Estimation and Control under Communication Network Constraints

Transcription

1 Etimatio ad Cotrol uder Commuicatio Network Cotrait A Thei Preeted by Bei Ya to The Departmet of Electrical ad Computer Egieerig i partial fulfillmet of the requiremet for the degree of Mater of Sciece i Electrical Egieerig i the field of Commuicatio ad Sigal Proceig Northeater Uiverity Boto, Maachuett May 00

2 Copyright 00 by Bei Ya All Right Reerved

3 Abtract Motivated by ditributed cotrol ad eor etwork applicatio i Electric Eergy Sytem, we coider the problem of etimatio via a commuicatio etwork. Whe data i et via commuicatio chael i a large, wirele, multi-eor etwork, the effect of commuicatio cotrait o etimatio performace, uch a commuicatio delay ad aychroou irregular amplig, he to be coidered. I thi thei we formulate a delay mitigatio method baed o a Kalma filter with time-tampig techology, which tramute commuicatio delay ito icreaed etimatio error, o that the cloed-loop cotrol ytem remai table eve i the preece of igificat delay. The reultig igal to etimatio error ratio (SEER), at ay poit i the etimatio-cotrol loop, i a mootoe decreaig fuctio of the erage commuicatio delay. Whe the eor amplig patter (SSP) are irregular ad aychroou, the SEER i a mootoe decreaig fuctio i each oe of the erage amplig iterval, with very mior depedece o higher order momet of thi iterval. The bet performace i achieved whe the idividual SSP are regular ad aliged i uch a way that the uperpoed et of amplig itat i a cloe to regular a poible. I particular, ychroized regular amplig (i.e., all eor ampled regularly at the ame time itat) i iferior to uiformly taggered regular amplig, i which the amplig itat of idividual eor are paced evely withi a igle amplig iterval. i

4 Ackowledgemet It took almot two year to write thi thei, ad fortuately the upport ad guidace from may ource durig thi log time helped me fiih it a well a poible. I my mid, the aitace ha bee academic ad emotioal. To thee people I am thakful. I would like to expre my gratitude to my advior, Profeor Haoch Lev-Ari, whoe guidace ad upport from the iitial to the fial level eabled me to make progre o reearch ad fiih the thei a well. I he a great fortue to work with him, ad hi patiece ad motivatio helped me to overcome the may challege I ecoutered i the proce of carryig out my reearch ad writig thi thei. I am really thakful for the time he pet o my reearch ad thei ad all the help he offered. Beide my advior, I would like to thak Profeor Alekadar Stakovic who itroduced me to a ew ad excitig reearch area: etworked etimatio ad cotrol, which became my reearch ubject for a log time. Latly, I offer my regard ad bleig to all my fried who upported me i ay repect durig the completio of the thei. ii

5 Cotet Abtract...i Ackowledgemet... ii Lit of Figure...v Lit of Table...vii Chapter Itroductio.... Spatially-Ditributed Cyber-Phyical Sytem Cotiuou-Dicrete Kalma Filter Cotiuou-Dicrete Cotroller....4 Thei Cotributio ad Structure... 4 Chapter Delay-Mitigatig Etimatio ad Cotrol...6. Delay i Obervatio Path Delay-Mitigatig Cotrol Example: cotrollig a imple utable ytem....4 Cocludig Remark... 3 Chapter 3 Effect of Seor Samplig Patter Seor Samplig Patter Claificatio The Effect of SSP o Kalma Filter Performace The Dyamic of Pt () Kow Performace Boud Simulatio Reult: igle-eor etimatio with irregular SSP Regular Samplig Irregular Samplig iii

6 3.3.3 The Effect of Samplig Iterval Variace Idex The Effect of Kurtoi of The Samplig Iterval Simulatio Reult: two-eor etimatio with irregular SSP Effect of SSP Irregularity Effect of Seor Differece Summary of Reult... 6 Chapter 4 Cocludig Remark Summary of Reult Further Reearch Appedix A Averaged Time Update...68 Appedix B Diagoal Scalig Lemma...7 Appedix C The Characteritic Fuctio ad Statitic...73 Appedix D Perturbatio Aalyi of P,...78 Referece...80 iv

7 Lit of Figure Fig.. Structure of hared-etwork cotrol ytem... 5 Fig.. The firt approach of remote cotrol ytem via a etwork... 6 Fig..3 The ecod approach of remote cotrol ytem via a etwork... 6 Fig..4 The flow figure of remote cotrol ytem via a etwork... 7 Fig..5 The evolutio of Pt ()... Fig..6 State etimator baed feedback cotrol loop... 4 Fig.. Networked etimatio/cotrol cofiguratio... 0 Fig.. Effect of delay mitigatio o SEER, i the preece of obervatio oie. Fig..3 Effect of delay mitigatio o SEER, i the preece of ytem parameter iaccuracy..3 Fig. 3. Periodically-variat SSP... 8 Fig. 3. Probability deity fuctio of uiform ditributio... 8 Fig. 3.3 Probability deity fuctio for the geometric ditributio... 9 Fig. 3.4 Fig. 3.5 P, from equatio v F i irregular SSP with calar parameter P, v F i regular SSP with utable ytem Fig. 3.6 SEER v. T (regular amplig) Fig. 3.7 Effect of eor choice o etimatio performace (regular amplig) Fig. 3.8 SEER v. T (uiform radom ditributio amplig with 0.3 ) Fig. 3.9 Effect of relative variace idex o SEER (top) ad zoom-i (bottom): periodically-variat SSP Fig. 3.0 Effect of relative variace idex o SEER (top) ad zoom-i (bottom): uiformly-ditributed radom SSP Fig. 3. Effect of relative variace idex o SEER (top) ad zoom-i (bottom): comparig both SSP choice Fig. 3. Kurtoi of the mixed SSP... 5 v

8 Fig. 3.3 Effect of amplig iterval kurtoi o SEER (top) ad zoom-i (bottom): covex combiatio of two periodically-variat SSP... 5 Fig. 3.4 Etimatio performace ( trace( P, ) ) a a fuctio of F ad F Fig. 3.5 Cotour plot of Fig Fig. 3.6 Effect of relative hift o etimatio performace: FF Fig. 3.7 Effect of relative hift o etimatio performace: F F Fig. 3.8 The effect of eor differece o etimatio performace Fig. 3.9 The effect of eor differece i aother apect... 6 vi

9 Lit of Table Table. The cotiuou-dicrete Kalma filter... Table 3. Statitic of the SSP Table 3. Characteritic fuctio of the SSP Table 3.3 Kurtoi with chagig ad fixed variace idex Table 3.4 Combiatio of amplig rate i eor ad vii

