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1 Informaton Scences 225 (2013) Contents lsts avalable at ScVerse ScenceDrect Informaton Scences journal homepage: Dynamc output feedback control for a class of swtched delay systems under asynchronous swtchng Ru Wang a,, Zh-Gang Wu a, Peng Sh b,c a School of Aeronautcs and Astronautcs, State Key Laboratory of Structural Analyss of Industral Equpment, Dalan Unversty of Technology, Dalan , Chna b School of Electrcal and Electronc Engneerng, The Unversty of Adelade, Adelade, SA 5005, Australa c School of Engneerng and Scence, Vctora Unversty, Melbourne, 8001 Vc, Australa artcle nfo abstract Artcle hstory: Receved 24 May 2012 Receved n revsed form 20 September 2012 Accepted 17 October 2012 Avalable onlne 10 November 2012 Keywords: Swtched delay systems Dynamc output feedback control Average dwell tme Asynchronous swtchng In ths paper, the problem of output feedback stablzaton s consdered for a class of swtched delay systems under asynchronous swtchng. When the swtchng sgnal of the swtched controller nvolves delay, by constructng a novel Lyapunov functonal whch s allowed to ncrease durng the runnng tme of actve subsystems wth the msmatched controller, suffcent condtons for exponental stablty are developed for a class of swtchng sgnals based on the average dwell tme method. Moreover the stablzng output feedback controllers are desgned. Fnally, an example s gven to demonstrate the feasblty and effectveness of the proposed desgn technques. Ó 2012 Elsever Inc. All rghts reserved. 1. Introducton Swtchng systems, as a class of hybrd dynamcal systems, consst of a set of tme-varyng subsystems and a swtchng sgnal that orchestrates the swtchng between them. Such control systems appear n many applcatons, such as communcaton networks, flght and ar traffc control and robot manpulators [6,15,3]. Therefore swtched systems have drawn consderable attenton n recent years [1,7,8,19,21,27,31,32]. As s well known, tme-delay phenomenon s very common n practcal engneerng control and s frequently a source of nstablty and performance deteroraton [4,5,16]. At present, there has been ncreasng nterest n swtched delay systems [2,12,13,18,20]. On the other hand, n the deal case, the swtchng of the controllers concdes exactly wth that of correspondng subsystems, that s to say, the controllers are swtched synchronously wth the subsystems. In actual operaton, however, snce t takes tme to dentfy the actve subsystem and apply the matched controller, the swtchng tme of controllers may lag behnd that of practcal subsystems, whch results n asynchronous swtchng between the controllers and system modes. Therefore, t s sgnfcant to study the problem of asynchronous swtchng and some valuable results have been obtaned [17,23,24,26]. In[28], the asynchronously swtched control problem for a class of swtched lnear systems wth average dwell tme was nvestgated. [30] studed the problems of stablty, L 2 -gan and asynchronous H 1 control for a class of dscrete-tme swtched systems. The robust control problem for uncertan swtched delay systems under asynchronous Correspondng author. E-mal address: ruwang@dlut.edu.cn (R. Wang) /$ - see front matter Ó 2012 Elsever Inc. All rghts reserved.

