Accuracy Aspects of Parameter Estimation in Linear Dynamical Systems

Copyright © IFAC Identification and System Para meter Estimation 1982, Washington D,C .. USA 1982

ACCURACY ASPECTS OF PARAMETER ESTIMATION IN LINEAR DYNAMICAL SYSTEMS G. A. Elkobrosy* and

J.

Hrusak**

*Department of Applied Mathematics and Physics Sciences, Faculty of Engin eering, University of Alexandria, Alexandria, Egypt * *Department of Cyb ern etics, Technical Institut e of M echanical and Electrical Engin eering, Plzen, Czechoslovakia

Abstract. The pa per considers the influence of feedback on the accuracy of parameter estimates in discrete-time stochastic dynamical systems. It was shown by Goodwin and Payne (1977), and others, that the accuracy of noise parameters is independent of the input signal used for a certain classes of systems. This result has been considered in the design of optimal input signal by some authors, e.g. Zarrop (l979) • .. However, it will be shown that for a commonly used model, Astrom model, in system identification and adaptive control, the accuracy of noise parameters are dependent of experimental concitions. The relation between the accuracies of system and noise parameter estimates will be derived. Results concerning the role of feedback in identification experiment will be presented and illustrated by examples. Keywords. Uptimal experiment design; identification; parameter estimation; spectral analySiS; discrete time systems. INTHODUCTION Experiment design for dynamic system identification has attracted conSiderable attention because most industrial, biological, economic and sociological systems are dynamic in nature. Different aspects of identification and experiment design problems, e.g., input signal synthesis, sampling instants, feedback settings, etc., have been discussed, see for example, (lstr~m and Exkhoff, 1971~ hlehra,1976; Goodwin and Payne, 1977; Gustavsson and others, 1976; Strejc, 1978; Peterka, 1981; Evkhoff, 1981). One of the most important factors in any system identification experiment is the relationship between the experimental conditions and the achievable accuracy of the parameter estimates. By experimental condition we mean the manner in which the input to the system is determined. It can, as in open-loop experiments , be chosen freely by the experiment designer. It can also be determined partly from output feedback by a regulator a given structure, etc. In this concept other conditions such as the sampling rate, the experiment length, etc., can be included. These will, however, not be considered here. It is then an important question how 933

the experimental condition influences the accuracy of the parameter estimates In this paper we proceed to optimize the experimental conditions in order to obtain as good accuracy of the parameter estimates as possible. We consider a linear single-input singleoutput discrete-time stochastic system given the general form Yt = Gl(z)u t + G2 (z)e t (1.1) and essentially, we will be concerned with, a commonly used model in system identification and adaptive control, Astrom model given by A(z

-1

)Yt=B(z

-1

)ut_d+C(z

-1

let

(1.2)

and Box and Jenkins model given by = B(z-l) u

Yt

A(z-l)

+

t-d

t

C(z-l) D(z-l) et

(1.3)

where tUts, Ytl are the input and output sequences respectively, tetl is a sequence of i.i.d. Gaussian variable having varianceov 2 , z-l deno~ tes the backward shift opera tor, d ij 1

G. A. Elkobrosy and J.

934

is the time delay and A(z C( z

-1

-1

n

)=l +.Laz i =l 1 q

)=l+~

i= 1

c .z

-i

-i

,

B(z

, l)(z

1

-1

-1

m -i ) =L:> . z i=O 1 r

)=l·.L: dz i=l 1

-i

( 1.4)

