The problem of model order determination, which is an integral part of system identification for dynamical systems of auto-regressive and least square structures is investigated. Twelve model order testing criteria available in the literature are critically evaluated from simulated data and from two different physical processes: (1) human operator responses in compensatory tracking manual control; and (2) wind tunnel mounted model responses to tunnel flow instabilities. The study revealed that of all the criteria tested, only a subset is adequate to establish reliable model order. Based on this observation, a computationally effective working rule is proposed for model order determination for practical dynamical systems

Balakrishna, S

Jategaonkar, RV

Raol, JR

NAL-IR

Part I Modelling and System Identification Determination of Model Order for Dynamical System R. V. JATEGAONKAR. I. R. RAOL. AND S. BALAKRISHNA Ah.,warr-Thhe problem of model order determination. nhieh is an inlea,rrl pan of system identifiration for dynamid system, of uuIo.regms- sive and I P ~ S I  square smeNrer 1s inre,ligard Twelve model order resting criteria araihhle in the IItwhNm are ~ril ically wdwA1ed from S i m d m d  dnra 2nd lrom MIO different physical prnccnses iiz., I) human opcntor responses in rompensaloty mrl ing mmud mnfrol, 2nd 2) wind runnel mounted model rerponrer 10 munnel flow inrrrbilitier. Tl~e rmdy revellled that of all the criteria terted. only B rubre! i s  adequate to esrrblish reliabk mod4 order. Based on this obrprration, a comptationally effective w d -  ing NIO is popwd for model order determination fw practical dynarnied r y r t e m r I. INrRooucnoN Time series methods are gaining considerable acceptance in system identiiication in new of their inherent simplicity and flexibility [ 11-16], These techques provide external descnptions of svstems under study and lead to parsimonious represenlation of the process. The acccur~te detcrmination of the dynmieal order of time series models is a neccssar). first srep in identifica- tion and continues to be a subject of extensive contempomy research. A literature study reveals that many statistical tests are availa- ble to find the model order for any given process. Selection of a reliablc and efficient test ci te ion has been elusive since most critena xe Sensitive to statistical properties of the process. Vali- dation of mosr of the available criteria has generally been through simulated data. However. these order determination techniques have to be used with practical systems with unknown stmctures and finite data length. It is therefore necessaq to validate any model order cnienon using a wide variety of data base. The work reported ID h i s  paper was motivated by the need to determine dynamic denvatives from free response of a wind tunnel model 10 tunnel flow [El. and human operator model determination from a fixed base simulator 191. The data sets are from two disinctly different real processes and are considered to be good candidates for critically evaluating and comparing vari~ 0"s critena WET. the recently suggested whiteness test-2 [ I S ]  does consider the cumulative effects of autocorrelations of residuals. The pole- zero cancellations 131, 1121, [ I  I ]  are made visually and are subjec- tive. A systematic exact pole-zero cancellation is possible [ 161, but is computationally complex. Fit error methods are useful but again subjective. Some of these tests have been comparcd for their quality in [ I  I ]  and 1121 using simulated second-order plants. Table I1 summarizes the objective-type tests. in which a local minimum of a c i te ion  function is usually sought. The final prediction error (FPE) criterion due to Akaike [I71 is based on one step ahead prediction and is essentially designed for white noise corrupted processes. The Akaikc information criterion (AIC) [I71 i s  a generalized concept based on a mean log likelihood function. Both the FPE and AIC depend only on residual v a r -  ance and numbb of estimated parameters and may yield residu- als which are conelated. Sometimes these tests yield multiple minima. The criterion autoregessive transfer function (CAT) by P a n e n  has been proposed as the best finite AR model derived from finite sample data generated by AR model of infinite