2 R. Wang et al. / Informaton Scences 225 (2013) swtchng was consdered n [24]. However, the references mentoned above dd not consder the dynamc output feedback control. In practcal applcatons, the system states may be not measured due to some reasons, therefore they cannot be used for feedback control. Therefore t s very sgnfcant to desgn the dynamc output feedback control for ths case. To the best of our knowledge, no attenton has been pad to the asynchronously swtched control problem of swtched delay systems va dynamc output feedback controllers. In ths paper, we study the dynamc output feedback stablzaton problem for a class of swtched delay systems under asynchronous swtchng. Through constructng a pecewse Lyapunov functonal whch can be allowed to ncrease durng the runnng tme of the actve subsystem wth the msmatched controller, based on the average dwell tme method, a soluton for dynamc output feedback controllers are derved n terms of LMIs such that the resultng closed-loop system s exponentally stable. The man contrbutons of the paper are as follows: Frst, the dynamc output feedback controllers are desgned whle on exstng work, the state feedback controller desgn problem was consdered. Second, both the delayed state and the delayed swtchng sgnal are consdered. Snce ths two knd of delays le n two dfferent types of sets, how to deal wth the case, the state delays and swtchng delays coexst s a challengng ssue. The paper s organzed as follows. In secton 2, prelmnares and problem formulaton are ntroduced. Secton 3 gves the suffcent condtons of exponental stablty and the controller desgn algorthm of the system. It s the man result of ths paper. In Secton 4, an example s gven to llustrate the effectveness of the proposed approach. The conclusons are summarzed n Secton 5. Notatons: Throughout ths paper, R n denotes the n-dmensonal Eucldean space, P > 0 means that P s a postve defnte, k max (P) and k mn (P) denote the maxmum and mnmum egenvalues of P, I s the dentty matrx wth approprate dmensons, kk denotes Eucldean vector norm, denotes the symmetrc block n one symmetrc matrx, dag{...} stands for a block-dagonal matrx. 2. Prelmnares and problem formulaton Consder a class of swtched delay systems of the form _xðtþ ¼A rðtþ xðtþþb rðtþ uðtþþe rðtþ xðt hþ xðhþ ¼wðhÞ h 2½ h 0Š yðtþ ¼C rðtþ xðtþ ð1þ x 2 R n s the state, u 2 R q s the control nput, y 2 R p s the measurement output, r(t):[0, + 1)? M = {1,2,...,m} s the swtchng sgnal. Specfcally, denote r(t):{(t 0,r (t 0 )),,(t k,r(t k )),, jk =0,1,2,...}, t 0 s the ntal swtchng nstant, and t k s the kth swtchng nstant. A, B, C, E are constant matrces wth approprate dmensons, w(h) s a dfferentable vector-valued ntal functon on [ h,0], h > 0 denotes the state delay. If delay h s neglected, system (1) wll reduce the model presented n [10]. When the controllers are swtched synchronously wth the subsystems, the dynamc output feedback controllers are formed as _nðtþ ¼G rðtþ nðtþþl rðtþ yðtþ uðtþ ¼K rðtþ nðtþ: ð2þ n s the state of the controllers, G, L, K are constant matrces. However, n practcal engneerng, snce t takes tme to dentfy the actve subsystem and apply the matched controller, the swtchng tme of controllers may lag behnd that of practcal subsystems, whch results n asynchronous swtchng between the controllers and system modes. Thus, we need to take the swtchng delay nto account. Remark 1. Because we may not know the ntal mode and the subsequent modes of the system n advance, the swtchngs of the controllers may not concde exactly wth those of system modes. If a wrong controller s used over a specfed amount of tme, the soluton to the system mght escape to nfnty before a correct controller s swtched nto acton [25]. We now consder the dynamc output feedback controllers of the followng form: _nðtþ ¼G rðtþ nðtþþl rðtþ yðtþ uðtþ ¼K rðt sd ÞnðtÞ ð3þ s d s the delay of swtched controllers to system modes. The followng defntons wll be used n the sequel. Defnton 1 [13]. The equlbrum x = 0 of system (1) s sad to be exponentally stable under r(t) f the soluton x(t) of system (1) satsfes kxðtþk 6 kkxðt 0 Þke kðt t 0Þ 8t P t 0 for constants k P 1 and k >0.

