MODE-INDEPENDENT GUARANTEED COST CONTROL OF SINGULAR MARKOVIAN DELAY JUMP SYSTEMS WITH SWITCHING PROBABILITY RATE DESIGN

Internatonal Journal of Innovatve Computng, Informaton and Control ICIC Internatonal c 214 ISSN 1349-4198 Volume 1, Number 4, August 214 pp. 1291 133 MODE-INDEPENDENT GUARANTEED COST CONTROL OF SINGULAR MARKOVIAN DELAY JUMP SYSTEMS WITH SWITCHING PROBABILITY RATE DESIGN Peng Zhang, Jangtao Cao and Guolang Wang School of Informaton and Control Engneerng Laonng Shhua Unversty No. 1, Dandong Road, Wanghua Dst., Fushun 1131, P. R. Chna zhangpeng1577@126.com Receved July 213; revsed November 213 Abstract. Ths paper consders the guaranteed cost control of sngular Markovan jump systems (SMJSs) wth tme delay whose mode sgnal s naccessble. The man contrbuton s to develop an approach to mode-ndependent guaranteed control, the swtchng probablty rate s also desgned. New suffcent condtons of such controller are proposed n terms of lnear matrx nequaltes wth some equaton constrants, whch also characterze the swtchng probablty rate. Fnally, numercal examples are used to demonstrate the effectveness and advantage of the proposed method. Keywords: Sngular Markovan jump systems, Mode-ndependent control, Guaranteed cost control, Lnear matrx nequaltes (LMIs) 1. Introducton. As we know, sngular system was ntroduced by H. H. Rosenbrock [1 n 1974. Many practcal systems such as electrc power system, economc systems, mechancal engneerng systems and robotcs, can be descrbed by sngular systems. Durng the past years, a lot of attenton has been pad to ths system. Many mportant results have emerged, such as [2, 3, and the references cted theren. Due to delay exstng n many ndustral systems, t has been a hot topc. In the past decades, all knds of delayed sngular systems were studed, and many nvaluable results were obtaned, see, e.g., [4, 5, 6, 7. It s seen that there are no jumpng parameters. When the structure of sngular systems changes abruptly, a knd of systems named to be SMJSs s very apprecated to descrbe such phenomenon, whch s an mportant branch n stochastc system. Up to now, many mportant results were reported n [8, 9, 1, 11, 12, 13, 14. On the other hand, many nevtable factors n the actual control systems wll result n system havng uncertan parameters. In ths case, we usually desre to acheve a mnmum value of performance ndex. However, f the uncertanty of a system s consdered, the concluson wll be more conservatve, and the robustness of system performance wll be destroyed. Instead, a strategy named as guaranteed cost control [15, 16, 17 s an effectve method to overcome ths defect. However, by nvestng the exstng references, t s seen that the guaranteed cost control s realzed by mode-dependent controllers. Such controllers have an deal assumpton that ther modes need to be avalable onlne. Ths assumpton wll have the applcaton scope lmted. Compared wth mode-dependent controller, mode-ndependent control s very approprate to deal wth such general problem, n whch system mode s not used. Moreover, t s seen that almost the exstng results of MJSs see, e.g., [18, 19, 2, and the references cted there n, have an assumpton that the swtchng probablty rate s gven beforehand. In some practcal cases, one may have more freedom to choose an approprate swtchng probablty rate. For normal state-space 1291

