-ica vol 42 no 2 pp 309ndash314 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Theorem 8 Given a scalar 120574 gt 0 and a state feedback con-troller 119906 = 119870(119905)119909 the close-loop system (9) is said to be FTBwith respect to (119879 119877(sdot) 119889) and satisfies 119869 = int

119879

1199050

(119910119879

(119905)119910(119905) minus

1205742

120596119879

(119905)120596(119905))119889119905 lt 119881(1199090 1199050) If there exists a nonsingular and

piecewise continuously differential matrix-valued function Γ(sdot)and a continuously differential matrix-valued function 119871(sdot)

such that

Γ

119879

(119905) 119864119879

= 119864Γ (119905) ge 119877 (119905) ge 0 (42a)

[

[

[

[

[

[

[

[

[

[

[

Π (119905) 119866 (119905) 0 Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905) Γ

119879

(119905)119873119879

119886+ 119871119879

(119905)119873119879

119887

119866119879

(119905) minus

1

1 + 119889

119868 0 0 0

0 0 minus1205742

119868 119865119879

(119905) 0

119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905) 0 119865 (119905) minus119868 0

119873119886Γ (119905) + 119873

119887119871 (119905) 0 0 0 minus120576119868

]

]

]

]

]

]

]

]

]

]

]

lt 0 (42b)

Γ

119879

(1199050) 119864119879

= 119864Γ (1199050) le

1

1 + 119889

119877 (1199050) (42c)

are fulfilled over [1199050 119879] the state feedback gain is obtained with

119870(119905) = 119871(119905)Γ

minus1

(119905) where

Π (119905) = minus 119864Γ (119905) + Γ

119879

(119905) 119860

119879

(119905) + 119860 (119905) Γ (119905) + 119871119879

(119905) 119861

119879

(119905)

+ 119861 (119905) 119871 (119905) + 120576119872119872119879

(43)

Proof By Schur complement and (42b) we have

[

[

[

[

[

[

[

[

Π (119905) 119866 (119905) 0 Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)

119866119879

(119905) minus

1

1 + 119889

119868 0 0

0 0 minus1205742

119868 119865119879

(119905)

119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905) 0 119865 (119905) minus119868

]

]

]

]

]

]

]

]

+

1

120576

[

[

[

[

[

[

Γ119879

(119905)119873119879

119886+ 119871119879

(119905)119873119879

119887

0

0

0

]

]

]

]

]

]

[119873119886Γ (119905) + 119873

119887119871 (119905) 0 0 0] lt 0

(44)

Continue using Schur complement and then (44) is equiva-lent to

[

[

[

Π0(119905) ⋆ ⋆

⋆ ⋆ ⋆

⋆ ⋆ ⋆

]

]

]

+

[

[

[

[

Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)

0

119865119879

(119905)

]

]

]

]

[119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905) 0 119865 (119905)]

=

[

[

[

[

Π0(119905) ⋆ (Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)) 119865 (119905)

⋆ ⋆ ⋆

119865119879

(119905) (119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905)) ⋆ 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

lt 0

(45)

Mathematical Problems in Engineering 7

where Π0(119905) = Π(119905) + (1120576)(Γ

119879

(119905)119873119879

119886+ 119871119879

(119905)119873119879

119887)(119873119886Γ(119905) +

119873119887119871(119905))

We rewrite (45) as

[

[

[

[

[

[

[

Π1(119905) 119866 (119905) (Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)) 119865 (119905)

119866119879

(119905) minus

1

1 + 119889

119868 0

119865119879

(119905) (119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905)) 0 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

]

]

lt 0 (46)

since119870(119905) = 119871(119905)Γ

minus1

(119905) and

Π1(119905) = minus119864

Γ (119905) + Γ

119879

(119905) (119860

119879

(119905) + 119870119879

(119905) 119861

119879

(119905))

+ (119860 (119905) + 119861 (119905)119870 (119905)) Γ (119905) + 120576119872119872119879

+

1

120576

Γ

119879

(119905) (119873119886+ 119873119887119870 (119905))

119879

(119873119886+ 119873119887119870 (119905)) Γ (119905)

+ Γ

119879

(119905) (119862119879

(119905) + 119870119879

(119905) 119863119879

(119905))

sdot (119862 (119905) + 119863 (119905)119870 (119905)) Γ (119905)

(47)

By Lemma 3 we have

Π1(119905) = minus119864

Γ (119905) + Γ

119879

(119905) 119862119879

119888(119905) 119862119888(119905) Γ (119905)

+ Γ

119879

(119905) (119860

119879

(119905) + 119870119879

(119905) 119861

119879

(119905))

+ (119860 (119905) + 119861 (119905)119870 (119905)) Γ (119905)

+ Γ

119879

(119905) (119872Δ (119905)119873119886+119872Δ (119905)119873

119887119870 (119905))

119879

+ (119872Δ (119905)119873119886+119872Δ (119905)119873

119887119870 (119905)) Γ

= minus119864Γ (119905) + Γ

119879

(119905) 119862119879

119888(119905) 119862119888(119905) Γ (119905) + Γ

119879

(119905) 119860119879

119888(119905)

+ 119860119888(119905) Γ (119905)

(48)

Taking Γ(119905) = Γ

minus1

(119905) since Γ119879(119905)119864 = 119864119879

Γ(119905) and 119868 =

Γ(119905)Γ(119905) 0 = Γ(119905)Γ(119905) + Γ(119905)Γ(119905) it is easy to have

minusΓ

minus119879

(119905) 119864Γ (119905) Γ

minus1

(119905) = minus119864119879

Γ (119905)Γ (119905) Γ (119905) = 119864

119879

Γ (119905)

(49)

Pre- and postmultiplying (46) by diag(Γminus119879(119905) 119868 119868) anddiag(Γminus1(119905) 119868 119868) we have