However, the Box and Jenkins model, eq. (1.3), can be considered as a special case of the model of eq. (1.1) where the transfer functions Gl and G2 have no common parameters. This has been studied, e.g. by Goodwin and Payne (1977 ) and the synthesis of optimal input signals, for the open-loop case, has been studied by many authors, e.g. Zarrop (1979). General results, however, concerning Astrom model, eq. (1.2), have no t been addressed to. In Soderstrom, Ljung and Gustavsson (1974, 1975), SOderstrom (1975) and Gustavsson, Ljung and Soderstrom (1976) a first-order system has been examined in detail with constraint input and output variances. This study has raised a number of interesting conjectures regarding the role of feedback in optimal experiment design. In this paper we give a new results concerning the optimal experimental conditions of Istrom model and we will compare these results with that of Box and Jenkins model. The paper is organized as follows: In section 2, we derive the information matrix for a general SISO system and discuss criteria of accuracy. In section 3, achievable accuracy in an open-loop experiment as well as the accuracy with which the noise parameters are determined in the two models will be discussed. In section 4, achievable accuracy in a closed-loop experiment, the role of feedback to enhance or deteriorate parameter estimation accuracy will be discussed and examples will be given. Conclusions are given in section 5. THE INFORMATION MATRIX AND CUITERIA OF ACCURACY Consider a general linear system Yt = Gl(z)u t + G2 (z)e t Ut = G3 (z)Yt + St

t

t 1

(2.1) (2.2)

where Ut \ and y t are the input and output sequences respectively, tet1 is a sequence of i.i.d. Gaussian variables having variance qV~. Gl(z) and G (z) are unknown transfer functions. 2 I t is assumed that Gl(z) and G (z) and 2

Hru~ak

its inverses are stable. tSt1 is a sequence of set-point perturbations or external inputs. G ( z) is the trans3 fer function of the feedback control law that is causal. It is assumed that G3 (z ) and the sequence {St) may be chosen, at will hy the exper1menter. Gl (z) and · G ( z) and (7-l are para2 meterized by an identifiable vector Q (c.f. Gustavsson, and others, 1976). Following Elkobrosy (1982) the average information matrix per sample is given by

where N is the number of data pOints, Ey/~ denotes conditional expectation over the data given the parameters and 'r/. t is the residual sequence given by

'Y( t = G;l [Yt - GlUt]

(2.4)

where the shorthand notation G.1 =G1 (z) has been used. Substituting (2.2) into (2.4) gives ( 2.5) '? t=G;1[n-G I G3 )Yt- Gl s tJ where from (2.1), ( 2.2), (2.4) and (2.5) to'?t/o~k! isgivenby

d'7 t

-1

G ) -1

~9k ",-G 2 (l-G 1 3

-G

-1

2

[0o9

G1

G (1- G1 G3 ) k 3

'd

G1

c

~gk "' t

_ l o G2 G2+ d9k

(2 . 6 )

1Je t

substituting \~.6) into ( 2.3), 1ettin! N~- and using Parseval's Theorem gives the following asymptotic expression for the information matrix.

