2.6 System Identifition, Estimtion, nd Lerning Leture otes o. Mrh 3, 26 6 Model Struture of Liner ime Invrint Systems 6. Model Struture In representing dynmil system, the first step is to find n pproprite struture of the model. Depending on the hoie of model struture, effiieny nd ury of modeling re signifintly different. he following exmple illustrtes this. Consider the impulse response of stle, liner time-invrint system, s shown elow. Impulse Response is generi representtion tht n represent lrge lss of systems, ut is not neessrily effiient, i.e. it often needs lot of prmeters for representing the sme input-output reltionship thn other model strutures. g ( k u(t g ( g (2 y(t G( A liner time-invrint system is ompletely hrterized y its impulse response,, 2, g ( k G ( = g ( k k k = g (, g (2, oo mny prmeters lthough trunted. 2 3 4 k Cn we represent the system with fewer prmeters? Consider k g ( k = k =,2,3,... G ( = k = k k Multiplying : G ( = = k k = G ( k = k = 2 k k G ( = G ( = = herefore, G ( is represented y only one prmeter: one pole when using rtionl funtion. he numer of prmeters redues if one finds proper model struture. he following setion desries vrious strutures of liner time-invrint systems.
6.2 A Fmily of rnsfer Funtion Models 6.2. ARX Model Struture Consider rtionl funtion for G ( : y ( t = B ( u (t ( A ( where A ( nd B ( re polynomils of : A ( + +... + n n, (2 B +... + ( u ( t G ( = B ( y (t A ( he input-output reltionship is then desried s y ( t + y ( t +... + y ( t n n = u ( t +... + u ( t n See the lok digrm elow. (3 u ( t e (t e ( t e ( t + + + + _ + + + 2 + + n + + 2 + + n y ( t exogenous input Auto Regression 2
ow let us onsider n unorrelted noise input e ( t entering the system. As long s the noise enters nywhere etween the output y ( t nd the lok of, i.e. e ( t, e ( t, e ( t in the ove lok digrm, the dynmi eution remins the sme nd is givey: y ( t + y ( t +... + y ( t n n = u ( t +... + u ( t n + e ( t (4 Inluding the noise term, this model is lled Auto Regressive with exogenous input model, or ARX Model for short. Using the polynomils A ( nd B (, (4 redues to A ( y ( t = B ( u ( t + e ( t (5 he djustle prmeters involved in the ARX model re θ = (, 2...,, n,, 2,..., n (6 Compring (5 with ( of Leture otes 9 yields G (,θ = B ( H (,θ = (7 A ( A ( See the lok digrm elow. u ( t G ( = B ( y ( t y ( t A ( H ( = A ( e ( t ote tht the unorrelted noise term e ( t enters s diret error in the dynmi eution. his lss of model strutures, lled Eution Error Model, hs fvorle hrteristi leding to liner regression, whih is esy to identify. ote tht if n = then y ( t = B ( u ( t + e ( t. his is lled Finite Impulse Response (FIR Model, s we hve seeefore. 3
6.2.2 Liner Regressions Consider the one-step-hed predition model in Leture otes 9-(8 y ı( t t = H (G( u ( t + [ H (] y ( t 9-(8 From (7, the preditor of ARX model is givey yı( t θ = B ( u ( t + [ A (] y ( t (8 his e more diretly otined from (4, sine the noise is unorrelted. he regression vetor ssoited with this predition model of ARX is defined s: ϕ(t [ y ( t,..., y ( t n, u ( t,..., u ( t ] (9 using this regression vetor, (8 redues to y ı( t θ = ϕ (t θ ( ote tht the preditor y ı( t θ is slr funtion of ϕ(t : known vetor, nd θ : djustle prmeters. ϕ(t does not inlude ny informtion of θ. he known nd unknown prts re seprted, nd θ is linerly involved in the preditor. Liner Regression. 6.2.3 ARMAX Model Struture Liner regressions e otined only for lss of model strutures. Mny others nnot e written in suh mnner where prmeter vetor is linerly involved in the preditor model. Consider the following input-output reltionship: y (t + y ( t,..., + n y ( t n = u ( t,..., u ( t ( + e ( t + e ( t +... + e ( t n n Using n ( = + +... + n, ( redues to A ( y ( t = B ( u ( t + C( e ( t (2 herefore G (, θ = B ( A ( C ( H (, θ = A ( (3 4