10 Chapter Itroductio Proce i commuicatio techology ha opeed up the poibility of large cale cotrol ytem i which the cotrol tak i ditributed amog everal proceor, eor, etimator ad cotroller itercoected via commuicatio chael. Such cotrol ytem may be ditributed over log ditace ad may ue a large umber of actuator ad eor. The combiatio of a patially-ditributed phyical (aalog) ytem with a digital commuicatio etwork impoe uique cotrait that affect the deig of uch a cyber-phyical ytem (CPS). Seor data eed to be ampled ad quatized o it ca be tramitted over digital commuicatio lik with limited capacity. Commuicatio delay [] ha igificat effect o the quality of etimatio, degrade the performace of etworked cotroller, ad ofte reult i lo of tability. Data packet may get lot, or i cae of retramiio, are received with a added delay. Fially, the propoed ue of large umber of eor i patially-ditributed cyber-phyical ytem may require accommodatio of differet, aychroou, amplig rate. Motivated by ditributed cotrol ad Electric Eergy Sytem applicatio, we coider the problem of etimatio ad cotrol via commuicatio etwork uig Kalma filterig []. Whe data i et via ureliable commuicatio chael i a large, wirele, multi-hop eor etwork, the effect of commuicatio delay ad lo of iformatio i the cotrol loop caot be eglected. We aalyze thi problem uig a cotiuou-time dicrete-meauremet Kalma filter, meaure the delay with time-tamp techology ad model the arrival of obervatio a a radom proce. We tudy the tatitical covergece propertie of the etimatio error covariace, howig ome reult about the relatiohip betwee commuicatio delay, eor amplig patter ad etimatio error covariace. Numerou reult he bee publihed withi the broad area of etworked etimatio ad cotrol. Mot of thee he focued o the effect of dropped commuicatio packet or, more geerally, the impact of irregular eor amplig time patter [3]. Oly a few publihed reult addre the challege of irregular obervatio delay ad it effect o the quality of tate etimatio ([4], [5]). Ad eve

11 fewer he coidered the poibility of overcomig the detabilizig effect of delay i a etworked cotrol loop. I Ch. we aalyze the effect of delay i electric power ytem a a pecial applicatio. Electric power ytem are amog mot patially exteded egieered ytem. Their table operatio i the preece of diturbace ad outage i made poible by real-time cotrol over commuicatio etwork. Thi mode of operatio i likely to expad i the future, a mai eabler for developmet of utaiable eergy ytem are i the iformatio layer. Example iclude eig, etimatio ad cotrol of compoet i the eergy layer. We propoe a delay cacelig approach baed o time-tampig of tramitted igal, which ca uccefully mitigate the detabilizig effect of delay if we commuicate tate etimate rather tha cotrol igal from the etimatio/cotrol ceter to the actuator. By time-tampig we mea that every eor ample xk () t i tramitted over the etwork a a ordered triplet ( x, t, k ), coitig of the ample value ' x ', the correpodig amplig time itat ' t ' ad the idetifier ' k ' of the eor. Time tampig i eabled by the ailability of precie time igal derived from the Global Poitioig Sytem (GPS) i variou ode of the power ytem. Such igal are part of Phaor Meauremet Uit (PMU) that are icreaigly beig deployed a a key igrediet of ytem-wide moitorig ad cotrol. Our delay cacelig method tramute commuicatio delay ito icreaed etimatio error, o that the erage igal to oie ratio of ay igal i the etimatio-cotrol loop i a mootoe decreaig (ad, i fact early liear) fuctio of the erage commuicatio delay. We demotrate the utility of our approach via a example. I particular, we how that our time-tampig-baed delay cacelig method make it poible to overcome very large delay i the cotrol path while oidig advere effect o the tability of the ytem. I our example the delay ca be 0-0 time higher tha the mallet value that will caue itability i the origial cotrol path. Moreover we aalyze the effect of eor amplig patter (SSP) o the etimatio performace. A regular SSP i completely characterized i term of it amplig iterval T. A irregular SSP ha a time-varyig amplig iterval T ( ), which we chooe to characterize i term of it momet: the erage ad the variace of T ( ), a well a higher-order momet, a eeded. We how i Ch. 3 that the erage igal to etimatio error ratio (SEER) deped primarily o the

12 erage amplig iterval, with a very mior depedece o the variace ad a egligible depedece o higher-order momet of the time-varyig amplig iterval. The bet performace (i.e., mallet etimatio error) for a igle-eor ytem i achieved whe the SSP i regular, o that all it higher-order momet vaih. The ame cocluio hold alo for a Kalma filter etimatio that employ multiple eor: the erage etimatio error covariace i a mootoe decreaig fuctio i each oe of the erage amplig rate. I additio, the erage etimatio error covariace deped alo o the relative aligmet of the idividual SSP. The bet performace i achieved whe the idividual SSP are regular ad aliged i uch a way that the uperpoed et of amplig itat i a cloe to regular a poible.. Spatially-Ditributed Cyber-Phyical Sytem The broad ubject of thi thei i etimatio ad cotrol i Spatially-Ditributed Cyber-Phyical Sytem. Cyber-Phyical Sytem (CPS) are itegratio of computatio with phyical procee, uch a the electrical power grid. Cyber-Phyical here mea that the ytem coit of two layer, oe i phyical (eergy flow) layer which i aalog ad the other i cyber (iformatio fl ow) layer which i digital. The iterface betwee the two layer i provided by eor ad actuator. Etimatio ad cotrol algorithm are implemeted digitally i the cyber layer [6]. Spatially-Ditributed mea that the ytem i pread over a large phyical ditace [7]. Thi eceitate to ue a commuicatio etwork a part of the cyber layer to coect betwee eor, etimatio/cotrol ceter ad actuator. It alo mea that the ytem, i geeral, would exhibit we-propagatio pheomea ad may eed to be decribed by partial differetial equatio. I thi thei, we oly coider ytem that ca be decribed by a lumped, liear model. Specifically we aume a tate pace model a below x( t) Ax( t) Bu( t) Gw( t) (.) where xt (), ut () ad wt () are the tate, iput ad proce oie, repectively. We hall aume that ut () ad wt () are ucorrelated, ad that wt () i white 3

13 Gauia, o that E{ w( t) w( )} Q ( t ). Due to the patial ditributio, we ca ay the eor ad actuator are ot co-located, ad i geeral the A/D tep (amplig) may he differet, o-ychroou rate at differet eor. Furthermore, the commuicatio etwork may caue occaioal lo of iformatio (packet drop), which mea that ome of the ample could be miig i the etimatio tep. Thu, i geeral, amplig patter could be mildly irregular. We will deote the amplig itat for eor k a ( k ) t, o that the meauremet received at the cetral etimator are give by y t C x t (.) ( k) ( k) k ( ) k ( ) k ( ) So eve if each eor ue regular amplig, the tream of meauremet received at the cetral etimator i till irregular, i geeral. The ychrophaor tadard (a evolvig IEEE tadard for power ytem) will force all ampler to ue a GPS to get ychroized meauremet, reultig i a regular, ychroized tream of obervatio from all eor. Alo uig a TCP-IP protocol which ivolve retramiio of dropped packet will elimiate the pheomeo of packet droppig at the expee of icreaed commuicatio delay. Neverthele, oe of our objective i thi thei i to demotrate that reliable tate etimatio ca be obtaied eve i the preece of o-ychroou, irregular, multirate eor amplig. I particular, we how that etimatio performace deped maily o the erage amplig rate, ad i oly mildly affected by the irregularitie i the amplig patter. Spatially-Ditributed Cyber-Phyical Sytem are kow i the cotrol ytem literature a Networked Cotrol Sytem (NCS) [8]. Geeral peakig, there are two major type of NCS architecture, hared-etwork cotrol ytem ad remote cotrol ytem. Uig hared-etwork to trafer data from eor to cotroller ad from cotroller to actuator ca greatly reduce the complexity of the whole ytem. Thi tructure, a how i Fig.., provide more flexibility i itallatio ad troublehootig. Furthermore thi tructure i epecially ueful whe a cotrol loop eed to exchage iformatio with other cotrol loop. 4

