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Refined instrumental variable methods of recursivetime-series analysis Part III. ExtensionsPETER YOUNG

a & ANTHONY JAKEMAN

a

a Centre for Resource and Environmental Studies, Australian National University, Canberra

Australia

Available online: 21 May 2007

To cite this article: PETER YOUNG & ANTHONY JAKEMAN (1980): Refined instrumental variable methods of recursive time-

series analysis Part III. Extensions, International Journal of Control, 31:4, 741-764

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Refined instrumental variable methods of recursive

time-series n lysis

Part 111. Extensions

PETER YOUNGtS and ANTHONY JAKEMANt

This is th e final paper in a series of t hre e which have been concerned w i t h th e compre-

hensive evalua tion of the refined instrumental variab le

I V )

method of recursive

time-series analysis. The paper shows how the refined

IV

procedure can be extended

in various important directions and how it can provide the basis for the synthesis of

optim al generalized equat ion error GE E) algor ithms for a wide class of etochastic

dynamic systems. The topics discussed include the estimation of param eters in

continuous-time differential equation models from continuous o r discrete d at a

the estimation of time-variable parameters in continuous o r discret e-tim e models of

dynamic systems the design of stochestic state reconstruction Wiener-Kalmen)

filters direct from da ta the estimation of parametere in multi-input, single ou tp ut

MISO)

transfer function models the design

of

simple stochastic approximation

SA)

implementatio ns of t he refined

I V

algorithms and the use of the recursive algorithms

in self-adaptive self tuning) control.

1 ntroduction

I n the first two parts of this paper Young and Jake man

1979

a, Jakeman

and Young 1979 a ) we have been concerned with th e description an d compre-

hensive evaluation of th e refined instrumental variable

I V )

approach to

time-series analysis for single input, single output SISO) and multivariable

dynamic systems described by discrete-time series models. In

this, the third

and final part of th e paper, we consider how the refined I V method can be

extended in various directions to handle continuous time-series models and

discrete or continuous time-series models with time-variable parameters. We

also discuss briefly other extensions including off-line and on-line adaptive

methods of designing stat e reconstruction Kalm an) filters for stochastic

systems

;

th e development of

IV

estimation procedures for specific time-series

models, such as the multiple input-single ou tput transfer function model an d

finally the estimation of parameters in multivariable system models in those

situations where the observation space is less tha n the dimension of th e model

space. Fo r convenience, in a11 cases except the latt er , we shall consider refined

I V estimation algorithms with non-symmetric matrix gains. Bearing in mind

th e results of th e first two parts of the paper, however, it is clear tha t symmetr ic

matrix gain alternatives could be implemented and i t is likely th at, a t least for

reasonable sample size, they would perform in a similar manner.

Received 10 August, 1979

t Centre for Resource and Environmental Studies, Australian National University,

Canberra, Australia.

1Currently Visiting Professor, Control and Management Systems Division,

Engineering Department, University

of

Cambridge

002&7179/80/3104 0741 02.00 1980 Taylor Francls Ltd

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742

P Young and

A

akeman

2

Continuous time dynamic systems described by ordinary differential equations

models

The refined

IV

procedure can be applied to both SISO and multivariable

continuous time-series models bu t, for simplicity of exposition, we will describe

here only the SISO implementation. The extension of the SISO procedures

to the multivariable situation is, however, quite obvious by analogy with the

discrete-time case discussed in Part

I1

of the paper (Jakeman an d Young

1979 a) . Using nomenclature similar to tha t used previously, the continuous-

time SISO model is illustrated in block

M

(within dotted lines) of Fig. and

can be written as

(ii)

(iii

YV

= x t )

f

1)

where

s

is the differential operator,

i

.e. sx(t

=

dx t /dl (Ioosel

y

interpreted here

as the Laplace operator)

; A B,

C and

D

are polynomials in s of the following

.

form,

and

[ t)

is a continuous-time white noise process.

It is well known (e.g.

AstrGm 1970, Jazwinski 1970) th at theoretical and analytical difficul ties

arise because of the use of continuous white noise in mathematical models of

dynamic systems, particularly transfer function formulations such as eqn.

1 ) (ii). In the present context , this difficulty is manifested in the form of

practical problems associated with the recursive estimation of t he parameters in

the

C

and

D

polynomials characterizing the noise model. For the moment ,

however, i t is convenient to assume a continuous-time model of the form

1 (ii)

although, as we shall see, i t is necessary in practice to evaluate t he noise-

components of the model in discrete-time

in

order to circumvent estimation

problems.

2.1. Discrete and continuous time recursive

algorithms

It

is clear that the model

(1)

is algebraically equivalent to the discrete-time

SISO model discussed in previous parts of this paper.

Let us consider,

therefore, the situation whcre we wish to implement the estimation algorithm

in discrete-time using sampled da ta from the continuous-time system we will

refer to this as CD (continuous-discrete) analysis (Young 1979 a). Using an

approach similar to th at used in previous parts of the paper, it is then possible

to obtain estimates of t he parameters in the continuous-time model polynomials

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Refined instrumental variable

methods

of recursive time-se ries an aly sis 743

A

and B by minimizing a least squares cost function of the form

Here

y

=

[yo yo

yOTlTand

u

=

[u

uo2

,

u O T I T

where the first zero

subscript on u and y indicates that the variables are, respectively, the basic

input and out put variables i.e. the

'

zeroth

'

derivatives of u and y ), while the

second subscript

i

= l

2

, T denotes the sampled values of the variables

at

time ti i.e. y t,) and

u t,).

