Output Feedback Stabilization of Networked Control Systems With Packet Dropouts

Transcription

1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 9, SEPTEMBER Output Feedback Stablzaton of Networked Control Systems Wth Packet Dropouts Wen-An Zhang and L Yu Abstract In ths paper, we dscuss the modelng and control of a class of networked control systems (NCSs) wth packet dropouts. For the cases that there may be packet dropouts n both the backward and the forward channels n the communcaton network, and that the network-nduced delays are shorter than one samplng perod, the closed-loop NCS s modeled as a dscrete-tme swtched system wth four subsystems. By usng the asynchronous dynamcal systems approach and the average dwell-tme method, suffcent condtons for the exponental stablty of the closed-loop NCS are presented n terms of nonlnear matrx nequaltes, and the relaton between the packet dropout rate and the stablty of the closed-loop NCS s explctly establshed. A procedure nvolvng an teratve algorthm s proposed to desgn the observer-based output feedback controllers. Lastly, an llustratve example s gven to demonstrate the effectveness of the proposed results. Index Terms Exponental stablty, networked control systems, output feedback, packet dropouts, swtched systems. I. INTRODUCTION Control systems n whch control loops are closed through a seral network are called networked control systems (NCSs). Recently, NCSs have receved ncreasng attenton because of ther many advantages n practcal applcatons, such as ncreased system flexblty and the great benefts from sharng of the resources, as well as lower cost and smpler nstallaton and mantenance [] [3]. As a result, modelng, analyss, and control of NCSs wth many new emerged problems, such as network-nduced delay, packet dropout, and quantzaton, has appeared to be a topc of sgnfcant nterest to the control communty []; see, e.g., [5] [7] and references theren. One of the man ssues n NCSs s network-nduced delays, whch are usually the major causes for the deteroraton of system dynamc performance and potental system nstablty. Therefore, varous approaches for the delay ssue n NCSs have been presented n the lterature; see, e.g., [] and [9] [2]. Another mportant problem n NCSs s the packet dropout phenomenon, whch s also commonly known as one of the major causes of deteroraton n system performance. There may exst unavodable errors or losses n the transmsson of packets due to sharng a common network medum and the presence of the uncertantes and noses n the communcaton channel. If ths happens, the corrupted packet s dropped []. We may call ths knd of packet dropout the network-nduced packet dropout, snce n ths case the packet dropout s caused by the network tself. The NCSs are dstrbuted control systems that contan a number of controlled plants, and all the sensors, controllers, and actuators are nterconnected by one communcaton network. Due to the lmtaton of network bandwdth, only some of these controlled plants can gan access to the network at each samplng nstant. Therefore, some schedulng algorthms should be nvolved to assgn a transmsson schedule to each transmsson entty. For a set of faster samplng NCSs, the message collsons are more lkely to happen Manuscrpt receved October, 200; revsed February, Recommended by Assocate Edtor C. T. Abdallah. Ths work was supported by the Natonal Natural Scence Funds for Dstngushed Young Scholars under Grant The authors are wth the Department of Automaton, Zhejang Unversty of Technology, Hangzhou 30032, Chna (e-mal wazhg@hotmal.com; lyu@zjut.edu.cn). Dgtal Object Identfer 0.09/TAC f every data packet was guaranteed to be delvered. Therefore, some specfed system performance may not be acheved for the NCSs under the prevously desgned schedulng polcy, that s, the NCSs may be unschedulable n ths case. As has been ponted out n [5], f a set of NCSs s unschedulable, we mght consder droppng some of the packets n the faster samplng NCSs to ensure that the set of NCSs