935

Accuracy Aspec ts of Parameter Estimation

racy with which the noise parameters is determined in the two models will be discussed. Lemma 3.1. For Box and Jenkins model described by eq. (1.3) and (1.4), and G -* = Gi -If" ' To form a basis for satisfying i the comparison of di fferent experime1. A constraint is imposed on the innts, a measure of accuracy is required. put variance or the output variance. 2. The determinant of the informatIon However, the par ameter accuracy is a function of both the experimental con- matrix i s used as a design criterion. ditions and the form of th~ estimator. Then, the accuracy with which the noise parameters are estimated is indepenFor any unbiased estimate Q , of a parameter Q, it is well known that a lo- dent of the input signal. wer bound of th e covariance of the es- Proof. (c.f. Goodwin and Payne, 1973, timate exi~ts, and is given by th e so- ~ Zarrop, 1979). called Craner - Rao inequality Define the (n+m+q+r+2) - vector Q of the unknown parameters as (2.8) (3.11 QT = [T a , b T , c T , d T ,QJ J Any unbiased estimate that satisfies the bound with an equality sign in eq. the corresponding partitioning of the (2 .H) is called an efficient estimate. information matrix yields Therefore, by assumIng an unh ~a s e d nnd efficient estimator, for example the maximum lik eli h oo d estimate has these M = (3.2) properties under reasonable general o M22 c ondi tions , the error covariance matrix of th e estimate, which is g iven hy where, the matrix Mll is of order th e inverse of the Fisher's informa(m+n+l) and is a function of the spection matrix, can be determined without tral distribution function of the inhaving to know the actual form of the put sequence and corresponds to the syestimator (see th e dis c ussion of in~tem parameters (aT, b T ). The matrix flu ence of th e identification method on identifiability and accuracy in M22 is of order (q+r+l) and is a conGustavsson Hnd others, 1976 ) . We adopt stant matrix independent of the input the standard approach of defining opsignal and corresponds to the noise timality via a suitably chosen scalar parameters (c T , dT,QJ), and is given function of the information matrix. We shall principally use the determiby nant of the information matrix as the 2 design crit erion leading to D-optimal =qJL 0 Q . ?)9k designs. An advantage of this crite11 J rion is that the optimal design is not 1 aG~ -1 '?>G 2 affected by parameter scaling.When no + 2T 1 ~Q . G2 G2 ~ dw restriction is put on the variances -1r J k of the input or the output, it would j=I,2, •.• ,q+r+l be favourable to have very large sig(3.3) nals. Ther e fore the cases of constrai- k=I,2 .•.• ,q+r+l ned output variance where -1 -q :'Il" C(z) l+clz + ... +c z G2 ( z)=D(z) =--~~-~l------~g~--r (3.4) Eb \ = d Fy ( w ) 1 ' 1 ) 0 ( 2. 9 ) l+d1z + ... +drz -'TT or constrained input variance For finite (large) N, the standard deviations (s.d.) of the noise parame:1r Etu~~ =~ dFu(w)~ ~ 2 ' ~2) 0 (2.10) ters are given by -lr s.d. (Q . )=N- 1/ 2 will be considered. In the sequel we I 11 shall be concerned with the following i=l,2, ••• ,q+r+l categories of experimental conditions. This completes the proof. 1. Upen loop (i.e. G3 =0, St40) Lemma 3.2. For Astrom model described 2. A Single feedback (G 3+O) together by eq. (1.2) and (1.4), satisfying with set point perturbations (StfO). 1. A constraint is imposed on the input variance or output variance. 2. The determinant of the information ACHIEVABLE ACCURACY IN AN matrix is used as a design criteria. OPEN-LOOP EXPERIMENT Then, the accuracy with which the noiIn the following two Lemmas, the accu- se parameters, exceptqJ , are estimated is dependent of the input sigtlal.

where fs(w) is the spectral distribut ion function of the set point sequr ,1\ _ ence tSt ,Gi--Gi(e jw ), Gi -G i ( e - jw )

1

-

l

[Mll : J

[1JfJ'Jll)q-.J]

r

~

~1\"

f

f

~~

-*

~

V[M;~l ..

G. A. Elkobrosy and J.

936

Proof. (c.f. Elkobrosy. 191:11a) TIe1Ine the (n+m+q+2) - vector ~ of the unknown parameters as ] (3.5) QT= [T a • bT • CT .f7V the corresponding partitioning of the

inforrnat:on=m[at~:: Yie~:: J - T 1112

(3.6)

M22

where 1. The matrix Mll is of order (m+n+l) and is a function of the spectral distribution function of the input sequence and corresponds to the syT T stem parame~ers (a • b ). 2. The matrix M22 is of order (q+l) and is a constant matrix independent of the input signal and corresponds to the noise parameters T (c .0'-'). 3. The matrix M12 is a (n+m+l) x (q+l) constant matrix. independent of the input signal and is given by

LM 12 ]J"k

1 rT)G; _~ -1 - 2lr) d a " G2 G2

~G2

~ck

J

-11'

= 0 otherwise j=1.2 ••••• n k=1.2 ••••• q

dw

( ~-1)

(3-8)

Tho corresponding partitioning of the covariance matrix p. say. of the parameter ,s:;m:t'[SP::eldSp12]

(3.9)

1'22

where I'll and 1'22 correspond to system and noise parameters respectively. Then 1 [-

-

T -

1- 1

-1 -

P 22 =~ M22 -M 12 "11 "12 (3.10) The ~atrix "11 is a function of the spectral distribution function of the input signal. This completes the proof. Remark 3.1. Note that the accuracy of (]V is independent of the input signal in the two models. Note also that 1 [-

P11 = N

Mll - M12 "2;

let Ai11 = NMl!

