Transition Probability Bounds for the Stochastic Stability Robustness of Continuous- and Discrete-Time Markovian Jump Linear Systems

Transcription

1 Transton Probablty Bounds for the Stochastc Stablty Robustness of Contnuous- and Dscrete-Tme Markovan Jump Lnear Systems Mehmet Karan, Peng Sh, and C Yalçın Kaya June 17, 2006 Abstract Ths paper consders the robustness of stochastc stablty of Markovan jump lnear systems n contnuous- and dscrete-tme wth respect to ther transton rates and probabltes respectvely The contnuous-tme (dscrete-tme) system s descrbed va a contnuous-valued state vector and a dscrete-valued mode whch vares accordng to a Markov process (chan) By usng stochastc Lyapunov functon approach and Kronecker product transformaton technques, suffcent condtons are obtaned for the robust stochastc stablty of the underlyng systems, whch are n terms of upper bounds on the perturbed transton rates and probabltes Analytcal expressons are derved for scalar systems, whch are straghtforward to use Numercal examples are presented to show the potental of the proposed technques 1 Introducton A large class of physcal systems have varable structures subject to random changes, whch may result from the abrupt phenomena such as component and nterconnecton falures, parameters shftng Systems wth ths character may be modelled as hybrd ones, that s, the state space of the systems contans both dscrete- and contnuousvalued states Among ths knd of systems, jumpng lnear systems have been a subject of great practcal mportance whch has attracted a lot of nterest In jumpng lnear Boeng Australa Lmted Current address: The Boeng Company, PO Box 3707, Mal Stop: 8A-48, Seattle WA USA Emal: MehmetKaran@boengcom School of Technology, Unversty of Glamorgan, Pontyprdd, CF37 1DL, UK Emal: psh@glamacuk School of Mathematcs and Statstcs, Unversty of South Australa, Mawson Lakes, SA 5095, Australa Emal: yalcnkaya@unsaeduau 1

2 systems, the dynamcs of the dscrete and contnuous states are modelled, respectvely, by a fnte state Markov process/chan and lnear dfferental/dfference equatons subject to the dscrete process/chan Dscrete-valued states, referred to also as system modes, am to capture sgnfcant changes n the system For example, Interactng Multple Model (IMM) algorthm for radar trackng, as descrbed n [Bar-Shalom and L, 1993], employs multple models to descrbe dfferent trajectory legs and a Markov chan s used to model the transton between these models Stablty propertes of systems descrbed by multple models swtchng accordng to Markov processes/chans can be analysed va the noton of stochastc stablty ntroduced n [J and Chzeck, 1990] and [Feng et al, 1992] When the parameters of the Markov process/chan descrbng the transton between dfferent models are not completely known, t s mportant to know how much uncertanty can be tolerated for the system to be stochastcally stable The am of ths paper s to provde perturbaton bounds on Markov process/chan transton rates/probabltes so that the perturbed jumpng lnear system remans stable In jumpng lnear quadratc (JLQ) control theory, there has been a dramatc progress snce the poneerng work on JLQ control by [Krasovsk and Ldsk, 1961] The JLQ control problem was solved by [Sworder, 1969] usng stochastc maxmum prncple for state feedback n fnte horzon case [Wonham, 1971] also obtaned the same results usng dynamc programmng for both fnte and nfnte horzon cases [Marton and Bertrand, 1985] provded an approach to output feedback JLQ control problem The contnuous-tme partally observable stuaton was studed by [Fragoso, 1988] An analyss of the dscrete-tme verson of JLQ control problem was gven n [Chzeck et al, 1986] for the case wthout drvng nose, and [Fragoso, 1989] for the case wth drvng nose, respectvely [Costa et al, 1999] studed the dscrete-tme JLQ control problem n the presence of constrants on the state and control varables The problems of controllablty, stablzablty, and contnuous-tme JLQ control have been theoretcally addressed by [J and Chzeck, 1990] and [Boukas, 2005] and the references theren [Feng and Loparo, 1990] has studed the problem of almost sure nstablty of the random harmonc oscllator The stochastc stablty propertes of jumpng lnear systems has been systematcally nvestgated n [Feng et al, 1992] and shown that several stablty concepts are equvalent [Kalmanovch and Haddad, 1994] tackled the problem of JLQ control when the dscrete Markov process n systems s not drectly observable and obtaned necessary condtons for optmalty [Pan and Bar-Shalom, 1996] derved a suffcent condton for the stablzaton for a class of adaptve controllers when the mode s not drectly observed The counterpart of H control of jump lnear systems was nvestgated by [Pan and Basar, 1994] va zero-sum dfferental games Also, [Srchander and Walker, 1989] made the stochastc stablty analyss for contnuous-tme fault tolerant control systems, n whch the system has two random processes wth Markovan transton characterstcs (one representng the random falures of the system and the other representng the falure detecton and dentfcaton (FDI) decson behavour) On the other hand, desgn of control systems that can handle model uncertantes has 2