[

[

[

[

[

Π2(119905) Γ

119879

(119905) 119866 (119905) 119862119879

119888(119905) 119865 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0

119865119879

(119905) 119862119888(119905) 0 119865

119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

lt 0 (50)

where Π2(119905) = 119864Γ(119905) + 119862

119879

119888(119905)119862119888(119905) + Γ

119879

(119905)119860119888(119905) + 119860

119879

119888(119905)Γ(119905)

By Theorem 6 close-loop system is FTB and satisfies 119869 lt119881(1199090 1199050)

For norminal system it is easy to obtain the followingresult

Corollary 9 Given a scalar 120574 gt 0 and a state feedback con-troller 119906 = 119870(119905)119909 the norminal close-loop system is said to beFTBwith respect to (119879 119877(sdot) 119889) and satisfies 119869 = int

119879

1199050

(119910119879

(119905)119910(119905)minus

1205742

120596119879

(119905)120596(119905))119889119905 lt 119881(1199090 1199050) If there exists a nonsingular and

piecewise continuously differential matrix-valued function Γ(sdot)and a continuously differential matrix-valued function 119871(sdot)

such that (50) (42a) and (42c) are fulfilled over [1199050 119879] the

state feedback gain is obtained with 119870(119905) = 119871(119905)Γ

minus1

(119905)

All the conditions in this paper are expressed in terms oftime-varyingDDLMIs However with an appropriate choiceof the structure of the unknown matrix function 119875(119905) it canbe turned into a ldquostandardrdquo LMI problem The unknownmatrix function119875(119905) has been assumed to be piecewise affinethat is

119875 (0) (or (119875 (119905))) = Π0

1

119875 (119905) (or (119875 (119905))) = Π0

119896+ Π119904

119896(119905 minus (119896 minus 1) 119879

119904)

(119905) (or (119875 (119905))) = Π119904

119896

119896 isin 119873 119896 lt 119896 119905 isin [(119896 minus 1) 119879119904 119896119879119904]

119875 (119905) (or (119875 (119905))) = Π0

119896+1

+ Π0

119896+1

(119905 minus 119896119879119904)

(119905) (or (119875 (119905))) = Π0

119896+1

119905 isin [119896119879119904 119879]

(51)

where 119896 = max 119896 isin 119873+ 119896 lt 119879119879119904

Hence the conditions are reduced to a set of LMIs Notethat the conditions are not strict LMI conditions due to119864119879

Γ(119905) = Γ119879

(119905)119864 ge 0 and this may cause a big troublein checking the conditions numerically In order to translatethe nonstrict LMI into strict LMI we can use the followinglemma

8 Mathematical Problems in Engineering

Lemma 10 (see [27 28]) Let 119883 isin 119877119899times119899 be symmetric such

that 119864119879119871119883119864119871gt 0 and let 119879 isin 119877

(119899minus119903)times(119899minus119903) be nonsingular Then119883119864 + 119867

119879

119879119878119879 is nonsingular and its inverse is expressed as

(119883119864 + 119867119879

119879119878119879

)

minus1

= X119864119879

+ 119878T119867 (52)

whereX is symmetric and T is a nonsingular matrix with

119864119879

119877X119864119877= (119864119879

119871119883119864119871)

minus1

T = (119878119879

119878)

minus1

119879minus1

(119867119867119879

)

minus1

(53)

where 119867 and 119878 are any matrix with full row rank and satisfy119872119864 = 0 and 119864119878 = 0 respectively 119864 is decomposed as 119864 =

119864119871119864119879

119877with 119864

119871isin 119877119899times119903 and 119864

119877isin 119877119899times119903 are of full column rank

LetΠ0119896= 1198830

119896119864+119867

119879

1198790

119896119878119879 andΠ119904

119896= 119883119904

119896119864+119867

119879

119879119904

119896119878119879 Using

Lemma 10 we can get (1198830119896119864+119867119879

1198790

119896119878119879

)minus1

= X0119896119864119879

+119878T0119896119867 and

(119883119904

119896119864+119867

119879

119879119904

119896119878119879

)minus1

= X119904119896119864119879

+119878T0119896119867 In this way the condition

119864119879

Γ(119905) = Γ119879

(119905)119864 ge 0 is satisfied so the nonstrict LMIs havebeen translated into strict LMIs Exploiting the MATLABLMI toolbox it is possible to find matrices 119883119904

119896 1198790

119896 119879119904

119896 1198830

119896(or

X0119896X119904119896 T0119896 T 119904119896) such that Γ(119905) (or Γ(119905)) and 119871(119905) verify the

conditions

4 Numerical Example

Consider the linear time-varying descriptor system withjumps defined by

119864 = [

1 0

0 0

] 119860 = [

0 1

2 (119905 + 1) 3 (119905 + 1)2]

119861 = [

1

1

] 119865 = [05 1] 119860119896= [

001 0

0 0

]

119866 = [

1 0

0 1

] 119863 = 1 119862 = [1 1]

119872 = [

01

01

] 119873119886= [02 01] 119873

119887= [01 01]

(54)

Given 120596(119905) = [sin(2120587119905 + 1) cos(2120587119905 minus 2)]119879 Ω = [1525] 119877 = diag(12 1) 120574 = 15 and 120576 = 1654 It is easy tosee

119864119877= 119864119871= [

1

0

] (55)

We choose119867119879 = 119878 = [0 1]119879 Solving the LMIs by the numer-

ical algorithm in the previous section using MATLAB LMItoolbox it is possible to find two piecewise affinematrix func-tions Γ(119905) and 119871(119905) which verify the conditions ofTheorem 8Therefore the following state feedback control law can beobtained (see Figure 1)

15 2 25

0

05

1

15

2

25

3

minus05

minus15

minus1

Figure 1 1198701(119905) (solid line) and 119870

2(119905) (dashed line)

5 Conclusions

In this paper we have formulated and studied the finite-time119867infin

control problem of uncertain time-varying descriptorsystem with jumps at fixed times A sufficient condition forFTB and a sufficient condition for finite-time 119867

infinof the

system have been presented in terms of DLMIs and LMIsThen the state feedback controller has been designed toguarantee the closed-loop systemrsquos FTB and satisfy the 119871

