Hybrid State Feedback Controller Design of Networked Switched Control Systems with Packet Dropout

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1 Amercan Control Conference Marrott Waterfront, Baltmore, MD, USA June 3-July, WeB6.6 Hybrd State Feedbac Controller Degn of etwored Swtched Control Sytem wth Pacet Dropout Dan Ma, Georg M. Dmrov and Jun Zhao Abtract It ha become a common practce to employ networ n control ytem for connectng controller and enor/actuator on controlled plant and procee. A networed wtched control ytem, a a pecal cae of networed control ytem, tuded. Such a ytem wth pacet dropout repreented a an aynchronou dynamcal wtched ytem. Both the wtched control ytem tate dropout and the wtchng gnal dropout are condered n the ytem dynamc multaneouly. Suffcent condton for the exponental tablty and control ynthe of networed wtched control ytem at a fxed nown data pacet tranmon rate under arbtrary wtchng equence are derved. Swtched Lyapunov-Kraov technque ued n analyzng the tablty of the ytem accordngly reduce the conervatvene of ung common Lyapunov functon. Fnally, a lower bound on pacet tranmon rate that guarantee the tablty of networed wtched control ytem dcued. I. ITRODUCTIO ETWORKED Control Sytem (CS a an mportant cla of feedbac control ytem, whoe feedbac path are mplemented by a real-tme operatonal networ, have drawn conderable attenton n the control and computer communty n the lat decade. Th type of ytem allow for reduced wrng a well a for lower ntallaton cot. It alo permt greater aglty n dagno and mantenance and flexblty for ndutral feld applcaton. Example of uch ytem can be found n arcraft and automoble ndutry [-], automated manufacturng ytem [4-5], and tele-robotc [7]. Th new technology ha alo poed a theoretcal control problem of how to degn the controller when control nfratructure networ baed mplementaton. The lmtaton of networ bandwdth and the dfference of varou tranmon protocol have affected the tranmon tmng and accuracy tmely and exactly. In partcular, networ-nduced delay [] and pacet dropout [, 6] often occur. Generally peang, dependng on the networ Th wor wa upported by Mntry of Educaton& Scence of the R.Macedona (grant o 4-354/-7..7, the SFC under Grant 9868, the atonal Potdoctoral Foundaton of Chna (84489 and the ortheatern Unverty Potdoctoral Foundaton of Chna (845. Dan Ma and Jun Zhao are wth Key Laboratory of Integrated Automaton of Proce Indutry, Mntry of Educaton, ortheatern Unverty, Shenyang 4, Chna (e-mal: madan@e.neu.edu.cn; zhaoun@e.neu.edu.cn. Georg M. Dmrov wth Department of Computer Engneerng, Dogu Unverty, Kadoy, TR-347, Itanbul, Turey (e-mal: gdmrov@dogu.edu.tr. protocol employed, pacet dropout reult from tranmon error n phycal networ ln (whch far more common n wrele than n wred networ or from buffer overflow due to congeton, or even from node falure occaonally. In real tme control ytem, networ-nduced delay and pacet dropout degrade the control performance and even can render the overall ytem untable. The tablty and tablzaton of wtched ytem have attracted much attenton n lat decade [4-]. The extence of a common Lyapunov functon for all ubytem wa hown to guarantee a wtched ytem aymptotcally table under an arbtrary wtchng law [5]. A common Lyapunov functon for all ubytem, whch obvouly qualfe a a uual Lyapunov functon, may not ext or be dffcult to fnd. Therefore, the wtched Lyapunov functon technque propoed by Daafouz, Rednger and Iung [4] ha proven to be a powerful and effectve tool to reduce the conervatvene. However, thee mportant method for wtched ytem mut be reevaluated before they become applcable to the networed wtched control ytem. Recently, networed wtched control ytem (SCS, a a pecal cla of networed control ytem, whoe wtched plant