Research Article Robust Delay Independent Stability Analysis...
6
Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 410349, 5 pages http://dx.doi.org/10.1155/2013/410349 Research Article Robust Delay Independent Stability Analysis for the Switched Interval Time-Delay Systems with Time-Driven Switching Strategy Juing-Shian Chiou 1 and Jen-Hsing Li 2 1 Department of Electrical Engineering, Southern Taiwan University of Science and Technology, Tainan, Taiwan 2 Department of Electrical Engineering, Kun Shan University, Tainan, Taiwan Correspondence should be addressed to Juing-Shian Chiou; [email protected] Received 13 July 2013; Accepted 23 September 2013 Academic Editor: Jui-Sheng Lin Copyright © 2013 J.-S. Chiou and J.-H. Li. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Some new criteria of delay independent stability for the switched interval time-delay systems are deduced. e switching structure does depend on time-driven switching strategies. e total activation time ratio of the switching law can be determined to guarantee that the switched interval time-delay system is exponentially stable. 1. Introduction Switched systems constitute an important class of hybrid systems. Such systems can be described by a family of contin- uous-time subsystems (or discrete-time subsystems) and a rule that orchestrates the switching between them. It is well known that a wide class of physical systems in power systems, chemical process control systems, navigation sys- tems, automobile speed change system, and so forth may be appropriately described by the switched model [1–7]. In the study of switched systems, most works have been centralized on the problem of stability. In the last two decades, there has been increasing interest in the stability analysis for such switched systems; see, for example, [8, 9] and the references cited therein. Two important methods are used to construct the switching law for the stability analysis of the switched systems. One is the state-driven switching strategy [9]; the other is the time-driven switching strategy [8]. e state-driven switching method is that if all subsys- tems have the common Lyapunov function or the multiple Lyapunov functions, there are many choices of switching strategy to make the whole system stable. However, using these kinds of methods, the system must meet conditions completely. erefore, the common Lyapunov function or the multi-Lyapunov function is difficult to construct for practical systems; even if we can construct the function, it is more complicated and not easy to implement on practical systems. e time-driven switching method is based on the con- cept of dwell time [2] that when all subsystem matrices are Hurwitz stable, then the entire switched system is exponen- tially stable for any switching signal if the time between consecutive switching (dwell time) is sufficiently large. [10] that switching among stable linear systems results in a stable system provided that switching is slow-on-the-average. But in many applications, unstable subsystems of switched systems cannot be avoided in fact [11]. If the average dwell time is cho- sen sufficiently large, and the total activation time of unstable subsystems is relatively small compared with that of Hurwitz stable subsystems, then exponential stability of a desire degree is guaranteed. Furthermore, the time-delay phenomenon also cannot be avoided in practical systems, for instance, chemical process, long distance transmission line, hybrid procedure, electron network, and so forth. e problem of time-delay may cause instability and poor performance of practical systems [12– 14]. erefore, the stability analysis of switched systems with time delay is very worthy to be researched. In a control system, uncertainties may be due to measure errors, modeling errors, linearization approximations, and so forth. ere seem to be some alternatives in formulating uncertainties or
Transcript of Research Article Robust Delay Independent Stability Analysis...
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 410349 5 pageshttpdxdoiorg1011552013410349
Research ArticleRobust Delay Independent Stability Analysis for theSwitched Interval Time-Delay Systems with Time-DrivenSwitching Strategy
Juing-Shian Chiou1 and Jen-Hsing Li2
1 Department of Electrical Engineering Southern Taiwan University of Science and Technology Tainan Taiwan2Department of Electrical Engineering Kun Shan University Tainan Taiwan
Correspondence should be addressed to Juing-Shian Chiou jschioumailstustedutw
Received 13 July 2013 Accepted 23 September 2013
Academic Editor Jui-Sheng Lin
Copyright copy 2013 J-S Chiou and J-H Li This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Some new criteria of delay independent stability for the switched interval time-delay systems are deducedThe switching structuredoes depend on time-driven switching strategiesThe total activation time ratio of the switching law can be determined to guaranteethat the switched interval time-delay system is exponentially stable
1 Introduction
Switched systems constitute an important class of hybridsystems Such systems can be described by a family of contin-uous-time subsystems (or discrete-time subsystems) and arule that orchestrates the switching between them It iswell known that a wide class of physical systems in powersystems chemical process control systems navigation sys-tems automobile speed change system and so forth may beappropriately described by the switched model [1ndash7] In thestudy of switched systems most works have been centralizedon the problem of stability In the last two decades therehas been increasing interest in the stability analysis for suchswitched systems see for example [8 9] and the referencescited therein Two important methods are used to constructthe switching law for the stability analysis of the switchedsystems One is the state-driven switching strategy [9] theother is the time-driven switching strategy [8]
The state-driven switching method is that if all subsys-tems have the common Lyapunov function or the multipleLyapunov functions there are many choices of switchingstrategy to make the whole system stable However usingthese kinds of methods the system must meet conditionscompletelyTherefore the commonLyapunov function or themulti-Lyapunov function is difficult to construct for practical
systems even if we can construct the function it is morecomplicated and not easy to implement on practical systems
The time-driven switching method is based on the con-cept of dwell time [2] that when all subsystem matrices areHurwitz stable then the entire switched system is exponen-tially stable for any switching signal if the time betweenconsecutive switching (dwell time) is sufficiently large [10]that switching among stable linear systems results in a stablesystemprovided that switching is slow-on-the-average But inmany applications unstable subsystems of switched systemscannot be avoided in fact [11] If the average dwell time is cho-sen sufficiently large and the total activation time of unstablesubsystems is relatively small compared with that of Hurwitzstable subsystems then exponential stability of a desiredegree is guaranteed
Furthermore the time-delay phenomenon also cannot beavoided in practical systems for instance chemical processlong distance transmission line hybrid procedure electronnetwork and so forth The problem of time-delay may causeinstability and poor performance of practical systems [12ndash14] Therefore the stability analysis of switched systems withtime delay is very worthy to be researched In a controlsystem uncertaintiesmay be due tomeasure errorsmodelingerrors linearization approximations and so forth Thereseem to be some alternatives in formulating uncertainties or