Figure

1.

Refined IV

algorithm for continuous-time systems.

'noise

M

:

w .

I

~ s i

,

~ t )

,

YO*

t

-

r.5thte.--

I

1) REFIND

reconstruct~oni

I A s I

o r

1

- - tate

I

fi lter

Now, by direct analogy with the analysis in the DD discrete-discrete) case

of Part I, the recursive estimate d of the unknown parameter vector

a

[a a ..

,

a b b , , ..., b - IT can be obtained from th e following discrete-time

algorithm,

i a = a,-,

P,-,ak*[sz

+z , * ~ k - l ~ k* ] -l ~ k *T,-, Ok*)

or

ii) a

=

-P,lt,*{~,*~

-

Ok*)

and

iii) P, =

P,-,- Pk-,g,*[e2 + z ~ * ~k-lltk*]-l~k*TPk-l

u(t)

where

Here 6 is an estimate of the variance of

e t ) ; i k

s the outpu t of an adaptive

' auxiliary model ' as shown in Fig. 1 and t he st ar superscript indicates that

z

ariable

...

I

i

~ l t e r s

* ,

a c

kit)

~ S I -

recurswe

or

u*.dt)

-

a u x ~ lary

model

iteraiive update

,

I

v

C sl

- GoRITw

[s)A s)

a

k i t )

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744

P. Young and

A .

Jakeman

the Jariables are filtered by adaptive prefilters

' C B A

again as shown in

Fig. 1. The i =

1, 2 ,

n subscript on the variables within th e square brackets

in

5)

denotes th e ith time-derivative of the variabIe, while the

k

subscript

outside th e brackets indicates th at the enclosed variables are all sampled a t th e

kth sampling instant.

.

This algorithm has close similarity with the

I V

algorithm suggested some

years ago by Young 1969), h e only difference lies in the na ture of the prefilters ;

in the previous algorithm, these were termed ' sta te variable filters an d were

introduced mainly to avoid direct differentiation of noisy signals. I n th is

sense, the function of the present filters is identical

:

their presence means that

it is not the direct derivatives of the variables

y l ) , i t )

and

u t )

hat are required

for estimation b ut t he derivatives of the filtered variables y* t), j . * t ) and u* t).

And these filtered derivatives, unlike th e direct derivatives, are physically

realizable s a product of th e filtering operation Young 1964, 1969). Of

course the prefilters here do more than just avoid differentiat ion of noisy

signals ; they also represent the mechanism for inducing asymptot ic statisti cal

efficiency.

In the present case, the ' optimal prefilters are defined in terms of estimates

of t he a

priori

unknown polynomials

A ,

C and

D. t

is necessary, therefore,

to define some adaptive procedure for synthesizing the prefilters as t he estima-

tion proceeds. In the situation where C .0, i.e.

t )

is white noise, the

adaptive synthesis of the prefilters 1 A is fairly straightforward

:

both the

prefilter and auxiliary model parameters can be updated either recursively or

itera tively as shown in Fig. 1, exactly as .in the discrete-time model case

described in Part I of this paper. When th e noise

[ t )

is coloured i.e. C 1.0

and/or

D

1.0), however, the situation is no t so straightforward : in contrast to

the discrete-time model situation, i t is not easy to construct a similarly moti-

vated recursive estimator for the continuous-time noise model parameters since

the derivatives of the white noise e t ) do not exist in theory.

While i t may be possible to solve this noise estimation problem by con-

sidering either band-limited noise or purely autoregressive noise where

derivatives of

e t )

do not occur), we feel that it may be better to consider a

hybrid approach. Here, the noise is estimated in purely discrete-time DD)

terms by the use of the

A M L

or refined

A M L

algorithms described previously.

This does not create any implementation problems because the noise model is

only required for adaptive prefiltering operations, which can easily be carried

out in discrete-time when using D analysis. The general implementat ion

in this case is shown in Fig. 2 a) and the detailed structure of the deriva tive

generating filters

l / A s )

is illustrated in

Fig.

2 b).

t

should be noted here

th at t he filter in Fig. 2 b ) s similar to the ' stat e variable filter suggested by

Kohr see, e.g. Kohr and Hoberock 1966) : the only difference is that the

coefficients ti i = 1, 2, n are not constant, as in the Kohr case, but are

aduplively

adjusted, either iteratively or recursively, as the estimation proceeds.

U p to this point we have assumed that, while the algorithm

4)

is imple-

mentcd in discrete-time, the signals y t),4 t)and

u t )

are available in continuous-

time form so that they can be passed through the continuous-time prefilters

prior to sampling. In practice, however, it could well be th at both in put an d

outp ut signals are naturally in sampled data form. This difficulty can be

circumvented, albeit in an approximate manner, by assuming th at t he signals

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Refined instrumental variable methods

of

recursive time series analysis

7 4 5

remain constant over the sampling interval and passing them directly into the

continuous-time filters. In other words, the sampled dat a ar e converted to a

continuous time staircase form prior to filtering. I n this manner, the

prefilters perform an additional, useful, interpolation role and provide

estimates of the continuous-time filtered variables.

and I

and

I hold I

I hold i

x

J i. .Y J

a1

Figure

2.

Refined I V algorithm

: D

implementation.

a) verall implementation

X

closed : continuous-time data available

;

X operative

:

discrete data only

available ; b ) state variable filter l / A s ) applied to u t) .

r - - . -

Of course th e estimates of the filtered variables emerging from the prefilters

are in no sense optimal and th e efficacy of this approach is clearly dependent

upon the sampling period T the approximation will be good for small T

and will become progressively worse as T s lengthened. Fortunately the

estimation results do not appear particula.rly sensitive to the choice of T nd

acceptable performance can be obtained from qui te coarse sampling frequencies.