14 Fig.. Structure of hared-etwork cotrol ytem O the other had, a remote cotrol ytem i a ytem cotrolled by a cotroller located far away from it. There are two geeral approache to deigig a remote cotrol ytem through a etwork. The firt tructure i to he everal ubytem, i which each of the ubytem cotai a et of eor, a et of actuator, ad a cotroller by itelf. Thee ytem compoet are attached to the ame cotrol plat, a how i Fig... I thi cae, a ubytem cotroller receive a et poit from the cetral cotroller. Aother tructure i to coect a et of eor ad a et of actuator to a etwork directly. Seor ad actuator i thi cae are attached to a plat (phyical layer), while a cetral cotroller i eparated from the plat via a etwork coectio to perform a cloed-loop cotrol over the etwork, a how i Fig..3. 5

15 Fig.. The firt approach of remote cotrol ytem via a etwork Fig..3 The ecod approach of remote cotrol ytem via a etwork Both tructure he their pro ad co. The firt oe (Fig..) i more modular. A cotrol loop i impler to be recofigured. The ecod oe (Fig..3) ha better iteractio becaue data are tramitted to compoet directly. The mai 6

16 differece of thee two type of architecture i iformatio layer, a the firt oe ha the hierarchical iformatio layer ad the ecod oe ha the cetralized. A cotroller i the ecod tructure ca oberve ad proce every meauremet, wherea a cotroller i the firt tructure may he to wait util the et poit i atified to trafer the complete meauremet, tatue igal, or alarm igal [9]. For the ake of better performace, we ue the ecod tructure to realize our cotrol ytem. I additio to a irregular tream of obervatio, the ue of a commuicatio etwork withi the cyber layer may caue delay both i the path from eor to cetral cotroller ad i the path from cetral cotroller to actuator, a how i Fig I thi thei, we focu o the effect of delay i the cotrol path, ad we how, i Ch., that uch effect ca be mitigated by Kalma filterig with time-tamp techology which we will explai i detail i Ch.. Fig..4 The flow figure of remote cotrol ytem via a etwork. Cotiuou-Dicrete Kalma Filter The tate pace model decribed i (.) ad (.) i kow a a cotiuou-dicrete model i which the tate equatio i i cotiuou time ad the meauremet equatio i i dicrete time. The reao for a cotiuou-dicrete model i that we 7

17 are itereted i CPS with their characteritic two-layer tructure: ) A phyical ytem that live i the real world, i cotiuou time. ) A etimatio ad cotrol ytem that live i cyberpace, amely i implemeted i a computer/digital proceor, i dicrete time. Seor ad actuator act a iterface betwee the two layer, ad they tur aalog igal ito a digital format ad vice vera. The Cotiuou-Dicrete Kalma filter addree the geeral problem of tryig to etimate the tate of a cotiuou time ytem from dicrete-time meauremet. We will ue the cotiuou time liear tate equatio (.), a dicued i ectio.. It will be eaier to decribe the operatio of the cotiuou-dicrete Kalma filter if we iterpret the meauremet equatio (.) a a equivalet igle-eor time variat meauremet equatio, viz., equatio (.3). y( t ) C( t ) x( t ) ( ), (.3) where { t } i the uperpoitio [0] of the idividual amplig equece ( k ) t. A uperpoitio of two ordered equece i the ordered uio of the two idividual et, alog with a et of label that idicated the origi of each elemet i the combied equece. For itace, the uperpoitio of S {, 3, 7,8} ad S {,5,6,9} i the two-row matrix The label below each elemet of the combied equece tell u whether it came from S or S. I cae of a tie, whe two or more elemet i the firt row he the ame value, their label ca be arraged i ay order. Whe we uperpoe the amplig equece ( k) { ti ; i }, we will refer to the elemet of the combied equece a { t ; } ad to their label a { S S( t ); }. Alo ( ) repreet the meauremet oie, aumed to be idepedet, white, with a Gauia Ditributio, o that E{ ( ) ( k)} R( ) ( k) Sice ( ) (i.3) repreet the meauremet oie at time itat t, ad each eor may he a differet level of meauremet oie, R ( ) deped o the eor, amely R( ) Rk whe S( t) k. I other word, we aume 8

18 eor-depedet but tatioary meauremet oie i our model. The cotiuou-dicrete tate pace model (.) ad (.3) give rie to a tadard cotiuou-dicrete Kalma filter, a decribed i [], which alterate betwee a time-update proce (i cotiuou time) ad a meauremet-update tep (i dicrete time), a follow. Meauremet update: Whe a meauremet become ailable at t t, we carry out the update: x( t ) x( t ) K( t ) v( t ) (.4) Where x( t ) E[ x( t ) { y( t ), t t }] i the poterior etimate of xt ( ) from all i i meauremet acquired up to (ad icludig) the time itat t, ad x( t ) E[ x( t ) { y( t ), t t }] i the prior etimate of the ame xt ( ) baed o all i i meauremet precedig the time itat t. Alo vt ( ) i the iovatio proce aociated with the meauremet yt ( ), ad i determied via v( t ) y( t ) C( t ) x( t ). Fially, the Kalma gai Kt ( ) i give by K t P t C t t, where P ( t) E{[ x( t) x( t)][ x( t) x( t)] } * ( ) ( ) ( ) ( ) * ad ( t ) E v( t ) v( t ) C( t ) P ( t ) C ( t ) R( ). To complete the meauremet update tep, we update alo the error covariace, viz., P t P t K t t K t (.5) * ( ) ( ) ( ) ( ) ( ) where P ( t) E{[ x( t) x( t)][ x( t) x( t)] } Notice that if everal meauremet occur at the ame time itat, ay t t t t t, the the Kalma filter ha to carry out m m m coecutive meauremet update at each time itat { t ; i m} without ay time update i betwee. i Time update: Betwee meauremet update, the Kalma filter carrie out a cotiuou-time time update a follow: 9

