Output Feedback Stabilization of Networked Control Systems With Random Delays Modeled by Markov Chains

Transcription

1 668 IEEE RANSACIONS ON AUOMAIC CONROL, VOL 54, NO 7, JULY 29 Output Feeback Stabilization of Networke Control Systems With Ranom Delays Moele by Markov Chains Yang Shi, Member, IEEE, an Bo Yu, Stuent Member, IEEE Abstract his note investigates the output feeback stabilization of networke control systems (NCSs) he sensor-to-controller (S-C) an controller-to-actuator (C-A) ranom network-inuceelays are moele as Markov chains he focus is on the esign of a two-moe-epenent controller that epens on not only the current S-C elay but also the most recent available C-A elay at the controller noe he resulting close-loop system is transforme to a special iscrete-time jump linear system hen, the sufficient an necessary conitions for the stochastic stability are establishe Further, the output feeback controller is esigne via the iterative linear matrix inequality (LMI) approach Simulation examples illustrate the effectiveness of the propose metho Inex erms Jump linear system, Markov chains, networke control systems, packet ropout, ranom elays, stochastic stability I INRODUCION Networke control systems (NCSs) are a type of istribute control systems, where the information of control system components (reference input, plant output, control input, etc) is exchange via communication networks NCSs have many attractive avantages, such as reuce system wiring, low weight an space, ease of system iagnosis an maintenance, an increase system agility, which motivate the research in NCSs On the other han, the introuction of networks also presents some constraints such as time elays an packet ropouts which bring ifficulties for analysis anesign of NCSs he stuy of NCSs has been an active research area in the past several years, see [] [8], to name a few One of the constraints is the network-inuce time elays, which can egrae the performance or even cause instability Various methoologies have been propose for moeling, stability analysis, an controller esign for NCSs in the presence of network-inuce time elays an packet ropouts Generally, existing results can be classifie into two main categories: ) to esign a controller first, an then etermine the network conitions such as the maximum allowable transfer interval to guarantee the stability an maintain certain performance [2], [9], []; 2) to explicitly incorporate the network-inuceelays using certain moels, eg, Markov process, into the controller esign [], [] [7] he methoevelope in this note belongs to the latter he Markov chain, a iscrete-time stochastic process with the Markov property, can be effectively use to moel the network-inuce elays in NCSs In [], the time elays of NCSs are moele by using the Markov chains, an further an LQG optimal controller esign metho is propose Xiao et al [] propose two types of controller esign methos for NCSs moele as finite-imensional, iscrete-time jump linear systems: One is the state feeback controller that only epens on the elays from sensor to controller (S-C elays), Manuscript receive December 8, 27; revise July 29, 28, February 6, 29, an March 3, 29 First publishe June 23, 29; current version publishe July 9, 29 his work was supporte by the Natural Sciences an Engineering Research Council of Canaa Recommene by Associate Eitor S Mascolo he authors are with the Department of Mechanical Engineering, University of Saskatchewan, Saskatoon, SK S7N 5A9, Canaa ( yangshi@usaskca; yangshi@ieeeorg; boyu@usaskca) Color versions of one or more of the figures in this technical note are available online at Digital Object Ientifier 9/AC Fig Diagram of a networke control system an is calle the one-moe-epenent controller; the other is the output feeback controller that oes not epen on