State-Space Model Based Generalized Predictive Control for Networked Control Systems

Transcription

1 Prceedngs f the 7th Wrld Cngress he Internatnal Federatn f Autmatc Cntrl State-Space Mdel Based Generalzed Predctve Cntrl fr Netwred Cntrl Systems Bn ang* Gu-Png Lu** We-Hua Gu*** and Ya-Ln Wang**** *Schl f Infrmatn Scence and Engneerng Central Suth Unversty Changsha Chna e-mal: tangbn36@63.cm. **Department f Engneerng Unversty f Glamrgan Pntyprdd CF37 DL UK. e-mal: gplu@glam.ac.u. ***Schl f Infrmatn Scence and Engneerng Central Suth Unversty Changsha Chna e-mal: gwh@csu.edu.cn. ****Schl f Infrmatn Scence and Engneerng Central Suth Unversty Changsha Chna e-mal: ylwang@mal.csu.edu.cn. Abstract: hs paper s cncerned wth generalzed predctve cntrl based n state-space mdel fr netwred cntrl systems. Bth the netwr-nduced tme delay and pacet lss are taen nt accunt. he netwred predctve cntrller desgn s dussed wth respect t three netwr cases whch are related t pacet lss netwr-nduced delay and bth pacet lss and netwr-nduced delay respectvely. Smulatn and experment results shw the effectveness f ur netwred predctve cntrl algrthms.. INRODUCION Recently the research fr netwred cntrl systems NCSs has attracted much attentn. One f the man prblems n NCSs s t cmpensate fr the nfluence f netwr-nduced tme delay and pacet drput whch have been prved t serusly degrade the system perfrmance and even lead t unstablty. Fr netwr-nduced tme delay cmpensatn prblem please refer t Zhang Sh Chen & Huang 005 Yang Wang Huang & Gan 006 Hu Ba Sh and Wu 007 and Lam Ga & Wang 007 Zhang & L 007 Mercangz & Dyle III 007 Lu Mu Rees & Cha 006. Fr pacet lss cmpensatn prblem please refer t Imer Yusel & Basar 006 Gupta Hassb & Murray 007 Huang & Dey 007 Xng & Lam 007 Hu & Yam 007. It s seen frm the abve-mentned lteratures that mdel predctve cntrl has nt been ntensvely studed especally fr the case f pacet lss. In ths paper we cnsder generalzed predctve cntrl based n state-space mdel fr netwred cntrl systems wth bth the netwr-nduced tme delay and pacet lss. he netwred predctve cntrller desgn s cncerned n three netwr cases whch are related t pacet lss cnstant netwr-nduced tme delay and bth randm pacet lss and randm netwr-nduced tme delay respectvely. Furthermre smulatn and experment results are gven t shw the effectveness f ur cntrller desgn methd.. PROBLEM FORMULAION he cntrlled plant s mdelled by a lnear drete-tme system x t = Ax t Bu t B t B vv t yt = Cx t t n m t Z s the tme step x t R and ut R are respectvely the plant state and cntrl nput. v t and t are respectvely the plant dsturbance and measurement nse. A B Bv and C are cnstant matrces f apprprate dmensns. We assumed that v t and t are whte Gaussan and zer-mean nses wth cvarance matrces Q and respectvely and are v Q ndependent each ther. he ntal cndtn x 0 s assumed t be ndependent f v t and t have mean zer and cvarance matrx Q. Actuatr ca 0 Fg. he structure f netwred cntrl systems he framewr f the NCS cnsdered n ths paper s shwn n Fg.. Netwrs exst between sampler and cntrller and between cntrller and actuatr. he sampler s tme-drven the cntrller and actuatr are event-drven and the data are transmtted n a sngle pacet at each tme step. Let Φ { = } a subsequence f { 3 } dente the U Cntnuus Plant Netwr Cntrller y Sampler /08/$ IFAC / KR