1

det P 11

> det

-1

Mll

. det

-1 1'2 2 > det M22 ( 3.12)

but. det M22 det I'll = det Mll de t 1'22

(3.13)

which is the same as in Box an d Jenkins model. see (3.2). ACIII EVABLE ACCliRACY Jl\ CLO SED-LOOP EXI'EHH! Et-.T In this section the influenc e of feedback on the accuracy of parameter estimates will be discussed. However. in the case of Uox and J e nk I ns mod e l. in the !"ense of lJ-optimalJ ty and WIth input power constraints. the presence of feedback will. in general deterIOrate the achievable accuracy and maximal accuracy can always be achieved with an open-loop experiment (c.f. Goodwin and Payne. 1977; ~g and others, 1977a). In the sequel . wc w l ~ l discuss the case for Astr~m model. Theorem 4.]. (Equivalenc e Theorem ) For Astrom model described by eq. (1.2) and (1.4), satisfying 1. A constraint is imposed on the output variance. see eq. (2.9). 2. The determinant of th e information matrix is used as a desi gn criter i a. 3. The time delay d=l. th e polynomIals in (1.4) are such that C(z-l) 1 A(z-l) # 1 Then, a closed-loop experim e nt when the system input is generated by a minimum variance control law with an external signal satisfying (4.4) gives the same maximum accuracy as an opti~al open-loop experim ent. Proof. (c.f. Elkohrosy. 1981a) Define the (m +n T 2) - vector Q of the unknown parameters as ijT = [aT, bT,O'-'J

=

whore.

Pl~

Hru~ak

T]-l "12 (3.11)

!l22 = NM22

Then. from (3.11) and (3.10)

The corrc!"ponuing partitioning of the information matrix yields Maa

M=

Mab

MaO'"" ( 4.1)

where in both closed-loop a s well as Qpen-loop experiments th e matrices " afJ" and ~1 )(JV are equa l to z e ro, an~ the matrix M(J'-'c1" i s Cl cons tant. In the sequal we will show that th e matrices Maa , Mbb and Mab are the s a me for both open-loop anG closed-loop exp e riments. In the open-loop case, the output power constraint can be written. using (2.1). as

937

Accuracy Aspects of Parameter Estimation "It

2

it

dFy(w i -= GlGldfu(w)-'-G2G20V' dw

(4.2)

substituting (4.2) into (2.7), note that G3 =0, yields :1f

open

[M aa

6 * J "k =-.l. -.£ G-"G- l 21\ ) 'd a " 2 2 G

"dG _2 dw

oak

J . . -li J 1 )11" 'dG1 , _ It -1 ~ G1 - -- G G - - dw +

1 :1\

1t"

'0 G1 -~2"1T .,(]V 1T ?la J" (4.3) 1

211" -11 6 a J" 1 1 0 a k 'G -it -'*: -1 -1 0 1 Gl G2 G2 Gl ~ak dFy(w) in equutlon (4.3) the first and second terms are equal. In the closed-loop case , the output power constraints can he written, using (2.1) and (2.2) with G the min i mum variance control law 3 (Astrom, 1970), as it -1 2 dF y (w) :.:: G-*" G1GIG'J (4.4) _ dF s (w) + qv dw 2 . substituting (4.4) into (2.7), note that G3 is the minimum variance control law, yields ,

( 'dG* 1 J __l G-IIG-*"x 2l\ov2 -11" ?>a j 1 2

-=

,··1

G

2

from ( 4.5 ) and

-1

Gl

(4.3)

_

lM J

•

ua open Mhb und Mab • Let

)G l

'?>a

dFy(w) (4.5)

k

l McHI ] closed =

S irrilar Procedures for

[

Maa

Mab]

hla~

Mbb

-y

"-

Hemark 4.2. For the output power constraint, eq. (2.9) note that in the closed-loop case, eq. (4.4), that ~ 1> 0'-'2 must be satisfied, whereas in the open-loop case, eq. (4.2) lr 2 ~ _1_ q-.J d (4.7) 1 2~ A(ejw)A(e- jw ) w

>

f

o

must be satisfied. This means, if the constraint ~ 1 is restricted to lie in the range 11'" (J'J

2

<:

J

b l~ 1"..... ",11

(7V2

"

,..