2

gain At last two numerical examples have been used toillustrate the main result

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by the National Nature ScienceFoundation of China (no 61074005) and the Talent Project oftheHighEducation of Liaoning province China underGrantno LR2012005

References

[1] G V Kamenkov ldquoOn stability of motion over a finite intervalof timerdquo Journal of Applied Mathematics and Mechanics vol 17pp 529ndash540 1953

[2] A A Lebedev ldquoOn stability of motion during a given intervalof timerdquo Journal of Applied Mathematics and Mechanics vol 18pp 139ndash148 1954

[3] A A Lebedev ldquoOn the problem of stability of motion overa finite interval of timerdquo Journal of Applied Mathematics andMechanics vol 18 pp 75ndash94 1954

[4] P Dorato ldquoShort time stability in linear time-varying systemsrdquoin Proceedings of the IRE International Convention Record Part4 pp 83ndash87 1961

[5] LWeiss andE F Infante ldquoFinite time stability under perturbingforces and on product spacesrdquo IEEE Transactions on AutomaticControl vol 12 no 1 pp 54ndash59 1967

Mathematical Problems in Engineering 9

[6] F Amato R Ambrosino M Ariola and C Cosentino ldquoFinite-time stability of linear time-varying systems with jumpsrdquoAutomatica vol 45 no 5 pp 1354ndash1358 2009

[7] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoIEEETransactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[8] F Amato M Ariola and C Cosentino ldquoFinite-time stabiliza-tion via dynamic output feedbackrdquo Automatica vol 42 no 2pp 337ndash342 2006

[9] G Garcia S Tarbouriech and J Bernussou ldquoFinite-timestabilization of linear time-varying continuous systemsrdquo IEEETransactions on Automatic Control vol 54 no 2 pp 364ndash3692009

[10] Y Shen ldquoFinite-time control of linear parameter-varying sys-tems with norm-bounded exogenous disturbancerdquo Journal ofControlTheory and Applications vol 6 no 2 pp 184ndash188 2008

[11] L Liu and J T Sun ldquoFinite-time stabilization of linear systemsvia impulsive controlrdquo International Journal of Control vol 81no 6 pp 905ndash909 2008

[12] F Amato M Ariola and C Cosentino ldquoFinite-time control oflinear time-varying systems via output feedbackrdquo in Proceedingsof the American Control Conference (ACC rsquo05) pp 4722ndash4726Portland OR USA June 2005

[13] C-J Wang ldquoControllability and observability of linear time-varying singular systemsrdquo IEEE Transactions on AutomaticControl vol 44 no 10 pp 1901ndash1905 1999

[14] C-J Wang and H-E Liao ldquoImpulse observability and impulsecontrollability of linear time-varying singular systemsrdquo Auto-matica vol 37 no 11 pp 1867ndash1872 2001

[15] N A Kablar ldquoFinite-time stability of time-varying linearsingular systemsrdquo in Proceedings of the 37th IEEE Conference onDecision and Control vol 4 pp 3831ndash3836 Tampa Fla USADecember 1998

[16] NAKablar andD LDebeljkovic ldquoFinite-time stability robust-ness of time-varying linear singular systemsrdquo in Proceedings ofthe 3rd Asian Control Conference Shanghai China July 2000

[17] N A Kablar and D L Debeljkovic ldquoFinite-time instabilityof time-varying linear singular systemsrdquo in Proceedings of theAmerican Control Conference vol 3 pp 1796ndash1800 San DiegoCalif USA June 1999

[18] G Zhang Y Xia and P Shi ldquoNew bounded real lemma fordiscrete-time singular systemsrdquo Automatica vol 44 no 3 pp886ndash890 2008

[19] M Chadli H R Karimi and P Shi ldquoOn stability and stabi-lization of singular uncertain Takagi-Sugeno fuzzy systemsrdquoJournal of the Franklin Institute vol 351 no 3 pp 1453ndash14632014

[20] M Chadli A Abdo and S X Ding ldquoH minus Hinfin fault detectionfilter design for discrete-time Takagi-Sugeno fuzzy systemrdquoAutomatica vol 49 no 7 pp 1996ndash2005 2013

[21] J Yao Z-H Guan G R Chen and D W C Ho ldquoStabilityrobust stabilization and 119867

infincontrol of singular-impulsive

systems via switching controlrdquo Systems amp Control Letters vol55 no 11 pp 879ndash886 2006

[22] J Raouf and E K Boukas ldquoStabilisation of singular Markovianjump systems with discontinuities and saturating inputsrdquo IETControl Theory amp Applications vol 3 no 7 pp 971ndash982 2009

[23] Z-H Guan C W Chan A Y T Leung and G ChenldquoRobust stabilization of singular-impulsive-delayed systemswith nonlinear perturbationsrdquo IEEE Transactions on Circuitsand Systems vol 48 no 8 pp 1011ndash1019 2001

[24] S W Zhao J T Sun and L Liu ldquoFinite-time stability oflinear time-varying singular systems with impulsive effectsrdquoInternational Journal of Control vol 81 no 11 pp 1824ndash18292008

[25] J Xu and J Sun ldquoFinite-time stability of linear time-varyingsingular impulsive systemsrdquo IET ControlTheory amp Applicationsvol 4 no 10 pp 2239ndash2244 2010

[26] L G Wu and D W C Ho ldquoSliding mode control of singularstochastic hybrid systemsrdquo Automatica vol 46 no 4 pp 779ndash783 2010

[27] E Uezato and M Ikeda ldquoStrict LMI conditions for stabilityrobust stabilization and 119867

infincontrol of descriptor systemsrdquo in

Proceedings of the 38th IEEE Conference on Decision amp Control(CDC rsquo99) vol 4 pp 4092ndash4097 IEEE Phoenix Ariz USADecember 1999

