Output Feedback Stabilization of Networked Control Systems with Random Delays

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1 Preprnts of the 8th IFAC Worl Congress Mlano (Italy) August 28 - September 2 2 Output Feebac Stablzaton of Networe Control Systems wth Ranom Delays Shou-Wan Gao Gong-You ang College of Informaton Scence an Engneerng Ocean Unversty of Chna Qngao 266 Chna (e-mal:gaoshouwan@gmal.com; gtang@ouc.eu.cn) Abstract: hs paper s concerne wth the output feebac stablzaton problem of networe control systems (NCSs) wth ranom communton elays. Mean-square stablty s erve for the close-loop NCSs. Base on ths the one-moe-epenent output feebac controller s esgne an the explct expressons of the controller gans are obtane n terms of lnear matrx nequaltes (LMIs). A numerl example s prove to show the effectveness an applblty of the propose results.. INRODUCION Networe control systems (NCSs) are feebac control loops close through a real tme networ. hat s n NCSs sensors controllers actuators an other system components communte over communton networs (Zhang Brancy & Phllps 2). he use of communton networs offers many avantages n terms of low cost reuce weght smple nstallaton an mantenance an hgh relablty. As a result NCSs have great potental n appltons n manufacturng plants vehcles arcrafts an spacecrafts (Walsh Ye & Bushnell 22). However the ntroucton of networs also presents some constrants such as tme elays pacet ropouts an so on whch mae the NCS analyss an synthess complex. herefore the control problem of NCSs has attracte conserable research nterests. Among a few others networ-nuce elay s probably the most mportant ssue to be aresse for NCSs. An t s usually use by the lmte bt rate of the communton channel by a noe watng to sen out a pacet va a busy channel or by sgnal processng an propagaton. he elays may be constant (Yu et al. 24a) tme-varyng (Yu et al. 24b) an n most ses ranom (Gao Meng & Chen 28; He Wang & Zhou 29; Nlsson 998; Sh & Yu 29; sa & Ray 997; Wang Ho & Lu 24; Xao Hassb & How 2; Yang et al. 26; Zhang et al. 25). In the lterature there have been baally two approaches for moellng the ranom elay phenomenon n the NCSs. he frst approach s to vew the elay as a bnary swtchng sequence whch obeys a Bernoull strbute whte sequence tang on values of zero an one wth certan probablty (Gao Meng & Chen 28; sa & Ray 997; Wang Ho & Lu 24; Yang et al. 26). he other way to moel the elay s to employ the fnte state Marov chan (He Wang & Zhou 29; Nlsson 998; Sh & Yu 29; Xao Hassb & How 2; Zhang et al. 25). Compare wth the frst approach the Marov moel has two man avantages: () hs wor was supporte by the Natonal Natural Scence Founaton of Chna (Grant No ) an the Natural Scence Founaton of Shanong Provnce (Grant No. ZR2FM9). the epenences between elays are taen nto account snce n real networs the current tme elays are usually relate wth the prevous elays (Nlsson 998); (2) the pacet ropout coul be nclue naturally (Xao Hassb & How 2). Recently there have been some results publshe on such a moel (He Wang & Zhou 29; Sh & Yu 29; Zhang et al. 25). Zhang et al. (25) moelle the sensorto-controller (S-C) an the controller-to-actuator (C-A) elays as two fferent Marov chans an base on ths esgne a two-moe-epenent state feebac controller. However they nherently assume that at each samplng nstant the current S-C elay an prevous C-A elay coul be obtane at the controller noe. Practlly the prevous C-A elay s not always avalable beuse the nformaton about C-A elays nees to be transmtte through the S-C communton ln before reachng the controller as suggeste by Sh & Yu (29). It s worth notng that f the tme elay exsts the nformaton of at tme nstant n be nown at the controller noe. Hence for NCSs wth S-C an C-A elays moelle as two Marov chans Sh an Yu (29) propose a twomoe-epenent output feebac control heme whch epens on an. Moreover by employng a sequence varyng n a Marovan fashon n the measurement moel the robust fault etecton problem was stue for a class of networe systems wth unnown nput multple state elays an ata mssng (He Wang & Zhou 29). Although NCSs wth ranom elays moelle as Marov chan have been stue for a few years there are stll some nterestng problems that eserve further research. o the best of authors nowlege the esgn of one-moe-epenent output feebac controller for NCSs wth both S-C elay an C-A elay has not been fully nvestgate whch s the focus of ths wor. By ang a buffer to the actuator noe (Fg. ) the S-C elay an the C-A elay n Copyrght by the Internatonal Feeraton of Automatc Control (IFAC) 325

