BUMPLESS SWITCHING SCHEME DESIGN AND ITS APPLICATION TO HYPERSONIC VEHICLE MODEL. Received October 2010; revised March 2011

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1 Internatonal Journal of Innovatve Computng, Informaton and Control ICIC Internatonal c 212 ISSN Volume 8, Number 1(B), January 212 pp BUMPLESS SWITCHING SCHEME DESIGN AND ITS APPLICATION TO HYPERSONIC VEHICLE MODEL Wen Bao 1, Ywen Q 1, Daren Yu 1, Zhaohu Yao 1 and Jun Zhao 2 1 Hypersonc Technology Research Center Harbn Insttute of Technology No. 92, West Da-Zh Street, Harbn 151, P. R. Chna baowen@ht.edu.cn; qywen@gmal.com 2 College of Informaton Scence and Engneerng Northeastern Unversty No. 11, Lane 3, Wenhua Road, Hepng Dstrct, Shenyang 114, P. R. Chna zhaojun@se.neu.edu.cn Receved October 21; revsed March 211 Abstract. Ths paper nvestgates the problem of swtchng performance optmal control for contnuous-tme controller swtched systems. An extended bumpless swtchng technque s proposed va lnear quadratc (LQ) optmal control and nternal model prncple. Then a suffcent condton for guaranteeng the swtched closed-loop systems to be stable under bumpless swtchng s developed. Dwell-tme and multple Lyapunov functons methodologes are utlzed for the stablty analyss and swtchng law desgn. Smulatons for the generc hypersonc vehcle model show the effectveness of the proposed desgn method. Keywords: Controller swtched systems, Bumpless swtchng, Dwell tme, Hypersonc vehcle model 1. Introducton. In recent years, there has been ncreasng nterest n the stablty analyss and desgn methodology of swtched systems due to ther sgnfcance both n theory and applcatons [1. Many researches have been presented on a wde range of topcs, ncludng the modelng, optmzaton, stablty analyss and control, among whch the stablty ssues have been a major focus [2-7. In contrast, less attenton has been pad to the ssue of swtchng control performance, whch s very mportant n practcal applcatons and s also a very challengng problem. For nstance, n flght control, multple controllers are often used to control the same hghly nonlnear flght vehcle, one for each operatng pont or task. Nevertheless, swtchng among controllers mples control dscontnutes and undesred transents. Moreover, these dscontnutes mght drectly lead to performance degradaton, actuator saturaton and even nstablty of the closed-loop system. The soluton of ths problem s called bumpless swtchng (transfer). Generally, bumpless swtchng arses n several cases of practcal nterests. The frst case s swtchng between manual and automatc mode [8. Another case s swtchng among several lnear controllers by gan schedulng [9, 1. The thrd case s the mprovement of system performance va swtchng between controllers wth dfferent propertes [11. Fnally, swtchng can result from the need for onlne testng and evaluaton of dfferent control law desgns. Bumpless swtchng s often performed n the steady state to meet safety requrements [12. Approaches that focus on bumpless swtchng are manly as follows: ant-wndup bumpless transfer (AWBT) scheme [13, optmal lnear quadratc control methods [12, 14, L 2 norm methods [8, 15, nterpolaton [9, 11, and observerbased technque [16, etc. In these mentoned works, the AWBT method [13 represents 677

