Research Article Stabilization Analysis for the Switched...

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2013 Article ID 630545 5 pageshttpdxdoiorg1011552013630545

Research ArticleStabilization Analysis for the Switched Large-ScaleDiscrete-Time Systems via the State-Driven Switching

Chi-Jo Wang and Juing-Shian Chiou

Department of Electrical Engineering Southern Taiwan University of Science and Technology Tainan Taiwan

Correspondence should be addressed to Juing-Shian Chiou jschioumailstustedutw

Received 15 February 2013 Accepted 7 April 2013

Academic Editor Her-Terng Yau

Copyright copy 2013 C-J Wang and J-S Chiou 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

A criterion of stabilization for the switched large-scale discrete-time system is deduced by employing state-driven switchingmethodand the Lyapunov stability theorem This particular method especially can be applied to cases when all individual subsystems areunstable

1 Introduction

Recently switched systems have attracted great researchattention in academic and industrial applications They alsohave been widely and successfully applied to a variety ofindustrial processes such as the automotive industry chem-ical procedure control systems navigation systems automo-bile speed change systems aircraft control systems air trafficcontrol and many other fields [1ndash7] Switched systems area class of hybrid dynamical systems consisting of severalcontinuous-time or discrete-time subsystems and a rule thatorchestrates the switching sequences among them In thestudy of switched systems most of the works have been cen-tralized on the problem of stability see for example [8ndash17]and the references cited therein Two important methods areused to construct the switching law for the stability analysisof the switched systems One is the state-driven switchingstrategy [8ndash13] and the other is the time-driven switchingstrategy [14ndash17] The time-driven switching method is basedon the concept of a dwell time If there exists at least anindividual stable subsystem then the switched system is sta-ble with appropriate dwell time switching law But in manyapplications all individual subsystems are unstable Whereasfor systems with the state-driven switching strategy therehave been many choices of switching strategy to make thewhole system stable even when all individual subsystems areunstable

Recently some stability conditions and stabilization ap-proaches have been proposed for the switched discrete-time

systems Reference [18] studied the quadratic stabilizationof discrete-time switched linear systems when a designedswitching rule is imposed upon the feedback controller ofsubsystems and studied quadratic stabilization of switchedsystem with norm-bounded time-varying uncertainties Theevent-driven scheduling strategy for constructing switchinglaw to stabilize the switched system is presented in [19]Reference [20] studied stability property for the switched sys-tems which are composed of a continuous-time LTI sub-system and a discrete-time LTI subsystem when the twosubsystems are Hurwitz and Schur stable respectively It wasshown that if the subsystem matrices are symmetric then acommon Lyapunov function exists for these two subsystemsand that the switched system is exponentially stable underarbitrary switching There exists a switched quadratic Lya-punov function to check asymptotic stability of the switcheddiscrete-time system in [21] References [22 23] studiedrobust stability analysis and control synthesis of uncertainswitched systems Reference [24] has shown that to achievecontrollability for a switched linear system it is sufficientto use cyclic and synchronous switching paths and constantcontrol laws

Large-scale systems the mathematical models of manyphysical and engineering systems are frequently of high di-mension or possess interacting dynamic phenomena Suchsystems consist of a number of interdependent subsystemswhich serve particular functions share resources and aregoverned by a set of interrelated goals and constraints [25 26]

2 Discrete Dynamics in Nature and Society

In real systems the large-scale systems include electric powersystems nuclear reactors aerospace systems large electricnetworks economic systems process control systems chem-ical and petroleum industries different types of societalsystems and ecological systems Recently many approacheshave been used to investigate the stability and stabilization oflarge-scale systems [27ndash33]

As we have indicated very few research articles haveever discussed the stabilization of the switched large-scalediscrete-time system Therefore we will investigate the sta-bilization analysis of this new class of switched large-scalediscrete-time systems In this paper we will present noveland general approaches by Lyapunov stability theorem toconstruct a state-driven switching strategy such that theswitched large-scale discrete-time system is asymptoticallystable Particularly this method can be applied to cases whenall individual subsystems are unstableNotationThe following notationswill be used throughout thepaper 120582(119860) stands for the eigenvalues of matrix 119860 119860

120582=

120582max(119860) (119860120582 = 120582min(119860)) means the maximum (minimum)eigenvalue in matrix 119860 119860 denotes the norm of matrix 119860that is 119860 = Max[120582(119860119879119860)]12 119860119879 is the transpose of matrix119860 and diagsdot sdot sdot denotes a block-diagonal matrix

2 System Description and Problem Statement

Consider the switched large-scale discrete-time systems with119903 individual systems and each individual system composed of119873 interconnected subsystems

119909119894 (119896 + 1) = 119860120590(119909(119896))119894119894119909119894 (119896) +

119873

sum

119895 = 119894

119895=1

119860120590(119909(119896))119894119895119909119895 (119896) (1)

where 119909119894(119896) isin 119877119899119894 is the state vectors of the 119894th subsystem

