research article robust finite-time control for...
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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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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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
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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
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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
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
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
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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