2 Mathematical Problems in Engineering
perturbations [13 15] shared the formulation which systemsmatrices are assumed to be perturbationHowever it does nottake the position that the system matrices are expressed inas the sum of the two terms but that the bounds for themare somehow known These systems are called interval time-delay systems The goal of this paper is to derive some robustsufficient stability conditions for the switched interval time-delay system
Basically current efforts to achieve stability in time-delaysystems can be divided into two categories namely delay-independent criteria and delay-dependent criteria In thispaper in view of delay-independent analysis we expect toaid in studying stability and designing time-driven switchinglaw to achieve and implement in a practical switched intervaltime-delay system
The following notations will be used throughout thepaper 120582(119860) stands for the eigenvalues of matrix 119860119860 denotes the norm of matrix 119860 that is 119860 =
Max[120582(119860119879119860)]12 and 120583(119860) means the matrix measure of
matrix 119860 that is 120583(119860) = Max[120582((119860 + 119860119879)2)] 119860 =
Max[120582(119860119879119860)]12
2 System Description and Problem Statement
First consider the following switched time-delay system
(119905) = 119860120590(119905)
119909 (119905) + 119861120590(119905)
119909 (119905 minus 120591)
119909 (1199050) = 1199090 119909 (119905) = 120595 (119905) 119905 isin [minus120591 0]
(1)
where 119909(119905) isin R119899 is state 119860120590(119905)
isin R119899times119899 119861120590(119905)
isin R119899times1198991199050
ge 0 is the initial time 1199090is the initial state and 120590(119905)
[1199050infin) rarr 1 2 119873 is a piecewise constant function of
time called a switch signal that is the matrix 119860120590(119905)
switchesbetween matrices 119860
1 1198602 119860
119873belonging to the set Α equiv
1198601 1198602 119860
119873 and 119860
119894 119894 isin 1 2 119873 the matrix 119861
120590(119905)
switches betweenmatrices1198611 1198612 119861
119873belonging to the set
119861 equiv 1198611 1198612 119861
119873 and 119861
119894 119894 isin 1 2 119873 120591 gt 0 is the
time-delay duration120595(119905) is a vector-valued initial continuousfunction defined on the interval [minus120591 0] and finally 120595(119905)defined on minus120591 le 119905 le 0 is the initial condition of the state
Let us consider the switched interval time-delay systemdescribed by
(119905) = 119860119868
119894119909 (119905) + 119861
119868
119894119909 (119905 minus 120591) 119894 isin 1 2 119873 (2)
where 119860119868
119894and 119861
119868
119894are matrices whose elements vary in
prescribed defined as
119860119868
119894= [119886119896119897 119894
] 119861119868
119894= [119887119896119897 119894
] (3)
where 119896 119897 = 1 2 119899 119886119896119897 119894
le 119886119896119897 119894
le 119886119896119897 119894
and 119887119896119897 119894
le 119887119896119897 119894
le
119887119896119897 119894
Denote
119860119894= [119886119896119897 119894
] 119860119894= [119886119896119897 119894
]
119861119894= [119887119896119897 119894
] 119861119894= [119887119896119897 119894
]
(4)
and let
119860119894=
119860119894+ 119860119894
2 119861
119894=
119861119894+ 119861119894
2 (5)
where119860119894and 119861
119894are the average matrices between119860
119894119860119894 and
119861119894 119861119894 respectively Furthermore
119860119887
119894= 119860119868
119894minus 119860119894 119861
119887
119894= 119861119868
119894minus 119861119894 (6)
where119860119887119894and 119861
119887
119894are the bias matrices between119860
119894119860119894 and 119861
119894
119861119894 respectively Also
119860119898
119894= 119860119894minus 119860119894 119861
119898
119894= 119861119894minus 119861119894 (7)
where 119860119898
119894and 119861
119898
119894are the maximal bias matrices between 119860
119894
119860119894 and 119861
119894 119861119894 respectively
From the properties of matrix norm we have10038171003817100381710038171003817119860119887
119894
10038171003817100381710038171003817le
1003817100381710038171003817119860119898
119894
1003817100381710038171003817 = 120572119894
10038171003817100381710038171003817119861119887
119894
10038171003817100381710038171003817le
1003817100381710038171003817119861119898
119894
1003817100381710038171003817 = 120573119894 (8)
and denote
120572119868= max (120572
119894) 120573
119868= max (120573
119894) (9)
where 1 le 119894 le 119873In this paper we study the robust stability analysis and
switching law design for the switched interval time-delaysystems
3 Delay-independent Stability Analysis
Some helpful lemmas and definitions are given below
Lemma 1 (see [16]) Consider the time-delay system
(119905) = 119860119909 (119905) + 119861119909 (119905 minus 120591) (10)
where 119909 isin 119877119899119860 and 119861 are matrices in proper dimensions and
120591 is the delay duration The stability of the time-delay systemimplies the stability for the following systems
(119905) = (119860 + 119911119861)119908 (119905) forall |119911| = 1 (11)
and vice versa
In the light of Lemma 1 for the switched time-delaysystem (1) all individual subsystems can be implied
(119905) = (119860119894+ 119911119861119894) 119908 (119905) = 119860
119894 (119911) 119908 (119905) forall |119911| = 1 (12)
Therefore the system (12) is exponentially stable if and onlyif the switched time-delay system (1) is exponentially stable
Lemma 2 (see [17]) For matrices119860 isin R119899times119899 and 119861 isin R119899times119899 thefollowing relation holds
1003817100381710038171003817exp [(119860 + 119911119861) 119905]1003817100381710038171003817 le exp [120583 (119860 + 119911119861) 119905]
le exp [(120583 (119860) + 119861) 119905] forall |119911| = 1
(13)
Mathematical Problems in Engineering 3
Without loss of generality we assume that the switchedinterval time-delay system (2) at least has one individualsubsystem whose 120583(119860
119894) + 119861
119894 + 120572119894+ 120573119894values are less than
zero the that of remaining individual system are not less thanzero that is
120583 (119860119894) +
10038171003817100381710038171198611198941003817100381710038171003817 + 120572119894+ 120573119894lt 0 1 le 119894 le 119903 (14a)
120583 (119860119894) +
10038171003817100381710038171198611198941003817100381710038171003817 + 120572119894+ 120573119894ge 0 119903 + 1 le 119894 le 119873 (14b)
Definition 3 Consider
120582minus
doi = min (1003816100381610038161003816120583 (119860119894) +
10038171003817100381710038171198611198941003817100381710038171003817 + 120572119894+ 120573119894
1003816100381610038161003816) 1 le 119894 le 119903 (15a)
120582+
doi = max (120583 (119860119894) +
10038171003817100381710038171198611198941003817100381710038171003817 + 120572119894+ 120573119894) 119903 + 1 le 119894 le 119873
(15b)
Furthermore we assume that 119879+(119905) (or 119879minus(119905)) is the totalactivation time of individual subsystems whose 120583(119860
119894)+119861119894+
120572119894+ 120573119894values are not less than zero (total activation time of
individual subsystems whose 120583(119860119894) + 119861
119894 + 120572119894+ 120573119894values
are less than zero) The total activation time ratio between119879minus(119905) and 119879
+(119905) can be called a switching law of the switched