119894 = 1 2 119873 119899 = sum119873

119894=1119899119894 119860120590(119909(119896))119894119894 and 119860120590(119909(119896))119894119895

aresome constant matrices of compatible dimensions 119909(119896) =

[119909119879

1(119896) 119909

119879

2(119896) sdot sdot sdot 119909

119879

119873(119896)]119879

isin 119877119899 120590(119909(119896)) 119877119899 rarr 1 2

119903 is a piecewise constant scalar function of state called aswitch signal that is120590(119909(119896)) = 119897 implies the subsystemmatri-ces 119860 119897119894119894 and 119860 119897119894119895

119897 isin 1 2 119903 Essentially the switcheddiscrete-time system (1) can be described as follows

Individual system 119897

119909119894 (119896 + 1) = 119860 119897119894119894119909119894 (119896) +

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896) (2)

For the stabilization of switched large-scale discrete-timesystem (2) some helpful lemmas are given subsequently

Lemma 1 (see [34]) For any matrices119860 and 119861 with appropri-ate dimensions one has

119860119879119861 + 119861119879119860 le 120574119860

119879119860 +

1

120574119861119879119861 (3)

for any constant 120574 gt 0

Lemma 2 For any matrices 1198601 1198602 119860119873 with the samedimensions the following inequality holds for any positiveconstant 120576

(

119873

sum

119894=1

119860 119894)

119879

(

119873

sum

119894=1

119860 119894)

le (1 + 120576) 119860119879

11198601 + (1 + 120576

minus1) (1 + 120576) 119860

119879

21198602

+ (1 + 120576minus1)2

(1 + 120576) 119860119879

31198603

+ sdot sdot sdot + (1 + 120576minus1)119873minus2

(1 + 120576) 119860119879

119873minus1119860119873minus1

+ (1 + 120576minus1)119873minus1

119860119879

119873119860119873

(4)

Proof For a positive constant 120576 119860 and 119861 have the samedimension such that

(119860 + 119861)119879(119860 + 119861) le (1 + 120576) 119860

119879119860 + (1 + 120576

minus1) 119861119879119861 (5)

In view of inequality (5) we have

(

119873

sum

119894=1

119860 119894)

119879

(

119873

sum

119894=1

119860 119894)

le (1 + 120576)119860119879

11198601 + (1 + 120576

minus1)(

119873

sum

119894=2

119860 119894)

119879

(

119873

sum

119894=2

119860 119894)

le (1 + 120576)119860119879

11198601 + (1 + 120576) (1 + 120576

minus1)119860119879

21198602

+ (1 + 120576minus1)2

(

119873

sum

119894=3

119860 119894)

119879

(

119873

sum

119894=3

119860 119894)

le (1 + 120576)119860119879

11198601 + (1 + 120576) (1 + 120576

minus1)119860119879

21198602

+ sdot sdot sdot (1 + 120576) (1 + 120576minus1)119873minus2

119860119879

119873minus1119860119873minus1

+ (1 + 120576minus1)119873minus1

119860119879

119873119860119873

(6)

Lemma 3 Let 120572119897 = 1(1+120576minus1)119897minus1(1+120576) 119897 = 1 2 119903minus1 and

120572119903 = 1(1+120576minus1)119903minus1 then 120572119897 isin [0 1] (1 le 119897 le 119903) andsum

119903

119897=1120572119897 = 1

for any positive constant 120576

Proof Obviously if there exists a positive constant 120576 then 0 lt120572119897 lt 1 119897 = 1 2 119903 and

119903

sum

119897=1

120572119897 =1

1 + 120576+

1

(1 + 120576minus1) (1 + 120576)

+ sdot sdot sdot +1

(1 + 120576minus1)119903minus2

(1 + 120576)

+1

(1 + 120576minus1)119903minus1

Discrete Dynamics in Nature and Society 3

=1 (1 + 120576) minus 1 (1 + 120576) (1 + 120576

minus1)119903minus1

1 minus 1 (1 + 120576minus1)

+1

(1 + 120576minus1)119903minus1

= 1

(1205721 + 1205722 + sdot sdot sdot + 120572119903minus1 is a geometric progression) (7)

Remark 4 By Lemmas 2 and 3 it is an obvious fact that

(

119903

sum

119897=1

120572119897119860119879

119897)(

119903

sum

119897=1

120572119897119860 119897) le

119903

sum

119897=1

120572119897119860119879

119897119860 119897 (8)

First we discuss the stability of the nominal switcheddiscrete-time system For the nominal switched discrete-timesystem

119909 (119896 + 1) = 119860 119897119909 (119896) 119897 = 1 2 119903 (9)

Lemma 5 There exists a switching law for the nominalswitched discrete-time system (9) such that the system (9) isasymptotically stable if there exist a symmetric matrix 119875 gt 0

and positive constants 120572119897 (1 le 119897 le 119903) satisfyingsum119903

119897=1120572119897 = 1 such

that119903

sum

119897=1

120572119897 (119860119879

119897119875119860 119897) minus 119875 lt 0 (10)

