Stabilization and Performance Analysis for a Class of Switched Systems

Transcription

1 43rd IEEE Conference on Decson and Control December 14-17, 2004 Atlants, Paradse Island, Bahamas ThB02.6 Stablzaton and Performance Analyss for a Class of Swtched Systems Le Fang, Ha Ln, and Panos J. Antsakls Abstract Ths paper nvestgates stablty and control desgn problems wth performance analyss for dscrete-tme swtched lnear systems. The swtched Lyapunov functon method s combned wth Fnsler s Lemma to generate varous tests n the enlarged space contanng both the state and ts tme dfference, allowng extra degree of freedom for stablty analyss and control desgn. Two performance measures beng consdered are the decay rate and the nputoutput performance. A new LMI based stablty test for the exstence of swtched Lyapunov functons s frst developed. If a swtched Lyapunov functon exsts, asymptotc stablty of the swtched system also mples ts exponental stablty. An LMI optmzaton problem s then formulated to fnd a bound on the decay rate of the system. To attan the bound, state feedback control gans are desgned. Usng the same framework and the well-known S-procedure, a generalzed suffcent LMI condton s obtaned whch guarantees a γ- performance of the closed-loop swtched systems subject to nput dsturbances. I. INTRODUCTION In recent years, consderable research efforts have been devoted to the study of swtched systems. The motvaton for studyng swtched systems comes from the fact that many practcal systems are nherently multmodal 8, and the fact that some of ntellgent control methods are based on the dea of swtchng between dfferent controllers 9, 11, 18, 20. The exstence of systems that cannot be asymptotcally stablzed by a sngle statc contnuous feedback controller 4 also motvates the study. A survey of basc problems n stablty and desgn of swtched systems s gven n 15. One of the basc problems s to fnd condtons whch guarantee that swtched systems are asymptotcally stable under arbtrary swtchng sequences. Stablty analyss of swtched systems s usually carred out n the Lyapunov framework 3, 9. For swtched lnear systems, stablty under arbtrary swtchng s equvalent to the exstence of a common Lyapunov functon 15. Although progress has been recently made 5, 16, 19, 25, fndng a common Lyapunov functon s stll an open problem. Qute often, a lnear matrx nequalty (LMI) problem formulaton s used to obtan suffcent stablty condtons by constructng a set of quadratc Lyapunov-lke functons 12, 23. More recently, the swtched Lyapunov functon (SLF) method and less conservatve LMI based condtons were developed n 6 for stablty analyss and control desgn for swtched lnear systems. The partal support of the Natonal Scence Foundaton (NSF CCR ) s gratefully acknowledged. L. Fang, H. Ln and P. Antsakls are wth Department of Electrcal Engneerng, Unversty of Notre Dame, Notre Dame, IN E-mal: {lfang, hln1, antsakls.1}@nd.edu /04/$ IEEE 3265 For swtched lnear systems, the exstence of SLF s a weaker condton than the solvablty of Le algebra, whch mples the exstence of a common quadratc Lyapunov functon 1. The Le algebra approach, however, can be generalzed to study swtched nonlnear systems 17 whle the extenson of SLF approach to the nonlnear context s not straghtforward. In the SLF method, we also assume that the SLF strctly decrease along the solutons of the systems for all tme nstances. Ths restrctve assumpton can be relaxed for certan classes of swtched lnear systems. One can deduce asymptotc stablty usng multple Lyapunov functons whose Le dervatve are only negatve semdefnte 10. The SLF method has been appled to solve dfferent problems of swtched lnear systems 7, 14, 28. In partcular, the nput-output performance problem was studed n 7. There are some related works n the lterature on analyzng the nput-output propertes of swtched systems. For example, the L 2 analyss 29 and the l dsturbance attenuaton problem 13 have been studed. Motvated by the work n 6, we combne the SLF method wth Fnsler s Lemma 21, 26 to study stablty and control desgn problems wth performance analyss for dscrete-tme arbtrarly swtchng