2 678 W. BAO, Y. QI, D. YU, Z. YAO AND J. ZHAO an AWBT operator forcng the output of the off-lne controller to track that of the onlne one. Nevertheless, the nput of off-lne controller s condtoned by the operator and thus s not necessarly close to that of the onlne one at the swtchng nstant [17. The L 2 method [8, 15 s to desgn a compensator ensurng an L 2 bound on the msmatch between the actual plant output and that of the deal closed-loop behavor. Ths method, unfortunately, s only applcable to lnear plant wth a suffcently accurate model [1. Moreover, as the authors themselves pont out n [11, the nterpolaton algorthm reles only on stablty consderatons wth the controller swtched suffcently slowly. In fact, the LQ technque [14 stands out as a convenent tool for the steady-state bumpless swtchng synthess. It has been appled to many physcal systems such as vertcal/short take-off land arcraft [1, coal-fred boler/turbne [18 and magnetc bearngs [19. The central feature of LQ approach s the compensator F (n [14), whch s a statc matrx and can drve the output of the off-lne controller to track that of the onlne controller. However, because the matrx F s obtaned by adjustng two weghtng matrces and they have only lmted ampltude of adjustment, the method does not produce an adequate soluton. Nevertheless, the conceptual clarty and the computatonal smplcty of ths method gve a strong motvaton to extend t. In ths paper, we present a new extended scheme for the bumpless swtchng of a controller swtched system. Frstly, we wll desgn a set of compensatory controllers (CCs) for the off-lne controllers respectvely. Wth the effect of CC, at the nstant of swtchng, the output of onlne controller and that of the off-lne controller are as close as possble so as to reduce the magntude of the dscontnuty (or error). We refer to [14, the LQ framework and the nput weghtng of off-lne controller s used for desgn. However, the proposed method s manly dfferent from that n [14. In our scheme, these CCs, desgned based on nternal model prncple, can make the off-lne controllers track the output of onlne controller wth nearly zero steady state error such that the contnuty of plant nput becomes much strcter at swtchng nstants. Furthermore, an ntegral acton s ntroduced to elmnate the (trackng) error. By consderng the ntegral of error as an extra set of state, the error mnmzaton problem s transformed nto an augmented system stablty problem. It ndcates that once the augmented system s stable, the trackng error wll asymptotcally converge to zero, or be reduced to an acceptable value as close to zero as possble n fnte tme. And based on ths fact, a CC can be obtaned. In addton, based on the multple Lyapunov functons combned wth the dwell tme technque, a swtchng law desgn and a condton renderng swtched closed-loop systems exponentally stable are gven. Note that ths condton s developed n the swtched systems framework, whch only a few artcles have addressed [2. The paper s organzed as follows. Secton 2 gves the problem formulaton and some prelmnares. In Secton 3, we present the CC desgn. The swtchng law desgn and stablty analyss are gven n Secton 4. An algorthm s then presented n Secton 5. Smulatons are shown n Secton 6. Conclusons are gven n Secton Problem Statement and Prelmnares. Consder the controller swtched system ẋ(t) = Ax(t) + Bu(t), u(t) = u σ(t) (t), y(t) = Cx(t), x(t ) = x(), where σ(t) : [, + ) I N = {1, 2,..., N} s the swtchng sgnal to be desgned, x(t) R n s the state, u(t) R m s the control nput, u (t) R m are the sub-controllers outputs, y(t) R l s the vector of plant measurements. A, B and C are real constant matrces of approprate dmensons. Correspondng to the swtchng sgnal σ, we have the swtchng (1)