Proof If there exist a symmetric matrix 119875 gt 0 and positiveconstants 120572119897 (1 le 119897 le 119903) satisfying sum119903

119897=1120572119897 = 1 such that the

inequality (10) holds the inequality (10) is equivalent to thefollowing inequality

119903

sum

119897=1

120572119897 (119860119879

119897119875119860 119897 minus 119875) lt 0 119897 = 1 2 119903 (11)

Then for all 119909(119896) isin R119899 119909(119896) = 0

119909119879(119896) [

119903

sum

119897=1

120572119897 (119860119879

119897119875119860 119897 minus 119875)]119909 (119896) lt 0 (12)

Therefore it follows that for any 119896 at least there exists an 119897 isin1 2 119903 such that

119909119879(119896) [119860

119879

119897119875119860 119897 minus 119875] 119909 (119896) lt 0 (13)

From (13) it implies that a convex combination of the corre-sponding Lyapunov function (12) is negative along the trajec-tory and from (13) at least one must be negative Thus thenominal switched discrete-time system (9) is asymptoticallystable

Lemma 6 (see [34]) Chebyshev inequality holds for anymatrix119882119894 isin 119877119899times119899

(

119873

sum

119894=1

119860 119894)

119879

(

119873

sum

119894=1

119860 119894) le 119873(

119873

sum

119894=1

119860119879

119894119860 119894)

(14)

In the study of switched large-scale discrete-time systemsthe main problems are how to constructively design a switch-ing rule and how to derive sufficient stability conditionswhich can guarantee the stability of the switched large-scalediscrete-time systems under the switching rule

3 Stabilization Analysis

For the switched large-scale discrete-time system (2) with 119903individual systems and each individual system composed of119873 interconnected subsystems assume that sum119903

119897=1radic212057212

119897119860 119897119894119894

are Schur matrices To investigate the stability of system(1) we denote 119909

119879(119896) = [119909

119879

1119909119879

2sdot sdot sdot 119909

119879

119873]119879 and 119875 =

diag1198751 1198752 119875119873 where

2(

119903

sum

119897=1

12057212

119897119860 119897119894119894

)

119879

119875119894(

119903

sum

119897=1

12057212

119897119860 119897119894119894

) minus 119875119894 = minus119868 (15)

Theorem 7 Consider the switched large-scale discrete-timesystem (1) and assume that sum119903

119897=112057212

119897119860 119897119894119894

are Schur matricesThere exists a switching law such that the switched large-scalediscrete-time system (1) is asymptotically stable if matrices1198751 1198752 119875119873 gt 0 satisfy (15) and the following inequality (16)holds

119903

sum

119897=1

119873

sum

119895 = 119894

119895=1

120572119897

100381710038171003817100381710038171003817119860119879

119897119895119894119875119895119860 119897119895119894

100381710038171003817100381710038171003817le 05(119873 minus 1)

minus1 (16)

where 120572119897 = 1(1 + 120576minus1)119897minus1(1 + 120576) (119897 = 1 2 119903 minus 1) 120572119903 =

1(1 + 120576minus1)119903minus1 and 120576 gt 0

Proof By inequality (12) we obtain

119873

sum

119894=1

[[[

[

119903

sum

119897=1

120572119897(119860 119897119894119894119909119894(119896) +

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

119879

times119875119894(119860 119897119894119894119909119894 (119896) +

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

]]]

]

minus119909119879

119894(119896) 119875119894119909119894 (119896)

le

119873

sum

119894=1

119903

sum

119897=1

120572119897(119860 119897119894119894119909119894 (119896))

119879

119875119894 (119860 119897119894119894119909119894 (119896))

+

119903

sum

119897=1

120572119897(119860 119897119894119894119909119894 (119896))

119879

119875119894(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))


+

119903

sum

119897=1

120572119897(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

119879

119875119894 (119860 119897119894119894119909119894 (119896))

+

119903

sum

119897=1

120572119897(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

119879

119875119894(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

minus119909119879

119894(119896) 119875119894119909119894 (119896)

le

119873

sum

119894=1

119903

sum

119897=1

120572119897

[[[

[

2(119860 119897119894119894(119896))119879

119875119894 (119860 119897119894119894119909119894 (119896))

+

119903

sum

119897=1

2120572119897(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

119879

times119875119894(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

]]]

]

minus119909119879

119894(119896) 119875119894119909119894 (119896)

le

119873

sum

119894=1

119903

sum

119897=1

2120572119897(119860 119897119894119894119909119894(119896))

119879

119875119894 (119860 119897119894119894119909119894 (119896))

+

119903

sum

119897=1

2120572119897 (119873 minus 1)

times(

119873

sum

119895 = 119894

119895=1

(119860 119897119894119895119909119895 (119896))

119879

119875119894 (119860 119897119894119895119909119895 (119896)))

minus119909119879

119894(119896) 119875119894119909119894 (119896)

le

119873

sum

119894=1

119909119879

119894(119896)

minus 119868 + 2 (119873 minus 1)

times(

119903

sum

119897=1

119873

sum

119895 = 119894

119895=1

120572119897

100381710038171003817100381710038171003817119860119879

119897119895119894119875119895119860 119897119895119894

100381710038171003817100381710038171003817)

