Model-Based Networked Control System Stability Based on Packet Drop Distributions

Transcription

1 Model-Based Networed Cotrol System Stablty Based o Pacet Drop Dstrbutos Laz Teg, Peg We, ad We Xag Faculty of Egeerg ad Surveyg Uversty of Souter Queeslad West Street, Toowoomba, QLD 435, Australa {teg,we,xagwe}@usq.edu.au Abstract Ts paper studes te system stablty of a Model- Based Networed Cotrol System te cases were pacet losses follow a certa dstrbutos. I ts study, te urelable ature of etwor ls s modelled as a stocastc process. Ts process provdes us two system structures, represetg pacets dropped ad receved respectvely. Ts ew system wt two structures s asymptotcally stable, f te plat model s updated wt te data from plat wt te mum terval ad te pacet drop follows dscrete dstrbutos wt fte expectatos suc as Uform Dstrbuto ad Beroull dstrbuto. If te pacet loss follows dscrete dstrbutos wt fte expectato suc as Possoa Dstrbuto, te stocastc system s stable we te bggest terval s lmted to te mal update tme terval. Tese results are verfed smulatos. Keywords Pacet Drop Dstrbuto, Model-Based Networed Cotrol System, System Stablty I. INTRODUCTION Networed cotrol systems (NCSs ave attracted cosderable amout of atteto te past decade. Compared wt tradtoal feedbac cotrol systems, NCSs reduce te system wrg, mae te system easy to operate, mata ad dagose case of malfuctog. I spte of te great advatages tat te etwored cotrol arctecture brgs, sertg a etwor betwee te plat ad te cotroller troduces may problems as well. Networ duced delays are uavodable because of te scedulg scemes. Pacet drops occur sometmes because of etwor cogestos. Ulmted data rate s ot possble because of fte badwdt avalable. I [-], system stablty as bee studed wle etwor tme delays are cosdered. Vjay Gupta, Baba Hassb ad Rcard M. Murry [3], vestgated te system performace wt pacet drops, ad cocluded tat pacet drops degrade a system s performace ad possbly cause system stablty. I [4], Jo K. Yoo, Daw M. Tlbury ad Nadt R. Soparar used state estmator tecques to reduce te commucato volume a etwored cotrol system. Pacet drop over a etwor exbts stocastc beavor. Te etwor ca be descrbed as tere s some correlato betwee cosecutve pacets term of Marov Ca. I [5-6] H flterg for a class of ucerta Marova jump lear systems s vestgated. A Marova jumper lear flter s gve terms of lear matrx equaltes. Optmal Kalma flters are used Marov jump lear systems as te estmator, ad te lear matrx equalty te bouded real lemma s gve as bot ecessary ad suffcet [7-8]. I NCS, we cosder te commucato betwee te sesor ad te cotroller or estmator s subject to upredctable pacet loss. We assume tat f a pacet dropped, a ew observato s tae. I [9], Motestruque proposed a Model-based NCS, ad provded te ecessary ad suffcet codtos for stablty terms of te update tme ad te parameters of te plat ad ts model, assumg tat te frequecy at wc te etwor updates te state te cotroller s costat. I te autors best owledge, te pacet drop dstrbuto as ot bee fully vestgated. Ts wor studes system stablty te cases were pacet drops follow dfferet dstrbutos. We model te urelable ature of te etwor ls as a stocastc process, ad assume tat ts stocastc process s depedet of te system tal codto ad te plat model state s updated wt te plat state at te tme we pacet arrves. Te, a model for te model-based NCS s bult up ad a ew system matrx s obtaed regardg te tervals betwee te arrved pacets followg radom dstrbutos. Te result of our study sows tat te system s stable as log as te system error s reset wt te mum update tme. Our furter study also sows tat te dstrbutos of te pacet drops affect te system stablty. Ts cocluso s demostrated examples at te ed. Ts paper s orgazed as follows. I secto, system model s set up te form of pacet losses. I secto 3, system stablty s aalyzed te cases were pacet drops follow dfferet dstrbutos. I secto 4, example s provded to verfy our cocluso. Cocluso s draw secto 5. II. SYSTEM DESCRIPTION A model-based cotrol system Fg. s cosdered, were te plat s gve as: x ( + = Ax( + Bu( ( were x( s te plat state vector. A, B are system parameter matrces. A model of te plat s bult up to provde te estmated plat state vector. Te plat model dyamcs s gve by: x ˆ( + = Ax ˆ ˆ( + Bu ˆ ( ( were x ˆ( s te estmate of te plat state, Â, Bˆ are te model matrces. We defe te modellg error matrces