I

v F

(see block

above 1n a1)

I

u:(t)

I

uE-, t )

I

u:-l t)

I :

I

: 4u; ( t )

I

u. ( t )

I

I

I

I

m > I

=-

I

I

_

- .

(b

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746

P Young

and

A

Jakeman

Finally, i t is worth noting tha t, if continuous-time measurements of y(t ) an d

u(t)are available, then it is possible to consider a continuous-time implementa-

tion.of the estimation algorithm itself (i.e. using CC analysis). This is a logical

development of early continuous-time gradient procedures for estimat ing

dynamic system parameters (see, e.g. Young 1965 a, Levadi 1964, Ka ya an d

Yamamura 1962, Young 1976). The most obvious impleinentation would be

an estimation algorithm of the form

(i)

(ii)

which is a continuous-time equivalent of the discrete-time recursive algorithm.

Algorithms of thi s form are also discussed by Solo (1978).

Note that it would be difficult to implement the estimation algorithm (6)

for other than

D

1.0 because of the difficulty in estimating th e an d D

polynomials (unless we once again consider some hybrid mechanization which

would be rather impractical). Thus when f(t ) s not white noise, th e estimates

produced by the algorithm will not have any optimal properties. They will,

however,. be consistent, asymptotically unbiased and, on the basis of previous

experience, they should be reasonably efficient (see Jakeman 1979). Note also

th at we can reduce the computational complexity furthe r by replacing (t) in

(6) by a simpler stochastic approximation

SA)

gain (e.g. Young 1976). Thi s

would be a continuous-time equivalent of t he SA algorithms discussed in

7

2.2. Experimental results

The CD approach to t he continuous-time model estimation discussed in t he

previous section has been evaluated by Monte Ca;rlo simulation analysis applied

to two systems described by second order differential equations.

In

the first

case, the sys tem was of th e form

with u (t ) chosen as a random binary signal with levels plus and minus 1.0.

In the second, the system was modified to

with K 0-781, w 1-6,

0.5

and u(t) chosen as the following combination

of three sinusoidal signals,

u(t ) =sin (0.5wdt) sin (w,t) +sin (l.5wdt)

where w is the damped natura l frequency of the system i.e. w o,Z/(1-

c2 .

In both of these examples, the noise l ( t) was simulated white noise adju ste d

to give several different signal/noise ratios S (defined as in Young and Ja keman

1979 a) .

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Refined instrumental variable methods. of recursive time-series analysis

747

Tables 1

a )

nd 1 b )are typical of the results obtained during the analysis.

For each sample size, column represents the average parameter value over 10

experiments, while columns and 3 represent standard deviation from th e true

and average parameter value, respectively. I n Table 1 , th e sampling interval

T,

is chosen to be quite rapid a t

0.1

sec which represents

1/31.4

of th e Shannon

maximum sampling period, P ,

=

n /w , In Table 1 b) , wo different sampling

intervals are compared one fairly coarse

P, /8) ,

he other quite small

P, /40)

there seems to be some bias on a,

in

the coarse sampling situation.

Number

o

samples

Parameter True

value

100 500

Table

a). S=10, T,=0.1.

Sampling rate

Parameter True

value PJ40 Pel8

Table b ) .

S= 10, 500 samples.

Other simulations have tended to confirm that quite coarse sampling

intervals can be tolerated but suggest that the degree of bias is, no t surprisingly,

a function of the system dynamic characteristics and the value of S As a

result, the algorithm should always be used with great care if th e sampling ra te

is low and, as

a

rule of thumb, sampling intervals should always be chosen less

than P,/10. But there is clearly a need for more research on this topic before

the algorithm can be used with confidence with coarsely sampled data.

The algorithm 4) has also been applied with, some success both to multi-

variable systems Jakeman

1979

and to real data. Typical of the latt er are

the results shown in Table c) and Fig.

3.

Table c)compares the estimates

obtained when carrying out

CD

and DD time-series analysis on da ta obtained

during fluorescence decay experiments on 1-naphthol Jakeman e t al 1978).

I t is clear th a t the continuous-time and discrete-time models have virtually

identical dynamic characteristics in this case, where the sampling period was

short in relation to P , approximately P,/113).

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P .

Young nnd

1. akernan

6 dye

tr cer

data

model a t ) = b , l ~ ( t - ~ l

+

b,stGs

odel

vtput

.

C t )

time

hours)

Figure 3 .

Results from model of dye tracer concentration in Murrumbidgee River,

Australia.

Dynamic characteristics

Parameter

Model estimates Time constant Steady state.

(nsec) gain

Discrete time T 0.212 nsec a

=

.9724 7.575 1.0

bo= 0.0276

Continuous time; time unit = 0- 212 a

=

35.9622 7.624 1-0

nsec bo= 1.00

Table c).

Figure 3 shows the observed and estimated dye concentration in a river,

where the estimated concentration is generated by

a

second order differential

equation model estimated using algorithm 4. The data used in this exercise

wcre collected during dye tracer experiments carried out on t he Murrumbidgee

River system in Australia (Whitehead et al 1978). Here, as can be seen from

Fig.

3 ,

i t was not possible to maintain a completely regular sampling interval

but T s approximately

half

an hour P , / 3 0 ) . This demonstrates how data

with irregular sampling intervals can be used, provided the longest sampling

interval does not lead to serious interpolation errors and estimation bias.