θ= [, n ] 2,, n,, 2,, n,,,, (4 2 his model struture onsists of the Moving Averge prt (MA, C ( e ( t, the Auto Regressive(AR prt, A ( y ( t, nd the exogenous input prt (X. his model struture is lled n ARMAX model for short. An ARMAX model nnot e written s liner regression. 6.2.4 Pseudo-liner Regressions he one-step-hed preditor for the ove ARMAX model is givey GH H B ( A ( yı(t θ = u ( t + [ ] y ( t C ( C ( (5 his nnot e written in the sme form s the liner regression, ut e written in similr (pprently liner inner produt. Multiplying ( to oth sides of (5 nd dding [ ( ] y ı( t θ yields yı ( t θ =B( u ( t + [ A ( ] y ( t + [ ( ] ε (t, θ (6 Define Predition Error ε (t, θ = y ( t y ı( t θ (7 nd vetor ϕ ( t s ϕ ( t, θ [ y ( t,..., y ( t n, u ( t,..., u ( t ε ( t, θ,... ε ( t n, θ ] n (8 hen (5 redues to y ı ( t θ = ϕ ( t, θ θ (9 ote tht ε ( t, θ inludes θ nd thereforeϕ depends onθ. Stritly speking (5 is not liner funtion of θ.. A Pseudo Liner Regression his will mke signifint differene in prmeter estimtion s disussed lter. 5
6.2.5 Output Error Model Struture Eution Error Model C ( e ( t Output Error Model e ( t u ( t y ( t B ( A ( u ( t B( F ( y ( t oise enter the proess. his resemles Preoess oise in the Klmn Filter. oise dynmis is independent of the proess dynmis his resemles the mesurement noise of KF Let z ( t e undistured output drivey u ( t lone, (2 z ( t + f z ( t,..., + f n f z ( t n f u ( t B( z ( t = u ( t,..., u ( t n f F ( F ( = + f +... + f n n ote tht z ( t is not n oservle output. Wht we oserved is y ( t (2 y ( t B ( = u ( t + e ( t F ( he prmeters to e determined re olletively givey (22 θ =[ ] 2 n f f 2 f n f ote tht z ( t is vrile to e omputed (estimted sed on the prmeter vetor θ ; therefore, z ( t,θ. he one-step-hed preditor is (23 y ı( t θ = B ( u ( t = z ( t,θ F ( Whih is nothing ut z ( t,θ. herefore, y ı( t θ = z ( t, θ 6
y ı(t θ = f z ( t, θ f z ( t 2, θ 2 f n f z ( t n f,θ (24 + u ( t + + u ( t where ϕ (t,θ is = ϕ (t, θ θ (25 ϕ (t, θ = [ u ( t u ( t n z ( t, θ z( t n,θ ] herefore this is Pseudo-Liner Regression. Box-Jenkins Model Struture his simple output error (OE model e extended to the one hving n ARMA model for the error dynmis e ( t B( C ( (26 y ( t = u ( t + e ( t F ( D ( f C D u ( t B F y ( t 6.3 Stte Spe Model Stte vriles x ( t = [ 2 x ( t, x ( t, x n ( t] Sttionry ime-invrint (27 x (t + = A ( θ x ( t + B ( θ u ( t + w (t (28 y (t = C ( θ x ( t + v ( t A (θ R B (θ R n n n m l n C (θ R Mtrix A (θ, B (θ nd C (θ ontin prmeters to e determined, θ { w ( t}nd { u ( t }re proess nd output noises, respetively, with zero men vlues nd ovrine mtries: E [ w ( tw (t] = R (θ (29 E [ v ( tv (t] = R (θ E [ w ( tv (t] = R (θ 2 2 Usully R 2 (θ = 7
Using forwrd shift opertion, we n rewrite (27 s [I A (θ ] x ( t = B (θ u ( t + w ( t herefore the output y ( t is givey (3 (t = C (θ [I A ( θ ] B ( θ u( t + C(θ [I A (θ ] ( w ( t + v t from (28 Compring this with y ( t = G (,θ u ( t + H (,θ e ( t (2 (3 H (,θ e ( t C(θ [I A ( θ ] w ( t + v ( t Eution Error Model w/ A ( OE Model Innovtions representtion of the Klmn filter. Let x ı( t,θ e the estimted stte using the Klmn filter. he predition error givey (32 y ( t C (θ x ı( t, θ = C ( θ [ x ( t x ı( t,θ ] + v (t e ( t represents the prt of [I A (θ ] Bu + I A (θ ] w] y ( t tht nnot e predited form pst dt. his prt is lled, the innovtion, denoted e ( t. Using this innovtion, K-F is ritten s (33 x ı( t +, θ = A (θ x ı( t,θ + B (θ u ( t + k (θ e ( t (34 y ( t = C (θ x ı( t,θ + e ( t he ovrine of innovtion e ( t is (35 E [ e ( t e (t] = (θ P ( θ C ( θ + R 2 ( θ Error ovrine of e ( t nd x ı re not orrelted stte estimtion Comining (33 nd (34, nd ompring it with (2, 8