either the S-C elays or the C-A elays (elays from controller to actuator), an calle the moe-inepenent controller In [2], H control for NCSs is esigne uner the framework of Markovian jump linear systems (MJLSs) [8] [2] base on the Boune Real Lemma, but only the S-C elays are consiere In [5], a moe-inepenent state feeback controller is esigne for NCSs subject to Markovian packet loss In [6], the Markov chain is employe to moel the packet loss an a one-moe-epenent state feeback controller is evelope by introucing buffers to eal with the C-A packet loss In all the aforementione references, the evelope controllers are either moe-inepenent or one-moe-epenent, an the esign problem can thus be reaily converte into a stanar MJLS problem [8] [2] o reuce the conservativeness of the stabilization conitions of an NCS, it is esirable to incorporate not only the S-C elay but also the C-A elay into the esign However, involving the C-A elay is complicate an challenging because the controller an actuator noes are istributively locate Smart sensor technology [6], by aing a cost-effective embee processor to the actuator noe (Fig ), can process an calculate the C-A elay in real-time at the actuator noe, an sen this information to the controller noe via the S-C communication meium It is worthwhile noting that the transmission of the C-A elay information will also suffer from the S-C elay Zhang et al [3] propose a promising two-moe-epenent state feeback scheme to stabilize NCSs with S-C an C-A elays moele as two Markov chains In [3], it is assume that at each sampling instant, the current S-C elay ( k ) an previous C-A elay ( k) can be obtaine by the time-stamping technique However, practically the previous C-A elay ( k) is not always available because the information about C-A elays nees to be transmitte through the S-C communication link before reaching the controller, as shown in Fig In aition, when the full state information is not available, the state feeback controller in [3] cannot be irectly applie o the best of authors knowlege, involving two network-inuceelay moes to esign the controller that simultaneously epens on both k an k has not been fully investigate, which is the focus of this work When consiering both k an k, the resulting close-loop system cannot be transforme to a stanar MJLS, an thus the well-evelope results on MJLSs [8] [2] cannot be irectly applie he rest of the note is organize as follows In Section II, we analyze the available elay information an formulate the output feeback controller esign problem In Section III, the sufficient an necessary conitions to guarantee the stochastic stability are presente first an the /$25 29 IEEE

2 IEEE RANSACIONS ON AUOMAIC CONROL, VOL 54, NO 7, JULY equivalent LMI conitions with constraints are erive Simulation examples are given to illustrate the effectiveness of the propose metho in Section IV he conclusion remarks are aresse in Section V C Close-Loop System Consiering the time elay in the S-C link, we have II PROBLEM FORMULAION Consier the NCS setup in Fig he iscrete-time linear time-invariant plant moel is x(k +)Ax(k) +B ~u(k) y(k) Cx(k) (a) (b) where x(k) 2 n, ~u(k) 2 m, y(k) 2 p, an A, B, an C are known real matrices with appropriate imensions Boune ranom elays exist in both links from sensor to controller an controller to actuator, as shown in Fig Here, k represents the S-C elay an k stans for the C-A elay he output feeback controller is to be esigne A Delays Moele by Markov Chains One way to moel the elays k an k is to use the finite state Markov chain as in [], [], [3] he main avantages of the Markov moel are: ) the epenencies between elays are taken into account since in real networks the current time elays are usually relate with the previous elays [] an 2) the packet ropout coul be inclue