3 been one of the most challengng problems and receved consderable attenton from control engneers and scentsts n the past decades There are two major ssues n robust controller desgn The frst s concerned wth the robust stablty of the uncertan closed-loop system (see for example, [Khargonekar et al, 1990] and the references theren), and the other s robust performance (see for example, [Karan et al, 1994] and [Xe et al, 1992] The problems of robust stochastc stablty, stablzaton, and control have been extensvely studed For some representatve pror work on ths general topc, we refer the reader to [Boukas, 1993, 1995, Boukas and Sh, 1998, Sh and Boukas, 1997, Sh et al, 1998, 1999, 2000] and the references theren However, to the best of our knowledge, robust stochastc stablty for jumpng lnear contnuoustme and dscrete-tme systems wth perturbed transton rates and probabltes has not yet been fully studed Ths problem s qute mportant n real physcal systems descrbed by jumpng systems, smply because the probablty of the system mode jumpng from one state to another may not be known exactly In ths paper, we consder the stablty robustness of contnuous- and dscrete-tme Markovan jump lnear systems (MJLS) wth respect to uncertanty appearng n system mode transton rates and probabltes respectvely Stablty of MJLS has been nvestgated va the noton of stochastc stablty ntroduced n [J and Chzeck, 1990], and suffcent condtons of perturbaton bound for the stochastc stablty of MJLS have been obtaned usng stochastc Lyapunov functons n the work by Feng et al ([J and Chzeck, 1990], [Feng et al, 1992]) The structure of the paper s as follows The Sectons 2 and 3 gve new reformulatons of exstng relevant results n the lterature These new reformulatons are used n Secton 4 to obtan perturbaton bounds on the transton rates and probabltes for the stochastc stablty of contnuous- and dscrete-tme MJLS Numercal results are provded n Secton 5 for scalar contnuous-tme and dscrete-tme MJLS Notaton The notaton n ths paper s qute standard R n and R n m denote, respectvely, the n dmensonal Eucldean space and the set of all n m real matrces The superscrpt T denotes the transpose and the notaton X Y (respectvely, X>Y) where X and Y are symmetrc matrces, means that X Y s postve sem-defnte (respectvely, postve defnte) I s the dentty matrx wth compatble dmenson E } s the expectaton operator wth respect to some probablty measure P The trace and -th egenvalue of a matrx M are denoted by tr(m) and ρ (M) respectvely Mnmum and maxmum sngular values of a matrx M are denoted by σ mn (M) and σ max (M) respectvely where σ max (M) =ρ max (M T M) where ρ 1/2 max( ) denotes the maxmum real-valued egenvalue of ( ) 3

4 2 Problem Defnton Consder the followng autonomous contnuous- and dscrete-tme Markovan jump lnear systems (MJLS) S and Σ wth the state vectors x(t) R N and x[k] R N respectvely S : ẋ(t) = A(η t ) x(t) (21) Σ : x[k +1] = F (γ k ) x[k] (22) For the contnuous-tme system S, the system mode η t,t 0} s a tme homogeneous Markov process wth rght contnuous trajectores and takng values n a fnte set M = 1, 2,, M} wth statonary transton probabltes π j h + o(h), j Prob (η t + h = j η t = ) = (23) 1+π h + o(h), = j o(h) where h>0, lm h 0 h =0and π j 0 s the transton rate from mode at tme t to mode j at tme t + h, and π = M j π j The ntal system mode probablty vector s defned by π(0) = [π 1 (0) π M (0)] T (24) Note that the system mode probablty vector π(t) can be found va π(t) = Π T π(t) (25) The Markov process transton rate matrx Π s defned by π 11 π 12 π 1M π 21 π 22 π 2M Π = π M1 π M2 π MM Here, we assume that the Markov process s rreducble (26) For the dscrete-tme system Σ, the system mode γ k,k=0, 1,} s a tme homogeneous Markov chan takng values n a fnte set M = 1, 2,, M} wth statonary transton probabltes λ j = Prob(γ k+1 = j γ k = ) 4