1292 P. ZHANG, J. T. CAO AND G. L. WANG MJSs, [21 frstly studed the stablzaton problem of MJSs, whose matrx of the adopted Lyapunov functon should be postve-defnte. Although the proposed result s necessary and suffcent, t s not appled to SMJSs. That s because the dervatve matrx of SMJSs s sngular, and t makes the matrx of Lyapunov functon only nonsngular nstead of beng postve-defnte. In ths case, t s sad that the method of [21 cannot be used to deal wth SMJSs smlarly. Based on the mentoned facts, t s meanngful to study the guaranteed cost control of SMJSs realzed by mode-ndependent controller the swtchng probablty rate s also desgned. To the best of our knowledge, there are stll no results avalable n the lterature. In ths paper, the mode-ndependent guaranteed cost control of sngular Markovan delay jump systems wth swtchng probablty rate desgn s frstly consdered. Under the swtchng probablty rate and mode-ndependent controller gven beforehand, a suffcent condton guaranteeng the closed-loop system stochastcally admssble n addton to cost functon bounded s proposed. Based on ths, an exstence condton to solve such problem s developed n terms of LMIs wth some equaton constrants. In order to make the condton wthn the desred framework ultmately, some novel technques for dealng wth swtchng probablty rate are exploted. Because of the swtchng probablty rate desgned, t wll be less conservatve. Fnally, numercal example demonstrates the utlty and superorty of the presented methods. Notaton: R n m denotes n m dmenson matrx. In symmetrc block matrces, stands for the correspondng poston about matrx s transpose. dag{ } s used to ndcate block-dagonal matrx, consderng deg( ) as a symbol of the bggest polynomal order number and also det(d) s a sgn of the determnant of phalanx D, and (D) = D + D T. 2. Problem Formulaton. Consderng a class of SMJSs wth tme delay as follows: { Eẋ(t) = A(ηt )x(t) + A d (η t )x(t τ) + B(η t )u(t) (1) x(t) = ϕ(t), t [ τ, x(t) R n s state varables, u(t) R n s the control nput, and tme delay τ satsfes τ. Matrx E s sngular wth rank E = r n, and A(η t ), A d (η t ), B(η t ) R n n are known matrx. The jump sgnal η t = S = {1, 2, 3,..., N} s defned as { πj h + o(h) j Pr{η t+h = j η t = } = (2) 1 + π h + o(h) = j h >, lm h +(o(h)/h) =, and π j, f j, π = N, j π j, and Π = (π j ) N N. In ths paper, we wll dscuss the problem of guaranteed cost control realzed by a mode-ndependent controller u(t) = Kx(t) (3) K s controller gan to be determned, and the swtchng probablty rate wth property (2) s also desgned. Defnton 2.1. For SMJS (1), defne a cost functon ( + ) J = E [x T (t)s x(t) + u T (t)r u(t)dt S, R are gven postve-defnte matrx. J s called an upper bound of the cost functon, f there s a constant J to make the closed-loop system stable and J J. Namely, u(t) s called the guaranteed cost controller. (4)

GUARANTEED COST CONTROL OF SMJSS 1293 Defnton 2.2. Sngular Markovan jump system (1) wth u(t) = s sad to be regular and mpulse free for any constant tme delay τ, f pars (E, A(η t )) and (E, A(η t ) + A d (η t )) are regular and mpulse free for every η t S. 3. Man Results. Proposton 3.1. Consder system (1) wth cost functon (4), f there exst P, Z, Q Q and µ π max, such that the followng condtons hold for all S M 11 M 12 P T E = E T P (5) M 11 M 12 Z I K T M 22 τa T d Z Z S 1 < R 1 (6) = (P T Ā ) + Q < Q (7) π j E T P j + Q + µτq E T ZE = P T A d + E T ZE, M 22 = Q E T ZE, Ā = A + B K System (1) s stochastcally stable, and cost functon (4) satsfes J ϕ T ()E T P ϕ() + + τ β τ x T (α)q x(α)dα [ẋ(α)e T τzeẋ(α) + µx T (α)qx(α) dαdβ Proof: Frstly, we prove system (1) s regular and mpulse-free. From (6), one has [ P T (Ā + A d ) N + π j E T P j + µτq < (9) by premultplyng and postmultplyng (6) wth [ I I and ts transform respectvely. Moreover, there are two nonsngular matrces M and N satsfyng Ir A 11 A Ē = MEN =, M(Ā + A d )N = 12 A 21 A 22, (1) P M T 11 P P N = 12 P 21 P 22 Based on condton (5) by premultplyng and postmultplyng (5) by N T and N, t s known that P 12 =. Let Q N T 11 Q QN = 12 Q 21 Q 22 > (11) whch llustrates that Q 22 >. Based on (1) and (11), (9) becomes Υ 11 Υ 12 Υ 21 [(A 22 ) T P 22 + µτq 22 < (12) (8)