wth all enor/actuator and hybrd controller connected by communcaton channel, have attracted reearcher n control theory and control engneerng. For SCS, nce the wtched control ytem tate and the wtchng gnal can be tranmtted n a ngle pacet through networ, they hould be condered multaneouly, thu the reult for general CS can not be appled to SCS drectly. The SCS wth networ-nduced delay n the ytem tate under arbtrary wtchng gnal have been propoed n the control lterature. The control ynthe for contnuou tme wtched ytem wth unnown tme varyng delay ha been nvetgated a a problem of tablzablty for uncertan ytem wth polytopc uncertante n [8]. The networ-nduced delay whch le than a nown contant ha been tuded n []. The robut exponentally tablzng condton for SCS wth networ-nduced delay and pacet dropout n the ytem tate under average dwell tme wtchng gnal have been gven n [3]. The SCS wth both the wtchng gnal and control nput affected by networed nduced delay alo dcued. See, for ntance, [9,, ], and the reference theren. However, no lterature about SCS wth wtchng gnal pacet dropout gven yet. How to analyze the tablty and degn the hybrd controller for the SCS wth pacet dropout, epecally wth wtchng gnal pacet dropout of great mportance. Th motve our //$6. AACC 368

2 preent wor. In th paper, we are ntereted n tablty analy and control ynthe of networed wtched control ytem wth pacet dropout under arbtrary wtchng equence. Frtly, a dcrete-tme wtched control ytem wth pacet dropout formulated. Both the wtched control ytem tate pacet dropout and the wtchng gnal pacet dropout are condered n the ytem dynamc multaneouly. Aynchronou dynamcal wtched ytem theory and wtched Lyapunov-Kraov technque are ued to analyze th type of ytem. A uffcent condton on exponental tablzaton of the SCS wth pacet dropout gven. Then an algorthm for hybrd tate feedbac controller degn gven. Fnally, a lower bound on pacet tranmon rate that guarantee the tablty of networed wtched control ytem dcued. II. PROBLEM FORMULATIO The ytem to be tuded n th paper can be depcted n Fg.. The wtched plant va enor nteract wth the hybrd controller through a networ tranmon channel. Thu, the cloed-loop ytem can be modeled a a dcrete-tme wtched ytem x = A ( x σ Bσ( u ( wth a hybrd tate feedbac controller where u = K x ( ˆ σ ( ˆ x R n the wtched ytem σ = ϒ= the wtchng gnal whch a pecewe contant functon dependng on tme and/or tate x. σ ( = mean that the th ubytem actvated. A and B, ϒ, are contant matrce. K, ϒ, are hybrd tate feedbac controller gan to be degned. ote that for ϒ, σ ( = denote that the th ubytem tate, ( : R {,,,... } {,,..., } x = Aσ( x Bσ( u u K x = ˆ σ ( ˆ xˆ ˆ( σ x σ ( S S actvated at tme, otherwe, σ ( =, we can obtan = σ ( =. Thu the dcrete-tme wtched ytem ( can be re-wrtten a follow x = σ ( ( Ax Bu (3 = A depcted n Fg., we aume that both x and σ ( can be tranmtted n a ngle pacet through the networ. If networ congeton or node falure occur, dependng on the networ protocol employed, pacet dropout happen nevtably. xˆ and ˆ( σ are wtched ytem tate and wtchng gnal receved by the hybrd controller over the aynchronou communcaton networ, repectvely. The networ wth pacet dropout can be modeled a a wtch that cloe at a certan rate r. When the wtch cloed, the networ pacet contanng x and σ ( are tranmtted, wherea when t open, the output of the wtch held at the prevou value and the pacet lot. Defne the tranmon ndcator functon a, ample tranmtted = (4, ample not tranmtted Thu the dynamc of the wtch can be modeled a xˆ = β x ( β xˆ (5 ˆ σ ( = βσ( ( β ˆ σ( (6 where β are wtch varable, β =, β =. Combnng (-(3 gve the cloed-loop networed wtched control ytem wth the hybrd tate feedbac controller can be decrbed by x = σ ( ( Ax B( ˆ σ ( K xˆ = = (7 T T Letς [ ˆ = x x ] be the augmented tate vector, the cloed-loop networed wtched control ytem wth the networ pacet dropout effect repreented by ς ς =Φ (8 When the wtch n poton S, =, β = and Fg.. etwored wtched control ytem wth pacet dropout 369

