Internatonal Journal of Automaton and Computng 14(5), October 2017, 615-625 DOI: 101007/s11633-015-0920-z Stablty Analyss and Ant-wndup Desgn of Swtched Systems wth Actuator Saturaton Xn-Quan Zhang 1 Xao-Yn L 2 Jun Zhao 3 1 School of Informaton and Control Engneerng, Laonng Shhua Unversty, Fushun 113001, Chna 2 School of Foregn Languages, Laonng Shhua Unversty, Fushun 113001, Chna 3 State Key Laboratory of Synthetcal Automaton for Process Industres, Northeastern Unversty, Shenyang 110819, Chna Abstract: The stablty analyss and ant-wndup desgn problem s nvestgated for two lnear swtched systems wth saturatng actuators by usng the sngle Lyapunov functon approach Our purpose s to desgn a swtchng law and the ant-wndup compensaton gans such that the maxmzng estmaton of the doman of attracton s obtaned for the closed-loop system n the presence of saturaton Frstly, some suffcent condtons of asymptotc stablty are obtaned under gven ant-wndup compensaton gans based on the sngle Lyapunov functon method Then, the ant-wndup compensaton gans as desgn varables are presented by solvng a convex optmzaton problem wth lnear matrx nequalty (LMI) constrants Two numercal examples are gven to show the effectveness of the proposed method Keywords: Ant-wndup, swtched systems, saturatng actuators, sngle Lyapunov functon, doman of attracton, lnear matrx nequalty (LMI) 1 Introducton A swtched system as an mportant hybrd dynamcal system s composed of a famly of contnuous-tme or dscretetme subsystems and a swtchng law that governs whch subsystem s actvated along the system trajectory durng a certan nterval of tme Many practcal systems such as computer controlled systems, power systems and network control systems can be modeled as swtched systems As a result, the analyss and synthess of swtched systems have attracted consderable attenton n the past few years 1 7 As ponted out by 1, stablty s of great mportance n the analyss and control for swtched systems Because stablty under arbtrary swtchngs s a preferable property whch allows us to pursue for other performances wth stablty mantaned It s well known that the stablty under arbtrary swtchngs can be guaranteed by a common Lyapunov functon 8 10, however, to fnd such a common Lyapunov functon s often dffcult, yet swtched system stll may be stable under certan swtchng laws The multple Lyapunov functons method 11, 12, the sngle Lyapunov functon method 13, 14 and the average dwell-tme technque 15 are effectve tools for choosng such swtchng laws On the other hand, almost every physcal actuator s subject to saturaton for practcal control systems It s generally known that actuator saturaton can lead to performance deteroraton of the system and even make the Research Artcle Manuscrpt receved August 7, 2014; accepted November 18, 2014; publshed onlne October 27, 2016 Ths work was supported by Scentfc Research Fund of Educaton Department of Laonng Provnce (No L2014159) Recommended by Assocate Edtor Qng-Long Han c Insttute of Automaton, Sprnger-Verlag GmbH Germany 2016 otherwse stable system unstable Thus, more and more attenton has been focused on the analyss and synthess for systems subject to actuator saturaton and many methods have been developed to deal wth actuator saturaton 16 21 In general, there are two major methods to deal wth actuator saturaton The frst method s to take actuator saturaton nto account at the outset of the control desgn, and then desgn a lnear controller whch drves the system to stablty 19 The second method s to neglect actuator saturaton and desgn a controller whch meets the performance specfcatons n the frst stage of the control desgn process, and then add an ant-wndup compensator that weakens the nfluence of saturaton 21 However, the antwndup approach s a popular method and effcent technque to deal wth actuator saturaton from the practcal perspectve The study of swtched systems subject to actuator saturaton becomes more dffcult due to the phenomena of nteractng swtchng and actuator saturaton nonlneartes Thus, the stablty results about such swtched systems are few 22 25 For a class of saturated swtched lnear systems, Zhang et al 22 addressed the robust stablzaton problem based on the multple Lyapunov functons method Va the multple quadratc Lyapunov functons method, a swtchng ant-wndup desgn s proposed for a lnear system wth actuator saturaton n 23 For a class of dscrete-tme saturated swtched systems, Benzaoua et al 24 nvestgated the stablzaton problem va the swtched Lyapunov functon method In 25, by usng multple Lyapunov functons method, the desgn of swtchng scheme s consdered for a