3

Syst em s described by time variable parameter models

The idea of modifying recursive algorithms to allow for the estim ation of

time-variable model parameters has been exploited many times since th e

publication of R. E. Kalman s seminal papers on st ate variable filter-estimation

theory in the early nineteen sixties (Kalman

1960,

Kalman and Bucy

1 9 6 1 ) .

In the case of V algorithms, thi s particular extension has so far been heuristic

(see Young 1 9 6 9 ) . Bu t now, with the advent of the refined V algorithm, i t is

possible

to

pu t such modifications on a sounder theoretical base and t o const ruct

algorithms which have greater practical potential.

There are a num ber of ways in which th e time variable parameter modifica-

tions can be introduced. The most straightforward is simply to take no te of

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Refined instrumental variable nzetho s of recursive t ime series ana lysis

749

the relationship between the refined I V algorithm and the Kalman estimation

algorithms and introduce additional a priori information in the form of a

stochastic model for the parameter variations. The general form of this model

is the following discrete-time, Gauss-Markov model,

Here

@

and I are assumed

known

and possibly time variable matrices, while

qk is a discrete white noise vector with zero mean .and covariance matrix Q

which is independent of the observational white noise source e i.e.

where S,, is th e Kronecker delta function.

This device is now well known in the recursive parameter estimation litera-

ture and it is straightforward to show (see, e.g. Young 1974) how the simple

recursive linear least squares regression equation can be modified in the light

of the additional a priori information inherent in (7) to include additional

prediction equations which allow for the update between samples of both the

parameter estima tes and the covariance matrix of th e parametric estimation

errors,

P .

Unlike the basic

I V

algorithm, th e refined I V algorithm would appear to

have certain optimal properties. In particular, the theoretical results of Pierce

(1972), together with the stochastic simulation results reported in Pa rt s and

I1

of this paper, have shown th at the pk matrix generated

by

the refined

algorithm provides a good empirical estimate of t he covariance matr ix

P of

the estimation errors, where

P

=

E

piT

and

B =

a

A It

is possible, therefore, to employ the same approach used to

modify the recursive linear regression equations to similarly modify the refined

I V

algorithms. The resulting algorithmt takes the following prediction-

correction form (see, e.g. Young

1974,

p.

214)

i )

klkvl

(ii) Pklk-

~ f i - @ ~

r Q F T

ii

a k ~ ~ ~ ~ - ~ - p ~ ~ ~ - ~ ~ ~ * [ 6 ~z ~ ~klk-l~k*] -l

correction

on receipt { ~ k * ~k~k-l-~k*)

( i ~ ) i PkIk-l- kIk-I~k*[@ bkIk-I~k*]-l

sample

~ k * ~

klk-l

Equations 8) (i) to (iv) constitute the refined I V algorithm for estimating

stochastically variable parameters in a discrete time-series model of

a

S SO

t t will be noted tha t the derivation of this algorithm is made a little more

obvious if the symmetric gain matrix form of the refined IV algorithm is utilized.

? from this algorithm is a somewhat closer approximation to P than

f ,

n the non-

,

symmetric gain

c se

(see Young and Jakeman 1979 a).

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750 P

Young and

A

Jakeman

system.

At first sight, this algorithm appears somewhat restrictive since i t

requires

a priori

knowledge of th e matrices nd

I

in the stochastic model of

th e parameter variations. Bu t pas t experience with similar algori thms e.g.

Young 1969, Norton 1975) has indicated th at t he assumption is no t as limiting

as i t might appear. First , it is possible to consider a class of simple random

walk models which represent special cases of 7) with very simple and

I

matrices and which seem to offer some considerable practical potential.

Second,

priori

information on the natur e of parameter variations can some-

times be utilized to arrive

at

simple Gauss-Markov models.

In the first case, the three random walk models that have proven most

useful in practice are as follows :

The pure random walk

RW)

The smoothed random walk SRW)

where a is a constant scalar with 0

< a <

1.0.

The

integrated random walk

IRW)

Th e R W model 9) was first used in the early nineteen sixties see, e.g. Kop p

and Orford

1963,

Lee

1964).

The

IRW

model 11) was suggested in t he

parameter estimation con text by Norton 1975) who has used it successfully

in a

number of practical applications. The SRW model 10) is of more rec ent

origin Young and .Kaldor

1978

and seems to provide .a good compromise

between models 9) and

l l ) ,

lthough it requires the specification of one addi-

tional parameter, th e smoothing constant u

=

1/ ~ , , here i- is the approximate

exponential smoothing constant in sampling intervalst.

All of the models 9) to

1

1) are non-stationary in a statistical sense and so

they allow for wide variation in the parameters. Their different characteris tics

are described fully by Norton 1975) and Young and Kaldor 1978). P u t

simply, the progression from model 9) through 10) to 11)allows for grea ter

overall variation in the estimated parameters for any specified covariance

matrix Q accompanied by greater smoothing of the short -term variations.

Jn th e case where more general and

I

matrices are considered it may

often be possible to a ss l~m e ha t, for physical reasons, the variations in th e

parameter are correlated with the variations in other

measured

variables

affecting th e system. For example, the parameters in an aerospace vehicle are

known t o be functions of variables such as dynamic pressure, Mach number,

altitude, etc. Young 1979 b) . Or again, th e numerator coefficients in a

t

Strictly, the time constant, T = -T,/log,

1

-a time units, where

T,

s the

sampling interval in time units.