[28] G P Lu and D W C Ho ldquoContinuous stabilization controllersfor singular bilinear systems the state feedback caserdquo Automat-ica vol 42 no 2 pp 309ndash314 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

where Π0(119905) = Π(119905) + (1120576)(Γ

119879

(119905)119873119879

119886+ 119871119879

(119905)119873119879

119887)(119873119886Γ(119905) +

119873119887119871(119905))

We rewrite (45) as

[

[

[

[

[

[

[

Π1(119905) 119866 (119905) (Γ

119879

(119905) 119862119879

(119905) + 119871119879

(119905) 119863119879

(119905)) 119865 (119905)

119866119879

(119905) minus

1

1 + 119889

119868 0

119865119879

(119905) (119862 (119905) Γ (119905) + 119863 (119905) 119871 (119905)) 0 119865119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

]

]

lt 0 (46)

since119870(119905) = 119871(119905)Γ

minus1

(119905) and

Π1(119905) = minus119864

Γ (119905) + Γ

119879

(119905) (119860

119879

(119905) + 119870119879

(119905) 119861

119879

(119905))

+ (119860 (119905) + 119861 (119905)119870 (119905)) Γ (119905) + 120576119872119872119879

+

1

120576

Γ

119879

(119905) (119873119886+ 119873119887119870 (119905))

119879

(119873119886+ 119873119887119870 (119905)) Γ (119905)

+ Γ

119879

(119905) (119862119879

(119905) + 119870119879

(119905) 119863119879

(119905))

sdot (119862 (119905) + 119863 (119905)119870 (119905)) Γ (119905)

(47)

By Lemma 3 we have

Π1(119905) = minus119864

Γ (119905) + Γ

119879

(119905) 119862119879

119888(119905) 119862119888(119905) Γ (119905)

+ Γ

119879

(119905) (119860

119879

(119905) + 119870119879

(119905) 119861

119879

(119905))

+ (119860 (119905) + 119861 (119905)119870 (119905)) Γ (119905)

+ Γ

119879

(119905) (119872Δ (119905)119873119886+119872Δ (119905)119873

119887119870 (119905))

119879

+ (119872Δ (119905)119873119886+119872Δ (119905)119873

119887119870 (119905)) Γ

= minus119864Γ (119905) + Γ

119879

(119905) 119862119879

119888(119905) 119862119888(119905) Γ (119905) + Γ

119879

(119905) 119860119879

119888(119905)

+ 119860119888(119905) Γ (119905)

(48)

Taking Γ(119905) = Γ

minus1

(119905) since Γ119879(119905)119864 = 119864119879

Γ(119905) and 119868 =

Γ(119905)Γ(119905) 0 = Γ(119905)Γ(119905) + Γ(119905)Γ(119905) it is easy to have

minusΓ

minus119879

(119905) 119864Γ (119905) Γ

minus1

(119905) = minus119864119879

Γ (119905)Γ (119905) Γ (119905) = 119864

119879

Γ (119905)

(49)

Pre- and postmultiplying (46) by diag(Γminus119879(119905) 119868 119868) anddiag(Γminus1(119905) 119868 119868) we have

[

[

[

[

[

Π2(119905) Γ

119879

(119905) 119866 (119905) 119862119879

119888(119905) 119865 (119905)

119866119879

(119905) Γ (119905) minus

1

1 + 119889

119868 0

119865119879

(119905) 119862119888(119905) 0 119865

119879

(119905) 119865 (119905) minus 1205742

119868

]

]

]

]

]

lt 0 (50)

where Π2(119905) = 119864Γ(119905) + 119862

119879

119888(119905)119862119888(119905) + Γ

119879

(119905)119860119888(119905) + 119860

119879

119888(119905)Γ(119905)

By Theorem 6 close-loop system is FTB and satisfies 119869 lt119881(1199090 1199050)

For norminal system it is easy to obtain the followingresult

Corollary 9 Given a scalar 120574 gt 0 and a state feedback con-troller 119906 = 119870(119905)119909 the norminal close-loop system is said to beFTBwith respect to (119879 119877(sdot) 119889) and satisfies 119869 = int

119879

1199050

(119910119879

(119905)119910(119905)minus

1205742

120596119879

(119905)120596(119905))119889119905 lt 119881(1199090 1199050) If there exists a nonsingular and

piecewise continuously differential matrix-valued function Γ(sdot)and a continuously differential matrix-valued function 119871(sdot)

such that (50) (42a) and (42c) are fulfilled over [1199050 119879] the

state feedback gain is obtained with 119870(119905) = 119871(119905)Γ

minus1

(119905)

All the conditions in this paper are expressed in terms oftime-varyingDDLMIs However with an appropriate choiceof the structure of the unknown matrix function 119875(119905) it canbe turned into a ldquostandardrdquo LMI problem The unknownmatrix function119875(119905) has been assumed to be piecewise affinethat is

119875 (0) (or (119875 (119905))) = Π0

1

119875 (119905) (or (119875 (119905))) = Π0

119896+ Π119904

119896(119905 minus (119896 minus 1) 119879

119904)

(119905) (or (119875 (119905))) = Π119904

119896

119896 isin 119873 119896 lt 119896 119905 isin [(119896 minus 1) 119879119904 119896119879119904]

119875 (119905) (or (119875 (119905))) = Π0

119896+1

+ Π0

119896+1

(119905 minus 119896119879119904)

(119905) (or (119875 (119905))) = Π0

119896+1

119905 isin [119896119879119904 119879]

(51)

where 119896 = max 119896 isin 119873+ 119896 lt 119879119879119904

Hence the conditions are reduced to a set of LMIs Notethat the conditions are not strict LMI conditions due to119864119879

Γ(119905) = Γ119879

(119905)119864 ge 0 and this may cause a big troublein checking the conditions numerically In order to translatethe nonstrict LMI into strict LMI we can use the followinglemma

8 Mathematical Problems in Engineering

Lemma 10 (see [27 28]) Let 119883 isin 119877119899times119899 be symmetric such

that 119864119879119871119883119864119871gt 0 and let 119879 isin 119877

(119899minus119903)times(119899minus119903) be nonsingular Then119883119864 + 119867

