Fuzzy Control for Nonlinear Singular Markovian JumpSystems with Time Delay
Li Li12 Qingling Zhang2 Yi Zhang3 and Baoyan Zhu4
1College of Mathematics and Physics Bohai University Jinzhou Liaoning 121013 China2Institute of Systems Science Northeastern University Shenyang Liaoning 110819 China3School of Science Shenyang University of Technology Shenyang Liaoning 110870 China4College of Science Shenyang Jianzhu University Shenyang Liaoning 110168 China
Correspondence should be addressed to Qingling Zhang qlzhangmailneueducn
Received 7 January 2015 Accepted 20 April 2015
Academic Editor P Balasubramaniam
Copyright copy 2015 Li Li et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper investigates the problem of119867infinfuzzy control for a class of nonlinear singular Markovian jump systems with time delay
This class of systems under consideration is described by Takagi-Sugeno (T-S) fuzzy models Firstly sufficient condition of thestochastic stabilization by the method of the augmented matrix is obtained by the state feedback And a designed algorithm for thestate feedback controller is provided to guarantee that the closed-loop system not only is regular impulse-free and stochasticallystable but also satisfies a prescribed 119867
infinperformance for all delays not larger than a given upper bound in terms of linear matrix
inequalities Then 119867infin
fuzzy control for this kind of systems is also discussed by the static output feedback Finally numericalexamples are given to illustrate the validity of the developed methodology
1 Introduction
Singular systems also known as descriptor systems havebeen widely studied in the past several decades They havebroad applications and can be found in many practicalsystems such as electrical circuits power systems networkeconomics and other systems [1 2] Due to their extensiveapplications many research topics on singular systems havebeen extensively investigated such as the stability and sta-bilization [3 4] and 119867
infincontrol problem [5 6] A lot of
attention has been paid to the investigation of Markovianjump systems (MJSs) over the past decades Applicationsof such class of systems can be found representing manyphysical systems with random changes in their structuresand parameters Many important issues have been studiedfor this kind of physical systems such as the stabilityanalysis stabilization and119867
infincontrol [7ndash10]When singular
systems experience abrupt changes in their structures it isnatural to model them as singular Markovian jump systems(SMJSs) [11ndash13] Time delay is one of the instability sourcesfor dynamical systems and is a common phenomenon in
many industrial and engineering systems such as those incommunication networks manufacturing and biology [14]So the study of SMJSs with time delay is of theoretical andpractical importance [15 16]
The fuzzy control has been proved to be a powerfulmethod for the control problem of complex nonlinear sys-tems Specially the Takagi-Sugeno (T-S) fuzzy model hasattracted much attention due to the fact that it providesan efficient approach to take full advantage of the linearcontrol theory to the nonlinear control In recent years thisfuzzy-model-based technique has been used to deal withnonlinear time delay systems [17 18] and nonlinear MJSs [1920] But singular Markovian jump fuzzy systems (SMJFSs)are not fully studied [21 22] which motivates the mainpurpose of our study In this paper a new method using theaugmented matrix will be given to the control of SMJFSsBy this method the number of LMIs will be decreased sothe complexity of the calculation will be greatly reducedwhen the number of fuzzy rulers is relatively large Andat the same time some new relaxation matrices added willreduce the conservation of control conditions compared with
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 896515 13 pageshttpdxdoiorg1011552015896515
2 Mathematical Problems in Engineering
previous literatures And when using the augmented matrixto design the static output feedback control there are notany crossing terms between system matrices and controllergains so assumptions for the output matrix [23] the equalityconstraint for the output matrix [24] and the boundingtechnique for crossing terms are not necessary therefore theconservatism brought by them will not exist
In this paper the119867infinfuzzy control problem for a class of
nonlinear SMJSs with time delay which can be representedby T-S fuzzy models is considered Our aim is to designfuzzy state feedback controllers and static output feedbackcontrollers for SMJFSs with time delay such that closed-loopsystems are stochastically admissible (regular impulse-freeand stochastically stable) with a prescribed119867
infinperformance
120574 Sufficient criterions are presented in forms of LMIs whichare simple and easy to implement compared with previousliteratures Finally numerical examples are given to illustratethemerit andusability of the approach proposed in this paper
Notations Throughout this paper notations used are fairlystandard for real symmetric matrices 119860 and 119861 the notation119860 ge 119861 (119860 gt 119861) means that the matrix 119860 minus 119861 is positivesemidefinite (positive definite) 119860119879 represents the transposeof the matrix 119860 and 119860minus1 represents the inverse of the matrix119860 120582max119861 (120582min119861) is themaximal (minimal) eigenvalue of thematrix 119861 diagsdot stands for a block-diagonal matrix 119868 is theunit matrix with appropriate dimensions and in a matrixthe term of symmetry is stated by the asterisk ldquolowastrdquo Let R119899
stand for the 119899-dimensional Euclidean space R119899times119898 is the setof all 119899 times 119898 real matrices and sdot denotes the Euclideannorm of vectors Esdot denotes the mathematics expectationof the stochastic process or vector 119871119899
2[0infin) stands for the
space of 119899-dimensional square integrable functions on [0infin)119862119899119889
= 119862([minus119889 0]R119899
) denotes Banach space of continuousvector functions mapping the interval [minus119889 0] into R119899 withthe norm 120601
119889= sup
minus119889le119904le0120601(119904)
2 Basic Definitions and Lemmas
Consider a SMJFS its 119894th fuzzy rule is given by
119877119894 if 1205851(119905) is1198721198941 1205852(119905) is1198721198942 and 120585119897(119905) is119872119894119897
where 119909(119905) isin R119899 is the state vector 119906(119905) isin R119898 is the controlinput119908(119905) isin RV is the exogenous disturbance which belongsto 119871V
2[0infin) and 119911(119905) isin R119901 is the controlled output 120601(119905) isin
119862119899119889
is a compatible vector-valued initial function and 119889 is anunknown but constant delay satisfying 119889 isin [0 119889] The scalar
119896 is the number of If-Then rules119872119894119895(119894 isin T 119895 = 1 2 119897)
are fuzzy sets 1205851(119905) minus 120585
119897(119905) are premise variables 119864 isin R119899times119899
may be a singular matrix with rank119864 = 119903 le 119899 119860119894(119903
119905)
119860119889119894(119903
119905) 119861
119894(119903
119905) 119861
119908119894(119903
119905) 119862
119894(119903
119905) 119862
119889119894(119903
119905) 119863
119894(119903
119905) and 119862
119908119894(119903
119905)
are known constant matrices with appropriate dimensions119903
119905 119905 ge 0 is a continuous-time Markovian process with right
continuous trajectories taking values in a finite set given byS = 1 2 119873 with the transition rate matrix Π ≜ 120587
Lemma 5 (see [29]) Suppose there are piecewise continuousreal square matrices 119860(119905) 119883 and 119876 gt 0 satisfying 119860119879
(119905)119883 +
119883119879
119860(119905) lt 0 for all 119905 Then the following conditions hold(i) 119860(119905) and 119883 are nonsingular(ii) 119860minus1
(119905) le 120575 for some 120575 gt 0
Lemma 6 (see [30]) If the following conditions hold
119872119894119894lt 0 1 le 119894 le 119903
1
119903 minus 1
119872119894119894+
1
2
(119872119894119895+119872
119895119894) lt 0 1 le 119894 = 119895 le 119903
(8)
then the following parameterized matrix inequality holds119903
sum
119894=1
119903
sum
119895=1
120572119894(119905) 120572
119895(119905)119872
119894119895lt 0 (9)
where 120572119894(119905) ge 0 and sum119903
119894=1120572119894(119905) = 1
Based on the parallel distributed compensation thefollowing state feedback controller will be considered here
119906119901(119905) =
119896
sum
119894=1
120582119894119870
119901119894119909 (119905) (10)
where119870119901119894(119901 isin S 119894 isin T) are local controller gains such that
the closed-loop system
119864 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895((119860
119901119894+ 119861
119901119894119870
119901119895) 119909 (119905)
+ 119860119889119901119894119909 (119905 minus 119889) + 119861
119908119901119894119908 (119905))
119911 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895((119862
119901119894+ 119863
119901119894119870
119901119895) 119909 (119905)
+ 119862119889119901119894119909 (119905 minus 119889) + 119862
119908119901119894119908 (119905))
119909 (119905) = 120601 (119905)
119905 isin [minus119889 0]
(11)
is stochastically admissible
3 The Design of the StateFeedback 119867
infinController
Firstly the sufficient condition will be given such thatsystem (11) is stochastically admissible Combining (4) and(10) fuzzy closed-loop system (11) can be rewritten in thefollowing form
119864 (119905)
=
119896
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119908119901119894119908 (119905))
119911 (119905) =
119896
sum
119894=1
120582119894(119862
119901119894119909 (119905) + 119862
119889119901119894119909 (119905 minus 119889) + 119862
119908119901119894119908 (119905))
119909 (119905) =120601 (119905)
119905 isin [minus119889 0]
(12)
where
119864 = [
119864 0
0 0
] isin R(119899+119898)times(119899+119898)
119860119901119894= [
119860119901119894
119861119901119894
119870119901119894minus119868
119898
] isin R(119899+119898)times(119899+119898)
119909 (119905) = [
119909 (119905)
119906 (119905)
] isin R119899+119898
119860119889119901119894
= [
119860119889119901119894
0
0 0
] isin R(119899+119898)times(119899+119898)
119861119908119901119894
= [
119861119908119901119894
0
] isin R(119899+119898)timesV
119862119901119894= [119862
119901119894119863
119901119894] isin R
119901times(119899+119898)
119862119889119901119894
= [119862119889119901119894
0] isin R119901times(119899+119898)
120601 (119905) =
119896
sum
119894=1
120582119894[
120601 (119905)
119870119901119894120601 (119905)
]
(13)
4 Mathematical Problems in Engineering
Remark 7 For systems (11) and (12) it can be seen that
det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(119860
119901119894+ 119861
119901119894119870
119901119895))
= det(119904[119864 0
0 0
] minus
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895[
119860119901119894
119861119901119894
119870119901119895minus119868
119898
])
= det(119904[119864 0
0 0
] minus
119896
sum
119894=1
120582119894[
119860119901119894
119861119901119894
119870119901119894minus119868
119898
])
= det(119904119864 minus119896
sum
119894=1
120582119894119860
119901119894)
det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(119860
119901119894+ 119861
119901119894119870
119901119895+ 119860
119889119901119894))
= det(119904[119864 0
0 0
] minus
119896
sum
119894=1
120582119894[
119860119901119894+ 119860
119889119901119894119861119901119894
119870119901119894
minus119868119898
])
= det(119904119864 minus119896
sum
119894=1
120582119894(119860
119901119894+ 119860
119889119901119894))
(14)
By rank119864 = rank119864 and Definition 1 it can be obtained thatthe regularity and nonimpulse of system (11) are equal to theregularity and nonimpulse of system (12) So the stochasticadmissibility of system (11) can be studied by system (12)
Theorem 8 For a prescribed scalar 119889 gt 0 there exists a statefeedback controller (10) with 119906
that system (11) when 119908(119905) = 0 is stochastically admissible forany constant time delay 119889 satisfying 119889 isin [0 119889] if there existmatrices 119875
119889 diag119876 0 minus 119864119879 diag119885 0119864 Pre- and postmultiplying bydiag119883119879
119901 119868
119903 119868
119903⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
119873minus1
and its transposition by Schur comple-
ment lemma it can be seen that
119873
sum
119902=1
120587119901119902119876
119902lt 119876 (21)
From (20) it can be concluded that
119896
sum
119894=1
120582119894(120587
119901119901119864119879
119883119901+ 119860
119879
119901119894119883
119901+ 119883
119879
119901119860
119901119894
minus 119864119879 diag 119885 0 119864) lt 0
(22)
On the other hand diag119866 119868119898119864 diag119867 119868
119898 = [
[1198681199030
0 0
] 0
0 0119898
]Then
119864119879
119883119901= 119883
119879
119901119864 = [
119864119879
0
0 0
][
119883119901
0
1198831199012119883
1199013
] = [
119864119879
1198831199010
0 0
]
= [
119864119879
119875119901119864 0
0 0
] ge 0
(23)
Denote 119860119901(119905) ≜ (sum
119896
119894=1120582119894119860
119901119894) = [
1198601199011(119905)
1198601199012(119905)
1198601199013(119905)
1198601199014(119905)
] from (22) itcan be obtained that
119883119879
1199013119860
1199014(119905) + 119860
119879
1199014(119905) 119883
1199013lt 0 (24)
for every 119901 isin S which implies that 1198601199014(119905) is nonsingular
Thus the pair (119864 sum119896
119894=1120582119894119860
119901119894) is regular and impulse-free for
every 119901 isin S By (20) it is easy to see that
[
Γ1119901119894
lowast
119864119879
[119885 0] 119864 + [119860119879
1198891199011198940]119883
119901minus119864
119879
119885119864 minus 119876119901
] lt 0 (25)
Pre- and postmultiplying (25) by [ 119868119899+119898
[119868119899
0
]
0 119868119899
] and its trans-pose it can be obtained that
119873
sum
119902=1
120587119901119902119864119879
119883119901+ 119883
119879
119901(119860
119901119894+ 119860
119889119901119894)
+ (119860119901119894+ 119860
119889119901119894)
119879
119883119901lt 0
(26)
Hence
119896
sum
119894=1
120582119894(120587
119901119901119864119879
119883119901+ 119883
119879
119901(119860
119901119894+ 119860
119889119901119894)
+ (119860119901119894+ 119860
119889119901119894)
119879
119883119901) lt 0
(27)
Equation (27) implies that the pair (119864 sum119896
119894=1120582119894(119860
119901119894+ 119860
119889119901119894))
is regular and impulse-free for every 119901 isin S Thus byDefinition 1 system (12) is regular and impulse-free ByRemark 7 this implies that system (11) is regular and impulse-free
Now it will be shown that system (11) is stochasticallystable Define a new process (119909
119905 119903
119905) 119905 ge 0 by 119909
119905= 119909(119905 +
120579) minus2119889 le 120579 le 0 then (119909119905 119903
119905) 119905 ge 119889 is a Markovian process
with the initial state (120601(sdot) 1199030) Now for 119905 ge 119889 choose the
following stochastic Lyapunov-Krasovskii candidate for thissystem
119881 (119909119905 119901 119905) =
4
sum
119898=1
119881119898(119909
119905 119901 119905) (28)
where
1198811(119909
119905 119901 119905) = 119909
119879
(119905) 119864119879
119875119901119864119909 (119905) = 119909
119879
(119905) 119864119879
119883119901119909 (119905)
= 119909119879
(119905) 119864119879
119883119901119909 (119905)
1198812(119909
119905 119901 119905) = int
119905
119905minus119889
119909119879
(119904) 119876119901119909 (119904) d119904
1198813(119909
119905 119901 119905) = 119889int
0
minus119889
int
119905
119905+120579
119879
(119904) 119864119879
119885119864 (119904) d119904 d120579
1198814(119909
119905 119901 119905) = int
0
minus119889
int
119905
119905+120579
119909119879
(119904) 119876119909 (119904) d119904 d120579
(29)
6 Mathematical Problems in Engineering
Let L be the weak infinitesimal generator of the randomprocess (119909
119905 119901) 119905 ge 0 Then for each 119901 isin S
L119881 (119909119905 119901 119905) le 2119909
119879
(119905) 119883119879
119901119864 (119905)
+ 119909119879
(119905) (
119873
sum
119902=1
120587119901119902119864119879
119883119902)119909 (119905)
+ 119909119879
(119905) 119876119901119909 (119905)
minus 119909119879
(119905 minus 119889)119876119901119909 (119905 minus 119889)
+ int
119905
119905minus119889
119909119879
(119904) (
119873
sum
119902=1
120587119901119902119876
119902)119909 (119904) d119904
+ 119889119909119879
(119905) 119876119909 (119905)
minus int
119905
119905minus119889
119909119879
(119904) 119876119909 (119904) d119904
+ 119889
2
119879
(119905) 119864119879
119885119864 (119905)
minus 119889int
119905
119905minus119889
119879
(119904) 119864119879
119885119864 (119904) d119904
(30)
From (21) it is clear that
int
119905
119905minus119889
119909119879
(119904) (
119873
sum
119902=1
120587119901119902119876
119902)119909 (119904) d119904
lt int
119905
119905minus119889
119909119879
(119904) 119876119909 (119904) d119904
(31)
From Lemma 4 it follows that
minus 119889int
119905
119905minus119889
119879
(119904) 119864119879
119885119864 (119904) d119904
le [
119909 (119905)
119909 (119905 minus 119889)
]
119879
[
minus119864119879
119885119864 119864119879
119885119864
119864119879
119885119864 minus119864119879
119885119864
][
119909 (119905)
119909 (119905 minus 119889)
]
(32)
So it can be concluded that
L119881 (119909119905 119901 119905) le
119896
sum
119894=1
120578119879
(119905) Φ119901119894120578 (119905) (33)
where
120578119879
(119905) = [119909119879
(119905) 119909119879
(119905 minus 119889)]
Φ119901119894= [
Υ1119901119894
lowast
Υ2119901119894
Υ3119901119894
]
Υ1119901119894= Γ
1119901119894+ [
119860119879
119901119894
119861119879
119901119894
]119889
2
119885 [119860119901119894119861119901119894]
Υ3119901119894= minus119876
119901minus 119864
119879
119885119864 + 119860119879
119889119901119894119889
2
119885119860119889119901119894
Υ2119901119894= 119864
119879
[119885 0] 119864 + [119860119879
1198891199011198940]119883
119901
+ 119860119879
119889119901119894119889
2
119885 [119860119901119894119861119901119894]
(34)
Using (20) it is easy to see that there exists a scalar 120581 gt 0
such that for every 119901 isin S L119881(119909119905 119901 119905) le minus120581119909(119905)
This together with (36) and (48) implies that there exists ascalar ] such that
Eint119905
0
119909 (119904)2 d119904 = Eint
119889
0
119909 (119904)2 d119904
+Eint119905
119889
119909 (119904)2 d119904
le ]E 10038171003817100381710038171206011003817100381710038171003817
2
119889
(51)
Considering this andDefinition 1 system (11) is stochasticallystable for any constant delay 119889 satisfying 119889 isin [0 119889]Thereforesystem (11) is stochastically admissible This completes theproof
In the following a set of sufficient conditions will bedeveloped under which the considered system is guaranteedto be stochastically admissible with an119867
infinperformance
Definition 9 System (11) is said to be stochastically admissiblewith an 119867
infinperformance 120574 if it is stochastically admissible
when 119908(119905) = 0 and under zero initial condition for nonzero119908(119905) isin 119871
V2[0infin)
Eintinfin
0
119911119879
(119905) 119911 (119905) d119905 le 1205742 intinfin
0
119908119879
(119905) 119908 (119905) d119905 (52)
The following result can be presented
8 Mathematical Problems in Engineering
Theorem 10 For a prescribed scalar 119889 gt 0 there exists astate feedback controller (10) with 119906
and the other notations are the same as in Theorem 8
Proof FromTheorem 8 when119908(119905) = 0 system (11) is stochas-tically admissible Let
119869119911119908(119905) = Eint
119905
0
[119911119879
(119904) 119911 (119904) minus 1205742
119908119879
(119904) 119908 (119904)] d119904 (55)
Under zero initial condition it is easy to see that
119869119911119908(119905) le Eint
119905
0
[119911119879
(119904) 119911 (119904) minus 1205742
119908119879
(119904) 119908 (119904)
+L119881 (119909119904 119901 119904)] d119904
le Eint119905
0
119896
sum
119894=1
120582119894[120589
119879
(119904) (Ω119901119894+ Θ
119879
119901119894Θ
119901119894) 120589 (119904)] d119904
(56)
where
120589119879
(119905) = [119909119879
(119905) 119909119879
(119905 minus 119889) 119908119879
(119905)]
Ω119901119894=
[
[
[
[
Υ1119901119894
lowast lowast
Υ2119901119894
Υ3119901119894
lowast
[119861119879
1199081199011198940]119883
1199010 minus120574
2
119868
]
]
]
]
Θ119901119894= [119862
119901119894119862119889119901119894
119862119908119901119894]
(57)
and notations of Υ1119901119894
Υ2119901119894
and Υ3119901119894
are the same as inTheorem 8 Hence by Schur complement lemma and using
the similar method in the proof of Theorem 8 from and(53) it can be obtained that 119869
119911119908(119905) lt 0 for all 119905 gt 0
Therefore for any nonzero 119908(119905) isin 119871V2[0infin) (52) holds
Hence according to Definition 9 the system is stochasticallyadmissible with an 119867
infinperformance 120574 This completes the
proof
Remark 11 Compared with methods in [21 22] because ofthe method of the augmented matrix adopted in Theorems8 and 10 the number of LMIs needed to solve is relativelysmall in this paper When the value of 119896 is relatively largethe quality of the computation is greatly reduced some newrelaxation matrices added will reduce the conservatism ofcontrol conditions compared with previous literatures whichcan be seen from Example 2
4 The Design of the Static OutputFeedback Controller
When 119903119905= 119901 isin S consider the overall SMJFS as follows
119864 (119905) =
119896
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119889119901119894119909 (119905 minus 119889) + 119862
119908119901119894119908 (119905))
(60)
It is difficult to drive LMI-based conditions of the stochas-tic stabilization by employing the static output feedbackcontrol approach due to the appearance of crossing termsbetween system matrices and control gains And system (60)can be rewritten in the following form
119864119909 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(Λ
119901119894119895119909 (119905) + 119861
119908119901119894119908 (119905))
119911 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(119862
119901119894119895119909 (119905) + 119862
119908119901119894119908 (119905))
(61)
where
119864 = [
119864 0
0 0
] isin R(119899+1199011)times(119899+119901
1)
119862119901119894119895= [119862
119901119894119863
119901119894119870
119901119895]
119860119889119901119894
= [
119860119889119901119894
0
0 0
] isin R(119899+1199011)times(119899+119901
1)
119909 (119905) = [
119909 (119905)
119910 (119905)
]
119861119908119901119894
= [
119861119908119901119894
0
] isin R(119899+1199011)timesV
Λ119901119894119895= [
119860119901119894
119861119901119894119870
119901119895
119862119910119901119894
minus119868
]
(62)
Remark 12 For systems (60) and (61) it can be seen that
det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
119896
sum
119904=1
120582119894120582119895120582119904(119860
119901119894+ 119861
119901119894119870
119901119895119862119910119901119904))
= det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895Λ
119901119894119895)
det(119904119864
minus
119896
sum
119894=1
119896
sum
119895=1
119896
sum
119904=1
120582119894120582119895120582119904(119860
119901119894+ 119861
119901119894119870
119901119895119862119910119901119904
+ 119860119889119901119894))
= det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(Λ
119901119894119895+ 119860
119889119901119894))
(63)
As the discussion in Remark 7 the stochastic admissibility ofsystem (60) can be studied by means of system (61)
Theorem 13 There exists an output feedback controller (59)with controller gains 119870
119901119894= 119871
119901119894119884minus1
1199012(119901 isin S 119894 isin T) such that
system (60) with 119908(119905) = 0 is stochastically admissible if thereexist matrices 119875
119901gt 0 119876
119901gt 0 119876 gt 0 119885 gt 0 119871
119901119894 119878
119901 and 119884
1199012
119901 isin S 1 le 119894 = 119895 le 119896 such that and
and the other notations are the same as in Theorem 13
Remark 15 Compared with the method in [31 32] becauseof the augmented matrix adopted in Theorems 13 and 14 thenumber of LMIs needed to solve is greatly decreased Whenthe value of 119896 is relatively large the computational complexitywill be reduced On the other hand by the augmentedmatrixthere are not any crossing terms between system matricesand controller gains so assumptions for the output matrix
[23] the equality constraint for the output matrix [24] andthe bounding technique for crossing terms are not necessaryhere therefore the conservatism brought by them will nothappen
5 Numerical Examples
Two examples will be given to illustrate the validity ofdeveloped methods
Mathematical Problems in Engineering 11
Example 1 To illustrate the 119867infin
controller synthesis thefollowing nonlinear time delay system is considered
(1 + 119886 cos 120579 (119905)) 120579 (119905) = minus119887 1205793 (119905) + 119888120579 (119905) + 119888119889(119905 minus 119889)
It can be expressed exactly by the following fuzzy singularMarkovian jump form
119864 (119905) =
3
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119901119894119906 (119905)
+ 119861119908119901119894119908 (119905))
119911 (119905) =
3
sum
119894=1
120582119894119862119901119894119909 (119905)
119909 (119905) = 120601 (119905)
119905 isin [minus119889 0] 119901 isin 1 2 3
(70)
where
119864 =[
[
[
1 0 0
0 1 0
0 0 0
]
]
]
1198601199011=
[
[
[
[
0 1 0
0 0 1
119888 minus119887 (1205952
+ 2) 119886 minus 1
]
]
]
]
1198601199012=
[
[
[
[
0 1 0
0 0 1
119888 0 minus119886 minus 1 minus 1198861205952
]
]
]
]
1198601199013=[
[
[
0 1 0
0 0 1
119888 0 119886 minus 1
]
]
]
1198601198891199011
= 1198601198891199012
= 1198601198891199013
=[
[
[
0 0 0
0 0 0
1198881198890 0
]
]
]
11986111= 119861
12= 119861
13=[
[
[
0
0
119890
]
]
]
11986121= 119861
22= 119861
23=[
[
[
0
0
08119890
]
]
]
11986131= 119861
32= 119861
33=[
[
[
0
0
05119890
]
]
]
1198611199081199011
= 1198611199081199012
= 1198611199081199013
=[
[
[
0
0
119891
]
]
]
1198621199011= 119862
1199012= 119862
1199013= [1 0 0]
1205821=
1199092
2(119905)
1205952+ 2
1205822=
1 + cos1199091(119905)
1205952+ 2
1205823=
1205952
minus 1199092
2(119905) + 1 minus cos119909
1(119905)
1205952+ 2
(71)
It is seen that 0 le 120582119894le 1sum3
119894=1120582119894= 1 Let Π = [ minus02 02 0
01 minus03 02
02 03 minus05
]120574 = 1 by solving and (53) in Theorem 10 controller gainsare given by
11987011= [minus142939 minus140620 minus36426]
11987012= [minus142911 minus144082 minus32860]
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8minus20
minus15
minus10
minus5
0
5
0 2 4 86minus1
0123
Time
Time
Ope
n-lo
op st
ate r
espo
nse
Mar
kov
chai
n
x1
x2
x3
Figure 1 State responses of the open-loop system
11987013= [minus142988 minus144111 minus36425]
11987021= [minus179535 minus177012 minus44471]
11987022= [minus179509 minus179806 minus41585]
11987023= [minus179578 minus179828 minus44470]
11987031= [minus286360 minus283641 minus69941]
11987032= [minus286344 minus285420 minus68118]
11987033= [minus286377 minus285436 minus69941]
(72)
To demonstrate the effectiveness assuming the initialcondition 120601(119905) = [minus12 08 minus05]
119879 Figures 1 and 2 showstate responses of the open-loop system and the closed-loopsystem controlled by (10) respectively From Figure 1 it canbe seen that the open-loop system is not stochastically admis-sible and Figure 2 shows that when the controller obtainedby Theorem 10 is exerted to this system it is stochasticallyadmissible
Example 2 Consider the example without uncertainties in[6]
Mode 1 1198601= [
15 14
minus35 minus45] 119860
1198891= [
02 119886
minus024 minus04] 119861
1= [
1
1]
1198611199081= [
15
14] 119862
1= [05 1] 119862
1198891= [minus02 02] 119863
1= 02 and
1198621199081= 02Mode 2 119860
2= [
17 15
minus13 minus25] 119860
1198892= [
119887 11
minus021 minus03] 119861
2= [
09
09]
1198611199082= [
14
15] 119862
2= [04 03] 119862
1198892= [minus01 02]119863
2= 03 and
1198621199082= 03Π = [
minus1 1
1 minus1] 119864 = [
1 0
0 0] 119889 = 03 120574 = 26 and in [6]
119886 = minus05 119887 = 21 but in this paper minus24 le 119886 le 2 minus2 le 119887 le 48are taken
In Figure 3 ldquoordquo represents the range of the feasiblesolutions using Theorem 10 in this paper and ldquolowastrdquo representsthe range of the feasible solutions using Theorem 3 in [6]
minus15minus1
minus050
051
152
253
354
Mar
kov
chai
n
0 1 2 3 4 5 6 7 8
0 2 4 6 8minus1
0123
Time
Time
Clos
ed-lo
op st
ate r
espo
nse
x1
x2
x3
Figure 2 State responses of the closed-loop system
minus2 minus15 minus1 minus05 0 05 1 15 2minus2
minus1
0
1
2
3
4
a
b
Figure 3 Comparison of the feasible regions
This illustrates that themethod obtained in this paper has lessconservatism
6 Conclusions
In this paper the problem of mode-dependent 119867infin
controlfor singular Markovian jump fuzzy systems with time delayis considered This class of systems under considerationis described by T-S fuzzy models The main contributionof this paper is to design state feedback controllers andstatic output feedback controllers which can guarantee thatresulting closed-loop systems are stochastically admissiblewith an119867
infinperformance 120574 by the method of the augmented
matrix Finally two examples are given to demonstrate theeffectiveness of main results obtained here
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Mathematical Problems in Engineering 13
Acknowledgments
This work was supported by The National Natural ScienceFoundation of China under Grants nos 61304054 61273003and 61273008 respectively and Science and TechnologyResearch Fund of Liaoning Education Department underGrant no L2013051
References
[1] L Dai Singular Control Systems vol 118 of Lecture Notes inControl and Information Sciences Springer Berlin Germany1989
[2] F L Lewis ldquoA survey of linear singular systemsrdquo CircuitsSystems and Signal Processing vol 5 no 1 pp 3ndash36 1986
[3] S Xu and J Lam ldquoRobust stability and stabilization of discretesingular systems an equivalent characterizationrdquo IEEE Trans-actions on Automatic Control vol 49 no 4 pp 568ndash574 2004
[4] Z Wu and W Zhou ldquoDelay-dependent robust stabilization foruncertain singular systemswith discrete and distributed delaysrdquoJournal of ControlTheory and Applications vol 6 no 2 pp 171ndash176 2008