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Refine d in strumen tal variable methods of recursive time -ser ies an aly sis 751

transfer function model between rainfall and runoff flow in hydrological systems

are known to be functions of soil moisture and evapo-transpiration (Young 1975,

Whitehead and Young i975).

I n such examples, t is often possible to define a in th e following form

where

T k

s a matrix (often diagonal) of t he relevant mea suredvari able s ;

a,*

is a vector of residual parameter variations which,

if

the T ransformation is

effective, will be only slowly variable and can be described, for example,. by

one of th e random walk models. I n the case of the RW (9), i.e.

ak*= akPl*

Tk -1

(13)

we see th at , upon subs titution from 13) into

12),

the variations

in

ak are given

by a Gauss-Markov m odel such as (7 ) with

@

=0

TkTk-,-I

and r

= I?,= Tk .

t is clearly possible, therefore, t o utilize the refined I V algorithm

(8)

with

0

and

Q

in the prediction eqns. (8) (i) and (ii) defined accordingly.

Such an

approach has been used previously with other recursive algorithms by Young

(1969, 1979 b).

The implementation of the algorithm defined by eqns. 8 (i) t o (iv) offers

several problems. I n particular, the equations imply th e parallel implementa-

tion of t he refined AML algorithm and its interactive use with th e refined I V

algorithm, as described by Young an d Jakeman (1979 a) . This introduces

considerable complexity and, for the present paper, we have once again

implemented only the special case where

El

is white noise, i.e. C(z-1) = D z-1) =

1.0. Here, the full refined AML is not required and the prefilters (nominally

e/Ab are defined

s 1/A^

This simpler algorithm works very well and

seems to give good results even if 5 is coloured noise. Moreover, the algori thm

in this form

has

also been modified furthe r to allow for an off-line smoothing

solution in which the recursive estimate a t any sampling ins tan t k is a condi-

tional estimate ti based on t he whole da ta se t of

N

sampIes. Th e smoothing

algori thm is an extension of Norton s work (Norton 1975) within a n I V context

and it requires both forward and backward recursive processing of t he dat a

(Young and Kaldor 1978, Kaldor 1978, Gelb 1974).

Finally, it should be remarked that the above approach to time variable

parameter estimation can be extended straightforwardly both to the multi-

variable and continuous-time situations. Such extensions are fairly obvious

and so they are not considered in detail in the present paper.

3.1. Experimental results

The IV algorithm (8) has been applied to the follpwing second order discrete

time system

where 5 is white noise chosen

to

make S = 20 ; while a, = 0.5, Vk

 

and both

b and

a

are time-variable with

0.3 ; k= 1,

..., 30

0.5 ; k=31, ... 60

0.4 ; k=61,

..., 100

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752

P oung nd A . Jake-

and

-0.35

;

k

1 ...,

60

a l k

=

-0.6 ; k=61, ... 100

Figures

4

(a),

b )

and c) show the estimation results obtained when an

IRW

model (11) is assumed, with

Q

selected as a diagonal matrix with elements

0.001, 0, and 0-001 respectively. Both t he recursive filtering an d smoothing

estimates are shown in all cases (for the constant parameter a,, the smoothed

estimate is, of course, itself a constant). I t is interesting to note th a t very

similar results to these were obtained using the SRW model with a = 0.9 nd

the diagonal elements of Q set

to

0.05, 0, and 0.05, respectively.

t is clear th at in thi s example where step changes in parameters occur, th e

smoothing algorithm is not pa,rticularly appropriate, since it attempts to

provide a smooth transition where abrupt changes are actually being en-

countered. I n practice, however, it is quite likely th at smoother changes in

parameters will often occur nd t is here th at the smoothing algorithm will have

maximum potential. Bu t it should be emphasized th at t he smoothing

algorithm used.here is

a n

off-line procedure and is computationally expensive

in comparison to

the

filtering algorithm (8). On line, fixed lag smoothing

algorithms (Gelb 1974) could be developed, however, if

c i r cu m s t an ~ s o

demanded.

I recursive es t im te

I S

0 5 0

1 O

number of

sarqdes

Figure

4.

ime variable parameter estimation for second order, stochastic system.

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Refined instrumental variable methods of recursive time-series analysis

75

Algorithms such as 8) with.parameter variation model 12) and 13) have

been used successfully for the estimation of the rapi d.1~ hanging parameters

of a simulated missile system e.g. Young 1979 b). I n this example, additional

flexibility was required to estimate the particularly rapid changes in parameters

th at occurred over the rocket boost period and this was introduced by making

the covariance matrix

Q

also a function of

k

The algorithm 8) has also been applied to many other sets of s imulated

and real data . These include time-series obtained from a large econometric

model of the Australian economy Young and Jakernan 19.79 b) real da ta

from the United States economy Young 1978) ; an d various sets of environ-

mental dat a Young 1975, Whitehead and Young 1975, Young 1978). In the

latter examples, the time-variable estimation was utilized specifically for the

identification of non-linearities in the model structure, a procedure for which it

seems singularly well suited.

4. Sta te rCconstruction

filter

design

Recently Young 1979 c) has shown how the estimates of th e st at e of

a

stochastic, discrete-time, SISO system can be obtained as a linear function

of the output s of t he adaptive prefilters used in the refined IV algorithm when

i t is applied t o t he following

ARMAX

model

In particular, i t can be shown tha t the sta te estimate jCk, is generated theoreti-

cally from a relationship of th e following form,

'jCk='k~

16)

where

z k = [ N l < l k :Nn

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764

P

oung nd

A J a k e m n

The s ta te reconstruction filter (16) can be implemented in practice by replacing

by its estimate obtained from a recursive IVAML algorithm and using t he

outputs of t he prefilters in the same algorithm to define

2 .

f th e fully recursive I V solution is utilized in the above fashion then t he

overall procedure constitutes an optimal adaptive Kalman filter, in the sense

th at th e asymptotically optimal state estimates are obtained as a by-product of

an asymptotically efficient parameter estimation scheme.