5 where λ j 0 s the transton probablty from mode at tme k to mode j at tme (k +1)and M λ j = 1 (27) Let λ[0] be the ntal system mode probablty vector defned by λ[0] = [ Probγ 0 =1}, Probγ 0 =2}, Probγ 0 = M} ] (28) Then the system mode probablty vector λ[k] at tme k can be found recursvely as λ[k] = Λ T λ[k 1] (29) where the transton probablty matrx Λ s defned by λ 11 λ 12 λ 1M λ 21 λ 22 λ 2M Λ = λ M1 λ M2 λ MM (210) Let x(t, x 0,η 0 ) and x[k, x 0,γ 0 ] denote the trajectory of the state x(t) and x[k], ofthe systems S n (21) and Σ n (22), from the ntal state x 0 wth an ntal system mode η 0 and γ 0 respectvely As ntroduced n [J and Chzeck, 1990] and [J et al, 1991], stochastc stablty of a system can be defned as follows Defnton 1 The systems S n (21) and Σ n (22) are sad to be stochastcally stable about the equlbrum pont 0, f for any ntal state x 0 R N and for any ntal mode η 0 = and γ 0 = where M, the followng nequaltes hold respectvely 0 E x(t, x 0,η 0 ) 2 } dt <, E x[k, x 0,γ 0 ] 2 } < k=0 (211) Now, consder systems S and Σ dentcal to S n (21) and Σ n (22), but wth jumpng rates π j and jumpng probabltes λ j respectvely such that S and Σ are stochastcally stable and π j = π j +Δπ j, λ j = λ j +Δλ j (212) Ths paper ams to fnd bounds on the transton perturbatons Δπ j and Δλ j for, j =1,,M such that the systems S and Σ reman stochastcally stable when S and Σ are stochastcally stable For ths purpose, frst exstng results n the lterature are reformulated for the stochastc stablty of the nomnal systems S and Σ These new reformulatons wll then be used n Secton 4 to obtan suffcent bounds on transton rate and probablty perturbatons 5

6 3 Reformulaton of Prevous Results Stochastc stablty of contnuous- and dscrete-tme Markovan jump lnear systems can be checked usng the followng result For brevty, we adopt the notaton A := A() and F := F () Lemma 1 ([Feng et al, 1992], [Fang and Loparo, 2002b] and [Fang and Loparo, 2002a]) The systems S n (21) and Σ n (22) are stochastcally stable f and only f there exst sets of symmetrc postve defnte matrces P and Φ satsfyng F T A T P + P A + M π j P j + Q = 0 (31) M λ j Φ j F Φ +Ξ = 0 (32) for any gven set of symmetrc postve defnte matrces Q and Ξ where M The above equatons can be rewrtten to obtan the followng result Lemma 2 The systems S n (21) and Σ n (22) are stochastcally stable f and only f there exst block dagonal symmetrc postve defnte matrces P and Φ satsfyng the followng equatons for any gven symmetrc block dagonal postve defnte matrx Q and Ξ respectvely A T P + PA +[Γ I][I (P E)] + Q = 0 (33) F T (Λ I)[I (Φ E)]} F Φ+ Ξ = 0 (34) where denotes the Kronecker product as defned n [Graham, 1981] and A P A 2 0 A = 0 P 2 0 RNM NM, P = RNM NM, 0 0 A M 0 0 P M v v Γ= v M Q Q R M M 2, Q = 0 0 Q M R NM NM, 6

7 F Φ F 2 0 F = 0 Φ 2 0 RNM NM, Φ= RNM NM, 0 0 F M 0 0 Φ M ū ū Λ= ū M Ξ Ξ R M M 2, Ξ= 0 0 Ξ M R NM NM, E = [ I I ] T R NM N where v and ū are defned by v = [ π 1 π 2 π M ] ū = [ λ 1 λ 2 λ M ] (35) (36) In the above lemma, I denotes ether the m m or the n n dentty matrx, approprately Remark 1 For dscrete-tme systems, [Fang and Loparo, 2002b] states an alternatve expresson, whch s equvalent to (32): The system Σ n (22) s stochastcally stable f and only f there exst sets of symmetrc postve defnte matrces Ψ satsfyng M λ j F T j Ψ j F j Ψ +Ξ = 0 (37) for any gven set of symmetrc postve defnte matrces Ξ where M A reformulaton of the above condton usng the Kronecker product s straghtforward In some practcal applcatons, although t s not qute clear how, one may fnd usng (37) more advantageous over (32) However n the sequel we wll not consder (37), smply because the stablty robustness expressons we obtan by usng (32) are dentcal to those we fnd by usng (37) 4 Man results From the lemmas n the prevous secton for the MJLS, we obtan the followng theorem whch gves bounds on the transton rates and probabltes for the stochastc stablty of Markovan jump lnear systems 7