2 7th IFAC Wrld Cngress IFAC'08 sequence f samplng nstants f successful data transmssns frm the sampler t the actuatr and s max be the maxmum pacet-lss upper Φ bund. Let and dente the netwr-nduced tme delays between sampler and cntrller and between cntrller and actuatr respectvely. We mdel the tw transmssn delays as nput delays. Wth - the cntrlled plant frm the vew f cntrller can be mdelled fr the nternal t [ as x t = A x t B u t B t ca B v t v yt = Cx t t 4 ca =. he netwred cntrller s a generalzed predctve cntrller based n an equvalent ncremental state-space mdel. he ncremental state-space mdel f 3-4 can be gven as ca x t = Axt BΔut B t Bvt 5 v yt = Cxt t 6 yt xt = Δx t Δ x t = x t x t Δ ut = ut ut v t = v t v t I C 0 t = t t A = 0 A B = B B I 0 = B Bv =. B C = [ I C ] v At every nstant t = a new cntrl ncrement vectr ΔU s prduced whch can be gven as fllw: ca Δ U = f x Δu Δu 7 c fc means that ΔU s a functn f ca x Δu Δu. Snce ths generalzed predctve cntrl GPC algrthm depends n the state x f the ncremental mdel and the avalable feedbac s the utput y f the rgnal plant t s necessary t estmate the state x usng y and the ncremental state-space mdel. In ths paper the ptmal estmatn f x s nt dussed n detal whch can be acheved wth the methd prpsed by Gupta Hassb & Murray 007. he cntrl vectr U gven as fllw: 3 sent frm cntrller t actuatr s u u U = McΔU 8 u Im Nu m Nu m Im I m Mc R m m = I m R. Im Im Im Fr the tme nterval t [ ] the elements f U act n sequence n the cntrlled plant whch can be gven frm the vew f cntrller as fllw: x l = Ax l Bu l B l B vv l y = Cx l and 0 {0} l. 3. NEWORKED PREDICIVE CONROLLER DESIGN In ths sectn the netwred predctve cntrller desgn s dussed n three cases fr clarty. he three cases are n the cmpensatn f the nfluences f the randm pacet lss the cnstant netwr-nduced delay and bth the randm pacet lss and randm netwr-nduced delay respectvely. 3. Case f randm pacet lss Fr the NCS cnsdered n ths case t s assumed that there exst the pacet lsses and every pacet sent s free f the transmssn delay.e. = 0. cmpensate fr the nfluence f pacet lsses n the NCS the cntrller creates a cntrl ncrement vectr Δ U = [ Δu Δ u Δ u N u ] f length N u > s usng the receved feedbac pacet whch s sampled at nstant and then the prduced cntrl vectr U = [ u u u N u ] s sent t actuatr n a sngle pacet. In the tme nterval [ ] the cntrls ul l are successvely mpsed n the cntrlled cntnuus plant. he cst functn n ths case s gven as fllw N J Δ U = E{ y w N u λ u } 9 Δ wt s reference lcus N s the maxmal step length f predctn s Nu N s the step length f predctve 3007

3 7th IFAC Wrld Cngress IFAC'08 cntrl ncrements. λ = Nu Frm the equatn abve t s seen that the thrd and furth terms at the rght hand are respectvely the whte plant nses at the nstant and ts future nstants and the whte measurement nse at nstant. S the ptmal predctn f y s gven as y ˆ = Ey [ ] = CAEx [ ] CA BΔ u 0 = CAEx [ ] CA BΔ u. = [ u ] Δ U = Δu Δ u N CB 0 CAB CB G = Nu Nu CA B CA B CB N N N Nu CA B CA B CA B are the weghts f cntrl Wth the ncremental mdel 5-6 we btan the > 0 step frward predctn y = CA x CA BΔ u = v = = CA B v CA B. = E{} dentes the mathematcal expectatn. Fr the utput feedbac case Ex [ ] can be estmated by Kalman flter. Fr every feedbac sampled at we btan frm 0 that y ˆ = Ey [ ] Let = Y ˆ [ yˆ yˆ N ] CA CA F = Ex [ ]. N CA hen we btan frm 0 the fllwng ptmal predctn n a vectr frm Y ˆ = GΔ U F Let λ = dag[ λ λ λ ] N u = [ ] W w w N N Nu Y = [ y y ] N. he cst functn 9 can be gven n the fllwng vectr frm Δ = {[ ] J U E Y W [ Y W ] ΔU ΔU } Substtutng nt and lettng s further gven as fllw: λ J ΔU = 0 7 ΔU Δ U = GG I G W F 3 Wth 3 and 8 the cntrl sequence fr the case f pacet lss s btaned. 3. Case f cnstant netwr-nduced tme delay Fr the NCS cnsdered n ths case t s assumed that there exst n pacet lss and every pacet sent experences sme ca cnstant netwr-nduced tme delay.e. = =. It s nted that = n ths case. cmpensate fr the nfluence f the cnstant delay the cntrller creates a cntrl ncrement vectr Δ U = [ Δu Δ u Δ u N u ] f length N u > s usng the receved feedbac pacet sampled at nstant and then prduces a cntrl vectr U = [ u u u Nu ] whch s sent frm cntrller t actuatr n a sngle pacet. Fr = n ths case nly the cntrl u s mpsed n the cntrlled cntnuus plant n the tme nterval [ ]. he cst functn n ths case s gven as fllw N J Δ U = E{ y w N0 Nu λ u } Δ 4 wt s reference lcus N0 and N N0 N u are respectvely the mnmal and maxmal step length f predctn s Nu N s the step length f predctve cntrl λ = Nu are the weghts f cntrl ncrements. In ths case the > 0 step frward predctn s dfferent frm that f the case f pacet lss whch s gven as fllw: y ca = CA x CA BΔ u = 3008