119909119894 (119896)

(17)

If conditions (16) are satisfied then the inequality (17) issmaller than zero for 119894 = 1 2 119873 and 119897 = 1 2 119903Therefore the switched large-scale discrete-time system isasymptotically stable

Remark 8 For the switched large-scale discrete-time system(1) with arbitrary 119903 individual systems and 119873 state vectorsTheorem 7 can be applied to cases when all individualsubsystems are unstable The particular method also can beapplied to cases when 119903 gt 119873 or 119903 = 119873 119903 lt 119873

Switching Law Switched large-scale discrete-time system (1)is switched to or stays at mode 119897 at sampling step 119896 if (18) issatisfied at 119896

119873

sum

119894=1

[[[

[

119860 119897119894119894119909119894 (119896) +

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896)

]]]

]

119879

times119875119894

[[[

[

119860 119897119894119894119909119894 (119896) +

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896)

]]]

]

minus 119875119894

lt 0

119897 = 1 2 119903

(18)

4 Conclusion

We have adopted the Lyapunov stability theorem to dealwith the stabilization analysis of switched large-scale discrete-time systems The results are straightforward By a simpleswitching law the sufficient stability conditions have beenderived for the switched large-scale discrete-time systems viathe state-driven switchingThe results can be applied to caseswhen 119903 gt 119873 119903 = 119873 or 119903 lt 119873 for the switched large-scalediscrete-time system with arbitrary 119903 individual systems and119873 state vectors In addition the particular method can beapplied to cases when all individual subsystems are unstable

Acknowledgment

This work is supported by the National Science Council Tai-wan underGrants nos NSC101-2221-E-218-027 andNSC100-2632-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] A S Morse ldquoSupervisory control of families of linear set-pointcontrollers I Exact matchingrdquo IEEE Transactions on AutomaticControl vol 41 no 10 pp 1413ndash1431 1996

[3] R Frasca L Iannelli and F Vasca ldquoDithered sliding-modecontrol for switched systemsrdquo IEEETransactions on Circuits andSystems II vol 53 no 9 pp 872ndash876 2006

[4] G Zhai D Liu J Imae and T Kobayashi ldquoLie algebraicstability analysis for switched systems with continuous-time


and discrete-time subsystemsrdquo IEEE Transactions on Circuitsand Systems II vol 53 no 2 pp 152ndash156 2006

[5] T-H Kim S-H Lee and J-T Lim ldquoStability analysis ofswitched systems via redundancy factorsrdquo International Journalof Systems Science vol 32 no 10 pp 1199ndash1204 2001

[6] X Zhang Z Q Xia and Y Gao ldquoExponential stabilizabilityof switched systems with polytopic uncertaintiesrdquo Journal ofAppliedMathematics vol 2012 Article ID 853170 15 pages 2012

[7] B Du and X Zhang ldquoDelay-dependent stability analysis andsynthesis for uncertain impulsive switched system with mixeddelaysrdquo Discrete Dynamics in Nature and Society vol 2011Article ID 381571 9 pages 2011

[8] Z Ji L Wang and G Xie ldquoQuadratic stabilization of switchedsystemsrdquo International Journal of Systems Science vol 36 no 7pp 395ndash404 2005

[9] D Cheng L Guo and J Huang ldquoOn quadratic Lyapunovfunctionsrdquo IEEE Transactions on Automatic Control vol 48 no5 pp 885ndash890 2003

[10] S Yang X Zhengrong C Qingwei and H Weili ldquoRobustreliable control of switched uncertain systems with time-varying delayrdquo International Journal of Systems Science vol 37no 15 pp 1077ndash1087 2006

[11] Z G Li C Y Wen and Y C Soh ldquoStabilization of a class ofswitched systems via designing switching lawsrdquo IEEE Transac-tions on Automatic Control vol 46 no 4 pp 665ndash670 2001

[12] Z Sun and S S Ge ldquoAnalysis and synthesis of switched linearcontrol systemsrdquo Automatica vol 41 no 2 pp 181ndash195 2005

[13] Z Ji L Wang G Xie and F Hao ldquoLinear matrix inequalityapproach to quadratic stabilisation of switched systemsrdquo IEEProceedings vol 151 no 3 pp 289ndash294 2004

[14] 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 2655ndash2660December 1999

[15] G Zhai B Hu K Yasuda and A N Michel ldquoStability analysisof switched systems with stable and unstable subsystems anaverage dwell time approachrdquo International Journal of SystemsScience vol 32 no 8 pp 1055ndash1061 2001

[16] B Lu and FWu ldquoSwitching LPV control designs using multipleparameter-dependent Lyapunov functionsrdquo Automatica vol40 no 11 pp 1973ndash1980 2004

[17] S-H Lee T-H Kim and J-T Lim ldquoA new stability analysis ofswitched systemsrdquoAutomatica vol 36 no 6 pp 917ndash922 2000