s schedulable and meanwhle to guarantee that the overall NCSs are stable. We may call ths knd of packet dropout the actve packet dropout snce, n ths case, the packet dropout s caused by the schedulers or the desgners of the NCSs. It s clear that a relaton between the packet dropout rate and the stablty of the closed-loop NCSs should be establshed before an applcable schedulng algorthm s desgned, and the correspondng desgn procedures for the controllers that render the closed-loop NCSs to be stable should also be presented. Some effectve approaches, such as the stochastc system approach and the swtched system approach, have been presented for the modelng and control of the NCSs under packet dropout effects; see, e.g., [], [5], [8], and [3] [5]. Among these results, stablty and dsturbance attenuaton results were presented n [] for a class of NCSs under both packet dropout and delay effects. State feedback stablzng controller was provded n [8] for a class of NCSs wth nonlnear perturbatons. The NCSs wth arbtrary but fnte packet dropout were modeled as a swtched system, and both state feedback and statc output feedback controller desgn methods were presented n []. However, all the NCSs consdered n [], [8], and [] assumed that there exst networks only n the backward channel (the channel between the sensor and the controller), whch s unrealstc n practcal applcatons. When consderng both-sde networks, the stablzaton results were gven n [5] for both the NCSs wth arbtrary packet dropout and the NCSs wth Markovan packet dropout; however, the desgned controller s an event-drven state feedback controller. It s notced that most of the exstng controllers desgned for the NCSs wth packet dropout are event-drven state feedback ones. However, the event-drven controllers are more dffcult to realze n practce compared wth the tme-drven ones. Moreover, the effect on the system dynamcs of the packet dropout n the backward channel s the same as that n the forward channel (the channel between the controller and the actuator) when an event-drven state feedback controller s used n the NCSs. However, the packet dropouts n the two dfferent channels have dfferent effects on the system dynamcs, and thus the resultng closed-loop systems wll be more complex when applyng the tme-drven output feedback controllers n the NCSs. Furthermore, none of the aforementoned results has ever establshed the relaton between the packet dropout rate and the stablty of the NCS except for [5], n whch suffcent condtons were presented for the exponental stablty of the NCS under packet dropout effect. Meanwhle the relaton between the packet dropout rate and the stablty of the NCS was mplctly establshed by usng the asynchronous dynamcal systems wth rate constrants on events (ADSs-RCE) approach. However, the NCS consdered n [5] has ts network only n the backward channel, and the desgned controller s a state feedback one. The above analyses motvate us to desgn the observer-based output feedback controllers to exponentally stablze a class of networked systems wth packet dropouts n both the backward and the forward channels, and meanwhle to explctly establsh the relaton between the packet dropout rate and the stablty of the closed-loop NCS by usng the ADSs-RCE approach and the average dwell-tme method [], [7]. Ths paper s organzed as follows. The NCSs wth short communcaton delay and packet dropouts are modeled as a class of dscrete-tme lnear swtched systems wth four subsystems n Secton II. The exponental stablty of such NCSs s studed n Secton III. A desgn procedure s proposed for the observer-based output feedback controllers /$ IEEE