4 7th IFAC Wrld Cngress IFAC'08 CA Bvv = CA B = ca = CA x CA BΔ u ca = CA BΔ u CA B v = ca ca = = CA B. = = v 5 It s seen frm 5 that y depends n bth the determned cntrl ncrements Δu ca Δu and the future nes Δ u Δ u. hen the ptmal predctn f y s gven as fllw: y ˆ = Ey [ ] ca ca = CA E[ x ] CA BΔ u ca ca CA BΔ u. Defne Yˆ N [ yˆ N yˆ N 0 = 0 ] Δ U = [ Δu Δ u N ] u CB 0 CAB CB G = N u Nu CA B CA B CB N N0 N N0 N N0 Nu CA B CA B CA B Δ U = [ Δu N Δu ] d 0 N0 CA N0 CA F N0 = Ex [ ] N CA N0 N0 CA B CA B CAB N 0 N0 CA B CA B CA B ΔUd. N N N N0 CA B CA B CA B hen we have frm 6 that Y ˆ N = GΔ U F N. 0 0 Smlarly 7 s further gven as fllw: λ N N0 Nu Δ U = GG I G W N F N 7 λ = dag[ λ λ λ ] N u 0 0 W N = [ w N w N]. Wth 7 and 8 the cntrl sequence fr the case f netwr-nduced tme delay s btaned. 3.3 Case f bth pacet lss and randm netwr-nduced tme delay Fr the NCS cnsdered n ths case t s assumed that there smultaneusly exst randm pacet lss and randm netwr-nduced tme delay. Fr every feedbac sampled at the cntrller prduces the related Δ U and U. Because f the randm netwr-nduce tme delay ΔU and U are same t thse f the case f cnstant netwrnduced tme delay. In the tme nterval [ ] t cmpensate fr the nfluence f the pacet lss the cntrls ul l are successvely mpsed n the cntrlled cntnuus plant. 4. SIMULAIONS he cnsdered drete-tme state-space mdel s dentfed frm a practcal crcut system. hs crcut system s a frstrder nertal lp. he mdel s gven as fllw: x t = Ax t Bu t B t yt = Cx t t A = B = B = C =. hen the equvalent ncremental state-space mdel s gven as fllw: x t = Ax t BΔ u t B t yt = Cxt t A = C = [ ]. 0 B = B = he nse t s assumed t be zer-mean and whte. Fr the netwred generalzed predctve cntrl algrthm the set pnt s 4 N N0 = 3 N u = 9 and the weghts λ = N u are set t be 0.. In the fllwng fg. ~7 the sld lne dentes the utput sgnal and the dtted the cntrl sgnal. 4. Case f cnstant pacet lss Fg. s the smulatn results fr the case f pacet lss. he sub fgures frm the tp t the bttm are related t the cases wth pacet lss 0 5 and 8 respectvely fr every nterval 3009