[5] Y Xia P Shi G Liu and D Rees ldquoRobust mixed 119867infin119867
2
state-feedback control for continuous-time descriptor systemswith parameter uncertaintiesrdquo Circuits Systems and SignalProcessing vol 24 no 4 pp 431ndash443 2005
[6] H Lu W Zhou C Duan and X Qi ldquoDelay-dependent robust119867
infincontrol for uncertain singular time-delay system with
Markovian jumping parametersrdquo Optimal Control Applicationsamp Methods vol 34 no 3 pp 296ndash307 2013
[7] Y Fang and K A Loparo ldquoStochastic stability of jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 7pp 1204ndash1208 2002
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] J Xiong and J Lam ldquoStabilization of discrete-time Markovianjump linear systems via time-delayed controllersrdquo Automaticavol 42 no 5 pp 747ndash753 2006
[10] W-H Chen Z-H Guan and P Yu ldquoDelay-dependent stabilityand 119867
infincontrol of uncertain discrete-time Markovian jump
systems with mode-dependent time delaysrdquo Systems amp ControlLetters vol 52 no 5 pp 361ndash376 2004
[11] E-K Boukas Control of Singular Systems with Random AbruptChanges Communications and Control Engineering SpringerBerlin Germany 2008
[12] J Zhang Y Xia and E K Boukas ldquoNew approach to Hinfin
control for Markovian jump singular systemsrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2273ndash2284 2010
[13] J Lam S Zhu S Xu and E-K Boukas ldquoRobust 119867infin
controlof descriptor discrete-time Markovian jump systemsrdquo Interna-tional Journal of Control vol 80 no 3 pp 374ndash385 2007
[14] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Basel Switzerland 2003
[15] G Wang Q Zhang and C Yang ldquoDissipative control forsingular Markovian jump systems with time delayrdquo OptimalControl ApplicationsampMethods vol 33 no 4 pp 415ndash432 2012
[16] Z Wu H Su and J Chu ldquoDelay-dependent 119867infin
control forsingular Markovian jump systems with time delayrdquo Optimal
Control Applications and Methods vol 30 no 5 pp 443ndash4612009
[17] C K Ahn ldquoT-S fuzzy119867infinsynchronization for chaotic systems
via delayed output feedback controlrdquo Nonlinear Dynamics vol59 no 4 pp 535ndash543 2010
[18] C K Ahn ldquoFuzzy delayed output feedback synchronizationfor time-delayed chaotic systemsrdquo Nonlinear Analysis HybridSystems vol 4 no 1 pp 16ndash24 2010
[19] H-N Wu and K-Y Cai ldquoMode-independent robust stabiliza-tion for uncertain Markovian jump nonlinear systems via fuzzycontrolrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 36 no 3 pp 509ndash519 2006
[20] Y Zhang S Xu Y Zou and J Lu ldquoDelay-dependent robuststabilization for uncertain discrete-time fuzzy Markovian jumpsystems with mode-dependent time delaysrdquo Fuzzy Sets andSystems vol 164 pp 66ndash81 2011
[21] A Zhang and H Fang ldquoRobustHinfinfuzzy control for uncertain
Markovian jump nonlinear singular systems with Wiener pro-cessrdquo Journal of Control Theory and Applications vol 8 no 2pp 205ndash210 2010
[22] L Li Q Zhang and B Zhu ldquo119867infin
fuzzy control for nonlin-ear time-delay singular Markovian jump systems with partlyunknown transition ratesrdquo Fuzzy Sets and Systems vol 254 pp106ndash125 2014
[23] J Dong and G-H Yang ldquoStatic output feedback 119867infin
controlof a class of nonlinear discrete-time systemsrdquo Fuzzy Sets andSystems vol 160 no 19 pp 2844ndash2859 2009
[24] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of
singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012
[25] J Wang H Wang A Xie and R Lu ldquoDelay-dependent 119867infin
control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2012
[26] D B Xu Q L Zhang and Y B Hu ldquoReduced-order Hinfin
controller design for uncertain descriptor systemsrdquo Acta Auto-matica Sinica vol 33 no 1 pp 44ndash47 2007
[27] L Xie and C E de Souza ldquoRobust 119867infin
control for linearsystems with norm bounder time-varying uncertaintiesrdquo IEEETransactions on Automatic Control vol 37 no 8 pp 1188ndash11911992
[28] Q-L Han ldquoAbsolute stability of time-delay systemswith sector-bounded nonlinearityrdquo Automatica vol 41 no 12 pp 2171ndash2176 2005
[29] D Yue J Lam and D W C Ho ldquoReliable 119867infin
control ofuncertain descriptor systems with multiple time delaysrdquo IEEProceedings ControlTheory and Applications vol 150 no 6 pp557ndash564 2003
[30] H D Tuan P Apkarian T Narikiyo and Y YamamotoldquoParameterized linear matrix inequality techniques in fuzzycontrol systemdesignrdquo IEEETransactions on Fuzzy Systems vol9 no 2 pp 324ndash332 2001
[31] S-W Kau H-J Lee C-M Yang C-H Lee L Hong and C-HFang ldquoRobust H
infinfuzzy static output feedback control of T-
S fuzzy systems with parametric uncertaintiesrdquo Fuzzy Sets andSystems vol 158 no 2 pp 135ndash146 2007
[32] D Saifia M Chadli S Labiod and T M Guerra ldquoRobust119867
infinstatic output feedback stabilization of T-S fuzzy systems
subject to actuator saturationrdquo International Journal of ControlAutomation and Systems vol 10 no 3 pp 613ndash622 2012
previous literatures And when using the augmented matrixto design the static output feedback control there are notany crossing terms between system matrices and controllergains so assumptions for the output matrix [23] the equalityconstraint for the output matrix [24] and the boundingtechnique for crossing terms are not necessary therefore theconservatism brought by them will not exist
In this paper the119867infinfuzzy control problem for a class of
nonlinear SMJSs with time delay which can be representedby T-S fuzzy models is considered Our aim is to designfuzzy state feedback controllers and static output feedbackcontrollers for SMJFSs with time delay such that closed-loopsystems are stochastically admissible (regular impulse-freeand stochastically stable) with a prescribed119867
infinperformance
120574 Sufficient criterions are presented in forms of LMIs whichare simple and easy to implement compared with previousliteratures Finally numerical examples are given to illustratethemerit andusability of the approach proposed in this paper
Notations Throughout this paper notations used are fairlystandard for real symmetric matrices 119860 and 119861 the notation119860 ge 119861 (119860 gt 119861) means that the matrix 119860 minus 119861 is positivesemidefinite (positive definite) 119860119879 represents the transposeof the matrix 119860 and 119860minus1 represents the inverse of the matrix119860 120582max119861 (120582min119861) is themaximal (minimal) eigenvalue of thematrix 119861 diagsdot stands for a block-diagonal matrix 119868 is theunit matrix with appropriate dimensions and in a matrixthe term of symmetry is stated by the asterisk ldquolowastrdquo Let R119899
stand for the 119899-dimensional Euclidean space R119899times119898 is the setof all 119899 times 119898 real matrices and sdot denotes the Euclideannorm of vectors Esdot denotes the mathematics expectationof the stochastic process or vector 119871119899
2[0infin) stands for the
space of 119899-dimensional square integrable functions on [0infin)119862119899119889
= 119862([minus119889 0]R119899
) denotes Banach space of continuousvector functions mapping the interval [minus119889 0] into R119899 withthe norm 120601
119889= sup
minus119889le119904le0120601(119904)
2 Basic Definitions and Lemmas
Consider a SMJFS its 119894th fuzzy rule is given by
119877119894 if 1205851(119905) is1198721198941 1205852(119905) is1198721198942 and 120585119897(119905) is119872119894119897
where 119909(119905) isin R119899 is the state vector 119906(119905) isin R119898 is the controlinput119908(119905) isin RV is the exogenous disturbance which belongsto 119871V
2[0infin) and 119911(119905) isin R119901 is the controlled output 120601(119905) isin
119862119899119889
is a compatible vector-valued initial function and 119889 is anunknown but constant delay satisfying 119889 isin [0 119889] The scalar
119896 is the number of If-Then rules119872119894119895(119894 isin T 119895 = 1 2 119897)
are fuzzy sets 1205851(119905) minus 120585
119897(119905) are premise variables 119864 isin R119899times119899
may be a singular matrix with rank119864 = 119903 le 119899 119860119894(119903
119905)
119860119889119894(119903
119905) 119861
119894(119903
119905) 119861
119908119894(119903
119905) 119862
119894(119903
119905) 119862
119889119894(119903
119905) 119863
119894(119903
119905) and 119862
119908119894(119903
119905)
are known constant matrices with appropriate dimensions119903
119905 119905 ge 0 is a continuous-time Markovian process with right
continuous trajectories taking values in a finite set given byS = 1 2 119873 with the transition rate matrix Π ≜ 120587
Lemma 5 (see [29]) Suppose there are piecewise continuousreal square matrices 119860(119905) 119883 and 119876 gt 0 satisfying 119860119879
(119905)119883 +
119883119879
119860(119905) lt 0 for all 119905 Then the following conditions hold(i) 119860(119905) and 119883 are nonsingular(ii) 119860minus1
(119905) le 120575 for some 120575 gt 0
Lemma 6 (see [30]) If the following conditions hold
119872119894119894lt 0 1 le 119894 le 119903
1
119903 minus 1
119872119894119894+
1
2
(119872119894119895+119872
119895119894) lt 0 1 le 119894 = 119895 le 119903
(8)
then the following parameterized matrix inequality holds119903
sum
119894=1
119903
sum
119895=1
120572119894(119905) 120572
119895(119905)119872
119894119895lt 0 (9)
where 120572119894(119905) ge 0 and sum119903
119894=1120572119894(119905) = 1
Based on the parallel distributed compensation thefollowing state feedback controller will be considered here
119906119901(119905) =
119896
sum
119894=1
120582119894119870
119901119894119909 (119905) (10)
where119870119901119894(119901 isin S 119894 isin T) are local controller gains such that
the closed-loop system
119864 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895((119860
119901119894+ 119861
119901119894119870
119901119895) 119909 (119905)
+ 119860119889119901119894119909 (119905 minus 119889) + 119861
119908119901119894119908 (119905))
119911 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895((119862
119901119894+ 119863
119901119894119870
119901119895) 119909 (119905)
+ 119862119889119901119894119909 (119905 minus 119889) + 119862
119908119901119894119908 (119905))
119909 (119905) = 120601 (119905)
119905 isin [minus119889 0]
(11)
is stochastically admissible
3 The Design of the StateFeedback 119867
infinController
Firstly the sufficient condition will be given such thatsystem (11) is stochastically admissible Combining (4) and(10) fuzzy closed-loop system (11) can be rewritten in thefollowing form
119864 (119905)
=
119896
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119908119901119894119908 (119905))
119911 (119905) =
119896
sum
119894=1
120582119894(119862
119901119894119909 (119905) + 119862
119889119901119894119909 (119905 minus 119889) + 119862
119908119901119894119908 (119905))
119909 (119905) =120601 (119905)
119905 isin [minus119889 0]
(12)
where
119864 = [
119864 0
0 0
] isin R(119899+119898)times(119899+119898)
119860119901119894= [
119860119901119894
119861119901119894
119870119901119894minus119868
119898
] isin R(119899+119898)times(119899+119898)
119909 (119905) = [
119909 (119905)
119906 (119905)
] isin R119899+119898
119860119889119901119894
= [
119860119889119901119894
0
0 0
] isin R(119899+119898)times(119899+119898)
119861119908119901119894
= [
119861119908119901119894
0
] isin R(119899+119898)timesV
119862119901119894= [119862
119901119894119863
119901119894] isin R
119901times(119899+119898)
119862119889119901119894
= [119862119889119901119894
0] isin R119901times(119899+119898)
120601 (119905) =
119896
sum
119894=1
120582119894[
120601 (119905)
119870119901119894120601 (119905)
]
(13)
4 Mathematical Problems in Engineering
Remark 7 For systems (11) and (12) it can be seen that
det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(119860
119901119894+ 119861
119901119894119870
119901119895))
= det(119904[119864 0
0 0
] minus
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895[
119860119901119894
119861119901119894
119870119901119895minus119868
119898
])
= det(119904[119864 0
0 0
] minus
119896
sum
119894=1
120582119894[
119860119901119894
119861119901119894
119870119901119894minus119868
119898
])
= det(119904119864 minus119896
sum
119894=1
120582119894119860
119901119894)
det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(119860
119901119894+ 119861
119901119894119870
119901119895+ 119860
119889119901119894))
= det(119904[119864 0
0 0
] minus
119896
sum
119894=1
120582119894[
119860119901119894+ 119860
119889119901119894119861119901119894
119870119901119894
minus119868119898
])
= det(119904119864 minus119896
sum
119894=1
120582119894(119860
119901119894+ 119860
119889119901119894))
(14)
By rank119864 = rank119864 and Definition 1 it can be obtained thatthe regularity and nonimpulse of system (11) are equal to theregularity and nonimpulse of system (12) So the stochasticadmissibility of system (11) can be studied by system (12)
Theorem 8 For a prescribed scalar 119889 gt 0 there exists a statefeedback controller (10) with 119906
that system (11) when 119908(119905) = 0 is stochastically admissible forany constant time delay 119889 satisfying 119889 isin [0 119889] if there existmatrices 119875
119889 diag119876 0 minus 119864119879 diag119885 0119864 Pre- and postmultiplying bydiag119883119879
119901 119868
119903 119868
119903⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
119873minus1
and its transposition by Schur comple-
ment lemma it can be seen that
119873
sum
119902=1
120587119901119902119876
119902lt 119876 (21)
From (20) it can be concluded that
119896
sum
119894=1
120582119894(120587
119901119901119864119879
119883119901+ 119860
119879
119901119894119883
119901+ 119883
119879
119901119860
119901119894
minus 119864119879 diag 119885 0 119864) lt 0
(22)
On the other hand diag119866 119868119898119864 diag119867 119868
119898 = [
[1198681199030
0 0
] 0
0 0119898
]Then
119864119879
119883119901= 119883
119879
119901119864 = [
119864119879
0
0 0
][
119883119901
0
1198831199012119883
1199013
] = [
119864119879
1198831199010
0 0
]
= [
119864119879
119875119901119864 0
0 0
] ge 0
(23)
Denote 119860119901(119905) ≜ (sum
119896
119894=1120582119894119860
119901119894) = [
1198601199011(119905)
1198601199012(119905)
1198601199013(119905)
1198601199014(119905)
] from (22) itcan be obtained that
119883119879
1199013119860
1199014(119905) + 119860
119879
1199014(119905) 119883
1199013lt 0 (24)
for every 119901 isin S which implies that 1198601199014(119905) is nonsingular
Thus the pair (119864 sum119896
119894=1120582119894119860
119901119894) is regular and impulse-free for
every 119901 isin S By (20) it is easy to see that
[
Γ1119901119894
lowast
119864119879
[119885 0] 119864 + [119860119879
1198891199011198940]119883
119901minus119864
119879
119885119864 minus 119876119901
] lt 0 (25)
Pre- and postmultiplying (25) by [ 119868119899+119898
[119868119899
0
]
0 119868119899
] and its trans-pose it can be obtained that
119873
sum
119902=1
120587119901119902119864119879
119883119901+ 119883
119879
119901(119860
119901119894+ 119860
119889119901119894)
+ (119860119901119894+ 119860
119889119901119894)
119879
119883119901lt 0
(26)
Hence
119896
sum
119894=1
120582119894(120587
119901119901119864119879
119883119901+ 119883
119879
119901(119860
119901119894+ 119860
119889119901119894)
+ (119860119901119894+ 119860
119889119901119894)
119879
119883119901) lt 0
(27)
Equation (27) implies that the pair (119864 sum119896
119894=1120582119894(119860
119901119894+ 119860
119889119901119894))
is regular and impulse-free for every 119901 isin S Thus byDefinition 1 system (12) is regular and impulse-free ByRemark 7 this implies that system (11) is regular and impulse-free
Now it will be shown that system (11) is stochasticallystable Define a new process (119909
119905 119903
119905) 119905 ge 0 by 119909
119905= 119909(119905 +
120579) minus2119889 le 120579 le 0 then (119909119905 119903
119905) 119905 ge 119889 is a Markovian process
with the initial state (120601(sdot) 1199030) Now for 119905 ge 119889 choose the
following stochastic Lyapunov-Krasovskii candidate for thissystem
119881 (119909119905 119901 119905) =
4
sum
119898=1
119881119898(119909
119905 119901 119905) (28)
where
1198811(119909
119905 119901 119905) = 119909
119879
(119905) 119864119879
119875119901119864119909 (119905) = 119909
119879
(119905) 119864119879
119883119901119909 (119905)
= 119909119879
(119905) 119864119879
119883119901119909 (119905)
1198812(119909
119905 119901 119905) = int
119905
119905minus119889
119909119879
(119904) 119876119901119909 (119904) d119904
1198813(119909
119905 119901 119905) = 119889int
0
minus119889
int
119905
119905+120579
119879
(119904) 119864119879
119885119864 (119904) d119904 d120579
1198814(119909
119905 119901 119905) = int
0
minus119889
int
119905
119905+120579
119909119879
(119904) 119876119909 (119904) d119904 d120579
(29)
6 Mathematical Problems in Engineering
Let L be the weak infinitesimal generator of the randomprocess (119909
119905 119901) 119905 ge 0 Then for each 119901 isin S
L119881 (119909119905 119901 119905) le 2119909
119879
(119905) 119883119879
119901119864 (119905)
+ 119909119879
(119905) (
119873
sum
119902=1
120587119901119902119864119879
119883119902)119909 (119905)
+ 119909119879
(119905) 119876119901119909 (119905)
minus 119909119879
(119905 minus 119889)119876119901119909 (119905 minus 119889)
+ int
119905
119905minus119889
119909119879
(119904) (
119873
sum
119902=1
120587119901119902119876
119902)119909 (119904) d119904
+ 119889119909119879
(119905) 119876119909 (119905)
minus int
119905
119905minus119889
119909119879
(119904) 119876119909 (119904) d119904
+ 119889
2
119879
(119905) 119864119879
119885119864 (119905)
minus 119889int
119905
119905minus119889
119879
(119904) 119864119879
119885119864 (119904) d119904
(30)
From (21) it is clear that
int
119905
119905minus119889
119909119879
(119904) (
119873
sum
119902=1
120587119901119902119876
119902)119909 (119904) d119904
lt int
119905
119905minus119889
119909119879
(119904) 119876119909 (119904) d119904
(31)
From Lemma 4 it follows that
minus 119889int
119905
119905minus119889
119879
(119904) 119864119879
119885119864 (119904) d119904
le [
119909 (119905)
119909 (119905 minus 119889)
]
119879
[
minus119864119879
119885119864 119864119879
119885119864
119864119879
119885119864 minus119864119879
119885119864
][
119909 (119905)
119909 (119905 minus 119889)
]
(32)
So it can be concluded that
L119881 (119909119905 119901 119905) le
119896
sum
119894=1
120578119879
(119905) Φ119901119894120578 (119905) (33)
where
120578119879
(119905) = [119909119879
(119905) 119909119879
(119905 minus 119889)]
Φ119901119894= [
Υ1119901119894
lowast
Υ2119901119894
Υ3119901119894
]
Υ1119901119894= Γ
1119901119894+ [
119860119879
119901119894
119861119879
119901119894
]119889
2
119885 [119860119901119894119861119901119894]
Υ3119901119894= minus119876
119901minus 119864
119879
119885119864 + 119860119879
119889119901119894119889
2
119885119860119889119901119894
Υ2119901119894= 119864
119879
[119885 0] 119864 + [119860119879
1198891199011198940]119883
119901
+ 119860119879
119889119901119894119889
2
119885 [119860119901119894119861119901119894]
(34)
Using (20) it is easy to see that there exists a scalar 120581 gt 0
such that for every 119901 isin S L119881(119909119905 119901 119905) le minus120581119909(119905)
This together with (36) and (48) implies that there exists ascalar ] such that
Eint119905
0
119909 (119904)2 d119904 = Eint
119889
0
119909 (119904)2 d119904
+Eint119905
119889
119909 (119904)2 d119904
le ]E 10038171003817100381710038171206011003817100381710038171003817
2
119889
(51)
Considering this andDefinition 1 system (11) is stochasticallystable for any constant delay 119889 satisfying 119889 isin [0 119889]Thereforesystem (11) is stochastically admissible This completes theproof
In the following a set of sufficient conditions will bedeveloped under which the considered system is guaranteedto be stochastically admissible with an119867
infinperformance
Definition 9 System (11) is said to be stochastically admissiblewith an 119867
infinperformance 120574 if it is stochastically admissible
when 119908(119905) = 0 and under zero initial condition for nonzero119908(119905) isin 119871
V2[0infin)
Eintinfin
0
119911119879
(119905) 119911 (119905) d119905 le 1205742 intinfin
0
119908119879
(119905) 119908 (119905) d119905 (52)
The following result can be presented
8 Mathematical Problems in Engineering
Theorem 10 For a prescribed scalar 119889 gt 0 there exists astate feedback controller (10) with 119906
and the other notations are the same as in Theorem 8
Proof FromTheorem 8 when119908(119905) = 0 system (11) is stochas-tically admissible Let
119869119911119908(119905) = Eint
119905
0
[119911119879
(119904) 119911 (119904) minus 1205742
119908119879
(119904) 119908 (119904)] d119904 (55)
Under zero initial condition it is easy to see that
119869119911119908(119905) le Eint
119905
0
[119911119879
(119904) 119911 (119904) minus 1205742
119908119879
(119904) 119908 (119904)
+L119881 (119909119904 119901 119904)] d119904
le Eint119905
0
119896
sum
119894=1
120582119894[120589
119879
(119904) (Ω119901119894+ Θ
119879
119901119894Θ
119901119894) 120589 (119904)] d119904
(56)
where
120589119879
(119905) = [119909119879
(119905) 119909119879
(119905 minus 119889) 119908119879
(119905)]
Ω119901119894=
[
[
[
[
Υ1119901119894
lowast lowast
Υ2119901119894
Υ3119901119894
lowast
[119861119879
1199081199011198940]119883
1199010 minus120574
2
119868
]
]
]
]
Θ119901119894= [119862
119901119894119862119889119901119894
119862119908119901119894]
(57)
and notations of Υ1119901119894
Υ2119901119894
and Υ3119901119894
are the same as inTheorem 8 Hence by Schur complement lemma and using
the similar method in the proof of Theorem 8 from and(53) it can be obtained that 119869
119911119908(119905) lt 0 for all 119905 gt 0
Therefore for any nonzero 119908(119905) isin 119871V2[0infin) (52) holds
Hence according to Definition 9 the system is stochasticallyadmissible with an 119867
infinperformance 120574 This completes the
proof
Remark 11 Compared with methods in [21 22] because ofthe method of the augmented matrix adopted in Theorems8 and 10 the number of LMIs needed to solve is relativelysmall in this paper When the value of 119896 is relatively largethe quality of the computation is greatly reduced some newrelaxation matrices added will reduce the conservatism ofcontrol conditions compared with previous literatures whichcan be seen from Example 2
4 The Design of the Static OutputFeedback Controller
When 119903119905= 119901 isin S consider the overall SMJFS as follows
119864 (119905) =
119896
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119889119901119894119909 (119905 minus 119889) + 119862
119908119901119894119908 (119905))
(60)
It is difficult to drive LMI-based conditions of the stochas-tic stabilization by employing the static output feedbackcontrol approach due to the appearance of crossing termsbetween system matrices and control gains And system (60)can be rewritten in the following form
119864119909 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(Λ
119901119894119895119909 (119905) + 119861
119908119901119894119908 (119905))
119911 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(119862
119901119894119895119909 (119905) + 119862
119908119901119894119908 (119905))
(61)
where
119864 = [
119864 0
0 0
] isin R(119899+1199011)times(119899+119901
1)
119862119901119894119895= [119862
119901119894119863
119901119894119870
119901119895]
119860119889119901119894
= [
119860119889119901119894
0
0 0
] isin R(119899+1199011)times(119899+119901
1)
119909 (119905) = [
119909 (119905)
119910 (119905)
]
119861119908119901119894
= [
119861119908119901119894
0
] isin R(119899+1199011)timesV
Λ119901119894119895= [
119860119901119894
119861119901119894119870
119901119895
119862119910119901119894
minus119868
]
(62)
Remark 12 For systems (60) and (61) it can be seen that
det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
119896
sum
119904=1
120582119894120582119895120582119904(119860
119901119894+ 119861
119901119894119870
119901119895119862119910119901119904))
= det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895Λ
119901119894119895)
det(119904119864
minus
119896
sum
119894=1
119896
sum
119895=1
119896
sum
119904=1
120582119894120582119895120582119904(119860
119901119894+ 119861
119901119894119870
119901119895119862119910119901119904
+ 119860119889119901119894))
= det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(Λ
119901119894119895+ 119860
119889119901119894))
(63)
As the discussion in Remark 7 the stochastic admissibility ofsystem (60) can be studied by means of system (61)
Theorem 13 There exists an output feedback controller (59)with controller gains 119870
119901119894= 119871
119901119894119884minus1
1199012(119901 isin S 119894 isin T) such that
system (60) with 119908(119905) = 0 is stochastically admissible if thereexist matrices 119875
119901gt 0 119876
119901gt 0 119876 gt 0 119885 gt 0 119871
119901119894 119878
119901 and 119884
1199012
119901 isin S 1 le 119894 = 119895 le 119896 such that and
and the other notations are the same as in Theorem 13
Remark 15 Compared with the method in [31 32] becauseof the augmented matrix adopted in Theorems 13 and 14 thenumber of LMIs needed to solve is greatly decreased Whenthe value of 119896 is relatively large the computational complexitywill be reduced On the other hand by the augmentedmatrixthere are not any crossing terms between system matricesand controller gains so assumptions for the output matrix
[23] the equality constraint for the output matrix [24] andthe bounding technique for crossing terms are not necessaryhere therefore the conservatism brought by them will nothappen
5 Numerical Examples
Two examples will be given to illustrate the validity ofdeveloped methods
Mathematical Problems in Engineering 11
Example 1 To illustrate the 119867infin
controller synthesis thefollowing nonlinear time delay system is considered
(1 + 119886 cos 120579 (119905)) 120579 (119905) = minus119887 1205793 (119905) + 119888120579 (119905) + 119888119889(119905 minus 119889)
It can be expressed exactly by the following fuzzy singularMarkovian jump form
119864 (119905) =
3
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119901119894119906 (119905)
+ 119861119908119901119894119908 (119905))
119911 (119905) =
3
sum
119894=1
120582119894119862119901119894119909 (119905)
119909 (119905) = 120601 (119905)
119905 isin [minus119889 0] 119901 isin 1 2 3
(70)
where
119864 =[
[
[
1 0 0
0 1 0
0 0 0
]
]
]
1198601199011=
[
[
[
[
0 1 0
0 0 1
119888 minus119887 (1205952
+ 2) 119886 minus 1
]
]
]
]
1198601199012=
[
[
[
[
0 1 0
0 0 1
119888 0 minus119886 minus 1 minus 1198861205952
]
]
]
]
1198601199013=[
[
[
0 1 0
0 0 1
119888 0 119886 minus 1
]
]
]
1198601198891199011
= 1198601198891199012
= 1198601198891199013
=[
[
[
0 0 0
0 0 0
1198881198890 0
]
]
]
11986111= 119861
12= 119861
13=[
[
[
0
0
119890
]
]
]
11986121= 119861
22= 119861
23=[
[
[
0
0
08119890
]
]
]
11986131= 119861
32= 119861
33=[
[
[
0
0
05119890
]
]
]
1198611199081199011
= 1198611199081199012
= 1198611199081199013
=[
[
[
0
0
119891
]
]
]
1198621199011= 119862
1199012= 119862
1199013= [1 0 0]
1205821=
1199092
2(119905)
1205952+ 2
1205822=
1 + cos1199091(119905)
1205952+ 2
1205823=
1205952
minus 1199092
2(119905) + 1 minus cos119909
1(119905)
1205952+ 2
(71)
It is seen that 0 le 120582119894le 1sum3
119894=1120582119894= 1 Let Π = [ minus02 02 0
01 minus03 02
02 03 minus05
]120574 = 1 by solving and (53) in Theorem 10 controller gainsare given by
11987011= [minus142939 minus140620 minus36426]
11987012= [minus142911 minus144082 minus32860]
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8minus20
minus15
minus10
minus5
0
5
0 2 4 86minus1
0123
Time
Time
Ope
n-lo
op st
ate r
espo
nse
Mar
kov
chai
n
x1
x2
x3
Figure 1 State responses of the open-loop system
11987013= [minus142988 minus144111 minus36425]
11987021= [minus179535 minus177012 minus44471]
11987022= [minus179509 minus179806 minus41585]
11987023= [minus179578 minus179828 minus44470]
11987031= [minus286360 minus283641 minus69941]
11987032= [minus286344 minus285420 minus68118]
11987033= [minus286377 minus285436 minus69941]
(72)
To demonstrate the effectiveness assuming the initialcondition 120601(119905) = [minus12 08 minus05]
119879 Figures 1 and 2 showstate responses of the open-loop system and the closed-loopsystem controlled by (10) respectively From Figure 1 it canbe seen that the open-loop system is not stochastically admis-sible and Figure 2 shows that when the controller obtainedby Theorem 10 is exerted to this system it is stochasticallyadmissible
Example 2 Consider the example without uncertainties in[6]
Mode 1 1198601= [
15 14
minus35 minus45] 119860
1198891= [
02 119886
minus024 minus04] 119861
1= [
1
1]
1198611199081= [
15
14] 119862
1= [05 1] 119862
1198891= [minus02 02] 119863
1= 02 and
1198621199081= 02Mode 2 119860
2= [
17 15
minus13 minus25] 119860
1198892= [
119887 11
minus021 minus03] 119861
2= [
09
09]
1198611199082= [
14
15] 119862
2= [04 03] 119862
1198892= [minus01 02]119863
2= 03 and
1198621199082= 03Π = [
minus1 1
1 minus1] 119864 = [
1 0
0 0] 119889 = 03 120574 = 26 and in [6]
119886 = minus05 119887 = 21 but in this paper minus24 le 119886 le 2 minus2 le 119887 le 48are taken
In Figure 3 ldquoordquo represents the range of the feasiblesolutions using Theorem 10 in this paper and ldquolowastrdquo representsthe range of the feasible solutions using Theorem 3 in [6]
minus15minus1
minus050
051
152
253
354
Mar
kov
chai
n
0 1 2 3 4 5 6 7 8
0 2 4 6 8minus1
0123
Time
Time
Clos
ed-lo
op st
ate r
espo
nse
x1
x2
x3
Figure 2 State responses of the closed-loop system
minus2 minus15 minus1 minus05 0 05 1 15 2minus2
minus1
0
1
2
3
4
a
b
Figure 3 Comparison of the feasible regions
This illustrates that themethod obtained in this paper has lessconservatism
6 Conclusions
In this paper the problem of mode-dependent 119867infin
controlfor singular Markovian jump fuzzy systems with time delayis considered This class of systems under considerationis described by T-S fuzzy models The main contributionof this paper is to design state feedback controllers andstatic output feedback controllers which can guarantee thatresulting closed-loop systems are stochastically admissiblewith an119867
