In this paper the theory of Equation Decoupling method for parameter estimation of unstable or augmented dynamically system is presented.If is also shown that the method is a generalisation of the so called Total least Squares method in which errors in measurements as well as states are taken into account

Gopalratnam, G

Raol, JR

In this paper the theory of Equation Decoupling method for parameter estimation of unstable or augmented dynamical system is presented. It is also shown that the method is a generalisation of the so called Total Least Squares method in which errors in measurements as well as states are taken into account

Girija, G

NAL-IR

Asymptotic and Generalisation Theory of Equation Decoupling Method for System Parameter Estimation

Asymptotic and generalisation theory of equation decoupling13; method for system parameter estimation

National Aerospace Laboratories Institutional Repository

Asymptotic and Generalisation Theory of Equation Decoupling Method for System Parameter Estimation r G Gopahdtnatn. Non-tzietnber Dr J R Pad. hct,- i t ietnber I n  rhis p u p r  the fhror) ofEquotion Decouplitig inerhod for pumwrcr  estbnuiion ofunsruble or iiiignirrired d)rfuniicol s)sreni is presenrcd. If is also showri r h u r  rhe merhod i~ u generuliAofim ojrhe so culled Torul Leusf Squares method it1 which errors in meusurcniesrs us wril us S I U ~ ~ S  are mken info uccoutif. Xqvordr; Asympouc. Uccouplixg, Nuirwicdl Siinulalion. Total k m l  Square INTRODUCTION T Th; output method (OEM)’ is the most widely used technique for estiination of parameters” of stab!.: dynmical system A 01 including ail-crali I-’. However the method pobes severe diffi- culties when applied to inhere’ntly unstable or augtnented systenis’. When the system is unstahle. numerical integration leads to direrzing solutions. A method called equatidn deiou- pling (EU)has been recently presented !is parameter estima- tion ot unstable systems’. This m,ihr,d uses measured states to decouple the state equations and iliiegraic thc system of ililferential equations independed of each other. The decou- piing 01  tlic equations may ;hang: the unstable system LO the ruble onz. Asjmptotic rheory of ED output emor method IEDOEM) and generalisation of the total lcast squares (TLS) solutinn we presented. Such Atudies are limited in open litera- ture’-‘. EQUATION DECOUPLING OUTPUT ERROR METHOD The dynamics of the system are given 3s r = Ax + Bu wirh x(0) = x., (1 )  ,l = c* + u u  (2) z ( k )  = ? ‘ ( k )  + . . ( k )  k = 1.2. .. N (3) Here. N is the number of data points and v is the mensurement noise assumed id be Gaussian with zero mean and covariance matrix R. The 0 { A ,  B, C, D )  represents the parameter vector to he estimated. The cost function to he ininiinised is defined as ,\ I T 2 E(O!  = - 1 I : ( k )  - y (k ) l  Here, 1 stands for iteration nuniher. The sci:sitivity imatrix a ( k )  needs to be computed. The insiabiliky caused due to ao nunierical divergence can be overcome by incoqmatlng sta- bilisuion into the OEM by using nxaauxd states. .rnt. The manncr of utilising tlie measured stares to stahilise the system equations leads to EDOEMX wherein the slates pertaining to the off diagonal elements x e  replaced by tlie cori~espoiiding measured states. The system matrix A is pxtirioned into two submatrices denoted by AD (diagonal matrix1 and Aon (with offdiagonal rlementsofA). Equation( 1)can be rewritten with x - .rn, as L J  The only integrated variables entering the diffcicnttal equn- tioii are in the first term. Thus each diffcrential equation can be integrated independently of the others. The cost function to be minimised is given by equation II). Hwever.  the computation ofthe sensitivity function involve> the decoupled matericcs AD and Ao,, and the state mcasurtmcnts augmenting the control input measurements. ASYMPTOTIC THEORY OF EDOEhl In this section the implications of the use of me;iwred states, in terms of sensitivity matrix computation arc studicd thercby providing analytical hasib t o  the working ( i f  the EDOM. Fol- the case wlicrc there is n o  DSOCCSS noise. euuiition ( I )  can be Minimisation of the above cost function with respect to 0 y ( k )  = Cx ( k )  + Did ( k )  ( 9 )  yrclds the estimates vf0 as n n where $ denotes the state transition matrix and v its integral. 110) y = e  d ~ = l A f + A - + + * - +  . . . . . .  ( 1 1 )  A? 0i+! = 0; t @ A @ /  (5) @ z p i  = I -t A A r  + A’ - + . .  . . . 2! 2 !  3! G (hpalratnarn and Dr J R Raol are with Flight Mechanics and eonlml .Diriaion. Narionnl Aemrpare Laburulory. Bangalom 560 017. t ~ : t p u i l l I x r c c c ~ v e d ~  iOctober31. 199h. AT A? Of; Thfi pnryw ~ a a  ix;ctvcO m Orccmber 20. 1995. W n i m  discussionon this where Af = r ( k  + I )  - ( k )  is the sampling interval. IE(1) Journtil-EL 2 tL Conipulation 111 ihc pilralneter incrclnent vector. A@, requires the wmpuration 01 tnu senhiilvity equations. Wtlich are oh- tiuned hy p3rti:iI ditfciciitiatwn oi tlic sybtcni equations with respect ti, each c l c l i im t  of thc unhnowii parameter vector Q. Sinw thc senitivit) equations have the matrixit as the system eqwitioris. thc xmic transition matrix. equation (10). can he u x d  to wlvc thc11i. By differentiating cquatiun ( X j  and (9) withrcquebtto (3. t l ie discrete fort~r of thescnsitivity equations are obr,iiiicli a>'' . (I .i) For siniplicity sccond order longitudinal dynamics of an air- cratt im v:msidercd. Thc dyn3iiaic5, i n  continuous and discreic fomi arc .