lnear systems. Two performance measures consdered n ths paper are the decay rate and the nput-output performance. Frst, a new LMIbased necessary and suffcent condton, whch generalzes the one n 6, s obtaned to check the exstence of a SLF. Our new method s conceptually smple. Here, dfference equatons are consdered as constrants and these dynamcal constrants are ncorporated nto the stablty analyss condton through the use of matrx Lagrange multplers. The key dea s to ncrease the dmenson of the LMIs and to ntroduce new matrx varables, allowng extra degree of freedom for stablty analyss and control desgn. We show that f a SLF exsts, the swtched system s not only asymptotcally stable but also exponentally stable. Of partcular nterest s the formulaton of an LMI optmzaton problem to fnd a sharp estmate of the decay rate to the orgn for swtched lnear systems. To attan a bound on the decay rate, swtched state feedback control desgn s nvestgated. By swtched control desgn, we mean the desgn of state feedback control gans for each subsystem such that the closed-loop swtched system s asymptotcally (or exponentally) stable. The closed-loop swtched system s the one correspondng to the closed-loop subsystems under an arbtrary swtchng rule. Fnally, we study the robustness of the feedback control law when the system s subject to nput dsturbances from a γ-performance pont of vew, whch provdes an upper bound on the

2 worst case energy ampltude gan for swtched systems over all possble nputs and swtchng sgnals. Usng the same mathematcal tool and the well-known S-procedure 2, a new LMI-based suffcent condton s obtaned to ensure the asymptotc stablty of the swtched system whle satsfyng a γ-performance condton. Better performance level than the one n 7 s guaranteed due to extra matrx varables ntroduced n the new LMI condton. Most of the ntroductory materal on SLF can be found n 6. The paper s organzed as follows. Secton II gves the problem formulaton. Stablty analyss of swtched systems under arbtrary swtchng s addressed n Secton III usng the SLF method combned wth Fnsler s Lemma. Exponental stablty of swtched systems s studed n Secton IV. A bound on the decay rate to the orgn s found by solvng an LMI optmzaton problem. Swtched state feedback control gans are then desgned to attan the bound. In Secton V, the nput-output performance and synthess problems are nvestgated. Secton VI concludes the paper. Notaton: The notaton s standard. Z s the set of nteger numbers. R m n denotes the set of m n real matrces. S n denotes the set of n n real symmetrc matrces and S n +, the set of n n real symmetrc postve-defnte matrces. λ(m) stands for the egenvalue of a matrx M. M T s the transpose of the matrx M. M>0(M<0) means that M s postve defnte (negatve defnte). II. PROBLEM FORMULATION Consder the class of swtched lnear systems =A σ(t) +B σ(t) u(t), t Z + (1) where R n s the state, u(t) R m s the control nput, and Z + = {0, 1, 2,...}. The swtchng sgnal σ(t) :Z + I = {1,...,N} s a pecewse constant functon of tme t, whch s unknown a pror. (For notatonal smplcty, we may not explctly menton the tme-dependence of the swtchng sgnal below.) Here, (A,B ) ( I) are constant matrces of approprate dmensons denotng the subsystems, and N 2 s the number of subsystems. We are nterested n the followng mportant questons. Q1) Suffcent condtons for asymptotc stablty: Is t possble to specfy condtons such that the autonomous swtched system s asymptotcally stable? Q2) Decay rate: If the answer to Q1) s yes, can a bound on the decay rate to the orgn be found? Q3) Control desgn: If the answer to Q2) s yes, does a control u(t) exst such that the bound s actually attaned? Q4) Robustness of the control law: Is the control robust when subject to nput dsturbances? In the sequel we show that answers to all the above questons are ndeed yes. III. STABILITY ANALYSIS In ths secton, we nvestgate the stablty of the orgn of an autonomous swtched system. The asymptotc stablty of 3266 the system s verfed by means of a set of LMIs formulated n terms of subsystem A-matrces. If feasble, these LMIs provde a set of