3 Φ = x ˆ = x, ˆ( σ = σ (, σ ( A σ ( BK ; σ ( A σ ( BK = = = = When the wtch n poton S, =, β = and Φ = xˆ = ˆ, x ˆ σ ( = ˆ σ (, σ( A σ( σl( BK l = = l=. I ote that ytem (8 an aynchronou dynamcal wtched ytem (ADSS, whch ncorporate contnuou and dcrete dynamc. It can be regarded a two wtched control ytem drven aynchronouly by external dcrete event wth fxed rate. For the ae of mplcty we aume that each of wtched control ytem tablzable under an arbtrary wtchng law. In order to analyze and degn ytem (8, we need the followng Lemma. Lemma [] : For the aynchronou dynamcal ytem (ADS x( = f ( x(, =,,, (9 wth contnuou-valued tate x ( R n. The rate of occurrence of dcrete event {,,, } are r,( =,,,. Thee rate repreent the fracton of tme that each dcrete event occur and = r =. If there ext a n Lyapunov functon V( x( :R R and calarα, α,, α > correpondng to each rate uch that and ( α α α α r r r > > Vx ( ( Vx ( ( ( α Vx ( (, =,,,, ( then the ADS exponentally table, wth decay rate greater thanα. Remar : Lemma requre the ADS to be table on the average. It doe not requre every dfference equaton of ADS to be table, but rather t guarantee the ADS to be table on the whole. Remar : For the ytem (8, f we tae the rate of occurrence of dcrete event = a r, then, the rate of occurrence of dcrete event = r. In th paper, we wll extend the reult of Lemma to the networed wtched control ytem wth pacet dropout, where the wtched control ytem tate and the wtchng gnal are affected by networ pacet dropout. We wll contruct wtched Lyapunov-Kraov functonal to analyze the tablty of SCS (8, accordngly reduce the conervatvene of ung common Lyapunov functon. Then the hybrd tate feedbac controller degn wll be preented. III. EXPOETIAL STABILITY OF SCS WITH PACKET DROPOUT Th ecton gve tablty condton of SCS (8. Theorem For ytem (8, aume that the plant tate and the wtchng gnal n a ngle pacet are tranmtted at the rate of r. If there ext ymmetrcal potve defnte matrce PQ,, ϒ and calarα, α > uch that α α > ( r r Φ Φ (3 T P Q α P Φ Φ (4 T P Q α P hold, then ytem (8 exponentally table. Proof Contruct the followng wtched Lyapunov- Kraov functonal T T ( ς = ς ( σ( ς ς ( σ( ς = = V P Q (5 where P, ϒ and Q, ϒ are ymmetrcal potve defnte matrce. For ytem (8, we have 37

4 V ( ς α V( ς T T ( ( P ( ( P = = = ς σ ς α ς σ ς T T ( ( Q ( l( Ql = l= ς σ ς α ς σ ς T T T ( ( P ( ( Q = = = ς Φ σ Φ ς ς σ ς T T ( ( P ( l( Ql ς = l= α ς σ ς α ς σ (6 Under the arbtrary wtchng law, aume that the th, th and l th ubytem are actvated at the tme, and repectvely, and we now that blnear term a the product of calar varable α, =, and matrx varable P ϒ, repectvely. Extenve numercal experence how that a combnaton of the path-followng method n [3] and the drect teraton wor very well on th type of BMI problem. Here, for the ae of mplcty, we only gve the degn method of K, ϒ n (3 and (4 whch are LMI n P, ϒ for fxed α, =,. IV. DESIG OF HYBRID STATE FEEDBACK COTROLLER In th ecton, we wll degn the hybrd tate feedbac controller for ytem (8. Theorem Aume the plant tate and the wtchng gnal are tranmtted n a ngle pacet at the rate of r. Gven calar α, α >, f there ext ymmetrcal potve defnte matrce PQ,, ϒ uch that σ ( = σ ( = σ ( =. l = = l= From (6, we obtan V ( ς α V( ς T σ( σ ( σl( η η, = = l= = Π (7 α α > (9 r r T Q α P Φ ( Φ W Q Φ T α P Φ W ( where η = [ ς ς ], T T T T Φ PΦ Q α P Π =,, =. α Ql From nequalte (3 and (4, we now that Π for =,. Therefore, under the arbtrary wtchng law V whch mple that V. (8 ( ς α ( ς V V V ( ς ( ς ( α ( ς, =, From Lemma, combnng ( and (8, we get the SCS (8 exponentally table. The proof completed. Remar 3: The matrx nequalte (3 and (4 nvolve hold, then ytem (8 exponentally tablzed by the hybrd tate feedbac controller (. Remar 4: In Theorem, the unnown hybrd tate feedbac controller gan K, ϒ are ncluded n the ytem matrce Φ, =,, thu K, ϒ n nequalte (3 and (4 are coupled wth the unnown ymmetrcal potve defnte matrce P, ϒ. Ung the Schur complement, we eparate Φ, =,from P, ϒ o the nequalte (3 and (4 can be re-wrtten by T Q α P Φ P Φ Q Φ Φ T α P P ( (3 We notce that the condton are not LMI becaue of the term P and P. A cone complementary lnearzaton algorthm [4] can be ued to convert the non-convex 37

5 optmzaton problem to a nonlnear mnmzaton problem. Replacng the term P n ( and ( byw, we get ( and (. Then we have the mnmzaton problem wth LMI contrant a follow Mnmze: trace( PW Subect to: P >, W > and (9-(, (4 P I, ϒ. (4 W Algorthm Step. Fnd a feable et ( PQ,, PW,, K,, ϒ whch meet the contrant of (9-( and (4, et = ; Step. Solve the followng LMI problem for the varable ( P, Q, P, W, K Mn: Trace( PW Subect to: (9-( and (4 Step 3. Subttute the obtaned matrx varable ( P, Q, P, W, K nto (9-. If they are atfed, then output the feable oluton. Ext. Step 4. If > where the maxmum number of teraton allowed, then ext. Step 5. Set = and ( PQ,,,, PW K = ( P, Q, P, W, K, go to Step. V. BOUD O PACKET TRASMISSIO RATE In the above two Theorem, we gve uffcent condton to guarantee the tablty of SCS (8 under the tranmttng rate r. However, the lower the tranmon rate, the le networ bandwdth ued. Moreover, large pacet dropout rate wll degrade the control performance and even can render the overall ytem untable. Therefore, the followng propoton wll preent the lower bound on pacet tranmttng rate r that tll guarantee the tablty of SCS. Propoton 3 Conder the framewor of Fg., aume that the cloed-loop wtched control ytem wth no dropout table under an arbtrary wtchng law (.e. A BK Schur when the wtch n poton S ( If the open-loop wtched ytem ( A margnally table under an arbtrary wtchng law, then the ytem exponentally table for all < r. ( If the open-loop wtched ytem untable under an arbtrary wtchng law, then the ytem exponentally table for all where r γ / γ <, λ ( P λ ( Q γ = log[ λ ( A BK ] ( ( P, max max max λmn P λmn λ ( P λ ( Q γ = log[ λ ( A ] ( ( P, max max max λmn P λmn Q, P, P,, ϒ are the oluton of nequalte ( and (. Ung Theorem 5.4 n the reference [], the proof of th propoton trval. Here omtted. VI. EXAMPLE In th ecton, an example gven to llutrate our man reult. Conder the followng wtched ytem wth two ubytem whoe feedbac path are mplemented by a networ. Subytem : Subytem :...5 x = x u... x = x u.. Suppoe that the networ bandwdth lmted and the pacet dropout happen. Alo, the ytem tate and the wtchng gnal n a ngle pacet are tranmtted at the rate r =.7, whch mean 7% of the pacet delvered to the hybrd tate feedbac controller. Settng α =.88, α =.755, we olve the nonlnear mnmzaton problem and have the hybrd tate feedbac controller gan K = [.88.], K = [ ]. 37

6 VII. COCLUSIOS In th paper, the tablty analy and control ynthe for a cla of networed wtched control ytem wth ytem tate and wtchng gnal pacet dropout are tuded. The dcrete-tme wtched ytem and the hybrd tate feedbac controller are connected va networ tranmon channel whch ubected to pacet dropout. The wtched ytem tate and the wtchng gnal ncludng n the networ pacet are condered multaneouly. The propoed oluton found by ung aynchronou dynamcal wtched ytem theory and wtched Lyapunov-Kraov technque. We have derved uffcent condton on exponental tablzaton of SCS n cloed-loop. The degn of hybrd controller for wtched ytem at a fxed nown pacet dropout rate repreented. The lower bound on pacet