[18] Z Ji and LWang ldquoQuadratic stabilization of uncertain discrete-time switched linear systemsrdquo in Proceedings of the IEEEInternational Conference on SystemsMan andCybernetics (SMCrsquo04) pp 1492ndash1497 October 2004

[19] G Xie and L Wang ldquoStabilization of a class of hybrid discrete-time systemsrdquo in Proceedings of the 2003 IEEE Conference onControl Applications pp 1404ndash1409 June 2003

[20] G Zhai H Lin A N Michel and K Yasuda ldquoStability analysisfor switched systems with continuous-time and discrete-timesubsystemsrdquo in Proceedings of the American Control Conference(AAC rsquo04) pp 4555ndash4560 July 2004

[21] J Daafouz P Riedinger and C Iung ldquoStability analysis andcontrol synthesis for switched systems a switched Lyapunovfunction approachrdquo IEEE Transactions on Automatic Controlvol 47 no 11 pp 1883ndash1887 2002

[22] D Xie L Wang F Hao and G Xie ldquoLMI approach to L 2-gainanalysis and control synthesis of uncertain switched systemsrdquoIEE Proceedings vol 151 no 1 pp 21ndash28 2004

[23] D Xie L Wang F Hao and G Xie ldquoRobust stability analysisand control synthesis for discrete-time uncertain switchedsystemsrdquo inProceedings of the 42nd IEEEConference onDecisionand Control pp 4812ndash4817 December 2003

[24] Z Sun ldquoSampling and control of switched linear systemsrdquoJournal of the Franklin Institute vol 341 no 7 pp 657ndash6742004

[25] M S Mahmoud M F Hassan and M G Darwish Large-ScaleControl Systems Marcel Dekker New York NY USA 1985

[26] M Y Waziri and Z A Majid ldquoAn improved diagonal Jacobianapproximation via a new Quasi-Cauchy condition for solvinglarge-scale systems of nonlinear equationsrdquo Journal of AppliedMathematics vol 2013 Article ID 875935 6 pages 2013

[27] B Labibi ldquoDecentralized control via disturbance attenuationand eigenstructure assignmentrdquo IEEE Transactions on Circuitsand Systems II vol 53 no 6 pp 468ndash472 2006

[28] S Xie and L Xie ldquoStabilization of a class of uncertain large-scalestochastic systems with time delaysrdquo Automatica vol 36 no 1pp 161ndash167 2000

[29] W-J Wang and L-G Mau ldquoStabilization and estimation forperturbed discrete time-delay large-scale systemsrdquo IEEE Trans-actions on Automatic Control vol 42 no 9 pp 1277ndash1282 1997

[30] S Xie and L Xie ldquoDecentralized stabilization of a class of inter-connected stochastic nonlinear systemsrdquo IEEE Transactions onAutomatic Control vol 45 no 1 pp 132ndash137 2000

[31] S Xie L Xie Y Wang and G Guo ldquoDecentralized controlof multimachine power systems with guaranteed performancerdquoIEE Proceedings vol 147 no 3 pp 355ndash365 2000

[32] H Wu ldquoDecentralized adaptive robust control for a class oflarge-scale systems including delayed state perturbations in theinterconnectionsrdquo IEEE Transactions on Automatic Control vol47 no 10 pp 1745ndash1751 2002

[33] W J Wang and L Luoh ldquoStability and stabilization of fuzzylarge-scale systemsrdquo IEEE Transactions on Fuzzy Systems vol12 no 3 pp 309ndash315 2004

[34] K Zhou and P P Khargonekar ldquoRobust stabilization of linearsystems with norm-bounded time-varying uncertaintyrdquo Sys-tems and Control Letters vol 10 no 1 pp 17ndash20 1988

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


In real systems the large-scale systems include electric powersystems nuclear reactors aerospace systems large electricnetworks economic systems process control systems chem-ical and petroleum industries different types of societalsystems and ecological systems Recently many approacheshave been used to investigate the stability and stabilization oflarge-scale systems [27ndash33]

As we have indicated very few research articles haveever discussed the stabilization of the switched large-scalediscrete-time system Therefore we will investigate the sta-bilization analysis of this new class of switched large-scalediscrete-time systems In this paper we will present noveland general approaches by Lyapunov stability theorem toconstruct a state-driven switching strategy such that theswitched large-scale discrete-time system is asymptoticallystable Particularly this method can be applied to cases whenall individual subsystems are unstableNotationThe following notationswill be used throughout thepaper 120582(119860) stands for the eigenvalues of matrix 119860 119860

120582=

120582max(119860) (119860120582 = 120582min(119860)) means the maximum (minimum)eigenvalue in matrix 119860 119860 denotes the norm of matrix 119860that is 119860 = Max[120582(119860119879119860)]12 119860119879 is the transpose of matrix119860 and diagsdot sdot sdot denotes a block-diagonal matrix

2 System Description and Problem Statement

Consider the switched large-scale discrete-time systems with119903 individual systems and each individual system composed of119873 interconnected subsystems

119909119894 (119896 + 1) = 119860120590(119909(119896))119894119894119909119894 (119896) +