In the off-line

recursive-iterative implementation, use of th e algorithm in this fashion

represents a Kalman filter design procedure in which the asymptotic gain

Kalm an (or Wiener) filter represented by eq n. ,( l6 ) s synthesized direct ly from

th e system data?. This should be particularly useful in practice because i t

obviates the need for specifying noise statistics and solving the covariance

matrix R,iccati equation, as required in normal Kalman filter design. Note also

that this approach to state estimation can also be applied in the purely

stochastic situation where no input variables

u

are present : here the refined

AML algorithm would provide the source of parameter estimates and prefiltered

variables.

An extension of t he above approach to continuous-time systems with

deterministic inputs is fairly obvious but involves some technical problems.

The expression for the estimate of th e continuous-t ime sta te -vec tor .jC(t) is of

the same basic form as

16)

(see Fig. I , i.e.

where, theoretica.lly, Z(t) is t he continuous equivalent of Zk. But the noise

model in th e continuous-time equivalent of eqn. (15) has equal order numerato r

and denominator polynomials which introduces estimation problems (see, e g

Phillips 1959, Phadke and Wu 1974).

We

will not discuss these problems in this present paper but will merely

note th at , in this situation, a straightforward ye t clearly suboptimal approach

is to use a more arbitrary state variable filter ( l / B ) . For example, th e choice

of

D

could be based on the heuristic notion t ha t its passband should encompass

the passband of the system under study (e.g. b = d ) . In this manner, the

filter will pass frequency components of interest in the estimation bu t at te nu at e

.high frequency noise. The resultant IV algorithm is then identical to that

suggested by Young (1969), and the s tate estimate obtained from (22) with

replaced by

p

although not optimal in any minimum variance sense, will be

asymptotically unbiased and consistent, i.e. jC(t)+x(t) for 1-co where is

the true stat e vector.

The I V algorithm in this latter case can be considered as

a

stochastic

equivalent of the adaptive observer suggested by Kreisselmeier 1 977 which

is based on th e deterministic equivalent of eqn. (22) and uses

a

continuous-time

deterministic estimation algorithm with .constant gains. Bu t unlike th e

Kreisselmeier observer, this stochastic sta te reconstruction filter will function

satisfactorily, albeit non-optimally, in the presence of even high levels of noise.

t

This would seem to satisfy Kalman s requirement (1960), hat the two problems

(parameter nd state estimation) should be optimized jointly if possible .

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Refined instrumental variable

metho s

of

recursive

time series analys is

7

Final ly, with the utilization of suitable multivariable canonical forms, it is

possible to extend the arguments in this section to multivariable systems

Jakeman and Young 1979 c . Such extensions will, however, suffer from the

disadvantages of complexity associated with all multivariable black box

methods see Jakernan and Young

1979

a) and they will need

to

be considered

in

detail before their practical potential can be evaluated.

4.1 . Experimental

results

Figures 5 a) nd b ) show the results obtained when the off-line Kalman

filter design procedure is applied to the following model

which is simply the nnovations or Kalman filter description

of

model 5,

considered

in

Par t of the paper. These results were obtkined using Monte

Carlo analysis with ten random simulations and the figures show the ensemble

averages of the two sta te variable estimates compared with the true st ate

variables generated by the model. The variance associated with the ensemble

averages was quite small as shown by the standard error bounds marked on t he

la1

ru

value

l o \

st imt

0 5 100

number of samples

Figure

5.

Joint parameter state estimation : output

of

adaptive state reconstruction

Kalman) filter for second order, stochastic system.

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756

P . Young

and

A . Jakernan

plots.

t

is interesting to observe th at t he estimate of the first element of

gk s the optimally filtered output of the system ik hich corresponds with t he

optimal one step ahead prediction of the output.

5.

The general refined

IV

approach to estimation and

ts

application to some

special model forms

So far in this paper we have talked mainly in terms of specific mathematical

model forms. One attraction of th e refined IV method, however, is t ha t it

suggests a general

approach

to stochastic model estimation that has some

similarities with the alternative prediction error

(PE)

minimization approach,

but which leads to algorithms with subtle bu t important practical differences of

mechanization. In this section, we will outline this approach and show tha t

it can be considered

to

arise from a conceptual basis which we will term

generalized equation-error (GE E) minimizationt. We will also demons trate

the efficacy of the approach by showing how i t can be applied to two specific

model forms, namely the multi-input, single output (MISO) transfer function

model ; and the tanks in series representation, as used in chemical engineering

and water resources modelling work.

5 . 1 . Generalized equution-error G E E )minimizat ion

The general refined I V approach

to

time-series analysis consists of th e

following three steps.

1 ) Formula te the stochastic, dynamic model so th at t he stochastic charac-

teristics are defined in relation t o a source term consisting of a white noise

innovations process

ek

(or k n the multivariable case), and then obtain an

expression for e, in terms of all the other model variables.