19 xˆ( t) Ax( t) Bu( t) P( t) AP( t) P( t) A GQG (.6) where x( t) E{ x( t) y( t ); t t} ad P( t) E{[ x( t) x( t)][ x( t) x( t)] }. Notice that for ay time t which i ot a meauremet itat, there i oly oe type of tate etimate x( t) x( t) x( t). The differetial equatio (.6) are iitialized with x ( t ) ad P ( t ) at time t, ad evolve i the iterval t t t. Notice that the time update ad meauremet update of Pt () caot be computed i advace, becaue they deped o the locatio of the amplig itat t ad thi iformatio i ot ailable apriori. Thi alo mea that the error covariace Pt () ha to be iterpreted a beig coditioal o the time patter, epecially whe t are radom. Thu, a more accurate defiitio of the tate error covariace matrix Pt () i i term of a coditioal mea, amely P( t) E{[ x( t) x( t)][ x( t) x( t)] t ; t t }. For computatioal purpoe, it i more coveiet to replace (.6) by equivalet itegral expreio a follow: ( ) A( tt ) tt A ( ) ( ) 0 x t e x t e Bu t d A( tt) tt A ( tt) A A 0 P( t) e P ( t ) e e GQG e d (.7) I particular, we will ue thee itegral equatio i our performace aalyi i Ch. 3 to evaluate P ( t ) ad t t A( tt ) A ( tt ) A A ( ) lim ( ) ( ) tt 0 P t P t e P t e e GQG e d. Notice that Pt (), a give by (.7), i a mootoe icreaig fuctio of time oce the iitializatio traiet he died off. O the other had, the meauremet update tep (.5) reduce the error covariace from P ( t ) to P ( t ), a illutrated i Fig..5. 0

20 Fig..5 The evolutio of Pt () A complete litig of the cotiuou-dicrete Kalma filter equatio i provided i Table.. The meauremet update tep are carried out at the time itat t ;. Each time update tep tart with t t, ad evolve i the cotiuou iterval t t t. Sice there are o meauremet i the iterval t 0 t t, the Kalma filter tart it operatio with a time update tep, uig the 0 iitializatio x( t0) 0 ad P t E x t x t * ( 0) { ( 0) ( 0)} 0. Table. The cotiuou-dicrete Kalma filter Meauremet Update v( t ) y( t ) C( t ) x( t ) * ( t) C( t) P ( t) C ( t) R( ) K t P t C t t * ( ) ( ) ( ) ( ) x( t ) x( t ) K( t ) v( t ) P t P t K t t K t * ( ) ( ) ( ) ( ) ( )

21 Time update t t t ( ) A( tt ) tt A ( ) ( ) 0 x t e x t e Bu t d A( tt) tt A ( tt) A A 0 P( t) e P ( t ) e e GQG e d I Ch. 3, our mai iteret i to obtai a compact characterizatio of the relatio betwee the eor amplig patter t ; ad the error covariace matrix Pt (). For thi purpoe we focu oly o the teady-tate value of the two equece P ( t); ad P ( t);, ice thee equece provide lower ad upper boud, repectively, o the itataeou Pt (). I ome cae (e.g., whe the amplig patter i regular) the equece P ( t ) coverge to a limit value a, ad o doe P ( t ). However, i geeral, thee equece may he a ocillatory behior eve whe, o we chooe to defie N P E{ P ( t )} lim E{ P ( t )} (.8), N N where idicate a log-term time-erage with repect to the dicrete idex (a o called Cearo limit []). Notice that the expectatio operatio E {} i carried out with repect to the radome i the amplig itat t. Whe the amplig patter i determiitic, oly the time erage i eeded. The time-ivariat P, are ued i Ch.3 to provide a meaigful meaure of etimatio performace with repect to amplig patter..3 Cotiuou-Dicrete Cotroller I cotrol theory, oe baic cotrol law i feedback cotrol. The cotrol iput uc() t

22 i aumed to be a itataeou fuctio of the plat tate xt (), ad we could write the cotrol law a u ( t) k( x( t), t) to idicate the depedece of u () t o c both xt () ad t. A we aume a liear ytem model i thi thei, the optimal c olutio to the LQG (Liear Quadratic Gauia) cotrol problem i u ( ) ( ) ˆ ( ) c t K t x t, where Kt () i the cotroller' gai ad xt ˆ( ) i the optimal (i MSE ee) tate etimatio becaue we ca t get the tate directly, ad what we he i the etimated tate itead of real tate. Alo I thi thei, becaue the ytem parameter are cotat, the cotroller' gai i time-ivariat, a K. What make the problem here differet from LQG i that we itroduce delay i the cotrol path. Coequetly, it i o loger obviou what the optimal cotrol olutio i. I ay cae, i thi thei, we aim to mitigate the effect of delay i the cotrol path, o that the LQG optimal cotrol olutio u ( ) ( ) ˆ ( ) c t K t x t i till applicable (optimal or early optimal). I reality, there will alo be delay i the eig path, ad it will caue additioal error i the tate etimatio xt ˆ( ). However, i thi thei, we will aume that there i o delay i the eig path, o that we ca focu o the effect of delay i the cotrol path ad itroduce our techique to mitigate thi delay. We alo coider the quetio which igal we are goig to ed back from cetral etimator to actuator: the tate etimate xt ˆ( ), or the cotrol igal u ( ) ( ) ˆ ( ) c t K t x t. There will be a igificat differece betwee thee two igal, epecially i dimeioality. For a igle actuator cae, uc() t i a calar, but xt ˆ( ) i a vector. I a complicated ytem like the power ytem, xt ˆ( ) may coit of te, eve hudred of tate. However, we chooe to tramit xt ˆ( ) itead of u () t. Firt of all, our delay mitigatio cheme relie o uig xt ˆ( ) ad hig a reaoable accurate ytem model. Secodly, the proceig ca be combied with the actuator D/A uit which covert digital tramiio ito a cotiuou-time cotrol igal. The architecture of State-Etimator-Baed Feedback Cotrol i howed i Fig..6. c 3

23 Fig..6 State etimator baed feedback cotrol loop The preece of delay i the cotrol path degrade the quality of the cotrol igal ad may evetually lead to lo of tability. The delay mitigatio techique that we propoe i Ch. make it poible to overcome very large delay i the cotrol path without lo of tability. I fact, we how that the SEER i the cloed loop ytem decreae almot liearly (i db/ec) a the erage delay icreae..4 Thei Cotributio ad Structure I thi thei we propoe a ew method for mitigatio of delay i the cotrol path of a patially-ditributed cyber-phyical ytem. I particular, thi approach of delay cacelig baed o time-tampig of tramitted igal ca uccefully mitigate the detabilizig effect of delay. Actually the method tramute commuicatio delay ito icreaed etimatio error, o that the erage SEER of ay igal i the etimatio-cotrol loop i a mootoe decreaig (ad, i fact early liear) fuctio of the erage commuicatio delay. We demotrated the utility of our approach via a imple example. I particular, we howed that the value of delay i the cotrol loop ca be icreaed by a factor of 0 beyod the critical value 0.8 with oly a moderate degradatio (0 db) i SEER. While loger delay ( alo be tolerated, the reultig degradatio i etimator quality ( cr 0 cr ) ca 0dB ) may ot be 4