Lyapunov matrces that can be combned to form a swtched quadratc Lyapunov functon. The autonomous swtched system s gven by =A σ(t). (2) Defne the ndcaton functon ξ(t) =ξ 1 (t),...,ξ N (t) T (3) wth { 1, σ(t) = ξ (t) = 0, otherwse Then, the swtched system (2) can also be wrtten as N = ξ (t)a. (4) =1 To check asymptotc stablty of system (2), a SLF wth a structure smlar to that of the system descrpton was used 6: ( N ) V (t, ) = x T (t) ξ (t)p, (5) =1 where P S n +,( =1,...,N). If such a postve-defnte Lyapunov functon exsts and V (t, ) = V (t +1,) V (t, ) (6) s negatve defnte along the solutons of (2), then the orgn of the swtched system (2) s asymptotcally stable. In the followng, a new LMI-based necessary and suffcent condton s obtaned by combnng SLF method wth Fnsler s Lemma 21, 26. Ths condton s more general than those condtons of Theorem 2 n 6 and no matrx nverson s nvolved n the constructon of SLF. We frst ntroduce Fnsler s Lemma, whch has been prevously used n the control lterature manly wth the purpose of elmnatng desgn varables n matrx nequaltes. In ths context, Fnsler s Lemma s usually referred to as Elmnaton Lemma. Lemma 1 (Fnsler s Lemma): Let x R n, P S n, and H R m n such that rank(h) =r<n. The followng statements are equvalent: 1) x T Px < 0, Hx =0,x 0. 2) X R n m : P + XH + H T X T < 0. In Lemma 1, tem 1) has a constraned quadratc form n R n whle tem 2) provdes an unconstraned quadratc form, where the constrant s taken nto account by ntroducng multpler X. Recall that the requrement V (t, ) < 0, 0 can be stated as P,P j S n + such that T T P 0 0 P j A I =0, < 0, 0,

3 assumng that σ(t) = and σ(t +1)=j. It now becomes clear that Lemma 1 can be appled to the swtched system under study. In the new procedure, the dynamc dfference equatons that characterze the system are seen as constrants, whch are naturally ncorporated nto the stablty condton usng Fnsler s Lemma. In contrast wth standard space methods, where stablty s carred n the space of the state vector, the stablty test s generated n the enlarged space contanng both the state and ts tme dfference. Theorem 1: There exsts a Lyapunov functon of the form (5) whose dfference s negatve defnte, provng asymptotc stablty of (2) f and only f there exst P S n +, and matrces F,G R n n ( =1,...,N), satsfyng A F T + F A T P A G F G T AT F T P j G G T < 0, (7) (, j) I I. The Lyapunov functon s then gven by (5). Proof: Apply Lemma 1 wth P 0 x,p, 0 P j A H T T F,X I G T, P S n +,F,G R n n ( I). The equvalence between asymptotc stablty of (2) and the followng feasblty test s then establshed: P S n +,F,G R n n ( I): A T F T + F A P A T G F G T A F T P j G G T < 0. The result follows by transposng A n (8). The key dea behnd Theorem 1 s to ncrease the dmenson of the LMIs and to ntroduce new matrx varables F and G, here dentfed as Lagrange multplers, allowng some degree of freedom to verfy (5) and (6). Wth specal choces of F and G, we have the followng corollary. Corollary 1: The followng statements are equvalent. ) There exsts a Lyapunov functon of the form (5) whose dfference s negatve defnte, provng asymptotc stablty of (2). ) There exst P S n +,( =1,...,N), satsfyng P A T P j > 0, (, j) I I. (9) P j A P j The Lyapunov functon s then gven by (5). ) There exst P S n + and G R n n ( =1,...,N), satsfyng P A G G T AT P j G G T < 0, (, j) I I. (10) The Lyapunov functon s gven by (5). (8) 3267 TABLE I NUMERICAL COMPLEXITY ASSOCIATED WITH THREE STABILITY TESTS Stablty Tests K (scalar varables) L (LMI rows) Nn(n+1) Cor. 1-) 2 Nn+2N 2 n Nn(n+1) Cor. 1-) + Nn 2 2 Nn+2N 2 n Nn(n+1) Theorem 1 +2Nn 2 2 Nn+2N 2 n Proof: The equvalence of )-) follows from Theorem 1 by makng F = G =0for tem ) and F =0for tem ), respectvely n condton (8). It s not dffcult to see that Corollary 1 s essentally equvalent to Theorem 2 n 6 whle the proof here s more straghtforward. Moreover, wth no matrx nverson nvolved n the Lyapunov functon, Theorem 1 allows us to formulate an LMI optmzaton problem to fnd a sharp estmate of exponental convergence rate of (2) as llustrated n Secton IV, provded that (7) s feasble. Remark 1: The numercal