616 Internatonal Journal of Automaton and Computng 14(5), October 2017 class of swtched lnear systems n the presence of actuator saturaton To the best of the authors knowledge, for the problem of the stablty analyss and ant-wndup desgn wth resort to the sngle Lyapunov functon method nearly no results have been reported for dscrete-tme saturated swtched systems n the exstng lterature, whch motvates the present study In ths paper, we study the stablty analyss and antwndup desgn problem for two swtched lnear systems wth actuator saturaton based on the sngle Lyapunov functon approach We frst obtan some suffcent condtons of asymptotc stablty when ant-wndup compensaton gans are gven Then, the ant-wndup compensators are desgned whch am to enlarge the doman of attracton of the consdered system Fnally, all the results are formulated and solved as a convex optmzaton problem wth lnear matrx nequalty (LMI) constrants In ths paper, compared wth the exstng results on swtched systems wth actuator saturaton, the results have two features Frst of all, the stablty analyss and antwndup desgn problem s addressed for the swtched systems wth saturatng actuator, whle most exstng works consdered only the stablzaton problem; second, we use the sngle Lyapunov functons method for desgnng a swtchng law, whle the exstng works mostly amed at arbtrary swtchngs for dscrete-tme swtched systems subject to actuator saturaton The paper s organzed as follows The system descrpton and relevant prelmnares are gven n Secton 2 Secton 3 provdes the stablty condtons for the consdered systems The desgn problem of ant-wndup compensators s proposed n Secton 4 Two examples llustrate the effectveness of the proposed method n Secton 5 Conclusons are n Secton 6 Notatons Throughout ths paper, the followng notatons are used I denotes the dentty matrx wth compatble dmenson A T s the transpose of the matrx A denotes the symmetry elements n symmetrc matrces, that s Q 11 Q 12 Q 11 Q 12 = Q 22 Q T 12 Q 22 2 Problem descrpton and prelmnares 21 The contnuous tme system We consder the followng class of swtched lnear systems wth actuator saturaton ẋ = A σx + B σsat(u) (1) y = C σx where x R n s the state vector, u R m s the control nput vector and y R p s the measured output vector The functon sat : R m R m s the standard, vector-valued, saturaton functon: sat(u) = sat(u 1 ) sat(u 2 ) sat(u m ) T sat(u j )=sgn(u j )mn 1, u j } j Q m = 1,,m} It s well known that t s wthout loss of generalty to assume unty saturaton level 16 Functon σ : 0, ) I N = 1,,N} s a pecewse constant swtchng sgnal; σ = means that the -th subsystem s actve A,B and C are constant matrces wth approprate dmensons For system (1), we wll consder that a set of n c-order dynamc output feedback controllers are of the form ẋ c = A cx c + B cu c v c = C cx c + D cu c (2) I N where x c R nc, u c = y and v c = u are the vector of state, nput and output of the controller respectvely Due to our focus on analyss and desgn of ant-wndup compensaton gans, as commonly adopted n the lterature (see, for example 20 ), we assume that the dynamc compensators have been desgned that stablze the system (1) and (2) wthout nput saturaton and satsfy performance requrements In order to allevate the undesrable effects of the wndup caused by saturatng actuators, a typcal ant-wndup compensator nvolves addng to the controller dynamcs a correcton term of the form E c(sat(v c) v c) Then, the fnal controller structure has the form ẋ c = A cx c + B cu c + E c(sat(v c) v c) v c = C cx c + D cu c (3) I N Obvously, by usng such the correcton terms, the dynamc controllers (3) contnue to operate n the lnear doman n the absence of saturaton, whch does not affect the consdered systems performance, and the controller state of the system under nput saturaton are able to be corrected through the ant-wndup compensators whch recover the nomnal performance of the system as much as possble Now, under the above dynamc controllers and antwndup strategy, the closed-loop system can be wrtten as ẋ = A x + B sat(v c) y = C x ẋ c = A cx c + B cc x + E c(sat(v c) v c) (4) v c = C cx c + D cc x I N Then, we defne a new state vector as x = R n+nc (5) and the matrces x c