119879

119879119878119879 is nonsingular and its inverse is expressed as

(119883119864 + 119867119879

119879119878119879

)

minus1

= X119864119879

+ 119878T119867 (52)

whereX is symmetric and T is a nonsingular matrix with

119864119879

119877X119864119877= (119864119879

119871119883119864119871)

minus1

T = (119878119879

119878)

minus1

119879minus1

(119867119867119879

)

minus1

(53)

where 119867 and 119878 are any matrix with full row rank and satisfy119872119864 = 0 and 119864119878 = 0 respectively 119864 is decomposed as 119864 =

119864119871119864119879

119877with 119864

119871isin 119877119899times119903 and 119864

119877isin 119877119899times119903 are of full column rank

LetΠ0119896= 1198830

119896119864+119867

119879

1198790

119896119878119879 andΠ119904

119896= 119883119904

119896119864+119867

119879

119879119904

119896119878119879 Using

Lemma 10 we can get (1198830119896119864+119867119879

1198790

119896119878119879

)minus1

= X0119896119864119879

+119878T0119896119867 and

(119883119904

119896119864+119867

119879

119879119904

119896119878119879

)minus1

= X119904119896119864119879

+119878T0119896119867 In this way the condition

119864119879

Γ(119905) = Γ119879

(119905)119864 ge 0 is satisfied so the nonstrict LMIs havebeen translated into strict LMIs Exploiting the MATLABLMI toolbox it is possible to find matrices 119883119904

119896 1198790

119896 119879119904

119896 1198830

119896(or

X0119896X119904119896 T0119896 T 119904119896) such that Γ(119905) (or Γ(119905)) and 119871(119905) verify the

conditions

4 Numerical Example

Consider the linear time-varying descriptor system withjumps defined by

119864 = [

1 0

0 0

] 119860 = [

0 1

2 (119905 + 1) 3 (119905 + 1)2]

119861 = [

1

1

] 119865 = [05 1] 119860119896= [

001 0

0 0

]

119866 = [

1 0

0 1

] 119863 = 1 119862 = [1 1]

119872 = [

01

01

] 119873119886= [02 01] 119873

119887= [01 01]

(54)

Given 120596(119905) = [sin(2120587119905 + 1) cos(2120587119905 minus 2)]119879 Ω = [1525] 119877 = diag(12 1) 120574 = 15 and 120576 = 1654 It is easy tosee

119864119877= 119864119871= [

1

0

] (55)

We choose119867119879 = 119878 = [0 1]119879 Solving the LMIs by the numer-

ical algorithm in the previous section using MATLAB LMItoolbox it is possible to find two piecewise affinematrix func-tions Γ(119905) and 119871(119905) which verify the conditions ofTheorem 8Therefore the following state feedback control law can beobtained (see Figure 1)

15 2 25

0

05

1

15

2

25

3

minus05

minus15

minus1

Figure 1 1198701(119905) (solid line) and 119870

2(119905) (dashed line)

5 Conclusions

In this paper we have formulated and studied the finite-time119867infin

control problem of uncertain time-varying descriptorsystem with jumps at fixed times A sufficient condition forFTB and a sufficient condition for finite-time 119867

infinof the

system have been presented in terms of DLMIs and LMIsThen the state feedback controller has been designed toguarantee the closed-loop systemrsquos FTB and satisfy the 119871

2

gain At last two numerical examples have been used toillustrate the main result

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by the National Nature ScienceFoundation of China (no 61074005) and the Talent Project oftheHighEducation of Liaoning province China underGrantno LR2012005

References

[1] G V Kamenkov ldquoOn stability of motion over a finite intervalof timerdquo Journal of Applied Mathematics and Mechanics vol 17pp 529ndash540 1953

[2] A A Lebedev ldquoOn stability of motion during a given intervalof timerdquo Journal of Applied Mathematics and Mechanics vol 18pp 139ndash148 1954

[3] A A Lebedev ldquoOn the problem of stability of motion overa finite interval of timerdquo Journal of Applied Mathematics andMechanics vol 18 pp 75ndash94 1954

[4] P Dorato ldquoShort time stability in linear time-varying systemsrdquoin Proceedings of the IRE International Convention Record Part4 pp 83ndash87 1961

[5] LWeiss andE F Infante ldquoFinite time stability under perturbingforces and on product spacesrdquo IEEE Transactions on AutomaticControl vol 12 no 1 pp 54ndash59 1967

Mathematical Problems in Engineering 9

[6] F Amato R Ambrosino M Ariola and C Cosentino ldquoFinite-time stability of linear time-varying systems with jumpsrdquoAutomatica vol 45 no 5 pp 1354ndash1358 2009

[7] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoIEEETransactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[8] F Amato M Ariola and C Cosentino ldquoFinite-time stabiliza-tion via dynamic output feedbackrdquo Automatica vol 42 no 2pp 337ndash342 2006

[9] G Garcia S Tarbouriech and J Bernussou ldquoFinite-timestabilization of linear time-varying continuous systemsrdquo IEEETransactions on Automatic Control vol 54 no 2 pp 364ndash3692009

[10] Y Shen ldquoFinite-time control of linear parameter-varying sys-tems with norm-bounded exogenous disturbancerdquo Journal ofControlTheory and Applications vol 6 no 2 pp 184ndash188 2008

[11] L Liu and J T Sun ldquoFinite-time stabilization of linear systemsvia impulsive controlrdquo International Journal of Control vol 81no 6 pp 905ndash909 2008

[12] F Amato M Ariola and C Cosentino ldquoFinite-time control oflinear time-varying systems via output feedbackrdquo in Proceedingsof the American Control Conference (ACC rsquo05) pp 4722ndash4726Portland OR USA June 2005

[13] C-J Wang ldquoControllability and observability of linear time-varying singular systemsrdquo IEEE Transactions on AutomaticControl vol 44 no 10 pp 1901ndash1905 1999