3 BUMPLESS SWITCHING SCHEME DESIGN 679 sequence Λ = {x ; (, t ), ( 1, t 1 ),..., ( k, t k ),..., k I N }. When t [t k, t k+1 ), σ(t) = k, that s, the k -th subsystem s actvated. Assumpton 2.1. The par (A, B) s stablzable. Remark 2.1. It should be notced that ths assumpton s necessary for stablzablty of a lnear system. If a system s stablzable, then ts all unstable egenvalues can be made stable by means of a feedback gan. A set of controllers can be desgned n advance based on the method n [21 as follows: { ẋc (t) = A c x c (t) + B c1 C : x(t) + B c r 2 c (t), (2) u (t) = C c x c (t) + D c x(t), where x c (t) R n c s the -th controller state. For the onlne controller, r c = r(t) R l s the reference command sgnal. For the off-lne one, r c (t) = r (t) s the CC s output to be desgned (see Fgure 1). An off-lne controller s closed by ts CC to drve the control sgnal close to the onlne one. Snce the purpose of ths artcle s to extend the results of lterature [14, the scope of ths paper s to dscuss the steady-state swtchng. CC Compensator + + w q 1 Sub-trackng s Controller - + r r q C q C p x cq u q u p u x Plant y Fgure 1. The proposed bumpless swtchng scheme wth I N = {p, q} Assumpton 2.2. Throughout the paper, we consder the stuaton that the system (1) s under steady-state swtchng. Remark 2.2. The closed-loop system under the ntal mplemented controller performs a lnear behavor before controller swtchng. Thus the steady-state swtchng here means to start controller swtchng after the closed-loop system under the ntal mplemented controller has reached ts steady state. Here t s assumed that steady state s reached once the closed-loop system outputs reman wthn 99% and 11% of the steady state values. Integral acton s used n classcal control to elmnate steady state errors when trackng sgnals. It can be ntroduced nto the LQ framework by consderng the ntegral of the trackng error as an extra set of state varables. For any off-lne controller C, I N, defne the error state as ẇ (t) = u(t) u (t) = u(t) C c x c (t) D c x(t). (3)

4 68 W. BAO, Y. QI, D. YU, Z. YAO AND J. ZHAO Defnton 2.1. Gven a postve scalar ε >, a swtchng (onlne) controller C p ( p I N ) s sad to perform a bumpless swtchng f, whenever controller s swtched, there exsts a fnte tme T ε > such that the output of a controller C q ( q p I N ) to be swtched satsfes the condton lm t Tε u(t) u q (t) ε, where u(t) = u p (t). In partcular, C p s sad to perform a strctly bumpless swtchng f lm t u(t) u q (t) =. Combnng (2) and (3), we have the augmented system for the th off-lne controller: [ [ [ [ [ [ ẋc (t) Ac xc (t) Bc1 Bc2 = + x(t) + r (t) + u(t). (4) ẇ (t) C c w (t) I D c For smplcty, the augmented system (4) s rewrtten as x (t) = Ā x (t) + B 1 x(t) + B 2 r (t) + F u(t), (5) where x = [ x T c w T T s the augmented system state vector. The cost performance ndex for the desgn of the th bumpless swtchng CC s defned as J = 1 2 T ( x T Q x + ϑ T R ϑ ) dt, (6) where Q, R >, and ϑ = r r s the dfference between CC output and swtched systems reference sgnal. Q for state varables s standard n optmal control, whle the R s used for scalng ϑ. The sze of ϑ wll serously nfluence the error sze (3). Remark 2.3. In a vewpont of an optmal control, the relatve magntudes of Q and R may be selected to trade off requrements on the smallness of the state x aganst that of ϑ. For nstance, a larger R wll make t necessary for ϑ to be smaller. On the other hand, to make x go to zero more quckly wth tme, we may select a larger Q. Assumpton 2.3. [ Ā B2 s stablzable, Q = Π T Π and [ Ā Π s observable. Remark 