2 70 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 9, SEPTEMBER 2007 Then by the above analyss and assumptons, and makng use of the models n () (3), the dynamcs of the closed-loop NCS can be descrbed by the followng four subsystems. a) There s no packet dropout n ether the backward channel or the forward channel A + BK 0BK A 0 LC 0 0 K 0K 0 0 Fg.. Structure of a networked control system wth packet dropouts. S z(k +)=Az(k);A = n Secton IV. The smulaton results of an llustratve example are presented n Secton V. Lastly, conclusons are gven. II. A SWITCHED SYSTEM MODEL FOR NCSS The NCS wth packet dropouts n both the forward and the backward channels s llustrated n Fg., where the plant s descrbed by the followng dscrete-tme lnear tme-nvarant system model x(k +)=Ax(k) +Bu(k) y(k) =Cx(k) where x(k) 2< n s the system state, u(k) 2< m s the control nput, and y(k) 2< p s the measured output. The controller s an observerbased output feedback controller wth ts model descrbed as follows ^x(k+)=a^x(k)+bu(k)+l[w(k)0 ^y(k)] Observer ^y(k) =C ^x(k) (2) Controller v(k) =K ^x(k) (3) where ^x(k) 2< n s the estmated state of the system () and ^y(k) 2 < p s the estmated output. L 2< n2p and K 2< m2n are the observer gan and the controller gan, respectvely. It s assumed that the pars (A; B) and (C; A) are controllable and observable, respectvely. In Fg., we use two swtches T and T2 to denote the states of the forward channel and the backward channel, respectvely. For example, when T s closed, the controller output v(k) s successfully transmtted to the actuator, and we have u(k) =v(k); whereas when t s open, the output of the swtch s held at the prevous value and a packet s lost, and we have n ths case u(k) =u(k 0 ). The stuaton s the same for the backward channel, that s, w(k) =y(k) when T2 s closed and w(k) = w(k 0 ) when T2 s open. Denote sc and ca the sensor-to-controller and the controller-to-actuator delay, respectvely; then we further make the followng assumptons for the NCS under consderaton. A) The sensor, the controller, and the actuator have the same samplng perod T ; they are all tme-drven and are synchronzed. A2) sc <Tand ca <T. Under assumpton A2), there s no need to consder the nfluence of the network-nduced delays. At the tme step k, the sensor sends out the measured output y(k) whle the controller sends out the controller output v(k), and they arrve at (or just drop out) the controller and the actuator at tme k + sc and tme k + ca, respectvely. Snce sc <T and ca <T, the controller and the actuator do not take any actons and just wat untl the next samplng nstant begns. Then, at the samplng nstant k+, the controller and the actuator read n w(k) and u(k) to update the dynamcs of the observer and the plant, respectvely, and mmedately after that, the sensor and the controller send out the new measured output y(k+) and the new controller output v(k+), respectvely, to begn a new control task. Defne the estmaton error by e(k) =x(k) 0 ^x(k) and let z(k) =[x T (k) e T (k) u T (k 0 ) w T (k 0 ) ] T () b) Packet dropout only occurs n the backward channel S2 z(k +)=A2z(k);A2 = A+BK 0BK 0 0 LC A0LC 0 0L K 0K 0 0 c) Packet dropout only occurs n the forward channel S3 z(k +)=A3z(k);A3 = A 0 B 0 0 A 0 LC 0 0 d) There are packet dropouts n both the backward channel and the forward channel S z(k +)=Az(k);A = A 0 B 0 LC A 0 LC 0 0L It can be seen from the above analyss that durng each samplng perod there are four dfferent cases that may arse. Now takng all the subsystems nto consderaton, S S can be lumped nto a general framework that s descrbed by the followng dscrete-tme swtched system z(k +)=A (k)z(k) () where (k) 2 M = f; 2; 3; g s a pecewse constant functon, called a swtchng sgnal. Subsystem S2 s such a system that the packets contanng the measured outputs of the plant are always lost. Therefore, t s magnable that S2 s an unstable system f the plant s unstable. Smlarly, subsystems S3 and S are also unstable f the plant s unstable. On the other hand, snce system () s controllable and observable, by properly desgnng the observer-based controller of the form (2) and (3), the closed-loop subsystem S can be a stable one. So, swtched system () contans stable and unstable subsystems, and t s reasonable to conclude that f system () resded n the subsystem S for a long enough tme (whch ndcates that