8 Theorem 1 Consder systems S n (21) and Σ n (22) dentcal to the nomnal systems S and Σ, but wth jumpng rates π j and jumpng probabltes λ j respectvely such that π j = π j +Δπ j λ j = λ j +Δλ j where S and Σ are stochastcally stable, e there exst postve defnte solutons P and Φ for =1,,M for gven postve defnte matrces Q and Ξ to the followng equatons A T P + P A + F T M π j P j + Q = 0 (41) M λ j Φ j F Φ + Ξ = 0 (42) Then the perturbed systems S and Σ are also stochastcally stable f the followng nequaltes are satsfed respectvely 1/2 M max Δπj 2 mn σmn (Q ) } ( M ) ρ 1/2 max P 2 max M Δλ 2 j 1/2 mn σmn (Ξ ) } where the perturbatons satsfy the followng addtonal constrants =1 } 2 max σ max (F ) ρ 1/2 max ( M =1 (43) Φ 2 ) (44) M Δπ j = 0, M Δλ j = 0 for all =1,,M and for all, j =1,,M where j Δπ j > π j, Δλ j > λ j (45) Proof For the perturbaton bound n (43) gven for the contnuous-tme MJLS S, defne an energy functon E(x, η) for the system n (21) as E(x(t),η t ) = x T (t) P ηt x(t) 8

9 Then the system s stochastcally stable f L(E)(x(t),η t ) = x T (t) < 0 ( A(η t ) T P ηt + P ηt A(η t ) + M π ηt,j P j ) x(t) where L s the nfntesmal generator actng on E(x, η) Then, usng Lemma 2, we can wrte A T P + PA +[ ( Γ+ΔΓ ) I][I (P E)] 0 e the matrx on the left-hand sde s negatve sem-defnte Manpulatons yeld A T P + PA +(Γ I)[I (P E)] + (ΔΓ I) [I (P E)] 0 Q +(ΔΓ I) [I (P E)] 0 where ṽ ṽ Γ = ṽ M Δv Δv ΔΓ = Δv M ṽ and Δv are defned by ṽ = [ π 1 π 2 π M ] Δv = [Δπ 1 Δπ 2 Δπ M ] Thus, a suffcent condton can be gven as } σ max (ΔΓ I) [I (P E)] σmn (Q) = mn σmn (Q ) } (46) where σ max ( ) and σ mn ( ) denote the maxmum and mnmum sngular values of ( ) respectvely Note that σ max (ΔΓ I) [I (P E)] } σmax (ΔΓ I) σ max [I (P E)] Now, we can choose ΔΓ such that σ max (ΔΓ I) σ max [I (PE)] mn σmn (Q ) } (47) 9

10 We should note that the above nequalty provdes a more conservatve bound on the permssble perturbaton ΔΓ snce any value of ΔΓ whch satsfes the bound n (47) wll also satsfy the nequalty n (46) On the other hand, ( ( ) M ) σ max [I (P E)] = σ max PE = ρ 1/2 max P 2 where ρ max ( ) denotes the maxmum real egenvalue of ( ) Moreover, =1 σ max (ΔΓ I) = ρmax(δγ 1/2 T ΔΓ) = ρ 1/2 max(δγ ΔΓ T ) = max Δv Δv T M } 1/2 =max M M Δπj 2 1/2 (48) Consequently, the perturbed system s stable f the bound n (43) s satsfed Smlarly, the dscrete-tme system Σ s stable f F T ( (Λ+ΔΛ) I ) [I (Φ E)] F Φ 0 (49) where ũ ũ Λ = ũ M Δu Δu ΔΛ = Δu M ũ and Δu are defned by ũ = [ λ 1 λ2 λm ] Δu = [Δλ 1 Δλ 2 Δλ M ] Hence, F T (Λ I)[I (Φ E)] F Φ+F T (ΔΛ I) [I (Φ E)] F 0 (410) From (42) and Lemma 2, we then obtan Ξ+F T ((ΔΛ) I) [I (Φ E)]F 0 10