2.4. It s well known n the optmal control lterature (for specfc detal see [22) that n order for the algebrac Rccat Equaton (17) (n next Secton), to have a postve defnte stablzng soluton, (Ā, B 2, Π) s requred to be stablzable and observable. In addton, t wll be shown that the stablty of the off-lne closed-loop system composed of (5) and (2) s ensured as (17) has a constant, postve defnte soluton. In ths paper, we are nterested n the followng two problems. Problem 1 (Bumpless Swtchng CC Desgn Problem) Gven a set of controllers (2), fnd a set of CCs respectvely, each of whch ncludes a compensator and a sub-trackng controller such that augmented system (5) s stablzed and cost functonal J s mnmzed. Problem 2 (Closed-loop System Stablty Analyss Problem) Gven a set of controllers (2) and ther CCs, construct a swtchng law σ and derve a suffcent condton such that the closed-loop system composed of (1), (2) and CC s asymptotcally stable and smultaneously guarantees the bumpless swtchng by swtchng controllers. 3. Bumpless Swtchng CC Desgn. Now based on the above dscusson, we gve a soluton to Problem 1. Suppose there exsts a vector functon λ(t), such that the condtonal extremum problem of (6) subject to constrants (5) can be transformed nto a non-condtonal extremum problem wth the performance ndex J = 1 2 T ( H(t) λt x ) dt, (7)

5 BUMPLESS SWITCHING SCHEME DESIGN 681 where the Hamltonan functon s gven by H(t) = 1 T [ x 2 Q x + ( r r) T R ( r r) + λ ( T Ā x + B 1 x + B 2 r + F u ). A necessary condton for mnmzng the cost ndex (7) s H/ λ = x, H/ x = λ, H/ r =. Furthermore, H/ x = Q x + ĀT λ = λ, (8) H/ r = R ( r r) + B T 2 λ =. (9) From (9), t follows that r = r R 1 B T 2 λ. (1) Substtutng (1) nto (5) yelds x = Ā x + B 1 x + B ( 2 r R 1 B T 2 λ ) + F u. (11) Defnng λ(t) = P (t) x (t) g(t), (12) then we can get λ(t) = P (t) x (t) + P (t) x (t) ġ(t). (13) Moreover, by (12), Equaton (8) can be wrtten as λ(t) = Q x (t) ĀT (P (t) x (t) g(t)) = ( Q + ĀT P (t) ) x (t) + ĀT g(t). (14) In addton, applyng (11) and (12) to (13) results n λ(t) = P (t) x (t) + P (t) [ Ā x (t) + B 1 x(t) + B 2 r B 2 R 1 B T 2 (P (t) x (t) g(t)) + F u ġ(t). Then, from (14) and (15), t follows: [ P (t) + P (t)ā + ĀT P (t) P (t) B 2 R 1 B T 2 P (t) + Q x (t) = ġ(t) + ( Ā T P (t) B 2 R 1 B T 2 ) g(t) P (t) B 1 x(t) P (t) B 2 r P (t)f u(t). Note that the left sde of (16) s a product of a functon of tme and state varables x (t), whle the rght sde s only a functon of tme. It means that for arbtrary t and x (t), the followng two equatons: P (t) + P (t)ā + ĀT P (t) P (t) B 2 R 1 B T 2 P (t) + Q =, ġ(t) = (ĀT P (t) B 2 R 1 B T 2 )g(t) P (t) B 1 x(t) P (t) B 2 r P (t)f u(t). must be satsfed. These two equatons should be extended to nfnte horzon (.e., T ) for practcal engneerng consderaton [22, whch are P Ā + ĀT P P B2 R 1 B T 2 P + Q =, (17) (ĀT ) P B2 R 1 B T 2 g P B1 x P B2 r P F u =. (18) It follows from (17) and (18) that g = ( ) Ā T P B2 R 1 B T 1 ( P B1 2 x + P B2 r + P F u ), (19) n whch the symmetrc postve defnte matrx P s the soluton of algebrac Rccat Equaton (17). Thus usng (1), (12) and (19), we have the CC of off-lne controller C, I N for bumpless swtchng as follows: (15) (16) r = K r r + K c x c + K w w + K x x + K u u, (2) where K r = I + R 1 B T 2 Σ 1, K x = R 1 B T 2 P, K x = R 1 B T 2 Σ 2, K u = R 1 B T 2 Σ 3, Σ 1 = Γ B 2, Σ 2 = Γ B 1, Σ 3 = ΓF, K x = [K c K w and Γ = ( ) Ā T P B2 R 1 B T 1 P 2. K c and K w are the sub-trackng controller gans, K r, K x and K u are the compensator gans.