X Q Zhang et al / Stablty Analyss and Ant-wndup Desgn of Swtched Systems wth Actuator Saturaton 617 à = G = A + B D cc B cc 0 I nc B C c A c,k =, B = D cc C c Therefore, accordng to (4) and (5), the closed-loop system can be rewrtten as = à ( B + GE c)(v c), I N (6) where v c = K, (v c)=v c sat(v c) 22 The dscrete tme system We consder the followng class of dscrete-tme swtched systems subject to actuator saturaton x(k +1)=A σx(k)+b σsat(u(k)) (7) y(k) =C σx(k) where k Z +, x(k) R n s the state vector, u(k) R m s the control nput vector and y(k) R p s the measured output vector The functon sat : R m R m s the standard, vector-valued, saturaton functon sat(u) = sat(u 1 ) sat(u 2 ) sat(u m ) T sat(u j )=sgn(u j )mn 1, u j } j Q m = 1,,m} The functon σ(k) s a swtchng law that takes ts values n the fnte set I N = 1,, N}; σ(k) = means that the -th subsystem s actve A, B and C are constant matrces of approprate dmensons For system (7), we wll assume that a set of n c-order dynamc compensators are of the form x c(k +1)=A cx c(k)+b cu c(k) v c(k) =C cx c(k)+d cu c(k) (8) I N where x c(k) R nc, u c(k) =y(k) andv c(k) =u(k) arethe state, nput and output of the controller respectvely We assume that the dynamc compensators have been desgned whch can stablze the systems (7) and (8) wthout actuator saturaton Smlarly, n order to weaken the effects of the wndup caused by actuator saturaton, a typcal ant-wndup compensator nvolves addng to the controller dynamcs a correcton term of the form E c(sat(v c(k)) v c(k)) Then, the revsed compensators have the form x c(k +1)=A cx c(k)+b cu c(k)+ E c(sat(v c(k)) v c(k)) (9) v c(k) =C cx c(k)+d cu c(k), I N Clearly, the compensators (9) would contnue to run n the lnear doman wthout saturaton by usng such modfed terms whch does not affect the systems nomnal performance, and the ant-wndup compensators can also amend B 0 the controller state n order to recover the nomnal performance of the system wth nput saturaton as much as possble Then, n combnaton wth the above dynamc controllers and ant-wndup strategy, the closed-loop system can be wrtten as x(k +1)=A x(k)+b sat(v c(k)) y(k) =C x(k) x c(k +1)=A cx c(k)+b cc x(k)+ (10) E c(sat(v c(k)) v c(k)) v c(k) =C cx c(k)+d cc x(k) I N Now, defne a new state vector (k) = x(k) x c(k) R n+nc (11) and the matrces à = G = A + B D cc B cc 0 I nc B C c A c,k =, B = D cc C c Thus, from (10) and (11), the closed-loop system can be rewrtten as (k +1)=Ã(k) ( B + GE c)(v c) I N (12) where v c = K (k), (v c)=v c sat(v c) The objectve of ths paper s to desgn the ant-wndup compensaton gans and the swtchng law such that the system (6) or (12) s locally asymptotcally stable n the orgn of the state space and meanwhle the estmaton of doman of attracton of the closed-loop system (6) or (12) s maxmzed The followng lemmas wll be needed n the development of the man results For a postve defnte matrx P R (n+nc) (n+nc) and a scalar ρ>0, an ellpsod Ω(P, ρ) s defned as } Ω(P, ρ) = R n+nc : T P ρ Consder matrces K, H R m (n+nc) and defne the followng polyhedral set: L(K,H )= R n+nc : B 0 (K j H j ) 1, IN, j Q m } where K j, Hj are the j -th row of matrces K and H respectvely 20, 21 Lemma 1 Consder the functon (v c) defned above If L(K,H ), then the relaton T (K )J (K ) H 0 (13) I N