F/w) -= Fy
[d~t MU] ~ [det MU(Fy)] Having F (w), inverting the eq. (4.2) ,.y we get F (w) in an open-loop experiu ment or equivalently, inverting the "eq. (4.4) we get Fs (w) in a c1osedloop experiment. This completes the proof. Hemark 4.1. Theorem 4.1. can be considered as a generalization of the results of Goodwin and Payne (1977); ~g, Goodwin and Payne (1977); Ng and Qureshi (1981), and others, concerning autoregressive model. Yt=alYt_l +a2Yt_2+··· +anYt_n+hut_l+et with o utput power constraints,eq.(2.9)

.

A(eJw)A(e-Jw)

dw

(4.8)

c

only closed-loop experiments are poSSible to achieve maximum accuracy. Hemark 4.3. ~ote that, in theorem 4.1., the two designs are equivalent with any measure of accuracy based on the information matrix. Example 4.1. (c.f. Goodwin and Payne, 1977; Ng and Qureshi, 1981). Consider the second order system 2 y t =a lY t-l +a 2 y t_2+ bu t_l +e t ' e t IV G( 0.0"-' ) b-=l.O (]V=l

(4.6 )

Then det Mll may he maximized subject to the constraint on F (w) by choosing "

For this model, it can be shown that det M , eq. (4.6), will be maximized ll by a white noise sequence ~Yt" having variance CO l' From eq. (4.4), the set point perturbations will be a white noise sequence having variance

and wlth output power constraint ~1=5 (such that eq. (4.7) will be satisfied). In closed-loop experiment. in the sense of D-optimality. maximal accuracy can be achieved when the system input is a minimum variance control law with external white noise,i.e. ut=-O.5Yt+O.lYt_I+ E t '

E trvG(O,4)

In an open-loop experiment, maximal accuracy can be achieved with an input signal comprising of two sinusoidal frequencies, where Ut = ml coswlt + m2 cosw 2 t and ml =O.8252 m2 =2.0459 Exam~le

wl =O.7854 w2 =2.3212

4.2._(c.f. Gustavsson. Ljung and oderstrom, 1977). Consider the first order system

G. A. Elkobrosy and J. Hrusak

938

where il =U.5 h=l.U (]V =1 l and with output power constraint ~1=3 (such that eq.(4.7 ) will be satisfied) In closed-loop exper 1Ment, in the sense of D-optimality, maximal accuracy can be achieved when the input is a minimum \tariance control law with external whit e no i se, i. e . u t = -O. 5Y t ... t t t t fVG (O,2) In an open-loop experiment, maximal accuracy can be achieved with a first order autoregressive input, i.e. u = -O.75u _ +C c t rvG(O,1.2) t t l t However, when the polynomial C(z)=Fl the accuracy with which the unknown parameters in the Model (1.2) and (1.4) are estimated is given by the following result,for details see(Elkohrosy,1981a) Theorem 4.2. For Xstrom wodel described bye q • ( 1. 2 ) and (1. 4) sat 1 s f Yi ng 1. A constraint is imposed on the output variance, eq.(2.9) 2. determinant of the inforMation matrix is used as a design criteria 3. The polynomials in (1.4) are such that A(zl C(Z) # 1 Then, there exists an optimal experiment which is closed-loop. Examyle 4.3.(c.f. Gustavsson and others 1977 • Consider the first order system