interval time-delay system (2) Therefore we will find theratio for the total activation time such that the switchedinterval time-delay system (2) is globally and exponentiallystable with stability margin 120582
Theorem 4 Suppose that the switched interval time-delaysystem (2) exists in at least one individual subsystem whose120583(119860119894)+119861119894+120572119894+120573119894value is less than zeroThe switched interval
time-delay system (2) is globally and exponentially stablewith stability margin 120582 if the system (2) satisfies the followingswitching law
inf119905ge1199050
[119879minus(119905)
119879+ (119905)] ge
(120582+
doi + 120582lowast)
(120582minus
doi minus 120582lowast) (16)
where 120582 isin (0 120582minus
doi) and 120582lowastisin (120582 120582
minus
doi)
Proof By Lemma 1 the stability of the switched intervaltime-delay system (2) can be transformed into the followingsystem
(119905) = (119860119868
119894+ 119911119861119868
119894)119908 (119905) = 119860
119868
119894119908 (119905) (17)
The trajectory response of system (17) is written as follows
119908 (119905) = 119890119860119868
119901119894+1(119905minus119905119894)119890119860119868
119901119894(119905119894minus119905119894minus1) sdot sdot sdot 119890
119860119868
1199011(1199051minus1199050)119908 (119905
0) (18)
In view of Lemma 2 we can obtain the inequality
119908 (119905) le 119890(119860119868
119901119894+1+119911119861119868
119901119894+1) (119905minus119905119894) sdot 119890
(119860119868
119901119894+119911119861119868
119901119894)(119905119894minus119905119894minus1)
sdot sdot sdot 119890119860119868
1199011+119911119861119868
1199011(1199051minus1199050) 1003817100381710038171003817119908 (119905
0)1003817100381710038171003817
le 119890[120583(119860119868
119901119894+1)+119861119868
119901119894+1](119905minus119905119894) sdot 119890
[120583(119860119868
119901119894)+119861119868
119901119894](119905119894minus119905119894minus1)
sdot sdot sdot 119890[120583(119860119868
1199011)+119861119868
1199011](1199051minus1199050) 1003817100381710038171003817119908 (119905
0)1003817100381710038171003817
(19)
From the properties of matrix measure we have
120583 (119860119868
119894) le 120583 (119860
119887
119894+ 119860119894)
le 120583 (119860119887
119894) + 120583 (119860
119894) le 120583 (119860
119894) + 120572119894
10038171003817100381710038171003817119861119868
119894
10038171003817100381710038171003817le
10038171003817100381710038171198611198941003817100381710038171003817 + 120573119894
(20)
Hence the inequality (19) can be written as
119908 (119905) le 119890[120583(119860119901119894+1
)+120572119901119894+1+119861119901119894+1
+120573119901119894+1](119905minus119905119894)
sdot sdot sdot 119890[120583(119860119901119894)+120572119901119894+119861119901119894+120573119901119894](119905119894minus119905119894minus1)
sdot sdot sdot 119890[120583(1198601199011)+1205721199011+1198611199011+1205731199011](1199051minus1199050) 1003817100381710038171003817119908 (119905
0)1003817100381710038171003817
le 119890120582+
doi119879+minus120582minus
doi119879minus 1003817100381710038171003817119908 (119905
0)1003817100381710038171003817
(21)
Furthermore the switching law (16) means that
120582+
doi119879+(119905) minus 120582
minus
doi119879minus(119905) le minus120582
lowast(119879+(119905) + 119879
minus(119905))
= minus120582lowast(119905 minus 1199050)
(22)
Finally if we choose 120582 isin (0 120582minus
doi) and 120582lowast
isin (0 120582minus
doi) thefollowing inequality can be obtained
119908 (119905) le 119890minus120582lowast(119905minus1199050) 1003817100381710038171003817119908 (119905
0)1003817100381710038171003817
le 119890minus120582(119905minus1199050) 1003817100381710038171003817119908 (119905
0)1003817100381710038171003817
(23)
From the previous inequality (23) the system (17) isglobally and exponentially stable with stability margin 120582 andimplies that the system (2) is also stable as the systems (17)and (2) have same stability as properties Hence the switchedinterval time-delay system (2) is also globally and exponen-tially stable with stability margin 120582
Remark 5 By Theorem 4 the stability condition of theswitched interval time-delay system (1) is independent oftime-delay
4 Example
Example 1 Consider the switched interval time-delay systemwith interval matricesSubsystem 1 Consider
119860119868
1= [
[minus03 03] [08 12]
[08 12] [minus03 03]]
119861119868
1= [
[minus01 03] [minus01 01]
[minus01 01] [minus01 03]]
(24a)
Subsystem 2 Consider
119860119868
2= [
[minus31 minus29] [minus02 02]
[09 11] [minus33 minus27]]
119861119868
2= [
[minus02 0] [minus01 01]
[minus04 0] [minus04 minus02]]
(24b)
4 Mathematical Problems in Engineering
x0
0
05
1
15
2
25
0 02 04 06 08 1 12 14
State x1
Statex2
Sw1 01 s Sw
2 03 s
Figure 1 Trajectory response in Example
From (5) and (7) we obtain the average matrices andmaximal bias matricesSubsystem 1 Consider
1198601= [
0 1
1 0] 119861
1= [
01 0
0 01]
119860119898
1= [
03 02
02 03] 119861
119898
1= [
02 01
01 02]
(25)
Subsystem 2 Consider
1198602= [
minus3 0
1 minus3] 119861
2= [
minus01 0
minus02 minus03]
119860119898
2= [
01 02
01 03] 119861
119898
2= [
01 01
02 01]
(26)
From (15a) and (15b) we can calculate 120582+
doi = 19 and120582minus
doi = 14867 Finally the total activation time ratio for theswitching law is (with 120582 = 03 120582lowast = 06)
119879minus(119905)
119879+ (119905)ge
(120582+
doi + 120582lowast)
(120582minus
doi minus 120582lowast)= 28194 (27)
In order to satisfy the switching law (27) we choosethe total activation time ratio 3 1 The activation time ofsubsystem 1 is 01 sec and the activation time of subsystem 2is 03 sec respectively The trajectory of the switched intervaltime-delay system (for the average matrices 119860
1 1198611 1198602 and
1198612) is shown in Figure 1 with initial state [1 2]
119879 and time-delay 01 sec
5 Conclusion
Wehave developedmethodologies for the delay-independentstability criteria of switched interval time-delay systems
with time-driven switching strategy On delay-independentstability analysis the sufficient conditions of the switchedlaws are presented and the total activation time ratio underthe switching laws is required to be not less than a specifiedconstant such that the switched interval time-delay systemis delay-independent and exponentially stable with stabilitymargin In addition the main advantages of our approachshowed that we can quantify the region of stability extendto arbitrary subsystems of switched time-delay systems anddevelop the simple time-driven switching rule to stabilize theswitched interval time-delay systems
Acknowledgment
This work is supported by the National Science Council Tai-wan under Grants no NSC 102-2221-E-218-017 and NSC100-2632-E-218-001-MY3
References
[1] D Liberzon and A S Morse ldquoBasic problems in stability anddesign of switched systemsrdquo IEEE Control Systems Magazinevol 19 no 5 pp 59ndash70 1999
[2] J S Chiou and C M Cheng ldquoStabilization analysis of theswitched discrete-time systems using Lyapunov stability theo-rem and genetic algorithmrdquo Chaos Solitons and Fractals vol42 no 2 pp 751ndash759 2009
[3] C J Wang and J S Chiou ldquoA stability condition with delay-dependence for a class of switched large-scale time-delay sys-temsrdquo Journal of Applied Mathematics vol 2013 Article ID360170 7 pages 2013
[4] C J Wang and J S Chiou ldquoStabilization analysis for theswitched large-scale discrete-time systems via the state-drivenswitchingrdquo Discrete Dynamics in Nature and Society vol 2013Article ID 630545 5 pages 2013
[5] H Sun and L Hou ldquoComposite disturbance observer-basedcontrol and 119867
infinoutput tracking control for discrete-time
