Tokyo Institute of Technology Periodic Sequencing Control over Multi Communication Channels with Packet Losses

Transcription

1 oyo Insttute of echnology Fujta Laboratory oyo Insttute of echnology erodc Sequencng Control over Mult Communcaton Channels wth acet Losses FL6-7- /8/6 zwrman Gusrald

2 oyo Insttute of echnology Fujta Laboratory Outlne oyo Insttute of echnology Introducton roblem Formulaton erodc System Results Concluson & Future Wors

3 oyo Insttute of echnology Fujta Laboratory lant Introducton. lant oyo Insttute of echnology Enc Dec Enc Dec Controller Networ Controller Challenges : Band-lmted channels me delay acet loss QoS (Qualty of Servce) control: -Wred networs use low pacet loss channel s epensve n term of networ costs -Wreless networs use hgh power s epensve n term of energy consumpton

4 oyo Insttute of echnology Fujta Laboratory Introducton oyo Insttute of echnology o Deal wth ths trade off : Introduce networ model wth many channels and dfferent pacet loss probabltes Introduce a perodc sequencng control scheme. Desgned offlne smplcty of the desgn Can be easly mplemented

5 oyo Insttute of echnology Fujta Laboratory roblem Formulaton oyo Insttute of echnology z H w u q θ Comm. Channel v u Q q y K Bw B Q Θ v y C D w v System (H ) : q : [ ] R p Swtchng pattern Only one channel can be used at one tme Θ θ M θ p q p θ { } : prob θ acet loss ndcator acet loss probablty

6 oyo Insttute of echnology roblem Formulaton ssume that there s p channels wth pacet loss probabltes {... p} ssume the perodc sequencng q {. p p} each channel s used n tmes n one perod. Goal : Necessary condton on for the stablzaton of the plant. (note: apply state-feedbac controller) For system θ ( ) whch s N-perodc the orgn s sad to be : Mean-square stable f for every ntal state [ ] θ lm E ( ) θ Stochastcally stable f for every ntal state E [ ] θ < ( ) θ oyo Insttute of echnology Fujta Laboratory

7 oyo Insttute of echnology Fujta Laboratory erodc System oyo Insttute of echnology erodc System : ( ) Where s a perodc matr of perod N.e N Lemma : he followng statements are equvalent each other. ( ) s stable. here ests a N-perodc postve defnte soluton of the Lyapunov nequalty ( ) < S. Bttant and. Colaner IFC erodc control systems

8 oyo Insttute of echnology Fujta Laboratory Results oyo Insttute of echnology roposton : For the system H the orgn s mean square stable (stochastcally stable) ff there ests an N- n n perodc matr such that l l > and l R l l l l l < for l IN () (roof) Defne the Lyapunov functon as : V Where s tme varyng and he system s stochastcally stable ff l l l l l N If < for l I holds for l then t also holds for ln E V θ <

9 Fujta Laboratory oyo Insttute of echnology oyo Insttute of echnology roof System H can be wrtten as : pplyng lemma and compute the epected value of the dfference : [ ] [ ] [ ] [ ] [ ] ( ) ) ( < E E E V E V V E θ θ θ θ ( ) B w K B Q D I C K B Q B w ) ( Θ Θ

10 oyo Insttute of echnology Fujta Laboratory Results oyo Insttute of echnology roposton : he necessary condton on the pacet loss probabltes so that there ests a controller that stablzes the plant s gven by p n N ma λ( ) < Where denotes an egenvalue λ () (roof) By proposton he system s stable ff there ests an N-perodc > such that N nequaltes n () hold. If the uncontrolled system s stable we also can guarantee that the controlled system s also stable. Straghtforward calculaton leads to the nequalty :

11 oyo Insttute of echnology Fujta Laboratory roof oyo Insttute of echnology n n n p ( ) N N < > p... p he soluton of ests ff s a stable matr.e. p n ma λ < N ( ) n n... n p N

12 oyo Insttute of echnology Fujta Laboratory erformance-guaranteed Decay Rate ssume there are two channels : and Fnd the largest perodc N such that : sup (for any pacet loss realzaton) he mamum perodc N s gven by : ( ) c c oyo Insttute of echnology N log log c ( c )

13 Results for Scalar Case oyo Insttute of echnology By pplyng the same computaton as n proposton the necessary and suffcent condton for the system to be stable s gven by ssume that we have two channels ( and ) whch are used n and n tmes respectvely. he relaton between n and n s gven by ssume that we have two channels ( and ) whch are swtched every tme step. he necessary & suffcent condton for the stablty s gven by oyo Insttute of echnology n p ( ) ( ) a ac n < < log n < ( a ( ) a ) log( a ( ) c a ac ac c ) a c Fujta Laboratory

14 oyo Insttute of echnology Fujta Laboratory Conclusons & Future Wor oyo Insttute of echnology he necessary condton for the stablty of perodc sequencng system and also the performance measure are ntroduced. Net step consder schedulng scheme and apply the model to mult-agent or sensor networs problems