*

Yt+aYt_l=but_l+et+cet_l ' etNG(u,~2) with ouput power constraints.eq.(2.9). It can he shown that maximal accuracy can be ach1eved only, in the sense of D-optimality, when the system input is generated hy a minimum variance control law with a white noise sequence with variance( SI-~2)(1-c2) / ,/ Remark 4.4. For the lack of space simulation results cannot be considered. CONCLUSIONS The paper has been concerned with the influence of experimental condition on the accuracy of parameter estimates in discrete tiMe stochastic dynamical systems. Essentially'. two models have been conSidered, Astrom model, eq.(1.2), and ~ox and Jenkins model, eq.(1.3). For AstroM Model, it has been shown that the accuracy of noise parameters, in contrary to Box and Jenkins model, is dependent of the input signal. Concerning the role of feedback, the analysis and simulations show that, for the constra1nt output power case and in the sense of D-optlmal~ty, closed loop exper1ments for Astrom model are not necessarily, in contrary to Box and Jenkins model, inferior to open loop experiments. We have presented an "Equivalence Theorem", Theorem 4.1, under certain conditions, when the system input is generated by a minimum

variance control law and a suitably chosen external s e t-pOint perturbation ,eq. ( 4.4 ) , gi. v es the same maximum ace ... uracy as an o pt1mal open l oop experiment, if the latter exists. Moreover, It was shown that wi thin a c erta i n range. e q. ( 4.~ ) , o f the c onstraint,maximal accuracy can be achieved only w1th cl osed loop experiment~. It has been shown als o that, th e results of earlier investlgators f or autoregressive model can he derived us i ng Theorem 4.1. Further, i t has been indicated that , Theorem 4.2 •• in th e case where MaxiMUM accuracy canno t be achieved in an open lo o p experiment, feedback will give us the second de g ree of freedom to reach maximal acc'uracy. Finally, three nontrivial examples has been considered. The r e sults 1n this paper have been developed for SISO case for simplicity of exposition. However, the extension to MIMO case is straightforward taking in conSideration identifiability conditions. HEFWENCES ~str~m,K.J.,and P. Eykhoff(1971). • Au~omatica, 7, 123-162. AstroM,K.J.(1970).Introduction to stochastic control theory , Academic. Elkobrosy,G.A.(1981).The 8th IFAC World Congress.Kyoto-Japan. Elkobros!,G.A.(1981a).Ph.D. Dissertation.VSSE,Plze~.Czechoslovakia.

Elkohrosy,G.A.(1982).IFAC Workshop, Ankara-Turkey. Eykhoff,P.(Ed.,1981).Trends and progress in sys.ident •• Pergamon Press. Fedorov,V.V.(1972).Theory of optimal experiments.Academic Press. Goodwin,G.C.,and H.L. Payne(1977). Dynamic system ident.,Academic. Goodwin,G.C.,and H.L. Payne(1973). 3rd IFAC symp.ident.The Netherlands. Gustavsson,I.L. and others(1976).7th IFAC symp.ident. ,Also Aut. ,13,1977. Mehra,H.K.,and D.G. Lainiotis(1976). System ident.,Academic Press. Ng,T.S.,and Z.H.Qureshi{198l).The 8th . IFA~ World Congress,Kyoto-Japan. ~g,T.S.,and others(1977).IEEE Trans. Autom.Control, AC-22. ~g,T.S.,and others(1977a). Automatica 13, 571-577. ~!terka,_P.(lY81).Automatica,17,41.

Soderstrom,T.(1975).Report UPTEC 756 _3H, UPEsala University, Sweden. Soderstrorn,T. ,and others(1975).Proc. _Bth IF!C Congress,paper 18-1. Soderstrom,T.,and others(1974). Report 7428, Lund institute of Technology. Strejc, V.(1978). The 7th IFAC Congress.Helsinki, Finland. Viort. B.(1972). D-optirnal design for dynamic rnodels.Dept.statist.,University Wisconsin,Madison,TR314,316. Zarrop,M.B.(1979). Optimal experiment design for dynamic system ident. Springer - Verlag.