switched systems with time-varying delayrdquoMathematical Prob-lems in Engineering vol 2013 Article ID 698935 12 pages 2013
[6] J Wei P Shi H R Karimi and B Wang ldquoBIBO stability anal-ysis for delay switched systems with nonlinear perturbationrdquoAbstract and Applied Analysis vol 2013 Article ID 738653 8pages 2013
[7] Z He X Wang Z Gao and J Bai ldquoSliding mode control basedon observer for a class of state-delayed switched systems withuncertain perturbationrdquoMathematical Problems in Engineeringvol 2013 Article ID 614878 9 pages 2013
[8] J S Chiou ldquoStability analysis for a class of switched large-scale time-delay systems via time-switched methodrdquo IEEProceedingsmdashControl Theory amp Applications vol 153 no 6 pp684ndash688 2006
[9] J S Chiou C J Wang C M Cheng and C C Wang ldquoAnalysisand synthesis of switched nonlinear systems using the T-S fuzzymodelrdquoAppliedMathematicalModelling vol 34 no 6 pp 1467ndash1481 2010
[10] J P Hespanha and A S Morse ldquoStability of switched systemswith average dwell-timerdquo in Proceedings of the 38th IEEEConference on Decision and Control (CDC rsquo99) pp 2655ndash2660Phoenix Ariz USA December 1999
[11] G Zhai B Hu K Yasuda and A N Michel ldquoStability analysisof switched systems with stable and unstable subsystems an
Mathematical Problems in Engineering 5
average dwell time approachrdquo International Journal of SystemsScience vol 32 no 8 pp 1055ndash1061 2001
[12] F H Hsiao J D Hwang and S P Pan ldquo119863-stability analysisfor discrete uncertain time-delay systemsrdquoAppliedMathematicsLetters vol 11 no 2 pp 109ndash114 1998
[13] J H Park ldquoSimple criterion for asymptotic stability of intervalneutral delay-differential systemsrdquoAppliedMathematics Lettersvol 16 no 7 pp 1063ndash1068 2003
[14] G Zhai Y Sun X Chen and A N Michel ldquoStability and L2
gain analysis for switched symmetric systems with time delayrdquoin Proceedings of the American Control Conference pp 2682ndash2687 Denver Colo USA June 2003
[15] E Tissir and A Hmamed ldquoStability tests of interval time delaysystemsrdquo Systems amp Control Letters vol 23 no 4 pp 263ndash2701994
[16] AHmamed ldquoFurther results on the robust stability of uncertaintime-delay systemsrdquo International Journal of Systems Sciencevol 22 no 3 pp 605ndash614 1991
[17] P LancasterTheory ofMatrices Academic Press NewYork NYUSA 1969
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Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
perturbations [13 15] shared the formulation which systemsmatrices are assumed to be perturbationHowever it does nottake the position that the system matrices are expressed inas the sum of the two terms but that the bounds for themare somehow known These systems are called interval time-delay systems The goal of this paper is to derive some robustsufficient stability conditions for the switched interval time-delay system
Basically current efforts to achieve stability in time-delaysystems can be divided into two categories namely delay-independent criteria and delay-dependent criteria In thispaper in view of delay-independent analysis we expect toaid in studying stability and designing time-driven switchinglaw to achieve and implement in a practical switched intervaltime-delay system
The following notations will be used throughout thepaper 120582(119860) stands for the eigenvalues of matrix 119860119860 denotes the norm of matrix 119860 that is 119860 =
Max[120582(119860119879119860)]12 and 120583(119860) means the matrix measure of
matrix 119860 that is 120583(119860) = Max[120582((119860 + 119860119879)2)] 119860 =
Max[120582(119860119879119860)]12
2 System Description and Problem Statement
First consider the following switched time-delay system
(119905) = 119860120590(119905)
119909 (119905) + 119861120590(119905)
119909 (119905 minus 120591)
119909 (1199050) = 1199090 119909 (119905) = 120595 (119905) 119905 isin [minus120591 0]
(1)
where 119909(119905) isin R119899 is state 119860120590(119905)
isin R119899times119899 119861120590(119905)
isin R119899times1198991199050
ge 0 is the initial time 1199090is the initial state and 120590(119905)
[1199050infin) rarr 1 2 119873 is a piecewise constant function of
time called a switch signal that is the matrix 119860120590(119905)
switchesbetween matrices 119860
1 1198602 119860
119873belonging to the set Α equiv
1198601 1198602 119860
119873 and 119860
119894 119894 isin 1 2 119873 the matrix 119861
120590(119905)
switches betweenmatrices1198611 1198612 119861
119873belonging to the set
119861 equiv 1198611 1198612 119861
119873 and 119861
119894 119894 isin 1 2 119873 120591 gt 0 is the
time-delay duration120595(119905) is a vector-valued initial continuousfunction defined on the interval [minus120591 0] and finally 120595(119905)defined on minus120591 le 119905 le 0 is the initial condition of the state
Let us consider the switched interval time-delay systemdescribed by
(119905) = 119860119868
119894119909 (119905) + 119861
119868
119894119909 (119905 minus 120591) 119894 isin 1 2 119873 (2)
where 119860119868
119894and 119861
119868
119894are matrices whose elements vary in
prescribed defined as
119860119868
119894= [119886119896119897 119894
] 119861119868
119894= [119887119896119897 119894
] (3)
where 119896 119897 = 1 2 119899 119886119896119897 119894
le 119886119896119897 119894
le 119886119896119897 119894
and 119887119896119897 119894
le 119887119896119897 119894
le
119887119896119897 119894
Denote
119860119894= [119886119896119897 119894
] 119860119894= [119886119896119897 119894
]
119861119894= [119887119896119897 119894
] 119861119894= [119887119896119897 119894
]
(4)
and let
119860119894=
119860119894+ 119860119894
2 119861
119894=
119861119894+ 119861119894
2 (5)
where119860119894and 119861
119894are the average matrices between119860
119894119860119894 and
119861119894 119861119894 respectively Furthermore
119860119887
119894= 119860119868
119894minus 119860119894 119861
119887
119894= 119861119868
119894minus 119861119894 (6)
where119860119887119894and 119861
119887
119894are the bias matrices between119860
119894119860119894 and 119861
119894
119861119894 respectively Also
119860119898
119894= 119860119894minus 119860119894 119861
119898
119894= 119861119894minus 119861119894 (7)
where 119860119898
119894and 119861
119898
119894are the maximal bias matrices between 119860
119894
119860119894 and 119861
119894 119861119894 respectively
From the properties of matrix norm we have10038171003817100381710038171003817119860119887
119894
10038171003817100381710038171003817le
1003817100381710038171003817119860119898
119894
1003817100381710038171003817 = 120572119894
10038171003817100381710038171003817119861119887
119894
10038171003817100381710038171003817le
1003817100381710038171003817119861119898
119894
1003817100381710038171003817 = 120573119894 (8)
and denote
120572119868= max (120572
119894) 120573
119868= max (120573
119894) (9)
where 1 le 119894 le 119873In this paper we study the robust stability analysis and
switching law design for the switched interval time-delaysystems
3 Delay-independent Stability Analysis
Some helpful lemmas and definitions are given below
Lemma 1 (see [16]) Consider the time-delay system
(119905) = 119860119909 (119905) + 119861119909 (119905 minus 120591) (10)
where 119909 isin 119877119899119860 and 119861 are matrices in proper dimensions and
120591 is the delay duration The stability of the time-delay systemimplies the stability for the following systems
(119905) = (119860 + 119911119861)119908 (119905) forall |119911| = 1 (11)
and vice versa