the packet dropouts do not occur so frequently), the closed-loop system () then tends to be stable. The above analyss motvates us to use the ADSs-RCE approach and the average dwell-tme method to study the stablty and stablzaton of the NCS. Wthout causng confuson, we use [0,k) to denote the tme nterval [0,kT ). Let n, = ; 2; 3; ; denote the tmes that the subsystems S are actvated over the nterval [0,k). then we have k = n, and the event rates of those events descrbed by the subsystems S can be defned as r = n =k, = ; 2; 3;. As llustrated n Fg., two data transmssons are actually nvolved to complete a control task at each tme step, that s, n one control task a packet contanng the measured output s transmtted from the sensor to the controller; meanwhle a packet contanng the controller output s transmtted from the controller to the actuator. Therefore, there are n total 2k data transmssons over the tme nterval [0,k), and a packet dropout occurs when such a

3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 9, SEPTEMBER data transmsson fals. So, the packet dropout rate can be defned as follows = n 2 + n 3 +2n = 2k 2 (r 2 + r 3 +2r ) (5) The objectve of ths paper s to desgn an observer-based output feedback controller of the form (2) and (3) such that the closed-loop NCS () s exponentally stable n the presence of a certan amount of packet dropouts n the communcaton networks, and meanwhle to establsh the relaton between the packet dropout rate and the stablty of the closed-loop NCS (). To ths end, we frst ntroduce the defnton for the exponental stablty of the NCS (). Defnton For any gven ntal condtons (k 0 ;) 2< + 2 C n, () s globally exponentally stable f the solutons of () satsfy kx(k)k c 0(k0k ) kx(k 0)k; 8 k k 0 where c>0 s a constant and > s the decay rate. Defnton 2 [] For any swtchng sgnal (k) and any k, let N [0;k) denote the number of swtchng of (k) over the tme nterval [0;k). IfN [0;k) N 0 + k=t a holds for N 0 0 and T a > 0, then T a s called the average dwell tme and N 0 the chatter bound. For smplcty, but wthout loss of generalty, we choose N 0 =0. Remark The actuator and the controller n the NCS shown n Fg. are ntellgent nodes, that s, they are desgned wth certan network protocols. We further assume that there s a memory unt nsde the controller and all past controller outputs are stored n t. At each samplng nstant, the actuator reads n the control nput and then mmedately sends an ACK sgnal to the controller to tell t whether the current control nput s receved or not. The observer-based controller wll then select the correspondng control nput, whch s the same as the actuator actually adopted to update the state of the observer. In the case that such a controller wth a memory unt s not avalable, the control nputs used to update the states of the observer are not guaranteed to be the same ones the actuator adopts. In such a case, an alternatve s to desgn a dynamc output feedback controller of the followng form ^x(k +)=F^x(k) +Gw(k) v(k) =H ^x(k) where F, G, and H are controller parameter matrces wth approprate dmensons to be determned, and some results smlar to those proposed n ths paper can be obtaned by usng such a controller. Remark 2 For smplcty, the plant consdered n ths paper s descrbed by a dscrete-tme system model, and the closed-loop NCS s fnally modeled as a dscrete-tme swtched system wth four subsystems. A relatvely more complex case s to consder a contnuous-tme system model. Under the same network envronment and by adoptng the same dscrete-tme observer-based output feedback controller, the correspondng closed-loop NCS can be modeled as a dscrete-tme onestep state-delay swtched system wth eght subsystems by followng a process smlar to that presented n ths secton. The stablty analyss and controller desgn of such an NCS can also be dealt wth by adoptng the processes proposed n the followng sectons. III. EXPONENTIAL STABILITY ANALYSIS Let us frst consder some useful results for the subsystems of system () z(k +)=A z(k); 2 M = f; 2; 3; g () Choose the followng Lyapunov functon for subsystem () V (k) =z T (k)p z(k) (7) An exponental decay or ncrease estmaton of the Lyapunov functon (7), whch plays a key role n the dervaton of the man results, wll frst be presented n the followng lemma. Lemma For gven scalar " > 0, f there exsts symmetrc postve-defnte matrx P such that the followng matrx nequalty holds 2 = 0P " A T P " P A 0P < 0; 2 M = f; 2; 3; g (8) then, along any trajectory of subsystem (), the Lyapunov functon V (k) n (7) has the followng decay or ncrease property V (k) " 02(k0k ) V (k 0 ) (9) where k 0 s the ntal tme step. Proof Defne (k) =" k0k z(k), and by (), we obtan (k +)=" A (k) (0) Choose the Lyapunov functon W (k) = T (k)p (k) for subsystem (0), where P s the soluton of (8). Then the forward dfference for W (k) along any trajectory of subsystem (0) s gven by W (k) =W (k +)0 W (k) = T (k) (k) where = " 2 A T P A 0 P. For nonzero (k), < 0 ndcates that W (k) < 0, whch mples W (k) <W (k 0). Thus, we have V (k) =" 02(k0k ) W (k) <" 02(k0k ) W (k 0) = " 02(k0k ) V (k 0 ) Furthermore, by Schur s complement, t follows that < 0 s equvalent to (8). Ths completes the proof. The suffcent condtons for the exponental stablty of the closedloop NCS () are now presented n the followng theorem. Theorem For gven postve scalars ", r, and >, f there exst symmetrc postve-defnte matrces P, =; 2; 3; such that the followng nequaltes hold " r >> () T a >T 3 a = ln (2 ln ) (2) 2 < 0; =; 2; 3; (3) P P ; 8 ; 2 M () where 2 are gven n (8) and T a s the average dwell tme defned n Defnton 2, then the closed-loop NCS () s globally exponentally stable wth a decay rate, where = 0 ln =(2T a ln ) +. Proof Choose the followng pecewse Lyapunov functon for system () V (k) (k) =z T (k)p (k) z(k) (5) where P, =; 2; 3; are the solutons of () (). For an arbtrary pecewse constant swtchng sgnal (k) and any gven nteger k>0, we let k < < k, denote the swtchng ponts of (k) over the nterval [0,k). By () and the expresson of V (k) (k), we

4 708 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 9, SEPTEMBER 2007 have V V, 8 ; 2 M. Therefore, by Lemma, we obtan by nducton that V (k) (k) " 02(k0k ) V (k ) " 02(k0k ) V (k 0 ) 02(k 0k ) " " 02(k0k ). V (k 0 ) N [0; k) " 02(k0k ) 02(k 0k ) " " 02k (0) V (0)(0) = N [0; k) (" 02 ) n V (0) (0) = N [0; k) = N [0; k) 0n =k22k " V (0) (0) " r 02k V (0) (0) < N [0; k) 02k V (0) (0) () In the last nequalty, condton () s used. By () and the expresson of N [0; k), we can further obtan V (k) (k)< N [0; k) 02k V (0) (0) = N [0; k)(ln =ln ) 02k V (0) (0) = 02k(ln =02T ln ) 02k V (0) (0)=( ) 02k V (0) (0) By (5) and (7), we have (7) kx(k)k 2 V (k) (k)<( ) 02k V (0) (0) ( ) 02k 2kx(0)k 2 (8) and thus kx(k)k 2 = ( ) 0k kx(0)k, where = mn mn(p), 2 2M = max max(p), and mn(p) and max(p) 2M are, respectvely, the mnmum and maxmum egenvalues of P. Besdes, condton (2) and >guarantee that >. Therefore, closed-loop NCS () s globally exponentally stable wth a decay rate accordng to Defnton. Ths completes the proof. Remark 3 By Theorem, we have to know the exact values of the event rates r ( =; 2; 3; ) to decde whether condton () s satsfed or not. However, n most of the cases only the packet dropout rate s known for us. In such case, one may expect to have the exponental stablty condtons dependng on for the closed-loop NCS (), and such condtons can be obtaned by approprately choosng the parameters " n " r. One possble way s to choose " j = p " ", j =2; 3; then condton () s modfed to be " 0 " >>. Remark Snce subsystem S s stable and the subsystems Sm (m = 2; 3; ) are unstable, we have " > and 0 < " m <. Therefore, a larger r and some smaller r m result n a larger, whch ultmately results n a smaller Ta 3. Meanwhle, by (5), some smaller r m result n a smaller packet dropout rate, whch ndcates that the swtched system () does not swtch so frequently, and thus a larger average dwell tme T a can be obtaned. Therefore, condton (2) s easer to