2 ~ A= A Aˆ ~, ad B = B Bˆ, represetg te dfferece betwee te plat ad te model. u( Plat L u( ^ x( We cosder oly state feedbac cotroller, wc s gve by: x( Model Fgure Model-based Networed Cotrol System Model u ( = Lxˆ( (3 were L s te cotroller feedbac ga matrx. We assume tat te dyamc model s subject to te same cotrol sgal u( as te orgal system. III. STABILITY ANALYSIS As te pacets are set troug a etwor from te plat to te cotroller, te pacet drops appe radomly. Perods of pacet streams are terrupted by perods wtout pacets receved. Te tervals betwee te receved pacets vary wt a dstrbuto. Te legts of terval deped o te QoS of te etwor. Te stocastc process { γ } models te urelable ature of te etwor ls. We assume tat γ s depedet of te tal codto, x(. We reuse te prevous pacet f a pacet s ot receved. Te vector x ( s curret state x( wt probablty p f a pacet s receved. Tat gves us te followg equato: x( = γ x( (4 Altoug te cotrol pacets may ot be receved by te plat, we mae te assumpto tat te model wll always ow wtout delay weter or ot te cotrol pacet was receved. We defe te state error as: e( = x( xˆ( (5 We assume tat te plat model state x ˆ( be updated wt te plat s state x( at every, were =, s te terval betwee te receved pacets, =,,,... Te, e ( =. Now we ca wrte te evoluto of te closed loop NCS, + = A( γ (6 e( + e( - Networ x( were We defe A BL A =, Aˆ + BL ˆ A( γ = A+ BL BL A = ~ ~ ~, A+ BL Aˆ BL γ = γ = z ( =, (6 ca be represeted by e( (7 z( + = A( γ z( (8 We modeled te system as a set of lear systems, wc te system jumps from oe mode represetg by A to aoter represetg by A. We defe matrx Λ as te fucto of A, A, p ad α Λ = f A, A, p, (9 ( α were p s te probablty of te terval betwee te receved pacets, α s te pacet drop rate. O te terval, [, +, te system descrbed by (8 as te followg respose e( z( = =Λ =Λ z( e( Note tat at tme, z ( =, tat s te error s reset to zero. We ca represet ts by z ( = z(, ere I s te ut matrx wt proper dmesos ad Λ z z(, we ave ( = z( = Λ z(. = te tal codto z ( = If at system respose s: z( =Λ =Λ... =Λ Λ Λ Λ z( Λ... z( Λ, te z(

3 .e. z( =Λ Λ... Λ z( ( Te update tme vary radomly. Te frequecy at wc te etwor updates te state s ot costat. Te pacet loss may obey dfferet dstrbuted statstcal processes. Te dstrbuto of tervals, represetg by, may follow Beroull Dstrbuto process, Uform Dstrbuto, Possoa Dstrbuto, etc. We categorze te case update tme > as tme delay, ad > as pacet loss, were s data terval wtout pacet loss, mal update tme. stable. A. Beroull Dstrbuto s te > wll cause te system Bascally, radom varable, γ =, we ts l fals,.e. te pacet s lost, γ =, oterwse. γ taes value wt small probablty α, represetg pacets dropped to yeld a terval betwee te receved pacets, ad γ taes value wt bg probablty α, represetg o pacet dropout. α s a ow costat. Fg. sows te probablty fucto of a Beroull dstrbuto. -α α Probablty Te probablty mass fucto of ts dstrbuto s p K K Fgure Probablty Mass Fucto Beroull Dstrbuto = Pr( α, = = α, = γ ( = Fg. 3 sows tat a stream of pacets s terrupted wt a terval of certa legt wtout pacets. Pacets γ We defe Λ = αa + ( α A to model te jump system case te pacet loss obeys Beroull Dstrbuto process. I (, f te terval betwee te receved pacets s less ta te mum update tme,, tere are pacets arrved from te plat to update te model states before te system become stable. Te, te system eeps stable. We, pacet loss, te system becomes stable. > B. Uform Dstrbuto Te adjacet pacets may be receved a perod, followg cosecutve pacets drop out as sow Fg. 4. Pacets Te legt of tervals betwee te receved pacets may vary. We assume tat te terval varable X as ay of possble values,,,, tat are equally possble. Te probablty of ay outcome, s /, were =,,,. Te probablty fucto s defed oly at teger values of as followg. p = Pr( X = = ( Fg. 5 s te probablty mass fucto Uform Dstrbuto. Te coectg les are oly gudes for te eye ad do ot dcate cotuty. / Fgure 4 A Pacet Stream Uform Dstrbuto Probablty τ K K X τ Fgure 5 Probablty Mass Fucto Uform Dstrbuto Tme Fgure 3 A Pacet Stream Beroull Dstrbuto Tme We defe Λ= α( A + A A + ( α A to model te jump system case te pacet loss obeys Uform Dstrbuto process. We ave Λ = αa + ( α A, were α s te data dropout rate. I