In the light of Lemma 1 for the switched time-delaysystem (1) all individual subsystems can be implied
(119905) = (119860119894+ 119911119861119894) 119908 (119905) = 119860
119894 (119911) 119908 (119905) forall |119911| = 1 (12)
Therefore the system (12) is exponentially stable if and onlyif the switched time-delay system (1) is exponentially stable
Lemma 2 (see [17]) For matrices119860 isin R119899times119899 and 119861 isin R119899times119899 thefollowing relation holds
1003817100381710038171003817exp [(119860 + 119911119861) 119905]1003817100381710038171003817 le exp [120583 (119860 + 119911119861) 119905]
le exp [(120583 (119860) + 119861) 119905] forall |119911| = 1
(13)
Mathematical Problems in Engineering 3
Without loss of generality we assume that the switchedinterval time-delay system (2) at least has one individualsubsystem whose 120583(119860
119894) + 119861
119894 + 120572119894+ 120573119894values are less than
zero the that of remaining individual system are not less thanzero that is
120583 (119860119894) +
10038171003817100381710038171198611198941003817100381710038171003817 + 120572119894+ 120573119894lt 0 1 le 119894 le 119903 (14a)
120583 (119860119894) +
10038171003817100381710038171198611198941003817100381710038171003817 + 120572119894+ 120573119894ge 0 119903 + 1 le 119894 le 119873 (14b)
Definition 3 Consider
120582minus
doi = min (1003816100381610038161003816120583 (119860119894) +
10038171003817100381710038171198611198941003817100381710038171003817 + 120572119894+ 120573119894
1003816100381610038161003816) 1 le 119894 le 119903 (15a)
120582+
doi = max (120583 (119860119894) +
10038171003817100381710038171198611198941003817100381710038171003817 + 120572119894+ 120573119894) 119903 + 1 le 119894 le 119873
(15b)
Furthermore we assume that 119879+(119905) (or 119879minus(119905)) is the totalactivation time of individual subsystems whose 120583(119860
119894)+119861119894+
120572119894+ 120573119894values are not less than zero (total activation time of
individual subsystems whose 120583(119860119894) + 119861
119894 + 120572119894+ 120573119894values
are less than zero) The total activation time ratio between119879minus(119905) and 119879
+(119905) can be called a switching law of the switched
interval time-delay system (2) Therefore we will find theratio for the total activation time such that the switchedinterval time-delay system (2) is globally and exponentiallystable with stability margin 120582
Theorem 4 Suppose that the switched interval time-delaysystem (2) exists in at least one individual subsystem whose120583(119860119894)+119861119894+120572119894+120573119894value is less than zeroThe switched interval
time-delay system (2) is globally and exponentially stablewith stability margin 120582 if the system (2) satisfies the followingswitching law
inf119905ge1199050
[119879minus(119905)
119879+ (119905)] ge
(120582+
doi + 120582lowast)
(120582minus
doi minus 120582lowast) (16)
where 120582 isin (0 120582minus
doi) and 120582lowastisin (120582 120582
minus
doi)
Proof By Lemma 1 the stability of the switched intervaltime-delay system (2) can be transformed into the followingsystem
(119905) = (119860119868
119894+ 119911119861119868
119894)119908 (119905) = 119860
119868
119894119908 (119905) (17)
The trajectory response of system (17) is written as follows
119908 (119905) = 119890119860119868
119901119894+1(119905minus119905119894)119890119860119868
119901119894(119905119894minus119905119894minus1) sdot sdot sdot 119890
119860119868
1199011(1199051minus1199050)119908 (119905
0) (18)
In view of Lemma 2 we can obtain the inequality
119908 (119905) le 119890(119860119868
119901119894+1+119911119861119868
119901119894+1) (119905minus119905119894) sdot 119890
(119860119868
119901119894+119911119861119868
119901119894)(119905119894minus119905119894minus1)
sdot sdot sdot 119890119860119868
1199011+119911119861119868
1199011(1199051minus1199050) 1003817100381710038171003817119908 (119905
0)1003817100381710038171003817
le 119890[120583(119860119868
119901119894+1)+119861119868
119901119894+1](119905minus119905119894) sdot 119890
[120583(119860119868
119901119894)+119861119868
119901119894](119905119894minus119905119894minus1)
sdot sdot sdot 119890[120583(119860119868
1199011)+119861119868
1199011](1199051minus1199050) 1003817100381710038171003817119908 (119905
0)1003817100381710038171003817
(19)
From the properties of matrix measure we have
120583 (119860119868
119894) le 120583 (119860
119887
119894+ 119860119894)
le 120583 (119860119887
119894) + 120583 (119860
119894) le 120583 (119860
119894) + 120572119894
10038171003817100381710038171003817119861119868
119894
10038171003817100381710038171003817le
10038171003817100381710038171198611198941003817100381710038171003817 + 120573119894
(20)
Hence the inequality (19) can be written as
119908 (119905) le 119890[120583(119860119901119894+1
)+120572119901119894+1+119861119901119894+1
+120573119901119894+1](119905minus119905119894)
sdot sdot sdot 119890[120583(119860119901119894)+120572119901119894+119861119901119894+120573119901119894](119905119894minus119905119894minus1)
sdot sdot sdot 119890[120583(1198601199011)+1205721199011+1198611199011+1205731199011](1199051minus1199050) 1003817100381710038171003817119908 (119905
0)1003817100381710038171003817
le 119890120582+
doi119879+minus120582minus
doi119879minus 1003817100381710038171003817119908 (119905
0)1003817100381710038171003817
(21)
Furthermore the switching law (16) means that
120582+
doi119879+(119905) minus 120582
minus
doi119879minus(119905) le minus120582
lowast(119879+(119905) + 119879
minus(119905))
= minus120582lowast(119905 minus 1199050)
(22)
Finally if we choose 120582 isin (0 120582minus
doi) and 120582lowast
isin (0 120582minus
doi) thefollowing inequality can be obtained
119908 (119905) le 119890minus120582lowast(119905minus1199050) 1003817100381710038171003817119908 (119905
0)1003817100381710038171003817
le 119890minus120582(119905minus1199050) 1003817100381710038171003817119908 (119905
0)1003817100381710038171003817
(23)
From the previous inequality (23) the system (17) isglobally and exponentially stable with stability margin 120582 andimplies that the system (2) is also stable as the systems (17)and (2) have same stability as properties Hence the switchedinterval time-delay system (2) is also globally and exponen-tially stable with stability margin 120582
Remark 5 By Theorem 4 the stability condition of theswitched interval time-delay system (1) is independent oftime-delay
4 Example
Example 1 Consider the switched interval time-delay systemwith interval matricesSubsystem 1 Consider
119860119868
1= [
[minus03 03] [08 12]
[08 12] [minus03 03]]
119861119868
1= [
[minus01 03] [minus01 01]
[minus01 01] [minus01 03]]
(24a)
Subsystem 2 Consider
119860119868
2= [
[minus31 minus29] [minus02 02]
[09 11] [minus33 minus27]]
119861119868
2= [
[minus02 0] [minus01 01]
[minus04 0] [minus04 minus02]]
(24b)
4 Mathematical Problems in Engineering
x0
0
05
1
15
2
25
0 02 04 06 08 1 12 14
State x1
Statex2
Sw1 01 s Sw
2 03 s
Figure 1 Trajectory response in Example
From (5) and (7) we obtain the average matrices andmaximal bias matricesSubsystem 1 Consider
1198601= [
0 1
1 0] 119861
1= [
01 0
0 01]
119860119898
1= [
03 02
02 03] 119861
119898
1= [
02 01
01 02]
(25)
Subsystem 2 Consider
1198602= [
minus3 0
1 minus3] 119861
2= [
minus01 0
minus02 minus03]
119860119898
2= [
01 02
01 03] 119861
119898
2= [
01 01
02 01]
(26)
From (15a) and (15b) we can calculate 120582+
doi = 19 and120582minus
doi = 14867 Finally the total activation time ratio for theswitching law is (with 120582 = 03 120582lowast = 06)
119879minus(119905)
119879+ (119905)ge