1294 P. ZHANG, J. T. CAO AND G. L. WANG Υ 11 = (A 11 ) T P 11 + (A 21 ) T P 21 + µτq 11 + Υ 12 = (A 21 ) T P 22 + (P 11 ) T A 22 + (P 21 ) T A 22 + π j P 11 j π j P 12 j + µτq 12 Υ 21 = (A 12 ) T P 11 + (A 22 ) T P 21 + (P 22 ) T A 21 + µτq 21 Obvously, we get (A 22 ) T P 22 + (P 22 ) T A 22 < (13) Then, system (1) s regular and mpulse-free. Now, choose a Lyapunov functon V (t, η t ) = V 1 (t, η t ) + V 2 (t, η t ) + V 3 (t, η t ) + V 4 (t, η t ) (14) V 1 (t, η t ) = x T (t)e T P x(t), V 2 (t, η t ) = V 3 (t, η t ) = t τ t+β t t τ x T (α)q x(α)dα ẋ T (α)e T τzeẋ(α)dαdβ, V 4 (t, η t ) = µ Lettng the weak nfntesmal generator L, one has 1 LV (t, η t ) = lm h h E (V (t + h, η t+h) V (t, η t )) = ẋ T (t)e T P x(t) + x T (t)p T Eẋ(t) + x T (t) x T (t τ)q x(t τ) + t t τ t t τ x T (α) t τ t+β x T (α)qx(α)dαdβ π j E T P j x(t) + x T (t)q x(t) π j Q j x(α)dα + τ 2 ẋ T (t)e T ZEẋ(t) ẋ(α)e T ZEẋ(α)dα + x T (t)µτqx(t) µ ξ T (t)ψ ξ(t) < ξ(t) = [ x T (t) x T (t d) T, Ψ = Ψ 11 Ψ 12 [ Ψ 11 Ψ 12 Ψ 22 t = (P T Ā ) + ĀT τ 2 ZĀ + Q + µτq E T ZE + t τ x T (α)qx(α)dα π j E T P j = P T A d + ĀT τ 2 ZA d + E T ZE, Ψ 22 = Q + A T dτ 2 ZA d E T ZE By [23, t s concluded that system s stochastcally stable and stochastcally admssble. Moreover, by computng, t s obtaned that J T = E E T T [ x T (t)s x(t) + u T (t)r u(t) + LV (t, η t ) dt E ξ T (t) Ψ ξ(t)dt + V () V () T LV (t, η t )dt (15) (16)

GUARANTEED COST CONTROL OF SMJSS 1295 Ψ = Ψ + dag{s + K T R K, }. Lettng T, one has J = E + [ ξ T (t) Ψ ξ(t) + V () dt V () (17) whch mples the upper bound of the cost functon s J = V (). By condton (6), t s concluded that Ψ <. Ths completes the proof. Remark 3.1. Proposton 3.1 gves an exstence condton wth guaranteed cost controller (3) gven beforehand, an upper bound of the system performance ndex s found. However, t should be ponted out that the unknown swtchng probablty rate of system s related to some varable, whch makes π j E T P j nonlnear. Moreover, mode-ndependent control gan K and mode-dependent Lyapunov matrx P have strong couplng whch should be decoupled frst. Such nonlneartes cannot be solved drectly, whch should be consdered carefully. Theorem 3.1. Consder system (1) wth cost functon (4), f there exst matrces W, H, ˆQ, ˇQ, Q, Q, P, Z, Q, scalars µ and π j wth j, such that the followng condtons hold for all W Φ 11 Φ 12 Φ 13 Φ 14 Φ 15 Φ 16 (G) Φ 24 Φ 25 Φ 33 τx T A T d Z < (18) Φ 55 Φ 66 [ E T P E T W E P T E R + U T QT V T E R ER T P < (19) j E R W H = I, ˆQ ˇQ = I (2) ˆQ µi Q < (21) µ π j (22),j Q < Q (23) Φ 11 = (A G) + (B Y ) E P E T E P T E T + Z, Φ 12 = A G + B Y + X T G T Φ 13 = A d X + E P E T + E P T E T Z, Φ 14 = τ(a G) T + τ(b Y ) T Φ 15 = [ X T Y T X T X T Φ 24 = τg T A T + τy T B T, Φ 25 = [ Y T, Φ 16 = [ π 1 I... π ( 1) I π (+1) I... π N I Φ 33 = X T X + Q E P E T E P T E T + Z, Φ 55 = dag Φ 66 = dag{h,..., H,..., H }, X = P E T + V Q U { S, R, Q, 1 } τ ˇQ E = E L E T R wth E R R n r and E L R n r. V R n (n r) and U R (n r) n are any gven and satsfy EV = UE =, system (1) wth controller (3) s stochastcally stable, and cost functon (4) satsfes (8). In ths case, the control gan and swtchng probablty rate are gven as K = Y G 1 (24)