naturally [] In this note, k an k are moele as two homogeneous Markov chains that take values in M f; ; ;g an N f; ; ;g, an their transition probability matrices are 3[ ij ] an 5[ rs ], respectively hat means k an k jump from moe i to j an from moe r to s, respectively, with probabilities ij an rs: ij Pr( k+ jj k i); rs Pr( k+ sj k r) x(k +)Ax(k) +B ~u(k); (3a) ~y(k) y(k k )Cx(k k ): Augment the plant s state an output as ~X(k) x(k) y(k ) y(k 2) y(k ) (3b) then we have ~X(k +) A ~ X(k) ~ + B ~ ~u(k) (4a) ~y(k) C( ~ k ) X(k) ~ (4b) where ~A ~C( k ) Similarly, efine then A C I ; B ~ I B [C ]; for (k) [ I ] (+ )th block being ientity ; for (k) > ~Z(k) z(k) u(k ) u(k 2) u(k ) where ij ; rs an ij ; j for all i; j 2Man r; s 2N s rs where ~Z(k +) F ~ ( k ; k ) Z(k) ~ + G ~ ( k ; k ) ~y(k) (5a) ~u(k) H ~ ( k ; k ; k ) Z(k) ~ + J ~ ( k ; k ; k ) ~y(k) (5b) B wo-moe-depenent Output Feeback Controller In NCSs, the elay information is important for the controller esign For the controller noe, at time instant k, k can be obtaine by comparing the current time an the time-stamp of the sensor information receive Similarly, at the actuator noe, the embee processor can compare the current time with the time-stamp of the control signal receive to calculate the C-A elay information k at current time k However, this information cannot be receive by the controller immeiately, because it nees to be transmitte through the network from sensor to controller So if the time elay k exists, the information of k at time instant k woul be known at the controller noe Specially, if k, k is available at the controller noe Consequently, it is esirable to esign the output feeback controller that simultaneously epens on both k an k he ynamic output feeback control law is z(k +)F ( k ; k ) z(k)+g ( k ; k ) ~y(k) u(k)h ( k ; k ) z(k)+j ( k ; k ) ~y(k) (2a) (2b) where z(k) 2 n is the state vector of the output feeback controller; an F, G, H, an J are appropriately imensione matrices to be esigne Clearly, the controller in (2) is two-moe-epenent ~F ( k ; k ) F ( k ; k ) H ( k ; k ) ~G ( k ; k ) I I G ( k ; k ) J ( k ; k ) ~H ( k ; k ; k ) [H ( k ; k ) ] ; for (k) [ I ] (+ )th block being ientity ~J ( k ; k ; k ) J ( k; k ) ; for (k) ; for (k) > ; for (k) >,

3 67 IEEE RANSACIONS ON AUOMAIC CONROL, VOL 54, NO 7, JULY 29 By combining (4) an (5) anefining X(k) ~ X(k) ~ Z(k) the close-loop system can be written as where X(k +) A + BK ( k ; k ; k ) C( k ) X(k) (6) A ~A ; B ~ B I ; C(k ) I ~C( k ) K ( k ; k ; k ) ~F ( k ; k ) G ~ (k ; k ) ~H ( k ; k ; k ) J ~ (k ; k ; k ) Remark : By applying the propose two-moe-epenent controller in (2), the resulting close-loop system in (6) cannot be transforme to a stanar MJLS, because the close-loop system epens on k, k, an k In aition, k is relate with both k an k his makes the analysis anesign more challenging, an this two-moe-epenent controller esign has not been investigate in the literature he objective of this note is: Design the output feeback controller to guarantee the stochastic stability of the NCS in (6) For the stochastic stability, we aopt the efinition in [3] Definition : he system in (6) is stochastically stable if for every finite X X(), initial moe () 2M, an ( ) 2N, there exists a finite W > such that the following hols: E k kx(k)k 2 j X ; ; <X WX : (7) III MAIN RESULS In this section, we first give the sufficient an necessary conitions for the output feeback stabilization of system in (6), an then erive the equivalent conitions of LMIs with nonconvex constraints As k is to be incorporate into the controller esign, the multi-step jump of Markov chains is involve in the system evolvement hen, a natural question is: What is the transition probability matrix for the multi-step elay moe jump? his is of great importance for the erivation of