tranmon rate that tll guarantee the tablty of SCS dcued. REFERECES [] W. Zhang. Stablty Analy of etwored Control Sytem, USA: Cae Wetern Reerve Unverty,. [] J. lon. Real-tme Control Sytem wth Delay. Dept. of Automatc Control, Lund Inttute of Technology, Lund, Sweden, Jan [3] M. Yu, L. Wang, T. Chu, and F. Gao. "An LMI approach to networed control ytem wth data pacet dropout and tranmon delay", Intl. J. of Hybrd Sytem, vol. 3 no., pp. 9-33, 3. [4] W. S. Wong, R. W. Brocet. "Sytem wth fnte communcaton bandwdth contrant - II: tablzaton wth lmted nformaton feedbac", IEEE Tran. on Automatc Control, vol. 44, no. 5, pp , May 999. [5] P. Seler and R. Sengupta. "An H approach to networed control", IEEE Tran. on Automatc Control, vol. 5, no. 3, pp , Mar. 5. [6] Q. Lng and M. D. Lemmon. "Optmal dropout compenaton n networed control ytem", n Proceedng of 4 nd IEEE Conference on Decon and Control, Mau, Hawa USA, pp , Dec. 3. [7] Y. Tpuwan and M. Y. Chow. "Control methodologe n networed control ytem", Control Engneerng Practce, vol., no., pp. 99-, 3. [8] L. Hetel, J. Daafouz, and C. Iung. "Stablzaton of arbtrary wtched lnear ytem wth unnown tme-varyng delay", IEEE Tran. on Automatc Control, vol. 5, no., pp , Oct. 6. [9] L. Hetel, J. Daafouz, and C. Iung. "LMI control degn for a cla of exponental uncertan ytem wth applcaton to networ controlled wtched ytem", n Proceedng of Amercan Control Conference, ew Yor Cty, USA, pp. 4-46, July 7. [] L. Hetel, J. Daafouz, and C. Iung. "Stablty analy for dcrete tme wtched ytem wth temporary uncertan wtchng gnal", n Proceedng of the 46 th IEEE Conference on Decon and Control, ew Orean, LA, USA, Dec.-4, pp , 7. [] L. Hetel, J. Daafouz, and C. Iung. "Analy and control of LTI and wtched ytem n dgtal loop va an event-baed modelng", Internatonal Journal of Control, vol. 8, no. 7, pp.5 38, 8. [] Z. H. Guan, H. Zhang and S. H. Yang. "Robut pave control for nternet-baed wtchng ytem wth tme-delay", Chao, Solton and Fractal, vol. 36, , 8. [3] D. Ma, G. M. Dmrov, T. Lu and J. Zhao. "Robut exponental tablzaton of wtched ytem wth networ tme-varyng delay and pacet dropout", n Proceedng of the 7 th Chnee Control Conference, Kunmng, Chna, pp.96-, July 8. [4] J. Daafouz, P.Rednger and C. Iung. "Stablty analy and control ynthe for wtched ytem: a wtched Lyapunov functon approach", IEEE Tran. On Automatc Control, vol.47, no., pp ,. [5] D. Lberzon. Swtchng n Sytem and Control, Boton, MA: Brhauer, 3. [6] Z. D. Sun, S. S. Ge. Swtched Lnear Sytem-Control and Degn, ew Yor: Sprnger Verlag, 4. [7] J. P. Hepanha, A. S. More. "Stablty of wtched ytem wth average dwell-tme", n Proceedng of the 38 th IEEE Conference on Decon and Control, Phoenx, Arzona, Omnpre, pp , Dec [8] Y. J. Wang, G. M. Xe, L. Wang. "Controllablty of wtched tme-delay ytem under contraned wtchng", J. Math. Analy. Appl., vol. 86, pp [9] X. R. Mao. "Exponental Stablty of tochatc delay nterval ytem wth Marovan wtchng", IEEE Tran. on Automatc Control, vol. 47, no., pp. 64-6, Oct.. [] J. Zhao and D. J. Hll. "On tablty, L -Gan and H control for wtched ytem", Automatca, vol. 44, pp. 3, 8. [] A. Hab, S. P. Boyd and J. P. How. "Control of aynchronou dynamcal ytem wth rate contrant on event", n Proceedng of the 38 th IEEE Conference on Decon and Control, Phoenx, Arzona, Omnpre, pp , Dec [] S. Boyd, L. El GhaouL, E. Feron, Balarhnan. Lnear Matrx Inequalte n Sytem and Control Theory. Phladelpha, PA: SIAM, 994. [3] A. Hab, J. P. How and S. P. Boyd. "A path-followng method for olvng BMI problem n control", n Proceedng of Amercan Control Conference, pp , 999. [4] L. E. Ghaou, F. Outry, and M. AtRam. "A cone complementarty lnearzaton algorthm for tatc output-feedbac and related problem", IEEE Tran. On Automatc Control, vol.4, no.8, pp. 7-76,