order 1181. The modified final prediction error (AMFPE) 1211 has heen developed for anloregressive moving average (ARM) models in terms of estimated variances of the process correlation function m d  estimated parameters. A simplified version of AMFPE has been used in this paper to be applicable to AR models. CAT3 i s  a modification of CAT2 (201 to account for any ambiguity which may arise for "true" first-order AR processes due to omission of s: terms. The methods discussed so far involve rxpticit parameter identi- fication before criteria arc evaluated. The entropy test 1221 I S  an exception and is based on an information measure which is related to time st ies  of the process. In this study the entropy test has  been used for real process data. Ill. APPLICATIONS Three applications of model order estimation are no\* consid- ered through various criteria detailed in Tables I and 11. The data chains for the tests were derived from ( I )  a simulated sscund order system, (2) human activity from a fixed base simulator and (3) forces on a wind tunnel model exposed to mildly tuibulcnt flows. The AR model identiiication was carried out as in P! and . . .  11. MODEI. ORDER DETERMINATION the LS model idcntification as in [ I ) .  The results of the cnlcna as applied lo model Order estimat'on In the abience of (1 onon knowledre. any s w e m  that IS anal\zed. Sirnuhied SecwdOrdrr Swrcrn A second-order rystem of known w,, and i pntchcd on an analop computer was cxoited by a 7 0 0 - H ~  low-pas li!tzrcd Gaussian white noise. and the input-output pair wcrr diyrized at a sampling rate of I ms. The AR pararneteri wcrc evalustcd in a model ordci xlin m i n g  only rhe output dam Fig. I >how a plot of various Indice3 (,I the criteria as a function of model order. Thc entropy. FPE. I \ IFPE.  CATL and CAT3 explicitly indicate n nummum at a model <order of 2 .  The prediction l i t  error ihoivs a mimimum at model oidu: of 1 and above. The F-rest almost ncadv stislies a 5 pencmt m k  hypothehi, 31 F:,?. Thus. the objrctwe tern clearly i n u i x r .  llx correct modcl order. md  the sub!?Jcctm teiis strongly hu?:cbt thc correct modcl urdcr. Thc  LS inodd pxmicrcrs 'wrc walualed next. 3s.m !I/ .m I . I  ,LEE rq*NsACTlONr ON I I ~ T E M S .  M N .  AND CYBERNETICS. VOL. u(c-I2, NO. I, IANVARY/FEBRUARY 1982 TABLE II OBJECTIVE TEKS FOR MODEL DETERM~NAT~ON TCSt CnreliO" Features Comments Minimization of av- Developed for uni. error N - n - I eragc error for one variatc procerscs step prediction con- corrupted by white ridenng both the er- noise. The pendr) ran due to innova- for degrees of free. tion pan of the p m  dom is significannlly cess and errors due reduced for large N .  to inaccuracies in estimating parame- 2) Informationcntc- AIC(,,) = N I ~ ~ , ;  + 2" Generalized concept no" of FPE. Estimate of mean log likelihood. I) Final prediction FPYn. N )  - N + o + I g 2  ten. 3 )  Modified final AMFPE("). ='+A Minimization of one predlcllo" Error I - A step ahead predicr- ion error from c o ~  vanancc m a i m  and esiimatcd vafianoce of the process i tself. 4) c n r r n a  auto-rc ' Minimization of in- greisme transfer a) CAT1 = 1 - % f - iegraied ie lsr ivr  between nrh order AR model and rheo re t ical  A R ( m 1  mor(<1 function 0"- * mean square error Doer not explicitly depend on numberof data N .  Could be used for processes corrupted by color Compuiaiiondly dif~ ficui,. noise. 1 bl CAT2 = - " Modified CATI. For m y  arbitray 7 Compuiauonally I U ~  model order n. al l  II i = i  "i nenOi, models to be esri- ,531-, I L I . $00 Fig 5 .  AR model for human performance-order dc!crminauon time history prediction, test residuals for whiteness V. CONCLUSION The ambiguous problem of the determination of the model order of practical dynamical systems with opriorirj, assumed AR or LS StNctUre was analyzed using I2 contemporary model order test criteria. The data base was derived from physical situations with widely different dynamical characteristics. This model de- termination effort has provided a basis on which the utility of the n: ' ":S X POLES 0 ZEROS MODEL OROEP N = two. Fig U Cn1e"a lor AR malrl  order dcwminruon 01 random response. different test criteria can , b e  compared and evaluated. 