[14] C-J Wang and H-E Liao ldquoImpulse observability and impulsecontrollability of linear time-varying singular systemsrdquo Auto-matica vol 37 no 11 pp 1867ndash1872 2001

[15] N A Kablar ldquoFinite-time stability of time-varying linearsingular systemsrdquo in Proceedings of the 37th IEEE Conference onDecision and Control vol 4 pp 3831ndash3836 Tampa Fla USADecember 1998

[16] NAKablar andD LDebeljkovic ldquoFinite-time stability robust-ness of time-varying linear singular systemsrdquo in Proceedings ofthe 3rd Asian Control Conference Shanghai China July 2000

[17] N A Kablar and D L Debeljkovic ldquoFinite-time instabilityof time-varying linear singular systemsrdquo in Proceedings of theAmerican Control Conference vol 3 pp 1796ndash1800 San DiegoCalif USA June 1999

[18] G Zhang Y Xia and P Shi ldquoNew bounded real lemma fordiscrete-time singular systemsrdquo Automatica vol 44 no 3 pp886ndash890 2008

[19] M Chadli H R Karimi and P Shi ldquoOn stability and stabi-lization of singular uncertain Takagi-Sugeno fuzzy systemsrdquoJournal of the Franklin Institute vol 351 no 3 pp 1453ndash14632014

[20] M Chadli A Abdo and S X Ding ldquoH minus Hinfin fault detectionfilter design for discrete-time Takagi-Sugeno fuzzy systemrdquoAutomatica vol 49 no 7 pp 1996ndash2005 2013

[21] J Yao Z-H Guan G R Chen and D W C Ho ldquoStabilityrobust stabilization and 119867

infincontrol of singular-impulsive

systems via switching controlrdquo Systems amp Control Letters vol55 no 11 pp 879ndash886 2006

[22] J Raouf and E K Boukas ldquoStabilisation of singular Markovianjump systems with discontinuities and saturating inputsrdquo IETControl Theory amp Applications vol 3 no 7 pp 971ndash982 2009

[23] Z-H Guan C W Chan A Y T Leung and G ChenldquoRobust stabilization of singular-impulsive-delayed systemswith nonlinear perturbationsrdquo IEEE Transactions on Circuitsand Systems vol 48 no 8 pp 1011ndash1019 2001

[24] S W Zhao J T Sun and L Liu ldquoFinite-time stability oflinear time-varying singular systems with impulsive effectsrdquoInternational Journal of Control vol 81 no 11 pp 1824ndash18292008

[25] J Xu and J Sun ldquoFinite-time stability of linear time-varyingsingular impulsive systemsrdquo IET ControlTheory amp Applicationsvol 4 no 10 pp 2239ndash2244 2010

[26] L G Wu and D W C Ho ldquoSliding mode control of singularstochastic hybrid systemsrdquo Automatica vol 46 no 4 pp 779ndash783 2010

[27] E Uezato and M Ikeda ldquoStrict LMI conditions for stabilityrobust stabilization and 119867

infincontrol of descriptor systemsrdquo in

Proceedings of the 38th IEEE Conference on Decision amp Control(CDC rsquo99) vol 4 pp 4092ndash4097 IEEE Phoenix Ariz USADecember 1999

[28] G P Lu and D W C Ho ldquoContinuous stabilization controllersfor singular bilinear systems the state feedback caserdquo Automat-ica vol 42 no 2 pp 309ndash314 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Lemma 10 (see [27 28]) Let 119883 isin 119877119899times119899 be symmetric such

that 119864119879119871119883119864119871gt 0 and let 119879 isin 119877

(119899minus119903)times(119899minus119903) be nonsingular Then119883119864 + 119867

119879

119879119878119879 is nonsingular and its inverse is expressed as

(119883119864 + 119867119879

119879119878119879

)

minus1

= X119864119879

+ 119878T119867 (52)

whereX is symmetric and T is a nonsingular matrix with

119864119879

119877X119864119877= (119864119879

119871119883119864119871)

minus1

T = (119878119879

119878)

minus1

119879minus1

(119867119867119879

)

minus1

(53)

where 119867 and 119878 are any matrix with full row rank and satisfy119872119864 = 0 and 119864119878 = 0 respectively 119864 is decomposed as 119864 =

119864119871119864119879

119877with 119864

119871isin 119877119899times119903 and 119864

119877isin 119877119899times119903 are of full column rank

LetΠ0119896= 1198830

119896119864+119867

119879

1198790

119896119878119879 andΠ119904

119896= 119883119904

119896119864+119867

119879

119879119904

119896119878119879 Using

Lemma 10 we can get (1198830119896119864+119867119879

1198790

119896119878119879

)minus1

= X0119896119864119879

+119878T0119896119867 and

(119883119904

119896119864+119867

119879

119879119904

119896119878119879

)minus1

= X119904119896119864119879

+119878T0119896119867 In this way the condition

119864119879

Γ(119905) = Γ119879

(119905)119864 ge 0 is satisfied so the nonstrict LMIs havebeen translated into strict LMIs Exploiting the MATLABLMI toolbox it is possible to find matrices 119883119904

119896 1198790

119896 119879119904

119896 1198830

119896(or

X0119896X119904119896 T0119896 T 119904119896) such that Γ(119905) (or Γ(119905)) and 119871(119905) verify the

conditions

4 Numerical Example

Consider the linear time-varying descriptor system withjumps defined by

119864 = [

1 0

0 0

] 119860 = [

0 1

2 (119905 + 1) 3 (119905 + 1)2]

119861 = [

1

1

] 119865 = [05 1] 119860119896= [

001 0

0 0

]

119866 = [

1 0

0 1

] 119863 = 1 119862 = [1 1]

119872 = [

01

01

] 119873119886= [02 01] 119873

119887= [01 01]

(54)

Given 120596(119905) = [sin(2120587119905 + 1) cos(2120587119905 minus 2)]119879 Ω = [1525] 119877 = diag(12 1) 120574 = 15 and 120576 = 1654 It is easy tosee