(120582+
doi + 120582lowast)
(120582minus
doi minus 120582lowast)= 28194 (27)
In order to satisfy the switching law (27) we choosethe total activation time ratio 3 1 The activation time ofsubsystem 1 is 01 sec and the activation time of subsystem 2is 03 sec respectively The trajectory of the switched intervaltime-delay system (for the average matrices 119860
1 1198611 1198602 and
1198612) is shown in Figure 1 with initial state [1 2]
119879 and time-delay 01 sec
5 Conclusion
Wehave developedmethodologies for the delay-independentstability criteria of switched interval time-delay systems
with time-driven switching strategy On delay-independentstability analysis the sufficient conditions of the switchedlaws are presented and the total activation time ratio underthe switching laws is required to be not less than a specifiedconstant such that the switched interval time-delay systemis delay-independent and exponentially stable with stabilitymargin In addition the main advantages of our approachshowed that we can quantify the region of stability extendto arbitrary subsystems of switched time-delay systems anddevelop the simple time-driven switching rule to stabilize theswitched interval time-delay systems
Acknowledgment
This work is supported by the National Science Council Tai-wan under Grants no NSC 102-2221-E-218-017 and NSC100-2632-E-218-001-MY3
References
[1] D Liberzon and A S Morse ldquoBasic problems in stability anddesign of switched systemsrdquo IEEE Control Systems Magazinevol 19 no 5 pp 59ndash70 1999
[2] J S Chiou and C M Cheng ldquoStabilization analysis of theswitched discrete-time systems using Lyapunov stability theo-rem and genetic algorithmrdquo Chaos Solitons and Fractals vol42 no 2 pp 751ndash759 2009
[3] C J Wang and J S Chiou ldquoA stability condition with delay-dependence for a class of switched large-scale time-delay sys-temsrdquo Journal of Applied Mathematics vol 2013 Article ID360170 7 pages 2013
[4] C J Wang and J S Chiou ldquoStabilization analysis for theswitched large-scale discrete-time systems via the state-drivenswitchingrdquo Discrete Dynamics in Nature and Society vol 2013Article ID 630545 5 pages 2013
[5] H Sun and L Hou ldquoComposite disturbance observer-basedcontrol and 119867
infinoutput tracking control for discrete-time
switched systems with time-varying delayrdquoMathematical Prob-lems in Engineering vol 2013 Article ID 698935 12 pages 2013
[6] J Wei P Shi H R Karimi and B Wang ldquoBIBO stability anal-ysis for delay switched systems with nonlinear perturbationrdquoAbstract and Applied Analysis vol 2013 Article ID 738653 8pages 2013
[7] Z He X Wang Z Gao and J Bai ldquoSliding mode control basedon observer for a class of state-delayed switched systems withuncertain perturbationrdquoMathematical Problems in Engineeringvol 2013 Article ID 614878 9 pages 2013
[8] J S Chiou ldquoStability analysis for a class of switched large-scale time-delay systems via time-switched methodrdquo IEEProceedingsmdashControl Theory amp Applications vol 153 no 6 pp684ndash688 2006
[9] J S Chiou C J Wang C M Cheng and C C Wang ldquoAnalysisand synthesis of switched nonlinear systems using the T-S fuzzymodelrdquoAppliedMathematicalModelling vol 34 no 6 pp 1467ndash1481 2010
[10] J P Hespanha and A S Morse ldquoStability of switched systemswith average dwell-timerdquo in Proceedings of the 38th IEEEConference on Decision and Control (CDC rsquo99) pp 2655ndash2660Phoenix Ariz USA December 1999
[11] G Zhai B Hu K Yasuda and A N Michel ldquoStability analysisof switched systems with stable and unstable subsystems an
Mathematical Problems in Engineering 5
average dwell time approachrdquo International Journal of SystemsScience vol 32 no 8 pp 1055ndash1061 2001
[12] F H Hsiao J D Hwang and S P Pan ldquo119863-stability analysisfor discrete uncertain time-delay systemsrdquoAppliedMathematicsLetters vol 11 no 2 pp 109ndash114 1998
[13] J H Park ldquoSimple criterion for asymptotic stability of intervalneutral delay-differential systemsrdquoAppliedMathematics Lettersvol 16 no 7 pp 1063ndash1068 2003
[14] G Zhai Y Sun X Chen and A N Michel ldquoStability and L2
gain analysis for switched symmetric systems with time delayrdquoin Proceedings of the American Control Conference pp 2682ndash2687 Denver Colo USA June 2003
[15] E Tissir and A Hmamed ldquoStability tests of interval time delaysystemsrdquo Systems amp Control Letters vol 23 no 4 pp 263ndash2701994
[16] AHmamed ldquoFurther results on the robust stability of uncertaintime-delay systemsrdquo International Journal of Systems Sciencevol 22 no 3 pp 605ndash614 1991
[17] P LancasterTheory ofMatrices Academic Press NewYork NYUSA 1969
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
Without loss of generality we assume that the switchedinterval time-delay system (2) at least has one individualsubsystem whose 120583(119860
119894) + 119861
119894 + 120572119894+ 120573119894values are less than
zero the that of remaining individual system are not less thanzero that is
120583 (119860119894) +
10038171003817100381710038171198611198941003817100381710038171003817 + 120572119894+ 120573119894lt 0 1 le 119894 le 119903 (14a)
120583 (119860119894) +
10038171003817100381710038171198611198941003817100381710038171003817 + 120572119894+ 120573119894ge 0 119903 + 1 le 119894 le 119873 (14b)
Definition 3 Consider
120582minus
doi = min (1003816100381610038161003816120583 (119860119894) +
10038171003817100381710038171198611198941003817100381710038171003817 + 120572119894+ 120573119894
1003816100381610038161003816) 1 le 119894 le 119903 (15a)
120582+
doi = max (120583 (119860119894) +
10038171003817100381710038171198611198941003817100381710038171003817 + 120572119894+ 120573119894) 119903 + 1 le 119894 le 119873
(15b)
Furthermore we assume that 119879+(119905) (or 119879minus(119905)) is the totalactivation time of individual subsystems whose 120583(119860
119894)+119861119894+
120572119894+ 120573119894values are not less than zero (total activation time of
individual subsystems whose 120583(119860119894) + 119861
119894 + 120572119894+ 120573119894values
are less than zero) The total activation time ratio between119879minus(119905) and 119879
+(119905) can be called a switching law of the switched
interval time-delay system (2) Therefore we will find theratio for the total activation time such that the switchedinterval time-delay system (2) is globally and exponentiallystable with stability margin 120582
Theorem 4 Suppose that the switched interval time-delaysystem (2) exists in at least one individual subsystem whose120583(119860119894)+119861119894+120572119894+120573119894value is less than zeroThe switched interval
time-delay system (2) is globally and exponentially stablewith stability margin 120582 if the system (2) satisfies the followingswitching law
inf119905ge1199050
[119879minus(119905)
119879+ (119905)] ge
(120582+
doi + 120582lowast)
(120582minus
doi minus 120582lowast) (16)
where 120582 isin (0 120582minus
doi) and 120582lowastisin (120582 120582
minus
doi)
Proof By Lemma 1 the stability of the switched intervaltime-delay system (2) can be transformed into the followingsystem
(119905) = (119860119868
119894+ 119911119861119868
119894)119908 (119905) = 119860
119868
119894119908 (119905) (17)
The trajectory response of system (17) is written as follows
119908 (119905) = 119890119860119868
119901119894+1(119905minus119905119894)119890119860119868
119901119894(119905119894minus119905119894minus1) sdot sdot sdot 119890
119860119868
1199011(1199051minus1199050)119908 (119905
0) (18)
In view of Lemma 2 we can obtain the inequality
119908 (119905) le 119890(119860119868