3 74 R. Wang et al. / Informaton Scences 225 (2013) Defnton 2 ([1,7]). For swtchng sgnal r and any t P t 0 P 0, let N r (t 0,t) denote the number of swtchng of r over the tme nterval (t 0,t). If N r ðt 0 tþ 6 N 0 þ t t 0 s a : ð4þ holds for N 0 P 0, s a > 0, then s a s called the average dwell-tme and N 0 s the chatter bound. As commonly used n the lterature, for convenence, we choose N 0 = 0 n ths paper. 3. Man results In ths secton, we wll gve stablty analyss, synthess condtons and a desgn algorthm Stablty analyss Applyng the dynamc output feedback controllers (3) to system (1),we have the closed-loop system _xðtþ ¼A rðtþ xðtþþe rðtþ xðt hþ ð5þ x ¼ x A rðtþ ¼ A r B r K rðt sd Þ n L r C r G r E rðtþ ¼ E rðtþ The followng result presents a suffcent condton of exponentally stablty for system (5). Theorem 1. For gven postve constants a and b, f there exst matrces P > 0, Q > 0, " 2 M such that " # R ¼ AT P þ P A þ Q þ ap P E < 0 ð6þ e ah Q " P ¼ ATP # j þ P A j þ Q bp P E < 0 ð7þ e bh Q then dynamc output feedback controllers (3) make system (5) exponentally stable under asynchronous swtchng for any swtchng sgnal satsfyng average dwell tme s a P s a ¼ ln l þðaþbþs d ð8þ a l P 1 satsfes P 6 lp j Q 6 lq j 8 j 2 M: ð9þ Proof. Due to the swtchng delay, the jth subsystem has been swtched to the th subsystem, and the controller K j s stll actve for s d. Thus, we have ( _xðtþ ¼ A j xðtþþe xðt hþ 8t 2½t t þ s d Þ A xðtþþe xðt hþ 8t 2½t þ s d t þ1 Þ: ð10þ A j ¼ A B K j L C G A ¼ A B K L C G When "t 2 [t k + s d,t k+1 ), the Lyapunov functonal canddate V 1r ðtþ ¼x T ðtþp r xðtþþ : x T ðsþe aðs tþ Q r xðsþds P, Q are postve defnte matrces satsfyng (6), (7) and (9). Along the trajectory of (10) we have ð11þ _V 1 þ av 1 ¼ x T ðtþ½p A þ A T P þ Q þ ap ŠxðtÞþ2x T ðtþp E xðt hþ x T ðt hþe ah Q xðt hþ ¼f T ðtþr fðtþ fðtþ ¼ x T ðtþ x T T ðt hþ.

4 R. Wang et al. / Informaton Scences 225 (2013) From (6), we can get _V 1 þ av 1 6 0: When "t 2 [t k,t k + s d ), the Lyapunov functonal canddate V 2r ðtþ ¼x T ðtþp r xðtþþ x T ðsþe bðt sþ Q r xðsþds P, Q are postve defnte matrces satsfyng (6), (7) and (9). Along the trajectory of (10), we have ð12þ ð13þ _V 2 bv 2 ¼ x T ðtþ½p A j þ A T j P þ Q bp ŠxðtÞþ2x T ðtþp E xðt hþ x T ðt hþe bh Q xðt hþ 6 f T ðtþp fðtþ: From (7), we can get _V 2 bv 2 6 0: Obvously x T ðsþe aðs tþ Q xðsþds 6 Thus, combnng (11), (13) and (15), t holds that V 1 ðtþ 6 V 2 ðtþ: x T ðsþq xðsþds 6 x T ðsþe bðt sþ Q xðsþds Consderng the whole nterval [t 0,t), the Lyapunov functonal canddate s the combnaton of (11) and (13) VðtÞ ¼ V 1rðtÞ t 2½t k þ s d t kþ1 Þ k ¼ V 2r ðtþ t 2½t k t k þ s d Þ k ¼ : For t 2 [t k + s d,t k+1 ), ntegratng both sdes of (12) from t k + s d to t, and combnng (4), (9) and (16), we have VðtÞ 6 e aðt ðt k þs d ÞÞ V 1 ððt k þ s d Þ þ Þ 6 e aðt ðt kþs d ÞÞ V 2 ððt k þ s d Þ Þ 6 e aðt ðt kþs d ÞÞ e bs d V 2 t þ k 6 le aðt ðt kþs d ÞÞ e bs d V 2 t þ k l k e ðkþ1þbs d e a½t t 0 ðkþ1þs d Š Vðt 0 Þ 6 e ðaþbþs ln lþðaþbþs d d e s a a ðt t 0 Þ Vðt 0 Þ: Smlarly, for t 2 [t k,t k + s d ), we obtan VðtÞ 6 e bðt