119864119877= 119864119871= [

1

0

] (55)

We choose119867119879 = 119878 = [0 1]119879 Solving the LMIs by the numer-

ical algorithm in the previous section using MATLAB LMItoolbox it is possible to find two piecewise affinematrix func-tions Γ(119905) and 119871(119905) which verify the conditions ofTheorem 8Therefore the following state feedback control law can beobtained (see Figure 1)

15 2 25

0

05

1

15

2

25

3

minus05

minus15

minus1

Figure 1 1198701(119905) (solid line) and 119870

2(119905) (dashed line)

5 Conclusions

In this paper we have formulated and studied the finite-time119867infin

control problem of uncertain time-varying descriptorsystem with jumps at fixed times A sufficient condition forFTB and a sufficient condition for finite-time 119867

infinof the

system have been presented in terms of DLMIs and LMIsThen the state feedback controller has been designed toguarantee the closed-loop systemrsquos FTB and satisfy the 119871

2

gain At last two numerical examples have been used toillustrate the main result

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This project is supported by the National Nature ScienceFoundation of China (no 61074005) and the Talent Project oftheHighEducation of Liaoning province China underGrantno LR2012005

References

[1] G V Kamenkov ldquoOn stability of motion over a finite intervalof timerdquo Journal of Applied Mathematics and Mechanics vol 17pp 529ndash540 1953

[2] A A Lebedev ldquoOn stability of motion during a given intervalof timerdquo Journal of Applied Mathematics and Mechanics vol 18pp 139ndash148 1954

[3] A A Lebedev ldquoOn the problem of stability of motion overa finite interval of timerdquo Journal of Applied Mathematics andMechanics vol 18 pp 75ndash94 1954

[4] P Dorato ldquoShort time stability in linear time-varying systemsrdquoin Proceedings of the IRE International Convention Record Part4 pp 83ndash87 1961

[5] LWeiss andE F Infante ldquoFinite time stability under perturbingforces and on product spacesrdquo IEEE Transactions on AutomaticControl vol 12 no 1 pp 54ndash59 1967

Mathematical Problems in Engineering 9

[6] F Amato R Ambrosino M Ariola and C Cosentino ldquoFinite-time stability of linear time-varying systems with jumpsrdquoAutomatica vol 45 no 5 pp 1354ndash1358 2009

[7] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoIEEETransactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[8] F Amato M Ariola and C Cosentino ldquoFinite-time stabiliza-tion via dynamic output feedbackrdquo Automatica vol 42 no 2pp 337ndash342 2006

[9] G Garcia S Tarbouriech and J Bernussou ldquoFinite-timestabilization of linear time-varying continuous systemsrdquo IEEETransactions on Automatic Control vol 54 no 2 pp 364ndash3692009

[10] Y Shen ldquoFinite-time control of linear parameter-varying sys-tems with norm-bounded exogenous disturbancerdquo Journal ofControlTheory and Applications vol 6 no 2 pp 184ndash188 2008

[11] L Liu and J T Sun ldquoFinite-time stabilization of linear systemsvia impulsive controlrdquo International Journal of Control vol 81no 6 pp 905ndash909 2008

[12] F Amato M Ariola and C Cosentino ldquoFinite-time control oflinear time-varying systems via output feedbackrdquo in Proceedingsof the American Control Conference (ACC rsquo05) pp 4722ndash4726Portland OR USA June 2005

[13] C-J Wang ldquoControllability and observability of linear time-varying singular systemsrdquo IEEE Transactions on AutomaticControl vol 44 no 10 pp 1901ndash1905 1999

[14] C-J Wang and H-E Liao ldquoImpulse observability and impulsecontrollability of linear time-varying singular systemsrdquo Auto-matica vol 37 no 11 pp 1867ndash1872 2001

[15] N A Kablar ldquoFinite-time stability of time-varying linearsingular systemsrdquo in Proceedings of the 37th IEEE Conference onDecision and Control vol 4 pp 3831ndash3836 Tampa Fla USADecember 1998

[16] NAKablar andD LDebeljkovic ldquoFinite-time stability robust-ness of time-varying linear singular systemsrdquo in Proceedings ofthe 3rd Asian Control Conference Shanghai China July 2000

[17] N A Kablar and D L Debeljkovic ldquoFinite-time instabilityof time-varying linear singular systemsrdquo in Proceedings of theAmerican Control Conference vol 3 pp 1796ndash1800 San DiegoCalif USA June 1999

[18] G Zhang Y Xia and P Shi ldquoNew bounded real lemma fordiscrete-time singular systemsrdquo Automatica vol 44 no 3 pp886ndash890 2008

[19] M Chadli H R Karimi and P Shi ldquoOn stability and stabi-lization of singular uncertain Takagi-Sugeno fuzzy systemsrdquoJournal of the Franklin Institute vol 351 no 3 pp 1453ndash14632014

[20] M Chadli A Abdo and S X Ding ldquoH minus Hinfin fault detectionfilter design for discrete-time Takagi-Sugeno fuzzy systemrdquoAutomatica vol 49 no 7 pp 1996ndash2005 2013

[21] J Yao Z-H Guan G R Chen and D W C Ho ldquoStabilityrobust stabilization and 119867

infincontrol of singular-impulsive

systems via switching controlrdquo Systems amp Control Letters vol55 no 11 pp 879ndash886 2006

[22] J Raouf and E K Boukas ldquoStabilisation of singular Markovianjump systems with discontinuities and saturating inputsrdquo IETControl Theory amp Applications vol 3 no 7 pp 971ndash982 2009

[23] Z-H Guan C W Chan A Y T Leung and G ChenldquoRobust stabilization of singular-impulsive-delayed systemswith nonlinear perturbationsrdquo IEEE Transactions on Circuitsand Systems vol 48 no 8 pp 1011ndash1019 2001

[24] S W Zhao J T Sun and L Liu ldquoFinite-time stability oflinear time-varying singular systems with impulsive effectsrdquoInternational Journal of Control vol 81 no 11 pp 1824ndash18292008