infinperformance 120574 by the method of the augmented
matrix Finally two examples are given to demonstrate theeffectiveness of main results obtained here
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Mathematical Problems in Engineering 13
Acknowledgments
This work was supported by The National Natural ScienceFoundation of China under Grants nos 61304054 61273003and 61273008 respectively and Science and TechnologyResearch Fund of Liaoning Education Department underGrant no L2013051
References
[1] L Dai Singular Control Systems vol 118 of Lecture Notes inControl and Information Sciences Springer Berlin Germany1989
[2] F L Lewis ldquoA survey of linear singular systemsrdquo CircuitsSystems and Signal Processing vol 5 no 1 pp 3ndash36 1986
[3] S Xu and J Lam ldquoRobust stability and stabilization of discretesingular systems an equivalent characterizationrdquo IEEE Trans-actions on Automatic Control vol 49 no 4 pp 568ndash574 2004
[4] Z Wu and W Zhou ldquoDelay-dependent robust stabilization foruncertain singular systemswith discrete and distributed delaysrdquoJournal of ControlTheory and Applications vol 6 no 2 pp 171ndash176 2008
[5] Y Xia P Shi G Liu and D Rees ldquoRobust mixed 119867infin119867
2
state-feedback control for continuous-time descriptor systemswith parameter uncertaintiesrdquo Circuits Systems and SignalProcessing vol 24 no 4 pp 431ndash443 2005
[6] H Lu W Zhou C Duan and X Qi ldquoDelay-dependent robust119867
infincontrol for uncertain singular time-delay system with
Markovian jumping parametersrdquo Optimal Control Applicationsamp Methods vol 34 no 3 pp 296ndash307 2013
[7] Y Fang and K A Loparo ldquoStochastic stability of jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 7pp 1204ndash1208 2002
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] J Xiong and J Lam ldquoStabilization of discrete-time Markovianjump linear systems via time-delayed controllersrdquo Automaticavol 42 no 5 pp 747ndash753 2006
[10] W-H Chen Z-H Guan and P Yu ldquoDelay-dependent stabilityand 119867
infincontrol of uncertain discrete-time Markovian jump
systems with mode-dependent time delaysrdquo Systems amp ControlLetters vol 52 no 5 pp 361ndash376 2004
[11] E-K Boukas Control of Singular Systems with Random AbruptChanges Communications and Control Engineering SpringerBerlin Germany 2008
[12] J Zhang Y Xia and E K Boukas ldquoNew approach to Hinfin
control for Markovian jump singular systemsrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2273ndash2284 2010
[13] J Lam S Zhu S Xu and E-K Boukas ldquoRobust 119867infin
controlof descriptor discrete-time Markovian jump systemsrdquo Interna-tional Journal of Control vol 80 no 3 pp 374ndash385 2007
[14] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Basel Switzerland 2003
[15] G Wang Q Zhang and C Yang ldquoDissipative control forsingular Markovian jump systems with time delayrdquo OptimalControl ApplicationsampMethods vol 33 no 4 pp 415ndash432 2012
[16] Z Wu H Su and J Chu ldquoDelay-dependent 119867infin
control forsingular Markovian jump systems with time delayrdquo Optimal
Control Applications and Methods vol 30 no 5 pp 443ndash4612009
[17] C K Ahn ldquoT-S fuzzy119867infinsynchronization for chaotic systems
via delayed output feedback controlrdquo Nonlinear Dynamics vol59 no 4 pp 535ndash543 2010
[18] C K Ahn ldquoFuzzy delayed output feedback synchronizationfor time-delayed chaotic systemsrdquo Nonlinear Analysis HybridSystems vol 4 no 1 pp 16ndash24 2010
[19] H-N Wu and K-Y Cai ldquoMode-independent robust stabiliza-tion for uncertain Markovian jump nonlinear systems via fuzzycontrolrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 36 no 3 pp 509ndash519 2006
[20] Y Zhang S Xu Y Zou and J Lu ldquoDelay-dependent robuststabilization for uncertain discrete-time fuzzy Markovian jumpsystems with mode-dependent time delaysrdquo Fuzzy Sets andSystems vol 164 pp 66ndash81 2011
[21] A Zhang and H Fang ldquoRobustHinfinfuzzy control for uncertain
Markovian jump nonlinear singular systems with Wiener pro-cessrdquo Journal of Control Theory and Applications vol 8 no 2pp 205ndash210 2010
[22] L Li Q Zhang and B Zhu ldquo119867infin
fuzzy control for nonlin-ear time-delay singular Markovian jump systems with partlyunknown transition ratesrdquo Fuzzy Sets and Systems vol 254 pp106ndash125 2014
[23] J Dong and G-H Yang ldquoStatic output feedback 119867infin
controlof a class of nonlinear discrete-time systemsrdquo Fuzzy Sets andSystems vol 160 no 19 pp 2844ndash2859 2009
[24] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of
singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012
[25] J Wang H Wang A Xie and R Lu ldquoDelay-dependent 119867infin
control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2012
[26] D B Xu Q L Zhang and Y B Hu ldquoReduced-order Hinfin
controller design for uncertain descriptor systemsrdquo Acta Auto-matica Sinica vol 33 no 1 pp 44ndash47 2007
[27] L Xie and C E de Souza ldquoRobust 119867infin
control for linearsystems with norm bounder time-varying uncertaintiesrdquo IEEETransactions on Automatic Control vol 37 no 8 pp 1188ndash11911992
[28] Q-L Han ldquoAbsolute stability of time-delay systemswith sector-bounded nonlinearityrdquo Automatica vol 41 no 12 pp 2171ndash2176 2005
[29] D Yue J Lam and D W C Ho ldquoReliable 119867infin
control ofuncertain descriptor systems with multiple time delaysrdquo IEEProceedings ControlTheory and Applications vol 150 no 6 pp557ndash564 2003
[30] H D Tuan P Apkarian T Narikiyo and Y YamamotoldquoParameterized linear matrix inequality techniques in fuzzycontrol systemdesignrdquo IEEETransactions on Fuzzy Systems vol9 no 2 pp 324ndash332 2001
[31] S-W Kau H-J Lee C-M Yang C-H Lee L Hong and C-HFang ldquoRobust H
infinfuzzy static output feedback control of T-
S fuzzy systems with parametric uncertaintiesrdquo Fuzzy Sets andSystems vol 158 no 2 pp 135ndash146 2007
[32] D Saifia M Chadli S Labiod and T M Guerra ldquoRobust119867
infinstatic output feedback stabilization of T-S fuzzy systems
subject to actuator saturationrdquo International Journal of ControlAutomation and Systems vol 10 no 3 pp 613ndash622 2012
Lemma 5 (see [29]) Suppose there are piecewise continuousreal square matrices 119860(119905) 119883 and 119876 gt 0 satisfying 119860119879
(119905)119883 +
119883119879
119860(119905) lt 0 for all 119905 Then the following conditions hold(i) 119860(119905) and 119883 are nonsingular(ii) 119860minus1
(119905) le 120575 for some 120575 gt 0
Lemma 6 (see [30]) If the following conditions hold
119872119894119894lt 0 1 le 119894 le 119903
1
119903 minus 1
119872119894119894+
1
2
(119872119894119895+119872
119895119894) lt 0 1 le 119894 = 119895 le 119903
(8)
then the following parameterized matrix inequality holds119903
sum
119894=1
119903
sum
119895=1
120572119894(119905) 120572
119895(119905)119872
119894119895lt 0 (9)
where 120572119894(119905) ge 0 and sum119903
119894=1120572119894(119905) = 1
Based on the parallel distributed compensation thefollowing state feedback controller will be considered here
119906119901(119905) =
119896
sum
119894=1
120582119894119870
119901119894119909 (119905) (10)
where119870119901119894(119901 isin S 119894 isin T) are local controller gains such that
the closed-loop system
119864 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895((119860
119901119894+ 119861
119901119894119870
119901119895) 119909 (119905)
+ 119860119889119901119894119909 (119905 minus 119889) + 119861
119908119901119894119908 (119905))
119911 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895((119862
119901119894+ 119863
119901119894119870
119901119895) 119909 (119905)
+ 119862119889119901119894119909 (119905 minus 119889) + 119862
119908119901119894119908 (119905))
119909 (119905) = 120601 (119905)
119905 isin [minus119889 0]
(11)
is stochastically admissible
3 The Design of the StateFeedback 119867
infinController
Firstly the sufficient condition will be given such thatsystem (11) is stochastically admissible Combining (4) and(10) fuzzy closed-loop system (11) can be rewritten in thefollowing form
119864 (119905)
=
119896
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119908119901119894119908 (119905))
119911 (119905) =
119896
sum
119894=1
120582119894(119862
119901119894119909 (119905) + 119862
119889119901119894119909 (119905 minus 119889) + 119862
119908119901119894119908 (119905))
119909 (119905) =120601 (119905)
119905 isin [minus119889 0]
(12)
where
119864 = [
119864 0
0 0
] isin R(119899+119898)times(119899+119898)
119860119901119894= [
119860119901119894
119861119901119894
119870119901119894minus119868
119898
] isin R(119899+119898)times(119899+119898)
119909 (119905) = [
119909 (119905)
119906 (119905)
] isin R119899+119898
119860119889119901119894
= [
119860119889119901119894
0
0 0
] isin R(119899+119898)times(119899+119898)
119861119908119901119894
= [
119861119908119901119894
0
] isin R(119899+119898)timesV
119862119901119894= [119862
119901119894119863
119901119894] isin R
119901times(119899+119898)
119862119889119901119894
= [119862119889119901119894
0] isin R119901times(119899+119898)
120601 (119905) =
119896
sum
119894=1
120582119894[
120601 (119905)
119870119901119894120601 (119905)
]
(13)
4 Mathematical Problems in Engineering
Remark 7 For systems (11) and (12) it can be seen that
det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(119860
119901119894+ 119861
119901119894119870
119901119895))
= det(119904[119864 0
0 0
] minus
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895[
119860119901119894
119861119901119894
119870119901119895minus119868
119898
])
= det(119904[119864 0
0 0
] minus
119896
sum
119894=1
120582119894[
119860119901119894
119861119901119894
119870119901119894minus119868
119898
])
= det(119904119864 minus119896
sum
119894=1
120582119894119860
119901119894)
det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(119860
119901119894+ 119861
119901119894119870
119901119895+ 119860
119889119901119894))
= det(119904[119864 0
0 0
] minus
119896
sum
119894=1
120582119894[
119860119901119894+ 119860
119889119901119894119861119901119894
119870119901119894
minus119868119898
])
= det(119904119864 minus119896
sum
119894=1
120582119894(119860
119901119894+ 119860
119889119901119894))
(14)
By rank119864 = rank119864 and Definition 1 it can be obtained thatthe regularity and nonimpulse of system (11) are equal to theregularity and nonimpulse of system (12) So the stochasticadmissibility of system (11) can be studied by system (12)
Theorem 8 For a prescribed scalar 119889 gt 0 there exists a statefeedback controller (10) with 119906
that system (11) when 119908(119905) = 0 is stochastically admissible forany constant time delay 119889 satisfying 119889 isin [0 119889] if there existmatrices 119875
119889 diag119876 0 minus 119864119879 diag119885 0119864 Pre- and postmultiplying bydiag119883119879
119901 119868
119903 119868
119903⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
119873minus1
and its transposition by Schur comple-
ment lemma it can be seen that
119873
sum
119902=1
120587119901119902119876
119902lt 119876 (21)
From (20) it can be concluded that
119896
sum
119894=1
120582119894(120587
119901119901119864119879
119883119901+ 119860
119879
119901119894119883
119901+ 119883
119879
119901119860
119901119894
minus 119864119879 diag 119885 0 119864) lt 0
(22)
On the other hand diag119866 119868119898119864 diag119867 119868
119898 = [
[1198681199030
0 0
] 0
0 0119898
]Then
119864119879
119883119901= 119883
119879
119901119864 = [
119864119879
0
0 0
][
119883119901
0
1198831199012119883
1199013
] = [
119864119879
1198831199010
0 0
]
= [
119864119879
119875119901119864 0
0 0
] ge 0
(23)
Denote 119860119901(119905) ≜ (sum
119896
119894=1120582119894119860
119901119894) = [
1198601199011(119905)
1198601199012(119905)
1198601199013(119905)
1198601199014(119905)
] from (22) itcan be obtained that
119883119879
1199013119860
1199014(119905) + 119860
119879
1199014(119905) 119883
1199013lt 0 (24)
for every 119901 isin S which implies that 1198601199014(119905) is nonsingular
Thus the pair (119864 sum119896
119894=1120582119894119860
119901119894) is regular and impulse-free for
every 119901 isin S By (20) it is easy to see that
[
Γ1119901119894
lowast
119864119879
[119885 0] 119864 + [119860119879
1198891199011198940]119883
119901minus119864
119879
119885119864 minus 119876119901
] lt 0 (25)
Pre- and postmultiplying (25) by [ 119868119899+119898
[119868119899
0
]
0 119868119899
] and its trans-pose it can be obtained that
119873
sum
119902=1
120587119901119902119864119879
119883119901+ 119883
119879
119901(119860
119901119894+ 119860
119889119901119894)
+ (119860119901119894+ 119860
119889119901119894)
119879
119883119901lt 0
(26)
Hence
119896
sum
119894=1
120582119894(120587
119901119901119864119879
119883119901+ 119883
119879
119901(119860
119901119894+ 119860
119889119901119894)
+ (119860119901119894+ 119860
119889119901119894)
119879
119883119901) lt 0
(27)
Equation (27) implies that the pair (119864 sum119896
119894=1120582119894(119860
119901119894+ 119860
119889119901119894))
is regular and impulse-free for every 119901 isin S Thus byDefinition 1 system (12) is regular and impulse-free ByRemark 7 this implies that system (11) is regular and impulse-free
Now it will be shown that system (11) is stochasticallystable Define a new process (119909
119905 119903
119905) 119905 ge 0 by 119909
119905= 119909(119905 +
120579) minus2119889 le 120579 le 0 then (119909119905 119903
119905) 119905 ge 119889 is a Markovian process
with the initial state (120601(sdot) 1199030) Now for 119905 ge 119889 choose the
following stochastic Lyapunov-Krasovskii candidate for thissystem
119881 (119909119905 119901 119905) =
4
sum
119898=1
119881119898(119909
119905 119901 119905) (28)
where
1198811(119909
119905 119901 119905) = 119909
119879
(119905) 119864119879
119875119901119864119909 (119905) = 119909
119879
(119905) 119864119879
119883119901119909 (119905)
= 119909119879
(119905) 119864119879
119883119901119909 (119905)
1198812(119909
119905 119901 119905) = int
119905
119905minus119889
119909119879
(119904) 119876119901119909 (119904) d119904
1198813(119909
119905 119901 119905) = 119889int
0
minus119889
int
119905
119905+120579
119879
(119904) 119864119879
119885119864 (119904) d119904 d120579
1198814(119909
119905 119901 119905) = int
0
minus119889
int
119905
119905+120579
119909119879
(119904) 119876119909 (119904) d119904 d120579
(29)
6 Mathematical Problems in Engineering
Let L be the weak infinitesimal generator of the randomprocess (119909
119905 119901) 119905 ge 0 Then for each 119901 isin S
L119881 (119909119905 119901 119905) le 2119909
119879
(119905) 119883119879
119901119864 (119905)
+ 119909119879
(119905) (
119873
sum
119902=1
120587119901119902119864119879
119883119902)119909 (119905)
+ 119909119879
(119905) 119876119901119909 (119905)
minus 119909119879
(119905 minus 119889)119876119901119909 (119905 minus 119889)
+ int
119905
119905minus119889
119909119879
(119904) (
119873
sum
119902=1
120587119901119902119876
119902)119909 (119904) d119904
+ 119889119909119879
(119905) 119876119909 (119905)
minus int
119905
119905minus119889
119909119879
(119904) 119876119909 (119904) d119904
+ 119889
2
119879
(119905) 119864119879
119885119864 (119905)
minus 119889int
119905
119905minus119889
119879
(119904) 119864119879
119885119864 (119904) d119904
(30)
From (21) it is clear that
int
119905
119905minus119889
119909119879
(119904) (
119873
sum
119902=1
120587119901119902119876
119902)119909 (119904) d119904
lt int
119905
119905minus119889
119909119879
(119904) 119876119909 (119904) d119904
(31)
From Lemma 4 it follows that
minus 119889int
119905
119905minus119889
119879
(119904) 119864119879
119885119864 (119904) d119904
le [
119909 (119905)
119909 (119905 minus 119889)
]
119879
[
minus119864119879
119885119864 119864119879
119885119864
119864119879
119885119864 minus119864119879
119885119864
][
119909 (119905)
119909 (119905 minus 119889)
]
(32)
So it can be concluded that
L119881 (119909119905 119901 119905) le
119896
sum
119894=1
120578119879
(119905) Φ119901119894120578 (119905) (33)
where
120578119879
(119905) = [119909119879
(119905) 119909119879
(119905 minus 119889)]
Φ119901119894= [
Υ1119901119894
lowast
Υ2119901119894
Υ3119901119894
]
Υ1119901119894= Γ
1119901119894+ [
119860119879
119901119894
119861119879
119901119894
]119889
2
119885 [119860119901119894119861119901119894]
Υ3119901119894= minus119876
119901minus 119864
119879
119885119864 + 119860119879
119889119901119894119889
2
119885119860119889119901119894
Υ2119901119894= 119864
119879
[119885 0] 119864 + [119860119879
1198891199011198940]119883
119901
+ 119860119879
119889119901119894119889
2
119885 [119860119901119894119861119901119894]
(34)
Using (20) it is easy to see that there exists a scalar 120581 gt 0
such that for every 119901 isin S L119881(119909119905 119901 119905) le minus120581119909(119905)
This together with (36) and (48) implies that there exists ascalar ] such that
Eint119905
0
119909 (119904)2 d119904 = Eint
119889
0
119909 (119904)2 d119904
+Eint119905
119889
119909 (119904)2 d119904
le ]E 10038171003817100381710038171206011003817100381710038171003817
2
119889
(51)
Considering this andDefinition 1 system (11) is stochasticallystable for any constant delay 119889 satisfying 119889 isin [0 119889]Thereforesystem (11) is stochastically admissible This completes theproof
In the following a set of sufficient conditions will bedeveloped under which the considered system is guaranteedto be stochastically admissible with an119867
infinperformance
Definition 9 System (11) is said to be stochastically admissiblewith an 119867
infinperformance 120574 if it is stochastically admissible
when 119908(119905) = 0 and under zero initial condition for nonzero119908(119905) isin 119871
V2[0infin)
Eintinfin
0
119911119879
(119905) 119911 (119905) d119905 le 1205742 intinfin
0
119908119879
(119905) 119908 (119905) d119905 (52)
The following result can be presented
8 Mathematical Problems in Engineering
Theorem 10 For a prescribed scalar 119889 gt 0 there exists astate feedback controller (10) with 119906
and the other notations are the same as in Theorem 8
Proof FromTheorem 8 when119908(119905) = 0 system (11) is stochas-tically admissible Let
119869119911119908(119905) = Eint
119905
0
[119911119879
(119904) 119911 (119904) minus 1205742
119908119879
(119904) 119908 (119904)] d119904 (55)
Under zero initial condition it is easy to see that
119869119911119908(119905) le Eint
119905
0
[119911119879
(119904) 119911 (119904) minus 1205742
119908119879
(119904) 119908 (119904)
+L119881 (119909119904 119901 119904)] d119904
le Eint119905
0
119896
sum
119894=1
120582119894[120589
119879
(119904) (Ω119901119894+ Θ
119879
119901119894Θ
119901119894) 120589 (119904)] d119904
(56)
where
120589119879
(119905) = [119909119879
(119905) 119909119879
(119905 minus 119889) 119908119879
(119905)]
Ω119901119894=
[
[
[
[
Υ1119901119894
lowast lowast
Υ2119901119894
Υ3119901119894
lowast
[119861119879
1199081199011198940]119883
1199010 minus120574
2
119868
]
]
]
]
Θ119901119894= [119862
119901119894119862119889119901119894
119862119908119901119894]
(57)
and notations of Υ1119901119894
Υ2119901119894
and Υ3119901119894
are the same as inTheorem 8 Hence by Schur complement lemma and using
the similar method in the proof of Theorem 8 from and(53) it can be obtained that 119869
119911119908(119905) lt 0 for all 119905 gt 0
Therefore for any nonzero 119908(119905) isin 119871V2[0infin) (52) holds
Hence according to Definition 9 the system is stochasticallyadmissible with an 119867
infinperformance 120574 This completes the
proof
Remark 11 Compared with methods in [21 22] because ofthe method of the augmented matrix adopted in Theorems8 and 10 the number of LMIs needed to solve is relativelysmall in this paper When the value of 119896 is relatively largethe quality of the computation is greatly reduced some newrelaxation matrices added will reduce the conservatism ofcontrol conditions compared with previous literatures whichcan be seen from Example 2
4 The Design of the Static OutputFeedback Controller
When 119903119905= 119901 isin S consider the overall SMJFS as follows
119864 (119905) =
119896
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119889119901119894119909 (119905 minus 119889) + 119862
119908119901119894119908 (119905))
(60)
It is difficult to drive LMI-based conditions of the stochas-tic stabilization by employing the static output feedbackcontrol approach due to the appearance of crossing termsbetween system matrices and control gains And system (60)can be rewritten in the following form
119864119909 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(Λ
119901119894119895119909 (119905) + 119861
119908119901119894119908 (119905))
119911 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(119862
119901119894119895119909 (119905) + 119862
119908119901119894119908 (119905))
(61)
where
119864 = [
119864 0
0 0
] isin R(119899+1199011)times(119899+119901
1)
119862119901119894119895= [119862
119901119894119863
119901119894119870
119901119895]
119860119889119901119894
= [
119860119889119901119894
0
0 0
] isin R(119899+1199011)times(119899+119901
1)
119909 (119905) = [
119909 (119905)
119910 (119905)
]
119861119908119901119894
= [
119861119908119901119894
0
] isin R(119899+1199011)timesV
Λ119901119894119895= [
119860119901119894
119861119901119894119870
119901119895
119862119910119901119894
minus119868
]
(62)
Remark 12 For systems (60) and (61) it can be seen that
det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
119896
sum
119904=1
120582119894120582119895120582119904(119860
119901119894+ 119861
119901119894119870
119901119895119862119910119901119904))
= det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895Λ
119901119894119895)
det(119904119864
minus
119896
sum
119894=1
119896
sum
119895=1
119896
sum
119904=1
120582119894120582119895120582119904(119860
119901119894+ 119861
119901119894119870
119901119895119862119910119901119904
+ 119860119889119901119894))
= det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(Λ
119901119894119895+ 119860
119889119901119894))
(63)
As the discussion in Remark 7 the stochastic admissibility ofsystem (60) can be studied by means of system (61)
Theorem 13 There exists an output feedback controller (59)with controller gains 119870
119901119894= 119871
119901119894119884minus1
1199012(119901 isin S 119894 isin T) such that
system (60) with 119908(119905) = 0 is stochastically admissible if thereexist matrices 119875
119901gt 0 119876
119901gt 0 119876 gt 0 119885 gt 0 119871
119901119894 119878
119901 and 119884
1199012
119901 isin S 1 le 119894 = 119895 le 119896 such that and
and the other notations are the same as in Theorem 13
Remark 15 Compared with the method in [31 32] becauseof the augmented matrix adopted in Theorems 13 and 14 thenumber of LMIs needed to solve is greatly decreased Whenthe value of 119896 is relatively large the computational complexitywill be reduced On the other hand by the augmentedmatrixthere are not any crossing terms between system matricesand controller gains so assumptions for the output matrix
[23] the equality constraint for the output matrix [24] andthe bounding technique for crossing terms are not necessaryhere therefore the conservatism brought by them will nothappen
5 Numerical Examples
Two examples will be given to illustrate the validity ofdeveloped methods
Mathematical Problems in Engineering 11
Example 1 To illustrate the 119867infin
controller synthesis thefollowing nonlinear time delay system is considered
(1 + 119886 cos 120579 (119905)) 120579 (119905) = minus119887 1205793 (119905) + 119888120579 (119905) + 119888119889(119905 minus 119889)
It can be expressed exactly by the following fuzzy singularMarkovian jump form
119864 (119905) =
3
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119901119894119906 (119905)
+ 119861119908119901119894119908 (119905))
119911 (119905) =
3
sum
119894=1
120582119894119862119901119894119909 (119905)
119909 (119905) = 120601 (119905)
119905 isin [minus119889 0] 119901 isin 1 2 3
(70)
where
119864 =[
[
[
1 0 0
0 1 0
0 0 0
]
]
]
1198601199011=
[
[
[
[
0 1 0
0 0 1
119888 minus119887 (1205952
+ 2) 119886 minus 1
]
]
]
]
1198601199012=
[
[
[
[
0 1 0
0 0 1
119888 0 minus119886 minus 1 minus 1198861205952
]
]
]
]
1198601199013=[
[
[
0 1 0
0 0 1
119888 0 119886 minus 1
]
]
]
1198601198891199011
= 1198601198891199012
= 1198601198891199013
=[
[
[
0 0 0
0 0 0
1198881198890 0
]
]
]
11986111= 119861
12= 119861
13=[
[
[
0
0
119890
]
]
]
11986121= 119861
22= 119861
23=[
[
[
0
0
08119890
]
]
]
11986131= 119861
32= 119861
33=[
[
[
0
0
05119890
]
]
]
1198611199081199011
= 1198611199081199012
= 1198611199081199013
=[
[
[
0
0
119891
]
]
]
1198621199011= 119862
1199012= 119862
1199013= [1 0 0]
1205821=
1199092
2(119905)
1205952+ 2
1205822=
1 + cos1199091(119905)
1205952+ 2
1205823=
1205952
minus 1199092
2(119905) + 1 minus cos119909
1(119905)
1205952+ 2
(71)
It is seen that 0 le 120582119894le 1sum3
119894=1120582119894= 1 Let Π = [ minus02 02 0
01 minus03 02
02 03 minus05
]120574 = 1 by solving and (53) in Theorem 10 controller gainsare given by
11987011= [minus142939 minus140620 minus36426]
11987012= [minus142911 minus144082 minus32860]
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8minus20
minus15
minus10
minus5
0
5
0 2 4 86minus1
0123
Time
Time
Ope
n-lo
op st
ate r
espo
nse
Mar
kov
chai
n
x1
x2
x3
Figure 1 State responses of the open-loop system
11987013= [minus142988 minus144111 minus36425]
11987021= [minus179535 minus177012 minus44471]
11987022= [minus179509 minus179806 minus41585]
11987023= [minus179578 minus179828 minus44470]
11987031= [minus286360 minus283641 minus69941]
11987032= [minus286344 minus285420 minus68118]
11987033= [minus286377 minus285436 minus69941]
(72)
To demonstrate the effectiveness assuming the initialcondition 120601(119905) = [minus12 08 minus05]
119879 Figures 1 and 2 showstate responses of the open-loop system and the closed-loopsystem controlled by (10) respectively From Figure 1 it canbe seen that the open-loop system is not stochastically admis-sible and Figure 2 shows that when the controller obtainedby Theorem 10 is exerted to this system it is stochasticallyadmissible
Example 2 Consider the example without uncertainties in[6]
Mode 1 1198601= [
15 14
minus35 minus45] 119860
1198891= [
02 119886
minus024 minus04] 119861
1= [
1
1]
1198611199081= [
15
14] 119862
1= [05 1] 119862
1198891= [minus02 02] 119863
1= 02 and
1198621199081= 02Mode 2 119860
2= [
17 15
minus13 minus25] 119860
1198892= [
119887 11
minus021 minus03] 119861
2= [
09
09]
1198611199082= [
14
15] 119862
2= [04 03] 119862
1198892= [minus01 02]119863
2= 03 and
1198621199082= 03Π = [
minus1 1
1 minus1] 119864 = [
1 0
0 0] 119889 = 03 120574 = 26 and in [6]
119886 = minus05 119887 = 21 but in this paper minus24 le 119886 le 2 minus2 le 119887 le 48are taken
In Figure 3 ldquoordquo represents the range of the feasiblesolutions using Theorem 10 in this paper and ldquolowastrdquo representsthe range of the feasible solutions using Theorem 3 in [6]
minus15minus1
minus050
051
152
253
354
Mar
kov
chai
n
0 1 2 3 4 5 6 7 8
0 2 4 6 8minus1
0123
Time
Time
Clos
ed-lo
op st
ate r
espo
nse
x1
x2
x3
Figure 2 State responses of the closed-loop system
minus2 minus15 minus1 minus05 0 05 1 15 2minus2
minus1
0
1
2
3
4
a
b
Figure 3 Comparison of the feasible regions
This illustrates that themethod obtained in this paper has lessconservatism
6 Conclusions
In this paper the problem of mode-dependent 119867infin
controlfor singular Markovian jump fuzzy systems with time delayis considered This class of systems under considerationis described by T-S fuzzy models The main contributionof this paper is to design state feedback controllers andstatic output feedback controllers which can guarantee thatresulting closed-loop systems are stochastically admissiblewith an119867
infinperformance 120574 by the method of the augmented
matrix Finally two examples are given to demonstrate theeffectiveness of main results obtained here
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Mathematical Problems in Engineering 13
Acknowledgments
This work was supported by The National Natural ScienceFoundation of China under Grants nos 61304054 61273003and 61273008 respectively and Science and TechnologyResearch Fund of Liaoning Education Department underGrant no L2013051
References
[1] L Dai Singular Control Systems vol 118 of Lecture Notes inControl and Information Sciences Springer Berlin Germany1989
[2] F L Lewis ldquoA survey of linear singular systemsrdquo CircuitsSystems and Signal Processing vol 5 no 1 pp 3ndash36 1986
[3] S Xu and J Lam ldquoRobust stability and stabilization of discretesingular systems an equivalent characterizationrdquo IEEE Trans-actions on Automatic Control vol 49 no 4 pp 568ndash574 2004
[4] Z Wu and W Zhou ldquoDelay-dependent robust stabilization foruncertain singular systemswith discrete and distributed delaysrdquoJournal of ControlTheory and Applications vol 6 no 2 pp 171ndash176 2008
[5] Y Xia P Shi G Liu and D Rees ldquoRobust mixed 119867infin119867
2
state-feedback control for continuous-time descriptor systemswith parameter uncertaintiesrdquo Circuits Systems and SignalProcessing vol 24 no 4 pp 431ndash443 2005
[6] H Lu W Zhou C Duan and X Qi ldquoDelay-dependent robust119867
infincontrol for uncertain singular time-delay system with
Markovian jumping parametersrdquo Optimal Control Applicationsamp Methods vol 34 no 3 pp 296ndash307 2013
[7] Y Fang and K A Loparo ldquoStochastic stability of jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 7pp 1204ndash1208 2002
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] J Xiong and J Lam ldquoStabilization of discrete-time Markovianjump linear systems via time-delayed controllersrdquo Automaticavol 42 no 5 pp 747ndash753 2006
[10] W-H Chen Z-H Guan and P Yu ldquoDelay-dependent stabilityand 119867
infincontrol of uncertain discrete-time Markovian jump
systems with mode-dependent time delaysrdquo Systems amp ControlLetters vol 52 no 5 pp 361ndash376 2004
[11] E-K Boukas Control of Singular Systems with Random AbruptChanges Communications and Control Engineering SpringerBerlin Germany 2008
[12] J Zhang Y Xia and E K Boukas ldquoNew approach to Hinfin
control for Markovian jump singular systemsrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2273ndash2284 2010