'For the typcal dynmical  system considered, and similar situations, a minimal set of test criteria has been proposed as a worklng procedure to determine and validate realistic model orders without expending significaot computational effort. NOMENCLATURE AIC Akaike's information criterion. AMFPE Modified final prediction error. AR Autoregressive. ARMA Autoregressive motlng average. autoregressive transfer function. 5 LS Least squares. n Model order of AR/LS description N Number of samples. q- '  Back shift operator. r Residual of estimation procedure. R, Autocorrelation of residual. V,  Loss function. Y System O U I D U ~ .  a,, Natural frequency, radians, second. f Damping ratio. a' Variance. ( A 1 q - I ) .  B ( y - ' ) )  Pol!nomials with regression coefficients 01 the oulpui and Input srquences. APPENDIX Thc :niodjlisd f i n d  prediction urmr (AMFPE) is evalualud i,>r AR :nodcis 3s [ ? I ]  and q = 111:11 ( I . , ) Near-Nash Feedback Control of a Contposite System with a Time-Scale Hierarchy UMIT OZGUNER. MEMBER. ILLE A h w u -  4 large-reale conlpoaitr system 5meNre encountered in em- noniic \>\tcrw. nvanufacrurin~ proreses, pouer \!aemr. PIC.. uhich has P rno-Iitne-,cAe hierarch\. i \  eomidered. A dynamic g m r  wirh quadratic cost rrilrris a w x i a t e d  with each subsystem is ICI up and the Nash feedback matcg: annI!md. I f  i) rhoan that a near-hrh d u i i o n  cm he "brained from Ion order independent eonmol problem, in the fast time- scille and an independent gante. related to a loa order subrptem. played in the \IOU time wale. Funhermore. P decentralized inionnition SUCN~P turns out to be sufficient to iniplemrnt the feedback controls. 1. INTRODUCTION An important concept in the area 01 analysis and control of large-scale interconnected systems is that 01 hrerurch,,. Although this concept can be viewed in a number of ways. one special class, that of a rime-scuk hierarch,v, is especially suited to mathematical formulation. Interconnected systems with an inher- ent time-scale hierarchy also appear a lot in practice. especially in be supplied from above and a direct miormatinn llnh ui!l ni,i I have to bc set up. i Recent results in sinpularly perturbed x s r e r n s  [ i ] ,  12; h t  provided a suitable framework lor a mathematical rnodc! lor ti lsi. above. Sandsll P I  01. 131 iiaw indicated that two'-iime-scaie f i l tcnnq might be \!eived as a herarch!. A presentation 01 the quadraui:  repulaloi for multi-time-scale Interconnected swtcms with dub consideration of the hierarchy in\;oived and the ncccsian. 1n1i1r-c. mation transfer ivas given b) Orgiiner 141. [j]. The obsrncn, problem us also in.wstigated 161. The main extension providcoii to the results of Cho,i\-Kokorovic [ I ] .  121 is based on assumin: ' r  I structure uherr uncoupled subsystems exist in any time s c d c  s This results in the anal>,sis/control problent also decoupllng i each time scale. Furthermore, the necessav information tinnil$, ~ to implemenl the near-optimal control is clearly identified. Especially in view of the applications in economics as in<  cated. for a number 01 problems 3 dynamic eame seuing scef ~ niore natural than theoprimal control problem solved by Or$i in (51. Some interest has recently been shown in the area 01 problerns lor singularly perturbed systems. Gardncr and C r u :  and Khalil and KokoIpvic [ R ]  investigated the well-posedness Nash solutions and Orgiiner 191 considered the \irnpliiioat provided b? specilic cost criteria. In this correspondence we will consider near-Nash feedb solutions to a quadratic game for a two-time-scale system n = A  multiple subsystems in the lower (laster) level in the hierar T h e  subsystems in the lower level will he shown 10 have cox ~ (regulator) problems to solve in the fast time scale. wherca: -. actual game is in fact played in the slow time scalc. All reg" = -~ problems and the game will be independent of each other ar much lower order than the original problem. Furthermore" -< feedback solutions will require an inlormation pattern comp:, with rhe interconnection structure. I -  ' R, ' )her ha 11. THE TWO-TIME-SCALE MULTISUBSYSTEM GAME denhfv liii ! 

Determination of model order for dynamical system

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Determination of model order for dynamical system

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