(36 y ( t = G (,θ u ( t + H (,θ e ( t (37 G (,θ = C (θ [I A (θ ] B (θ H (,θ = C (θ [I A (θ ] K (θ + I his shows the reltionship etween the stte spe model nd the input-output model. hey re onneted through the innovtion proess. (I A x ı( t,θ = Bu (t + K e (t x ı( t,θ = (I A Bu (t + (I A Ke (t (38 y t ( = C( I A Bu( t +C( I A Ke t + ( ( e t 6.4 Consistent nd Unised Estimtion: Preview of Prt 3, System ID his setioriefly desries some importnt issues on model struture in estimting the prmeters involved in the model. Detils will e disussed in Prt 3, System Identifition. Let Z e set of dt otined over the period of time: t. One of the ritil issues in system identifition is whether the estimted ı model prmeters θ sed on the dt set Z pprohes the true vlues θ, s the numer of dt points tends to infinity. Severl onditions must e met to gurntee this importnt property, lled Consisteny. First, the model struture must e the orret one. Seond, the dt set Z must e rih enough to identify ll the prmeters involved in the model. Furthermore, it depends on whether the noise term v ( t entering the system is orrelted, whih estimtion lgorithm is used for determining ı θ, nd how the prmeters of the model re involved in the preditor y ı(t θ. Consider the following sured norm of predition error: V (θ, Z = ( t t = 2 y ( y ı( t θ 2 Assume tht the one-step-hed preditor is givey liner regression: (39 y ı( t θ = ϕ (t, θ θ (9 he Lest Sure Estimte (LSE provides the optiml prmeters minimizing the men sured error V (θ, Z : where ı LS θ θ = rg min V (θ, Z = (R( f ( (4 R( = ϕ ( tϕ ( t nd f ( = ϕ ( t y ( t (4 t = t = 9
Suppose tht the model struture is orret, nd rel dt re generted from the true proess with the true prmeter vlues θ : y ( t = ϕ (t θ + v (t (42 Whether the estimte ı θ LS is onsistent or not depends on the dt set nd the stohsti properties of the noise term v ( t. Sustituting the expression of the true proess into f ( yields (t( (t f ( = ϕ ϕ θ + v (t = R ( θ + ϕ (t v (t (43 t = t = ı f *( LS θ θ = (R( [R( θ + f * ( ] θ = (R( f * ( (44 o e onsistent, i.e. lim ı LS θ = θ, the following onditions must e met: (I Mtrix lim ( R must e non-singulr. he dt series, ϕ (, ϕ ( 2, ϕ ( 3,, must e le to Persistently Exiting the system. (II lim f *( =. his e hieved in two wys: Cse A: v ( t is n unorrelted rndom proess with zero men vlues. hen, v ( t is not orrelted with y ( t, y ( t 2, y ( t 3, nd u ( t, u ( t 2, u ( t 3,, i.e. ll the omponents of ϕ (t. herefore, lim ϕ (t v (t = * t = Cse B: he model struture is FIR, i.e. n =, nd inputs u ( t, u ( t 2, re unorrelted with v ( t. he noise term v ( t itself my e orrelted, for exmple, v (t = H (,θ e( t. If the model struture is FIR with unorrelted inputs, then ϕ (t v ( re unorrelted, hene lim f *( =. t he ove two re strightforwrd ses; Consistent estimtes re gurnteed with simple LSE, s long s the dt re persistently exiting. Cre must e tken for other model strutures nd orrelted noise term. For exmple, ARMAX model is used, the liner regression nnot e used, nd the output seuene involved in ϕ (t my e orrelted with v ( : t ϕ (t = [ y( t, y( t 2, ] y ( t = ϕ ( t θ ( + v t his my e orrelted with v ( t * Ergodiity of the rndom proess is ssumed.