24 acceptable i ome applicatio. We alo how that the relatio betwee erage SEER ad erage commuicatio delay doe ot chage whe additioal white oie i applied at eor output ad commuicatio chael. I particular, 0dB oie brig aroud 0dB decreae i SEER for each erage delay. Moreover, we obtai imilar reult if there i iaccuracy i the ytem parameter. Eve more, we fid that the SEER decreae i db i liear with the percetage of iaccuracy of ytem parameter. I additio, we alo aalyze the performace of a (Kalma filter) tate etimator i the preece of irregular eor amplig. I particular, for oe eor amplig, we fid that the erage SEER i a mootoe decreaig fuctio of the erage amplig iterval. More pecifically, we how i detail that the erage SEER maily deped o the erage amplig rate, with very weak depedece o higher order momet (i.e. variace ad kurtoi) of the amplig iterval. Furthermore, for the two-eor cae, we alo fid that the erage error covariace i a mootoe decreaig fuctio of the erage amplig rate of the two eor. From above, we ca ay that the mot igificat factor that decide the etimatio performace i the erage amplig rate of each eor. Thi thei i arraged i the followig order: Ch. propoe our delay cacelig method that tramute commuicatio delay ito icreaed etimatio error. With thi method, we improve the cotrol performace whe delay i hort ad tabilize the ytem whe delay i log. Ch. 3 demotrate the relatio betwee etimatio performace ad the tatitic of amplig patter (regular ad irregular). Fially, Ch. 4 i dedicated to cocluio ad a brief dicuio o poible directio for future reearch. 5

25 Chapter Delay-Mitigatig Etimatio ad Cotrol Commuicatio delay i oe of the uoidable ide-effect of uig a computer etwork to itercoect eor, actuator ad cotrol ceter/ode. The time to read a eor meauremet ad to ed a cotrol igal to a actuator through the etwork deped o etwork characteritic uch a topology, routig cheme ad etwork coditio. The overall performace of a etwork-baed cotrol ytem ca be igificatly affected by etwork delay. The everity of the delay problem i aggrated whe data lo occur durig a tramiio. Moreover, delay ot oly degrade the performace of a etwork-baed cotrol ytem, but ca alo reult i itability. I thi chapter, we aalyze the effect of delay i electric power ytem a a pecial applicatio. Electric power ytem are amog mot patially exteded egieered ytem. Their table operatio i the preece of diturbace ad outage i made poible by real-time cotrol over commuicatio etwork. Thi mode of operatio i likely to expad i the future, a mai eabler for developmet of utaiable eergy ytem are i the iformatio layer. Example iclude eig, etimatio ad cotrol of compoet i the eergy layer. We propoe here a delay cacelig approach baed o time-tampig of tramitted igal, which ca uccefully mitigate the detabilizig effect of delay if we commuicate tate etimate rather tha cotrol igal from the etimatio/cotrol ceter to the actuator. By time-tampig we mea that every eor ample xk () t i tramitted over the etwork a a ordered triplet ( x, t, k ), coitig of the ample value ' x ', the correpodig amplig time itat ' t ' ad the idetifier ' k ' of the eor. Time tampig i eabled by the ailability of precie time igal derived from the Global Poitioig Sytem (GPS) i variou ode of the power ytem. Such igal are part of Phaor Meauremet Uit (PMU) that are icreaigly beig deployed a a key igrediet of ytem-wide moitorig ad cotrol. Our delay cacelig method tramute commuicatio delay ito icreaed 6

26 etimatio error, o that the erage igal to oie ratio of ay igal i the etimatio-cotrol loop i a mootoe decreaig (ad, i fact early liear) fuctio of the erage commuicatio delay. We demotrate the utility of our approach via a example. I particular, we how that our time-tampig-baed delay cacelig method make it poible to overcome very large delay i the cotrol path while oidig advere effect o the tability of the ytem. I our example the delay ca be 0-0 time higher tha the mallet value that will caue itability i the origial cotrol path. I additio we examie the effect of modelig error ad additive oie o our delay mitigatio cheme. Both cotribute a further degradatio i the overall SEER.. Delay i Obervatio Path The optimum MSE etimate of the tate xt () i xˆ ( t) xˆ ( t y ( t)), amely the projectio of the tate xt () o the meauremet ubpace y. ( k) ( k) ( t) pa{ yk ( t ); k,... K, t t} I the preece of delay i the obervatio path, the meauremet y t ( k ) k( ) become ailable oly at the time itat t, where ( k) ( k) i the ( k ) (geerally time -variat) delay experieced i the tramiio of y t from the ( k ) k( ) k -th eor to the etimatio/cotrol ceter. A a reult, the optimum MSE etimate of xt () i the preece of kow delay i ˆ ( ) ˆ( ( )), amely the ( d) ( d) x t x t y t projectio of the tate xt () o the delayed meauremet ubpace y ( t) pa{ y ( t ); k,... K, t t}. (d) ( k ) ( k ) ( k ) k Sice ( y d ) () t i a ubpace of y () t, it follow that ( d ) xˆ () t ha a larger error covariace, amely ( d P ) ( t) P( t) for all t. We ca make a more detailed tatemet about the relatio betwee P ( d ) () t ad Pt () by carefully examiig the relatio betwee ( d ) xˆ () t ad xt ˆ( ). I 7

27 particular, whe i ot too large, i.e., t t t, we ca explicitly olve the differetial equatio (.), viz., d x ( d) ( d) ˆ t Ax ˆ t dx ( ) ( ) d P ( d ) ( ) ( ) ( t ) AP d d ( t ) P ( t ) A BQB dx So that ˆ ( d ) A x t e x t (.) ( ) ˆ ( ) (.) ( d ) A A ( ) ( ) A A 0 P t e P t e e GQG e d (.3) Thi reult allow u to compare the erage value of ( d P ) ( ) with the erage value of P ( t ). A we metioed i Sec.., P, N lim E[ P( t)], N N amely, it i both time eraged a well a eemble eraged with repect to poible radome of the amplig time itat. Similarly, we defie the erage delayed error covariace lim [ ( )]. To oid ueceary N ( d) ( d), E P N N P complicatio, we aume here that the matrix A i diagoalizable, o that 0 A T T 0 N, where i are the eigevalue of A ad T i a o-igular (quare) matrix. Uig the approach decribed i App. A, we ca obtai a imple relatio betwee ( d ) P, ad P,, viz., ( d ) * T P, T T P, T ( T GQG T ) (.4) where [ ( )] M deote the Schur-Hadamard product, ad l m l, m [ ( ( ) )] M l m l, m l m Here () i the eraged characteritic fuctio aociated with the ditributio 8