complexty assocated wth the LMI condtons can be computed n terms of the number K of scalar varables and number L of LMI rows. As dscussed n 2, the number of floatng pont operaton or the tme requred to test the feasblty of the set of LMIs s proportonal to K 3 L. Table I shows K and L as a functon of n (states) and N (subsystems) for three tests presented here. (For a practcal purpose, only those nstances wth N 10, n 10 s tractable.) In the case of restrctve swtchng sgnals, we can modfy these condtons or nvoke the S-procedure to mprove the conservatsm of these condtons. Take for nstance, a system whch does not allow arbtrary transtons between subsystems wll have the set of all ordered pars (, j) of subsystem ndces much smaller than I I. IV. ATTAINABLE BOUNDS ON DECAY RATE A. A Bound on Decay Rate From Theorem 1, the asymptotc stablty of a swtched lnear system under arbtrary swtchng can be checked wth the feasblty test (7). Indeed, we can say more: f the swtched system (2) s asymptotcally stable, t s also exponentally stable about the orgn,.e., κ > 0 and 0 ξ<1, such that κ ξ t x(0). (11) for all ntal condtons x(0) and for all t 0. In fact, the Lyapunov functon (5) s postve defnte, decrescent, and radally unbounded snce V (t, 0) = 0, t 0, and η 2 V (t, ) ρ 2 (12) for all R n wth η = mn I λ mn (P ) and ρ = max I λ max (P ) postve scalars. Furthermore, V ν 2 (13)

4 wth ν =mn (,j) I I λ mn (P A T P ja ) <ρ. Wth the above observatons, the exponental stablty of (2) about the orgn mmedately follows from the well-known fact 24: Theorem 2: The sequence s exponentally stable about the orgn f there exsts a Lyapunov functon V (t, ) such that t 0 η 2 V (t, ) ρ 2, V (t +1,) V (t, ) ν 2, (14) η, ρ, ν > 0. Then κ ξ t x(0), where κ 2 = ρ/η and ξ 2 =1 ν/ρ. We obtan the followng corollary. Corollary 2: If the LMIs gven n Theorem 1 or Corollary 1 are satsfed, the autonomous swtched system (2) s exponentally stable about the orgn. In the case of verfyng exponental stablty of (2), t may be desrable not only fnd feasble solutons to (10) or (7) but to search for solutons that gve an estmate of the decay rate ξ n Theorem 2. To ths end, we want to maxmze the rato ν/ρ. Snce V = x T (t) ( A T P ja P ), constrants (12) and (13) can be rewrtten as: ηi < P <ρi, A F T + F A T P + νi A G F G T AT F T P j G G T (, j) I I, < 0, Note that nequaltes (15) represent strct LMIs but the constrants (12) and (13) are non-strct. Recall that mnmzaton under non-strct LMI constrants gves the same result as mnmzaton under strct LMI constrants when both strct and non-strct LMI constrants are feasble 2. Ths s the case for (15). To obtan a well-posed optmzaton problem, we should normalze ρ to 1 snce the rato ν/ρ can be made arbtrarly small by choosng suffcently large ρ wthout volatng constrants (15). Wth 1 ρ, ν ν/ρ, η η/ρ, G G /ρ, P P /ρ, the followng optmzaton problem s proposed: maxmze ν subject to ηi < P <I, AF T + F A T P + νi A G F G T A T F T P j G G T ν, η > 0, (, j) I I, < 0, (15) (16) Remark 2: Constrants ηi < P < I ( I) lmt the condton number of P to 1/η. The advantage of the condton number lmt s that t wll prevent the LMI soluton algorthm from convergng to P that could lead to roundoff problems. The followng theorem answers the Q2): 3268 Theorem 3: Swtched system (2) s exponentally stable wth a decay rate ξ = (1 ν) 1 2 f the problem (16) s feasble. The assured bound on the decay rate s gven by (1 ν opt ) 1 2 wth νopt the optmal value of optmzaton problem (16). Remark 3: It s possble to produce a better estmate of the decay rate of the swtched system. However, a fner parttonng s needed 27. A SLF havng the same swtchng sgnals as the swtched system may not be suffcent. B. Swtched State Feedback An mportant aspect of the new condtons (7) gven n Theorem 1 s that they are LMIs n P, F and G where there s no cross product between the matrx A and the Lyapunov matrx P ( I). Ths fact has an mpact on the synthess problem consdered below. Let us consder the swtched systems =A σ +B σ u(t) (17) where u(t) s the control and the swtchng sgnal s avalable n real-tme. The stablzng state feedback