6 682 W. BAO, Y. QI, D. YU, Z. YAO AND J. ZHAO Remark 3.1. If Assumpton 2.3 holds, then (17) has a unque soluton, and the augmented closed-loop system composed of (5) and (2) s stable [22 whch guarantees the bumpless swtchng of onlne controller. Ths wll be dscussed n the next secton. Remark 3.2. The dervaton of compensator F n [14 s carred out through the mnmzaton of a functonal that ncludes two dfferences. One s the dfference between the nput sgnals of the controllers (both onlne and off-lne), the other s the dfference between the output sgnals. The functonal s gven by J = 1 T (z 2 u(t) T W u z u (t)+z e (t) T W e z e (t))dt, where z u (t) = u(t) u (t), z e (t) = r r, and W u, W e are weghtng matrces. Especally, z u (t) represents the error between the off-lne controller output and the onlne one, thus the smaller z u (t) s, the better the bumpless performance wll be. Then, the key pont of [14 s to choose approprate weghtng matrces W u and W e such that the J can be mnmzed under the effect of F. However, n Secton 6, ths technque wll be found to provde an ncomplete convergence of the z u (t) and produce a pronounced non-vanshng error of z u (t). In other words, the weghtng matrces W u and W e have lmted regulaton range or ablty, they can not be regulated arbtrarly to further reduce the ndex J value (.e., J s bounded). That s the drawback or lmtatons of the LQ bumpless swtchng desgn. For ths reason, the focus of our work s on mnmzng the error to the maxmum extent, whch s dfferent from [14, and ths s also the major contrbuton of our work. From (3) and (4), the proposed approach reduces the realzaton of error mnmzng to the solvng of the augmented system stablty problem based on the nternal model prncple. By ntroducng an ntegral acton, the method can make the off-lne controller acheve zero-error trackng or even completely elmnate the error n theory (f t ). 4. Swtchng Law Desgn and Stablty Analyss. The objectve of ths secton s to derve a global exponental stablty condton for the swtched closed-loop systems under bumpless swtchng and to desgn a dwell tme dependent swtchng law. Wthout loss of generalty, we assume that two controllers C p and C q ( q p I N ) are used for bdrectonal swtchng. Substtutng (2) nto (2), we have the descrpton of an off-lne controller C, I 2 = {p, q} wth compensatory effect. That s ẋ c (t) = A x c (t) + B v(t), u (t) = C x c (t) + D v(t), where A = A c + B c2 K c, B = [ Ξ 1 Ξ 2 Ξ 3 Ξ 4, C = C c, D = [ D c, Ξ 1 = B c1 +B c2 K x, Ξ 2 = B c2 K r, Ξ 3 = B c2 K w, Ξ 4 = B c2 K u and v = [ x T r T w T u T T. Then, from (5) and (21), the augmented system for the off-lne controller C s gven by x off (t) = à x off (t) + B x x(t) + B u u(t) + B r r(t), (22) where x off = [ [ [ [ [ x T c w T T A Ξ, à = 3, C B Ξ1 x =, D B Ξ4 u =, c I B Ξ2 r =. It follows from Remark 3.1 that the matrx à s Hurwtz stable. On the other hand, from (1) and (2), we obtan the augmented system for a controller C j, j I 2 when t s onlne. That s x onj (t) = Âj x onj (t) + ˆB rj r(t), (23) where x onj = [ [ [ x T x T T A + BDcj BC c j, Âj = cj, ˆBrj =. B cj1 A cj B cj2 (21)

7 BUMPLESS SWITCHING SCHEME DESIGN 683 Remark 4.1. Matrx Âj s supposed to be Hurwtz,.e., n the absence of controller swtchng, the ndvdual closed-loop systems under (1) and (2) would be globally stable. Consequently, when C p (j = p) s onlne and C q ( = q) s off-lne, the followng ndvdual closed-loop system s obtaned by augmentng (22) wth (23): ẋ cl (t) = A clp x cl (t) + B clp r(t), where x cl = [ [ [ x T on p x T T Âp ˆBrp off q, matrces Aclp =, B clp = and Ξ 5 = Ξ 5 Ã q B [ rq Bxq + B uq D cp Buq C cp. In the same way, when Cq (j = q) s onlne and C p ( = p) s off-lne, we can have the other closed-loop system ẋ cl (t) = A clq x cl (t) + B clq r(t). Thus, the swtched closed-loop system when r(t) = can be gven by ẋ cl (t) = A clσ(t) x cl (t), σ(t) I 2. (24) Defnton 4.1. System (24) s sad to be globally exponentally stable under a swtchng sgnal σ(t), f all solutons x