satsfy. Furthermore, a larger T a together wth a larger ultmately results n a larger exponental decay rate. In a word, t can be derved by the condtons presented n Theorem that the exponental decay rate ncreases when the packet dropout rate s decreasng, and decreases when s ncreasng. Thus, the relaton between the packet dropout rate and the stablty of the closed-loop NCS () s readly establshed by Theorem. Remark 5 Snce " >, t s possble for condton () to be satsfed. Moreover, for gven r, =; 2; 3;, " r s a monotonc ncreasng functon n ". Therefore, on the condton that the matrx nequaltes (3) and () are feasble, " should be chosen as large as possble to obtan a large, and consequently to obtan a large decay rate. In addton, parameter should be chosen as small as possble to obtan a large and a small Ta 3, and consequently to obtan a large decay rate and meanwhle to make condton (2) easer to satsfy. Nevertheless, a feasble soluton to condtons (), (3), and () does not guarantee that the closed-loop NCS () s exponentally stable. A very small T a wll be obtaned f the NCS s subject to a large amount of packet dropouts. For gven,, and ", the worst case s that T a s too small to satsfy condton (2) and that ths very small T a results n a nonpostve, and ultmately leads to. In ths case, the exponental stablty of the closed-loop NCS s no longer guaranteed. IV. OUTPUT FEEDBACK CONTROLLER DESIGN In ths secton, we wll present a procedure to desgn the observerbased output feedback controllers of the form (2) and (3). It s noted that A ( = ; 2; 3; ) can be wrtten as A = A 0 + B 0HC 0, where A 0 = ;A 02 = A 03 = ;A 0 = H = K 0 0 L C 02 = C 0 = I 0I 0 0 C 0C 0 0I B 0 = B 02 = BT 0 I 0 T 0 I 0 0 B 03 = B 0 = T 0 I 0 0 ;C0 = C03 = I 0I C 0 0 Pre- and postmultplyng 2 n (8) by dagfi; P 0 g, t can be concluded that 2 < 0 are equvalent to the followng matrx nequaltes 0P " A0 T + " C0H T T B0 T 3 0P 0 < 0; ; 2; 3; (9) Introducng the new varables Q, condton (9) can be represented as 0P " A0 T +" C0H T T B0 T < 0; Q =P 0 ;; 2; 3; 3 0Q Now, by usng a cone complementarty problem [8], the observerbased output feedback controllers can be obtaned by solvng the followng nonlnear mnmzaton problem wth LMI constrants subject to (), (2), (), and mnmze Tr( P Q ) 0P " A0 T + " C0H T T B0 T < 0 3 0Q P I 0; ; 2; 3; (20) I Q

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 9, SEPTEMBER The above nonlnear mnmzaton problem can be solved by the followng teratve algorthm. Algorthm Step Choose some small ntal " ( =; 2; 3; ) such that (), (2), (), and (20) are feasble. Set " 0 = ". Step 2 Fnd a feasble soluton set (P 0, Q 0, K 0, L 0, =; 2; 3; )to (), (2), (), and (20). Set k =0. Step 3 Solve the followng LMI problem for varables (P, Q, K, L, =; 2; 3; ) mnmze Tr( (P k Q + Q k P )) subject to (); (2); (); and (20) Fg. 2. State trajectores. Set P k+ = P, Q k+ = Q, K k+ = K, and L k+ = L. Step If condtons 2 < 0, =; 2; 3; are satsfed, then set " 0 = " and return to Step 2 after ncreasng " to some extent. If condtons 2 < 0 are not satsfed wthn a specfed number of teratons, then ext. Otherwse, set k = k +and go to Step 3. V. ILLUSTRATIVE EXAMPLE In ths secton, an example s gven to llustrate the effectveness of the proposed results. Consder the plant descrbed by system model (), where A = ; B = C = (2) Snce A has egenvalues at ,.0000, 2.73, and , the above dscrete-tme system s open-loop unstable. The samplng perod T s chosen to be T = s, and sc < T, ca < Thold. Choosng =2, and by applyng Algorthm, the parameters ", observer gan L, and controller gan K are obtaned as follows after 35 teratons " =; " 2 = " 3 =03; " =030 L = K = Suppose that the events descrbed by subsystem S occurs n tmes over the nterval [0, 50), where n = and n 2 = n 3 = n = 2; then we have r =088 and r 2 = r 3 = r =00. By (5), the packet dropout rate can be calculated as =05(r 2 + r 3 +2r )=8%. Snce n ths case " r = 205, we choose = 2 to satsfy condton (), then we obtan Ta 3 = ln=(2 ln ) = On the other hand, the maxmum