11 Thus, a suffcent condton can be gven as σ max F T (ΔΛ I) [I (Φ E)]F } σ mn (Ξ) = mn σmn (Ξ ) } (411) Note that σ max F T (ΔΛ I) [I (Φ E)]F } σ max (ΔΛ I)[I (Φ E)]} σmax 2 (F ) σ max (ΔΛ I)} σ max [I (Φ E)]} σmax(f 2 ) and On the other hand, σ max [I (Φ E)] = σ max ( Φ E ) = ρ 1/2 max ( M =1 Φ 2 ) σ max ΔΛ I)} = σ max ΔΛ)} = ρ 1/2 max(δλ ΔΛ T ) = max Δv Δv T } 1/2 M 1/2 M = max Δλ 2 j M Hence, the perturbed system s stable f the bound n (44) s satsfed 5 Examples In ths secton, we provde examples to llustrate the above results for the case of scalar Markovan jump lnear systems n contnuous- and dscrete-tme 51 Scalar contnuous-tme MJLS wth two modes Consder the robust stablty of a scalar Markovan jump lnear system gven by ẋ(t) = a(η t ) x(t), t 0 S : x(0) = x 0 R (51) where the transton rate matrx Π, assocated wth the system mode η t, s gven by Π = [ π1 ] π 1 π 2 π 2 (52) 11

12 where π > 0 Our am s to nvestgate the stablty of the system S whch s dentcal to the nomnal system S except that the transton rates are perturbed as π = π +Δπ (53) Note from (31) that the perturbed system S s stable f and only f 2a 1 p 1 + π 1 (p 2 p 1 )+q 1 = 0, (54) 2a 2 p 2 + π 2 (p 1 p 2 )+q 2 = 0 (55) where a = a(), and q 1, q 2, p 1 and p 2 are postve Let q 1 = q 2 =1n what follows Then the above equatons can be solved as p 1 = n 1 d, p 2 = n 2 d (56) n 1 = π 1 + π 2 2a 2, (57) n 2 = π 1 + π 2 2a 1, (58) d = 4a 1 a 2 2a 2 π 1 2a 1 π 2 (59) Therefore the condtons p 1 > 0 and p 2 > 0 are satsfed f and only f n 1 > 0, n 2 > 0, d>0 or n 1 < 0, n 2 < 0, d<0 The frst set of constrants can be rewrtten wth the addtonal constrants on π 1,π 2 π 1 + π 2 > 2maxa 1,a 2 }, π 1 > 0, (510) 2a 1 a 2 >a 2 π 1 + a 1 π 2, π 2 > 0 (511) and the second set of constrants becomes π 1 + π 2 < 2mna 1,a 2 }, π 1 > 0, (512) a 2 π 1 + a 1 π 2 > 2a 1 a 2, π 2 > 0 (513) However, (512) cannot be satsfed f ether a 1 or a 2 s negatve Thus, the constrants n (510) and (511) become suffcent condtons for the stablty of the system n terms of π 1 and π 2 when ether of the subsystems s stable Moreover, the suffcent condtons of the knd gven n (510)-(513), whch are obtaned by choosng q 1 = q 2 =1, are, n general, very dffcult to calculate for hgher dmensonal systems We can also use (43) to obtan a bound on Δπ n (53) for =1, 2 for the stochastc stablty of the perturbed system S as max Δπ } < 1 mn =1,2 ( q ) =1,2 2 p p 2 2 (514) where p 1 and p 2 can be calculated for the system S from (54) wth the transton rates π 1 and π 2 12

13 Now, let us consder the example gven n [J and Chzeck, 1990] where the system s a scalar MJLS as gven n (51) where a 1 = 1/3, a 2 = 4/3 (515) and the probablty transton rate matrx π n (52) s gven by π 1 = 1, π 2 = 1 It can be found that ths system s stable snce from (56) and (57), p 1 and p 2 can be calculated wth the above values of the transton rates π as p 1 = 21, p 2 = 6 The admssble values of (π 1,π 2 ) for the stochastc stablty of the perturbed system S wth the parameter values n (515) can be found from (510)-(511) as R = (π 1,π 2 ) π 1 + π 2 > 2 3, 4π 1 π 2 > 8 } 3,π 1 > 0, π 2 > 0 Note that the above bounds are necessary and suffcent for the stochastc stablty of the perturbed system S, however these bounds are n general very dffcult to compute for hgher dmensonal systems On the other hand, we can also obtan bounds from (514) on (Δπ 1, Δπ 2 ), or equvalently on (π 1,π 2 ) also usng (53), as max Δπ } < 1 1 =1, = Thus, the above result yelds an admssble (π 1,π 2 )-regon R 1 for the stochastc stablty of the perturbed system S as } R 1 = (π 1,π 2 ) π 1 1 < , π 2 1 < , π 1 > 0, π 2 > 0 These two admssble regons R and R 1 can be seen n Fgure 1 It s not surprsng that R 1 allows relatvely small perturbaton bounds on the transton rates snce the values of the nomnal system transton rates are close to the boundary of the admssble transton rate regon R On the other hand, f we choose the nomnal system transton rates as π 1 = 2, π 2 = 1 then from (56) and (57) we obtan p 1 = 51 26, p 2 = Now, from (514) we obtan the followng bound max Δπ } < =1,2 13