cl (t) of (24) startng from any ntal condton x cl (t ) satsfy x cl (t) κ x cl (t ) e η(t t ), t t for some constants κ 1 and η >. The swtchng law desgn n terms of dwell tme prncples. We mpose restrctons on the set of admssble swtchng sgnals by defnng the set S T = {σ(t) : t k+1 t k T }, where t k are the commutaton nstants and T >. In other words, for the postve constant T, S T denotes the set of all swtchng sgnals wth nterval between consecutve dscontnutes no smaller than T. The constant T s called the dwell tme [1. Then fnd the mnmum T for whch (24) s exponentally stable for all possble σ(t) S T. Theorem 4.1. (Bumpless Swtchng) Consder the system (24). Gven controllers C, I 2, ther assocated closed-loop system matrces A cl and a scalar ϵ >, f there exsts a scalar β > such that the followng nequaltes dv (x cl (t)) dt A cl x cl < βv (x cl (t)), I 2 (25) have postve defnte matrces P >, then for every σ S τd wth the dwell tme of C j, j I 2 satsfyng τ d = max{t s, T ϵ }, the swtched closed-loop system (24) s globally exponentally stable and the controller C j performs a bumpless swtchng. Proof: () For any swtchng sgnal σ S T, let = t, t 1, t 2,... be the correspondng swtchng tme seres. For any t >, there exsts an nteger such that t [t, t +1 ). Let N σ (t, t) denote the number of swtchngs of σ over the nternal [t, t. Also suppose that durng [t, t +1 ), mode s actve, where I 2. For the gven postve defnte matrces P, I 2, let V (x cl ) = x T cl P x cl be a Lyapunov canddate correspondng to mode. It s obvous that there exst postve constants a > and b > such that a x cl 2 V (x cl ) b x cl 2, (26) where a = nf I 2 {λ mn (P )}, b = sup I 2 {λ max (P )}. Then t follows from (25) that V (x cl ) < βv (x cl ), whch yelds V (x cl (t)) e β(t t ) V (x cl (t )). Therefore, from (26), we obtan x cl (t) 2 χe β(t t ) x cl (t ) 2, (27) where χ = b a. By ths fact, a smlar nequalty can be derved as x cl(t ) 2 χe β(t t 1 ) x cl (t 1 ) 2. Iteratng the nequalty results n x cl (t ) 2 χ Nσ(,t) e β(t t 1 ) e β(t 1 t 2) e β(t 1 t ) x cl (t ) 2 χ Nσ(,t) e Nσ(,t)βT x cl (t ) 2. (28)

8 684 W. BAO, Y. QI, D. YU, Z. YAO AND J. ZHAO In vew of the nequaltes (27) and (28), we deduce the followng nequalty x cl (t) χ (N σ(,t)+1) e (Nσ(,t)+1)βT 2 x cl (t ). Hence, when χ (N σ(,t)+1) e (Nσ(,t)+1)βT 2 < 1, that s, the mnmum dwell tme satsfes τ d T s > T = lnχ β, (29) where T s = nf{t : T > T }, the system (24) s globally exponentally stable. () In addton, for steady-state swtchng, due to the fact that the sgnals x(t), u(t) and r(t) have reached ther steady states, then takng the dervatve on both sdes of Equaton (22) wth respect to tme, we have x off (t) = à x off (t). (3) Snce matrx à s stable, the state of (3) must satsfy x off (t) = [ ẋ c (t) T ẇ (t) T T, t for I 2. Then by (3) and Defnton 2.1, the onlne controller C j, j I 2 performs a strctly bumpless swtchng. However, we also see that the swtched system acheves strctly bumpless swtchng only when tme approaches nfnty. In order to facltate mplementaton n practce, choose an approprate scalar ϵ > that can meets the needs of C j for performng bumpless swtchng n a fnte tme nterval T ϵ. Thus, we have ẇ (t) ϵ τd, t τ d, where < ϵ τd ϵ. Ths completes the proof. 5. Algorthm of Bumpless Swtchng. Accordng to Theorem 4.1, the detaled algorthm s gven as follows. Step 1: Gven a scalar β >. Fnd two postve defnte matrces P, I 2 satsfyng A T cl P + P A cl + βp <, (31) then calculate λ mn (P ) and λ max (P ), to obtan a χ. Step 2: Calculate T accordng to (29). Step 3: Repeat Steps 1 to 2 untl a satsfactory small value of T s obtaned and then gve an estmate T s = nf{t : T > T }. Step 4: Based on lnear system theory, the state response of system (3) s x off (t) = eãt x off, where x off s the ntal state of (3) for the off-lne controller C. Accordngly, the error state s ẇ (t) = [ ẇ 1 (t) ẇ m (t) T, where ẇ h (t) = C h [eãt x off and C h, h = 1, 2,..., m are approprate dmensonal matrces whose elements are ether one or