[13] J Lam S Zhu S Xu and E-K Boukas ldquoRobust 119867infin
controlof descriptor discrete-time Markovian jump systemsrdquo Interna-tional Journal of Control vol 80 no 3 pp 374ndash385 2007
[14] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Basel Switzerland 2003
[15] G Wang Q Zhang and C Yang ldquoDissipative control forsingular Markovian jump systems with time delayrdquo OptimalControl ApplicationsampMethods vol 33 no 4 pp 415ndash432 2012
[16] Z Wu H Su and J Chu ldquoDelay-dependent 119867infin
control forsingular Markovian jump systems with time delayrdquo Optimal
Control Applications and Methods vol 30 no 5 pp 443ndash4612009
[17] C K Ahn ldquoT-S fuzzy119867infinsynchronization for chaotic systems
via delayed output feedback controlrdquo Nonlinear Dynamics vol59 no 4 pp 535ndash543 2010
[18] C K Ahn ldquoFuzzy delayed output feedback synchronizationfor time-delayed chaotic systemsrdquo Nonlinear Analysis HybridSystems vol 4 no 1 pp 16ndash24 2010
[19] H-N Wu and K-Y Cai ldquoMode-independent robust stabiliza-tion for uncertain Markovian jump nonlinear systems via fuzzycontrolrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 36 no 3 pp 509ndash519 2006
[20] Y Zhang S Xu Y Zou and J Lu ldquoDelay-dependent robuststabilization for uncertain discrete-time fuzzy Markovian jumpsystems with mode-dependent time delaysrdquo Fuzzy Sets andSystems vol 164 pp 66ndash81 2011
[21] A Zhang and H Fang ldquoRobustHinfinfuzzy control for uncertain
Markovian jump nonlinear singular systems with Wiener pro-cessrdquo Journal of Control Theory and Applications vol 8 no 2pp 205ndash210 2010
[22] L Li Q Zhang and B Zhu ldquo119867infin
fuzzy control for nonlin-ear time-delay singular Markovian jump systems with partlyunknown transition ratesrdquo Fuzzy Sets and Systems vol 254 pp106ndash125 2014
[23] J Dong and G-H Yang ldquoStatic output feedback 119867infin
controlof a class of nonlinear discrete-time systemsrdquo Fuzzy Sets andSystems vol 160 no 19 pp 2844ndash2859 2009
[24] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of
singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012
[25] J Wang H Wang A Xie and R Lu ldquoDelay-dependent 119867infin
control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2012
[26] D B Xu Q L Zhang and Y B Hu ldquoReduced-order Hinfin
controller design for uncertain descriptor systemsrdquo Acta Auto-matica Sinica vol 33 no 1 pp 44ndash47 2007
[27] L Xie and C E de Souza ldquoRobust 119867infin
control for linearsystems with norm bounder time-varying uncertaintiesrdquo IEEETransactions on Automatic Control vol 37 no 8 pp 1188ndash11911992
[28] Q-L Han ldquoAbsolute stability of time-delay systemswith sector-bounded nonlinearityrdquo Automatica vol 41 no 12 pp 2171ndash2176 2005
[29] D Yue J Lam and D W C Ho ldquoReliable 119867infin
control ofuncertain descriptor systems with multiple time delaysrdquo IEEProceedings ControlTheory and Applications vol 150 no 6 pp557ndash564 2003
[30] H D Tuan P Apkarian T Narikiyo and Y YamamotoldquoParameterized linear matrix inequality techniques in fuzzycontrol systemdesignrdquo IEEETransactions on Fuzzy Systems vol9 no 2 pp 324ndash332 2001
[31] S-W Kau H-J Lee C-M Yang C-H Lee L Hong and C-HFang ldquoRobust H
infinfuzzy static output feedback control of T-
S fuzzy systems with parametric uncertaintiesrdquo Fuzzy Sets andSystems vol 158 no 2 pp 135ndash146 2007
[32] D Saifia M Chadli S Labiod and T M Guerra ldquoRobust119867
infinstatic output feedback stabilization of T-S fuzzy systems
subject to actuator saturationrdquo International Journal of ControlAutomation and Systems vol 10 no 3 pp 613ndash622 2012
Remark 7 For systems (11) and (12) it can be seen that
det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(119860
119901119894+ 119861
119901119894119870
119901119895))
= det(119904[119864 0
0 0
] minus
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895[
119860119901119894
119861119901119894
119870119901119895minus119868
119898
])
= det(119904[119864 0
0 0
] minus
119896
sum
119894=1
120582119894[
119860119901119894
119861119901119894
119870119901119894minus119868
119898
])
= det(119904119864 minus119896
sum
119894=1
120582119894119860
119901119894)
det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(119860
119901119894+ 119861
119901119894119870
119901119895+ 119860
119889119901119894))
= det(119904[119864 0
0 0
] minus
119896
sum
119894=1
120582119894[
119860119901119894+ 119860
119889119901119894119861119901119894
119870119901119894
minus119868119898
])
= det(119904119864 minus119896
sum
119894=1
120582119894(119860
119901119894+ 119860
119889119901119894))
(14)
By rank119864 = rank119864 and Definition 1 it can be obtained thatthe regularity and nonimpulse of system (11) are equal to theregularity and nonimpulse of system (12) So the stochasticadmissibility of system (11) can be studied by system (12)
Theorem 8 For a prescribed scalar 119889 gt 0 there exists a statefeedback controller (10) with 119906
that system (11) when 119908(119905) = 0 is stochastically admissible forany constant time delay 119889 satisfying 119889 isin [0 119889] if there existmatrices 119875
119889 diag119876 0 minus 119864119879 diag119885 0119864 Pre- and postmultiplying bydiag119883119879
119901 119868
119903 119868
119903⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
119873minus1
and its transposition by Schur comple-
ment lemma it can be seen that
119873
sum
119902=1
120587119901119902119876
119902lt 119876 (21)
From (20) it can be concluded that
119896
sum
119894=1
120582119894(120587
119901119901119864119879
119883119901+ 119860
119879
119901119894119883
119901+ 119883
119879
119901119860
119901119894
minus 119864119879 diag 119885 0 119864) lt 0
(22)
On the other hand diag119866 119868119898119864 diag119867 119868
119898 = [
[1198681199030
0 0
] 0
0 0119898
]Then
119864119879
119883119901= 119883
119879
119901119864 = [
119864119879
0
0 0
][
119883119901
0
1198831199012119883
1199013
] = [
119864119879
1198831199010
0 0
]
= [
119864119879
119875119901119864 0
0 0
] ge 0
(23)
Denote 119860119901(119905) ≜ (sum
119896
119894=1120582119894119860
119901119894) = [
1198601199011(119905)
1198601199012(119905)
1198601199013(119905)
1198601199014(119905)
] from (22) itcan be obtained that
119883119879
1199013119860
1199014(119905) + 119860
119879
1199014(119905) 119883
1199013lt 0 (24)
for every 119901 isin S which implies that 1198601199014(119905) is nonsingular
Thus the pair (119864 sum119896
119894=1120582119894119860
119901119894) is regular and impulse-free for
every 119901 isin S By (20) it is easy to see that
[
Γ1119901119894
lowast
119864119879
[119885 0] 119864 + [119860119879
1198891199011198940]119883
119901minus119864
119879
119885119864 minus 119876119901
] lt 0 (25)
Pre- and postmultiplying (25) by [ 119868119899+119898
[119868119899
0
]
0 119868119899
] and its trans-pose it can be obtained that
119873
sum
119902=1
120587119901119902119864119879
119883119901+ 119883
119879
119901(119860
119901119894+ 119860
119889119901119894)
+ (119860119901119894+ 119860
119889119901119894)
119879
119883119901lt 0
(26)
Hence
119896
sum
119894=1
120582119894(120587
119901119901119864119879
119883119901+ 119883
119879
119901(119860
119901119894+ 119860
119889119901119894)
+ (119860119901119894+ 119860
119889119901119894)
119879
119883119901) lt 0
(27)
Equation (27) implies that the pair (119864 sum119896
119894=1120582119894(119860
119901119894+ 119860
119889119901119894))
is regular and impulse-free for every 119901 isin S Thus byDefinition 1 system (12) is regular and impulse-free ByRemark 7 this implies that system (11) is regular and impulse-free
Now it will be shown that system (11) is stochasticallystable Define a new process (119909
119905 119903
119905) 119905 ge 0 by 119909
119905= 119909(119905 +
120579) minus2119889 le 120579 le 0 then (119909119905 119903
119905) 119905 ge 119889 is a Markovian process
with the initial state (120601(sdot) 1199030) Now for 119905 ge 119889 choose the
following stochastic Lyapunov-Krasovskii candidate for thissystem
119881 (119909119905 119901 119905) =
4
sum
119898=1
119881119898(119909
119905 119901 119905) (28)
where
1198811(119909
119905 119901 119905) = 119909
119879
(119905) 119864119879
119875119901119864119909 (119905) = 119909
119879
(119905) 119864119879
119883119901119909 (119905)
= 119909119879
(119905) 119864119879
119883119901119909 (119905)
1198812(119909
119905 119901 119905) = int
119905
119905minus119889
119909119879
(119904) 119876119901119909 (119904) d119904
1198813(119909
119905 119901 119905) = 119889int
0
minus119889
int
119905
119905+120579
119879
(119904) 119864119879
119885119864 (119904) d119904 d120579
1198814(119909
119905 119901 119905) = int
0
minus119889
int
119905
119905+120579
119909119879
(119904) 119876119909 (119904) d119904 d120579
(29)
6 Mathematical Problems in Engineering
Let L be the weak infinitesimal generator of the randomprocess (119909
119905 119901) 119905 ge 0 Then for each 119901 isin S
L119881 (119909119905 119901 119905) le 2119909
119879
(119905) 119883119879
119901119864 (119905)
+ 119909119879
(119905) (
119873
sum
119902=1
120587119901119902119864119879
119883119902)119909 (119905)
+ 119909119879
(119905) 119876119901119909 (119905)
minus 119909119879
(119905 minus 119889)119876119901119909 (119905 minus 119889)
+ int
119905
119905minus119889
119909119879
(119904) (
119873
sum
119902=1
120587119901119902119876
119902)119909 (119904) d119904
+ 119889119909119879
(119905) 119876119909 (119905)
minus int
119905
119905minus119889
119909119879
(119904) 119876119909 (119904) d119904
+ 119889
2
119879
(119905) 119864119879
119885119864 (119905)
minus 119889int
119905
119905minus119889
119879
(119904) 119864119879
119885119864 (119904) d119904
(30)
From (21) it is clear that
int
119905
119905minus119889
119909119879
(119904) (
119873
sum
119902=1
120587119901119902119876
119902)119909 (119904) d119904
lt int
119905
119905minus119889
119909119879
(119904) 119876119909 (119904) d119904
(31)
From Lemma 4 it follows that
minus 119889int
119905
119905minus119889
119879
(119904) 119864119879
119885119864 (119904) d119904
le [
119909 (119905)
119909 (119905 minus 119889)
]
119879
[
minus119864119879
119885119864 119864119879
119885119864
119864119879
119885119864 minus119864119879
119885119864
][
119909 (119905)
119909 (119905 minus 119889)
]
(32)
So it can be concluded that
L119881 (119909119905 119901 119905) le
119896
sum
119894=1
120578119879
(119905) Φ119901119894120578 (119905) (33)
where
120578119879
(119905) = [119909119879
(119905) 119909119879
(119905 minus 119889)]
Φ119901119894= [
Υ1119901119894
lowast
Υ2119901119894
Υ3119901119894
]
Υ1119901119894= Γ
1119901119894+ [
119860119879
119901119894
119861119879
119901119894
]119889
2
119885 [119860119901119894119861119901119894]
Υ3119901119894= minus119876
119901minus 119864
119879
119885119864 + 119860119879
119889119901119894119889
2
119885119860119889119901119894
Υ2119901119894= 119864
119879
[119885 0] 119864 + [119860119879
1198891199011198940]119883
119901
+ 119860119879
119889119901119894119889
2
119885 [119860119901119894119861119901119894]
(34)
Using (20) it is easy to see that there exists a scalar 120581 gt 0
such that for every 119901 isin S L119881(119909119905 119901 119905) le minus120581119909(119905)
This together with (36) and (48) implies that there exists ascalar ] such that
Eint119905
0
119909 (119904)2 d119904 = Eint
119889
0
119909 (119904)2 d119904
+Eint119905
119889
119909 (119904)2 d119904
le ]E 10038171003817100381710038171206011003817100381710038171003817
2
119889
(51)
Considering this andDefinition 1 system (11) is stochasticallystable for any constant delay 119889 satisfying 119889 isin [0 119889]Thereforesystem (11) is stochastically admissible This completes theproof
In the following a set of sufficient conditions will bedeveloped under which the considered system is guaranteedto be stochastically admissible with an119867
infinperformance
Definition 9 System (11) is said to be stochastically admissiblewith an 119867
infinperformance 120574 if it is stochastically admissible
when 119908(119905) = 0 and under zero initial condition for nonzero119908(119905) isin 119871
V2[0infin)
Eintinfin
0
119911119879
(119905) 119911 (119905) d119905 le 1205742 intinfin
0
119908119879
(119905) 119908 (119905) d119905 (52)
The following result can be presented
8 Mathematical Problems in Engineering
Theorem 10 For a prescribed scalar 119889 gt 0 there exists astate feedback controller (10) with 119906
and the other notations are the same as in Theorem 8
Proof FromTheorem 8 when119908(119905) = 0 system (11) is stochas-tically admissible Let
119869119911119908(119905) = Eint
119905
0
[119911119879
(119904) 119911 (119904) minus 1205742
119908119879
(119904) 119908 (119904)] d119904 (55)
Under zero initial condition it is easy to see that
119869119911119908(119905) le Eint
119905
0
[119911119879
(119904) 119911 (119904) minus 1205742
119908119879
(119904) 119908 (119904)
+L119881 (119909119904 119901 119904)] d119904
le Eint119905
0
119896
sum
119894=1
120582119894[120589
119879
(119904) (Ω119901119894+ Θ
119879
119901119894Θ
119901119894) 120589 (119904)] d119904
(56)
where
120589119879
(119905) = [119909119879
(119905) 119909119879
(119905 minus 119889) 119908119879
(119905)]
Ω119901119894=
[
[
[
[
Υ1119901119894
lowast lowast
Υ2119901119894
Υ3119901119894
lowast
[119861119879
1199081199011198940]119883
1199010 minus120574
2
119868
]
]
]
]
Θ119901119894= [119862
119901119894119862119889119901119894
119862119908119901119894]
(57)
and notations of Υ1119901119894
Υ2119901119894
and Υ3119901119894
are the same as inTheorem 8 Hence by Schur complement lemma and using
the similar method in the proof of Theorem 8 from and(53) it can be obtained that 119869
119911119908(119905) lt 0 for all 119905 gt 0
Therefore for any nonzero 119908(119905) isin 119871V2[0infin) (52) holds
Hence according to Definition 9 the system is stochasticallyadmissible with an 119867
infinperformance 120574 This completes the
proof
Remark 11 Compared with methods in [21 22] because ofthe method of the augmented matrix adopted in Theorems8 and 10 the number of LMIs needed to solve is relativelysmall in this paper When the value of 119896 is relatively largethe quality of the computation is greatly reduced some newrelaxation matrices added will reduce the conservatism ofcontrol conditions compared with previous literatures whichcan be seen from Example 2
4 The Design of the Static OutputFeedback Controller
When 119903119905= 119901 isin S consider the overall SMJFS as follows
119864 (119905) =
119896
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119889119901119894119909 (119905 minus 119889) + 119862
119908119901119894119908 (119905))
(60)
It is difficult to drive LMI-based conditions of the stochas-tic stabilization by employing the static output feedbackcontrol approach due to the appearance of crossing termsbetween system matrices and control gains And system (60)can be rewritten in the following form
119864119909 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(Λ
119901119894119895119909 (119905) + 119861
119908119901119894119908 (119905))
119911 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(119862
119901119894119895119909 (119905) + 119862
119908119901119894119908 (119905))
(61)
where
119864 = [
119864 0
0 0
] isin R(119899+1199011)times(119899+119901
1)
119862119901119894119895= [119862
119901119894119863
119901119894119870
119901119895]
119860119889119901119894
= [
119860119889119901119894
0
0 0
] isin R(119899+1199011)times(119899+119901
1)
119909 (119905) = [
119909 (119905)
119910 (119905)
]
119861119908119901119894
= [
119861119908119901119894
0
] isin R(119899+1199011)timesV
Λ119901119894119895= [
119860119901119894
119861119901119894119870
119901119895
119862119910119901119894
minus119868
]
(62)
Remark 12 For systems (60) and (61) it can be seen that
det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
119896
sum
119904=1
120582119894120582119895120582119904(119860
119901119894+ 119861
119901119894119870
119901119895119862119910119901119904))
= det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895Λ
119901119894119895)
det(119904119864
minus
119896
sum
119894=1
119896
sum
119895=1
119896
sum
119904=1
120582119894120582119895120582119904(119860
119901119894+ 119861
119901119894119870
119901119895119862119910119901119904
+ 119860119889119901119894))
= det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(Λ
119901119894119895+ 119860
119889119901119894))
(63)
As the discussion in Remark 7 the stochastic admissibility ofsystem (60) can be studied by means of system (61)
Theorem 13 There exists an output feedback controller (59)with controller gains 119870
119901119894= 119871
119901119894119884minus1
1199012(119901 isin S 119894 isin T) such that
system (60) with 119908(119905) = 0 is stochastically admissible if thereexist matrices 119875
119901gt 0 119876
119901gt 0 119876 gt 0 119885 gt 0 119871
119901119894 119878
119901 and 119884
1199012
119901 isin S 1 le 119894 = 119895 le 119896 such that and
and the other notations are the same as in Theorem 13
Remark 15 Compared with the method in [31 32] becauseof the augmented matrix adopted in Theorems 13 and 14 thenumber of LMIs needed to solve is greatly decreased Whenthe value of 119896 is relatively large the computational complexitywill be reduced On the other hand by the augmentedmatrixthere are not any crossing terms between system matricesand controller gains so assumptions for the output matrix
[23] the equality constraint for the output matrix [24] andthe bounding technique for crossing terms are not necessaryhere therefore the conservatism brought by them will nothappen
5 Numerical Examples
Two examples will be given to illustrate the validity ofdeveloped methods
Mathematical Problems in Engineering 11
Example 1 To illustrate the 119867infin
controller synthesis thefollowing nonlinear time delay system is considered
(1 + 119886 cos 120579 (119905)) 120579 (119905) = minus119887 1205793 (119905) + 119888120579 (119905) + 119888119889(119905 minus 119889)
It can be expressed exactly by the following fuzzy singularMarkovian jump form
119864 (119905) =
3
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119901119894119906 (119905)
+ 119861119908119901119894119908 (119905))
119911 (119905) =
3
sum
119894=1
120582119894119862119901119894119909 (119905)
119909 (119905) = 120601 (119905)
119905 isin [minus119889 0] 119901 isin 1 2 3
(70)
where
119864 =[
[
[
1 0 0
0 1 0
0 0 0
]
]
]
1198601199011=
[
[
[
[
0 1 0
0 0 1
119888 minus119887 (1205952
+ 2) 119886 minus 1
]
]
]
]
1198601199012=
[
[
[
[
0 1 0
0 0 1
119888 0 minus119886 minus 1 minus 1198861205952
]
]
]
]
1198601199013=[
[
[
0 1 0
0 0 1
119888 0 119886 minus 1
]
]
]
1198601198891199011
= 1198601198891199012
= 1198601198891199013
=[
[
[
0 0 0
0 0 0
1198881198890 0
]
]
]
11986111= 119861
12= 119861
13=[
[
[
0
0
119890
]
]
]
11986121= 119861
22= 119861
23=[
[
[
0
0
08119890
]
]
]
11986131= 119861
32= 119861
33=[
[
[
0
0
05119890
]
]
]
1198611199081199011
= 1198611199081199012
= 1198611199081199013
=[
[
[
0
0
119891
]
]
]
1198621199011= 119862
1199012= 119862
1199013= [1 0 0]
1205821=
1199092
2(119905)
1205952+ 2
1205822=
1 + cos1199091(119905)
1205952+ 2
1205823=
1205952
minus 1199092
2(119905) + 1 minus cos119909
1(119905)
1205952+ 2
(71)
It is seen that 0 le 120582119894le 1sum3
119894=1120582119894= 1 Let Π = [ minus02 02 0
01 minus03 02
02 03 minus05
]120574 = 1 by solving and (53) in Theorem 10 controller gainsare given by
11987011= [minus142939 minus140620 minus36426]
11987012= [minus142911 minus144082 minus32860]
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8minus20
minus15
minus10
minus5
0
5
0 2 4 86minus1
0123
Time
Time
Ope
n-lo
op st
ate r
espo
nse
Mar
kov
chai
n
x1
x2
x3
Figure 1 State responses of the open-loop system
11987013= [minus142988 minus144111 minus36425]
11987021= [minus179535 minus177012 minus44471]
11987022= [minus179509 minus179806 minus41585]
11987023= [minus179578 minus179828 minus44470]
11987031= [minus286360 minus283641 minus69941]
11987032= [minus286344 minus285420 minus68118]
11987033= [minus286377 minus285436 minus69941]
(72)
To demonstrate the effectiveness assuming the initialcondition 120601(119905) = [minus12 08 minus05]
119879 Figures 1 and 2 showstate responses of the open-loop system and the closed-loopsystem controlled by (10) respectively From Figure 1 it canbe seen that the open-loop system is not stochastically admis-sible and Figure 2 shows that when the controller obtainedby Theorem 10 is exerted to this system it is stochasticallyadmissible
Example 2 Consider the example without uncertainties in[6]
Mode 1 1198601= [
15 14
minus35 minus45] 119860
1198891= [
02 119886
minus024 minus04] 119861
1= [
1
1]
1198611199081= [
15
14] 119862
1= [05 1] 119862
1198891= [minus02 02] 119863
1= 02 and
1198621199081= 02Mode 2 119860
2= [
17 15
minus13 minus25] 119860
1198892= [
119887 11
minus021 minus03] 119861
2= [
09
09]
1198611199082= [
14
15] 119862
2= [04 03] 119862
1198892= [minus01 02]119863
2= 03 and
1198621199082= 03Π = [
minus1 1
1 minus1] 119864 = [
1 0
0 0] 119889 = 03 120574 = 26 and in [6]
119886 = minus05 119887 = 21 but in this paper minus24 le 119886 le 2 minus2 le 119887 le 48are taken
In Figure 3 ldquoordquo represents the range of the feasiblesolutions using Theorem 10 in this paper and ldquolowastrdquo representsthe range of the feasible solutions using Theorem 3 in [6]
minus15minus1
minus050
051
152
253
354
Mar
kov
chai
n
0 1 2 3 4 5 6 7 8
0 2 4 6 8minus1
0123
Time
Time
Clos
ed-lo
op st
ate r
espo
nse
x1
x2
x3
Figure 2 State responses of the closed-loop system
minus2 minus15 minus1 minus05 0 05 1 15 2minus2
minus1
0
1
2
3
4
a
b
Figure 3 Comparison of the feasible regions
This illustrates that themethod obtained in this paper has lessconservatism
6 Conclusions
In this paper the problem of mode-dependent 119867infin
controlfor singular Markovian jump fuzzy systems with time delayis considered This class of systems under considerationis described by T-S fuzzy models The main contributionof this paper is to design state feedback controllers andstatic output feedback controllers which can guarantee thatresulting closed-loop systems are stochastically admissiblewith an119867
infinperformance 120574 by the method of the augmented
matrix Finally two examples are given to demonstrate theeffectiveness of main results obtained here
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Mathematical Problems in Engineering 13
Acknowledgments
This work was supported by The National Natural ScienceFoundation of China under Grants nos 61304054 61273003and 61273008 respectively and Science and TechnologyResearch Fund of Liaoning Education Department underGrant no L2013051
References
[1] L Dai Singular Control Systems vol 118 of Lecture Notes inControl and Information Sciences Springer Berlin Germany1989
[2] F L Lewis ldquoA survey of linear singular systemsrdquo CircuitsSystems and Signal Processing vol 5 no 1 pp 3ndash36 1986
[3] S Xu and J Lam ldquoRobust stability and stabilization of discretesingular systems an equivalent characterizationrdquo IEEE Trans-actions on Automatic Control vol 49 no 4 pp 568ndash574 2004
[4] Z Wu and W Zhou ldquoDelay-dependent robust stabilization foruncertain singular systemswith discrete and distributed delaysrdquoJournal of ControlTheory and Applications vol 6 no 2 pp 171ndash176 2008
[5] Y Xia P Shi G Liu and D Rees ldquoRobust mixed 119867infin119867
2
state-feedback control for continuous-time descriptor systemswith parameter uncertaintiesrdquo Circuits Systems and SignalProcessing vol 24 no 4 pp 431ndash443 2005
[6] H Lu W Zhou C Duan and X Qi ldquoDelay-dependent robust119867
infincontrol for uncertain singular time-delay system with
Markovian jumping parametersrdquo Optimal Control Applicationsamp Methods vol 34 no 3 pp 296ndash307 2013
[7] Y Fang and K A Loparo ldquoStochastic stability of jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 7pp 1204ndash1208 2002
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] J Xiong and J Lam ldquoStabilization of discrete-time Markovianjump linear systems via time-delayed controllersrdquo Automaticavol 42 no 5 pp 747ndash753 2006
[10] W-H Chen Z-H Guan and P Yu ldquoDelay-dependent stabilityand 119867
infincontrol of uncertain discrete-time Markovian jump
systems with mode-dependent time delaysrdquo Systems amp ControlLetters vol 52 no 5 pp 361ndash376 2004
[11] E-K Boukas Control of Singular Systems with Random AbruptChanges Communications and Control Engineering SpringerBerlin Germany 2008
[12] J Zhang Y Xia and E K Boukas ldquoNew approach to Hinfin
control for Markovian jump singular systemsrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2273ndash2284 2010
[13] J Lam S Zhu S Xu and E-K Boukas ldquoRobust 119867infin
controlof descriptor discrete-time Markovian jump systemsrdquo Interna-tional Journal of Control vol 80 no 3 pp 374ndash385 2007
[14] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Basel Switzerland 2003
[15] G Wang Q Zhang and C Yang ldquoDissipative control forsingular Markovian jump systems with time delayrdquo OptimalControl ApplicationsampMethods vol 33 no 4 pp 415ndash432 2012
[16] Z Wu H Su and J Chu ldquoDelay-dependent 119867infin
control forsingular Markovian jump systems with time delayrdquo Optimal
Control Applications and Methods vol 30 no 5 pp 443ndash4612009
[17] C K Ahn ldquoT-S fuzzy119867infinsynchronization for chaotic systems
via delayed output feedback controlrdquo Nonlinear Dynamics vol59 no 4 pp 535ndash543 2010
[18] C K Ahn ldquoFuzzy delayed output feedback synchronizationfor time-delayed chaotic systemsrdquo Nonlinear Analysis HybridSystems vol 4 no 1 pp 16ndash24 2010
[19] H-N Wu and K-Y Cai ldquoMode-independent robust stabiliza-tion for uncertain Markovian jump nonlinear systems via fuzzycontrolrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 36 no 3 pp 509ndash519 2006
[20] Y Zhang S Xu Y Zou and J Lu ldquoDelay-dependent robuststabilization for uncertain discrete-time fuzzy Markovian jumpsystems with mode-dependent time delaysrdquo Fuzzy Sets andSystems vol 164 pp 66ndash81 2011
[21] A Zhang and H Fang ldquoRobustHinfinfuzzy control for uncertain
Markovian jump nonlinear singular systems with Wiener pro-cessrdquo Journal of Control Theory and Applications vol 8 no 2pp 205ndash210 2010
[22] L Li Q Zhang and B Zhu ldquo119867infin
fuzzy control for nonlin-ear time-delay singular Markovian jump systems with partlyunknown transition ratesrdquo Fuzzy Sets and Systems vol 254 pp106ndash125 2014
[23] J Dong and G-H Yang ldquoStatic output feedback 119867infin
controlof a class of nonlinear discrete-time systemsrdquo Fuzzy Sets andSystems vol 160 no 19 pp 2844ndash2859 2009
[24] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of
singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012
[25] J Wang H Wang A Xie and R Lu ldquoDelay-dependent 119867infin
control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2012
[26] D B Xu Q L Zhang and Y B Hu ldquoReduced-order Hinfin
controller design for uncertain descriptor systemsrdquo Acta Auto-matica Sinica vol 33 no 1 pp 44ndash47 2007
[27] L Xie and C E de Souza ldquoRobust 119867infin
control for linearsystems with norm bounder time-varying uncertaintiesrdquo IEEETransactions on Automatic Control vol 37 no 8 pp 1188ndash11911992
[28] Q-L Han ldquoAbsolute stability of time-delay systemswith sector-bounded nonlinearityrdquo Automatica vol 41 no 12 pp 2171ndash2176 2005
[29] D Yue J Lam and D W C Ho ldquoReliable 119867infin
control ofuncertain descriptor systems with multiple time delaysrdquo IEEProceedings ControlTheory and Applications vol 150 no 6 pp557ndash564 2003
[30] H D Tuan P Apkarian T Narikiyo and Y YamamotoldquoParameterized linear matrix inequality techniques in fuzzycontrol systemdesignrdquo IEEETransactions on Fuzzy Systems vol9 no 2 pp 324ndash332 2001
[31] S-W Kau H-J Lee C-M Yang C-H Lee L Hong and C-HFang ldquoRobust H
infinfuzzy static output feedback control of T-
S fuzzy systems with parametric uncertaintiesrdquo Fuzzy Sets andSystems vol 158 no 2 pp 135ndash146 2007
[32] D Saifia M Chadli S Labiod and T M Guerra ldquoRobust119867
infinstatic output feedback stabilization of T-S fuzzy systems
subject to actuator saturationrdquo International Journal of ControlAutomation and Systems vol 10 no 3 pp 613ndash622 2012
119889 diag119876 0 minus 119864119879 diag119885 0119864 Pre- and postmultiplying bydiag119883119879
119901 119868
119903 119868
119903⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
119873minus1
and its transposition by Schur comple-
ment lemma it can be seen that
119873
sum
119902=1
120587119901119902119876
119902lt 119876 (21)
From (20) it can be concluded that
119896
sum
119894=1
120582119894(120587
119901119901119864119879
119883119901+ 119860
119879
119901119894119883
119901+ 119883
119879
119901119860
119901119894
minus 119864119879 diag 119885 0 119864) lt 0
(22)
On the other hand diag119866 119868119898119864 diag119867 119868
119898 = [
[1198681199030
0 0
] 0
0 0119898
]Then
119864119879
119883119901= 119883
119879
119901119864 = [
119864119879
0
0 0
][
119883119901
0
1198831199012119883
1199013
] = [
119864119879
1198831199010
0 0
]
= [
119864119879
119875119901119864 0
0 0
] ge 0
(23)
Denote 119860119901(119905) ≜ (sum
119896
119894=1120582119894119860
119901119894) = [
1198601199011(119905)
1198601199012(119905)
1198601199013(119905)
1198601199014(119905)
] from (22) itcan be obtained that
119883119879
1199013119860
1199014(119905) + 119860
119879
1199014(119905) 119883
1199013lt 0 (24)
for every 119901 isin S which implies that 1198601199014(119905) is nonsingular
Thus the pair (119864 sum119896
119894=1120582119894119860
119901119894) is regular and impulse-free for
every 119901 isin S By (20) it is easy to see that
[
Γ1119901119894
lowast
119864119879
[119885 0] 119864 + [119860119879
1198891199011198940]119883
119901minus119864
119879
119885119864 minus 119876119901
] lt 0 (25)
Pre- and postmultiplying (25) by [ 119868119899+119898
[119868119899
0
]
0 119868119899
] and its trans-pose it can be obtained that
119873
sum
119902=1
120587119901119902119864119879
119883119901+ 119883
119879
119901(119860
119901119894+ 119860
119889119901119894)
+ (119860119901119894+ 119860
119889119901119894)
119879
119883119901lt 0
(26)
Hence
119896
sum
119894=1
120582119894(120587
119901119901119864119879