t k Þ V 2 t þ k 6 le bðt tkþ V 1 t k 6 le bs d e aðt k t k 1 s d Þ V 1 ððt k 1 þ s d Þ Þ l k e ðkþ1þbs d e a½t t 0 ðkþ1þs d Š Vðt 0 Þ 6 e ðaþbþs ln lþðaþbþs d d e s a a ðt t 0 Þ Vðt 0 Þ: ð14þ ð15þ ð16þ ð17þ ð18þ ð19þ Notce (11) and (13), t obvously holds that akxðtþk 2 6 V 1 ðtþ 6 bkxðtþk 2 t 2½t k þ s d t kþ1 Þ akxðtþk 2 6 V 2 ðtþ 6 bkxðtþk 2 t 2½t k t k þ s d Þ ð20þ a ¼ mnfk mn ðp Þg b ¼ maxfb 1 b 2 g 82M b 1 ¼ maxfk max ðp Þg þ hmaxfk max ðq Þg 82M 82M b 2 ¼ maxfk max ðp Þg þ he bh maxfk max ðq Þg: 82M 82M

5 76 R. Wang et al. / Informaton Scences 225 (2013) Then, applyng (18) (20) yelds rffffff b kxðtþk 6 e ðaþbþs d 2 e lþðaþbþs ½ln d 2s a a 2 Šðt t0þ kxðt 0 Þk a 8t 2½t k t kþ1 Þ: ð21þ From (8), system (5) s exponentally stable. h Remark 2. Although the Lyapunov functonal constructed n Theorem 1 s allowed to ncrease both at the swtchng nstants t k and durng the runnng tme of actve subsystems wth the msmatched controllers [t k, t k + s d ), by restrctng the lower bound of the average dwell tme, the Lyapunov functonal s decreasng as a whole and hereby the system stablty s guaranteed. Remark 3. Theorem 1 provdes a suffcent condton for exponental stablty of system (1) (or for system (5) under control law (3)). However, nequaltes (6) and (7) are not n the form of LMIs f the controller gans are to be determned. We wll gve LMIs condtons for determnng the controller gans n the next subsecton Synthess condtons Ths secton wll gve some LMIs condtons for the controller desgn. Theorem 2. Gven postve numbers a, b and c, f there exst symmetrc matrces X, Y, T, Z and matrces ba b B b C b Aj ð8 j 2 MÞ such that the followng matrx nequaltes X hold, I Y > 0 ð22þ 2 N A þ A b 3 T þ cx þ ai E 0 X X þ ay Y E 0 0 e ah ci 0 0 < 0 ð23þ e ah T 0 5 c 1 I 2 N j A þ ^A T j þ cx 3 bi E 0 X X by Y E 0 0 e bh ci 0 0 < 0 ð24þ e bh T j 0 5 c 1 I N ¼ A X þ X A T þ B b C þ b C T BT þ ax þ Z X ¼ Y A þ b B C þ A T Y þ C T b B T þ ci N j ¼ A X þ X A T þ B b C j þ b C T j BT bx þ Z j then dynamc output feedback controllers (3) make the resultng swtched system exponentally stable under asynchronous swtchng correspondng to any swtchng sgnal wth average dwell tme s a satsfyng (8) and the controller parameters are gven by K ¼ C b ðm T Þ 1 L ¼ N 1 bb G ¼ N 1 ða b Y A X N L C X Y B K M T ÞðMT Þ 1 M and N satsfy the constrant ð25þ M N T ¼ I X Y ð26þ the constant l P 1 satsfes wth R 1 S 6 lr 1 j S j T 6 lt j ð27þ

6 R ¼ X I M T 0 S ¼ I Y 0 N T R. Wang et al. / Informaton Scences 225 (2013) j 2 M: Proof. Motvated by the method n [22,9,11], we defne matrces P ¼ Y N N T ð ¼ 1 2Þ W W > 0. Then, P 1 ¼ X M M T Z X wth Z > 0. We can easly obtan P I M T 0 P ¼ R 1 S. Here defne Q = dag{ci,t }, T > 0 and c s postve scalar to be chosen. We frst prove that matrx nequalty (6) s equvalent to LMI (23). ¼ I Y 0 N T, that s P R = S and thus Pre- and post-multplyng both sdes of nequalty (6) by dagfr T Ig and dag{r,i} yeld the followng matrx nequalty " # R T AT P R þ R T P A R þ R T Q R þ ar T P R R T P E < 0: ð28þ e ah Q A straght forward computaton gves the followng