618 Internatonal Journal of Automaton and Computng 14(5), October 2017 holds for any dagonal and postve defnte matrx J R m m Lemma 2 (Schur s complements) Gven the symmetrc matrx A = A 11 A 12 A T 12 A 22, the followng statements are equvalent: 1) A<0 2) A 11 < 0, A 22 A T 12A 1 11 A 12 < 0 3) A 22 < 0, A 11 A 12A 1 22 AT 12 < 0 3 Stablty condton In ths secton, under the gven ant-wndup compensaton gans, suffcent condtons for stablty of the closedloop system (6) or (12) are derved by way of the sngle Lyapunov functon approach 31 The contnuous tme system Theorem 1 Suppose that there exst symmetrc postve defnte matrces P R (n+nc) (n+nc),matrcesh R m (n+nc), E c R nc m, dagonal postve defnte matrces J R m m and a set of scalars ξ 0 such that =1 and ξ Ã T P + P Ã P ( B + GE C)+ H T J 2J Ω(P, 1) L(K,H ) I N < 0 (14) (15) Then the closed-loop swtched system (6) s asymptotcally stable at the orgn wth Ω(P, 1) contaned n the doman of attracton under the swtchng law T Ã T P + σ = arg mn P Ã P ( B + GE C)+ H T J 2J (16) Proof In vew of condton (15), f Ω(P, 1), then L(K,H ) Thus, n vew of Lemma 1, for Ω(P, 1) t follows that (K ) =K sat(k ) satsfes the sector condton (13) Then, Choose the Lyapunov functonal canddate for the system (6) as V () = T P (17) Then, the tme dervatve of V () alongthetrajectoryof the system (6) s V () = T P + T P = Ã ( B + GE c)(v c) T P+ T P Ã ( B + GE c)(v c) Thus, usng Lemma 1 and condton (15) gves V () Ã ( B + GE c)(k ) T P+ T P Ã ( B + GE c)(k ) 2 T (K )J (K ) H = T Ã T P + P ( B + GE C)+ P Ã H T J 2J T Multplyng (14) from the left by and then from the rght by,wehave T ξ =1 Ã T P + P Ã P ( B + GE C)+ H T J 2J Thus, usng the swtchng law (16) results n T Ã T P + V () P Ã < 0 < 0 P ( B + GE C)+ H T J 2J (18) Addtonally, accordng to (17), adjacent Lyapunov functons at swtchng ponts are equal, namely at the swtchng nstant V () =V j() whch meets wth the nonncreasng requrement on any Lyapunov functon over the swtched on tme sequence of the correspondng subsystem Therefore, the swtched system (6) s asymptotcally stable for all ntal states 0 Ω(P, 1) 32 The dscrete tme system Theorem 2 Suppose there exst symmetrc postve defnte matrx P R (n+nc) (n+nc), matrces H R m (n+nc),e c R nc m, a set of scalars ξ > 0 and dag-
X Q Zhang et al / Stablty Analyss and Ant-wndup Desgn of Swtched Systems wth Actuator Saturaton 619 onal postve defnte matrces J R m m satsfyng and =1 ξ P H T J Ã T P 2J ( B + GE c) T P < 0 P I N Ω(P, 1) L(K,H ) I N (19) (20) Then the closed-loop swtched system (12) s asymptotcally stable at the orgn wth Ω(P, 1) contaned n the doman of attracton under the swtchng law or equvalently T ΔV ((k)) Ã T P Ã P ÃT P ( B + GE c)+ H T J ( B + GE c) T P ( B + GE c) 2J Then, n vew of Lemma 2, (19) s equvalent to ξ =1 Ã T P Ã P (24) ÃT P ( B + GE c)+ H T J ( B + GE c) T < 0 (25) P ( B + GE c) 2J σ(k) = arg mn T Ã T P Ã P ÃT P ( B + GE c)+h T J ( B + GE c) T P ( B + GE c) 2J (21) Multplyng (25) from the left by T T andfromthe rght by T T T,wehave T ξ =1 Ã T P Ã P ÃT P ( B + GE c)+ H T J ( B + GE c) T < 0 P ( B + GE c) 2J Therefore, from the swtchng law (21) and (26), t s easy to have (26) Proof By condton (20), f Ω(P, 1), then L(K,H ) Therefore, by Lemma 1, for Ω(P, 1) t follows that (K (k)) = K (k) sat(k (k)) satsfes the condton (13) We choose the followng Lyapunov functon canddate for the system (12) V ((k)) = T (k)p(k) (22) The dfference of the Lyapunov functon canddate (22) along the trajectores of the consdered system (12) s gven by ΔV ((k)) = T (k +1)P(k +1) T (k)p(k) = Ã(k) ( B + GE c)(k (k)) T P Ã(k) ( B + GE c)(k (k)) T (k)p(k) (23) Therefore, for (k) Ω(P, 1), byusnglemma1and condton (20), we have ΔV ((k)) Ã(k) ( B + GE c)(k (k)) T P Ã(k) ( B + GE c)(k (k)) T (k)p(k) 2 T (K )J (K ) H ΔV ((k)) < 0 (27) Thus, the state trajectory of the consdered system (12) startng from nsde Ω(P, 1) wll reman nsde t Furthermore, the closed-loop system (12) s asymptotcally stable at the orgn wth Ω(P, 1) contaned n the doman of attracton under the swtchng law (21) 4 Ant-wndup desgn In order to guarantee that the consdered closed-loop system (6) or (12) s asymptotcally stable and meanwhle the estmated doman of attracton of the system (6) or (12) s maxmzed under the swtchng law, we wll gve the methods of desgnng the ant-wndup compensaton gans 41 The contnuous tme system Theorem 3 If there exst symmetrc postve defnte matrces X R (n+nc) (n+nc),matrcesm R m (n+nc), N R nc m, dagonal postve defnte matrces S R m m and a set of scalars ξ > 0 such that the followng condtons hold: ξ XÃT + ÃX B S GN + M T < 0 =1 2S (28) I N