4 (, f te terval betwee te receved pacets s less ta te mum update tme,, tere are pacets arrved from te plat to update te model states. Te system s stable. To eep te system stable, τ <, were τ s te bggest terval. C. Possoa Dstrbuto If pacet loss obeys a Posso dstrbuted statstcal process,.e. f successve pacet loss s depedet ad obeys ormal coutg statstcs, a sequece of pacets te terval varable X as ay of possble values,,,, probabltes p, were =,,,. Te probablty fucto s defed oly at teger values of as followg p λ λ e = Pr( X = = (3! were e s te base of te atural logartm (e = , s te factoral of, λ s te average value of X.! Fg. 6 s te probablty mass fucto Possoa Dstrbuto. m We defe Λ= α( p A + p A + ( α A = = m+ to model te jump system case te pacet loss obeys Possoa Dstrbuto process. Te, we ave Λ = αa + ( α A, were α s te data dropout rate. I (, f te terval betwee te receved pacets s less ta te mum update tme,, te system s stable. To eep te system stable, τ <, were τ s te bggest terval. However, te terval varable s attrbuted to a dscrete dstrbuto wt fte as sow Fg. 7, pacets drop out at radom pots tme. At some pots te adjacet pacets may be receved wtout terval betwee te pacets. At some pots pacets may be dropped wt bgger or smaller terval betwee te receved pacets. We, te terval may be fte wt a very small probablty..e. τ. We may fd a marg value amog te tervals Probablty K K Fgure 6 Probablty Mass Fucto Posso Dstrbuto betwee te receved pacets, τ m, we X = m. We te terval s less ta ts marg value, τ m, te λ Km X system stays stable. We te terval s greater ta te mum update tme, τ, te system becomes stable. Pacets From te dstrbutos we ca see tat f te radom varable X, wc represets te tervals betwee te pacets, s attrbuted to a dscrete dstrbuto wt fte expectato suc as Uform Dstrbuto ad Beroull Dstrbuto, te system eeps stable we te bggest terval s less ta te update tme, τ <. If X s attrbuted to a dscrete dstrbuto wt fte expectato suc as Possoa Dstrbuto, te system eeps stable for te perods ad becomes stable for te perods IV. SIMULATIONS τ > τ m. τ < τ I order to expermetally verfy te correctess of te above aalyss, we used a smple feedbac cotrol system to estmate te system respose case of pacet loss. We ow preset a example of a full state feedbac as followg: x( + = x( + u( wt te state feedbac law ( xˆ( u( = We desg a plat model usg a radom perturbato of te orgal plat matrces:.366 xˆ ( + =.4 We ave two matrces, ad A A = = τ xˆ( + u( Fgure 7 A Pacet Stream Possoa Dstrbuto Tme m

5 We assume tere s o pacet dropout ad te frequecy at wc te etwor updates te state s costat. Fg. 8 s a plot of magtude of te mum egevalues of I Λ agast update tme. From te grap t ca be see tat te mum value for s = 4. For > 4, te NCS as egevalues wt magtude larger ta oe ad terefore wll be ustable. Max egevalue Fg. 9- sow te plots of te system resposes wt tal codto z ( = case te pacet tervals are attrbuted to Beroull Dstrbuto, Uform Dstrbuto, ad Possoa Dstrbuto, respectvely. Te radom umbers, smulatg Beroull Dstrbuto pacet tervals, are as follows,, 4,,,,,,,,,,, 4,,,, 4,,, 4,,,, 4,,,,, Update Tme x(t - Fgure 8 Te Plot of Magtude of te Maxmum egevalues of te Test Matrx - -3 I te smulato, te system jumps from oe mode wt >4, represetg by A to aoter mode wt 4, represetg by A. Table sows te algortm. Table Te Algortm to Verfy te Aalyss Gve plat matrces, model matrces, ad correspodg dscrete dstrbuto. Form te matrces A ad A accordg to te cotrol sceme. Produce te radom umbers, smulatg te pacet tervals, accordg to te dfferet dscrete dstrbuto 3. Apply dfferet matrces to calculate te system respose accordg to dfferet pacet tervals f >4 else ed I A I A 4. Plot te system respose Tme Te radom umbers, smulatg Uform Dstrbuto pacet tervals, are as follows,, 3, 3,,, 4, 3, 3, 4, 3, 4,,, 3, 3,,,, 4,, 3,,, 3, 4, 3, 4, 3, 3,. x(t Fgure 9 Te System Respose wt Beroull Dstrbuto Pacet Itervals Tme Fgure Te System Respose wt Uform Dstrbuto Pacet Itervals Te radom umbers, smulatg Posso Dstrbuto pacet tervals, are as follows, (a, 3,, 3,,, 3, 3,,,,,, 4,,,,,,,,,,,, 3,,,, 3; (b 8, 8, 3, 3, 4, 4, 3, 3, 4, 4, 4, 5, 5, 5, 6, 4, 7, 5,, 4, 5,, 8, 8, 4, 5, 4, 6, 7,.