119873

sum

119895 = 119894

119895=1

119860120590(119909(119896))119894119895119909119895 (119896) (1)

where 119909119894(119896) isin 119877119899119894 is the state vectors of the 119894th subsystem

119894 = 1 2 119873 119899 = sum119873

119894=1119899119894 119860120590(119909(119896))119894119894 and 119860120590(119909(119896))119894119895

aresome constant matrices of compatible dimensions 119909(119896) =

[119909119879

1(119896) 119909

119879

2(119896) sdot sdot sdot 119909

119879

119873(119896)]119879

isin 119877119899 120590(119909(119896)) 119877119899 rarr 1 2

119903 is a piecewise constant scalar function of state called aswitch signal that is120590(119909(119896)) = 119897 implies the subsystemmatri-ces 119860 119897119894119894 and 119860 119897119894119895

119897 isin 1 2 119903 Essentially the switcheddiscrete-time system (1) can be described as follows

Individual system 119897

119909119894 (119896 + 1) = 119860 119897119894119894119909119894 (119896) +

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896) (2)

For the stabilization of switched large-scale discrete-timesystem (2) some helpful lemmas are given subsequently

Lemma 1 (see [34]) For any matrices119860 and 119861 with appropri-ate dimensions one has

119860119879119861 + 119861119879119860 le 120574119860

119879119860 +

1

120574119861119879119861 (3)

for any constant 120574 gt 0

Lemma 2 For any matrices 1198601 1198602 119860119873 with the samedimensions the following inequality holds for any positiveconstant 120576

(

119873

sum

119894=1

119860 119894)

119879

(

119873

sum

119894=1

119860 119894)

le (1 + 120576) 119860119879

11198601 + (1 + 120576

minus1) (1 + 120576) 119860

119879

21198602

+ (1 + 120576minus1)2

(1 + 120576) 119860119879

31198603

+ sdot sdot sdot + (1 + 120576minus1)119873minus2

(1 + 120576) 119860119879

119873minus1119860119873minus1

+ (1 + 120576minus1)119873minus1

119860119879

119873119860119873

(4)

Proof For a positive constant 120576 119860 and 119861 have the samedimension such that

(119860 + 119861)119879(119860 + 119861) le (1 + 120576) 119860

119879119860 + (1 + 120576

minus1) 119861119879119861 (5)

In view of inequality (5) we have

(

119873

sum

119894=1

119860 119894)

119879

(

119873

sum

119894=1

119860 119894)

le (1 + 120576)119860119879

11198601 + (1 + 120576

minus1)(

119873

sum

119894=2

119860 119894)

119879

(

119873

sum

119894=2

119860 119894)

le (1 + 120576)119860119879

11198601 + (1 + 120576) (1 + 120576

minus1)119860119879

21198602

+ (1 + 120576minus1)2

(

119873

sum

119894=3

119860 119894)

119879

(

119873

sum

119894=3

119860 119894)

le (1 + 120576)119860119879

11198601 + (1 + 120576) (1 + 120576

minus1)119860119879

21198602

+ sdot sdot sdot (1 + 120576) (1 + 120576minus1)119873minus2

119860119879

119873minus1119860119873minus1

+ (1 + 120576minus1)119873minus1

119860119879

119873119860119873

(6)

Lemma 3 Let 120572119897 = 1(1+120576minus1)119897minus1(1+120576) 119897 = 1 2 119903minus1 and

120572119903 = 1(1+120576minus1)119903minus1 then 120572119897 isin [0 1] (1 le 119897 le 119903) andsum

119903

119897=1120572119897 = 1

for any positive constant 120576

Proof Obviously if there exists a positive constant 120576 then 0 lt120572119897 lt 1 119897 = 1 2 119903 and

119903

sum

119897=1

120572119897 =1

1 + 120576+

1

(1 + 120576minus1) (1 + 120576)

+ sdot sdot sdot +1

(1 + 120576minus1)119903minus2

(1 + 120576)

+1

(1 + 120576minus1)119903minus1


=1 (1 + 120576) minus 1 (1 + 120576) (1 + 120576

minus1)119903minus1

1 minus 1 (1 + 120576minus1)

+1

(1 + 120576minus1)119903minus1

= 1



(

119903

sum

119897=1

120572119897119860119879

119897)(

119903

sum

119897=1

120572119897119860 119897) le

119903

sum

119897=1

120572119897119860119879

119897119860 119897 (8)


119909 (119896 + 1) = 119860 119897119909 (119896) 119897 = 1 2 119903 (9)



119897=1120572119897 = 1 such

that119903

sum

119897=1

120572119897 (119860119879

119897119875119860 119897) minus 119875 lt 0 (10)


119897=1120572119897 = 1 such that the


119903

sum

119897=1

120572119897 (119860119879

119897119875119860 119897 minus 119875) lt 0 119897 = 1 2 119903 (11)


119909119879(119896) [

119903

sum

119897=1

120572119897 (119860119879

119897119875119860 119897 minus 119875)]119909 (119896) lt 0 (12)