2) By defining appropriate prefilters, manipulate the expression for

e,

until i t is a linear relationship (or set of relationships) in the unknown para-

meters of the basic

deterministic

part

of

the

model

(i.e. the an d polynomial

coefficients in all examples considered so far). Because of their similarity to

the equation-error relationships used in ordinary IV analysis, these linear

expressions can be considered as generalized equation-error (GEE) functions.

3 )

Ap ply the recursive or recursive-iterative I V algorithms to the estima-

tion of th e parameters in the GEE model(s) obtained in st ep (2 ) , with t he I V s

generated in the usual manner as functions of the ou tpu t of a n adap tive

auxil iary model (in th e form of the deterministic par t of the system model).

f

prefilters are required in the definition of GEE , then they will also need to be

made adaptive, if necessary by reference to a noise model parameter estimation

algorithm (e.g. th e refined

A m

lgorithm) utilized

in

parallel a nd co-ordinated

with the IV algorithm.

This decomposition of t he estimation problem into parallel but co-ordinated

system and noise model estimation, as outlined in step

3 ) ,

is central to the

concept of

GEE

minimization

;

and it contributes

t

the robustness of t he

resultant algorithms in comparison with equivalent prediction error. (PE)

minimization algorithms (Ljung

1976,

Young and Jakeman 1979 c).

f

GEE

has been used previously

t

denote any EE function defined in terms of

prefiltered variables ; here we use i t more specifically to mean an optimal

GEE

function with prefilters chosen to induce asymptotic statistical efficiency.

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Refined instrumental variable metho s of recursive time series analysis 7 7

The robustness is enhanced further by the IV mechanization, which ensures

th at the algorithm is not susceptible to contravention of theoretical assumptions

about the nature of the noise process. In particular, the supposition th at the

noise arises from a white noise source, usually via some dynamic model (e.g. a

rational transfer function, as in the

ARMA

model) is not restrictive

:

provided

that the system input signals (u, or uk)are independent of the noise, then t he

refined IV algorithms will yield estimates which are asymptotically unbiased

and consistent

even if the noise ssumplions are imrrect.

This remains true

even if t he noise is highly structured, e.g. a periodic signal or d.c. bias. On th e

other hand, if the assumptions

are

valid, then the resulting estimates will, as

we have seen in this series of papers, have the additional desirable property of

asymptotic statistical efficiency.

5 . 2 .

The M I S transfer function model

To exemplify the

GEE

minimization approach, let us consider the MIS0

transfer function model. Most time-series research in the M I S 0 case has been

directed towards the so-called

ARMAX

representation, where the transfer

functions relating each separate input to the output have common denominator

polynomials. An alternative and potentially more useful model form is the

following MISO transfer function model where them individual input-output

transfer functions are defined independently (see, e.g. Box and Jenk ins 1970),

This model can be considered as the dynamic equivalent of regression

analysis, with the regression coefficients replaced by transfer functions. I n

this sense, such model has wide potential for application.

Considering the two input case for convenience, we note from (24) th at the

white noise source e is defined

as

I t is now straightforward to show that (25) can be written

in

two GEE forms.

First,

i

a single star superscript is utilized to denote prefiltering by CIDA,,

then

Here lk is defined as

t l k =Yk- k

where gZk s the output of the auxiliary model between the second input

u

and the output, i.e. i t is th at part of the output explained by the second input

alone. Similarly,

e

can be defined in terms of

t2

where

f2k =Yk

28)

In this case,

ek

=

A,(2k** B2u2,**

P9)

where the double star superscript denotes prefiltering by CIDA,.

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758

P. Young

and

A. Jakeman

By decomposing the problem into the two expressions (26) and (29), we have

been able to define two separate GEE S which are linear

in

t2ie unknown moae?

parameters for each transfer function in turn.

Now let us define,

where aif,j

= 1 2, ,

n

; and bij, j = 0,

1,

,

n

are th e j t h coefficients of

A

and Bi respectively. It is now possible to obtain estimates 8 and 6 of a

and a, from the following refined I V algorithm

where Qi and Kik* are defined as follows

-

*

Qik*= [gi,k-l

,

..,xi,k-n

uik*, . . . I Ui,k-n*]T

I

(ik*=[fi,k-L .., ' i,k-n*r ~ i k * ,.., Ui,k-n*IT

Algorithm (30) is used twice for

=

1 2 but when i = 2 the single star super-

scripts are replaced by double sta r superscripts. The adaptive prefiltering is

then executed in the same manner as for the SISO case, with th e refined AML

algorithm providing estimates of the and polynomial coefficients. The

extension to th e general case of inputs is obvious. There is also a symmetric

gain version of (30)with gik*Tbeing replaced by gik* everywhere except within

th e braces.

5.3. The tanks-in-series model

In chemical engineering and water resources research i t is quite often useful

to describe a dynamic system by means of a serial connection of identica l

tanks with each tank described by a first order differential equa tion with

transfer function b/(s+a) ; in other words the input u t ) and the output y t )

are related by

where m is the number of tanks in series and t (t ) s a noise term. Using GEE

approsch,it is possible to obtain refined I V estimates of a and b for different

and so identify and estimate the tanks-in-series model. We will not discuss

this in detail here since i t is done elsewhere (Jakeman and Young 1979 b), b u t

an example of it s use will be described

in

the next section.

5.4. Experimental results

Jakeman et a2 (1979) have evaluated the M I S 0 transfer function model

estimation procedure using both simulated and real da ta.

Figure 6 compares

the deterministic out put of a

MIS0

air pollution model obtained in thi s manne r

with the measured data. Here the d i t a are in the form of atmospheric ozone

measurements a t a downstream location in the San Joaquin Valley of Cali-

fornia. These are modelled in terms of two inpu t ozone measurements a t

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Refined instrumental variable methods of recursive time-series analysis

759

upstream in relation to th e prevailing wind) locations.