1296 P. ZHANG, J. T. CAO AND G. L. WANG π j = π 2 j, π = =1, j π j (25) Proof: Lettng X = P 1, pre- and post-multplyng (5) wth X T and (6) wth dag{x T, X T, I, I, I} and the correspondng transposes respectvely, one has EX = X T E T (26) Ω 11 Ω 12 τx T Ā T X T X T K T Ω 22 τx T A T d Z 1 S 1 R 1 < (27) Ω 11 =(ĀX ) + X T ( N ) π j E T P j X + X T Q X + X T τµqx X T E T ZEX Ω 12 =A d X + X T E T ZEX, Ω 22 = X T Q X X T E T ZEX Moreover, takng nto account (24) and lettng Z = G, Y = KG, t s concluded that Ω 11 Ω 12 Ω 12 τg T Ā T Ω 15 (Z ) τz T Ā T Ω 25 Ω 22 τx T A T d Z 1 < (28) Ω55 Ω 11 =(ĀG) X T E T ZEX + X T ( N ) π j E T P j X + X T Q X + X T τµqx Ω 12 =ĀZ + X T G T, Ω15 = [ X T G T K T, Ω 25 = [ Z T K T, Ω55 = dag{s 1, R 1 } mples (27) by pre- and post-multplyng (28) wth the followng matrx I Ā I τā I I K I (29) and ts transform respectvely. Secondly, due to the nature of the swtchng probablty rate, the followng equalty s always satsfed. That s π j W = (3)

GUARANTEED COST CONTROL OF SMJSS 1297 Based on (3), t s obtaned that (28) can be guaranteed by Φ 11 Φ 12 Φ 13 Φ 14 Φ 15 (G) Φ 24 Φ33 Φ 25 τx T A T d Z 1 Ω55 < (31) X T (E T P j )X X T E T W < (32) Φ 11 = (ĀG) + X T Q X + τx T µqx X T E T ZEX +,j π j W Φ 13 = A d X + X T E T ZEX, Φ15 = [ X T Y T, Φ25 = [ Y T Φ 33 = X T Q X X T E T ZEX Unfortunately, there are stll some problems n (27), such as nonlnear terms, for example, X T τµqx and X T ( N π je T P j ) X. Especally, µ π max must be satsfed, whch s a precondton. As for X T τµqx, by lettng µq < ˆQ, one has (21), Q = Q 1 and µ = µ. On the other hand, for any gven a m R +, R + = {x x > } s a gven set, and m s some postve nteger, t s known that a1 + a 2 + + a m a 1 + a 2 + + a m (33) Based on the establshed nequalty (33), we obtan that condton (22) mples µ π max holds. Let P = ˆP E + U T ˆQ V T (34) wth ˆP >, ˆQ beng nonsngular matrx, and accordng to [24, t s concluded that E T L ˆP E L >, UE =, EV =, E = E L E T R. Moreover, X T ( E T P j ) X = X T X = P 1 = ( ˆP E + U T ˆQ V T ) 1 = P E T + V Q U (35) [E R (E TL ˆP ) E L ER T X = X T [E R (E TR P E R ) 1 E TR X (36) X T Q X X X T Based on (35)-(37) and lettng Z 1 = Z, Q 1 = Q, R 1 (19) mples (32). Ths completes the proof. + Q 1 (37) = R, S 1 = S, we have that Remark 3.2. It s worth mentonng that compared wth some exstng results, Theorem 3.1 has the followng advantages: 1) Dfferent from [21, nonlnear terms such as π j E T P j are handled by technque (3), the swtchng probablty rate can be desgned. Especally, novelty technque (34) s used to further lnearze such related terms. In contrast to [25, 26, t s sad that the swtchng probablty rate plays an mportant role n systems synthess, whose effect s llustrated by numercal examples; 2) The couplng between mode-ndependent control gan K and mode-dependent matrx P s decoupled successfully; 3) The condton s expressed n terms of LMIs wth equaton constrants, whch could be solved by some exstng computng methods such as cone complementary lnearzaton. Based on the proposed method, we have the followng corollary smlarly.