119909119879(119896) [119860

119879

119897119875119860 119897 minus 119875] 119909 (119896) lt 0 (13)



(

119873

sum

119894=1

119860 119894)

119879

(

119873

sum

119894=1

119860 119894) le 119873(

119873

sum

119894=1

119860119879

119894119860 119894)

(14)




119897=1radic212057212

119897119860 119897119894119894


119879(119896) = [119909

119879

1119909119879


119879

119873]119879 and 119875 =

diag1198751 1198752 119875119873 where

2(

119903

sum

119897=1

12057212

119897119860 119897119894119894

)

119879

119875119894(

119903

sum

119897=1

12057212

119897119860 119897119894119894

) minus 119875119894 = minus119868 (15)


119897=112057212

119897119860 119897119894119894


119903

sum

119897=1

119873

sum

119895 = 119894

119895=1

120572119897

100381710038171003817100381710038171003817119860119879

119897119895119894119875119895119860 119897119895119894

100381710038171003817100381710038171003817le 05(119873 minus 1)

minus1 (16)




119873

sum

119894=1

[[[

[

119903

sum

119897=1

120572119897(119860 119897119894119894119909119894(119896) +

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

119879

times119875119894(119860 119897119894119894119909119894 (119896) +

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

]]]

]

minus119909119879

119894(119896) 119875119894119909119894 (119896)

le

119873

sum

119894=1

119903

sum

119897=1

120572119897(119860 119897119894119894119909119894 (119896))

119879

119875119894 (119860 119897119894119894119909119894 (119896))

+

119903

sum

119897=1

120572119897(119860 119897119894119894119909119894 (119896))

119879

119875119894(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))


+

119903

sum

119897=1

120572119897(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

119879

119875119894 (119860 119897119894119894119909119894 (119896))

+

119903

sum

119897=1

120572119897(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

119879

119875119894(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

minus119909119879

119894(119896) 119875119894119909119894 (119896)

le

119873

sum

119894=1

119903

sum

119897=1

120572119897

[[[

[

2(119860 119897119894119894(119896))119879

119875119894 (119860 119897119894119894119909119894 (119896))

+

119903

sum

119897=1

2120572119897(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

119879

times119875119894(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

]]]

]

minus119909119879

119894(119896) 119875119894119909119894 (119896)

le

119873

sum

119894=1

119903

sum

119897=1

2120572119897(119860 119897119894119894119909119894(119896))

119879

119875119894 (119860 119897119894119894119909119894 (119896))

+

119903

sum

119897=1

2120572119897 (119873 minus 1)

times(

119873

sum

119895 = 119894

119895=1

(119860 119897119894119895119909119895 (119896))

119879

119875119894 (119860 119897119894119895119909119895 (119896)))

minus119909119879

119894(119896) 119875119894119909119894 (119896)

le

119873

sum

119894=1

119909119879

119894(119896)

minus 119868 + 2 (119873 minus 1)

times(

119903

sum

119897=1

119873

sum

119895 = 119894

119895=1

120572119897

100381710038171003817100381710038171003817119860119879

119897119895119894119875119895119860 119897119895119894

100381710038171003817100381710038171003817)

119909119894 (119896)

(17)




119873

sum

119894=1

[[[

[

119860 119897119894119894119909119894 (119896) +

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896)

]]]

]

119879

times119875119894

[[[

[

119860 119897119894119894119909119894 (119896) +

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896)

]]]

]

minus 119875119894

lt 0

119897 = 1 2 119903

(18)

4 Conclusion


Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










=1 (1 + 120576) minus 1 (1 + 120576) (1 + 120576

minus1)119903minus1

1 minus 1 (1 + 120576minus1)

+1

(1 + 120576minus1)119903minus1

= 1



(

119903

sum

119897=1

120572119897119860119879

119897)(

119903

sum

119897=1

120572119897119860 119897) le

119903

sum

119897=1

120572119897119860119879

119897119860 119897 (8)


119909 (119896 + 1) = 119860 119897119909 (119896) 119897 = 1 2 119903 (9)



119897=1120572119897 = 1 such

that119903

sum

119897=1

120572119897 (119860119879

119897119875119860 119897) minus 119875 lt 0 (10)


119897=1120572119897 = 1 such that the


119903

sum

119897=1

120572119897 (119860119879

119897119875119860 119897 minus 119875) lt 0 119897 = 1 2 119903 (11)


119909119879(119896) [

119903

sum

119897=1

120572119897 (119860119879

119897119875119860 119897 minus 119875)]119909 (119896) lt 0 (12)


119909119879(119896) [119860

119879

119897119875119860 119897 minus 119875] 119909 (119896) lt 0 (13)



(

119873

sum

119894=1

119860 119894)

119879

(

119873

sum

119894=1

119860 119894) le 119873(

119873

sum

119894=1

119860119879

119894119860 119894)

(14)




119897=1radic212057212

119897119860 119897119894119894


119879(119896) = [119909

119879

1119909119879


119879

119873]119879 and 119875 =

diag1198751 1198752 119875119873 where

2(

119903

sum

119897=1

12057212

119897119860 119897119894119894

)