control problem s to fnd u(t) =K σ (18) such that the correspondng closed-loop swtched system =(A σ + B σ K σ ) (19) s stable. The followng theorem gves a suffcent condton to buld a swtched state feedback controller, whch ensures the exponental stablty of the closed-loop swtched system. Moreover, ths controller s optmal n the sense that the bound on the decay rate of the system s attaned. Q3) s thus solved. Theorem 4: If there s a soluton to maxmze ν subject to ηi < P <I, P + νi A G + B R G T AT + R T BT P j G G T < 0, ν, η > 0, (, j) I I, then the state feedback control gven by (18) wth K = R G 1, I (20) exponentally stablzes the system (17). The decay rate of the system s gven by ξ =(1 ν) 1 2. Proof: Condtons of Theorem 4 lead to ηi < P <I, P + νi (A + B K )G G T (A + B K ) T P j G G T < 0, ν, η > 0, (, j) I I, whch are equvalent to condton (16) wrtten for the closed-loop system (19) wth F = 0. The result then

5 follows from Theorem 3. Note that satsfyng (20) mples P j G G T < 0 and matrces G are non-sngular. Hence the followng feedback gan K = R G 1 s always avalable whenever (20) are feasble. Remark 4: The condton gven n Theorem 4 can also be adapted to the swtched statc output feedback control as shown n 6. V. INPUT-OUTPUT PERFORMANCE The next step s naturally to consder the robustness of the swtched state feedback control. Ths s the last queston, Q4) proposed n Secton II. We study the robustness from a γ-performance pont of vew, that s the desgned swtched feedback control ensures that the worst case energy ampltude gan of the closed loop system s less than or equal to some specfed postve level γ. A. γ-performance Consder an autonomous dscrete-tme swtched system gven by { = Aσ +Bσ w w(t) z(t) = Cσ z+dzw σ w(t) (21) where R n s the state, x(0) = 0, w(t) R q s the dsturbance and z(t) R p s an output vector. The swtchng rule σ s defned as prevously. Smlarly, the matrces (A σ,bσ w,cσ,d z σ zw ) are allowed to take values, at an arbtrary tme, n the fnte set {(A 1,B1 w,c1 z,d1 zw ),...,(A N,BN w,cn z,dn zw )}. Gven γ>0, the γ-performance for the swtched system (21) s defned as below. Defnton 1 (7): The autonomous system (21) s sad to have a γ-performance f t s asymptotcally stable and z T (t)z(t) <γ 2 t=0 w(t) L 2,.e., t=0 w T (t)w(t), (22) w T (t)w(t) <. t=0 Now, defne the same Lyapunov functon V (t, ) > 0, x 0consdered n Secton III, and the modfed Lyapunov stablty condtons V (t, ) < 0, γ 2 w T (t)w(t) z T (t)z(t), (,,w(t),z(t)) satsfyng (21), (,,w(t),z(t)) 0, (23) for a gven γ. The S-procedure 2 s nvoked to generate the equvalent condton V (t, ) <γ 2 w T (t)w(t) z T (t)z(t), (,,w(t),z(t)) satsfyng (21), (,,w(t),z(t)) 0. (24) If (24) s feasble for some 0 <γ< then t s possble to conclude that the system (21) s nternally asymptotcally 3269 stable and has a γ-performance snce 0 < V(t +1,) < γ 2 w T (k)w(k) t t z T (k)z(k), whch s vald for all t>0. In partcular, take t, 0 <V( ) <γ 2 w T (k)w(k) z T (k)z(k), (25) whch mples γ 2 wt (k)w(k) > zt (k)z(k). We are now ready to state the followng theorem whch gves a suffcent condton to check f the autonomous system (21) has a γ-performance. Theorem 5: The system (21) has a γ-performance f there exst P S n +, F 1, G 1 R n n, F 2,G 2 R n p, H 1 R p n, J 1 R q n, H 2 R p p, J 2 R q p, = {1,...,N}, such that where P = U = x P + U + U T < 0, (26) P P j I γ 2 I, (, j) (I I), (27) F 1A + F 2C z F 1 F 2 F 1B w + F 2D zw G 1A + G 2C z G 1 G 2 G 1B w + G 2D zw H 1A + H 2C z H 1 H 2 H 1B w + H 2D zw J 1A + J 2C z J 1 J 2 J 1B w + J 2D zw Proof: Assgn H T z(t) w(t) A T,P C T I 0 0 I B T D T,X and apply Lemma 1 on (24). P P j I γ 2 I F 1 G 1 H 1 J 1 F 2 G 2 H 2 J 2,, B. Swtched Feedback Control wth γ-performance In ths secton, we consder swtched lnear systems gven by { = Aσ +B σ u(t)+bσ w w(t) z(t) = Cσ z+dz σ u(t)+dzw σ w(t) (28) A γ-performance state feedback problem s to fnd a control (18) such that the correspondng closed-loop swtched system { = (Aσ + B σ K σ )+Bσ ww(t) z(t) = (Cσ z + DσK z σ )+Dσ zw (29) w(t) has a γ-performance. The followng theorem gves a suffcent condton to buld such a controller..