sufficient an necessary conitions for esigning the output feeback controller later soon o answer this, we give Proposition as follows Proposition : If the transition probability matrix from k to k is 5, then the transition probability matrix from k to k is 5, which is still a transition probability matrix of the Markov chain Specially, when k+, the transition probability matrix is 5 5 I Proof: he transition probability matrix from k to k is in the equation, shown at the bottom of the page A special case is that when k+, then 5 5 I his completes the proof : he sufficient an necessary conitions to guarantee the stochastic stability of system in (6) can be erive with Definition, which are shown in heorem For the ease of presentation, when the system is in moe i 2M an r 2N (ie, k i, k r), we enote F ( k ; k ), G( k ; k ), H( k ; k ), an J( k ; k ) as F (i; r), G(i; r), H(i; r), an J(i; r), respectively heorem : Uner the propose output feeback control law (2), the resulting close-loop system in (6) is stochastically stable if an only if there exists symmetric P (i; r) > such that the following matrix inequality: L(i; r) ij 5 +ij j s s 2 A + BK(i; s ;r) C(i) P (j; s 2 ) 2 A + BK(i; s ;r) C(i) P (i; r) < (8) hols for all i 2Man r 2N Proof: Sufficiency: For the close-loop system in (6), construct the Lyapunov function hen V (X(k);k)X(k) P ( k ; k ) X(k): Ef(V (X(k);k))g EfV (X(k +);k+)v (X(k);k)g E X(k +) P k+ ; k X(k +) j X ; i; r X(k) P ( k ; k ) X(k): (9) Define k+ j, k s, k s 2 o evaluate the first term in (9), we nee to apply the probability transition matrices for k! k+, ki! kj, an kj! k, respectively Accoring to Proposition, these three probability transition matrices are k! k+ :3; ki! kj :5 +ij ; kj! k :5 j : hen, (9) can be evaluate as Ef(V (X(k);k))g X(k) hus, if L(i; r) <, then j s s ij 5 +ij 2 A + BK(i; s ;r) C(i) P (j; s 2) 2 A + BK(i; s ;r) C(i) P (i; r) X(k): Ef(V (X(k);k))g X(k) L(i; r)x(k) min (L(i; r)) X(k) X(k) () kx(k)k 2 () j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j 5 :

4 IEEE RANSACIONS ON AUOMAIC CONROL, VOL 54, NO 7, JULY where inff min(l(i; r))g > From (), we can see that for any EfV (X( +); +)gefv (X ; )g Furthermore E t kx(t)k 2 (EfV (X ; g E t E fv (X( +); +)g) EfV (X; )g X() P ( ; ) X(): kx(t)k 2 : Accoring to Definition, the close-loop system in (6) is stochastically stable Necessity: For necessity, we nee to show that if the system in (6) is stochastically stable, then there exists symmetric P (i; r) > such that (8) hols It suffices to prove that for any boune an symmetric Q( k ; k ) >, there exists a set of P ( k ; k ) such that ij5 +ij A + BK(i; s ;r) C(i) 2 j s s P (j; s 2 ) A + BK(i; s ;r) C(i) P (i; r) Q(i; r): Define X(t) ~ P ( t; t; t ) X(t) E kt X(k) Q ( k ; k ) X(k)j X ; ; : Assuming that X(k) 6, since Q( k ; k ) >, as increases, X(t) ~ P ( t; t; t )X(t) is monotonically increasing, or else it increases monotonically until EfX(k) Q( k ; k )X(k)j X ; ; g for all t From (7), X(t) P ~ ( t; t ; t )X(t) is upper k k boune Furthermore, its limit exists an can be expresse as X(t) P (i; r)x(t) lim! X(t) ~ P ( t; t i; t r) X(t) lim! E kt Since this is vali for any X(t),wehave P (i; r) X(k) Q ( k ; k ) X(k) j X ; i; r : (2) lim ~P ( t; t i; t r) : (3)! From (2), we obtain P (i; r) > since Q( k ; k ) > Consier E X(t) ~ P ( t; t; t ) X(t) X(t +) ~ P 2 t; t+; t X(t+)j X ; i; r X(t) Q(i; r)x(t): (4) By using Proposition, the secon term in (4) can be evaluate as E X(t +) ~ P t ; t+ ; t X(t +) j X ; ; X(t) ij 5 +ij j s s 2 A+ BK(i; s ;r) C(i) ~ P ( t;j;s 2 ) 2 A + BK(i; s ;r) C(i) X(t): (5) Substituting (5) into (4) gives rise to X(t) P ~ ( t; t ; t ) ij 5 +ij j s s 2 A + BK(i; s ;r) C(i) ~ P ( t ;j;s 2 ) 2 A + BK(i; s ;r) C(i) 2 X(t) X(t) Q(i; r)x(t): (6) Letting! an noticing (3), it is shown that (8) hols his