620 Internatonal Journal of Automaton and Computng 14(5), October 2017 and XK jt M jt X 0 1 (29) I N,j Q m where K j,mj are the j -th row of matrces K and M respectvely, then the closed-loop saturated swtched system (6) wth ant-wndup compensaton gans E c = N S 1 s asymptotcally stable at the orgn wth Ω(X 1, 1) contaned n the doman of attracton under the state dependent swtchng law T Ã T X 1 + X 1 ( B + σ = arg mn X 1 Ã GE C)+H T J 2J (30) Proof Pre- and post-multplyng both sdes of nequalty (14) by the matrx P 1 0 J 1 we have ξ P 1 Ã T + ÃP 1 ( B + GE C)J 1 P 1 H T =1 2J 1 + < 0 and lettng X = P 1, S = J 1, M = H X, N = E cs, then we have ξ XÃT + ÃX B S GN + M T < 0 2S =1 whch s exactly (28) n Theorem 3 Applyng a smlar method to the nequalty (29), we can also obtan denotes the j-th row of K and H, respec- where K j,hj tvely 1 K j H j P 0 (31) Then, we can show that Ω(P, 1) L (K,H ) s mpled by (31) In fact, snce T P 1and(K j H j )P 1 (K j H j )T 1, t holds that 2 T (K j H j )T T P+ (K j H j )P 1 (K j H j )T 2 therefore, (31) mples Ω(P, 1) L (H,K ) Snce P = X 1, the swtchng rule (16) s the same as (30) of the theorem 1 By usng the sngle Lyapunov functons method, the suffcent condton s gven n Theorem 3 that allows to desgn the compensaton gans E c such that the closed-loop system s asymptotcally stable at the orgn wth Ω(X 1, 1) However, our objectve s to desgn the ant-wndup compensaton gans whch maxmze the estmaton of doman of attracton of the closed-loop system (6) whch means that the set Ω(X 1, 1) s maxmzed In general, there are two man methods to measure the largeness of a set The frst strategy s to measure the largeness of a set by ts volume The second strategy s to take ts shape nto consderaton In ths paper, by adoptng the latter method, the largeness of the set s measured wth respect to a gven shape reference set X R Let X R R n+nc be a prescrbed bounded convex set contanng the orgn For a set Ξ R n+nc whch contans the orgn, defne 17 : α R (Ξ) = sup α >0: αx R Ξ} Obvously, f α R(Ξ) 1, then X R Ξ Thus, α R(Ξ) provdes a knd of measure of the estmated doman of attracton Two typcal types of X R are the ellpsod X R = and the polyhedron } R n+nc : T R 1, R>0 X R = cov 1, 2,, l } where 1, 2,, l are a pror gven ponts n R n+nc As a result, the problem of maxmzng Ω(X 1, 1) wth respect to a gven shape reference set X R can be formulated as the followng constraned optmzaton problem: sup α, X, M,N,S,ξ st (a) αx R Ω(X 1, 1) (b) nequalty (28), I N (c) nequalty (29), I N,j Q m (32) If X R s an ellpsod, then, (a) s equvalent to 1 α R I 2 0 (33) I X If X R s a polyhedron of the form, then, (a) s equvalent to 1 T α 2 q 0, q 1, l (34) X q Let γ = 1 α 2 IfX R s an ellpsod, the optmzaton prob-
X Q Zhang et al / Stablty Analyss and Ant-wndup Desgn of Swtched Systems wth Actuator Saturaton 621 lem(34)canberewrttenas nf γ X, M,N,S,ξ γr I st (a) 0 I X (b) nequalty (28), I N (c) nequalty (29), I N,j Q m (35) If X R s a polyhedron of the form, we just need to replace (a) n (35) wth γ q T 0, q 1, l (36) X q 42 The dscrete tme system Theorem 4 Suppose that there exst symmetrc postve defnte matrx X R (n+nc) (n+nc),matrcesm R m (n+nc),n R nc m, dagonal postve defnte matrces S R m m and a set of scalars ξ > 0 such that the followng LMIs hold: and ξ =1 X M T XÃT 2S S BT N T G T X I N X XK jt M jt 0 1 I N,j Q m, < 0 (37) (38) where K j,mj are the j -th row of matrces K and M respectvely Then the closed-loop swtched system (12) wth ant-wndup compensaton gans E c = N S 1 s asymptotcally stable at the orgn wth Ω(X 1, 1) contaned n the doman of attracton under the swtchng law σ(k) =argmn T Ã T X 1 Ã X 1 ÃT X 1 ( B + GN S 1 )+ X 1 M T S 1 ( B + GN S 1 ) T X 1 ( B + GN S 1 ) 2S 1 (39) Proof Pre- and post-multplyng both sdes of nequalty (19) by the matrx P 1 0 0 J 1 0 P 1 and lettng X = P 1,S = J 1, M = H X, N = E cs, we have ξ =1 X M T XÃT 2S S BT N T G T X < 0 whch s exactly (37) n Theorem 4 In vew of 18, t s easy to see that the condton (19) s guaranteed