28 of the delay, amely N ( ) E{ e } lim e p( ) d N N, where we allow for both radom ad/or determiitic variatio i. The expreio (.4) provide a explicit relatio betwee ( d ) P,, the erage error covariace i the preece of delay, ad P,, the delay-free verio of the ame. Our prelimiary qualitative aalyi implie that P P ( d ),,, regardle of the ature of the delay equece. For itace, whe the delay i cotat (i.e., ) ad relatively mall, we fid that ( ) e. With the coditio max( Re( i ) ), we he ( l m ) ( l m ) ad i ( l m ). l m o that ad. Thi reduce (.4) to P P GQG (.5) ( d ) *,, which how clearly that the added covariace error equal * GQG, o that ( d ) P, icreae with icreaig delay. A a matter of fact, P will get cloe to ( d ), P iitial a the delay icreae if the ytem i table, otherwie P will ( d ), icreae to ifiity if the ytem i utable.. Delay-Mitigatig Cotrol I thi ectio, we coider the problem with delay i cotrol path ad a delay free i obervatio path. The etwork etimatio ad cotrol cofiguratio that we coider here coit of a igle etimatio ceter that collect obervatio from everal eor ad provide etimated tate etimatio to oe, or more, actuator. A commuicatio etwork i ued to tramit obervatio from the eor to the cetral etimatio ceter, ad to deliver tate etimate iformatio from thi etity 9

29 to idividual actuator. We aume that each actuator i equipped with a cotrol module that tralate the tate etimate iformatio received from the etimatio ceter ito a appropriate cotrol igal, a how i Fig... For the ake of preetatio implicity we limit our dicuio here to the cae of a igle actuator; however, our approach ca be ued with ay umber of eor ad actuator. Fig.. Networked etimatio/cotrol cofiguratio The etimatio ceter tramit a equece of tate etimate, ay xˆ( ), to the cotrol module attached to the actuator. The time itat {, } iclude, at leat, all the time whe xt ˆ( ) uderwet a meauremet update. The cotrol module receive, at the time itat, the pair { xˆ ( ), }, where deote the delay i the cotrol path (Fig..). By comparig the time-tamp with the phyical time of arrival the value of the delay., the cotrol module ca accurately determie Notice that the etimate delivered to the cotrol module at time refer to delay verio of the tate vector (i.e., at time ). To remedy thi dicrepacy, the 0

30 cotrol module carrie out a local time update, viz., A A( ) ˆ( ) ( ) ( ) x e x e Bu d (.6) Moreover, for all t, the tate etimate ued by the cotrol uit i ( ) t At ( ) ˆ( ) A( t) ( ) x ˆ t e x e Bu d (.7) It ca be how that for all t, xˆ ( t) xˆ ( t y ( )), o that the cotrol module form the optimum MSE etimate of the curret tate xt () from all the meauremet acquired by the etimatio ceter i the iterval [0, ]. Thi local etimate i the ued to cotruc t the cotrol igal u( t) Kxˆ ( t). The time-update equatio (.6), which map x( ) ito xˆ( ), iduce a imilar relatio betwee the aociated error covariace, viz., ˆ( ) A ( ) A A A P e P e e GQG e d (.8) 0 Thi i the ame relatio a (.3), with Pˆ( ), P ( ) replacig ( d P ) ( ), P ( t ), repectively. Thu our delay-mitigatio method reult i icreaed error covariace, but at the ame time, oid the detabilizig effect of delay i the cotrol path. The preece of delay i the cotrol path may reult i lo of tability of the overall ytem. Thi i true, i particular, whe oe of the fuctio of the cotrol loop i to tabilize a utable ope loop ytem. Thu the primary objective of our cotrol module i to mitigate the effect of delay i the cotrol path, o a to oid lo of tability of the cloed loop ytem. The method that we ue will allow u to overcome very large delay (at leat 0-0 time higher tha the mallet detabilizig delay i the origial cotrol loop) at the cot of icreaed tate etimatio error..3 Example: cotrollig a imple utable ytem We ue a igle-iput igle-output example to illutrate the capability of our delay-mitigatio method. The ytem parameter we ue i the imulatio are lited a below.

31 0 0 A, B G, C 0, Q 5 0 Thi i a utable ytem with (cotiuou-time) pole at 5. We tabilize it by applyig feedback cotrol with K [7 7], o that the pole of the cloed-loop ytem are table (located at 4 ad 3). Whe delay i preet i the cotrol feedback path, the cloed-loop ytem become utable for 0.8 cr. I cotrat, uig our delay mitigatio techique, the cloed-loop ytem remai table, with delay a high a 0. However, the level of etimatio error icreae with the icreae of delay, a how i Fig..: the SEER decreae whe delay icreae. Notice that we till he SEER 8dB whe, a delay value that i 0 time higher tha the critical delay 0.8, without delay mitigatio (top curve i Fig..). cr Fig.. Effect of delay mitigatio o SEER, i the preece of obervatio oie The effect of additive oie at the eor output or i the commuicatio chael i captured by the lower two curve i Fig... I oe cae (dahed lie), white oie i added to the acquired obervatio yt (). I the other (circle lie),

32 oie i added to the tramitted etimate xt (). The level of oie i each cae i adjuted to achieve a local igal to oie ratio of 0dB. All three curve i Fig.. diplay the ame depedece o the legth of delay ( ), ad illutrate the fact that our delay-mitigatio techique tramute delay ito icreaed etimatio error. Our method for mitigatig delay i the cotrol path relie o ailability of the ytem parameter A, B at the cotrol module, o that (.6) ad (.7) ca be ued to evaluate xt ˆ( ). Iaccuracie i ytem parameter will reult i further degradatio i the quality of the etimatio xt ˆ( ), a demotrated i Fig..3. The two lower curve i thi figure correpod to elemet wie iaccuracy i A ad B of % (dah lie) ad 5% (circle lie). Clearly, higher iaccuracy reult i higher etimatio error level (equivaletly lower SEER), ad thi relatio i approximately liear for mall level of iaccuracy. Fig..3 Effect of delay mitigatio o SEER, i the preece of ytem parameter iaccuracy.4 Cocludig Remark I thi chapter we he propoed a delay cacelig approach baed o 3

33 time-tampig of tramitted igal, which ca uccefully mitigate the detabilizig effect of delay. Time-tampig i eabled by the ailability of precie time igal derived from the Global Poitioig Sytem (GPS). Our delay cacelig method tramute commuicatio delay ito icreaed etimatio error, o that the erage SEER of ay igal i the etimatio-cotrol loop i a mootoe decreaig fuctio of the erage commuicatio delay. We demotrated the utility of our approach via a imple example. I particular, we he how (ee Fig..) that the value of delay i the cotrol loop ca be icreaed by a factor of 5-0 beyod the critical value cr with oly a moderate degradatio (0-0 db) i the igal-to-etimatio error ratio. While loger delay ca alo be tolerated, the reultig degradatio i etimator quality may ot be acceptable i ome applicatio. Fially we oberve that our time-tampig-baed approach ca be ued with ay tate-etimatio method ad ay feedback-cotrol trategy, ot jut Kalma filterig ad LQG tate-feedback cotrol. I particular, robut etimatio ad cotrol method ([3], [4], [5]) may be required to accommodate igificat ucertaity i ytem parameter (ad model). Edowig uch robut etimatio ad cotrol techique with delay mitigatio capability will be the key to ucceful implemetatio i patially-exteded ytem 4