2 Preprnts of the 8th IFAC Worl Congress Mlano (Italy) August 28 - September 2 2 be lumpe together an the new varable s efne as. he elay s moelle as a Marov chan then mean-square stablty s erve for the closeloop NCSs. Base on t the output feebac controller s esgne. A lnear matrx nequalty (LMI) approach s evelope to tacle aresse problem whch n be solve convenently by Matlab LMI oolbox. he metho propose n ths paper exhbts some smlartes wth the wely apple metho of gan heulng. At a samplng nstant the actuator selects the approprate controller gan accorng to. Fg.. Structure of a networe control system wth ranom communton elays he rest of the paper s organze as follows. Secton 2 erbes the NCS framewor an formulates the problem of the output feebac stablzaton. Secton 3 frst gves the necessary an suffcent contons to guarantee the meansquare stablty of the close-loop NCSs an then obtans the explct expressons of the controller parameters n terms of LMIs. Secton 4 proves a numerl example to llustrate the effectveness of our results. Fnally some conclung remars are gven n Secton PROBLEM FORMULAION Conser a class of rete-tme NCSs wth the structure shown n Fg.. he plant s assume to be of the form: Cx x Ax Bu y () n m where x R s the state vector u R p control nput y s the R s the output vector an A B an C are nown real matrces wth approprate mensons. Moreover the matrx C s of full-row ran.e. ran C p. As s shown n Fg. the controller s not co-lote wth the sensor an the actuator an t s place n a remote loton then both sensor measurement pacets an control pacets are transmtte through the networs. At each samplng pero the sample plant output y an ts tme stamp (.e. the tme the plant output s sample) are enpsulate nto a pacet an sent to the controller. Whenever a sensor pacet u. arrves the controller computes a control sgnal array l hen the tme stamp of the use plant output s ae to the control sgnal array u l an both of them are transmtte to the actuator. he buffer wll compare the tme stamps on the arrve control sgnal array an the one store n t an wll be upate only when the tme stamp on the arrve control sgnal array s newer. Conserng the effect of the ranom communton elay an the ata pacet ropout we erbe the one-moeepenent output feebac control law as: u K y (2) where enotes the step fference between the runnng step an the tme stamp of the use plant output an t epens on the ranom communton elay an the ata pacet ropout as erbe n Gao Meng & Chen (28). Assume that both the ranom communton elay an the ata pacet ropout are boune thus s boune. he step elay s moelle as a homogeneous Marov chan S 2 an ts whch taes values n transton probablty matrx s enote as ( j ). hat means jumps from moe to j wth probabltes where j an Pr j (3) j for all j S. j j Remar. he output feebac control law (2) epens on the step elay. However practlly s not always avalable at the controller noe beuse the control sgnals may suffer from C-A elay an C-A pacet ropout. hus at a samplng nstant f the controller receves a new sensor measurement t s just responsble for computng a control u K l y l sgnal array l u n whch l. Obvously vares at fferent samplng nstant so the number of control sgnals of u l vares. However t n not be larger than hence we n select the buffer wth the pacty of. hen the actuator wll select the approprate control sgnal n terms of. For example at samplng nstant f h h S the control sgnal uh Khy h wll be selecte an sent to the plant. It s worth notng that by comparng the current nstant wth the tme stamp on the control sgnal array the nformaton about n be obtane n real tme at the actuator noe whch ensures the actuator n select the approprate control sgnal. Remar 2. he buffer s upate only when the tme stamp on the arrve control sgnal array s newer. hus f at samplng nstant the buffer stores the control sgnal array l 325

3 Preprnts of the 8th IFAC Worl Congress Mlano (Italy) August 28 - September 2 2 l u wth tme stamp.e. the step elay s then at samplng nstant f the step elay s longer than the buffer wll not be upate. hs means that the step elay n ncrease at most at each step that s f j. j Applyng control law (2) to system () we n obtan the followng close-loop system x Ax BK Cx. (4) Moreover we augment the state vector as z x x x. (5) hen system (4) n be wrtten as where ˆ z A BK C z (6) A I Aˆ I R I B ˆ B R nn n m ˆ p n C C R an ˆ C has all elements beng zeros except for the th bloc beng C. he objectve of ths paper s to esgn the output feebac controller to guarantee the mean-square stablty of the NCS n (6). Defnton (He Wang & Zhou 29; Xong & Lam 26). System (6) s sa to be mean-square stable f 2 E z as for every ntal conton z an z S. 3. MAIN RESULS In ths secton we frst gve the necessary an suffcent contons for the mean-square stablty of close-loop system (6). hen we nvestgate how to esgn the controller such that close-loop system (6) s mean-square stable an the explct expressons of the controller gans are obtane n terms of LMIs. he followng lemma s frstly ntrouce whch guarantees the mean-square stablty of close-loop system (6). For the ease of presentaton when system (6) s n moe S (.e. ) we enote K an Cˆ as K an C ˆ respectvely. Lemma. he resultng close-loop system (6) s meansquare stable f an only f there exst matrces n n P R S satsfyng j j A BKC Pj ABKC P (7) for all S. Proof. For system (6) we construct the followng Lyapunov functon Hence V z P z. (8) V z V E z j A BKC Pj A BKC P z j. (9) he rest of the proof s smlar to that of heorem 9 n Xong & Lam (27) an s omtte here. Now we establsh a suffcent conton for the mean-square stablty of close-loop system (6) n the form of LMIs wth matrx equalty constrant from Lemma 2. Lemma 2. he close-loop system (6) s mean-square stable n n f there exst matrces X R n n G R m p M R an Q R S such that an QCˆ hol for all S where CG ˆ () G G X H H () H AGBMC AGBMC (2) 3252