1298 P. ZHANG, J. T. CAO AND G. L. WANG Corollary 3.1. The unforce system (1) s stochastcally stable, f there exst matrces W, H, Q, Q, Q, Z, P, scalars µ and π j wth j, satsfyng (21) and (22), and the followng LMIs hold for all S: Ῡ P T E = E T P (38) P T A d + E T ZE τa T Z Φ 16 Q E T ZE τa T d Z Z Φ 66 < (39) E T P j E T P W < (4) Q < Q (41) W H = I, Q Q = I (42) Ῡ = A T P + P T A + Q + τ ˆQ E T ZE Then, the swtchng probablty rate s computed by (25). 4. Numercal Example. In ths secton, several numercal examples are used to prove superorty and correctness of the provded method. Example 4.1. Consder an SMJS of form (1) s obtaned by [ 1.2.3.3.2.5 1.1 A 1 =, A 2 θ d1 =, A.1 2 = 1 1.5 θ takes dfferent values. Sngular matrx E s 1 E = [.5, A d2 =.22.1 When θ takes dfferent values, the comparsons between Corollary 3.1 and [25, 26 are lsted n Table 1. For ths example, t s seen that our results are less conservatve, f one can choose an approprate swtchng probablty rate. Table 1. The upper bounds of τ wth dfferent θ θ 1.8 1.8 2.5 2.5 [25 1.16 1.98 1.28 1.94 [26 1.95 2.4 2.8 2.36 Corollary 3.1 3.59 3.59 3.59 3.59 Example 4.2. Consder another SMJS, whose parameters are gven as: Mode 1: 1.1 1.2.5.1 4.3 A 1 =, A 2.3 2.1 d1 =, B.2.1 1 =, R 2 1 = 2.3, S 1 =.1 Mode 2: A 2 = [ 1.8 1 2.5 1.5 [.8.2, A d2 =.1 The tme delay s τ =.1, and matrx E s 1 E = [ 3.1, B 2 = 3, R 2 = 1.2, S 2 =.1

GUARANTEED COST CONTROL OF SMJSS 1299 Lettng ntal condton x() = [ 1.6 T, the state response of the open-loop system s shown n Fgure 1. It s seen that the open-loop system s unstable. By Theorem 3.1, one can desgn a mode-ndependent controller n addton to swtchng probablty rate matrx Π. That s K = [.4856.516.1263.1263 Π =.1632.1632 2.5 3 x 14 x 1 (t) x 2 (t) State varables x(t) 2 1.5 1.5 2 4 6 8 1 Tme(s) Fgure 1. The smulaton of open-loop system 1 x 1 (t) x 2 (t).8 State varables x(t).6.4.2.2.4 2 4 6 8 1 Tme(s) Fgure 2. State response of closed-loop system

13 P. ZHANG, J. T. CAO AND G. L. WANG Applyng these to the orgnal system, the smulaton of the closed-loop system s gven n Fgure 2, whch s stable. Example 4.3. Consder the DC motor drvng a load that changes randomly and abruptly, see Fgure 3. The swtchng s drven by a contnuous-tme Markov process {η t, t > } takng values n a fnte set N = {1, 2}. If we neglect the DC motor nductance L m and let (t), ϖ(t) and u(t) denote electrc current, the speed of the shaft at tme t and the voltage, respectvely, based on the basc electrcal and mechanc laws: ϖ(t) = b ϖ(t) + K t (t) J J (43) u(t) = K ϖ ϖ(t) + R(t) Fgure 3. The block dagram of a DC motor 3 25 x 1 (t) x 2 (t) 2 State varables x(t) 15 1 5 5 2 4 6 8 1 Tme(s) Fgure 4. State response of the closed-loop system

GUARANTEED COST CONTROL OF SMJSS 131 K ϖ, K t represent the electromotve force constant and the torque constant, respectvely. R s the electrc resstor, and J and b are defned by: J = J m + J c n 2 b = b m + b (44) c n 2 J m and J c are the moments of the motor and the load. b m and b c are the dampng ratos wth gear rato n. Now we let x 1 (t) = ϖ(t), x 2 (t) = (t), and one has [ 1 b K t [ ẋ(t) = J J x(t) + R 1 k ϖ u(t) (45) When there s tme delay τ =.1s, system (45) becomes (1). Wthout loss of generalty, t s assumed to be [ 1 b K t ẋ(t) = J J x(t) + x(t τ) + u(t) (46) R.5 1 k ϖ Lettng J m =.5kg m, J c1 = 5kg m, J c2 = 15kg m, b c1 = 1, b c2 = 24, R = 1Ω, b m = 1, R 1 =.3, S 1 =.1, R 2 =.4, S 2 =.1, K t = 3Nm/A, K ϖ = 1Vs/rad and n = 1, by Theorem 3.1, one can desgn a mode-ndependent controller n addton to swtchng probablty rate matrx Π. That s K = [ 1.2837 4.3221.193.193 Π =.37.37 Lettng ntal condton x() = [ 3 3 T, we have the state response of the closedloop system, whch s llustrated n Fgure 4. In addton, the curve of system mode s demonstrated n Fgure 5. Based on such smulatons, t shows the desred controller n addton to swtchng probablty rate s effectve. 2.5 2 Mode η t 1.5 1.5 2 4 6 8 1 Tme(s) Fgure 5. The curve of system mode

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