119883119901+ 119883
119879
119901(119860
119901119894+ 119860
119889119901119894)
+ (119860119901119894+ 119860
119889119901119894)
119879
119883119901) lt 0
(27)
Equation (27) implies that the pair (119864 sum119896
119894=1120582119894(119860
119901119894+ 119860
119889119901119894))
is regular and impulse-free for every 119901 isin S Thus byDefinition 1 system (12) is regular and impulse-free ByRemark 7 this implies that system (11) is regular and impulse-free
Now it will be shown that system (11) is stochasticallystable Define a new process (119909
119905 119903
119905) 119905 ge 0 by 119909
119905= 119909(119905 +
120579) minus2119889 le 120579 le 0 then (119909119905 119903
119905) 119905 ge 119889 is a Markovian process
with the initial state (120601(sdot) 1199030) Now for 119905 ge 119889 choose the
following stochastic Lyapunov-Krasovskii candidate for thissystem
119881 (119909119905 119901 119905) =
4
sum
119898=1
119881119898(119909
119905 119901 119905) (28)
where
1198811(119909
119905 119901 119905) = 119909
119879
(119905) 119864119879
119875119901119864119909 (119905) = 119909
119879
(119905) 119864119879
119883119901119909 (119905)
= 119909119879
(119905) 119864119879
119883119901119909 (119905)
1198812(119909
119905 119901 119905) = int
119905
119905minus119889
119909119879
(119904) 119876119901119909 (119904) d119904
1198813(119909
119905 119901 119905) = 119889int
0
minus119889
int
119905
119905+120579
119879
(119904) 119864119879
119885119864 (119904) d119904 d120579
1198814(119909
119905 119901 119905) = int
0
minus119889
int
119905
119905+120579
119909119879
(119904) 119876119909 (119904) d119904 d120579
(29)
6 Mathematical Problems in Engineering
Let L be the weak infinitesimal generator of the randomprocess (119909
119905 119901) 119905 ge 0 Then for each 119901 isin S
L119881 (119909119905 119901 119905) le 2119909
119879
(119905) 119883119879
119901119864 (119905)
+ 119909119879
(119905) (
119873
sum
119902=1
120587119901119902119864119879
119883119902)119909 (119905)
+ 119909119879
(119905) 119876119901119909 (119905)
minus 119909119879
(119905 minus 119889)119876119901119909 (119905 minus 119889)
+ int
119905
119905minus119889
119909119879
(119904) (
119873
sum
119902=1
120587119901119902119876
119902)119909 (119904) d119904
+ 119889119909119879
(119905) 119876119909 (119905)
minus int
119905
119905minus119889
119909119879
(119904) 119876119909 (119904) d119904
+ 119889
2
119879
(119905) 119864119879
119885119864 (119905)
minus 119889int
119905
119905minus119889
119879
(119904) 119864119879
119885119864 (119904) d119904
(30)
From (21) it is clear that
int
119905
119905minus119889
119909119879
(119904) (
119873
sum
119902=1
120587119901119902119876
119902)119909 (119904) d119904
lt int
119905
119905minus119889
119909119879
(119904) 119876119909 (119904) d119904
(31)
From Lemma 4 it follows that
minus 119889int
119905
119905minus119889
119879
(119904) 119864119879
119885119864 (119904) d119904
le [
119909 (119905)
119909 (119905 minus 119889)
]
119879
[
minus119864119879
119885119864 119864119879
119885119864
119864119879
119885119864 minus119864119879
119885119864
][
119909 (119905)
119909 (119905 minus 119889)
]
(32)
So it can be concluded that
L119881 (119909119905 119901 119905) le
119896
sum
119894=1
120578119879
(119905) Φ119901119894120578 (119905) (33)
where
120578119879
(119905) = [119909119879
(119905) 119909119879
(119905 minus 119889)]
Φ119901119894= [
Υ1119901119894
lowast
Υ2119901119894
Υ3119901119894
]
Υ1119901119894= Γ
1119901119894+ [
119860119879
119901119894
119861119879
119901119894
]119889
2
119885 [119860119901119894119861119901119894]
Υ3119901119894= minus119876
119901minus 119864
119879
119885119864 + 119860119879
119889119901119894119889
2
119885119860119889119901119894
Υ2119901119894= 119864
119879
[119885 0] 119864 + [119860119879
1198891199011198940]119883
119901
+ 119860119879
119889119901119894119889
2
119885 [119860119901119894119861119901119894]
(34)
Using (20) it is easy to see that there exists a scalar 120581 gt 0
such that for every 119901 isin S L119881(119909119905 119901 119905) le minus120581119909(119905)
This together with (36) and (48) implies that there exists ascalar ] such that
Eint119905
0
119909 (119904)2 d119904 = Eint
119889
0
119909 (119904)2 d119904
+Eint119905
119889
119909 (119904)2 d119904
le ]E 10038171003817100381710038171206011003817100381710038171003817
2
119889
(51)
Considering this andDefinition 1 system (11) is stochasticallystable for any constant delay 119889 satisfying 119889 isin [0 119889]Thereforesystem (11) is stochastically admissible This completes theproof
In the following a set of sufficient conditions will bedeveloped under which the considered system is guaranteedto be stochastically admissible with an119867
infinperformance
Definition 9 System (11) is said to be stochastically admissiblewith an 119867
infinperformance 120574 if it is stochastically admissible
when 119908(119905) = 0 and under zero initial condition for nonzero119908(119905) isin 119871
V2[0infin)
Eintinfin
0
119911119879
(119905) 119911 (119905) d119905 le 1205742 intinfin
0
119908119879
(119905) 119908 (119905) d119905 (52)
The following result can be presented
8 Mathematical Problems in Engineering
Theorem 10 For a prescribed scalar 119889 gt 0 there exists astate feedback controller (10) with 119906
and the other notations are the same as in Theorem 8
Proof FromTheorem 8 when119908(119905) = 0 system (11) is stochas-tically admissible Let
119869119911119908(119905) = Eint
119905
0
[119911119879
(119904) 119911 (119904) minus 1205742
119908119879
(119904) 119908 (119904)] d119904 (55)
Under zero initial condition it is easy to see that
119869119911119908(119905) le Eint
119905
0
[119911119879
(119904) 119911 (119904) minus 1205742
119908119879
(119904) 119908 (119904)
+L119881 (119909119904 119901 119904)] d119904
le Eint119905
0
119896
sum
119894=1
120582119894[120589
119879
(119904) (Ω119901119894+ Θ
119879
119901119894Θ
119901119894) 120589 (119904)] d119904
(56)
where
120589119879
(119905) = [119909119879
(119905) 119909119879
(119905 minus 119889) 119908119879
(119905)]
Ω119901119894=
[
[
[
[
Υ1119901119894
lowast lowast
Υ2119901119894
Υ3119901119894
lowast
[119861119879
1199081199011198940]119883
1199010 minus120574
2
119868
]
]
]
]
Θ119901119894= [119862
119901119894119862119889119901119894
119862119908119901119894]
(57)
and notations of Υ1119901119894
Υ2119901119894
and Υ3119901119894
are the same as inTheorem 8 Hence by Schur complement lemma and using
the similar method in the proof of Theorem 8 from and(53) it can be obtained that 119869
119911119908(119905) lt 0 for all 119905 gt 0
Therefore for any nonzero 119908(119905) isin 119871V2[0infin) (52) holds
Hence according to Definition 9 the system is stochasticallyadmissible with an 119867
infinperformance 120574 This completes the
proof
Remark 11 Compared with methods in [21 22] because ofthe method of the augmented matrix adopted in Theorems8 and 10 the number of LMIs needed to solve is relativelysmall in this paper When the value of 119896 is relatively largethe quality of the computation is greatly reduced some newrelaxation matrices added will reduce the conservatism ofcontrol conditions compared with previous literatures whichcan be seen from Example 2
4 The Design of the Static OutputFeedback Controller
When 119903119905= 119901 isin S consider the overall SMJFS as follows
119864 (119905) =
119896
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119889119901119894119909 (119905 minus 119889) + 119862
119908119901119894119908 (119905))
(60)
It is difficult to drive LMI-based conditions of the stochas-tic stabilization by employing the static output feedbackcontrol approach due to the appearance of crossing termsbetween system matrices and control gains And system (60)can be rewritten in the following form
119864119909 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(Λ
119901119894119895119909 (119905) + 119861
119908119901119894119908 (119905))
119911 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(119862
119901119894119895119909 (119905) + 119862
119908119901119894119908 (119905))
(61)
where
119864 = [
119864 0
0 0
] isin R(119899+1199011)times(119899+119901
1)
119862119901119894119895= [119862
119901119894119863
119901119894119870
119901119895]
119860119889119901119894
= [
119860119889119901119894
0
0 0
] isin R(119899+1199011)times(119899+119901
1)
119909 (119905) = [
119909 (119905)
119910 (119905)
]
119861119908119901119894
= [
119861119908119901119894
0
] isin R(119899+1199011)timesV
Λ119901119894119895= [
119860119901119894
119861119901119894119870
119901119895
119862119910119901119894
minus119868
]
(62)
Remark 12 For systems (60) and (61) it can be seen that
det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
119896
sum
119904=1
120582119894120582119895120582119904(119860
119901119894+ 119861
119901119894119870
119901119895119862119910119901119904))
= det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895Λ
119901119894119895)
det(119904119864
minus
119896
sum
119894=1
119896
sum
119895=1
119896
sum
119904=1
120582119894120582119895120582119904(119860
119901119894+ 119861
119901119894119870
119901119895119862119910119901119904
+ 119860119889119901119894))
= det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(Λ
119901119894119895+ 119860
119889119901119894))
(63)
As the discussion in Remark 7 the stochastic admissibility ofsystem (60) can be studied by means of system (61)
Theorem 13 There exists an output feedback controller (59)with controller gains 119870
119901119894= 119871
119901119894119884minus1
1199012(119901 isin S 119894 isin T) such that
system (60) with 119908(119905) = 0 is stochastically admissible if thereexist matrices 119875
119901gt 0 119876
119901gt 0 119876 gt 0 119885 gt 0 119871
119901119894 119878
119901 and 119884
1199012
119901 isin S 1 le 119894 = 119895 le 119896 such that and
and the other notations are the same as in Theorem 13
Remark 15 Compared with the method in [31 32] becauseof the augmented matrix adopted in Theorems 13 and 14 thenumber of LMIs needed to solve is greatly decreased Whenthe value of 119896 is relatively large the computational complexitywill be reduced On the other hand by the augmentedmatrixthere are not any crossing terms between system matricesand controller gains so assumptions for the output matrix
[23] the equality constraint for the output matrix [24] andthe bounding technique for crossing terms are not necessaryhere therefore the conservatism brought by them will nothappen
5 Numerical Examples
Two examples will be given to illustrate the validity ofdeveloped methods
Mathematical Problems in Engineering 11
Example 1 To illustrate the 119867infin
controller synthesis thefollowing nonlinear time delay system is considered
(1 + 119886 cos 120579 (119905)) 120579 (119905) = minus119887 1205793 (119905) + 119888120579 (119905) + 119888119889(119905 minus 119889)
It can be expressed exactly by the following fuzzy singularMarkovian jump form
119864 (119905) =
3
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119901119894119906 (119905)
+ 119861119908119901119894119908 (119905))
119911 (119905) =
3
sum
119894=1
120582119894119862119901119894119909 (119905)
119909 (119905) = 120601 (119905)
119905 isin [minus119889 0] 119901 isin 1 2 3
(70)
where
119864 =[
[
[
1 0 0
0 1 0
0 0 0
]
]
]
1198601199011=
[
[
[
[
0 1 0
0 0 1
119888 minus119887 (1205952
+ 2) 119886 minus 1
]
]
]
]
1198601199012=
[
[
[
[
0 1 0
0 0 1
119888 0 minus119886 minus 1 minus 1198861205952
]
]
]
]
1198601199013=[
[
[
0 1 0
0 0 1
119888 0 119886 minus 1
]
]
]
1198601198891199011
= 1198601198891199012
= 1198601198891199013
=[
[
[
0 0 0
0 0 0
1198881198890 0
]
]
]
11986111= 119861
12= 119861
13=[
[
[
0
0
119890
]
]
]
11986121= 119861
22= 119861
23=[
[
[
0
0
08119890
]
]
]
11986131= 119861
32= 119861
33=[
[
[
0
0
05119890
]
]
]
1198611199081199011
= 1198611199081199012
= 1198611199081199013
=[
[
[
0
0
119891
]
]
]
1198621199011= 119862
1199012= 119862
1199013= [1 0 0]
1205821=
1199092
2(119905)
1205952+ 2
1205822=
1 + cos1199091(119905)
1205952+ 2
1205823=
1205952
minus 1199092
2(119905) + 1 minus cos119909
1(119905)
1205952+ 2
(71)
It is seen that 0 le 120582119894le 1sum3
119894=1120582119894= 1 Let Π = [ minus02 02 0
01 minus03 02
02 03 minus05
]120574 = 1 by solving and (53) in Theorem 10 controller gainsare given by
11987011= [minus142939 minus140620 minus36426]
11987012= [minus142911 minus144082 minus32860]
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8minus20
minus15
minus10
minus5
0
5
0 2 4 86minus1
0123
Time
Time
Ope
n-lo
op st
ate r
espo
nse
Mar
kov
chai
n
x1
x2
x3
Figure 1 State responses of the open-loop system
11987013= [minus142988 minus144111 minus36425]
11987021= [minus179535 minus177012 minus44471]
11987022= [minus179509 minus179806 minus41585]
11987023= [minus179578 minus179828 minus44470]
11987031= [minus286360 minus283641 minus69941]
11987032= [minus286344 minus285420 minus68118]
11987033= [minus286377 minus285436 minus69941]
(72)
To demonstrate the effectiveness assuming the initialcondition 120601(119905) = [minus12 08 minus05]
119879 Figures 1 and 2 showstate responses of the open-loop system and the closed-loopsystem controlled by (10) respectively From Figure 1 it canbe seen that the open-loop system is not stochastically admis-sible and Figure 2 shows that when the controller obtainedby Theorem 10 is exerted to this system it is stochasticallyadmissible
Example 2 Consider the example without uncertainties in[6]
Mode 1 1198601= [
15 14
minus35 minus45] 119860
1198891= [
02 119886
minus024 minus04] 119861
1= [
1
1]
1198611199081= [
15
14] 119862
1= [05 1] 119862
1198891= [minus02 02] 119863
1= 02 and
1198621199081= 02Mode 2 119860
2= [
17 15
minus13 minus25] 119860
1198892= [
119887 11
minus021 minus03] 119861
2= [
09
09]
1198611199082= [
14
15] 119862
2= [04 03] 119862
1198892= [minus01 02]119863
2= 03 and
1198621199082= 03Π = [
minus1 1
1 minus1] 119864 = [
1 0
0 0] 119889 = 03 120574 = 26 and in [6]
119886 = minus05 119887 = 21 but in this paper minus24 le 119886 le 2 minus2 le 119887 le 48are taken
In Figure 3 ldquoordquo represents the range of the feasiblesolutions using Theorem 10 in this paper and ldquolowastrdquo representsthe range of the feasible solutions using Theorem 3 in [6]
minus15minus1
minus050
051
152
253
354
Mar
kov
chai
n
0 1 2 3 4 5 6 7 8
0 2 4 6 8minus1
0123
Time
Time
Clos
ed-lo
op st
ate r
espo
nse
x1
x2
x3
Figure 2 State responses of the closed-loop system
minus2 minus15 minus1 minus05 0 05 1 15 2minus2
minus1
0
1
2
3
4
a
b
Figure 3 Comparison of the feasible regions
This illustrates that themethod obtained in this paper has lessconservatism
6 Conclusions
In this paper the problem of mode-dependent 119867infin
controlfor singular Markovian jump fuzzy systems with time delayis considered This class of systems under considerationis described by T-S fuzzy models The main contributionof this paper is to design state feedback controllers andstatic output feedback controllers which can guarantee thatresulting closed-loop systems are stochastically admissiblewith an119867
infinperformance 120574 by the method of the augmented
matrix Finally two examples are given to demonstrate theeffectiveness of main results obtained here
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Mathematical Problems in Engineering 13
Acknowledgments
This work was supported by The National Natural ScienceFoundation of China under Grants nos 61304054 61273003and 61273008 respectively and Science and TechnologyResearch Fund of Liaoning Education Department underGrant no L2013051
References
[1] L Dai Singular Control Systems vol 118 of Lecture Notes inControl and Information Sciences Springer Berlin Germany1989
[2] F L Lewis ldquoA survey of linear singular systemsrdquo CircuitsSystems and Signal Processing vol 5 no 1 pp 3ndash36 1986
[3] S Xu and J Lam ldquoRobust stability and stabilization of discretesingular systems an equivalent characterizationrdquo IEEE Trans-actions on Automatic Control vol 49 no 4 pp 568ndash574 2004
[4] Z Wu and W Zhou ldquoDelay-dependent robust stabilization foruncertain singular systemswith discrete and distributed delaysrdquoJournal of ControlTheory and Applications vol 6 no 2 pp 171ndash176 2008
[5] Y Xia P Shi G Liu and D Rees ldquoRobust mixed 119867infin119867
2
state-feedback control for continuous-time descriptor systemswith parameter uncertaintiesrdquo Circuits Systems and SignalProcessing vol 24 no 4 pp 431ndash443 2005
[6] H Lu W Zhou C Duan and X Qi ldquoDelay-dependent robust119867
infincontrol for uncertain singular time-delay system with
Markovian jumping parametersrdquo Optimal Control Applicationsamp Methods vol 34 no 3 pp 296ndash307 2013
[7] Y Fang and K A Loparo ldquoStochastic stability of jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 7pp 1204ndash1208 2002
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] J Xiong and J Lam ldquoStabilization of discrete-time Markovianjump linear systems via time-delayed controllersrdquo Automaticavol 42 no 5 pp 747ndash753 2006
[10] W-H Chen Z-H Guan and P Yu ldquoDelay-dependent stabilityand 119867
infincontrol of uncertain discrete-time Markovian jump
systems with mode-dependent time delaysrdquo Systems amp ControlLetters vol 52 no 5 pp 361ndash376 2004
[11] E-K Boukas Control of Singular Systems with Random AbruptChanges Communications and Control Engineering SpringerBerlin Germany 2008
[12] J Zhang Y Xia and E K Boukas ldquoNew approach to Hinfin
control for Markovian jump singular systemsrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2273ndash2284 2010
[13] J Lam S Zhu S Xu and E-K Boukas ldquoRobust 119867infin
controlof descriptor discrete-time Markovian jump systemsrdquo Interna-tional Journal of Control vol 80 no 3 pp 374ndash385 2007
[14] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Basel Switzerland 2003
[15] G Wang Q Zhang and C Yang ldquoDissipative control forsingular Markovian jump systems with time delayrdquo OptimalControl ApplicationsampMethods vol 33 no 4 pp 415ndash432 2012
[16] Z Wu H Su and J Chu ldquoDelay-dependent 119867infin
control forsingular Markovian jump systems with time delayrdquo Optimal
Control Applications and Methods vol 30 no 5 pp 443ndash4612009
[17] C K Ahn ldquoT-S fuzzy119867infinsynchronization for chaotic systems
via delayed output feedback controlrdquo Nonlinear Dynamics vol59 no 4 pp 535ndash543 2010
[18] C K Ahn ldquoFuzzy delayed output feedback synchronizationfor time-delayed chaotic systemsrdquo Nonlinear Analysis HybridSystems vol 4 no 1 pp 16ndash24 2010
[19] H-N Wu and K-Y Cai ldquoMode-independent robust stabiliza-tion for uncertain Markovian jump nonlinear systems via fuzzycontrolrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 36 no 3 pp 509ndash519 2006
[20] Y Zhang S Xu Y Zou and J Lu ldquoDelay-dependent robuststabilization for uncertain discrete-time fuzzy Markovian jumpsystems with mode-dependent time delaysrdquo Fuzzy Sets andSystems vol 164 pp 66ndash81 2011
[21] A Zhang and H Fang ldquoRobustHinfinfuzzy control for uncertain
Markovian jump nonlinear singular systems with Wiener pro-cessrdquo Journal of Control Theory and Applications vol 8 no 2pp 205ndash210 2010
[22] L Li Q Zhang and B Zhu ldquo119867infin
fuzzy control for nonlin-ear time-delay singular Markovian jump systems with partlyunknown transition ratesrdquo Fuzzy Sets and Systems vol 254 pp106ndash125 2014
[23] J Dong and G-H Yang ldquoStatic output feedback 119867infin
controlof a class of nonlinear discrete-time systemsrdquo Fuzzy Sets andSystems vol 160 no 19 pp 2844ndash2859 2009
[24] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of
singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012
[25] J Wang H Wang A Xie and R Lu ldquoDelay-dependent 119867infin
control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2012
[26] D B Xu Q L Zhang and Y B Hu ldquoReduced-order Hinfin
controller design for uncertain descriptor systemsrdquo Acta Auto-matica Sinica vol 33 no 1 pp 44ndash47 2007
[27] L Xie and C E de Souza ldquoRobust 119867infin
control for linearsystems with norm bounder time-varying uncertaintiesrdquo IEEETransactions on Automatic Control vol 37 no 8 pp 1188ndash11911992
[28] Q-L Han ldquoAbsolute stability of time-delay systemswith sector-bounded nonlinearityrdquo Automatica vol 41 no 12 pp 2171ndash2176 2005
[29] D Yue J Lam and D W C Ho ldquoReliable 119867infin
control ofuncertain descriptor systems with multiple time delaysrdquo IEEProceedings ControlTheory and Applications vol 150 no 6 pp557ndash564 2003
[30] H D Tuan P Apkarian T Narikiyo and Y YamamotoldquoParameterized linear matrix inequality techniques in fuzzycontrol systemdesignrdquo IEEETransactions on Fuzzy Systems vol9 no 2 pp 324ndash332 2001
[31] S-W Kau H-J Lee C-M Yang C-H Lee L Hong and C-HFang ldquoRobust H
infinfuzzy static output feedback control of T-
S fuzzy systems with parametric uncertaintiesrdquo Fuzzy Sets andSystems vol 158 no 2 pp 135ndash146 2007
[32] D Saifia M Chadli S Labiod and T M Guerra ldquoRobust119867
infinstatic output feedback stabilization of T-S fuzzy systems
subject to actuator saturationrdquo International Journal of ControlAutomation and Systems vol 10 no 3 pp 613ndash622 2012
This together with (36) and (48) implies that there exists ascalar ] such that
Eint119905
0
119909 (119904)2 d119904 = Eint
119889
0
119909 (119904)2 d119904
+Eint119905
119889
119909 (119904)2 d119904
le ]E 10038171003817100381710038171206011003817100381710038171003817
2
119889
(51)
Considering this andDefinition 1 system (11) is stochasticallystable for any constant delay 119889 satisfying 119889 isin [0 119889]Thereforesystem (11) is stochastically admissible This completes theproof
In the following a set of sufficient conditions will bedeveloped under which the considered system is guaranteedto be stochastically admissible with an119867
infinperformance
Definition 9 System (11) is said to be stochastically admissiblewith an 119867
infinperformance 120574 if it is stochastically admissible
when 119908(119905) = 0 and under zero initial condition for nonzero119908(119905) isin 119871
V2[0infin)
Eintinfin
0
119911119879
(119905) 119911 (119905) d119905 le 1205742 intinfin
0
119908119879
(119905) 119908 (119905) d119905 (52)
The following result can be presented
8 Mathematical Problems in Engineering
Theorem 10 For a prescribed scalar 119889 gt 0 there exists astate feedback controller (10) with 119906
and the other notations are the same as in Theorem 8
Proof FromTheorem 8 when119908(119905) = 0 system (11) is stochas-tically admissible Let
119869119911119908(119905) = Eint
119905
0
[119911119879
(119904) 119911 (119904) minus 1205742
119908119879
(119904) 119908 (119904)] d119904 (55)
Under zero initial condition it is easy to see that
119869119911119908(119905) le Eint
119905
0
[119911119879
(119904) 119911 (119904) minus 1205742
119908119879
(119904) 119908 (119904)
+L119881 (119909119904 119901 119904)] d119904
le Eint119905
0
119896
sum
119894=1
120582119894[120589
119879
(119904) (Ω119901119894+ Θ
119879
119901119894Θ
119901119894) 120589 (119904)] d119904
(56)
where
120589119879
(119905) = [119909119879
(119905) 119909119879
(119905 minus 119889) 119908119879
(119905)]
Ω119901119894=
[
[
[
[
Υ1119901119894
lowast lowast
Υ2119901119894
Υ3119901119894
lowast
[119861119879
1199081199011198940]119883
1199010 minus120574
2
119868
]
]
]
]
Θ119901119894= [119862
119901119894119862119889119901119894
119862119908119901119894]
(57)
and notations of Υ1119901119894
Υ2119901119894
and Υ3119901119894
are the same as inTheorem 8 Hence by Schur complement lemma and using
the similar method in the proof of Theorem 8 from and(53) it can be obtained that 119869
119911119908(119905) lt 0 for all 119905 gt 0
Therefore for any nonzero 119908(119905) isin 119871V2[0infin) (52) holds
Hence according to Definition 9 the system is stochasticallyadmissible with an 119867
infinperformance 120574 This completes the
proof
Remark 11 Compared with methods in [21 22] because ofthe method of the augmented matrix adopted in Theorems8 and 10 the number of LMIs needed to solve is relativelysmall in this paper When the value of 119896 is relatively largethe quality of the computation is greatly reduced some newrelaxation matrices added will reduce the conservatism ofcontrol conditions compared with previous literatures whichcan be seen from Example 2
4 The Design of the Static OutputFeedback Controller
When 119903119905= 119901 isin S consider the overall SMJFS as follows
119864 (119905) =
119896
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119889119901119894119909 (119905 minus 119889) + 119862
119908119901119894119908 (119905))
(60)
It is difficult to drive LMI-based conditions of the stochas-tic stabilization by employing the static output feedbackcontrol approach due to the appearance of crossing termsbetween system matrices and control gains And system (60)can be rewritten in the following form
119864119909 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(Λ
119901119894119895119909 (119905) + 119861
119908119901119894119908 (119905))
119911 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(119862
119901119894119895119909 (119905) + 119862
119908119901119894119908 (119905))
(61)
where
119864 = [
119864 0
0 0
] isin R(119899+1199011)times(119899+119901
1)
119862119901119894119895= [119862
119901119894119863
119901119894119870
119901119895]
119860119889119901119894
= [
119860119889119901119894
0
0 0
] isin R(119899+1199011)times(119899+119901
1)
119909 (119905) = [
119909 (119905)
119910 (119905)
]
119861119908119901119894
= [
119861119908119901119894
0
] isin R(119899+1199011)timesV
Λ119901119894119895= [
119860119901119894
119861119901119894119870
119901119895
119862119910119901119894
minus119868
]
(62)
Remark 12 For systems (60) and (61) it can be seen that
det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
119896
sum
119904=1
120582119894120582119895120582119904(119860
119901119894+ 119861
119901119894119870
119901119895119862119910119901119904))
= det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895Λ
119901119894119895)
det(119904119864
minus
119896
sum
119894=1
119896
sum
119895=1
119896
sum
119904=1
120582119894120582119895120582119904(119860
119901119894+ 119861
119901119894119870
119901119895119862119910119901119904
+ 119860119889119901119894))
= det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(Λ
119901119894119895+ 119860
119889119901119894))
(63)
As the discussion in Remark 7 the stochastic admissibility ofsystem (60) can be studied by means of system (61)
Theorem 13 There exists an output feedback controller (59)with controller gains 119870
119901119894= 119871
119901119894119884minus1
1199012(119901 isin S 119894 isin T) such that
system (60) with 119908(119905) = 0 is stochastically admissible if thereexist matrices 119875
119901gt 0 119876
119901gt 0 119876 gt 0 119885 gt 0 119871
119901119894 119878
119901 and 119884
1199012
119901 isin S 1 le 119894 = 119895 le 119896 such that and
and the other notations are the same as in Theorem 13
Remark 15 Compared with the method in [31 32] becauseof the augmented matrix adopted in Theorems 13 and 14 thenumber of LMIs needed to solve is greatly decreased Whenthe value of 119896 is relatively large the computational complexitywill be reduced On the other hand by the augmentedmatrixthere are not any crossing terms between system matricesand controller gains so assumptions for the output matrix
[23] the equality constraint for the output matrix [24] andthe bounding technique for crossing terms are not necessaryhere therefore the conservatism brought by them will nothappen
5 Numerical Examples
Two examples will be given to illustrate the validity ofdeveloped methods
Mathematical Problems in Engineering 11
Example 1 To illustrate the 119867infin
controller synthesis thefollowing nonlinear time delay system is considered
(1 + 119886 cos 120579 (119905)) 120579 (119905) = minus119887 1205793 (119905) + 119888120579 (119905) + 119888119889(119905 minus 119889)
It can be expressed exactly by the following fuzzy singularMarkovian jump form
119864 (119905) =
3
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119901119894119906 (119905)
+ 119861119908119901119894119908 (119905))
119911 (119905) =
3
sum
119894=1
120582119894119862119901119894119909 (119905)
119909 (119905) = 120601 (119905)
119905 isin [minus119889 0] 119901 isin 1 2 3
(70)
where
119864 =[
[
[
1 0 0
0 1 0
0 0 0
]
]
]
1198601199011=
[
[
[
[
0 1 0
0 0 1
119888 minus119887 (1205952
+ 2) 119886 minus 1
]
]
]
]
1198601199012=
[
[
[
[
0 1 0
0 0 1
119888 0 minus119886 minus 1 minus 1198861205952
]
]
]
]
1198601199013=[
[
[
0 1 0
0 0 1
119888 0 119886 minus 1
]
]
]
1198601198891199011
= 1198601198891199012
= 1198601198891199013
=[
[
[
0 0 0
0 0 0
1198881198890 0
]
]
]
11986111= 119861
12= 119861
13=[
[
[
0
0
119890
]
]
]
11986121= 119861
22= 119861
23=[
[
[
0
0
08119890
]
]
]
11986131= 119861
32= 119861
33=[
[
[
0
0
05119890
]
]
]
1198611199081199011
= 1198611199081199012
= 1198611199081199013
=[
[
[
0
0
119891
]
]
]
1198621199011= 119862
1199012= 119862
1199013= [1 0 0]
1205821=
1199092
2(119905)
1205952+ 2
1205822=
1 + cos1199091(119905)
1205952+ 2
1205823=
1205952
minus 1199092
2(119905) + 1 minus cos119909
1(119905)
1205952+ 2
(71)
It is seen that 0 le 120582119894le 1sum3
119894=1120582119894= 1 Let Π = [ minus02 02 0
01 minus03 02
02 03 minus05
]120574 = 1 by solving and (53) in Theorem 10 controller gainsare given by
11987011= [minus142939 minus140620 minus36426]
11987012= [minus142911 minus144082 minus32860]
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8minus20
minus15
minus10
minus5
0
5
0 2 4 86minus1
0123
Time
Time
Ope
n-lo
op st
ate r
espo
nse
Mar
kov
chai
n
x1
x2
x3
Figure 1 State responses of the open-loop system
11987013= [minus142988 minus144111 minus36425]
11987021= [minus179535 minus177012 minus44471]
11987022= [minus179509 minus179806 minus41585]
11987023= [minus179578 minus179828 minus44470]
11987031= [minus286360 minus283641 minus69941]
11987032= [minus286344 minus285420 minus68118]
11987033= [minus286377 minus285436 minus69941]
(72)
To demonstrate the effectiveness assuming the initialcondition 120601(119905) = [minus12 08 minus05]
119879 Figures 1 and 2 showstate responses of the open-loop system and the closed-loopsystem controlled by (10) respectively From Figure 1 it canbe seen that the open-loop system is not stochastically admis-sible and Figure 2 shows that when the controller obtainedby Theorem 10 is exerted to this system it is stochasticallyadmissible