by P (K j H j )T (K j H j ) 0 (40) Then from the Lemma 2, (29) s equvalent to P (K j H j )T 0 (41) 1 Applyng a smlar method to the nequalty (41), t can be transformed nto (38) equvalently Addtonally, P = X 1, the swtchng rule (39) s the same as (21) of Theorem 2 Then, the problem of the maxmal Ω(X 1, 1) wth respect to a gven shape reference set X R can be presented as the followng constraned optmzaton problem: sup α, X, M,N,S,ξ st (a) αx R Ω(X 1, 1), I N (b) nequalty (37), I N (c) nequalty (38), I N,j Q m (42) If we choose X R as an ellpsod, then (a) s equvalent to 1 α R I 2 0, I N (43) I X If we chooose X R as a polyhedron of the form, then, (a) s equvalent to 1 T α 2 q 0, q 1, l, I N (44) X q Let γ = 1 α 2 IfX R s an ellpsod, the optmzaton problem(42)canberewrttenas nf γ, X, M,N,S,ξ γr I st (a) 0 I X (b) nequalty (37), I N (c) nequalty (38), I N,j Q m (45) If X R s a polyhedron of the form, we only need to replace (a) n (25) wth γ q T 0, q 1, l, I N (46) X q Remark 1 If = 0, namely wthout actuator saturaton, then the swtchng law (30) or (39) wll be reduced to the general swtchng law of swtched system n the absence of actuator saturaton Therefore, these results can be
622 Internatonal Journal of Automaton and Computng 14(5), October 2017 vewed as an extenson from the normal swtched systems to the swtched systems subject to actuator saturaton Remark 2 If the parameters ξ are gven n advance, the ant-wndup compensaton gans and the estmaton of doman of attracton are formulated and solved as a set of lnear matrx nequalty (LMI) optmzaton problem 5 Illustratve examples In the secton, we gve the followng examples to show the valdty of the results n Secton 4 Example 1 ẋ = A x + B sat(v c) (47) y = C x and the dynamc controllers wth the ant-wndup terms are gven as x c = A cx c + B cc x+ E c(sat(v c) v c) v c = C cx c + D cc x (48) where σ(k) I 2 = 1, 2} A 1 = 02 0 13 03,A 2 = 14 09 0 06 B 1 = 05 05,B 2 = 07 08 T T 03 03 C 1 =,C 2 = 06 08 A c1 = 09 03 06 03,A c2 = 06 07 03 09 B c1 = 06 06,B c2 = 08 15 T T 08 06 C c1 =,C c2 = 07 09 D c1 =15,D c2 =68 Then, the ant-wndup compensaton gans E c wll be desgned to stablze the swtched system (47) (48) wth actuator saturaton wth the maxmzed estmaton of doman of attracton n terms of the proposed method 1 0 0 0 0 1 0 0 Let R = We Solve the optmzaton 0 0 1 0 0 0 0 1 problem (35), then the optmal solutons s obtaned as follows: γ =0103 2,S 1 = 175482 5, S 2 =53269 1 11200 0 2900 0 4200 0 3700 0 95100 0 18800 0 9300 0 X = 135700 0 235000 672300 0 0091 0 0002 2 0002 6 0000 6 0010 9 0001 5 0000 2 P = 0007 7 0000 3 0001 5 119302 8 78476 9 N 1 =,N 2 = 352734 2 35592 6 M 1 = 23437 3 145902 2 63928 1 124664 3 M 2 = 17338 3 12619 3 48973 9 164549 3 1389 5 2982 2 E c1 =,E c2 = 2394 3 4278 9 Fg 1 gves the state response of the consdered system (47) (48) under the swtchng law and ant-wndup compensaton desgned gans wth the ntal state x(0) = 05 07 T The response of controller state wth x c(0) = 05 05 T s shown n Fg 2 Fg 3 provdes the control nput sgnal of the swtched system (47) (48) Fg 1 The state response of system (47) (48) On the other hand, t s worth notcng that f we let E c1 = E c2 = 0, the obtaned optmal soluton s γ = 12193 2, whch ndcates that the ant-wndup compensaton gans can enlarge the estmaton of doman of attracton of the consdered system Example 2 x(k +1)=A x(k)+b sat(v c(k)) (49) y(k) =C x(k) and the dynamc controllers wth the ant-wndup terms are gven as x c(k +1)=A cx c(k)+b cc x(k)+ E c(sat(v c(k)) v c(k)) (50) v c(k) =C cx c(k)+d cc x(k)