34 Chapter 3 Effect of Seor Samplig Patter The eed to tramit eor meauremet over digital commuicatio chael make it eceary to ample eor data at a dicrete-time equece of amplig itat. I thi chapter, we tudy the problem of tate etimatio of liear dyamic cotiuou ytem with dicrete-time meauremet. The equece of eor amplig time, which we call a eor amplig patter (SSP), could be irregular, i geeral. It i apparet that the amplig rate i a critical parameter i the applicatio of cotrol ytem. It i alo reaoable that a the amplig rate i icreaed, the performace of the cotrol ytem improve. However, computatio cot alo icreae becaue le time i ailable to do tate etimatio ad the cotroller equatio, ad thu high performace proceor mut be ued. Additioally, if the ytem ue A/D coverter, a higher amplig rate require fater A/D coverio peed which may alo icreae ytem cot. O the other had, reducig the amplig rate for the ake of reducig cot may degrade the performace or eve caue itability. I additio to performace ad computatioal cot coideratio, there are additioal cotrait impoed by the ue of a commuicatio etwork. I other word, we ca o loger rely o the Nyquit criterio i electig the eor amplig rate. Rather, we eed to take ito accout the (poibly time-varyig) cotrait impoed by the commuicatio etwork, ad the relatio betwee eor amplig patter ad the etimatio performace. We may eve coider chagig the amplig patter i repoe to chagig etwork coditio. A regular SSP i completely characterized i term of it amplig iterval T. A irregular SSP ha a time-varyig amplig iterval T ( ), which we chooe to characterize i term of it momet: the erage ad the variace of T ( ), a well a higher-order momet, a eeded. We how i thi chapter that the erage igal to etimatio error ratio (SEER) deped primarily o the erage amplig iterval, with a very mior depedece o the variace ad a egligible depedece 5

35 o higher-order momet of time-varyig amplig iterval. The bet performace (i.e., mallet etimatio error) for a igle-eor ytem i achieved whe the SSP i regular, o that all it higher-order momet vaih. The ame cocluio hold alo for a Kalma filter etimatio that employ multiple eor: the erage etimatio error covariace i a mootoe decreaig fuctio i each oe of the erage amplig rate. I additio, the erage etimatio error covariace deped alo o the relative aligmet of the idividual SSP. The bet performace i achieved whe the idividual SSP are regular ad aliged i uch a way that the uperpoed et of amplig itat i a cloe to regular a poible. We coider the ame dyamic model that we itroduced i Ch., viz., x( t) Ax( t) Bu( t) Gw( t) y( t ) C( t ) x( t ) ( ) where E{ w( t) w ( )} Q( t) ( t ), E{ ( ) ( k)} R( ) ( k) ad { t } 0: (3.) i the equece of amplig time itat. Several type of eor amplig patter are decribed i Sec. 3.. The performace of a cotiuou-dicrete Kalma filter, a decribed i Table., i a fuctio of the SSP we employ. A detailed aalyi of the igle-eor cae i provided i Sec. 3. ad 3.3. The multiple eor cae i dicued i Sec Seor Samplig Patter Claificatio A SSP for a igle eor i the equece t ; 0 of amplig time. We ca aume without lo of geerality that t0 0, ad cocer ourelve oly with the time variatio of the amplig iterval T ( ) t t for. Thi variatio could be determiitic, (e.., periodically-variat) or radom. To be pecific, we will coider 5 ditict SSP, a follow: () Regular amplig: T ( ) t t for all Regular amplig i the mot commoly ued amplig patter. Sample are evely paced i time with a fixed iterval, o the erage amplig rate: F /. 6

36 () Piecewie-regular amplig: t t N N N Thi type of amplig patter ca occur whe we itetioally adjut the amplig rate i repoe to igificat chage i operatig coditio. For itace, while a reaoable low amplig rate (uder 3000 ample/ec) may be ufficiet durig teady-tate operatio of a power ytem, a much higher rate (more tha 0K ample/ec) i eeded whe traiet occur. From a hort term perpective, the amplig i regular over the iterval Nr Nr, with rate r F. The erage amplig rate i a log term property, ad i give by N N N lim r N N N r r. 3 I other word, we he regular amplig with a occaioal tep chage i the amplig rate, i repoe to chage i ytem tatu. (3) Periodically-variat amplig: The ample iterval T ( ) ( i( )), a how i Fig. 3., where i the erage ample iterval, ( 0 ) i the relative amplitude of the T ( ) fluctuatio ad i the frequecy of fluctuatio. Whe 0, T ( ), o that periodic amplig iclude regular amplig a a pecial cae. 7

37 Fig. 3. Periodically-variat SSP (4) Uiformly ditributed radom amplig: T ( ) ( X ), where i the erage ample iterval, X i a i.i.d. equece with each X uiformly ditributed i the iterval [-,], ad ( 0 ) determie the variace of the fluctuatio of T ( ), a how i Fig. 3.. Whe 0, T ( ), o that uiform ditributed radom amplig iclude regular amplig a a pecial cae. Fig. 3. Probability deity fuctio of uiform ditributio 8

38 (5) Beroulli-drop radom amplig T ( ) t t i a i.i.d. equece with each T ( ) geometrically ditributed, amely, P( T ( ) k ) ( p) p ( k ), for {,,3...} k, a how i Fig. 3.3, where ( 0 p ). Thi amplig patter occur whe a regularly ampled eor output uffer a occaioal lo of a ample caued by a packet lo i a commuicatio etwork. If the packet droppig proce i Beroulli with drop probability p the the reultig ditributio i kow a geometric, a decribed above. The erage amplig iterval i, o that the erage p amplig rate with iterval. F p. Whe p 0, thi cae reduce to regular amplig Fig. 3.3 Probability deity fuctio for the geometric ditributio I thi thei we coider motly periodically-variat amplig ad uiformly ditributed radom amplig to tudy the effect of SSP o the etimatio error covariace of a (cotiuou-dicrete) Kalma filter. Both patter allow idepedet cotrol of the mea value of T ( ) via the parameter, ad of the variace of T ( ) via the parameter a how i Table 3.. 9