Example 2 Consider the example without uncertainties in[6]
Mode 1 1198601= [
15 14
minus35 minus45] 119860
1198891= [
02 119886
minus024 minus04] 119861
1= [
1
1]
1198611199081= [
15
14] 119862
1= [05 1] 119862
1198891= [minus02 02] 119863
1= 02 and
1198621199081= 02Mode 2 119860
2= [
17 15
minus13 minus25] 119860
1198892= [
119887 11
minus021 minus03] 119861
2= [
09
09]
1198611199082= [
14
15] 119862
2= [04 03] 119862
1198892= [minus01 02]119863
2= 03 and
1198621199082= 03Π = [
minus1 1
1 minus1] 119864 = [
1 0
0 0] 119889 = 03 120574 = 26 and in [6]
119886 = minus05 119887 = 21 but in this paper minus24 le 119886 le 2 minus2 le 119887 le 48are taken
In Figure 3 ldquoordquo represents the range of the feasiblesolutions using Theorem 10 in this paper and ldquolowastrdquo representsthe range of the feasible solutions using Theorem 3 in [6]
minus15minus1
minus050
051
152
253
354
Mar
kov
chai
n
0 1 2 3 4 5 6 7 8
0 2 4 6 8minus1
0123
Time
Time
Clos
ed-lo
op st
ate r
espo
nse
x1
x2
x3
Figure 2 State responses of the closed-loop system
minus2 minus15 minus1 minus05 0 05 1 15 2minus2
minus1
0
1
2
3
4
a
b
Figure 3 Comparison of the feasible regions
This illustrates that themethod obtained in this paper has lessconservatism
6 Conclusions
In this paper the problem of mode-dependent 119867infin
controlfor singular Markovian jump fuzzy systems with time delayis considered This class of systems under considerationis described by T-S fuzzy models The main contributionof this paper is to design state feedback controllers andstatic output feedback controllers which can guarantee thatresulting closed-loop systems are stochastically admissiblewith an119867
infinperformance 120574 by the method of the augmented
matrix Finally two examples are given to demonstrate theeffectiveness of main results obtained here
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Mathematical Problems in Engineering 13
Acknowledgments
This work was supported by The National Natural ScienceFoundation of China under Grants nos 61304054 61273003and 61273008 respectively and Science and TechnologyResearch Fund of Liaoning Education Department underGrant no L2013051
References
[1] L Dai Singular Control Systems vol 118 of Lecture Notes inControl and Information Sciences Springer Berlin Germany1989
[2] F L Lewis ldquoA survey of linear singular systemsrdquo CircuitsSystems and Signal Processing vol 5 no 1 pp 3ndash36 1986
[3] S Xu and J Lam ldquoRobust stability and stabilization of discretesingular systems an equivalent characterizationrdquo IEEE Trans-actions on Automatic Control vol 49 no 4 pp 568ndash574 2004
[4] Z Wu and W Zhou ldquoDelay-dependent robust stabilization foruncertain singular systemswith discrete and distributed delaysrdquoJournal of ControlTheory and Applications vol 6 no 2 pp 171ndash176 2008
[5] Y Xia P Shi G Liu and D Rees ldquoRobust mixed 119867infin119867
2
state-feedback control for continuous-time descriptor systemswith parameter uncertaintiesrdquo Circuits Systems and SignalProcessing vol 24 no 4 pp 431ndash443 2005
[6] H Lu W Zhou C Duan and X Qi ldquoDelay-dependent robust119867
infincontrol for uncertain singular time-delay system with
Markovian jumping parametersrdquo Optimal Control Applicationsamp Methods vol 34 no 3 pp 296ndash307 2013
[7] Y Fang and K A Loparo ldquoStochastic stability of jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 7pp 1204ndash1208 2002
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] J Xiong and J Lam ldquoStabilization of discrete-time Markovianjump linear systems via time-delayed controllersrdquo Automaticavol 42 no 5 pp 747ndash753 2006
[10] W-H Chen Z-H Guan and P Yu ldquoDelay-dependent stabilityand 119867
infincontrol of uncertain discrete-time Markovian jump
systems with mode-dependent time delaysrdquo Systems amp ControlLetters vol 52 no 5 pp 361ndash376 2004
[11] E-K Boukas Control of Singular Systems with Random AbruptChanges Communications and Control Engineering SpringerBerlin Germany 2008
[12] J Zhang Y Xia and E K Boukas ldquoNew approach to Hinfin
control for Markovian jump singular systemsrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2273ndash2284 2010
[13] J Lam S Zhu S Xu and E-K Boukas ldquoRobust 119867infin
controlof descriptor discrete-time Markovian jump systemsrdquo Interna-tional Journal of Control vol 80 no 3 pp 374ndash385 2007
[14] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Basel Switzerland 2003
[15] G Wang Q Zhang and C Yang ldquoDissipative control forsingular Markovian jump systems with time delayrdquo OptimalControl ApplicationsampMethods vol 33 no 4 pp 415ndash432 2012
[16] Z Wu H Su and J Chu ldquoDelay-dependent 119867infin
control forsingular Markovian jump systems with time delayrdquo Optimal
Control Applications and Methods vol 30 no 5 pp 443ndash4612009
[17] C K Ahn ldquoT-S fuzzy119867infinsynchronization for chaotic systems
via delayed output feedback controlrdquo Nonlinear Dynamics vol59 no 4 pp 535ndash543 2010
[18] C K Ahn ldquoFuzzy delayed output feedback synchronizationfor time-delayed chaotic systemsrdquo Nonlinear Analysis HybridSystems vol 4 no 1 pp 16ndash24 2010
[19] H-N Wu and K-Y Cai ldquoMode-independent robust stabiliza-tion for uncertain Markovian jump nonlinear systems via fuzzycontrolrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 36 no 3 pp 509ndash519 2006
[20] Y Zhang S Xu Y Zou and J Lu ldquoDelay-dependent robuststabilization for uncertain discrete-time fuzzy Markovian jumpsystems with mode-dependent time delaysrdquo Fuzzy Sets andSystems vol 164 pp 66ndash81 2011
[21] A Zhang and H Fang ldquoRobustHinfinfuzzy control for uncertain
Markovian jump nonlinear singular systems with Wiener pro-cessrdquo Journal of Control Theory and Applications vol 8 no 2pp 205ndash210 2010
[22] L Li Q Zhang and B Zhu ldquo119867infin
fuzzy control for nonlin-ear time-delay singular Markovian jump systems with partlyunknown transition ratesrdquo Fuzzy Sets and Systems vol 254 pp106ndash125 2014
[23] J Dong and G-H Yang ldquoStatic output feedback 119867infin
controlof a class of nonlinear discrete-time systemsrdquo Fuzzy Sets andSystems vol 160 no 19 pp 2844ndash2859 2009
[24] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of
singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012
[25] J Wang H Wang A Xie and R Lu ldquoDelay-dependent 119867infin
control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2012
[26] D B Xu Q L Zhang and Y B Hu ldquoReduced-order Hinfin
controller design for uncertain descriptor systemsrdquo Acta Auto-matica Sinica vol 33 no 1 pp 44ndash47 2007
[27] L Xie and C E de Souza ldquoRobust 119867infin
control for linearsystems with norm bounder time-varying uncertaintiesrdquo IEEETransactions on Automatic Control vol 37 no 8 pp 1188ndash11911992
[28] Q-L Han ldquoAbsolute stability of time-delay systemswith sector-bounded nonlinearityrdquo Automatica vol 41 no 12 pp 2171ndash2176 2005
[29] D Yue J Lam and D W C Ho ldquoReliable 119867infin
control ofuncertain descriptor systems with multiple time delaysrdquo IEEProceedings ControlTheory and Applications vol 150 no 6 pp557ndash564 2003
[30] H D Tuan P Apkarian T Narikiyo and Y YamamotoldquoParameterized linear matrix inequality techniques in fuzzycontrol systemdesignrdquo IEEETransactions on Fuzzy Systems vol9 no 2 pp 324ndash332 2001
[31] S-W Kau H-J Lee C-M Yang C-H Lee L Hong and C-HFang ldquoRobust H
infinfuzzy static output feedback control of T-
S fuzzy systems with parametric uncertaintiesrdquo Fuzzy Sets andSystems vol 158 no 2 pp 135ndash146 2007
[32] D Saifia M Chadli S Labiod and T M Guerra ldquoRobust119867
infinstatic output feedback stabilization of T-S fuzzy systems
subject to actuator saturationrdquo International Journal of ControlAutomation and Systems vol 10 no 3 pp 613ndash622 2012
This together with (36) and (48) implies that there exists ascalar ] such that
Eint119905
0
119909 (119904)2 d119904 = Eint
119889
0
119909 (119904)2 d119904
+Eint119905
119889
119909 (119904)2 d119904
le ]E 10038171003817100381710038171206011003817100381710038171003817
2
119889
(51)
Considering this andDefinition 1 system (11) is stochasticallystable for any constant delay 119889 satisfying 119889 isin [0 119889]Thereforesystem (11) is stochastically admissible This completes theproof
In the following a set of sufficient conditions will bedeveloped under which the considered system is guaranteedto be stochastically admissible with an119867
infinperformance
Definition 9 System (11) is said to be stochastically admissiblewith an 119867
infinperformance 120574 if it is stochastically admissible
when 119908(119905) = 0 and under zero initial condition for nonzero119908(119905) isin 119871
V2[0infin)
Eintinfin
0
119911119879
(119905) 119911 (119905) d119905 le 1205742 intinfin
0
119908119879
(119905) 119908 (119905) d119905 (52)
The following result can be presented
8 Mathematical Problems in Engineering
Theorem 10 For a prescribed scalar 119889 gt 0 there exists astate feedback controller (10) with 119906
and the other notations are the same as in Theorem 8
Proof FromTheorem 8 when119908(119905) = 0 system (11) is stochas-tically admissible Let
119869119911119908(119905) = Eint
119905
0
[119911119879
(119904) 119911 (119904) minus 1205742
119908119879
(119904) 119908 (119904)] d119904 (55)
Under zero initial condition it is easy to see that
119869119911119908(119905) le Eint
119905
0
[119911119879
(119904) 119911 (119904) minus 1205742
119908119879
(119904) 119908 (119904)
+L119881 (119909119904 119901 119904)] d119904
le Eint119905
0
119896
sum
119894=1
120582119894[120589
119879
(119904) (Ω119901119894+ Θ
119879
119901119894Θ
119901119894) 120589 (119904)] d119904
(56)
where
120589119879
(119905) = [119909119879
(119905) 119909119879
(119905 minus 119889) 119908119879
(119905)]
Ω119901119894=
[
[
[
[
Υ1119901119894
lowast lowast
Υ2119901119894
Υ3119901119894
lowast
[119861119879
1199081199011198940]119883
1199010 minus120574
2
119868
]
]
]
]
Θ119901119894= [119862
119901119894119862119889119901119894
119862119908119901119894]
(57)
and notations of Υ1119901119894
Υ2119901119894
and Υ3119901119894
are the same as inTheorem 8 Hence by Schur complement lemma and using
the similar method in the proof of Theorem 8 from and(53) it can be obtained that 119869
119911119908(119905) lt 0 for all 119905 gt 0
Therefore for any nonzero 119908(119905) isin 119871V2[0infin) (52) holds
Hence according to Definition 9 the system is stochasticallyadmissible with an 119867
infinperformance 120574 This completes the
proof
Remark 11 Compared with methods in [21 22] because ofthe method of the augmented matrix adopted in Theorems8 and 10 the number of LMIs needed to solve is relativelysmall in this paper When the value of 119896 is relatively largethe quality of the computation is greatly reduced some newrelaxation matrices added will reduce the conservatism ofcontrol conditions compared with previous literatures whichcan be seen from Example 2
4 The Design of the Static OutputFeedback Controller
When 119903119905= 119901 isin S consider the overall SMJFS as follows
119864 (119905) =
119896
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119889119901119894119909 (119905 minus 119889) + 119862
119908119901119894119908 (119905))
(60)
It is difficult to drive LMI-based conditions of the stochas-tic stabilization by employing the static output feedbackcontrol approach due to the appearance of crossing termsbetween system matrices and control gains And system (60)can be rewritten in the following form
119864119909 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(Λ
119901119894119895119909 (119905) + 119861
119908119901119894119908 (119905))
119911 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(119862
119901119894119895119909 (119905) + 119862
119908119901119894119908 (119905))
(61)
where
119864 = [
119864 0
0 0
] isin R(119899+1199011)times(119899+119901
1)
119862119901119894119895= [119862
119901119894119863
119901119894119870
119901119895]
119860119889119901119894
= [
119860119889119901119894
0
0 0
] isin R(119899+1199011)times(119899+119901
1)
119909 (119905) = [
119909 (119905)
119910 (119905)
]
119861119908119901119894
= [
119861119908119901119894
0
] isin R(119899+1199011)timesV
Λ119901119894119895= [
119860119901119894
119861119901119894119870
119901119895
119862119910119901119894
minus119868
]
(62)
Remark 12 For systems (60) and (61) it can be seen that
det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
119896
sum
119904=1
120582119894120582119895120582119904(119860
119901119894+ 119861
119901119894119870
119901119895119862119910119901119904))
= det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895Λ
119901119894119895)
det(119904119864
minus
119896
sum
119894=1
119896
sum
119895=1
119896
sum
119904=1
120582119894120582119895120582119904(119860
119901119894+ 119861
119901119894119870
119901119895119862119910119901119904
+ 119860119889119901119894))
= det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(Λ
119901119894119895+ 119860
119889119901119894))
(63)
As the discussion in Remark 7 the stochastic admissibility ofsystem (60) can be studied by means of system (61)
Theorem 13 There exists an output feedback controller (59)with controller gains 119870
119901119894= 119871
119901119894119884minus1
1199012(119901 isin S 119894 isin T) such that
system (60) with 119908(119905) = 0 is stochastically admissible if thereexist matrices 119875
119901gt 0 119876
119901gt 0 119876 gt 0 119885 gt 0 119871
119901119894 119878
119901 and 119884
1199012
119901 isin S 1 le 119894 = 119895 le 119896 such that and
and the other notations are the same as in Theorem 13
Remark 15 Compared with the method in [31 32] becauseof the augmented matrix adopted in Theorems 13 and 14 thenumber of LMIs needed to solve is greatly decreased Whenthe value of 119896 is relatively large the computational complexitywill be reduced On the other hand by the augmentedmatrixthere are not any crossing terms between system matricesand controller gains so assumptions for the output matrix
[23] the equality constraint for the output matrix [24] andthe bounding technique for crossing terms are not necessaryhere therefore the conservatism brought by them will nothappen
5 Numerical Examples
Two examples will be given to illustrate the validity ofdeveloped methods
Mathematical Problems in Engineering 11
Example 1 To illustrate the 119867infin
controller synthesis thefollowing nonlinear time delay system is considered
(1 + 119886 cos 120579 (119905)) 120579 (119905) = minus119887 1205793 (119905) + 119888120579 (119905) + 119888119889(119905 minus 119889)
It can be expressed exactly by the following fuzzy singularMarkovian jump form
119864 (119905) =
3
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119901119894119906 (119905)
+ 119861119908119901119894119908 (119905))
119911 (119905) =
3
sum
119894=1
120582119894119862119901119894119909 (119905)
119909 (119905) = 120601 (119905)
119905 isin [minus119889 0] 119901 isin 1 2 3
(70)
where
119864 =[
[
[
1 0 0
0 1 0
0 0 0
]
]
]
1198601199011=
[
[
[
[
0 1 0
0 0 1
119888 minus119887 (1205952
+ 2) 119886 minus 1
]
]
]
]
1198601199012=
[
[
[
[
0 1 0
0 0 1
119888 0 minus119886 minus 1 minus 1198861205952
]
]
]
]
1198601199013=[
[
[
0 1 0
0 0 1
119888 0 119886 minus 1
]
]
]
1198601198891199011
= 1198601198891199012
= 1198601198891199013
=[
[
[
0 0 0
0 0 0
1198881198890 0
]
]
]
11986111= 119861
12= 119861
13=[
[
[
0
0
119890
]
]
]
11986121= 119861
22= 119861
23=[
[
[
0
0
08119890
]
]
]
11986131= 119861
32= 119861
33=[
[
[
0
0
05119890
]
]
]
1198611199081199011
= 1198611199081199012
= 1198611199081199013
=[
[
[
0
0
119891
]
]
]
1198621199011= 119862
1199012= 119862
1199013= [1 0 0]
1205821=
1199092
2(119905)
1205952+ 2
1205822=
1 + cos1199091(119905)
1205952+ 2
1205823=
1205952
minus 1199092
2(119905) + 1 minus cos119909
1(119905)
1205952+ 2
(71)
It is seen that 0 le 120582119894le 1sum3
119894=1120582119894= 1 Let Π = [ minus02 02 0
01 minus03 02
02 03 minus05
]120574 = 1 by solving and (53) in Theorem 10 controller gainsare given by
11987011= [minus142939 minus140620 minus36426]
11987012= [minus142911 minus144082 minus32860]
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8minus20
minus15
minus10
minus5
0
5
0 2 4 86minus1
0123
Time
Time
Ope
n-lo
op st
ate r
espo
nse
Mar
kov
chai
n
x1
x2
x3
Figure 1 State responses of the open-loop system
11987013= [minus142988 minus144111 minus36425]
11987021= [minus179535 minus177012 minus44471]
11987022= [minus179509 minus179806 minus41585]
11987023= [minus179578 minus179828 minus44470]
11987031= [minus286360 minus283641 minus69941]
11987032= [minus286344 minus285420 minus68118]
11987033= [minus286377 minus285436 minus69941]
(72)
To demonstrate the effectiveness assuming the initialcondition 120601(119905) = [minus12 08 minus05]
119879 Figures 1 and 2 showstate responses of the open-loop system and the closed-loopsystem controlled by (10) respectively From Figure 1 it canbe seen that the open-loop system is not stochastically admis-sible and Figure 2 shows that when the controller obtainedby Theorem 10 is exerted to this system it is stochasticallyadmissible
Example 2 Consider the example without uncertainties in[6]
Mode 1 1198601= [
15 14
minus35 minus45] 119860
1198891= [
02 119886
minus024 minus04] 119861
1= [
1
1]
1198611199081= [
15
14] 119862
1= [05 1] 119862
1198891= [minus02 02] 119863
1= 02 and
1198621199081= 02Mode 2 119860
2= [
17 15
minus13 minus25] 119860
1198892= [
119887 11
minus021 minus03] 119861
2= [
09
09]
1198611199082= [
14
15] 119862
2= [04 03] 119862
1198892= [minus01 02]119863
2= 03 and
1198621199082= 03Π = [
minus1 1
1 minus1] 119864 = [
1 0
0 0] 119889 = 03 120574 = 26 and in [6]
119886 = minus05 119887 = 21 but in this paper minus24 le 119886 le 2 minus2 le 119887 le 48are taken
In Figure 3 ldquoordquo represents the range of the feasiblesolutions using Theorem 10 in this paper and ldquolowastrdquo representsthe range of the feasible solutions using Theorem 3 in [6]
minus15minus1
minus050
051
152
253
354
Mar
kov
chai
n
0 1 2 3 4 5 6 7 8
0 2 4 6 8minus1
0123
Time
Time
Clos
ed-lo
op st
ate r
espo
nse
x1
x2
x3
Figure 2 State responses of the closed-loop system
minus2 minus15 minus1 minus05 0 05 1 15 2minus2
minus1
0
1
2
3
4
a
b
Figure 3 Comparison of the feasible regions
This illustrates that themethod obtained in this paper has lessconservatism
6 Conclusions
In this paper the problem of mode-dependent 119867infin
controlfor singular Markovian jump fuzzy systems with time delayis considered This class of systems under considerationis described by T-S fuzzy models The main contributionof this paper is to design state feedback controllers andstatic output feedback controllers which can guarantee thatresulting closed-loop systems are stochastically admissiblewith an119867
infinperformance 120574 by the method of the augmented
matrix Finally two examples are given to demonstrate theeffectiveness of main results obtained here
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Mathematical Problems in Engineering 13
Acknowledgments
This work was supported by The National Natural ScienceFoundation of China under Grants nos 61304054 61273003and 61273008 respectively and Science and TechnologyResearch Fund of Liaoning Education Department underGrant no L2013051
References
[1] L Dai Singular Control Systems vol 118 of Lecture Notes inControl and Information Sciences Springer Berlin Germany1989
[2] F L Lewis ldquoA survey of linear singular systemsrdquo CircuitsSystems and Signal Processing vol 5 no 1 pp 3ndash36 1986
[3] S Xu and J Lam ldquoRobust stability and stabilization of discretesingular systems an equivalent characterizationrdquo IEEE Trans-actions on Automatic Control vol 49 no 4 pp 568ndash574 2004
[4] Z Wu and W Zhou ldquoDelay-dependent robust stabilization foruncertain singular systemswith discrete and distributed delaysrdquoJournal of ControlTheory and Applications vol 6 no 2 pp 171ndash176 2008
[5] Y Xia P Shi G Liu and D Rees ldquoRobust mixed 119867infin119867
2
state-feedback control for continuous-time descriptor systemswith parameter uncertaintiesrdquo Circuits Systems and SignalProcessing vol 24 no 4 pp 431ndash443 2005
[6] H Lu W Zhou C Duan and X Qi ldquoDelay-dependent robust119867
infincontrol for uncertain singular time-delay system with
Markovian jumping parametersrdquo Optimal Control Applicationsamp Methods vol 34 no 3 pp 296ndash307 2013
[7] Y Fang and K A Loparo ldquoStochastic stability of jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 7pp 1204ndash1208 2002
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] J Xiong and J Lam ldquoStabilization of discrete-time Markovianjump linear systems via time-delayed controllersrdquo Automaticavol 42 no 5 pp 747ndash753 2006
[10] W-H Chen Z-H Guan and P Yu ldquoDelay-dependent stabilityand 119867
infincontrol of uncertain discrete-time Markovian jump
systems with mode-dependent time delaysrdquo Systems amp ControlLetters vol 52 no 5 pp 361ndash376 2004
[11] E-K Boukas Control of Singular Systems with Random AbruptChanges Communications and Control Engineering SpringerBerlin Germany 2008
[12] J Zhang Y Xia and E K Boukas ldquoNew approach to Hinfin
control for Markovian jump singular systemsrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2273ndash2284 2010
[13] J Lam S Zhu S Xu and E-K Boukas ldquoRobust 119867infin
controlof descriptor discrete-time Markovian jump systemsrdquo Interna-tional Journal of Control vol 80 no 3 pp 374ndash385 2007
[14] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Basel Switzerland 2003
[15] G Wang Q Zhang and C Yang ldquoDissipative control forsingular Markovian jump systems with time delayrdquo OptimalControl ApplicationsampMethods vol 33 no 4 pp 415ndash432 2012
[16] Z Wu H Su and J Chu ldquoDelay-dependent 119867infin
control forsingular Markovian jump systems with time delayrdquo Optimal
Control Applications and Methods vol 30 no 5 pp 443ndash4612009
[17] C K Ahn ldquoT-S fuzzy119867infinsynchronization for chaotic systems
via delayed output feedback controlrdquo Nonlinear Dynamics vol59 no 4 pp 535ndash543 2010
[18] C K Ahn ldquoFuzzy delayed output feedback synchronizationfor time-delayed chaotic systemsrdquo Nonlinear Analysis HybridSystems vol 4 no 1 pp 16ndash24 2010
[19] H-N Wu and K-Y Cai ldquoMode-independent robust stabiliza-tion for uncertain Markovian jump nonlinear systems via fuzzycontrolrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 36 no 3 pp 509ndash519 2006
[20] Y Zhang S Xu Y Zou and J Lu ldquoDelay-dependent robuststabilization for uncertain discrete-time fuzzy Markovian jumpsystems with mode-dependent time delaysrdquo Fuzzy Sets andSystems vol 164 pp 66ndash81 2011
[21] A Zhang and H Fang ldquoRobustHinfinfuzzy control for uncertain
Markovian jump nonlinear singular systems with Wiener pro-cessrdquo Journal of Control Theory and Applications vol 8 no 2pp 205ndash210 2010
[22] L Li Q Zhang and B Zhu ldquo119867infin
fuzzy control for nonlin-ear time-delay singular Markovian jump systems with partlyunknown transition ratesrdquo Fuzzy Sets and Systems vol 254 pp106ndash125 2014
[23] J Dong and G-H Yang ldquoStatic output feedback 119867infin
controlof a class of nonlinear discrete-time systemsrdquo Fuzzy Sets andSystems vol 160 no 19 pp 2844ndash2859 2009
[24] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of
singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012
[25] J Wang H Wang A Xie and R Lu ldquoDelay-dependent 119867infin
control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2012
[26] D B Xu Q L Zhang and Y B Hu ldquoReduced-order Hinfin
controller design for uncertain descriptor systemsrdquo Acta Auto-matica Sinica vol 33 no 1 pp 44ndash47 2007
[27] L Xie and C E de Souza ldquoRobust 119867infin
control for linearsystems with norm bounder time-varying uncertaintiesrdquo IEEETransactions on Automatic Control vol 37 no 8 pp 1188ndash11911992
[28] Q-L Han ldquoAbsolute stability of time-delay systemswith sector-bounded nonlinearityrdquo Automatica vol 41 no 12 pp 2171ndash2176 2005
[29] D Yue J Lam and D W C Ho ldquoReliable 119867infin
control ofuncertain descriptor systems with multiple time delaysrdquo IEEProceedings ControlTheory and Applications vol 150 no 6 pp557ndash564 2003
[30] H D Tuan P Apkarian T Narikiyo and Y YamamotoldquoParameterized linear matrix inequality techniques in fuzzycontrol systemdesignrdquo IEEETransactions on Fuzzy Systems vol9 no 2 pp 324ndash332 2001
[31] S-W Kau H-J Lee C-M Yang C-H Lee L Hong and C-HFang ldquoRobust H
infinfuzzy static output feedback control of T-
S fuzzy systems with parametric uncertaintiesrdquo Fuzzy Sets andSystems vol 158 no 2 pp 135ndash146 2007
[32] D Saifia M Chadli S Labiod and T M Guerra ldquoRobust119867
infinstatic output feedback stabilization of T-S fuzzy systems
subject to actuator saturationrdquo International Journal of ControlAutomation and Systems vol 10 no 3 pp 613ndash622 2012
and the other notations are the same as in Theorem 8
Proof FromTheorem 8 when119908(119905) = 0 system (11) is stochas-tically admissible Let
119869119911119908(119905) = Eint
119905
0
[119911119879
(119904) 119911 (119904) minus 1205742
119908119879
(119904) 119908 (119904)] d119904 (55)
Under zero initial condition it is easy to see that
119869119911119908(119905) le Eint
119905
0
[119911119879
(119904) 119911 (119904) minus 1205742
119908119879
(119904) 119908 (119904)
+L119881 (119909119904 119901 119904)] d119904
le Eint119905
0
119896
sum
119894=1
120582119894[120589
119879
(119904) (Ω119901119894+ Θ
119879
119901119894Θ
119901119894) 120589 (119904)] d119904
(56)
where
120589119879
(119905) = [119909119879
(119905) 119909119879
(119905 minus 119889) 119908119879
(119905)]
Ω119901119894=
[
[
[
[
Υ1119901119894
lowast lowast
Υ2119901119894
Υ3119901119894
lowast
[119861119879
1199081199011198940]119883
1199010 minus120574
2
119868
]
]
]
]
Θ119901119894= [119862
119901119894119862119889119901119894
119862119908119901119894]
(57)
and notations of Υ1119901119894
Υ2119901119894
and Υ3119901119894
are the same as inTheorem 8 Hence by Schur complement lemma and using
the similar method in the proof of Theorem 8 from and(53) it can be obtained that 119869
119911119908(119905) lt 0 for all 119905 gt 0
Therefore for any nonzero 119908(119905) isin 119871V2[0infin) (52) holds
Hence according to Definition 9 the system is stochasticallyadmissible with an 119867
infinperformance 120574 This completes the
proof
Remark 11 Compared with methods in [21 22] because ofthe method of the augmented matrix adopted in Theorems8 and 10 the number of LMIs needed to solve is relativelysmall in this paper When the value of 119896 is relatively largethe quality of the computation is greatly reduced some newrelaxation matrices added will reduce the conservatism ofcontrol conditions compared with previous literatures whichcan be seen from Example 2
4 The Design of the Static OutputFeedback Controller
When 119903119905= 119901 isin S consider the overall SMJFS as follows
119864 (119905) =
119896
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119889119901119894119909 (119905 minus 119889) + 119862
119908119901119894119908 (119905))
(60)
It is difficult to drive LMI-based conditions of the stochas-tic stabilization by employing the static output feedbackcontrol approach due to the appearance of crossing termsbetween system matrices and control gains And system (60)can be rewritten in the following form
119864119909 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(Λ
119901119894119895119909 (119905) + 119861
119908119901119894119908 (119905))
119911 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(119862
119901119894119895119909 (119905) + 119862
119908119901119894119908 (119905))
(61)
where
119864 = [
119864 0
0 0
] isin R(119899+1199011)times(119899+119901
1)
119862119901119894119895= [119862
119901119894119863
119901119894119870
119901119895]
119860119889119901119894
= [
119860119889119901119894
0
0 0
] isin R(119899+1199011)times(119899+119901
1)
119909 (119905) = [
119909 (119905)
119910 (119905)
]
119861119908119901119894
= [
119861119908119901119894
0
] isin R(119899+1199011)timesV
Λ119901119894119895= [
119860119901119894
119861119901119894119870
119901119895
119862119910119901119894
minus119868
]
(62)
Remark 12 For systems (60) and (61) it can be seen that
det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
119896
sum
119904=1
120582119894120582119895120582119904(119860
119901119894+ 119861
119901119894119870
119901119895119862119910119901119904))
= det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895Λ
119901119894119895)
det(119904119864
minus
119896
sum
119894=1
119896
sum
119895=1
119896
sum
119904=1
120582119894120582119895120582119904(119860
119901119894+ 119861
119901119894119870
119901119895119862119910119901119904
+ 119860119889119901119894))
= det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(Λ
119901119894119895+ 119860
119889119901119894))
(63)
As the discussion in Remark 7 the stochastic admissibility ofsystem (60) can be studied by means of system (61)
Theorem 13 There exists an output feedback controller (59)with controller gains 119870
119901119894= 119871
119901119894119884minus1
1199012(119901 isin S 119894 isin T) such that
system (60) with 119908(119905) = 0 is stochastically admissible if thereexist matrices 119875
119901gt 0 119876
119901gt 0 119876 gt 0 119885 gt 0 119871
119901119894 119878
119901 and 119884
1199012
119901 isin S 1 le 119894 = 119895 le 119896 such that and
and the other notations are the same as in Theorem 13
Remark 15 Compared with the method in [31 32] becauseof the augmented matrix adopted in Theorems 13 and 14 thenumber of LMIs needed to solve is greatly decreased Whenthe value of 119896 is relatively large the computational complexitywill be reduced On the other hand by the augmentedmatrixthere are not any crossing terms between system matricesand controller gains so assumptions for the output matrix
[23] the equality constraint for the output matrix [24] andthe bounding technique for crossing terms are not necessaryhere therefore the conservatism brought by them will nothappen
5 Numerical Examples
Two examples will be given to illustrate the validity ofdeveloped methods
Mathematical Problems in Engineering 11
Example 1 To illustrate the 119867infin
