Controller Design for Networked Control Systems in Multiple-packet Transmission with Random Delays

Transcription

1 Appled Mehans and Materals Onlne: ISSN: , Vols , pp do:0.408/ 03 rans eh Publatons, Swtzerland H Controller Desgn for Networed Control Systems n Multple-paet ransmsson wth Random Delays Danl Wen, a and Yuan Gao, b Software College of Shenyang Normal Unversty, Shenyang, 0034, Chna a wdl09@63.om, b lyegaoyuan@63.om Keywords: networed ontrol systems; multple-paet transmsson; ommunaton random delays; H ontroller desgn Abstrat. Dsrete-tme NCSs n multple-paet transmsson wth random delays were dsussed. he random delays from the sensors to the ontrollers and from the ontrollers to the atuators were onsdered. A novel random delay model n multple-paet transmsson was proposed and the relatonshp of the delays and the system performane was studed. he state feedba H ontroller an be onstruted va solvng a lnear matrx nequalty, suh that the losed-loop system wth random delays exponentally mean-square stable and wth a presrbed H performane bound. he result shows that an example presented here llustrates the effetveness of the proposed method. Introduton Networed ontrol systems (NCSs) have reeved nreasng attenton n reent years as a result of many advantages nludng lower ost, easer nstallaton and mantenane, and hgher relablty, and are appled n many areas, e.g. automobles, manufaturng plants, arrafts and spaerafts. As we all now, data transmsson due to dfferent networs s lassfed as sngle-paet transmsson and multple-paet transmsson. here are many wors [-3] about NCSs based on sngle-paet transmsson model wth tme delay and paet dropout. he ontroller s desgned for NCSs wth sensor-to-ontroller and ontroller-to-atuator delays n sngle-paet n [3]. Due to the ommunaton bandwdth lmted and the dstrbuted networed, large amounts of data must be broen nto multple paets to be transmtted. Motvated by the above mentoned problem, ths paper s onerned wth the state feedba H ontrol problem for NCSs n multple-paet transmsson wth random ommunaton delays. he random delays from the sensor to the ontroller and from the ontroller to the atuator are onsdered, and they are modeled as a funton of stohast varables satsfyng Bernoull dstrbuted whte sequene. he state feedba H ontroller an be onstruted, suh that the losed-loop system wth random delays exponentally mean-square stable and wth a presrbed H performane bound. An example s presented to llustrate the effetveness of the proposed method. Problem Statement and Prelmnares Problem Statement. In ths paper, assume that the sensor, the ontroller and the atuator are lo-drven. he random ommunaton delays, whh are from the sensor to the ontroller and from the ontroller to the atuator, are less than one samplng perod. Consder the dsrete-tme LI model desrbed by x( ) Ax( ) Bu ( ) B( ) () z( ) Cx( ) Du( ) n q m where x( ) R s the state, z( ) R s the regulated output, u( ) R s the ontrol nput, and l ( ) R s the dsturbane nput. AB,, B, C and D are nown onstant matres of approprate dmensons. he par ( AB, ) s assumed to be stablzable. All rghts reserved. No part of ontents of ths paper may be reprodued or transmtted n any form or by any means wthout the wrtten permsson of rans eh Publatons, (ID: , Pennsylvana State Unversty, Unversty Par, USA-/05/6,0::47)

2 60 Advanes n Mehatrons and Control Engneerng We onsder the ase that the state s splt nto paets, that s x( ) X ( ), X ( ), X ( ) where X( ) x ( ) x ( ) x ( ) r, X ( ) xr ( ) xr ( ) xr r( ), X ( ) ( ) ( ) ( ) xr r xn xn. he random ommunaton delays onsdered n ths paper are modeled as a funton of stohast varables satsfyng Bernoull dstrbuted., ( N) denote the stohast varables, whh are ndependent and are Bernoull dstrbuted whte sequene tang the values of 0 and. Let the stohast varables ( ) be ours random delays Pr{ } E{ }, Pr{ 0} E{( )} ( ) then, 0 otherwse Pr{ } E{ }, Pr{ 0} E{( ) } where, ( N) are nown onstant. Denote dag, dag, for system (), the obtaned state wth random delays s x ( ) ( I ) x( ) x( ), and a ontrol nput wth random delays s u( ) ( I ) u ( ) u ( ) wth u ( ) Kx ( ), then u( ) ( I )( I ) Kx( ) ( ( I ) ( I )) x( ) x( ) () Combnng ontroller () wth system (), the followng losed-loop system wth random delays s obtaned: x( ) Ax( ) B Kx( ) B Kx( ) B ( ) 3 H B Kx( ) H B Kx( ) H B Kx( ) 3 z( ) Cx( ) DKx( ) DKx( ) H DKx( ) 3 HDKx( ) H3DKx( ) where A A B K, C C DK, ( I )( I ), ( I ) ( I ), 3 H ( )( ) ( I )( ) ( I )( ), H ( )( ) ( ) ( ) 3 H ( )( ) ( I )( ) ( I )( ) Defnton : he losed-loop system () s sad to be exponentally mean-square stable f wth ( ) 0, there exst onstants 0 and (0,), suh that E{ x( ) } E{ x(0) }. By Defnton, the problem addressed n ths paper s as follows: Desgn a H ontroller suh that the losed-loop system () s exponentally mean-square stable; for a gven salar 0 and all nonzero ( ), under the zero-ntal ondton, the ontrol output z ( ) satsfes E{ z( ) } ( ) (4) 0 0 Prelmnares. he followng lemma presented wll be used n the sequel. Lemma : [3] Let V ( ) be a Lyapunov funtonal. If there exst real salars 0, 0, 0 and 0 suh that V( ) and E{ V( ) } V( ) V( ), then the sequene satsfes E{ } 0 ( ). (3) H Controller Desgn wth Random Delays In ths seton, a desgn method of the ontroller gan s gven, whh guarantees the losed-loop system () exponentally mean-stable and wth the presrbed H performane bound.