39 Table 3. Statitic of the SSP Var( T ( )) Mea Variace Kurtoi Mea( T ( )) Periodically variat Uiformly Ditributed Radom Beroulli drop Radom p 3 p ( p) p 5 ( p) p 6 Oe coveiet way to characterize a eor amplig patter i i term of it T ( ) eraged characteritic fuctio T () Ee. For itace, the characteritic fuctio of the periodically-variat SSP i T ( ) e I0( ), where I () 0 i the modified zero-order Beel fuctio (ee App. C for a detailed derivatio). Variou momet of the amplig iterval ca be coveietly determied from the momet-geeratig property of the characteritic fuctio, viz., k d T ( ) 0 ET ( ) d k The characteritic fuctio of all the SSP that iteret u are give i Table 3. (ee derivatio i App. C). Table 3. Characteritic fuctio of the SSP characteritic fuctio () Periodically variat Samplig e I ( ) 0 Uiformly Ditributed Radom Samplig ih( ) e Beroulli drop Radom Samplig p e pe T 30

40 3. The Effect of SSP o Kalma Filter Performace The tate error covariace matrix Pt () of a Kalma filter decreae with every meauremet update, ad icreae i betwee meauremet update (recall Fig..5). Whe the amplig patter i regular, the pot-update covariace P ( t ) coverge to a teady-tate limit, ad imilarly for the pre-update covariace P ( t ). However, whe the SSP i irregular, the two equece P ( t);, ad P ( t);, may ocillate ad ot coverge. Thi motivated u to itroduce the erage error covariace P E P ( t ), P E P ( t ), a a meaure of teady tate performace. Our objective i thi ectio i to obtai a imple characterizatio of A, B, C, Q, R ad the SSP. P, ad P, i term of the ytem parameter 3.. The Dyamic of Pt () The time update of Pt () give rie to a explicit relatio betwee P, ad P,, imilar to the relatio we obtaied i (.3). Whe the amplig patter i regular, we ca alo obtai a dual relatio from the teady tate verio of the meauremet update. With two matrix equatio i the two (matrix) ukow P, ad P,, we ca, i priciple, obtai explicit expreio for the erage error covariace i term of the ytem parameter A, G, C, Q, R ad the characteritic fuctio () of the amplig iterval T ( ). However whe the SSP i irregular we ca oly deduce a approximate relatio from the meauremet update, ad oly uder the aumptio of relatively mall deviatio from regularity i the SSP. T 3

41 Average time update relatio: The time update relatio i the amplig iterval [ t, t) i (recall Table.) T ( ) * * AT( ) A T( ) A * A ( ) ( ) 0 P t e P t e e GQG e d (3.) where we ue the fact that T ( ) t t. Thi relatio ha the ame form a (.3), o that the aalyi of App. A ca be applied to deduce a erage relatio of the ame form a (.4). We obtai a relatio betwee *, T, T P, ad P,, amely T P T ( T P T ) ( T GQG T ) (3.3) where [ ( )] M T T l m l, m [ ( ( ) )] M T T l m l, m l m (3.4) are determied by T( ) T( ) T ( ) E{ e } e p( T ( )) ( ) dt, the characteritic fuctio of amplig iterval T ( ). Here deote a Schur-Hadamard product, ad i are the eigevalue of the ytem matrix A. A i Sec.. (ad App. A), we eed to aume here that the matrix A i diagoalizable, ay A T T. Notice that the relatio (3.3) i tatic, i the ee that G, Q, T,, ad P,, P, are all cotat (time ivariat) quatitie. The equatio (3.4) deped o the pecific SSP via the characteritic fuctio ( ). For itace, whe the SSP i periodically-variat (ee Table 3.), T l m, where we ue the Taylor erie! 4 T ( ) e I0( ) e [ ( ) ( ) ] expaio for the modified zero-order Beel fuctio I () 0 T T. Similarly, whe the SSP i uiformly ditributed radom, ih( ) 4 T ( ) e e [ ( ) ( ) ] 3! 5! Whe the fluctuatio i T ( ) i ot too large, i.e., whe max l m, lm, the both type of SSP ca be approximated by T ( ) e [ ( ) ]. 3

42 where for the periodically-variat cae ad 4 ditributed radom oe. Thi expreio how that for the uiformly 6 T ad T deped maily o (the erage value of T ( )), ad to a much leer extet o the fluctuatio amplitude. We will verify thi obervatio via umerical example i Sec Average meauremet update relatio: To complete the aalyi, we eed to obtai a erage verio of meauremet update relatio P t P t P t C t t t CP t * ( ) ( ) ( ) ( ) ( ) ( ) ( ) P t P t C t CP t * ( ) ( ) ( ) ( ) P t P t C t CP t * ( ) ( ) ( ) ( ) P t P t C CP t C R CP t * * ( ) ( ) ( ( ) ) ( ) Thi will provide a ecod equatio relatig the two ukow matrice, (3.5) P, ad P. Ufortuately, (3.5) i oliear i P () t, o that it appear difficult to, expre the erage P P t P t C CP t C R CP t i term of * *, ( ) ( ) ( ( ) ) ( ) P, P ( t ). A more compact form of thi meauremet update relatio ca be obtaied by applyig to (3.5) the matrix iverio lemma, viz., ( A BDC) A A B( D CA B) CA (3.6) with A P () t * B C C C D R We obtai P ( t ) ( P ( t ) C R C) *, o that P ( t ) P ( t ) C R C (3.7) which alo make it evidet that alway P ( t ) P ( t ). Averagig reult i E { P ( t )} E { P ( t )} C R C (3.8) Ufortuately, there i o imple relatio betwee E{ P ( t i )} ad E P { ( ti )}, 33

43 o that, i geeral, we caot get a imple erage verio of the meauremet update relatio. Notice, however that whe the SSP i regular we obtai (i teady tate) P( t) P, ad P( t) P, (tatic relatio),,, o that i thi cae (3.7) become the deired P P C R C (3.9) A perturbatio aalyi reveal that thi relatio alo hold, approximately, for mildly irregular SSP (ee App. D for detail). P P C R C (3.0),, I ummary, for mildly irregular, we obtai both a erage time update relatio ad a erage meauremet update relatio, viz., * T P, T T ( T P, ) ( ) T T T GQG T P, P, C R C where [ ( )] M T T l m l, m [ ( ( ) )] M T T l m l, m l m (3.) (3.) The accuracy of the approximatio (3.0) deped o the level of fluctuatio of P ( t ) aroud it erage value P,, a well a the fluctuatio of P ( t ) aroud it erage value P,. For itace, if we ue the uiformly-ditributed radom SSP, the the accuracy of (3.0) deped o the parameter. Perfect accuracy i achieved whe 0 (regular amplig), but a the value of icreae, the accuracy of (3.0) deteriorate. The followig calculatio of P, ad P, i baed o the approximatio (3.0), which aume a relatively mall fluctuatio of P ( t ). Explicit expreio for P, Now, we tur to tudy the behior of P, a a fuctio of erage amplig rate, ad the SSP uder the aumptio of mild irregularity. Thi will allow u to demotrate that fluctuatio of the amplig iterval T ( ) he a relatively mior 34