completes the proof heorem gives the sufficient an necessary conitions on the existence of the output feeback controller However, the conitions in (8) are nonlinear in the controller matrices o hanle this, the equivalent LMI conitions with nonconvex constraints are given in Proposition 2 Proposition 2: here exists a controller (2) such that the close-loop system in (6) is stochastically stable if an only if there exist matrices F (i; r), G(i; r), H(i; r), J(i; r), an symmetric matrices X(j; s 2 ) >, P (i; r) >, satisfying for all i; j 2Man r; s 2 2N, with P (i; r) V (i; r) < (7a) V (i; r) X(i; r) X(j; s 2)P (j; s 2)I (7b) V (i; r) V (i; r) V (i; r) ; V (i; r) V j (i; r) V j; (i; r) V j; (i; r) ; V j; (i; r) V j;s (i; r) ij 5 +ij rs 5 j s ij 5 +ij rs 5 j s ij 5 +ij rs 5 j s X(i; r)iagfx (i; r) X (i; r) X j(i; r)iagfx j;(i; r) X j;(i; r) A+ BK(i; ;r) C(i) A+ BK(i; ;r) C(i) A+ BK(i; ; r) C(i) X (i; r)g X j; (i; r)g X j;s (i; r)iag X(j; s2) X(j; s 2) X(j; s2) + Proof: By applying the Schur complement an letting X(j; s 2 )P (j; s 2 ), the proof can be reaily complete he conitions in Proposition 2 are a set of LMIs with nonconvex constraints his can be solve by several existing iterative LMI algorithms It was shown in [22] that the prouct reuction algorithm (PRA) is the best an selom fails to fin a global optimum hus, PRA [23] is employe to solve the conitions (7) :

5 672 IEEE RANSACIONS ON AUOMAIC CONROL, VOL 54, NO 7, JULY 29 Fig 3 S-C ranom elays Fig 2 Cart an inverte penulum system Remark 2: he two-moe-epenent controller (2) makes full use of the elay information by involving both the S-C an C-A elays Moreover, it inclues the one-moe-epenent an moe-inepenent controllers as special cases When F ( k ; k ) F ( k ), G( k ; k ) G ( k ), H( k ; k ) H ( k ), J( k ; k ) J ( k )8 k 2 N, the controller (2) is reuce to be one-moe-epenent When F ( k ; k ) F, G( k ; k ) G, H( k ; k ) H, J( k ; k ) J 8 k 2 M, an k 2 N, the controller (2) becomes a moe-inepenent one heorem an Proposition 2 can also hanle the one-moe-epenent an moe-inepenent controller esign problems as special cases Fig 4 C-A ranom elays IV NUMERICAL EXAMPLE o illustrate the effectiveness of the propose metho, we apply the results in Section III to a cart an inverte penulum system [], [3] shown in Fig 2, where x is the position of the cart, is the angular position of the penulum, an u is the input force he state variables are chosen as [x x_ ] _ he output is y [x ] We assume that the surface is frictionless an the system parameters are: m kg, m 2 :5 kg, L m he output feeback controller is esigne for the following linearizeiscrete-time moel with sampling time s :s: A B : : :66 :5 : :3374 :66 :996 :33 2:247 :996 :45 :896 :68 :377 ; C : he eigenvalues of A are,,5569, an 6423 Hence, the iscretetime system is unstable he ranom elays involve in this NCS are assume to be k 2 f; ; 2g an k 2f; g, an their transition probability matrices are given by 3 :6 :4 :5 :4 : :5 :4 : ; 5 :4 :6 :5 :5 : Figs 3 an 4 show part of the simulation run of the S-C elays k an C-A elays k governe by their corresponing transition probability matrices, respectively Base on Proposition, the transition probability matrix for the elay moe jumping from k2 to k is 5 2 By using Proposition 2, we esign the two-moe-epenent output feeback controller with the following matrices: F ( k ; k ), G( k ; k ), H( k ; k ), an J( k ; k ) F (; ) G(; ) :294 :57 :2873 :678 :454 :453 :663 :8348 :3473 :2992 :7629 :335 :2954 :59 :94 :689 :266 2:59 :64 6:9785 :25 :22 :46 2:7688 H(; ) [:866 :426 :488 4:6258] J(; ) [:355 34:4373] F (; ) G(; ) :28 :4 :299 :648 :628 :62 :69 :8579 :333 :28 :767 :79 :3 :56 :26 :693 :273 2:5833 :82 7:448 :3 :825 :32 2:837 H(; ) [:6793 :5327 :954 4:849]