equaltes. " # R T P A X þ B K M T A A R ¼ Y A X þ N L C X þ Y B K M T þ N G M T Y A þ N L C R T P E ¼ E 0 R T Y E 0 P R ¼ S T R X I ¼ Y X þ N M T ¼ X I Y I Y " # R T Q R ¼ cx X þ M T M T cx : cx ci Defne the followng transformaton of varables: ba ¼ Y A X þ N L C X þ Y B K M T þ N G M T bb ¼ N L C b ¼ K M T Z ¼ M T M T : ð29þ So, from (28) and (29), we have 2 N þ cx X A bt þ A þ ai þ cx E 0 X Y E e ah ci < 0: ð30þ 5 e ah T Accordng to Schur complement Lemma, matrx nequalty (30) s equvalent to LMI (23). Therefore, (6) s equvalent to (23). In the followng, we wll deduce (7) from matrx nequaltes (24). Pre- and post-multplyng both sdes of nequalty (7) by dagfr T Ig and dag{r,i} yeld the followng matrx nequalty " R T AT j P R þ R T P A j R þ R T Q R br T P R R T P # E < 0: ð31þ e bh Q From (31), we have 2 N j þ cx X A bt j þ A bi þ cx E 0 X by Y E e bh ci < 0 ð32þ 5 e bh T j ba j ¼ Y A X þ N L C X þ Y B K j M T j þ N G M T j bc j ¼ K j M T j Z j ¼ M j T j M T j : ð33þ Accordng to Schur complement Lemma, matrx nequalty (32) s equvalent to LMI (24). Therefore, (7) s equvalent to (24).

7 78 R. Wang et al. / Informaton Scences 225 (2013) If LMIs (22) (24) have feasble solutons b A b B b C X Y Z, then we can get matrces M, N from (26) and (29). Therefore, controller matrces (25) can be obtaned. From LMIs (22) (24) and Theorem 1, we know that system (1) wth dynamc output feedback controllers (3) s exponentally stable under asynchronous swtchng for any swtchng sgnal satsfyng (8) and (9). Ths completes the proof. If swtchng delay s d = 0, that s to say, the controllers are swtched synchronously wth the subsystems, we can derve the followng result. h. Corollary 1. Consder the swtched delay system (1). Gven postve numbers a and c, f there exst symmetrc matrces X,Y,T,Z and matrces A b B b C b ð8 2 MÞ such that (22), (23) hold, then dynamc output feedback controllers (2) make the resultng swtched system exponentally stable correspondng to any swtchng sgnal wth average dwell tme s a satsfyng s a > s a ¼ ln l a, the controller parameters are gven by (25) and constant l P 1 satsfes (27). If h = 0, swtched delay system (1) degenerates nto non-delay swtched system, we have the followng corollary. Corollary 2. Consder the swtched system (1) wth h = 0. Gven postve numbers a, b, f there exst symmetrc matrces X,Y and matrces A b B b C b Aj b ð8 j 2 MÞ such that LMIs (22), " N A þ A b # T þ ai < 0 X þ ay hold, " N j A þ A b # T j bi < 0 X by N ¼ A X þ X A T þ B C b þ C b T BT þ ax X ¼ Y A þ B b C þ A T Y þ C T b B T N j ¼ A X þ X A T þ B b C j þ b C T j BT bx then dynamc output feedback controllers (2) make the resultng swtched system exponentally stable under asynchronous swtchng correspondng to any swtchng sgnal wth average dwell tme s a satsfyng (8), the controller parameters are gven by (25) and constant l P 1 satsfes R 1 S 6 lr 1 j S j Algorthm Based on Theorem 2, we present an algorthm for the desgn of dynamc output controllers. Step I. Gven a, b and c, solve LMIs (22) (24) to obtan X Y T Z A b B b C b. Step II. Then obtan matrces M and N by (26) and (29). Step III. calculate matrces K, L and G accordng to (25). Step IV. From P ¼ R 1 S Q ¼ dagfci T g, calculate l by the followng optmzaton approach mnmze l s:t: P 6 lp j Q 6 lq j 8 j 2 M: ð34þ Step V. Calculate the average dwell tme bound based on (8). Then for any swtchng sgnal wth average dwell tme satsfyng (8), the dynamc output controllers gven by (3) make system (1) exponentally stable under asynchronous swtchng. 