number of swtchngs of (k) over the nterval [0, 50) s N m [0; 50) = 2 by the assumpton n 2 = n 3 = n = 2, and thus the mnmal average dwell tme can be calculated as ^T a =50=2 = 7. Therefore, condton (2) s also satsfed by the fact that T a ^T a >Ta 3. The closed-loop NCS wth a packet dropout rate =8%fnally has an exponental decay rate 3 = ^ = 0(ln =2 ^T ln )+ =838. Fg. 3. Dynamcs of estmaton errors. In the smulaton setup, the ntal condton s assumed to be x(0) = [ ] T. We suppose that the subsystems of the closed-loop NCS are actvated n the followng sequence S S S 2 S S S 3 S S S S S S 3 S S S 2 S S S S S In ths case, the exponental decay rate for the closed-loop NCS s = 3 =838. The state trajectores of the closed-loop NCS are shown n Fg. 2, and the dynamcs of the state estmaton errors are shown n Fg. 3. Smulaton results have demonstrated that the NCS wth a packet dropout rate =8%can be effectvely stablzed by the proposed observer-based output feedback controllers. VI. CONCLUSION In ths paper, we have studed the modelng and control problem for a class of NCSs under packet dropout effect. The networks n the NCSs were consdered to be n both the forward channel and the backward channel. The tme-drven observer-based output feedback controller was consdered, and the resultng closed-loop NCS was modeled as a dscrete-tme swtched system. A new method that combnes the ADSs-RCE approach and the average dwell-tme method was proposed to study the stablty of the obtaned closed-loop NCS. Suffcent condtons were presented for the exponental stablty of the closed-loop NCS, and the desgn procedures were also gven for the desred controllers. It s worth pontng out that ths newly proposed method enabled us to explctly establsh the relaton between the packet dropout rate and the stablty of the closed-loop NCS. The NCS s desgners can 8

6 70 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 52, NO. 9, SEPTEMBER 2007 then beneft from our results to desgn an applcable schedulng polcy for the fast-samplng NCSs, whch are unschedulable when every data packet s guaranteed to be delvered. Propertes of Stablzng PID Gan Set n Parameter Space Masam Saek, Member, IEEE REFERENCES [] F. L. Lan, Analyss, desgn, modelng and control of networked control systems, Ph.D. dssertaton, Unv. of Mchgan, Ann Arbor, 200. [2] L. Zhang, Access schedulng and controller desgn n networked control systems, Ph.D. dssertaton, Unv. of Maryland, College Park, [3] T. C. Yang, Networked control systems A bref survey, Proc. Inst. Elect. Eng. Contr. Theory Appl., vol. 53, no., pp. 03 2, 200. [] H. Ln and P. J. Antsakls, Stablty and persstent dsturbance attenuaton propertes for a class of networked control systems Swtched system approach, Int. J. Contr., vol. 78, no. 8, pp. 7 58, [5] W. Zhang, Stablty analyss of networked control systems, Ph.D. dssertaton, Case Western Reserve Unv., Cleveland, OH, 200. [] P. J. Antsakls and J. Balleul, Specal ssue on networked control systems, IEEE Trans. Autom. Control, vol. 9, no. 9, pp , 200. [7] G. Denel and M. T. Dawn, Packet-based control The H -optmal soluton, Automatca, vol. 2, no., pp. 37, 200. [8] M. Yu, L. Wang, and T. Chu, Sampled-data stablzaton of networked control systems wth nonlnearty, Proc. Inst. Elect. Eng. Contr. Theory Appl., vol. 52, no., pp. 09, [9] S. S. Hu and Q. X. Zhu, Stochastc optmal control and analyss of stablty of networked control systems wth long delay, Automatca, vol. 39, no., pp , [0] L. Q. Zhang, Y. Sh, T. W. Chen, and B. Huang, A new method for stablzaton of networked control systems wth random delays, IEEE Trans. Autom. Control, vol. 50, no. 8, pp. 77 8, [] F. W. Yang, Z. D. Wang, Y. S. Huang, and M. Gan, Control for networked systems wth random communcaton delays, IEEE Trans. Autom. Control, vol. 5, no. 3, pp , 200. [2] G. P. Lu, J. X. Mu, D. Rees, and S. C. Cha, Desgn and stablty of networked control systems wth random communcaton tme delay usng the modfed MPC, Int. J. Contr., vol. 79, no., pp , 200. [3] Q. Lng