119879

119875119894(

119903

sum

119897=1

12057212

119897119860 119897119894119894

) minus 119875119894 = minus119868 (15)


119897=112057212

119897119860 119897119894119894


119903

sum

119897=1

119873

sum

119895 = 119894

119895=1

120572119897

100381710038171003817100381710038171003817119860119879

119897119895119894119875119895119860 119897119895119894

100381710038171003817100381710038171003817le 05(119873 minus 1)

minus1 (16)




119873

sum

119894=1

[[[

[

119903

sum

119897=1

120572119897(119860 119897119894119894119909119894(119896) +

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

119879

times119875119894(119860 119897119894119894119909119894 (119896) +

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

]]]

]

minus119909119879

119894(119896) 119875119894119909119894 (119896)

le

119873

sum

119894=1

119903

sum

119897=1

120572119897(119860 119897119894119894119909119894 (119896))

119879

119875119894 (119860 119897119894119894119909119894 (119896))

+

119903

sum

119897=1

120572119897(119860 119897119894119894119909119894 (119896))

119879

119875119894(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))


+

119903

sum

119897=1

120572119897(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

119879

119875119894 (119860 119897119894119894119909119894 (119896))

+

119903

sum

119897=1

120572119897(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

119879

119875119894(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

minus119909119879

119894(119896) 119875119894119909119894 (119896)

le

119873

sum

119894=1

119903

sum

119897=1

120572119897

[[[

[

2(119860 119897119894119894(119896))119879

119875119894 (119860 119897119894119894119909119894 (119896))

+

119903

sum

119897=1

2120572119897(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

119879

times119875119894(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

]]]

]

minus119909119879

119894(119896) 119875119894119909119894 (119896)

le

119873

sum

119894=1

119903

sum

119897=1

2120572119897(119860 119897119894119894119909119894(119896))

119879

119875119894 (119860 119897119894119894119909119894 (119896))

+

119903

sum

119897=1

2120572119897 (119873 minus 1)

times(

119873

sum

119895 = 119894

119895=1

(119860 119897119894119895119909119895 (119896))

119879

119875119894 (119860 119897119894119895119909119895 (119896)))

minus119909119879

119894(119896) 119875119894119909119894 (119896)

le

119873

sum

119894=1

119909119879

119894(119896)

minus 119868 + 2 (119873 minus 1)

times(

119903

sum

119897=1

119873

sum

119895 = 119894

119895=1

120572119897

100381710038171003817100381710038171003817119860119879

119897119895119894119875119895119860 119897119895119894

100381710038171003817100381710038171003817)

119909119894 (119896)

(17)




119873

sum

119894=1

[[[

[

119860 119897119894119894119909119894 (119896) +

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896)

]]]

]

119879

times119875119894

[[[

[

119860 119897119894119894119909119894 (119896) +

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896)

]]]

]

minus 119875119894

lt 0

119897 = 1 2 119903

(18)

4 Conclusion


Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+

119903

sum

119897=1

120572119897(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

119879

119875119894 (119860 119897119894119894119909119894 (119896))

+

119903

sum

119897=1

120572119897(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

119879

119875119894(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

minus119909119879

119894(119896) 119875119894119909119894 (119896)

le

119873

sum

119894=1

119903

sum

119897=1

120572119897

[[[

[

2(119860 119897119894119894(119896))119879

119875119894 (119860 119897119894119894119909119894 (119896))

+

119903

sum

119897=1

2120572119897(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

119879

times119875119894(

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896))

]]]

]

minus119909119879

119894(119896) 119875119894119909119894 (119896)

le

119873

sum

119894=1

119903

sum

119897=1

2120572119897(119860 119897119894119894119909119894(119896))

119879

119875119894 (119860 119897119894119894119909119894 (119896))

+

119903

sum

119897=1

2120572119897 (119873 minus 1)

times(

119873

sum

119895 = 119894

119895=1

(119860 119897119894119895119909119895 (119896))

119879

119875119894 (119860 119897119894119895119909119895 (119896)))

minus119909119879

119894(119896) 119875119894119909119894 (119896)

le

119873

sum

119894=1

119909119879

119894(119896)

minus 119868 + 2 (119873 minus 1)

times(

119903

sum

119897=1

119873

sum

119895 = 119894

119895=1

120572119897

100381710038171003817100381710038171003817119860119879

119897119895119894119875119895119860 119897119895119894

100381710038171003817100381710038171003817)

119909119894 (119896)

(17)




119873

sum

119894=1

[[[

[

119860 119897119894119894119909119894 (119896) +

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896)

]]]

]

119879

times119875119894

[[[

[

119860 119897119894119894119909119894 (119896) +

119873

sum

119895 = 119894

119895=1

119860 119897119894119895119909119895 (119896)

]]]

]

minus 119875119894

lt 0

119897 = 1 2 119903

(18)

4 Conclusion


Acknowledgment


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Stabilization Analysis for the Switched...

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