6 Theorem 6: There exsts a swtched state feedback control (18) such that the closed-loop swtched system (29) swtched systems, Proc. IEEE 41st CDC, Las Vegas, NE, 2002, pp. 7 J. Daafouz and J. Bernussou, Robust dynamc output feedback for has a γ-performance f there exst P S n +, G 1 R n n , F 2,G 2 R m n, H 2 R m m, J 2 R m q 8 W. P. Dayawansa and C. F. Martn, A converse Lyapunov theorem for, R a class of dynamcal systems whch undergo swtchng, IEEE Trans. R m n, = {1,...,N}, such that AC, vol. 44, no. 4, pp , R. A. Decarlo, M. S. Brancky, S. Pettersson, and B. Lennartson, P + U + U T < 0, (30) Perspectves and results on the stablty and stablzablty of hybrd systems, Proc. of the IEEE, vol. 88, no. 7, pp , where P s gven by (27) and 10 J. Hespanha, Extendng LaSalle s Invarance Prncple to Swtched B wf 2 A G 1 + B R + B wg 2 B wh 2 B wj Lnear Systems, Proc. of the 40th CDC, Orlando, FL, 2001, pp G U = J. P. Hespanha and A. S. Morse, Swtchng between stablzng F 2 G 2 H 2 J. 2 D zw F 2 C G 1 + D R + D zw G 2 D zw H 2 D zw J 2 The γ-performance state feedback control law s gven by (18) wth K = R G 1 1. Proof: Use a transposed verson of Theorem 5 wth F 1 =0,H 1 =0,J 1 =0, (31) and R = K G 1. In order to obtan a convex condton (30), we made the choce n (31) but t s not unque. There are other choces that can also lead to a convex condton. Remark 5: By settng F 2 =0, G 2 =0, H 2 = I, and J 2 = 0, we can recover the correspondng condton n 7. Wthout restrctons on F 2, G 2, H 2, and J 2, a better γ-performance level wth respect to the one n 7 can be obtaned. However, ths came at expense of a more ntensve computaton; see also 22. Remark 6: The best upper bound on the L 2 -nduced gan can be acheved by mnmzng γ subject to the constrants defned by (30), whch s a classcal egenvalue problem 2. VI. CONCLUSION In ths paper the SLF method has been combned wth Fnsler s Lemma to study swtched lnear systems. New and less conservatve LMI condtons are developed for stablty and control desgn problems wth performance analyss. Output feedback control problem can also be treated n a smlar way by usng the technque developed n ths paper. ACKNOWLEDGMENT The authors would lke to thank Prof. Leond Faybusovch at Unversty of Notre Dame for helpful dscussons. REFERENCES 1 A. Agrachev and Danel Lberzon, Le Algebra Condtons for Exponental Stablty of Swtched Systems, Proc. of the 38th CDC, Phoenx, AZ, 1999, pp S. Boyd, L. EL Ghaou, E. Feron, and V. Balakrshnan, Lnear Matrx Inequaltes n System and Control Theory, SIAM, M. S. Brancky, Multple Lyapunov functons and other analyss tools for swtched and hybrd systems, IEEE Trans. Automat. Contr., vol. 43, no. 4, pp , R. W. Brockett, Asymptotc stablty and feedback stablzaton, n Dfferental Geometrc Control Theory, R. W. Brockett, R. S. Mllman, and H. J. Sussmann, Eds. Boston, MA: Brkhuser, 1983, pp D. Cheng, L. Guo, and J. 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