and M. D. Lemmon, Robust performance of soft real-tme networked control systems wth data dropouts, n Proc. st IEEE Conf. Decson Contr., Las Vegas, NV, Dec. 2002, pp [] M. Yu, L. Wang, G. M. Xe, and T. G. Chu, Stablzaton of networked control systems wth data packet dropout va swtched system approach, n Proc. IEEE Int. Symp. Computer Aded Contr. Syst. Desgn, Tape, Tawan, R.O.C., Sep. 200, pp [5] J. L. Xong and J. Lam, Stablzaton of lnear systems over networks wth bounded packet loss, Automatca, vol. 3, no., pp , [] G. S. Zha, B. Hu, K. Yasuda, and A. Mchel, Qualtatve analyss of dscrete-tme swtched systems, n Proc. Amer. Contr. Conf., Anchorage, AK, May 2002, pp [7] D. Lberzon, Swtchng n Systems and Control. Boston, MA Brkhauser, [8] L. E. Ghaou, F. Oustry, and M. AtRam, A cone complementarty lnearzaton algorthm for statc output-feedback and related problems, IEEE Trans. Autom. Control, vol. 2, no. 8, pp. 7 7, 997. Abstract The propertes of the PID gan set for whch all the closed-loop poles le n the left half of the complex plane are examned on the parameter plane. A new method of determnng the ncrease n the number of unstable poles when the gan crosses the boundary of stablty regons s proposed. Ths method s useful for determnng the number of unstable poles n each regon. These results are extended to 0-stablty based on conformal mappng. Index Terms Parameter space, PID control, stablty. I. INTRODUCTION In the last few years, fndng ways of descrbng all stablzng PID controllers has captured the attenton of several researchers. Snce the PID controller has just three tunng parameters, the parameter space approach s sutable for the control desgn [], [2]. In ths paper, we are nterested n the propertes of the set of stablzng PID gans on the three planes the PI, PD, and ID planes. The problem of drawng the set can be treated n the next two steps. Frst, the lnes that contan the boundary of the stablzng gan set are obtaned, and next the boundary of the stablzng gan sets s chosen based on the lnes. In [3], a method of drawng the lnes for absolute and relatve stablty s gven for the PI and PD planes based on the method of D-partton. In [] and [5], a method of computng the lnes s gven by extendng the Hermte Behler theorem for the ID plane, and t s shown that the stablzng set s composed of convex polygons on the ID plane. In [], a method of computng the lnes on the ID plane s gven by usng the boundary crossng theorem. Though the stablty of each regon dvded by the lnes can be determned by checkng the stablty of a pont chosen from the regon, more systematc methods have been studed. In [] and [5], such a method s gven for the ID plane, but t requres much computaton. A method of determnng the ncrease n the number of unstable poles when the gan moves across the lne s gven for the ID plane n [7]. However, ths number of ncrease of unstable poles has not been studed for the PI and PD planes, and t has not been extended to 0-stablty. There are other related problems and results. The effect of the two tme-delays on the stablty regon s treated nstead of the PID gans n [8]. The maxmum allowable tme-delay s gven analytcally n [9]. A method of estmatng the stablzng parameter ranges s developed for the multple-nput multple-output PID controllers n [0]. In ths paper, frst, prevous results regardng methods of drawng the lnes are summarzed. Next, the change of magnary axs poles caused by the PID gan perturbaton s estmated, and a few nterestng propertes concernng the ncrease n the number of unstable poles across the lnes are shown for each plane. Lastly, these results are extended to 0-stablty /$ IEEE Manuscrpt receved May 7, 200; revsed October 8, 200 and March 23, Recommended by Assocate Edtor X. Xe. The author s wth the Department of Mechancal System Engneerng, Hroshma Unversty, Hgash-Hroshma , Japan (e-mal saek@hroshma-u.ac.jp). Color versons of one or more of the fgures n ths paper are avalable onlne at http//eeexplore.eee.org. Dgtal Object Identfer 0.09/TAC