119901119894+1+119911119861119868
119901119894+1) (119905minus119905119894) sdot 119890
(119860119868
119901119894+119911119861119868
119901119894)(119905119894minus119905119894minus1)
sdot sdot sdot 119890119860119868
1199011+119911119861119868
1199011(1199051minus1199050) 1003817100381710038171003817119908 (119905
0)1003817100381710038171003817
le 119890[120583(119860119868
119901119894+1)+119861119868
119901119894+1](119905minus119905119894) sdot 119890
[120583(119860119868
119901119894)+119861119868
119901119894](119905119894minus119905119894minus1)
sdot sdot sdot 119890[120583(119860119868
1199011)+119861119868
1199011](1199051minus1199050) 1003817100381710038171003817119908 (119905
0)1003817100381710038171003817
(19)
From the properties of matrix measure we have
120583 (119860119868
119894) le 120583 (119860
119887
119894+ 119860119894)
le 120583 (119860119887
119894) + 120583 (119860
119894) le 120583 (119860
119894) + 120572119894
10038171003817100381710038171003817119861119868
119894
10038171003817100381710038171003817le
10038171003817100381710038171198611198941003817100381710038171003817 + 120573119894
(20)
Hence the inequality (19) can be written as
119908 (119905) le 119890[120583(119860119901119894+1
)+120572119901119894+1+119861119901119894+1
+120573119901119894+1](119905minus119905119894)
sdot sdot sdot 119890[120583(119860119901119894)+120572119901119894+119861119901119894+120573119901119894](119905119894minus119905119894minus1)
sdot sdot sdot 119890[120583(1198601199011)+1205721199011+1198611199011+1205731199011](1199051minus1199050) 1003817100381710038171003817119908 (119905
0)1003817100381710038171003817
le 119890120582+
doi119879+minus120582minus
doi119879minus 1003817100381710038171003817119908 (119905
0)1003817100381710038171003817
(21)
Furthermore the switching law (16) means that
120582+
doi119879+(119905) minus 120582
minus
doi119879minus(119905) le minus120582
lowast(119879+(119905) + 119879
minus(119905))
= minus120582lowast(119905 minus 1199050)
(22)
Finally if we choose 120582 isin (0 120582minus
doi) and 120582lowast
isin (0 120582minus
doi) thefollowing inequality can be obtained
119908 (119905) le 119890minus120582lowast(119905minus1199050) 1003817100381710038171003817119908 (119905
0)1003817100381710038171003817
le 119890minus120582(119905minus1199050) 1003817100381710038171003817119908 (119905
0)1003817100381710038171003817
(23)
From the previous inequality (23) the system (17) isglobally and exponentially stable with stability margin 120582 andimplies that the system (2) is also stable as the systems (17)and (2) have same stability as properties Hence the switchedinterval time-delay system (2) is also globally and exponen-tially stable with stability margin 120582
Remark 5 By Theorem 4 the stability condition of theswitched interval time-delay system (1) is independent oftime-delay
4 Example
Example 1 Consider the switched interval time-delay systemwith interval matricesSubsystem 1 Consider
119860119868
1= [
[minus03 03] [08 12]
[08 12] [minus03 03]]
119861119868
1= [
[minus01 03] [minus01 01]
[minus01 01] [minus01 03]]
(24a)
Subsystem 2 Consider
119860119868
2= [
[minus31 minus29] [minus02 02]
[09 11] [minus33 minus27]]
119861119868
2= [
[minus02 0] [minus01 01]
[minus04 0] [minus04 minus02]]
(24b)
4 Mathematical Problems in Engineering
x0
0
05
1
15
2
25
0 02 04 06 08 1 12 14
State x1
Statex2
Sw1 01 s Sw
2 03 s
Figure 1 Trajectory response in Example
From (5) and (7) we obtain the average matrices andmaximal bias matricesSubsystem 1 Consider
1198601= [
0 1
1 0] 119861
1= [
01 0
0 01]
119860119898
1= [
03 02
02 03] 119861
119898
1= [
02 01
01 02]
(25)
Subsystem 2 Consider
1198602= [
minus3 0
1 minus3] 119861
2= [
minus01 0
minus02 minus03]
119860119898
2= [
01 02
01 03] 119861
119898
2= [
01 01
02 01]
(26)
From (15a) and (15b) we can calculate 120582+
doi = 19 and120582minus
doi = 14867 Finally the total activation time ratio for theswitching law is (with 120582 = 03 120582lowast = 06)
119879minus(119905)
119879+ (119905)ge
(120582+
doi + 120582lowast)
(120582minus
doi minus 120582lowast)= 28194 (27)
In order to satisfy the switching law (27) we choosethe total activation time ratio 3 1 The activation time ofsubsystem 1 is 01 sec and the activation time of subsystem 2is 03 sec respectively The trajectory of the switched intervaltime-delay system (for the average matrices 119860
1 1198611 1198602 and
1198612) is shown in Figure 1 with initial state [1 2]
119879 and time-delay 01 sec
5 Conclusion
Wehave developedmethodologies for the delay-independentstability criteria of switched interval time-delay systems
with time-driven switching strategy On delay-independentstability analysis the sufficient conditions of the switchedlaws are presented and the total activation time ratio underthe switching laws is required to be not less than a specifiedconstant such that the switched interval time-delay systemis delay-independent and exponentially stable with stabilitymargin In addition the main advantages of our approachshowed that we can quantify the region of stability extendto arbitrary subsystems of switched time-delay systems anddevelop the simple time-driven switching rule to stabilize theswitched interval time-delay systems
Acknowledgment
This work is supported by the National Science Council Tai-wan under Grants no NSC 102-2221-E-218-017 and NSC100-2632-E-218-001-MY3
References
[1] D Liberzon and A S Morse ldquoBasic problems in stability anddesign of switched systemsrdquo IEEE Control Systems Magazinevol 19 no 5 pp 59ndash70 1999
[2] J S Chiou and C M Cheng ldquoStabilization analysis of theswitched discrete-time systems using Lyapunov stability theo-rem and genetic algorithmrdquo Chaos Solitons and Fractals vol42 no 2 pp 751ndash759 2009
[3] C J Wang and J S Chiou ldquoA stability condition with delay-dependence for a class of switched large-scale time-delay sys-temsrdquo Journal of Applied Mathematics vol 2013 Article ID360170 7 pages 2013
[4] C J Wang and J S Chiou ldquoStabilization analysis for theswitched large-scale discrete-time systems via the state-drivenswitchingrdquo Discrete Dynamics in Nature and Society vol 2013Article ID 630545 5 pages 2013
[5] H Sun and L Hou ldquoComposite disturbance observer-basedcontrol and 119867
infinoutput tracking control for discrete-time
switched systems with time-varying delayrdquoMathematical Prob-lems in Engineering vol 2013 Article ID 698935 12 pages 2013
[6] J Wei P Shi H R Karimi and B Wang ldquoBIBO stability anal-ysis for delay switched systems with nonlinear perturbationrdquoAbstract and Applied Analysis vol 2013 Article ID 738653 8pages 2013
[7] Z He X Wang Z Gao and J Bai ldquoSliding mode control basedon observer for a class of state-delayed switched systems withuncertain perturbationrdquoMathematical Problems in Engineeringvol 2013 Article ID 614878 9 pages 2013
[8] J S Chiou ldquoStability analysis for a class of switched large-scale time-delay systems via time-switched methodrdquo IEEProceedingsmdashControl Theory amp Applications vol 153 no 6 pp684ndash688 2006
[9] J S Chiou C J Wang C M Cheng and C C Wang ldquoAnalysisand synthesis of switched nonlinear systems using the T-S fuzzymodelrdquoAppliedMathematicalModelling vol 34 no 6 pp 1467ndash1481 2010
[10] J P Hespanha and A S Morse ldquoStability of switched systemswith average dwell-timerdquo in Proceedings of the 38th IEEEConference on Decision and Control (CDC rsquo99) pp 2655ndash2660Phoenix Ariz USA December 1999
[11] G Zhai B Hu K Yasuda and A N Michel ldquoStability analysisof switched systems with stable and unstable subsystems an