controller synthesis thefollowing nonlinear time delay system is considered
(1 + 119886 cos 120579 (119905)) 120579 (119905) = minus119887 1205793 (119905) + 119888120579 (119905) + 119888119889(119905 minus 119889)
It can be expressed exactly by the following fuzzy singularMarkovian jump form
119864 (119905) =
3
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119901119894119906 (119905)
+ 119861119908119901119894119908 (119905))
119911 (119905) =
3
sum
119894=1
120582119894119862119901119894119909 (119905)
119909 (119905) = 120601 (119905)
119905 isin [minus119889 0] 119901 isin 1 2 3
(70)
where
119864 =[
[
[
1 0 0
0 1 0
0 0 0
]
]
]
1198601199011=
[
[
[
[
0 1 0
0 0 1
119888 minus119887 (1205952
+ 2) 119886 minus 1
]
]
]
]
1198601199012=
[
[
[
[
0 1 0
0 0 1
119888 0 minus119886 minus 1 minus 1198861205952
]
]
]
]
1198601199013=[
[
[
0 1 0
0 0 1
119888 0 119886 minus 1
]
]
]
1198601198891199011
= 1198601198891199012
= 1198601198891199013
=[
[
[
0 0 0
0 0 0
1198881198890 0
]
]
]
11986111= 119861
12= 119861
13=[
[
[
0
0
119890
]
]
]
11986121= 119861
22= 119861
23=[
[
[
0
0
08119890
]
]
]
11986131= 119861
32= 119861
33=[
[
[
0
0
05119890
]
]
]
1198611199081199011
= 1198611199081199012
= 1198611199081199013
=[
[
[
0
0
119891
]
]
]
1198621199011= 119862
1199012= 119862
1199013= [1 0 0]
1205821=
1199092
2(119905)
1205952+ 2
1205822=
1 + cos1199091(119905)
1205952+ 2
1205823=
1205952
minus 1199092
2(119905) + 1 minus cos119909
1(119905)
1205952+ 2
(71)
It is seen that 0 le 120582119894le 1sum3
119894=1120582119894= 1 Let Π = [ minus02 02 0
01 minus03 02
02 03 minus05
]120574 = 1 by solving and (53) in Theorem 10 controller gainsare given by
11987011= [minus142939 minus140620 minus36426]
11987012= [minus142911 minus144082 minus32860]
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8minus20
minus15
minus10
minus5
0
5
0 2 4 86minus1
0123
Time
Time
Ope
n-lo
op st
ate r
espo
nse
Mar
kov
chai
n
x1
x2
x3
Figure 1 State responses of the open-loop system
11987013= [minus142988 minus144111 minus36425]
11987021= [minus179535 minus177012 minus44471]
11987022= [minus179509 minus179806 minus41585]
11987023= [minus179578 minus179828 minus44470]
11987031= [minus286360 minus283641 minus69941]
11987032= [minus286344 minus285420 minus68118]
11987033= [minus286377 minus285436 minus69941]
(72)
To demonstrate the effectiveness assuming the initialcondition 120601(119905) = [minus12 08 minus05]
119879 Figures 1 and 2 showstate responses of the open-loop system and the closed-loopsystem controlled by (10) respectively From Figure 1 it canbe seen that the open-loop system is not stochastically admis-sible and Figure 2 shows that when the controller obtainedby Theorem 10 is exerted to this system it is stochasticallyadmissible
Example 2 Consider the example without uncertainties in[6]
Mode 1 1198601= [
15 14
minus35 minus45] 119860
1198891= [
02 119886
minus024 minus04] 119861
1= [
1
1]
1198611199081= [
15
14] 119862
1= [05 1] 119862
1198891= [minus02 02] 119863
1= 02 and
1198621199081= 02Mode 2 119860
2= [
17 15
minus13 minus25] 119860
1198892= [
119887 11
minus021 minus03] 119861
2= [
09
09]
1198611199082= [
14
15] 119862
2= [04 03] 119862
1198892= [minus01 02]119863
2= 03 and
1198621199082= 03Π = [
minus1 1
1 minus1] 119864 = [
1 0
0 0] 119889 = 03 120574 = 26 and in [6]
119886 = minus05 119887 = 21 but in this paper minus24 le 119886 le 2 minus2 le 119887 le 48are taken
In Figure 3 ldquoordquo represents the range of the feasiblesolutions using Theorem 10 in this paper and ldquolowastrdquo representsthe range of the feasible solutions using Theorem 3 in [6]
minus15minus1
minus050
051
152
253
354
Mar
kov
chai
n
0 1 2 3 4 5 6 7 8
0 2 4 6 8minus1
0123
Time
Time
Clos
ed-lo
op st
ate r
espo
nse
x1
x2
x3
Figure 2 State responses of the closed-loop system
minus2 minus15 minus1 minus05 0 05 1 15 2minus2
minus1
0
1
2
3
4
a
b
Figure 3 Comparison of the feasible regions
This illustrates that themethod obtained in this paper has lessconservatism
6 Conclusions
In this paper the problem of mode-dependent 119867infin
controlfor singular Markovian jump fuzzy systems with time delayis considered This class of systems under considerationis described by T-S fuzzy models The main contributionof this paper is to design state feedback controllers andstatic output feedback controllers which can guarantee thatresulting closed-loop systems are stochastically admissiblewith an119867
infinperformance 120574 by the method of the augmented
matrix Finally two examples are given to demonstrate theeffectiveness of main results obtained here
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Mathematical Problems in Engineering 13
Acknowledgments
This work was supported by The National Natural ScienceFoundation of China under Grants nos 61304054 61273003and 61273008 respectively and Science and TechnologyResearch Fund of Liaoning Education Department underGrant no L2013051
References
[1] L Dai Singular Control Systems vol 118 of Lecture Notes inControl and Information Sciences Springer Berlin Germany1989
[2] F L Lewis ldquoA survey of linear singular systemsrdquo CircuitsSystems and Signal Processing vol 5 no 1 pp 3ndash36 1986
[3] S Xu and J Lam ldquoRobust stability and stabilization of discretesingular systems an equivalent characterizationrdquo IEEE Trans-actions on Automatic Control vol 49 no 4 pp 568ndash574 2004
[4] Z Wu and W Zhou ldquoDelay-dependent robust stabilization foruncertain singular systemswith discrete and distributed delaysrdquoJournal of ControlTheory and Applications vol 6 no 2 pp 171ndash176 2008
[5] Y Xia P Shi G Liu and D Rees ldquoRobust mixed 119867infin119867
2
state-feedback control for continuous-time descriptor systemswith parameter uncertaintiesrdquo Circuits Systems and SignalProcessing vol 24 no 4 pp 431ndash443 2005
[6] H Lu W Zhou C Duan and X Qi ldquoDelay-dependent robust119867
infincontrol for uncertain singular time-delay system with
Markovian jumping parametersrdquo Optimal Control Applicationsamp Methods vol 34 no 3 pp 296ndash307 2013
[7] Y Fang and K A Loparo ldquoStochastic stability of jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 7pp 1204ndash1208 2002
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] J Xiong and J Lam ldquoStabilization of discrete-time Markovianjump linear systems via time-delayed controllersrdquo Automaticavol 42 no 5 pp 747ndash753 2006
[10] W-H Chen Z-H Guan and P Yu ldquoDelay-dependent stabilityand 119867
infincontrol of uncertain discrete-time Markovian jump
systems with mode-dependent time delaysrdquo Systems amp ControlLetters vol 52 no 5 pp 361ndash376 2004
[11] E-K Boukas Control of Singular Systems with Random AbruptChanges Communications and Control Engineering SpringerBerlin Germany 2008
[12] J Zhang Y Xia and E K Boukas ldquoNew approach to Hinfin
control for Markovian jump singular systemsrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2273ndash2284 2010
[13] J Lam S Zhu S Xu and E-K Boukas ldquoRobust 119867infin
controlof descriptor discrete-time Markovian jump systemsrdquo Interna-tional Journal of Control vol 80 no 3 pp 374ndash385 2007
[14] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Basel Switzerland 2003
[15] G Wang Q Zhang and C Yang ldquoDissipative control forsingular Markovian jump systems with time delayrdquo OptimalControl ApplicationsampMethods vol 33 no 4 pp 415ndash432 2012
[16] Z Wu H Su and J Chu ldquoDelay-dependent 119867infin
control forsingular Markovian jump systems with time delayrdquo Optimal
Control Applications and Methods vol 30 no 5 pp 443ndash4612009
[17] C K Ahn ldquoT-S fuzzy119867infinsynchronization for chaotic systems
via delayed output feedback controlrdquo Nonlinear Dynamics vol59 no 4 pp 535ndash543 2010
[18] C K Ahn ldquoFuzzy delayed output feedback synchronizationfor time-delayed chaotic systemsrdquo Nonlinear Analysis HybridSystems vol 4 no 1 pp 16ndash24 2010
[19] H-N Wu and K-Y Cai ldquoMode-independent robust stabiliza-tion for uncertain Markovian jump nonlinear systems via fuzzycontrolrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 36 no 3 pp 509ndash519 2006
[20] Y Zhang S Xu Y Zou and J Lu ldquoDelay-dependent robuststabilization for uncertain discrete-time fuzzy Markovian jumpsystems with mode-dependent time delaysrdquo Fuzzy Sets andSystems vol 164 pp 66ndash81 2011
[21] A Zhang and H Fang ldquoRobustHinfinfuzzy control for uncertain
Markovian jump nonlinear singular systems with Wiener pro-cessrdquo Journal of Control Theory and Applications vol 8 no 2pp 205ndash210 2010
[22] L Li Q Zhang and B Zhu ldquo119867infin
fuzzy control for nonlin-ear time-delay singular Markovian jump systems with partlyunknown transition ratesrdquo Fuzzy Sets and Systems vol 254 pp106ndash125 2014
[23] J Dong and G-H Yang ldquoStatic output feedback 119867infin
controlof a class of nonlinear discrete-time systemsrdquo Fuzzy Sets andSystems vol 160 no 19 pp 2844ndash2859 2009
[24] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of
singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012
[25] J Wang H Wang A Xie and R Lu ldquoDelay-dependent 119867infin
control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2012
[26] D B Xu Q L Zhang and Y B Hu ldquoReduced-order Hinfin
controller design for uncertain descriptor systemsrdquo Acta Auto-matica Sinica vol 33 no 1 pp 44ndash47 2007
[27] L Xie and C E de Souza ldquoRobust 119867infin
control for linearsystems with norm bounder time-varying uncertaintiesrdquo IEEETransactions on Automatic Control vol 37 no 8 pp 1188ndash11911992
[28] Q-L Han ldquoAbsolute stability of time-delay systemswith sector-bounded nonlinearityrdquo Automatica vol 41 no 12 pp 2171ndash2176 2005
[29] D Yue J Lam and D W C Ho ldquoReliable 119867infin
control ofuncertain descriptor systems with multiple time delaysrdquo IEEProceedings ControlTheory and Applications vol 150 no 6 pp557ndash564 2003
[30] H D Tuan P Apkarian T Narikiyo and Y YamamotoldquoParameterized linear matrix inequality techniques in fuzzycontrol systemdesignrdquo IEEETransactions on Fuzzy Systems vol9 no 2 pp 324ndash332 2001
[31] S-W Kau H-J Lee C-M Yang C-H Lee L Hong and C-HFang ldquoRobust H
infinfuzzy static output feedback control of T-
S fuzzy systems with parametric uncertaintiesrdquo Fuzzy Sets andSystems vol 158 no 2 pp 135ndash146 2007
[32] D Saifia M Chadli S Labiod and T M Guerra ldquoRobust119867
infinstatic output feedback stabilization of T-S fuzzy systems
subject to actuator saturationrdquo International Journal of ControlAutomation and Systems vol 10 no 3 pp 613ndash622 2012
119889119901119894119909 (119905 minus 119889) + 119862
119908119901119894119908 (119905))
(60)
It is difficult to drive LMI-based conditions of the stochas-tic stabilization by employing the static output feedbackcontrol approach due to the appearance of crossing termsbetween system matrices and control gains And system (60)can be rewritten in the following form
119864119909 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(Λ
119901119894119895119909 (119905) + 119861
119908119901119894119908 (119905))
119911 (119905) =
119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(119862
119901119894119895119909 (119905) + 119862
119908119901119894119908 (119905))
(61)
where
119864 = [
119864 0
0 0
] isin R(119899+1199011)times(119899+119901
1)
119862119901119894119895= [119862
119901119894119863
119901119894119870
119901119895]
119860119889119901119894
= [
119860119889119901119894
0
0 0
] isin R(119899+1199011)times(119899+119901
1)
119909 (119905) = [
119909 (119905)
119910 (119905)
]
119861119908119901119894
= [
119861119908119901119894
0
] isin R(119899+1199011)timesV
Λ119901119894119895= [
119860119901119894
119861119901119894119870
119901119895
119862119910119901119894
minus119868
]
(62)
Remark 12 For systems (60) and (61) it can be seen that
det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
119896
sum
119904=1
120582119894120582119895120582119904(119860
119901119894+ 119861
119901119894119870
119901119895119862119910119901119904))
= det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895Λ
119901119894119895)
det(119904119864
minus
119896
sum
119894=1
119896
sum
119895=1
119896
sum
119904=1
120582119894120582119895120582119904(119860
119901119894+ 119861
119901119894119870
119901119895119862119910119901119904
+ 119860119889119901119894))
= det(119904119864 minus119896
sum
119894=1
119896
sum
119895=1
120582119894120582119895(Λ
119901119894119895+ 119860
119889119901119894))
(63)
As the discussion in Remark 7 the stochastic admissibility ofsystem (60) can be studied by means of system (61)
Theorem 13 There exists an output feedback controller (59)with controller gains 119870
119901119894= 119871
119901119894119884minus1
1199012(119901 isin S 119894 isin T) such that
system (60) with 119908(119905) = 0 is stochastically admissible if thereexist matrices 119875
119901gt 0 119876
119901gt 0 119876 gt 0 119885 gt 0 119871
119901119894 119878
119901 and 119884
1199012
119901 isin S 1 le 119894 = 119895 le 119896 such that and
and the other notations are the same as in Theorem 13
Remark 15 Compared with the method in [31 32] becauseof the augmented matrix adopted in Theorems 13 and 14 thenumber of LMIs needed to solve is greatly decreased Whenthe value of 119896 is relatively large the computational complexitywill be reduced On the other hand by the augmentedmatrixthere are not any crossing terms between system matricesand controller gains so assumptions for the output matrix
[23] the equality constraint for the output matrix [24] andthe bounding technique for crossing terms are not necessaryhere therefore the conservatism brought by them will nothappen
5 Numerical Examples
Two examples will be given to illustrate the validity ofdeveloped methods
Mathematical Problems in Engineering 11
Example 1 To illustrate the 119867infin
controller synthesis thefollowing nonlinear time delay system is considered
(1 + 119886 cos 120579 (119905)) 120579 (119905) = minus119887 1205793 (119905) + 119888120579 (119905) + 119888119889(119905 minus 119889)
It can be expressed exactly by the following fuzzy singularMarkovian jump form
119864 (119905) =
3
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119901119894119906 (119905)
+ 119861119908119901119894119908 (119905))
119911 (119905) =
3
sum
119894=1
120582119894119862119901119894119909 (119905)
119909 (119905) = 120601 (119905)
119905 isin [minus119889 0] 119901 isin 1 2 3
(70)
where
119864 =[
[
[
1 0 0
0 1 0
0 0 0
]
]
]
1198601199011=
[
[
[
[
0 1 0
0 0 1
119888 minus119887 (1205952
+ 2) 119886 minus 1
]
]
]
]
1198601199012=
[
[
[
[
0 1 0
0 0 1
119888 0 minus119886 minus 1 minus 1198861205952
]
]
]
]
1198601199013=[
[
[
0 1 0
0 0 1
119888 0 119886 minus 1
]
]
]
1198601198891199011
= 1198601198891199012
= 1198601198891199013
=[
[
[
0 0 0
0 0 0
1198881198890 0
]
]
]
11986111= 119861
12= 119861
13=[
[
[
0
0
119890
]
]
]
11986121= 119861
22= 119861
23=[
[
[
0
0
08119890
]
]
]
11986131= 119861
32= 119861
33=[
[
[
0
0
05119890
]
]
]
1198611199081199011
= 1198611199081199012
= 1198611199081199013
=[
[
[
0
0
119891
]
]
]
1198621199011= 119862
1199012= 119862
1199013= [1 0 0]
1205821=
1199092
2(119905)
1205952+ 2
1205822=
1 + cos1199091(119905)
1205952+ 2
1205823=
1205952
minus 1199092
2(119905) + 1 minus cos119909
1(119905)
1205952+ 2
(71)
It is seen that 0 le 120582119894le 1sum3
119894=1120582119894= 1 Let Π = [ minus02 02 0
01 minus03 02
02 03 minus05
]120574 = 1 by solving and (53) in Theorem 10 controller gainsare given by
11987011= [minus142939 minus140620 minus36426]
11987012= [minus142911 minus144082 minus32860]
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8minus20
minus15
minus10
minus5
0
5
0 2 4 86minus1
0123
Time
Time
Ope
n-lo
op st
ate r
espo
nse
Mar
kov
chai
n
x1
x2
x3
Figure 1 State responses of the open-loop system
11987013= [minus142988 minus144111 minus36425]
11987021= [minus179535 minus177012 minus44471]
11987022= [minus179509 minus179806 minus41585]
11987023= [minus179578 minus179828 minus44470]
11987031= [minus286360 minus283641 minus69941]
11987032= [minus286344 minus285420 minus68118]
11987033= [minus286377 minus285436 minus69941]
(72)
To demonstrate the effectiveness assuming the initialcondition 120601(119905) = [minus12 08 minus05]
119879 Figures 1 and 2 showstate responses of the open-loop system and the closed-loopsystem controlled by (10) respectively From Figure 1 it canbe seen that the open-loop system is not stochastically admis-sible and Figure 2 shows that when the controller obtainedby Theorem 10 is exerted to this system it is stochasticallyadmissible
Example 2 Consider the example without uncertainties in[6]
Mode 1 1198601= [
15 14
minus35 minus45] 119860
1198891= [
02 119886
minus024 minus04] 119861
1= [
1
1]
1198611199081= [
15
14] 119862
1= [05 1] 119862
1198891= [minus02 02] 119863
1= 02 and
1198621199081= 02Mode 2 119860
2= [
17 15
minus13 minus25] 119860
1198892= [
119887 11
minus021 minus03] 119861
2= [
09
09]
1198611199082= [
14
15] 119862
2= [04 03] 119862
1198892= [minus01 02]119863
2= 03 and
1198621199082= 03Π = [
minus1 1
1 minus1] 119864 = [
1 0
0 0] 119889 = 03 120574 = 26 and in [6]
119886 = minus05 119887 = 21 but in this paper minus24 le 119886 le 2 minus2 le 119887 le 48are taken
In Figure 3 ldquoordquo represents the range of the feasiblesolutions using Theorem 10 in this paper and ldquolowastrdquo representsthe range of the feasible solutions using Theorem 3 in [6]
minus15minus1
minus050
051
152
253
354
Mar
kov
chai
n
0 1 2 3 4 5 6 7 8
0 2 4 6 8minus1
0123
Time
Time
Clos
ed-lo
op st
ate r
espo
nse
x1
x2
x3
Figure 2 State responses of the closed-loop system
minus2 minus15 minus1 minus05 0 05 1 15 2minus2
minus1
0
1
2
3
4
a
b
Figure 3 Comparison of the feasible regions
This illustrates that themethod obtained in this paper has lessconservatism
6 Conclusions
In this paper the problem of mode-dependent 119867infin
controlfor singular Markovian jump fuzzy systems with time delayis considered This class of systems under considerationis described by T-S fuzzy models The main contributionof this paper is to design state feedback controllers andstatic output feedback controllers which can guarantee thatresulting closed-loop systems are stochastically admissiblewith an119867
infinperformance 120574 by the method of the augmented
matrix Finally two examples are given to demonstrate theeffectiveness of main results obtained here
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Mathematical Problems in Engineering 13
Acknowledgments
This work was supported by The National Natural ScienceFoundation of China under Grants nos 61304054 61273003and 61273008 respectively and Science and TechnologyResearch Fund of Liaoning Education Department underGrant no L2013051
References
[1] L Dai Singular Control Systems vol 118 of Lecture Notes inControl and Information Sciences Springer Berlin Germany1989
[2] F L Lewis ldquoA survey of linear singular systemsrdquo CircuitsSystems and Signal Processing vol 5 no 1 pp 3ndash36 1986
[3] S Xu and J Lam ldquoRobust stability and stabilization of discretesingular systems an equivalent characterizationrdquo IEEE Trans-actions on Automatic Control vol 49 no 4 pp 568ndash574 2004
[4] Z Wu and W Zhou ldquoDelay-dependent robust stabilization foruncertain singular systemswith discrete and distributed delaysrdquoJournal of ControlTheory and Applications vol 6 no 2 pp 171ndash176 2008
[5] Y Xia P Shi G Liu and D Rees ldquoRobust mixed 119867infin119867
2
state-feedback control for continuous-time descriptor systemswith parameter uncertaintiesrdquo Circuits Systems and SignalProcessing vol 24 no 4 pp 431ndash443 2005
[6] H Lu W Zhou C Duan and X Qi ldquoDelay-dependent robust119867
infincontrol for uncertain singular time-delay system with
Markovian jumping parametersrdquo Optimal Control Applicationsamp Methods vol 34 no 3 pp 296ndash307 2013
[7] Y Fang and K A Loparo ldquoStochastic stability of jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 7pp 1204ndash1208 2002
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] J Xiong and J Lam ldquoStabilization of discrete-time Markovianjump linear systems via time-delayed controllersrdquo Automaticavol 42 no 5 pp 747ndash753 2006
[10] W-H Chen Z-H Guan and P Yu ldquoDelay-dependent stabilityand 119867
infincontrol of uncertain discrete-time Markovian jump
systems with mode-dependent time delaysrdquo Systems amp ControlLetters vol 52 no 5 pp 361ndash376 2004
[11] E-K Boukas Control of Singular Systems with Random AbruptChanges Communications and Control Engineering SpringerBerlin Germany 2008
[12] J Zhang Y Xia and E K Boukas ldquoNew approach to Hinfin
control for Markovian jump singular systemsrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2273ndash2284 2010
[13] J Lam S Zhu S Xu and E-K Boukas ldquoRobust 119867infin
controlof descriptor discrete-time Markovian jump systemsrdquo Interna-tional Journal of Control vol 80 no 3 pp 374ndash385 2007
[14] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Basel Switzerland 2003
[15] G Wang Q Zhang and C Yang ldquoDissipative control forsingular Markovian jump systems with time delayrdquo OptimalControl ApplicationsampMethods vol 33 no 4 pp 415ndash432 2012
[16] Z Wu H Su and J Chu ldquoDelay-dependent 119867infin
control forsingular Markovian jump systems with time delayrdquo Optimal
Control Applications and Methods vol 30 no 5 pp 443ndash4612009
[17] C K Ahn ldquoT-S fuzzy119867infinsynchronization for chaotic systems
via delayed output feedback controlrdquo Nonlinear Dynamics vol59 no 4 pp 535ndash543 2010
[18] C K Ahn ldquoFuzzy delayed output feedback synchronizationfor time-delayed chaotic systemsrdquo Nonlinear Analysis HybridSystems vol 4 no 1 pp 16ndash24 2010
[19] H-N Wu and K-Y Cai ldquoMode-independent robust stabiliza-tion for uncertain Markovian jump nonlinear systems via fuzzycontrolrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 36 no 3 pp 509ndash519 2006
[20] Y Zhang S Xu Y Zou and J Lu ldquoDelay-dependent robuststabilization for uncertain discrete-time fuzzy Markovian jumpsystems with mode-dependent time delaysrdquo Fuzzy Sets andSystems vol 164 pp 66ndash81 2011
[21] A Zhang and H Fang ldquoRobustHinfinfuzzy control for uncertain
Markovian jump nonlinear singular systems with Wiener pro-cessrdquo Journal of Control Theory and Applications vol 8 no 2pp 205ndash210 2010
[22] L Li Q Zhang and B Zhu ldquo119867infin
fuzzy control for nonlin-ear time-delay singular Markovian jump systems with partlyunknown transition ratesrdquo Fuzzy Sets and Systems vol 254 pp106ndash125 2014
[23] J Dong and G-H Yang ldquoStatic output feedback 119867infin
controlof a class of nonlinear discrete-time systemsrdquo Fuzzy Sets andSystems vol 160 no 19 pp 2844ndash2859 2009
[24] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of
singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012
[25] J Wang H Wang A Xie and R Lu ldquoDelay-dependent 119867infin
control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2012
[26] D B Xu Q L Zhang and Y B Hu ldquoReduced-order Hinfin
controller design for uncertain descriptor systemsrdquo Acta Auto-matica Sinica vol 33 no 1 pp 44ndash47 2007
[27] L Xie and C E de Souza ldquoRobust 119867infin
control for linearsystems with norm bounder time-varying uncertaintiesrdquo IEEETransactions on Automatic Control vol 37 no 8 pp 1188ndash11911992
[28] Q-L Han ldquoAbsolute stability of time-delay systemswith sector-bounded nonlinearityrdquo Automatica vol 41 no 12 pp 2171ndash2176 2005
[29] D Yue J Lam and D W C Ho ldquoReliable 119867infin
control ofuncertain descriptor systems with multiple time delaysrdquo IEEProceedings ControlTheory and Applications vol 150 no 6 pp557ndash564 2003
[30] H D Tuan P Apkarian T Narikiyo and Y YamamotoldquoParameterized linear matrix inequality techniques in fuzzycontrol systemdesignrdquo IEEETransactions on Fuzzy Systems vol9 no 2 pp 324ndash332 2001
[31] S-W Kau H-J Lee C-M Yang C-H Lee L Hong and C-HFang ldquoRobust H
infinfuzzy static output feedback control of T-
S fuzzy systems with parametric uncertaintiesrdquo Fuzzy Sets andSystems vol 158 no 2 pp 135ndash146 2007
[32] D Saifia M Chadli S Labiod and T M Guerra ldquoRobust119867
infinstatic output feedback stabilization of T-S fuzzy systems
subject to actuator saturationrdquo International Journal of ControlAutomation and Systems vol 10 no 3 pp 613ndash622 2012
and the other notations are the same as in Theorem 13
Remark 15 Compared with the method in [31 32] becauseof the augmented matrix adopted in Theorems 13 and 14 thenumber of LMIs needed to solve is greatly decreased Whenthe value of 119896 is relatively large the computational complexitywill be reduced On the other hand by the augmentedmatrixthere are not any crossing terms between system matricesand controller gains so assumptions for the output matrix
[23] the equality constraint for the output matrix [24] andthe bounding technique for crossing terms are not necessaryhere therefore the conservatism brought by them will nothappen
5 Numerical Examples
Two examples will be given to illustrate the validity ofdeveloped methods
Mathematical Problems in Engineering 11
Example 1 To illustrate the 119867infin
controller synthesis thefollowing nonlinear time delay system is considered
(1 + 119886 cos 120579 (119905)) 120579 (119905) = minus119887 1205793 (119905) + 119888120579 (119905) + 119888119889(119905 minus 119889)
It can be expressed exactly by the following fuzzy singularMarkovian jump form
119864 (119905) =
3
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119901119894119906 (119905)
+ 119861119908119901119894119908 (119905))
119911 (119905) =
3
sum
119894=1
120582119894119862119901119894119909 (119905)
119909 (119905) = 120601 (119905)
119905 isin [minus119889 0] 119901 isin 1 2 3
(70)
where
119864 =[
[
[
1 0 0
0 1 0
0 0 0
]
]
]
1198601199011=
[
[
[
[
0 1 0
0 0 1
119888 minus119887 (1205952
+ 2) 119886 minus 1
]
]
]
]
1198601199012=
[
[
[
[
0 1 0
0 0 1
119888 0 minus119886 minus 1 minus 1198861205952
]
]
]
]
1198601199013=[
[
[
0 1 0
0 0 1
119888 0 119886 minus 1
]
]
]
1198601198891199011
= 1198601198891199012
= 1198601198891199013
=[
[
[
0 0 0
0 0 0
1198881198890 0
]
]
]
11986111= 119861
12= 119861
13=[
[
[
0
0
119890
]
]
]
11986121= 119861
22= 119861
23=[
[
[
0
0
08119890
]
]
]
11986131= 119861
32= 119861
33=[
[
[
0
0
05119890
]
]
]
1198611199081199011
= 1198611199081199012
= 1198611199081199013
=[
[
[
0
0
119891
]
]
]
1198621199011= 119862
1199012= 119862
1199013= [1 0 0]
1205821=
1199092
2(119905)
1205952+ 2
1205822=
1 + cos1199091(119905)
1205952+ 2
1205823=
1205952
minus 1199092
2(119905) + 1 minus cos119909
1(119905)
1205952+ 2
(71)
It is seen that 0 le 120582119894le 1sum3
119894=1120582119894= 1 Let Π = [ minus02 02 0
01 minus03 02
02 03 minus05
]120574 = 1 by solving and (53) in Theorem 10 controller gainsare given by
11987011= [minus142939 minus140620 minus36426]
11987012= [minus142911 minus144082 minus32860]
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8minus20
minus15
minus10
minus5
0
5
0 2 4 86minus1
0123
Time
Time
Ope
n-lo
op st
ate r
espo
nse
Mar
kov
chai
n
x1
x2
x3
Figure 1 State responses of the open-loop system
11987013= [minus142988 minus144111 minus36425]
11987021= [minus179535 minus177012 minus44471]
11987022= [minus179509 minus179806 minus41585]
11987023= [minus179578 minus179828 minus44470]
11987031= [minus286360 minus283641 minus69941]
11987032= [minus286344 minus285420 minus68118]
11987033= [minus286377 minus285436 minus69941]
(72)
To demonstrate the effectiveness assuming the initialcondition 120601(119905) = [minus12 08 minus05]
119879 Figures 1 and 2 showstate responses of the open-loop system and the closed-loopsystem controlled by (10) respectively From Figure 1 it canbe seen that the open-loop system is not stochastically admis-sible and Figure 2 shows that when the controller obtainedby Theorem 10 is exerted to this system it is stochasticallyadmissible
Example 2 Consider the example without uncertainties in[6]
Mode 1 1198601= [
15 14
minus35 minus45] 119860
1198891= [
02 119886
minus024 minus04] 119861
1= [
1
1]
1198611199081= [
15
14] 119862
1= [05 1] 119862
1198891= [minus02 02] 119863
1= 02 and
1198621199081= 02Mode 2 119860
2= [
17 15
minus13 minus25] 119860
1198892= [
119887 11
minus021 minus03] 119861
2= [
09
09]
1198611199082= [
14
15] 119862
2= [04 03] 119862
1198892= [minus01 02]119863
2= 03 and
1198621199082= 03Π = [
minus1 1
1 minus1] 119864 = [
1 0
0 0] 119889 = 03 120574 = 26 and in [6]
119886 = minus05 119887 = 21 but in this paper minus24 le 119886 le 2 minus2 le 119887 le 48are taken
In Figure 3 ldquoordquo represents the range of the feasiblesolutions using Theorem 10 in this paper and ldquolowastrdquo representsthe range of the feasible solutions using Theorem 3 in [6]
minus15minus1
minus050
051
152
253
354
Mar
kov
chai
n
0 1 2 3 4 5 6 7 8
0 2 4 6 8minus1
0123
Time
Time
Clos
ed-lo
op st
ate r
espo
nse
x1
x2
x3
Figure 2 State responses of the closed-loop system
minus2 minus15 minus1 minus05 0 05 1 15 2minus2
minus1
0
1
2
3
4
a
b
Figure 3 Comparison of the feasible regions
This illustrates that themethod obtained in this paper has lessconservatism
6 Conclusions
In this paper the problem of mode-dependent 119867infin
controlfor singular Markovian jump fuzzy systems with time delayis considered This class of systems under considerationis described by T-S fuzzy models The main contributionof this paper is to design state feedback controllers andstatic output feedback controllers which can guarantee thatresulting closed-loop systems are stochastically admissiblewith an119867
infinperformance 120574 by the method of the augmented
matrix Finally two examples are given to demonstrate theeffectiveness of main results obtained here
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Mathematical Problems in Engineering 13
Acknowledgments
This work was supported by The National Natural ScienceFoundation of China under Grants nos 61304054 61273003and 61273008 respectively and Science and TechnologyResearch Fund of Liaoning Education Department underGrant no L2013051
References
[1] L Dai Singular Control Systems vol 118 of Lecture Notes inControl and Information Sciences Springer Berlin Germany1989
[2] F L Lewis ldquoA survey of linear singular systemsrdquo CircuitsSystems and Signal Processing vol 5 no 1 pp 3ndash36 1986
[3] S Xu and J Lam ldquoRobust stability and stabilization of discretesingular systems an equivalent characterizationrdquo IEEE Trans-actions on Automatic Control vol 49 no 4 pp 568ndash574 2004