14 The above bound defnes an admssble transton rate regon R 2 as } R 2 = (π 1,π 2 ) π 1 2 < 03333, π 2 1 < 03333, π 1 > 0, π 2 > 0 Ths regon s also shown n Fgure 1 where t can be seen that R 2 s much larger than R 1 snce t s obtaned wth nomnal system transton rate values that are not close to the boundary of R We can conclude that the nomnal system S, wth π 1 = π 2 =1, s not as robust aganst transton rate perturbatons as the same system wth π 1 = 2, π 2 =1 3 2 R 1 R 2 1 R π π 1 Fgure 1: Robust stablty bounds on the transton rates for the scalar Markovan jump lnear system wth a 1 =1/3, a 2 = 4/3 R s the necessary and suffcent admssble regon (π 1,π 2 ) The nomnal system transton rate values (π 1 =1, π 2 =1)and (π 1 =2, π 2 =1)are used to obtan R 1 and R 2, respectvely 14

15 52 Scalar dscrete-tme MJLS wth two modes Let us consder the robust stablty of a scalar Markovan jump lnear system gven by x[k +1] = f(η k ) x[k], k =0, 1, S : (516) x[0] = x 0 R In (516), the transton probablty matrx Λ, assocated wth the system mode η k,s gven by [ ] [ ] λ11 λ Λ = 12 1 λ1 λ = 1 (517) λ 21 λ 22 λ 2 1 λ 2 where 1 > λ 1 > 0 and 1 > λ 2 > 0 Now consder a perturbed system S dentcal to the nomnal system S except that the transton probabltes are perturbed as where 1 >λ 1 > 0, and 1 >λ 2 > 0 λ = λ +Δλ, =1, 2 (518) Note from (32) that the perturbed system S s stable f and only f there exsts postve real numbers φ 1, φ 2, ξ 1 and ξ 2 whch satsfy the followng equatons where ξ 1 and ξ 2 can be chosen as 1 f1 2 [(1 λ 1)φ 1 + λ 1 φ 2 ] φ 1 + ξ 1 = 0, (519) f2 2 [(1 λ 2 )φ 2 + λ 2 φ 1 ] φ 2 + ξ 2 = 0 (520) Then the above equatons can be solved as where φ 1 = n 1 d, φ 2 = n 2 d n 1 = f1 2 λ 1 f2 2 λ 2 + f2 2 1 n 2 = f1 2 λ 1 f2 2 λ 2 + f1 2 1 (521) d = f 2 1 (f 2 2 1)λ 1 + f 2 2 (f 2 1 1)λ 2 + f f 2 2 f 2 1 f Therefore the condtons φ 1 > 0 and φ 2 > 0 are satsfed f and only f ether of the followng two sets of nequaltes are satsfed n 1 > 0, n 2 > 0, d > 0 (522) or n 1 < 0, n 2 < 0, d < 0 (523) 15

16 It can easly be shown that the frst set of nequaltes n (522) s satsfed f and only f f1 2 λ 1 + f2 2 λ 2 < mnf1 2,f2 2 } 1 (524) f1 2 (1 f 2 2 )λ 1 + f2 2 (1 f 1 2 )λ 2 < f1 2 + f 2 2 f 1 2 f (525) On the other hand, the set of nequaltes n (523) s satsfed f and only f f1 2 λ 1 + f2 2 λ 2 > maxf1 2,f2 2 } 1 (526) f1 2 (1 f2 2 )λ 1 + f2 2 (1 f1 2 )λ 2 > f1 2 + f2 2 f1 2 f2 2 1 (527) Hence, the system wll be stochastcally stable f the transton probablty parameters 0 <λ 1 < 1 and 0 <λ 2 < 1 belong to the polygon regon defned ether by (524)- (525) or by (526)-(527), n the unt square Note that f ether subsystem s stable, e f < 1 for some =1, 2, then (524) s not satsfed and (526)-(527) consttute the suffcent condtons to check for stochastc stablty Let the perturbaton on the transton probabltes be gven as or equvalently where, j =1, 2 From (44), 1/2 2 (2Δλ max Δλ 2 j = max ) } 2 1/2 λ j = λ j +Δλ j (528) λ = λ +Δλ (529) mn σmn (Ξ ) } = mnσ mn (1)} } 1 = mn f 2 } 1 = mn f 2 = 2max Δλ } } 2 max σ max (F ) ρ 1/2 max max f } 2 ρ 1/2 ( ) ρ 1/2 max φ φ2 2 1 ( φ φ2 2 max ( 2 =1 ( 2 φ 2 =1 ) Φ 2 ) ) (530) Hence, the perturbed system S s stable f max Δλ } } 1 1 mn 2 f 2 1 ( ) (531) φ φ