Mathematical Problems in Engineering 5
average dwell time approachrdquo International Journal of SystemsScience vol 32 no 8 pp 1055ndash1061 2001
[12] F H Hsiao J D Hwang and S P Pan ldquo119863-stability analysisfor discrete uncertain time-delay systemsrdquoAppliedMathematicsLetters vol 11 no 2 pp 109ndash114 1998
[13] J H Park ldquoSimple criterion for asymptotic stability of intervalneutral delay-differential systemsrdquoAppliedMathematics Lettersvol 16 no 7 pp 1063ndash1068 2003
[14] G Zhai Y Sun X Chen and A N Michel ldquoStability and L2
gain analysis for switched symmetric systems with time delayrdquoin Proceedings of the American Control Conference pp 2682ndash2687 Denver Colo USA June 2003
[15] E Tissir and A Hmamed ldquoStability tests of interval time delaysystemsrdquo Systems amp Control Letters vol 23 no 4 pp 263ndash2701994
[16] AHmamed ldquoFurther results on the robust stability of uncertaintime-delay systemsrdquo International Journal of Systems Sciencevol 22 no 3 pp 605ndash614 1991
[17] P LancasterTheory ofMatrices Academic Press NewYork NYUSA 1969
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
x0
0
05
1
15
2
25
0 02 04 06 08 1 12 14
State x1
Statex2
Sw1 01 s Sw
2 03 s
Figure 1 Trajectory response in Example
From (5) and (7) we obtain the average matrices andmaximal bias matricesSubsystem 1 Consider
1198601= [
0 1
1 0] 119861
1= [
01 0
0 01]
119860119898
1= [
03 02
02 03] 119861
119898
1= [
02 01
01 02]
(25)
Subsystem 2 Consider
1198602= [
minus3 0
1 minus3] 119861
2= [
minus01 0
minus02 minus03]
119860119898
2= [
01 02
01 03] 119861
119898
2= [
01 01
02 01]
(26)
From (15a) and (15b) we can calculate 120582+
doi = 19 and120582minus
doi = 14867 Finally the total activation time ratio for theswitching law is (with 120582 = 03 120582lowast = 06)
119879minus(119905)
119879+ (119905)ge
(120582+
doi + 120582lowast)
(120582minus
doi minus 120582lowast)= 28194 (27)
In order to satisfy the switching law (27) we choosethe total activation time ratio 3 1 The activation time ofsubsystem 1 is 01 sec and the activation time of subsystem 2is 03 sec respectively The trajectory of the switched intervaltime-delay system (for the average matrices 119860
1 1198611 1198602 and
1198612) is shown in Figure 1 with initial state [1 2]
119879 and time-delay 01 sec
5 Conclusion
Wehave developedmethodologies for the delay-independentstability criteria of switched interval time-delay systems
with time-driven switching strategy On delay-independentstability analysis the sufficient conditions of the switchedlaws are presented and the total activation time ratio underthe switching laws is required to be not less than a specifiedconstant such that the switched interval time-delay systemis delay-independent and exponentially stable with stabilitymargin In addition the main advantages of our approachshowed that we can quantify the region of stability extendto arbitrary subsystems of switched time-delay systems anddevelop the simple time-driven switching rule to stabilize theswitched interval time-delay systems
Acknowledgment
This work is supported by the National Science Council Tai-wan under Grants no NSC 102-2221-E-218-017 and NSC100-2632-E-218-001-MY3
References
[1] D Liberzon and A S Morse ldquoBasic problems in stability anddesign of switched systemsrdquo IEEE Control Systems Magazinevol 19 no 5 pp 59ndash70 1999
[2] J S Chiou and C M Cheng ldquoStabilization analysis of theswitched discrete-time systems using Lyapunov stability theo-rem and genetic algorithmrdquo Chaos Solitons and Fractals vol42 no 2 pp 751ndash759 2009
[3] C J Wang and J S Chiou ldquoA stability condition with delay-dependence for a class of switched large-scale time-delay sys-temsrdquo Journal of Applied Mathematics vol 2013 Article ID360170 7 pages 2013
[4] C J Wang and J S Chiou ldquoStabilization analysis for theswitched large-scale discrete-time systems via the state-drivenswitchingrdquo Discrete Dynamics in Nature and Society vol 2013Article ID 630545 5 pages 2013
[5] H Sun and L Hou ldquoComposite disturbance observer-basedcontrol and 119867
infinoutput tracking control for discrete-time
switched systems with time-varying delayrdquoMathematical Prob-lems in Engineering vol 2013 Article ID 698935 12 pages 2013
[6] J Wei P Shi H R Karimi and B Wang ldquoBIBO stability anal-ysis for delay switched systems with nonlinear perturbationrdquoAbstract and Applied Analysis vol 2013 Article ID 738653 8pages 2013
[7] Z He X Wang Z Gao and J Bai ldquoSliding mode control basedon observer for a class of state-delayed switched systems withuncertain perturbationrdquoMathematical Problems in Engineeringvol 2013 Article ID 614878 9 pages 2013
[8] J S Chiou ldquoStability analysis for a class of switched large-scale time-delay systems via time-switched methodrdquo IEEProceedingsmdashControl Theory amp Applications vol 153 no 6 pp684ndash688 2006
[9] J S Chiou C J Wang C M Cheng and C C Wang ldquoAnalysisand synthesis of switched nonlinear systems using the T-S fuzzymodelrdquoAppliedMathematicalModelling vol 34 no 6 pp 1467ndash1481 2010
[10] J P Hespanha and A S Morse ldquoStability of switched systemswith average dwell-timerdquo in Proceedings of the 38th IEEEConference on Decision and Control (CDC rsquo99) pp 2655ndash2660Phoenix Ariz USA December 1999
[11] G Zhai B Hu K Yasuda and A N Michel ldquoStability analysisof switched systems with stable and unstable subsystems an
Mathematical Problems in Engineering 5
average dwell time approachrdquo International Journal of SystemsScience vol 32 no 8 pp 1055ndash1061 2001
[12] F H Hsiao J D Hwang and S P Pan ldquo119863-stability analysisfor discrete uncertain time-delay systemsrdquoAppliedMathematicsLetters vol 11 no 2 pp 109ndash114 1998
[13] J H Park ldquoSimple criterion for asymptotic stability of intervalneutral delay-differential systemsrdquoAppliedMathematics Lettersvol 16 no 7 pp 1063ndash1068 2003
[14] G Zhai Y Sun X Chen and A N Michel ldquoStability and L2
gain analysis for switched symmetric systems with time delayrdquoin Proceedings of the American Control Conference pp 2682ndash2687 Denver Colo USA June 2003
[15] E Tissir and A Hmamed ldquoStability tests of interval time delaysystemsrdquo Systems amp Control Letters vol 23 no 4 pp 263ndash2701994
[16] AHmamed ldquoFurther results on the robust stability of uncertaintime-delay systemsrdquo International Journal of Systems Sciencevol 22 no 3 pp 605ndash614 1991
[17] P LancasterTheory ofMatrices Academic Press NewYork NYUSA 1969
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
average dwell time approachrdquo International Journal of SystemsScience vol 32 no 8 pp 1055ndash1061 2001
[12] F H Hsiao J D Hwang and S P Pan ldquo119863-stability analysisfor discrete uncertain time-delay systemsrdquoAppliedMathematicsLetters vol 11 no 2 pp 109ndash114 1998
[13] J H Park ldquoSimple criterion for asymptotic stability of intervalneutral delay-differential systemsrdquoAppliedMathematics Lettersvol 16 no 7 pp 1063ndash1068 2003
[14] G Zhai Y Sun X Chen and A N Michel ldquoStability and L2
gain analysis for switched symmetric systems with time delayrdquoin Proceedings of the American Control Conference pp 2682ndash2687 Denver Colo USA June 2003
[15] E Tissir and A Hmamed ldquoStability tests of interval time delaysystemsrdquo Systems amp Control Letters vol 23 no 4 pp 263ndash2701994
[16] AHmamed ldquoFurther results on the robust stability of uncertaintime-delay systemsrdquo International Journal of Systems Sciencevol 22 no 3 pp 605ndash614 1991
[17] P LancasterTheory ofMatrices Academic Press NewYork NYUSA 1969
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