*iveii hk (141 (15) 06!  Thc sc.nsil~\:it). i i i i itrix comput;~tim can he inadr using equa- t ion  i 121. The [xmineter vcct~)i- t o  be' estimated i s  given hy 0 = i L  . M>, . M,) . Z&. Ma,j, the  tlciiients CI.A and n. TO illusirate ihc coniputationh i i ivul\ed in the sensitivity iiiatrix. thz p;uual ditferentiation of the states with respect to the deriviitiw M,, I S  considered. f i rst  for OEM and then for EDOthl.  Usinp equatiim i I ?! and i 16) and the fact that the last three term> in equiitioli t 12) vanish. one obtains, (181 t J 3  In equatlon ib) i t ie second term rcprchents the first :r;id~cnr 01 the COSI functiim Expanding this first gradient ;uaJ uung equation (18). isuhscript Isrands for integrated vm:ihlc). 11ne ohtains; For thc prohlsm Jcscrihed hy equation i 13). thc Equatiu~i Decoupling fo~ i i iu l~ i t ion  i s  given hy Using cquarion (1 I j i n  equation i I21 Inr the sensitivity q u a -  tioii with ~espccl t i l M b , '  and;iinsidcrinp tile kict that terms 2nd .ind 5th in equ;itiun 1121 vanish. thus.. Fur the fist gradlent (wr f  M,) for EDOEM . To derive the asymptotic equivalence between EDOEM and OEM, I + w; (24) W," = wl + w, ; qni = qr + qn ; 'I,, = n. where subscript 1 denotes true values, n the measurement noise, I the integrated value, and i the errors due to integration. Substituting equation (24) into equation (19) and observing the fact that the integration errors (w i )  tend to zero as the iterations are increased. Here, the convergence of the OE algoi ithm . is assumed which. 2 generally guaranteed, for OEM, Substituting equation 124) into equation (23) and the fact that w, + 0 for EDOEM. one, gets, In equation (26) since the measurement noise [w,, , qn] is independent of the parameters to be estimated, the partials of q. and w,, wrt M, vanish. Next, the following term involving [w, , qn] are obtained, in equation (26) N - [ q , , ( k ) r  bvrdk) [ w n ( k -  I)] AI (27) N - l  N - l  k =  1 which tends to zero since the measurement noise yn and w. are uncorrelated. Hence, i n  the light ofthe aboveobservations, comparing equation (25) and (26), one gets. V E E ( 0 )  , VE,, (0) N -  1 N -  1 asymptotically for the two-state model. Thus, the asymptotic behaviow of the EDOEM is similar to that of OEM. However, the OEM does not work directly for unstable systems as noted earlier. In EDOEM, since the meas- ured states, obtained from the unstable plant operating in closed loop are stable, their use in the estimation process, tries to arrest this divergence and at the same time enables parame- ter estima.tion of the basic unstable plant directly. GENERALISATION THEORY OF EDOEM The EDOEM can be viewrd as a generalisation of the TLS approach for parameter estimation. TheTLS approach' is used to account for the errors in the measurements of X, the v a r -  ables in the regression equation, in addition to the errors due  to noise in Y. The general regression equation is given by y = x e + ~  (29) When the measurements X are noisy, the LS method does not explicitly account for these errors. The TLS method addresses this problem as follows: The equation (29) is restated as [XI v] [@'I - I f  = 0 (30) Due to the measurement errors, the compound data matrix [XI YI is full rank and there is no nonzero solution vector [@I - lf .  To get a solution, the rank of the measured data [XI YI is reduced with an estimate of data errors [AX I Ay1 * A  ([x I YI - [AX I A YI) [@'I - 11' = o (31) subject to the constraint of minimal approximation effort A A  1 1 [ A X  I A f l  C? 11: is minimal (32) where C is the square root of the convariance matrix of the row vectors of the dataerror matrix [AX I Afl .  The solution of equation (31) is given in terms of the singular value decom- position of the matirx [X I r] C-': r (33) The trasformed solution vector C [e' I - I ]  is found as the kernel of matrix [XI u] C-', which equals the last column vector vn + I of the singular matrix V. Equation (31) is solved for [hru I - I]', 1 is chosen such that the last element equals - 1. In case of EDOEM A c[errJ. I -  11 = iv,+, A A  (34) y = Cx, + rim (35) where x, stands for the integrated state generated from the integration of the state equations and rim stands for the meas- urement noise. If C = I, the identity matrix, y = x ,+nm (36) Introducing the integrated variables from equation (34) into the equation (35). (37) which is rewritten as equation (38) and in compact form as equation (39) 2. ?9) IE(1) Journal-EL . .  . pling erne of 0.05 s. Stafe equations: REFERENCES C S mwani. ‘Idemif . .  f %~ .-.; Vol77. August 19% ,. 

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Asymptotic and generalisation theory of equation decoupling13; method for system parameter estimation

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