[25] J Xu and J Sun ldquoFinite-time stability of linear time-varyingsingular impulsive systemsrdquo IET ControlTheory amp Applicationsvol 4 no 10 pp 2239ndash2244 2010

[26] L G Wu and D W C Ho ldquoSliding mode control of singularstochastic hybrid systemsrdquo Automatica vol 46 no 4 pp 779ndash783 2010

[27] E Uezato and M Ikeda ldquoStrict LMI conditions for stabilityrobust stabilization and 119867

infincontrol of descriptor systemsrdquo in

Proceedings of the 38th IEEE Conference on Decision amp Control(CDC rsquo99) vol 4 pp 4092ndash4097 IEEE Phoenix Ariz USADecember 1999

[28] G P Lu and D W C Ho ldquoContinuous stabilization controllersfor singular bilinear systems the state feedback caserdquo Automat-ica vol 42 no 2 pp 309ndash314 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

[6] F Amato R Ambrosino M Ariola and C Cosentino ldquoFinite-time stability of linear time-varying systems with jumpsrdquoAutomatica vol 45 no 5 pp 1354ndash1358 2009

[7] F Amato M Ariola and C Cosentino ldquoFinite-time stabilityof linear time-varying systems analysis and controller designrdquoIEEETransactions onAutomatic Control vol 55 no 4 pp 1003ndash1008 2010

[8] F Amato M Ariola and C Cosentino ldquoFinite-time stabiliza-tion via dynamic output feedbackrdquo Automatica vol 42 no 2pp 337ndash342 2006

[9] G Garcia S Tarbouriech and J Bernussou ldquoFinite-timestabilization of linear time-varying continuous systemsrdquo IEEETransactions on Automatic Control vol 54 no 2 pp 364ndash3692009

[10] Y Shen ldquoFinite-time control of linear parameter-varying sys-tems with norm-bounded exogenous disturbancerdquo Journal ofControlTheory and Applications vol 6 no 2 pp 184ndash188 2008

[11] L Liu and J T Sun ldquoFinite-time stabilization of linear systemsvia impulsive controlrdquo International Journal of Control vol 81no 6 pp 905ndash909 2008

[12] F Amato M Ariola and C Cosentino ldquoFinite-time control oflinear time-varying systems via output feedbackrdquo in Proceedingsof the American Control Conference (ACC rsquo05) pp 4722ndash4726Portland OR USA June 2005

[13] C-J Wang ldquoControllability and observability of linear time-varying singular systemsrdquo IEEE Transactions on AutomaticControl vol 44 no 10 pp 1901ndash1905 1999

[14] C-J Wang and H-E Liao ldquoImpulse observability and impulsecontrollability of linear time-varying singular systemsrdquo Auto-matica vol 37 no 11 pp 1867ndash1872 2001

[15] N A Kablar ldquoFinite-time stability of time-varying linearsingular systemsrdquo in Proceedings of the 37th IEEE Conference onDecision and Control vol 4 pp 3831ndash3836 Tampa Fla USADecember 1998

[16] NAKablar andD LDebeljkovic ldquoFinite-time stability robust-ness of time-varying linear singular systemsrdquo in Proceedings ofthe 3rd Asian Control Conference Shanghai China July 2000

[17] N A Kablar and D L Debeljkovic ldquoFinite-time instabilityof time-varying linear singular systemsrdquo in Proceedings of theAmerican Control Conference vol 3 pp 1796ndash1800 San DiegoCalif USA June 1999

[18] G Zhang Y Xia and P Shi ldquoNew bounded real lemma fordiscrete-time singular systemsrdquo Automatica vol 44 no 3 pp886ndash890 2008

[19] M Chadli H R Karimi and P Shi ldquoOn stability and stabi-lization of singular uncertain Takagi-Sugeno fuzzy systemsrdquoJournal of the Franklin Institute vol 351 no 3 pp 1453ndash14632014

[20] M Chadli A Abdo and S X Ding ldquoH minus Hinfin fault detectionfilter design for discrete-time Takagi-Sugeno fuzzy systemrdquoAutomatica vol 49 no 7 pp 1996ndash2005 2013

[21] J Yao Z-H Guan G R Chen and D W C Ho ldquoStabilityrobust stabilization and 119867

infincontrol of singular-impulsive

systems via switching controlrdquo Systems amp Control Letters vol55 no 11 pp 879ndash886 2006

[22] J Raouf and E K Boukas ldquoStabilisation of singular Markovianjump systems with discontinuities and saturating inputsrdquo IETControl Theory amp Applications vol 3 no 7 pp 971ndash982 2009

[23] Z-H Guan C W Chan A Y T Leung and G ChenldquoRobust stabilization of singular-impulsive-delayed systemswith nonlinear perturbationsrdquo IEEE Transactions on Circuitsand Systems vol 48 no 8 pp 1011ndash1019 2001

[24] S W Zhao J T Sun and L Liu ldquoFinite-time stability oflinear time-varying singular systems with impulsive effectsrdquoInternational Journal of Control vol 81 no 11 pp 1824ndash18292008

[25] J Xu and J Sun ldquoFinite-time stability of linear time-varyingsingular impulsive systemsrdquo IET ControlTheory amp Applicationsvol 4 no 10 pp 2239ndash2244 2010

[26] L G Wu and D W C Ho ldquoSliding mode control of singularstochastic hybrid systemsrdquo Automatica vol 46 no 4 pp 779ndash783 2010

[27] E Uezato and M Ikeda ldquoStrict LMI conditions for stabilityrobust stabilization and 119867

infincontrol of descriptor systemsrdquo in

Proceedings of the 38th IEEE Conference on Decision amp Control(CDC rsquo99) vol 4 pp 4092ndash4097 IEEE Phoenix Ariz USADecember 1999

[28] G P Lu and D W C Ho ldquoContinuous stabilization controllersfor singular bilinear systems the state feedback caserdquo Automat-ica vol 42 no 2 pp 309ndash314 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of