4. An example In ths secton, an example s presented to demonstrate the effectveness of proposed desgn method. Consder the swtched system (1) consstng of two subsystems descrbed by 9 0:2 A 1 ¼ B 1 ¼ 0:5 2 E 1 ¼ 0:1 0 C 1 ¼ 1 0 0:3 2 0:1 0:9 0:1 0:3 0 1 A 2 ¼ 1 0 B 2 ¼ 4 0: :2 E 2 ¼ 0: :1 C 2 ¼ h ¼ 0:4:

8 R. Wang et al. / Informaton Scences 225 (2013) swtchng sgnal tme (s) Fg. 1. swtchng sgnals (sold lne swtchng sgnal of subsystems, dashed lne swtchng sgnal of controllers) x 1 x state tme (s) Fg. 2. state trajectores of system (1) satsfyng the swtchng condton. We assume the delay of asynchronous swtchng s d = 0.3. Now, we desgn the output feedback controllers usng the algorthm. Choosng a =4,b = 2,c = 1.5, we can obtan postvedefnte matrces X Y T Z A b B b C b ð ¼ 1 2Þ by solvng LMIs (22) (24). Followng Step II, we get M and N from (26) and (29). Accordng to Step III, we can obtan controller gans K 1 ¼ 591:1 1308:3 67:6 328:1 0:2648 0:2835 L 1 ¼ 0:0948 2:9816 K 2 ¼ 41: :5746 0: :4389 L 2 ¼ 2: : :6939 4:2060 G 1 ¼ G 2 ¼ 2: : :3264 0:2892 0: :2327 : ð35þ Applyng (34) produces l = 2.9, and accordng to Step V, we have average dwell tme s a P s a ¼ ln lþðaþbþs d a ¼ 0:7162. Let s a = 0.8. Fg. 1 descrbes the swtchng sgnals, sold lne and dashed lne represent swtchng sgnals of subsystems and controllers, respectvely. Under ths swtchng sgnals and dynamc output feedback controllers wth parameters (35), the steady-state responses of the closed-loop system wth x 0 ¼ ½ 0:3 0:5 Š T are depcted n Fg. 2. Moreover, accordng to (21), we get kxðtþk 6 51:8189e 0:2096ðt t 0Þ kxðt 0 Þk: Therefore, t can be seen from Fg. 2 and (36) that the proposed dynamc output feedback controllers can guarantee that the closed-loop system s exponentally stable although there exsts asynchronous swtchng. ð36þ

9 80 R. Wang et al. / Informaton Scences 225 (2013) Concluson We have nvestgated the dynamc output feedback stablzaton problem for a class of swtched delay systems under asynchronous swtchng. Tme delays appear not only n the state, but also n the swtchng sgnal of the controller. Based on a novel Lyapunov functonal method combned wth the average dwell tme scheme, we have establshed suffcent condtons for exponental stablty n terms of LMIs. We have also desgned output feedback controllers and dentfed a class of swtchng sgnals satsfyng a specfc lower bound of the average dwell tme. In the future studes on ths topc, an extenson of these results to the case of nonlnear plant systems, networked control systems [29], or neural networks [14] would make a major step forward. 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