zero. Furthermore, choose a scalar ϵ > and solve ẇ h (T ϵ ) ε = to obtan m mnmum tme wth T ϵ1,, T ϵm that guarantees bumpless swtchng for the m components of onlne controller C j, j I 2 respectvely. Thus, defne T ϵ = sup{t ϵ1,..., T ϵm }. Step 5: Then τ d = max{t s, T ϵ } s a feasble mnmum dwell tme for both the swtched closed-loop system (24) stablty and the bumpless swtchng of C j. Remark 5.1. () In the algorthm, one can obtan T s by the stablty of A cl, I 2. Moreover, suppose that x off s avalable, then for a gven ε >, T ε can always be solved. () The key pont of the algorthm s the selecton of β n Step 1. Apparently, t dose not seem to be possble to know beforehand whch selecton features the best rate of convergence. One can practcally gve an ntal value of β > wth a small step length and decde to stop the process f there s no sgnfcant varaton of T. In addton, to select a scalar ϵ > experentally n Step 4, for nstance, one can gve < ϵ 1 2. (32)

9 BUMPLESS SWITCHING SCHEME DESIGN Smulatons. In ths secton, the nonlnear longtudnal model of the generc hypersonc flght vehcle [23 s consdered. The lnearzed equatons at the trm state and trm nput (V = 3m/s, γ = rad, h = 3m, α =.56rad, q = rad/s, δ T =.71, δ e =.1588rad) are obtaned as (1), where x(t) = [ V T γ T h T α T q T T, u(t) = [ δt T δe T T [, y(t) = V T h T T, V, γ, h, α and q are the velocty, flght path angle, alttude, angle of attack and ptch rate, respectvely. Fuel equvalence rato (δ T ) and ptch control surface deflecton (δ e ). And A = , B = [ T, C = [ 1 1 Remark 6.1. Note that (1) s a general lnear system. It can be smulated to llustrate the usefulness and effectveness of the obtaned results n general, and be verfed numercally. Here we consder the stuaton that the physcal system ncludng two controllers (one for the nomnal trackng control, one for the hot standby control). The control objectve s to make the output y(t) track the command r(t) even wth degraded control effectveness such as actuator fault. Suppose that the loss of control effectveness or falure of the actuator takes place at a certan tme. Then the prmary controller fals to work properly, meanwhle, a swtchng acton arses and forces the standby controller to take over Example: comparson between the non-bumpless swtchng and the proposed bumpless swtchng. For system (1), controllers (33) and (34) are obtaned by usng the algorthm n [21. We select two sets of weghtng matrces as Q 1 = dag(1, 12, 1, 8), R 1 = dag(2, 3) and Q 2 = dag(.1,.1, 1, 8), R 2 = 1 6 dag(2, 1), respectvely. Then by solvng (17) and usng Equaton (2), CCs (35) and (36) are obtaned, respectvely. Consequently, we can easly have the ndvdual closedloop system matrces of (24). The swtchng strategy σ(t) s: controller (33) s used frst and then authorty s swtched to controller (34) after the frst dwell tme τ d1 (suppose that the actuator fault happens at τ d1 s). Furthermore, to verfy the bdrectonal swtchng performance, the authorty s changed agan, and (34) s swtched to (33) after τ d2. Accordng to the algorthm n Secton 5, for β =.176, solvng (31) can gve two postve defnte matrces P 1 and P 2. Then calculatng (29) obtans T s > T = lnχ = Let β T T s = 123. Furthermore, choosng ϵ =.1, a smple calculaton shows that the dwell tme for bumpless swtchng n the two stages are T ϵ 1 = and T ϵ 2 = 1.64, by lettng the ntal state of (32) n the two swtchng moments be [ T and [ T respectvely. Consequently, the mnmum dwell tme for the above two goals s gven as τ d1 = and τ d2 = 123. The smulaton results by drect mplementaton of swtch are depcted n Fgure 2(a). The results ndcate that the system experences an undesrable transent n control nput, whch serously deterorates the trackng performance after swtchng. Fgure 2(b) shows the results by usng the proposed bumpless swtchng scheme. By forcng the dfference between off-lne controller and plant nput to a very small sze, a smooth transton s allowed and the trackng performance s assured wth mnmzed undesrable transents..