17 Let f 1 =1/2and f 2 =5/4 Note that one of the subsystems s stable Then the system wll be stochastcally stable f and only f (λ 1,λ 2 ) belongs to the regon furnshed by (526)-(527), denoted by R, namely R = (λ 1,λ 2 ) 0 <λ 1 < 1, 0 <λ 2 < 1, } 4λ 1 +25λ 2 > 9, 9λ 1 +75λ 2 > 27 (532) Let the nomnal values for transton probabltes be gven as λ 1 =02, λ2 =08 (533) It can be easly verfed that the system s stochastcally stable by checkng ( λ 1, λ 2 ) R One can also solve (519)-(520), wth the gven values of f and λ, and wth ξ 1 = ξ 2 =1,tofndφ 1 and φ 1 as φ > 0, φ > 0, (534) whch confrms that the system s stochastcally stable Now from (530), a suffcent condton for the stochastc stablty of the perturbed system can be gven as max Δλ } = 1 } 1 1 mn 2 f 2 φ φ (535) Thus, the above result yelds an admssble (λ 1,λ 2 )-regon R 1 for the stochastc stablty of the perturbed system S as R 1 = (λ 1,λ 2 ) λ 1 02 < 01013, λ 2 08 < 01013, } 0 < λ 1 < 1, 0 < λ 2 < 1 If the nomnal values of the transton probabltes are taken as λ 1 =06, λ2 =05 (536) then the perturbaton bound s gven by max Δλ } The above bound defnes an admssble transton probablty regon R 2 as R 2 = (λ 1,λ 2 ) λ 1 06 < 00210, λ 2 05 < 00210, } 0 < λ 1 < 1, 0 < λ 2 < 1 17

18 The stablty regon R and the robustness regons R 1 and R 2 about the two dfferent nomnal ponts dscussed above are depcted n Fgure 2 We observe a parallelsm to the example gven for the contnuous-tme MJLS: R 1 s much larger than R 2 snce t s obtaned wth nomnal system transton rate values that are not so close to the boundary of R 1 08 R 1 06 R λ 2 R λ 1 Fgure 2: Robust stablty bounds on the transton probabltes for the dscrete-tme MJLS wth f 1 =1/2and f 2 =5/4 R s the necessary and suffcent admssble regon (λ 1,λ 2 ), whose boundary s shown by the sold lne The nomnal system transton rate values (λ 1 =02, λ 2 =08) and (λ 1 =06, λ 2 =05) are used to obtan R 1 and R 2, respectvely 6 Concluson In ths paper, we consdered robust stablty of contnuous- and dscrete-tme Markovan jump lnear systems n terms of the perturbed transton rates and probabltes respectvely Usng stochastc Lyapunov functons, a bound on the transton rates and probabltes were gven so that the systems reman stochastcally stable Analytcal expressons were derved for scalar systems Examples were gven to llustrate the results for scalar systems where there exsts a necessary and suffcent condton on the robust stablty of the system 18