6 IEEE RANSACIONS ON AUOMAIC CONROL, VOL 54, NO 7, JULY Fig 5 Response of x Fig 7 Response of Fig 6 Response of x _ Fig 8 Response of _ J(; ) [:346 34:8353] :433 :9 :282 :2 :39 :432 :243 :9575 F (; ) :9 :295 :827 :88 :273 :59 :746 :658 :268 2:6224 G(; ) :277 6:7524 :865 :867 :93 2:7467 H(; ) [:2472 :582 :696 5:2399] J(; ) [:362 34:695] :444 :8 :279 :22 F (; ) :554 :654 :259 :9732 :373 :225 :82 :694 :24 :286 2:4783 :54 :84 :6543 G(; ) :293 6:8387 :95 :5 :7 2:879 H(; ) [:836 :579 :49 5:2547] J(; ) [: :938] :435 :2 :3265 :969 :6 :434 :99 :9299 F (2; ) :7 :798 :7885 :62 :2832 :532 :594 :6238 G(2; ) :44 2:686 :55 6:564 :98 :39 :57 2:6733 H(2; ) [:35 :5662 : 5:2564] J(2; ) [: :7758] F (2; ) G(2; ) :572 :423 :2994 :254 :345 :372 :99 :9492 :24 :588 :85 :574 :259 :5249 :479 :675 :37 2:655 :49 6:5676 :99 :45 :83 2:7559 H(2; ) [:887 :529 :882 5:969] J(2; ) [: :656]: he initial values of the iscrete-time moel an the output feeback controller are x(3) x(2) x() [ ], x() [ : ], an z(2) z() z() [ ] For the purpose of comparison, the moe-inepenent output feeback controller [] is applie to the same system Figs 5 8 illustrate the responses of four states using the propose two-moe-epenent controller an the moe-inepenent controller, respectively It is observe that the

7 674 IEEE RANSACIONS ON AUOMAIC CONROL, VOL 54, NO 7, JULY 29 propose two-moe-epenent controller outperforms the moe-inepenent one [] V CONCLUSION his note proposes an output feeback controller esign metho for NCSs with ranom network-inuceelays he S-C an C-A elays, moele by two Markov chains, are simultaneously incorporate into the controller esign in a general an practical way hen the resulting close-loop system is a special iscrete-time jump linear systems he sufficient an necessary conitions of the stochastic stability are erive in the form of a set of LMIs with nonconvex constraints he prouct reuction algorithm is employe to obtain the two-moe-epenent output feeback controller Simulation examples verify its effectiveness It is worth mentioning that the propose two-moe-epenent controller can be extene to consier the control performance an system uncertainties ACKNOWLEDGMEN he authors wish to thank the anonymous reviewers an the associate eitor for proviing constructive suggestions which have improve the presentation of the technical note [6] J Wu an Chen, Design of networke control systems with packet ropouts, IEEE rans Automat Control, vol 52, no 7, pp 34 39, Jul 27 [7] D Huang an S K Nguang, State feeback control of uncertain networke control systems with ranom time-elays, IEEE rans Automat Control, vol 53, no 3, pp , Apr 28 [8] Y Ji an H J Chizeck, Controllability, stabilizability, an continuous-time Markovian jump linear quaratic control, IEEE rans Automat Control, vol 35, no 7, pp , Jul 99 [9] P Shi an E-K Boukas, H control for Markovian jumping linear systems with parametric uncertainties, J Optimiz heory an Applic, vol 95, no, pp 75 99, 997 [2] P Shi, E-K Boukas, an R K Agarwal, Control of Markovian jump iscrete-time systems with norm boune uncertainty an unknown elay, IEEE rans Automat Control, vol 44, no, pp , Nov 999 [2] O L V Costa an R P Marques, Robust H -control for iscretetime markovian jump linear systems, Int J Control, vol 73, no, pp 2, Jan 2 [22] M C e Oliveira an J C Geromel, Numerical comparison of output feeback esign methos, in Proc American Control Conf, Albuquerque, NM, Jun 997, vol, pp [23] L Zhang, B Huang, an J Lam, H moel reuction of markovian jump linear systems, Syst Control Lett, vol 5, no 2, pp 3 8, Oct 23 REFERENCES [] J Nilsson, Real-ime Control Systems With Delays, PhD issertation, Lun Institute of echnology, Lun, Sween, 998 [2] W Zhang, M S Branicky, an S M Phillips, Stability of networke control systems, IEEE Control Syst Mag, vol 2, no, pp 84 99, Feb 2 [3] Y Shi, H Fang, an M Yan, Kalman filter base aaptive control for networke