3 Appled Mehans and Materals Vols heorem s presented to desgn the ontroller gan. Denote dag{ P P, P P, P, I B P B }, dag{ P, P, P, P, I, I, I, I} A B K 3B K B B K B K B K 0 ( I ) B K ( I ) B K B K 0 ( I ) 3BK ( I ) 3BK 3B K 0 C DK 3DK 0 DK DK DK 0 ( I ) DK ( I ) DK DK 0 ( I ) 3DK ( I ) 3DK 3DK 0 where ( I ) ( I ) ( I ) ( I ) ( I ) ( I ) ( I ) ( I ), ( I ) ( I ) 3 heorem : Consder plant () wth a gven salar 0, assume that there exst matres Y and X 0, P 0, P3 0 suh that the followng LMI holds P X * * * * * * * * * * * * 0 P3 P * * * * * * * * * * * 0 0 P3 * * * * * * * * * * I * * * * * * * * * AX B Y BY 3BY B X * * * * * * * * BY B Y BY 0 0 X * * * * * * * ( I ) BY ( I ) BY BY X * * * * * * 0 ( I ) 3BY ( I ) 3BY 3BY X * * * * * CX DY DY 3DY I * * * * DY DY DY I * * * ( I ) DY ( I ) DY DY I * * ( I ) 3DY ( I ) 3DY 3DY I * B X Denote K YX then, the losed-loop system () s exponentally mean-square stable and wth the H performane bound. Here, the proof of ths theorem s omtted due to the lmted wrtng. It s analogous to the one for heorem 3. n [4]. Example In ths seton, an example s presented to llustrate the effetveness of the ontroller desgn presented n ths paper. Consder the followng dsrete-tme networed ontrol systems studed n [5], and the ase that the state s splt nto paets, 0 0 x( ) ( ) ( ) ( ) 0.5 x u 0 z( ) x( ) u( ) 0 0

4 604 Advanes n Mehatrons and Control Engneerng By solvng the LMI n heorem, the values of the ontroller gan K and the H performane bound wth varous values of are, shown as n able. able Values of and K for varous values of, mn K [ ] [ ] [ ] he result of Row n able means no tme delays our; the result n Row due, s analogous to the one n sngle-paet transmsson n [5]. to Gven the ntal system state as x 0 [, ] and let the dsturbane nput ( ) be, 3 4(step) ( ) 0 otherwse he smulaton results of system () wth the ontroller gan and n Row 3 are depted n the followng fgures. From Fg. and Fg., we an see that the proposed method s effent. Fg. State responses of the system Fg. Regulated output responses of the system Conlusons We have studed the H ontrol problem for NCSs n multple-paet transmsson wth random ommunaton delays. he state feedba H ontroller s onstruted va solvng a lnear matrx nequalty, suh that the losed-loop system wth random delays exponentally mean-square stable and wth a presrbed H performane bound. An example has been presented to llustrate the effetveness of the proposed method. Referenes [] G. P. Lu: Predtve Controller Desgn of Networed Systems wth Communaton Delays and Data Loss, IEEE rans. Cruts Syst. II, Exp. Brefs, 57(00), pp [] H. L, Z. Sun, H. Lu and F. Sun: Stablzaton of Networed Control Systems Usng Delay-dependent Control Gans, IE Control heory Appl., 6 (0), pp [3] F. W. Yang, Z. D. Wang, Y. S. Hung and M. Gan: H Control for Networed Systems wth Random Communaton delays. IEEE ransatons on Automat Control, 5 (006), pp.5-58 [4] D. L. Wen and G. H. Yang: State Feedba H Control for Networed Control Systems wth Quantzaton and Random Communaton Delays. Internatonal Journal of Innovatve Computng, Informaton & Control. 7(0), pp [5] W. W. Che and G. H. Yang: State Feedba H Control for Quantzed Dsrete-tme Systems. Asan Journal of Control, 0 (008), pp

5 Advanes n Mehatrons and Control Engneerng 0.408/ H Controller Desgn for Networed Control Systems n Multple-Paet ransmsson wth Random Delays 0.408/