4 Preprnts of the 8th IFAC Worl Congress Mlano (Italy) August 28 - September 2 2 ag X X X. (3) Moreover the controller gans K S are gven by K M Q. (4) Proof. Assume that () s feasble then G G X. hs means that matrces G S are of full ran. Moreover as the matrx C s of full-row ran we n easly obtan that matrces C ˆ S are also of full-row ran. Hence t follows from () that matrces Q S are nonsngular. From () an (4) we have M Cˆ KCG ˆ. (5) Replacng M C ˆ n (2) by KCG ˆ we n rewrte (2) as H G ABKC ABKC GR (6) where R A BKC A BKC. hen substtutng (6) nto () yels As whch s equvalent to G G X G R RG. (7) X we have X G X X G G X G G G X. (8) hus t follows from (7) an (8) that G X G G R RG Pre- an post-multplyng (9) by. (9) G ag an ts transpose an applyng the Schur complement formula we obtan ˆ ˆ j ABKC X j ABKC X j (2) whch s equvalent to (7) wth P. X hs completes the proof. Next we uss how to solve the equalty constrant n (). Snce the matrces C ˆ S are of full-row ran there always exst orthogonal matrces U R an n n V R S such that where 2 C UCV ˆ UCˆ V V (2) n p an V R ag 2 p 2 n n p V R where r an r 2 p are nonzero sngular values of C ˆ. Notng the efnton of C ˆ (.e. Cˆ ) we now that r r 2 p are actually nonzero sngular values of C. Lemma 3. For gven ran Cˆ p ˆ p n C R S n n S f matrces G R of full ran an have the followng structure G G V V VGV V G V V GV wth G2 G 3 are (22) for all S where G R n G 2 R n p n p G 3 R an V an V 2 are efne n (2) there exst nonsngular matrces R satsfyng () for any S. Q he proof s smlar to that of Lemma 3 n Ho & Lu (23) an s omtte here. heorem. he close-loop system (6) s mean-square n n stable f there exst matrces X R G R n G 2 R n p n p G 3 R m p an M R S such that () hols for all S where G V GV V2G2V V2G3V2 an V an V 2 come from (2). Moreover the controller gans S are gven by K K MU G U (23) n whch U an are efne n (2). Proof. It has been shown n the proof of Lemma 2 that matrces G S are of full ran. By applyng Lemma 3 f there exst matrces G G 2 an G 3 such that 3253

5 Preprnts of the 8th IFAC Worl Congress Mlano (Italy) August 28 - September 2 2 G V G V V G V V G V where V an V 2 are efne n (2) there exst nonsngular matrces Q S satsfyng (). We n compute nonsngular matrces Q S as follows from () U V GV QU (24) shown n Fg. 3. It n be seen that close-loop system (4) s mean-square stable..e. G U QU G2 G 3 whch mples that (25) Q U G U. (26) hus we obtan (23) from (4) an (26). hs completes the proof. 4. NUMERICAL EXAMPLE Fg. 2. Step elay In ths secton for the purpose of llustratng the usefulness an flexblty of the propose metho we present a smulaton example. he parameters of the rete-tme NCS () are gven as follows: A B C. Snce matrx A has egenvalues at.2 an.4 the rete-tme system () s unstable. he ntal state values are set to be x 3 x 2 x x 3 the state space of the Marov chan obtane as S 2 3. Lettng n be. he transton probablty matrx of the Marov progress s gven by By heory we n obtan the controller gans as follows: K.4573 K.9 K2.92 K3.7. Fg. 2 shows one of the possble realzaton of the step elay. An the state trajectores of close-loop system (4) are. Fg. 3. States of the close-loop system 5. CONCLUSIONS In ths paper the output feebac stablzaton problem has been consere for NCSs wth ranom communton elays. A Marov chan s employe to erbe the ranom communton elays an base on the Marov moel the output feebac controller s esgne whch guarantees the mean-square stablty of the resultng close-loop systems. Furthermore t has been shown that the controller esgn problem uner conseraton s solvable f a set of LMIs s feasble. Smulaton results llustrate the feasblty an effectveness of our methos. REFERENCES Gao H.J. X.Y. Meng an.w. Chen (28). Stablzaton of networe control systems wth a new elay characterzaton. IEEE ransactons on Automatc Control 53 (9)

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