systems with unknown parameters an ranomly missing outputs, Int J Robust Nonlin Control (Special Issue on Control With Limite Information), 29, 2/rnc39 [4] G C Walsh, H Ye, an L G Bushnell, Stability analysis of networke control systems, IEEE rans Control Syst echnol, vol, no 3, pp , May 22 [5] Y ipsuwan an M-Y Chow, Control methoologies in networke control systems, Control Eng Practice, vol, no, pp 99, Oct 23 [6] D Hristu-Varsakelis an W S Levine, Hanbook of Networke an Embee Control Systems Cambrige, MA: Birkhäuser, 25 [7] Z Mao an B Jiang, Fault ientification an fault-tolerant control for a class of networke control systems, Int J Innov Comput, Inform, Control, vol 3, no 5, pp 2 3, 27 [8] Y Wang an Z Sun, H control of networke control system via LMI approach, Int J Innovative Comput, Inform, Control, vol 3, no 2, pp , 27 [9] D Yue, Q L Han, an C Peng, State feeback controller esign of networke control systems, IEEE rans Circuits Syst II, vol 5, no, pp , Nov 24 [] D Carnevale, A R eel, an D Nešić, A lyapunov proof of an improve maximum allowable transfer interval for networke control system, IEEE rans Automat Control, vol 52, no 5, pp , May 27 [] L Xiao, A Hassibi, an J P How, Control with ranom communication elays via a iscrete-time jump system approach, in Proc Amer Control Conf, Chicago, IL, Jun 2, vol 3, pp [2] P Seiler an R Sengupta, An H approach to networke control, IEEE rans Automat Control, vol 5, no 3, pp , Mar 25 [3] L Zhang, Y Shi, Chen, an B Huang, A new metho for stabilization of networke control systems with ranom elays, IEEE rans Automat Control, vol 5, no 8, pp 77 8, Aug 25 [4] F Yang, Z Wang, Y S Hung, an M Gani, H control for networke systems with ranom communication elays, IEEE rans Automat Control, vol 5, no 3, pp 5 58, Mar 26 [5] J Xiong an J Lam, Stabilization of linear systems over networks with boune packet loss, Automatica, vol 43, no, pp 8 87, Jan 27 Coorinating Batch Prouction an Pricing of a Make-to-Stock Prouct Liuxin Chen, Youyi Feng, an Jihong Ou Abstract Consier a make-to-stock prouct that is prouce in batches an sol in iniviual units he prouction process is stochastic with its mean time controllable in a fixe range; an the prouct is sol at either a high price with a low eman or a low price with a high eman Coorinating the ynamic ajustment of the prouction rate an the sale price is crucial for maximizing the total iscounte profit We erive in this note that, the optimal control of the prouction rate follows a critical stock policy an the optimal pricing follows a price-switch threshol policy, with both associate with the finishe goos inventory Inex erms Batch prouction, critical stock policy, ynamic pricing, make-to-stock, optimal prouction control, price-switch threshol policy I INRODUCION Most manufacturing systems make proucts in batches so as to more efficiently utilize machinery an labor resources, especially for the make-to-stock proucts In this note we consier a simple moel of a batch-prouction, make-to-stock manufacturing system he batch sizes are constant he prouction process is stochastic with its mean time controllable in a fixe range he prouct is sol at either a high price with a low eman or a low price with a high eman he costs Manuscript receive February 8, 28; revise February 9, 28, August 24, 28, December 2, 28, an March 6, 29 Current version publishe July 9, 29 his work was supporte by the Research Grants Council of the Hong Kong Special Aministrative Region (Project CUHK4235/3E) Recommene by Associate Eitor X Xie L Chen an Y Feng are with the Department of Management Sciences, City University of Hong Kong, Kowloon, Hong Kong ( linawuge@hotmail com; yyfeng@livecom) J Ou is with the Cheung Kong Grauate School of Business, Beijing, China ( jhou@ckgsbeucn) Digital Object Ientifier 9/AC /$25 29 IEEE