X Q Zhang et al / Stablty Analyss and Ant-wndup Desgn of Swtched Systems wth Actuator Saturaton 623 ant-wndup compensaton gans E c are desgned to stablze the system (49) and (50) subject to actuator saturaton wth the maxmal estmaton of doman of attracton under the desgned swtchng law 1 0 0 0 0 1 0 0 Let R = By solvng the optmzaton 0 0 1 0 0 0 0 1 problem (45), we obtan the optmal solutons as follows: α =19257 0,S 1 = 176502 3, S 2 =25298 4, Fg 2 The response of controller state of system (47) (48) X =10 3 0021 8 0039 2 0004 6 0192 0 0075 3 0005 5 0331 0 0039 3 0195 5 2335 4 Fg 3 The nput sgnal of system (47) (48) where σ(k) I 2 = 1, 2}, 3 2 2 2 A 1 =,A 2 = 11 4 02 06 08 01 B 1 =,B 2 = 153 05 T T 01 064 C 1 =,C 2 = 026 01 01 03 03 01 A c1 =,A c2 = 04 43 01 04 04 05 B c1 =,B c2 = 03 2 T 07 04 C c1 =,C c2 = 05 02 D c1 =24,D c2 =235 T 8208 9 1095 3 3050 8 0774 9 0289 8 0268 4 0071 4 P = 1311 3 0322 5 0081 0 3627 6 16020 9 N 1 =,N 2 = 42097 2 197315 9 M 1 = 90022 1 199539 7 101969 6 273032 0 M 2 = 1420 2 4695 2 15535 2 54867 0 0020 6 0633 3 E c1 =,E c2 = 0238 5 7799 5 The state response of the consdered system (49) (50) under the swtchng law and ant-wndup compensaton desgned gans wth the ntal state x(0) = 2 1 T s shown n Fg 4 Fg 5 gves the response of controller state wth x c(0) = 1 1 T The nput sgnal s depcted n Fg 6 It s easy to check that the closed-loop system (49) and (50) wthout actuator saturaton s asymptotcally stable at the orgn from the condtons (37) and (38) Now, the Fg 4 The state response of system (49) (50)
624 Internatonal Journal of Automaton and Computng 14(5), October 2017 In the same way, let E c1 = E c2 = 0, the obtaned optmal soluton s only α = 0089 7 Ths shows that the ant-wndup compensaton gans are able to enlarge the estmaton of doman of attracton of the closed-loop system obvously gans to mprove the performance of nonlnear swtched systems s a challengng ssue whch deserves further study n the future References 1 D Lberzon, A S Morse Basc problems n stablty and desgn of swtched systems IEEE Control Systems Magazne, vol 19, no 5, pp 59 70, 1999 2 W N, D Cheng Control of swtched lnear systems wth nput saturaton, Internatonal Journal of Systems Scence, vol 41, no 9, pp 1057 1065, 2010 3 Y C hen, S Fe, K Zhang, L Yu Control of swtched lnear systems wth actuator saturaton and ts applcatons Mathematcal and Computer Modellng, vol 56, no 1 2, pp 14 26, 2012 Fg 5 The response of controller state of system (49) (50) 4 H Ln, P J Antsakls Stablty and stablzablty of swtched lnear systems: A survey of recent results IEEE Transactons on Automatc Control, vol 54, no 2, pp 308 322, 2009 5 Z D Sun, S S Ge Analyss and synthess of swtched lnear control systems Automatca, vol 41, no 2, pp 181 195, 2005 6 G Zha, B Hu, K Yasuda, A N Mchel Dsturbance attenuaton propertes of tme-controlled swtched systems Journal of the Frankln Insttute, vol 338, no 7, pp 765 779, 2001 7 X M Sun, G P Lu, D Rees, W Wang Stablty of systems wth controller falure and tme-varyng delay IEEE Transactons on Automatc Control, vol 53, no 10, pp 2391 2396, 2008 Fg 6 The nput sgnal of system (49) (50) 6 Conclusons The stablty analyss and ant-wndup desgn problem has been studed for two swtched lnear systems subject to actuator saturaton n ths paper We ntroduce the sngle Lyapunov functon method and a sector condton to the desgn of the ant-wndup compensaton gans whch am at maxmzng the estmaton of doman of attracton of the closed-loop systems Then, the problem of desgnng the ant-wndup compensaton gans and the swtchng law s formulated and solved as a convex optmzaton problem wth a set of LMI constrants It should be noted that due to the complexty of dealng wth smultaneous swtchng and actuator saturaton nonlnearty, the proposed method s only sutable for lnear swtched systems wth saturatng actuators Thus, how to desgn a swtchng law and the ant-wndup compensaton 8 D Z Cheng Stablzaton of planar swtched systems Systems & Control Letters, vol 51, no 2, pp 79 88, 2004 9 D Lberzon, A S Morse Basc problems n stablty and desgn of swtched systems IEEE Control Systems Magazne, vol 19, no 5, pp 59 70, 1999 10 J Zhao, G M Dmrovsk Quadratc stablty of a class of swtched nonlnear systems IEEE Transactons on Automatc Control, vol 49, no 4, pp 574 578, 2004 11 M S Brancky Multple Lyapunov functons and other analyss tools for swtched and hybrd systems IEEE Transactons on Automatc Control, vol 43, no 4, pp 475 482, 1998 12 S Pettersson Synthess of swtched lnear systems In Proceedngs of the 42nd IEEE Conference on Decson and Control, IEEE, Mau, Hawa, USA, pp 5283 5288, 2003 13 D Z Cheng, L Guo, J Huang On quadratc Lyapunov functons IEEE Transactons on Automatc Control, vol 48, no 5, pp 885 890, 2003