5 7th IFAC Wrld Cngress IFAC'08 [ ]. It s seen that the cntrls are dfferent fr dfferent numbers f pacet lss and mre pacet lss leads t the bgger varance f cntrl sequence. pacet lss and the utput respnses are ust shfted ahead wth the steps f the transmssn delay. Fg.. he smulatn results fr the case f pacet lss 4. Case f cnstant netwr-nduced tme delay Fg. 3 s the smulatn results fr the case f cnstant netwr-nduced tme delay. he delays f three sub fgures frm the tp t the bttm are 0 0 and 0 respectvely. It s seen that the three cntrls are same t each ther thugh the transmssn delays are dfferent. Because f the transmssn delay every utput respnse s shfted ahead by the sze f delay. Fg. 4. he smulatn results fr the case f bth pacet lss and netwr-nduced tme delay 5. EXPERIMENS In ths sectn we apply the netwred predctve cntrl algrthm n the practcal crcut system t verfy ts effectveness. he predctve cntrl algrthm s mplemented n a netwred cntrller bard NCB and the sampler and actuatr are lcated at a netwred mplemented bard NIB. Fr mre nfrmatn f the bards refer t Lu Mu Rees & Cha Case f pacet lss Fg. 5 s the expermental results f the case f pacet lss. It s seen that the step respnse s very smlar t that f the smulatn results f Fg.. Fg. 3. he smulatn results fr the case f cnstant netwr-nduced tme delay 4.3 Case f pacet lss and netwr-nduced tme delay Fg. 4 s the smulatn results fr the case f bth pacet lss and netwr-nduced tme delay. he netwr-nduced tme delay s 0. he numbers f pacet lss frm the tp t the bttm are 0 5 and 8 respectvely fr every nterval [ ]. he cntrls are dfferent each ther wth respect t Fg. 5. he expermental results f the case f pacet lss 5. Case f cnstant netwr-nduced delay 300

6 7th IFAC Wrld Cngress IFAC'08 Fg. 6 s the expermental results f the case f cnstant netwr-nduced tme delay. It s seen that the step respnse s very smlar t that f smulatn results f Fg. 3. Fg. 6. he expermental results f the case f cnstant netwr-nduced tme delay 53. Case f bth pacet lss and netwr-nduced delay Fg. 7 s the expermental results f the case f bth pacet lss and netwr-nduced tme delay. It s seen that the step respnse s very smlar t that f the smulatn results f Fg. 4. Fg. 7. he expermental results f the case f bth pacet lss and netwr-nduced tme delay 6. CONCLUSIONS utput predctn t cnstruct the future respnse. By mnmzng the cst functn wth a mnmal predctn step related t netwr-nduced tme delay a cntrl sequence s cmputed t cunteract the nfluence f netwr-nduced tme delay. he smulatn and experment results have shwn the effectveness f ur netwred predctve cntrller desgn methd. REFERENCES Gupta V. B. Hassb and R.M. Murray 007. Optmal LQG cntrl acrss pacet-drppng lns. Systems & Cntrl Letters Hu L.S.. Ba P. Sh and Z.M. Wu 007. Sampled-data cntrl f netwred lnear cntrl systems. Autmatca Hu S. and W.Y. Yam 007. Stablty rbustness f netwred cntrl systems wth respect t pacet lss. Autmatca Huang M. and S. Dey 007. Stablty f Kalman flterng wth Marvan pacet lsses. Autmatca Imer O.C. S. Yusel and. Basar 006. Optmal cntrl f LI systems ver unrelable cmmuncatn lns. Autmatca Lam J. H.J. Ga and C.H. Wang 007. Stablty analyss fr cntnuus systems wth tw addtve tme-varyng delay cmpnents. Systems & Cntrl Letters Lu G.P. J.X. Mu D. Rees and S.C. Cha 006. Desgn and stablty analyss f netwred cntrl systems wth randm cmmuncatn tme delay usng the mdfed MPC. Internatnal urnal f cntrl Mercangz M. and F.J. Dyle III 007. Dstrbuted mdel predctve cntrl f an expermental fur-tan system. Jurnal f prcess cntrl Xng X.L. and J. Lam 007. Stablzatn f lnear systems ver netwrs wth bunded pacet lss. Autmatca Yang F.W. Z.D. Wang Y.S. Huang and M. Gan 007. H- nf cntrl fr netwred systems wth randm cmmuncatn delays. IEEE transactns n autmatc cntrl Zhang L.Q Y. Sh.W. Chen and B. Huang 005. A new methd fr stablzatn f netwred cntrl systems wth randm delays. IEEE transactns n autmatc cntrl Zhang Y. and S.Y. L 007. Netwred mdel predctve cntrl based n neghburhd ptmsatn fr serally cnnected large-ale prcess. Jurnal f prcess cntrl hs paper has presented a netwred predctve cntrller desgn methd fr NCS subect t pacet lss and netwrnduced tme delay. cmpensate fr the nfluence f pacet lss n NCS when the netwred cntrller receves a netwred feedbac t creates a cntrl sequence and sends t t actuatr. he elements f the cntrl sequence successvely act n the plant tll a new cntrl sequence s avalable fr actuatr. cmpensate fr the nfluence f netwr-nduced tme delay the cntrller uses the frward 30