19 7 Acknowledgments The authors would lke to thank the anonymous revewers for ther constructve comments whch mproved the paper References Y Bar-Shalom and X-R L Estmaton and Trackng: Prncples, Technques, and Software Artech House, Boston, MA, 1993 E K Boukas Control of systems wth controlled jump Markov dsturbances Control Theory and Advanced Technology, 9(2): , 1993 E K Boukas Robust stablty of lnear pecewse determnstc systems under matchng condtons Control Theory and Advanced Technology, 10(4): , 1995 E K Boukas Stablzaton of stochastc nonlnear hybrd systems Int J Innovatve Computng, Informaton and Control, 1(1): , 2005 E K Boukas and P Sh Stochastc stablty and guaranteed cost control of dscretetme uncertan systems wth Markovan jumpng parameters Int J Robust & Nonlnear Control, 8(13): , 1998 H J Chzeck, A S Wllsky, and D Castanon Dscrete-tme markovan-jump lnear quadratc optmal control Int J Control, 43(1): , 1986 O L V Costa, F E O Assumpção, E-K Boukas, and R P Marques Constraned quadratc state feedback control of dscrete-tme markovan jump lnear systems Automatca, 35(4): , 1999 Y Fang and K A Loparo Stablzaton of Contnuous-Tme Jump Lnear Systems IEEE Trans Automat Control, 47(10): , 2002a Y Fang and K A Loparo Stochastc Stablty of Jump Lnear Systems IEEE Trans Automat Control, 47(7): , 2002b X Feng and K Loparo Almost sure nstablty of the random harmonc oscllator SIAM J Appl Math, 50(3): , 1990 X Feng, K A Loparo, Y J, and H J Chzeck Stochastc stablty propertes of jump lnear systems IEEE Trans Automat Control, 37(1):38 53, 1992 M D Fragoso On a partally observable LQG problem for systems wth Markovan jumpng parameters and quadratc cost System & Control Letters, 10(5): , 1988 M D Fragoso Dscrete-tme jump LQG problem Int J Systems Scence, 20(12): ,

20 A Graham Kronecker products and matrx calculus: wth applcatons Halsted Press, 1981 Y J and H J Chzeck Controllablty, Stablzablty, and Contnuous-Tme Markovan Jump Lnear Quadratc Control IEEE Trans Automat Control, 35(7): , 1990 Y J, H J Chzeck, X Feng, and K A Loparo Stablty and control of dscretetme jump lnear systems Control Theory and Advanced Technology, 7(2): , 1991 G Kalmanovch and A H Haddad Optmal constant gan control of jump-lnear systems wth dscrete states uncertanty In Proc 33rd IEEE Conf Decson & Control, pages , Lake Buena Vsta, FL, 1994 M Karan, R C Wllamson, and B D O Anderson Performance of the Maxmum Lkelhood Constant Frequency Estmator for Frequency Trackng IEEE Trans Sgnal Processng, 42(10): , 1994 P P Khargonekar, I R Petersen, and K Zhou Robust stablzaton of uncertan lnear system: Quadratc stablzablty and H control theory IEEE Trans Automat Control, 35(3): , 1990 N N Krasovsk and E A Ldsk Analyss desgn of controller n systems wth random attrbutes Part 1 Automatc Remote Control, 22: , 1961 M Marton and P Bertrand Output feedback for a class of lnear systems wth stochastc jump parameters IEEE Trans Automat Control, 30(9): , 1985 G L Pan and Y Bar-Shalom Stablzaton of jump lnear gaussan systems wthout mode observatons Internatonal Journal of Control, 64(4): , 1996 Z Pan and T Basar Zero-sum dfferental games wth random structures and applcatons n H - control of jump lnear systems In Proc Sxth Int Symp on Dynamc Games and Applcatons, pages , St-Jovte, Quebec, Canada, 1994 P Sh and E K Boukas H control for Markovan jumpng lnear systems wth parametrc uncertanty J Optmzaton Theory and Applcatons, 95(1):75 99, 1997 P Sh, E K Boukas, and R K Agarwal Robust control for Markovan jumpng dscrete-tme systems Int J Systems Scence, 30(8): , 1998 P Sh, E K Boukas, and R K Agarwal Control of Markovan jump dscrete-tme systems wth norm bounded uncertanty and unknown delays IEEE Trans Automat Control, 44(11): , 1999 P Sh, M Karan, and C Kaya Robust Kalman flterng desgn for hybrd systems wth norm-bounded unknown nonlneartes In Proc 39th IEEE Conf on Decson & Control, pages , 2000 R Srchander and B K Walker Stochastc stablty analyss for contnuous-tme fault tolerant control systems Int J Control, 57(2): ,

21 D D Sworder Feedback control of a class of lnear systems wth jump parameters IEEE Trans Automat Control, 14(1):9 14, 1969 W M Wonham Probablstc Methods n Appled Mathematcs, chapter Random dfferental equatons n control theory, pages Academc Press, New York, 1971 L Xe, M Fu, and C E de Souza H control and quadratc stablzaton of systems wth parameter uncertanty va output feedback IEEE Trans Automat Control, 37: ,