X Q Zhang et al / Stablty Analyss and Ant-wndup Desgn of Swtched Systems wth Actuator Saturaton 625 14 G Zha Quadratc stablzablty of dscrete-tme swtched systems va state and output feedback In Proceedngs of the 40th IEEE Conference on Decson and Control, IEEE, Orlando, USA, pp 2165 2166, 2001 15 L Wang, C Shao Exponental stablsaton for tmevaryng delay system wth actuator faults: an average dwell tme method Internatonal Journal of Systems Scence, vol 41, no 4, pp 435 445, 2010 16 H Fang, Z Ln, T Hu Analyss of lnear systems n the presence of actuator saturaton and L 2-dsturbances Automatca, vol 40, no 7, pp 1229 1238, 2004 17 Q Zheng, F Wu Output feedback control of saturated dscrete-tme lnear systems usng parameter-dependent Lyapunov functons Systems & Control Letters, vol 57, no 11, pp 896 903, 2008 18 Y Cao, Z Ln, D G Ward An antwndup approach to enlargng doman of attracton for lnear systems subject to actuator saturaton IEEE Transactons on Automatc Control, vol 47, no 1, pp 140 145, 2002 19 T Hu, Z Ln, B M Chen Analyss and desgn for dscretetme lnear systems subject to actuator saturaton Systems & Control Letters, vol 45, no 2, pp 97 112, 2002 20 J M Gomes da Slva, S Tarbourech Ant-wndup desgn wth guaranteed regons of stablty for dscrete-tme lnear systems Systems & Control Letters, vol 55, no 3, pp 184 192, 2006 21 J M Gomes da Slva, S Tarbourech Antwndup desgn wth guaranteed regons of stablty: an LMI-based approach IEEE Transactons on Automatc Control, vol 50, no 1, pp 106 111, 2005 22 X Zhang, J Zhao, G M Dmrovsk Robust state feedback stablzaton of uncertan swtched lnear systems subject to actuator saturaton In Proceedngs of the 2010 Amercan Control Conference, IEEE, Marrott Waterfront Baltmore, USA, pp 3289 3274, 2010 23 L Lu, Z Ln, A swtchng ant-wndup desgn usng multple Lyapunov functons IEEE Transactons on Automatc Control, vol 55, no 1, pp 142 148, 2010 24 A Benzaoua, O Akhrf, L Saydy Stablzaton of swtched systems subject to actuator saturaton by output feedback In Proceedngs of the 45th IEEE Conference on Decson and Control, IEEE, San Dego, USA, pp 777 782, 2006 25 L Lu, Z Ln Desgn of swtched lnear systems n the presence of actuator saturaton IEEE Transactons on Automatc Control, vol 53, no 6, pp 1536 1542, 2008 Xn-Quan Zhang receved the B Sc degree n automaton and M Sc degree n control theory & engneerng from Laonng Techncal Unversty, Chna n 2003 and 2007, respectvely He receved the Ph D degree n control theory and control engneerng n 2012 at the College of Informaton Scence & Engneerng, of the Northeastern Unversty of Shenyang, Chna Snce 2012, as a lecturer, he has been wth School of Informaton and Control Engneerng, Laonng Shhua Unversty, Chna Hs research nterests nclude swtched systems, robust control and systems control under constrants E-mal: zxq 19800126@163com (Correspondng author) ORCID ID: 0000-0003-0712-3671 Xao-Yn L receved the B Sc degree n appled mathematcs from Shenyang Normal Unversty, Chna n 2007 She receved the M Sc degree n appled mathematcs n 2012 at Laonng Techncal Unversty of Fuxn, Chna Snce 2012, as a research ntern, she has been wth School of Foregn Languages, Laonng Shhua Unversty, Chna Her research nterests nclude systems optmzaton, constraned systems and ntellgent control E-mal: lxy1982 @163com Jun Zhao receved the B Sc and M Sc degrees n mathematcs n 1982 and 1984 respectvely, both from Laonng Unversty, Chna He receved the Ph D n control theory and applcatons n 1991 at Northeastern Unversty, Chna From 1992 to 1993, he was a postdoctoral fellow at the same Unversty Snce 1994, as a professor, he has been wth College of Informaton Scence and Engneerng, Northeastern Unversty, Chna From 1998 to 1999, he was a senor vstng scholar at the Coordnated Scence Laboratory, Unversty of Illnos at Urbana-Champagn, USA From November 2003 to May 2005, he was a research fellow at Department of Electronc Engneerng, Cty Unversty of Hong Kong Durng 2007 2010, he was a fellow at School of Engneerng, the Australan Natonal Unversty Hs research nterests nclude swtched systems, nonlnear systems and network synchronzaton E-mal: zhaojun@malneueducn