10 686 W. BAO, Y. QI, D. YU, Z. YAO AND J. ZHAO x x 14 Velocty (m/sec) Alttude (m) Velocty (m/sec) Alttude (m) tme (sec) tme (sec) tme (sec) tme (sec) Fuel equvalence rato tme (sec) (a) Ptch surface deflecton (rad) tme (sec) Fuel equvalence rato tme (sec) (b) Ptch surface deflecton (rad) tme (sec) Fgure 2. Response to the 5(m) alttude step order and 15(m/s) velocty step order (upper fgure); control nputs (lower fgure): (a) drectly swtch and (b) the proposed bumpless swtchng scheme 6.2. Example: comparson between the scheme of [14 and the proposed bumpless swtchng. For an accurate comparson between the method of [14 and the proposed method, we defne a bumpless performance ndex that s a measurement of the bump sze of plant control nput. ϱ = u(t max ) u(t s ) /u(t s ), where u(t max ) s the frst peak value of control nput at the swtchng moment, and u(t s ) s the steady-state value of control nput. The smaller the ndex s, the better the bumpless performance wll be. Wthout loss of generalty, we make the performance analyss only at the frst swtch. By method [14, we select two sets of weghtng matrces as W u1 = 1 5 dag(1, 1), W e1 = 1 3 dag(1, 1) and W u2 = 1 5 dag(1, 1), W e2 = 1 3 dag(2, 1), respectvely. Then, the bumpless swtchng compensatory gans (37) and (38) correspondng to (33) and (34) are obtaned, respectvely. The gan corresponds to the vector Θ = [ x T c x T u T T. [ [ [ B c11 =, B 1 c12 =, C 1 c1 =, (33) [ D c1 = ; [ [ [ B c21 =, B 1 c22 =, C 1 c2 =, (34) [ D c2 = [ [ K r1 = , K.22 c1 =, (35) [ [ [ K w1 =, K x1 =, K 1 u1 = ; [ [ K r2 = , K c2 =, (36).9614 [ [ [ K w2 =, K 2 2 x2 =, K 1 u2 =. 6 4

11 BUMPLESS SWITCHING SCHEME DESIGN 687 [ F 1 =, [ (37) F 2 = (38) The smulaton results by method [14 are shown n Fgure 3. Note that the trackng performance s not bad, whch, unfortunately, s at the cost of serous deteroraton of the plant control nput. Furthermore, we can obtan the detaled characterzaton. As shown n Tables 1 and 2, the bumpless performance ndces of fuel equvalence rato and ptch surface deflecton got by [14 are 44.24% and 242.2%, respectvely. However, the ndces obtaned by our method are much smaller, whch are 3.17% and 2.93%, respectvely. Thus, t clearly demonstrates that the proposed method acheves a better bumpless performance. Velocty (m/sec) tme (sec) Alttude (m) 3.7 x tme (sec) Fuel equvalence rato tme (sec) Ptch surface deflecton (rad) tme (sec) Fgure 3. Response to the 5(m) alttude step order and 15(m/s) velocty step order wth the bumpless swtchng scheme of [14 (upper fgure); control nputs wth the bumpless swtchng scheme of [14 (lower fgure) Table 1. Comparson of δ T Method u(t max ) u(t s ) ϱ Non-bumpless swtchng % Turner and Walker [ % The proposed method % 7. Conclusons. In ths paper, we have dealt wth the problem of bumpless swtchng performance optmal control for contnuous-tme swtched lnear systems. Regardng the bumpy phenomenon, frstly, a new extended bumpless swtchng scheme has been proposed

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