6 x(t x(t Tme (a Te Bggest Pacet Iterval s less ta 4.5 x Tme (b Te Bggest Pacet Iterval s Greater ta 4 Fgure Te System Respose wt Possoa Dstrbuto Pacet Itervals From te above smulato results we ca see tat f te pacet tervals are attrbuted to a dscrete dstrbuto wt fte support suc as Uform Dstrbuto ad Beroull Dstrbuto, te system eeps stable we te bggest terval s less ta te update tme. If tey are attrbuted to a dscrete dstrbuto wt fte support suc as Possoa Dstrbuto, te system eeps stable fte tme. We tme creases, te pacet terval rage creases. Te, te system becomes stable. V. CONCLUSION I ts paper, te stablty problem NCS wt upredctable pacet drops as bee vestgated. Te result of our study sows tat te system s stable as log as te system error s reset wt te mum update tme. Our furter study also sows tat te dstrbutos of te pacet drops affect te system stablty. If te pacet drop follows dscrete dstrbuto wt fte expectatos suc as Uform Dstrbuto ad Beroull dstrbuto, te system s asymptotcally stable we te mum tme terval betwee te receved pacets s uder te mum update tme. If te pacet loss follows dscrete dstrbuto wt fte expectato suc as Posso Dstrbuto, te stable stocastc system s asymptotcally stable, we te mum tme terval betwee te receved pacets s lmted to te mal update tme. Ts cocluso s demostrated examples at te ed. REFERENCES [] Mru Fe, Ju Y, ad Huoseg Hu, Robust Stablty Aalyss of a Ucerta Nolear Networed Cotrol System Category, Iteratoal Joural of Cotrol, Automato, ad Systems, Vol. 4, No., pp. 7-77, Apr. 6. [] Joa Nlsso, Real-Tme Cotrol Systems wt Delays, PD Tess, Uversty of Toroto, Caada, 3. [3] Vjay Gupta, Baba Hassb ad Rcard M. Murry, Optmal LQG Cotrol across Pacet droppg Ls, System & Cotrol Letters, Jue 7, Vol 56, pp [4] Jo K. Yoo, Daw M. Tlbury ad Nadt R. Soparar, Tradg Computato for Badwdt: Reducg Commucato Dstrbuted Cotrol Systems Usg State Estmators, IEEE Trasactos o Cotrol Systems Tecology, July, Vol, No. 4, pp [5] Carlos E. de Souza ad Marcelo D. Fragoso, Robust H flterg for ucerta Marova jump lear systems, Iteratoal Joural of Robust ad Nolear Cotrol, Vol., pp , Ja. [6] Marcelo D. Fragoso ad Carlos E. de Souza, H flterg for Marova jump lear systems, Te 35t coferece o Decso ad Cotrol, Kobe, Japa, December 996. [7] A. K. Fletcer, S. Raga, ad V. K Goyal, Estmato from Lossy Sesor Data: Jump Lear Modellg ad Kalma Flterg, Trd Iteratoal Symposum o Iformato Processg Sesor Networs, Bereley, Calfora, USA, 6-7 Apr. 4, pp [8] Alyso K. Fletcer, Sudeep Raga, Vve K Goyal, ad Kaa Ramcadra, Robust Predctve Quatzato: A New Aalyss ad Optmzato Framewor, ISIT 4, Ccago, USA, Jue 7-July, 4, pp. 49. [9] Lus A. Motestruque, Model-Based Networed Cotrol Systems, PD Tess, Uversty of Notre Dame, November 4.