[4] Z Wu and W Zhou ldquoDelay-dependent robust stabilization foruncertain singular systemswith discrete and distributed delaysrdquoJournal of ControlTheory and Applications vol 6 no 2 pp 171ndash176 2008
[5] Y Xia P Shi G Liu and D Rees ldquoRobust mixed 119867infin119867
2
state-feedback control for continuous-time descriptor systemswith parameter uncertaintiesrdquo Circuits Systems and SignalProcessing vol 24 no 4 pp 431ndash443 2005
[6] H Lu W Zhou C Duan and X Qi ldquoDelay-dependent robust119867
infincontrol for uncertain singular time-delay system with
Markovian jumping parametersrdquo Optimal Control Applicationsamp Methods vol 34 no 3 pp 296ndash307 2013
[7] Y Fang and K A Loparo ldquoStochastic stability of jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 7pp 1204ndash1208 2002
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] J Xiong and J Lam ldquoStabilization of discrete-time Markovianjump linear systems via time-delayed controllersrdquo Automaticavol 42 no 5 pp 747ndash753 2006
[10] W-H Chen Z-H Guan and P Yu ldquoDelay-dependent stabilityand 119867
infincontrol of uncertain discrete-time Markovian jump
systems with mode-dependent time delaysrdquo Systems amp ControlLetters vol 52 no 5 pp 361ndash376 2004
[11] E-K Boukas Control of Singular Systems with Random AbruptChanges Communications and Control Engineering SpringerBerlin Germany 2008
[12] J Zhang Y Xia and E K Boukas ldquoNew approach to Hinfin
control for Markovian jump singular systemsrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2273ndash2284 2010
[13] J Lam S Zhu S Xu and E-K Boukas ldquoRobust 119867infin
controlof descriptor discrete-time Markovian jump systemsrdquo Interna-tional Journal of Control vol 80 no 3 pp 374ndash385 2007
[14] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Basel Switzerland 2003
[15] G Wang Q Zhang and C Yang ldquoDissipative control forsingular Markovian jump systems with time delayrdquo OptimalControl ApplicationsampMethods vol 33 no 4 pp 415ndash432 2012
[16] Z Wu H Su and J Chu ldquoDelay-dependent 119867infin
control forsingular Markovian jump systems with time delayrdquo Optimal
Control Applications and Methods vol 30 no 5 pp 443ndash4612009
[17] C K Ahn ldquoT-S fuzzy119867infinsynchronization for chaotic systems
via delayed output feedback controlrdquo Nonlinear Dynamics vol59 no 4 pp 535ndash543 2010
[18] C K Ahn ldquoFuzzy delayed output feedback synchronizationfor time-delayed chaotic systemsrdquo Nonlinear Analysis HybridSystems vol 4 no 1 pp 16ndash24 2010
[19] H-N Wu and K-Y Cai ldquoMode-independent robust stabiliza-tion for uncertain Markovian jump nonlinear systems via fuzzycontrolrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 36 no 3 pp 509ndash519 2006
[20] Y Zhang S Xu Y Zou and J Lu ldquoDelay-dependent robuststabilization for uncertain discrete-time fuzzy Markovian jumpsystems with mode-dependent time delaysrdquo Fuzzy Sets andSystems vol 164 pp 66ndash81 2011
[21] A Zhang and H Fang ldquoRobustHinfinfuzzy control for uncertain
Markovian jump nonlinear singular systems with Wiener pro-cessrdquo Journal of Control Theory and Applications vol 8 no 2pp 205ndash210 2010
[22] L Li Q Zhang and B Zhu ldquo119867infin
fuzzy control for nonlin-ear time-delay singular Markovian jump systems with partlyunknown transition ratesrdquo Fuzzy Sets and Systems vol 254 pp106ndash125 2014
[23] J Dong and G-H Yang ldquoStatic output feedback 119867infin
controlof a class of nonlinear discrete-time systemsrdquo Fuzzy Sets andSystems vol 160 no 19 pp 2844ndash2859 2009
[24] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of
singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012
[25] J Wang H Wang A Xie and R Lu ldquoDelay-dependent 119867infin
control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2012
[26] D B Xu Q L Zhang and Y B Hu ldquoReduced-order Hinfin
controller design for uncertain descriptor systemsrdquo Acta Auto-matica Sinica vol 33 no 1 pp 44ndash47 2007
[27] L Xie and C E de Souza ldquoRobust 119867infin
control for linearsystems with norm bounder time-varying uncertaintiesrdquo IEEETransactions on Automatic Control vol 37 no 8 pp 1188ndash11911992
[28] Q-L Han ldquoAbsolute stability of time-delay systemswith sector-bounded nonlinearityrdquo Automatica vol 41 no 12 pp 2171ndash2176 2005
[29] D Yue J Lam and D W C Ho ldquoReliable 119867infin
control ofuncertain descriptor systems with multiple time delaysrdquo IEEProceedings ControlTheory and Applications vol 150 no 6 pp557ndash564 2003
[30] H D Tuan P Apkarian T Narikiyo and Y YamamotoldquoParameterized linear matrix inequality techniques in fuzzycontrol systemdesignrdquo IEEETransactions on Fuzzy Systems vol9 no 2 pp 324ndash332 2001
[31] S-W Kau H-J Lee C-M Yang C-H Lee L Hong and C-HFang ldquoRobust H
infinfuzzy static output feedback control of T-
S fuzzy systems with parametric uncertaintiesrdquo Fuzzy Sets andSystems vol 158 no 2 pp 135ndash146 2007
[32] D Saifia M Chadli S Labiod and T M Guerra ldquoRobust119867
infinstatic output feedback stabilization of T-S fuzzy systems
subject to actuator saturationrdquo International Journal of ControlAutomation and Systems vol 10 no 3 pp 613ndash622 2012
It can be expressed exactly by the following fuzzy singularMarkovian jump form
119864 (119905) =
3
sum
119894=1
120582119894(119860
119901119894119909 (119905) + 119860
119889119901119894119909 (119905 minus 119889) + 119861
119901119894119906 (119905)
+ 119861119908119901119894119908 (119905))
119911 (119905) =
3
sum
119894=1
120582119894119862119901119894119909 (119905)
119909 (119905) = 120601 (119905)
119905 isin [minus119889 0] 119901 isin 1 2 3
(70)
where
119864 =[
[
[
1 0 0
0 1 0
0 0 0
]
]
]
1198601199011=
[
[
[
[
0 1 0
0 0 1
119888 minus119887 (1205952
+ 2) 119886 minus 1
]
]
]
]
1198601199012=
[
[
[
[
0 1 0
0 0 1
119888 0 minus119886 minus 1 minus 1198861205952
]
]
]
]
1198601199013=[
[
[
0 1 0
0 0 1
119888 0 119886 minus 1
]
]
]
1198601198891199011
= 1198601198891199012
= 1198601198891199013
=[
[
[
0 0 0
0 0 0
1198881198890 0
]
]
]
11986111= 119861
12= 119861
13=[
[
[
0
0
119890
]
]
]
11986121= 119861
22= 119861
23=[
[
[
0
0
08119890
]
]
]
11986131= 119861
32= 119861
33=[
[
[
0
0
05119890
]
]
]
1198611199081199011
= 1198611199081199012
= 1198611199081199013
=[
[
[
0
0
119891
]
]
]
1198621199011= 119862
1199012= 119862
1199013= [1 0 0]
1205821=
1199092
2(119905)
1205952+ 2
1205822=
1 + cos1199091(119905)
1205952+ 2
1205823=
1205952
minus 1199092
2(119905) + 1 minus cos119909
1(119905)
1205952+ 2
(71)
It is seen that 0 le 120582119894le 1sum3
119894=1120582119894= 1 Let Π = [ minus02 02 0
01 minus03 02
02 03 minus05
]120574 = 1 by solving and (53) in Theorem 10 controller gainsare given by
11987011= [minus142939 minus140620 minus36426]
11987012= [minus142911 minus144082 minus32860]
12 Mathematical Problems in Engineering
0 1 2 3 4 5 6 7 8minus20
minus15
minus10
minus5
0
5
0 2 4 86minus1
0123
Time
Time
Ope
n-lo
op st
ate r
espo
nse
Mar
kov
chai
n
x1
x2
x3
Figure 1 State responses of the open-loop system
11987013= [minus142988 minus144111 minus36425]
11987021= [minus179535 minus177012 minus44471]
11987022= [minus179509 minus179806 minus41585]
11987023= [minus179578 minus179828 minus44470]
11987031= [minus286360 minus283641 minus69941]
11987032= [minus286344 minus285420 minus68118]
11987033= [minus286377 minus285436 minus69941]
(72)
To demonstrate the effectiveness assuming the initialcondition 120601(119905) = [minus12 08 minus05]
119879 Figures 1 and 2 showstate responses of the open-loop system and the closed-loopsystem controlled by (10) respectively From Figure 1 it canbe seen that the open-loop system is not stochastically admis-sible and Figure 2 shows that when the controller obtainedby Theorem 10 is exerted to this system it is stochasticallyadmissible
Example 2 Consider the example without uncertainties in[6]
Mode 1 1198601= [
15 14
minus35 minus45] 119860
1198891= [
02 119886
minus024 minus04] 119861
1= [
1
1]
1198611199081= [
15
14] 119862
1= [05 1] 119862
1198891= [minus02 02] 119863
1= 02 and
1198621199081= 02Mode 2 119860
2= [
17 15
minus13 minus25] 119860
1198892= [
119887 11
minus021 minus03] 119861
2= [
09
09]
1198611199082= [
14
15] 119862
2= [04 03] 119862
1198892= [minus01 02]119863
2= 03 and
1198621199082= 03Π = [
minus1 1
1 minus1] 119864 = [
1 0
0 0] 119889 = 03 120574 = 26 and in [6]
119886 = minus05 119887 = 21 but in this paper minus24 le 119886 le 2 minus2 le 119887 le 48are taken
In Figure 3 ldquoordquo represents the range of the feasiblesolutions using Theorem 10 in this paper and ldquolowastrdquo representsthe range of the feasible solutions using Theorem 3 in [6]
minus15minus1
minus050
051
152
253
354
Mar
kov
chai
n
0 1 2 3 4 5 6 7 8
0 2 4 6 8minus1
0123
Time
Time
Clos
ed-lo
op st
ate r
espo
nse
x1
x2
x3
Figure 2 State responses of the closed-loop system
minus2 minus15 minus1 minus05 0 05 1 15 2minus2
minus1
0
1
2
3
4
a
b
Figure 3 Comparison of the feasible regions
This illustrates that themethod obtained in this paper has lessconservatism
6 Conclusions
In this paper the problem of mode-dependent 119867infin
controlfor singular Markovian jump fuzzy systems with time delayis considered This class of systems under considerationis described by T-S fuzzy models The main contributionof this paper is to design state feedback controllers andstatic output feedback controllers which can guarantee thatresulting closed-loop systems are stochastically admissiblewith an119867
infinperformance 120574 by the method of the augmented
matrix Finally two examples are given to demonstrate theeffectiveness of main results obtained here
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Mathematical Problems in Engineering 13
Acknowledgments
This work was supported by The National Natural ScienceFoundation of China under Grants nos 61304054 61273003and 61273008 respectively and Science and TechnologyResearch Fund of Liaoning Education Department underGrant no L2013051
References
[1] L Dai Singular Control Systems vol 118 of Lecture Notes inControl and Information Sciences Springer Berlin Germany1989
[2] F L Lewis ldquoA survey of linear singular systemsrdquo CircuitsSystems and Signal Processing vol 5 no 1 pp 3ndash36 1986
[3] S Xu and J Lam ldquoRobust stability and stabilization of discretesingular systems an equivalent characterizationrdquo IEEE Trans-actions on Automatic Control vol 49 no 4 pp 568ndash574 2004
[4] Z Wu and W Zhou ldquoDelay-dependent robust stabilization foruncertain singular systemswith discrete and distributed delaysrdquoJournal of ControlTheory and Applications vol 6 no 2 pp 171ndash176 2008
[5] Y Xia P Shi G Liu and D Rees ldquoRobust mixed 119867infin119867
2
state-feedback control for continuous-time descriptor systemswith parameter uncertaintiesrdquo Circuits Systems and SignalProcessing vol 24 no 4 pp 431ndash443 2005
[6] H Lu W Zhou C Duan and X Qi ldquoDelay-dependent robust119867
infincontrol for uncertain singular time-delay system with
Markovian jumping parametersrdquo Optimal Control Applicationsamp Methods vol 34 no 3 pp 296ndash307 2013
[7] Y Fang and K A Loparo ldquoStochastic stability of jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 7pp 1204ndash1208 2002
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] J Xiong and J Lam ldquoStabilization of discrete-time Markovianjump linear systems via time-delayed controllersrdquo Automaticavol 42 no 5 pp 747ndash753 2006
[10] W-H Chen Z-H Guan and P Yu ldquoDelay-dependent stabilityand 119867
infincontrol of uncertain discrete-time Markovian jump
systems with mode-dependent time delaysrdquo Systems amp ControlLetters vol 52 no 5 pp 361ndash376 2004
[11] E-K Boukas Control of Singular Systems with Random AbruptChanges Communications and Control Engineering SpringerBerlin Germany 2008
[12] J Zhang Y Xia and E K Boukas ldquoNew approach to Hinfin
control for Markovian jump singular systemsrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2273ndash2284 2010
[13] J Lam S Zhu S Xu and E-K Boukas ldquoRobust 119867infin
controlof descriptor discrete-time Markovian jump systemsrdquo Interna-tional Journal of Control vol 80 no 3 pp 374ndash385 2007
[14] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Basel Switzerland 2003
[15] G Wang Q Zhang and C Yang ldquoDissipative control forsingular Markovian jump systems with time delayrdquo OptimalControl ApplicationsampMethods vol 33 no 4 pp 415ndash432 2012
[16] Z Wu H Su and J Chu ldquoDelay-dependent 119867infin
control forsingular Markovian jump systems with time delayrdquo Optimal
Control Applications and Methods vol 30 no 5 pp 443ndash4612009
[17] C K Ahn ldquoT-S fuzzy119867infinsynchronization for chaotic systems
via delayed output feedback controlrdquo Nonlinear Dynamics vol59 no 4 pp 535ndash543 2010
[18] C K Ahn ldquoFuzzy delayed output feedback synchronizationfor time-delayed chaotic systemsrdquo Nonlinear Analysis HybridSystems vol 4 no 1 pp 16ndash24 2010
[19] H-N Wu and K-Y Cai ldquoMode-independent robust stabiliza-tion for uncertain Markovian jump nonlinear systems via fuzzycontrolrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 36 no 3 pp 509ndash519 2006
[20] Y Zhang S Xu Y Zou and J Lu ldquoDelay-dependent robuststabilization for uncertain discrete-time fuzzy Markovian jumpsystems with mode-dependent time delaysrdquo Fuzzy Sets andSystems vol 164 pp 66ndash81 2011
[21] A Zhang and H Fang ldquoRobustHinfinfuzzy control for uncertain
Markovian jump nonlinear singular systems with Wiener pro-cessrdquo Journal of Control Theory and Applications vol 8 no 2pp 205ndash210 2010
[22] L Li Q Zhang and B Zhu ldquo119867infin
fuzzy control for nonlin-ear time-delay singular Markovian jump systems with partlyunknown transition ratesrdquo Fuzzy Sets and Systems vol 254 pp106ndash125 2014
[23] J Dong and G-H Yang ldquoStatic output feedback 119867infin
controlof a class of nonlinear discrete-time systemsrdquo Fuzzy Sets andSystems vol 160 no 19 pp 2844ndash2859 2009
[24] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of
singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012
[25] J Wang H Wang A Xie and R Lu ldquoDelay-dependent 119867infin
control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2012
[26] D B Xu Q L Zhang and Y B Hu ldquoReduced-order Hinfin
controller design for uncertain descriptor systemsrdquo Acta Auto-matica Sinica vol 33 no 1 pp 44ndash47 2007
[27] L Xie and C E de Souza ldquoRobust 119867infin
control for linearsystems with norm bounder time-varying uncertaintiesrdquo IEEETransactions on Automatic Control vol 37 no 8 pp 1188ndash11911992
[28] Q-L Han ldquoAbsolute stability of time-delay systemswith sector-bounded nonlinearityrdquo Automatica vol 41 no 12 pp 2171ndash2176 2005
[29] D Yue J Lam and D W C Ho ldquoReliable 119867infin
control ofuncertain descriptor systems with multiple time delaysrdquo IEEProceedings ControlTheory and Applications vol 150 no 6 pp557ndash564 2003
[30] H D Tuan P Apkarian T Narikiyo and Y YamamotoldquoParameterized linear matrix inequality techniques in fuzzycontrol systemdesignrdquo IEEETransactions on Fuzzy Systems vol9 no 2 pp 324ndash332 2001
[31] S-W Kau H-J Lee C-M Yang C-H Lee L Hong and C-HFang ldquoRobust H
infinfuzzy static output feedback control of T-
S fuzzy systems with parametric uncertaintiesrdquo Fuzzy Sets andSystems vol 158 no 2 pp 135ndash146 2007
[32] D Saifia M Chadli S Labiod and T M Guerra ldquoRobust119867
infinstatic output feedback stabilization of T-S fuzzy systems
subject to actuator saturationrdquo International Journal of ControlAutomation and Systems vol 10 no 3 pp 613ndash622 2012
To demonstrate the effectiveness assuming the initialcondition 120601(119905) = [minus12 08 minus05]
119879 Figures 1 and 2 showstate responses of the open-loop system and the closed-loopsystem controlled by (10) respectively From Figure 1 it canbe seen that the open-loop system is not stochastically admis-sible and Figure 2 shows that when the controller obtainedby Theorem 10 is exerted to this system it is stochasticallyadmissible
Example 2 Consider the example without uncertainties in[6]
Mode 1 1198601= [
15 14
minus35 minus45] 119860
1198891= [
02 119886
minus024 minus04] 119861
1= [
1
1]
1198611199081= [
15
14] 119862
1= [05 1] 119862
1198891= [minus02 02] 119863
1= 02 and
1198621199081= 02Mode 2 119860
2= [
17 15
minus13 minus25] 119860
1198892= [
119887 11
minus021 minus03] 119861
2= [
09
09]
1198611199082= [
14
15] 119862
2= [04 03] 119862
1198892= [minus01 02]119863
2= 03 and
1198621199082= 03Π = [
minus1 1
1 minus1] 119864 = [
1 0
0 0] 119889 = 03 120574 = 26 and in [6]
119886 = minus05 119887 = 21 but in this paper minus24 le 119886 le 2 minus2 le 119887 le 48are taken
In Figure 3 ldquoordquo represents the range of the feasiblesolutions using Theorem 10 in this paper and ldquolowastrdquo representsthe range of the feasible solutions using Theorem 3 in [6]
minus15minus1
minus050
051
152
253
354
Mar
kov
chai
n
0 1 2 3 4 5 6 7 8
0 2 4 6 8minus1
0123
Time
Time
Clos
ed-lo
op st
ate r
espo
nse
x1
x2
x3
Figure 2 State responses of the closed-loop system
minus2 minus15 minus1 minus05 0 05 1 15 2minus2
minus1
0
1
2
3
4
a
b
Figure 3 Comparison of the feasible regions
This illustrates that themethod obtained in this paper has lessconservatism
6 Conclusions
In this paper the problem of mode-dependent 119867infin
controlfor singular Markovian jump fuzzy systems with time delayis considered This class of systems under considerationis described by T-S fuzzy models The main contributionof this paper is to design state feedback controllers andstatic output feedback controllers which can guarantee thatresulting closed-loop systems are stochastically admissiblewith an119867
infinperformance 120574 by the method of the augmented
matrix Finally two examples are given to demonstrate theeffectiveness of main results obtained here
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Mathematical Problems in Engineering 13
Acknowledgments
This work was supported by The National Natural ScienceFoundation of China under Grants nos 61304054 61273003and 61273008 respectively and Science and TechnologyResearch Fund of Liaoning Education Department underGrant no L2013051
References
[1] L Dai Singular Control Systems vol 118 of Lecture Notes inControl and Information Sciences Springer Berlin Germany1989
[2] F L Lewis ldquoA survey of linear singular systemsrdquo CircuitsSystems and Signal Processing vol 5 no 1 pp 3ndash36 1986
[3] S Xu and J Lam ldquoRobust stability and stabilization of discretesingular systems an equivalent characterizationrdquo IEEE Trans-actions on Automatic Control vol 49 no 4 pp 568ndash574 2004
[4] Z Wu and W Zhou ldquoDelay-dependent robust stabilization foruncertain singular systemswith discrete and distributed delaysrdquoJournal of ControlTheory and Applications vol 6 no 2 pp 171ndash176 2008
[5] Y Xia P Shi G Liu and D Rees ldquoRobust mixed 119867infin119867
2
state-feedback control for continuous-time descriptor systemswith parameter uncertaintiesrdquo Circuits Systems and SignalProcessing vol 24 no 4 pp 431ndash443 2005
[6] H Lu W Zhou C Duan and X Qi ldquoDelay-dependent robust119867
infincontrol for uncertain singular time-delay system with
Markovian jumping parametersrdquo Optimal Control Applicationsamp Methods vol 34 no 3 pp 296ndash307 2013
[7] Y Fang and K A Loparo ldquoStochastic stability of jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 7pp 1204ndash1208 2002
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] J Xiong and J Lam ldquoStabilization of discrete-time Markovianjump linear systems via time-delayed controllersrdquo Automaticavol 42 no 5 pp 747ndash753 2006
[10] W-H Chen Z-H Guan and P Yu ldquoDelay-dependent stabilityand 119867
infincontrol of uncertain discrete-time Markovian jump
systems with mode-dependent time delaysrdquo Systems amp ControlLetters vol 52 no 5 pp 361ndash376 2004
[11] E-K Boukas Control of Singular Systems with Random AbruptChanges Communications and Control Engineering SpringerBerlin Germany 2008
[12] J Zhang Y Xia and E K Boukas ldquoNew approach to Hinfin
control for Markovian jump singular systemsrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2273ndash2284 2010
[13] J Lam S Zhu S Xu and E-K Boukas ldquoRobust 119867infin
controlof descriptor discrete-time Markovian jump systemsrdquo Interna-tional Journal of Control vol 80 no 3 pp 374ndash385 2007
[14] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Basel Switzerland 2003
[15] G Wang Q Zhang and C Yang ldquoDissipative control forsingular Markovian jump systems with time delayrdquo OptimalControl ApplicationsampMethods vol 33 no 4 pp 415ndash432 2012
[16] Z Wu H Su and J Chu ldquoDelay-dependent 119867infin
control forsingular Markovian jump systems with time delayrdquo Optimal
Control Applications and Methods vol 30 no 5 pp 443ndash4612009
[17] C K Ahn ldquoT-S fuzzy119867infinsynchronization for chaotic systems
via delayed output feedback controlrdquo Nonlinear Dynamics vol59 no 4 pp 535ndash543 2010
[18] C K Ahn ldquoFuzzy delayed output feedback synchronizationfor time-delayed chaotic systemsrdquo Nonlinear Analysis HybridSystems vol 4 no 1 pp 16ndash24 2010
[19] H-N Wu and K-Y Cai ldquoMode-independent robust stabiliza-tion for uncertain Markovian jump nonlinear systems via fuzzycontrolrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 36 no 3 pp 509ndash519 2006
[20] Y Zhang S Xu Y Zou and J Lu ldquoDelay-dependent robuststabilization for uncertain discrete-time fuzzy Markovian jumpsystems with mode-dependent time delaysrdquo Fuzzy Sets andSystems vol 164 pp 66ndash81 2011
[21] A Zhang and H Fang ldquoRobustHinfinfuzzy control for uncertain
Markovian jump nonlinear singular systems with Wiener pro-cessrdquo Journal of Control Theory and Applications vol 8 no 2pp 205ndash210 2010
[22] L Li Q Zhang and B Zhu ldquo119867infin
fuzzy control for nonlin-ear time-delay singular Markovian jump systems with partlyunknown transition ratesrdquo Fuzzy Sets and Systems vol 254 pp106ndash125 2014
[23] J Dong and G-H Yang ldquoStatic output feedback 119867infin
controlof a class of nonlinear discrete-time systemsrdquo Fuzzy Sets andSystems vol 160 no 19 pp 2844ndash2859 2009
[24] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of
singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012
[25] J Wang H Wang A Xie and R Lu ldquoDelay-dependent 119867infin
control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2012
[26] D B Xu Q L Zhang and Y B Hu ldquoReduced-order Hinfin
controller design for uncertain descriptor systemsrdquo Acta Auto-matica Sinica vol 33 no 1 pp 44ndash47 2007
[27] L Xie and C E de Souza ldquoRobust 119867infin
control for linearsystems with norm bounder time-varying uncertaintiesrdquo IEEETransactions on Automatic Control vol 37 no 8 pp 1188ndash11911992
[28] Q-L Han ldquoAbsolute stability of time-delay systemswith sector-bounded nonlinearityrdquo Automatica vol 41 no 12 pp 2171ndash2176 2005
[29] D Yue J Lam and D W C Ho ldquoReliable 119867infin
control ofuncertain descriptor systems with multiple time delaysrdquo IEEProceedings ControlTheory and Applications vol 150 no 6 pp557ndash564 2003
[30] H D Tuan P Apkarian T Narikiyo and Y YamamotoldquoParameterized linear matrix inequality techniques in fuzzycontrol systemdesignrdquo IEEETransactions on Fuzzy Systems vol9 no 2 pp 324ndash332 2001
[31] S-W Kau H-J Lee C-M Yang C-H Lee L Hong and C-HFang ldquoRobust H
infinfuzzy static output feedback control of T-
S fuzzy systems with parametric uncertaintiesrdquo Fuzzy Sets andSystems vol 158 no 2 pp 135ndash146 2007
[32] D Saifia M Chadli S Labiod and T M Guerra ldquoRobust119867
infinstatic output feedback stabilization of T-S fuzzy systems
subject to actuator saturationrdquo International Journal of ControlAutomation and Systems vol 10 no 3 pp 613ndash622 2012
This work was supported by The National Natural ScienceFoundation of China under Grants nos 61304054 61273003and 61273008 respectively and Science and TechnologyResearch Fund of Liaoning Education Department underGrant no L2013051
References
[1] L Dai Singular Control Systems vol 118 of Lecture Notes inControl and Information Sciences Springer Berlin Germany1989
[2] F L Lewis ldquoA survey of linear singular systemsrdquo CircuitsSystems and Signal Processing vol 5 no 1 pp 3ndash36 1986
[3] S Xu and J Lam ldquoRobust stability and stabilization of discretesingular systems an equivalent characterizationrdquo IEEE Trans-actions on Automatic Control vol 49 no 4 pp 568ndash574 2004
[4] Z Wu and W Zhou ldquoDelay-dependent robust stabilization foruncertain singular systemswith discrete and distributed delaysrdquoJournal of ControlTheory and Applications vol 6 no 2 pp 171ndash176 2008
[5] Y Xia P Shi G Liu and D Rees ldquoRobust mixed 119867infin119867
2
state-feedback control for continuous-time descriptor systemswith parameter uncertaintiesrdquo Circuits Systems and SignalProcessing vol 24 no 4 pp 431ndash443 2005
[6] H Lu W Zhou C Duan and X Qi ldquoDelay-dependent robust119867
infincontrol for uncertain singular time-delay system with
Markovian jumping parametersrdquo Optimal Control Applicationsamp Methods vol 34 no 3 pp 296ndash307 2013
[7] Y Fang and K A Loparo ldquoStochastic stability of jump linearsystemsrdquo IEEE Transactions on Automatic Control vol 47 no 7pp 1204ndash1208 2002
[8] D Yue and Q-L Han ldquoDelay-dependent exponential stabilityof stochastic systems with time-varying delay nonlinearity andMarkovian switchingrdquo IEEETransactions onAutomatic Controlvol 50 no 2 pp 217ndash222 2005
[9] J Xiong and J Lam ldquoStabilization of discrete-time Markovianjump linear systems via time-delayed controllersrdquo Automaticavol 42 no 5 pp 747ndash753 2006
[10] W-H Chen Z-H Guan and P Yu ldquoDelay-dependent stabilityand 119867
infincontrol of uncertain discrete-time Markovian jump
systems with mode-dependent time delaysrdquo Systems amp ControlLetters vol 52 no 5 pp 361ndash376 2004
[11] E-K Boukas Control of Singular Systems with Random AbruptChanges Communications and Control Engineering SpringerBerlin Germany 2008
[12] J Zhang Y Xia and E K Boukas ldquoNew approach to Hinfin
control for Markovian jump singular systemsrdquo IET ControlTheory amp Applications vol 4 no 11 pp 2273ndash2284 2010
[13] J Lam S Zhu S Xu and E-K Boukas ldquoRobust 119867infin
controlof descriptor discrete-time Markovian jump systemsrdquo Interna-tional Journal of Control vol 80 no 3 pp 374ndash385 2007
[14] K Gu V L Kharitonov and J Chen Stability of Time-DelaySystems Birkhauser Basel Switzerland 2003
[15] G Wang Q Zhang and C Yang ldquoDissipative control forsingular Markovian jump systems with time delayrdquo OptimalControl ApplicationsampMethods vol 33 no 4 pp 415ndash432 2012
[16] Z Wu H Su and J Chu ldquoDelay-dependent 119867infin
control forsingular Markovian jump systems with time delayrdquo Optimal
Control Applications and Methods vol 30 no 5 pp 443ndash4612009
[17] C K Ahn ldquoT-S fuzzy119867infinsynchronization for chaotic systems
via delayed output feedback controlrdquo Nonlinear Dynamics vol59 no 4 pp 535ndash543 2010
[18] C K Ahn ldquoFuzzy delayed output feedback synchronizationfor time-delayed chaotic systemsrdquo Nonlinear Analysis HybridSystems vol 4 no 1 pp 16ndash24 2010
[19] H-N Wu and K-Y Cai ldquoMode-independent robust stabiliza-tion for uncertain Markovian jump nonlinear systems via fuzzycontrolrdquo IEEE Transactions on Systems Man and CyberneticsPart B Cybernetics vol 36 no 3 pp 509ndash519 2006
[20] Y Zhang S Xu Y Zou and J Lu ldquoDelay-dependent robuststabilization for uncertain discrete-time fuzzy Markovian jumpsystems with mode-dependent time delaysrdquo Fuzzy Sets andSystems vol 164 pp 66ndash81 2011
[21] A Zhang and H Fang ldquoRobustHinfinfuzzy control for uncertain
Markovian jump nonlinear singular systems with Wiener pro-cessrdquo Journal of Control Theory and Applications vol 8 no 2pp 205ndash210 2010
[22] L Li Q Zhang and B Zhu ldquo119867infin
fuzzy control for nonlin-ear time-delay singular Markovian jump systems with partlyunknown transition ratesrdquo Fuzzy Sets and Systems vol 254 pp106ndash125 2014
[23] J Dong and G-H Yang ldquoStatic output feedback 119867infin
controlof a class of nonlinear discrete-time systemsrdquo Fuzzy Sets andSystems vol 160 no 19 pp 2844ndash2859 2009
[24] Y Zhang C Liu and X Mu ldquoRobust finite-time119867infincontrol of
singular stochastic systems via static output feedbackrdquo AppliedMathematics and Computation vol 218 no 9 pp 5629ndash56402012
[25] J Wang H Wang A Xie and R Lu ldquoDelay-dependent 119867infin
control for singular Markovian jump systems with time delayrdquoNonlinear Analysis Hybrid Systems vol 8 pp 1ndash12 2012
[26] D B Xu Q L Zhang and Y B Hu ldquoReduced-order Hinfin
controller design for uncertain descriptor systemsrdquo Acta Auto-matica Sinica vol 33 no 1 pp 44ndash47 2007
[27] L Xie and C E de Souza ldquoRobust 119867infin
control for linearsystems with norm bounder time-varying uncertaintiesrdquo IEEETransactions on Automatic Control vol 37 no 8 pp 1188ndash11911992
[28] Q-L Han ldquoAbsolute stability of time-delay systemswith sector-bounded nonlinearityrdquo Automatica vol 41 no 12 pp 2171ndash2176 2005
[29] D Yue J Lam and D W C Ho ldquoReliable 119867infin
control ofuncertain descriptor systems with multiple time delaysrdquo IEEProceedings ControlTheory and Applications vol 150 no 6 pp557ndash564 2003
[30] H D Tuan P Apkarian T Narikiyo and Y YamamotoldquoParameterized linear matrix inequality techniques in fuzzycontrol systemdesignrdquo IEEETransactions on Fuzzy Systems vol9 no 2 pp 324ndash332 2001
[31] S-W Kau H-J Lee C-M Yang C-H Lee L Hong and C-HFang ldquoRobust H
infinfuzzy static output feedback control of T-
S fuzzy systems with parametric uncertaintiesrdquo Fuzzy Sets andSystems vol 158 no 2 pp 135ndash146 2007
[32] D Saifia M Chadli S Labiod and T M Guerra ldquoRobust119867
infinstatic output feedback stabilization of T-S fuzzy systems
subject to actuator saturationrdquo International Journal of ControlAutomation and Systems vol 10 no 3 pp 613ndash622 2012
![Integral Sliding Mode Control for Markovian Jump T-S Fuzzy ... · for either T-S fuzzy descriptor systems [30] or Markovian jump descriptor systems [31, 32, 33]. There are, however,](https://static.documents.pub/doc/80x56/5e2ec3356373565cd5190be4/integral-sliding-mode-control-for-markovian-jump-t-s-fuzzy-for-either-t-s-fuzzy.jpg)