This analysis proved

particularly useful for interpreting across gaps in downstream da ta .

Table shows the results obtained when th e tanks-in-series estimation

procedure was applied t 100 samples of simulated dat a obtained f rom a second

order system

m

=

2). The results again include averages an d sta nda rd errors

over ten experiments.

mo el output i t

5

100

num er of mrrQles-

Figure 6. Results from model of ozone concentration in San Joaquin Valley,

California.

Signal to noise ratio,

Parameter True

value 10 30

Table 2 T,=0.2 sec P,/15-7).

6. ultivariable systems

with

limited dimension observation space

Another example of t he many possibilities which are opened up by th e

refined

V

approach

to

time-series analysis is the case where the number

of

observed variables in a multivariable system is less than th e number of out pu t

variables in the model. In theory, the symmetric matr ix gain refined

V

algorithm for multivariable systems described by Ja kem an an d Young 1979 a)

can be modified to allow for such a situation. This will bnly be possible, .of

course, provided the complete model is identifiable from the limited observa-

tions. The conditions for identifiability in these situations are not t he subject

of the present paper but, if we assume th at the model is identifiable i.e. unique

estimates of

all

the model parameters can be obtained -from the available

observat ions) then the modifications to t he symmetric gain refined

V

algorithm

are fairly straightforward.

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Refined instrumental variable methods of recursive time-series analysis 761

7. Stochastic approximation algorithms

t is well known that the recursive least squares and related algorithms,

such as those discussed in th is series of papers, can be interpre ted as special

examples of matr ix-gain , multi-dimensional, stochastic approximation (SA)

procedures (Tsypkin 1971, Young 1976). t is clearly possible, therefore, to

modify any of t he refined IV algorithms to form simpler, scalar gain al ter-

natives. While such SA procedures are computationally efficient, they will

not usually possess the rapid convergence characteristics and low sample

statistical efficiency of their matrix gain equivalents. They may prove

advantageous, however, where da ta are plentiful bu t computational load must

be kept t o a minimum.

I n the basic SA algorithms, the matrix gain

p is replaced by a scalar gain

which obeys the conditions of Dvoretzky (see, e.g. Tsypkin 1971). I n the

discrete-time case, the best known gain sequence of thi s type is y, = y/k when y

is a constant scalar

:

in other words, the gain sequence is made a monotonically

decreasing function of the recursive step number , k. I n the continuous-time

case the best known example is simply the continuous-time equivalent of y,.

Such SA algorithms can also be modified (normally heuristically) to allow

for variation in the estimated parameters

:

this is achieved by restricting the

monotonic decrease in gain in some manner, usually by making y, or y(t)

approach a constant yo exponentially as k or approaches infinity. This

modification is based on a partial analogjr with the behaviour of th e f , matrix

in algorithm

(8)

when a W model (9) for parameter variations is used to define

and I (Young 1979 d) .

The simple SA versions of the refined IV algorithms cannot be recom-

mended for general application since their rates of convergence can be intoler-

ably low (see, e.g. Ho and Blaydon 1966). Bu t i t is possible to consider

somewhat more complicated algorithms which represent a compromise between

the simplicity of basic SA and the complexity of the fully recursive matrix-gain

algorithms. simple example would be the following modification t o the

refined IV algorithm given by eqn. (4) (i) in Pa rt

I

of the paper (Young and

Jakeman 1979 a, p. 4)

ak

YkfjkD ik*{zk*T -yk*)

36) .

Here f , ~s a 2n 1, diagonal matrix with elements ( k - i*) -2 , i = 1 2 n,

and u , - ~ * ) - ~ , 0 1,

.

.

,

n ; while y, is a SA sequence, say y/k.

In other

words, the gain matrix pk s replaced by a purely diagonal approximation

y,P,D, so tha t the computational burden is proportional to n rather than n2

for the full refined IV algorithm.

Algorithms such as (36) seem to work reasonably well (see, e.g. Kuma r and

Moore 1979). As might be expected, their performance seems to

fall some-

where between th at of th e full algorithm and the scalar gain equivalent. I n

general, however, the simpler algorithms should only be used when this is

. necessitated by computational limitations, as for example in on-line applica-

tions using special low storage capacity microprocessors.

8.

Self adaptive control

Perhaps the prime motivation for th e development of recursive estimation

algorithms during the early nineteen sixties was their potential use in

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76

P

Young

and A Jakeman

self-adaptive control.

Of late, this, early stimulus has been revived with th e

development of the self-tuning regulator (STR) based on recursive least

squares (R LS) estimation (e.g. Astrom and Wittenmark 1973, Clarke an d

Gawthrop 1975):. In the STR th e effect of t he noise induced asymptotic bias

on the RLS parametric estimates is cleverly neutralized by embedding the

algorithm within a closed adaptive loop which automatically adjusts the

estimates and th e control law t o yield either minimum variance regulation or

some other objective, such as closed loop pole assignment (Wellstead

et a2

1979).

The concept of the STR can be contrasted with the earlier concept of self

adaptive control by identification and synthesis (SAIS),where the objective

is to explicitly obtain unbiased parameter estimates and then to separately

synthesize the control law using these estimates (e.g. Kalman 1958, Young

1965 b, Young 1979 b) .

While the STR seems to possess good practical potential, there a re certa in

situations where the alternative of SAIS will have some advantages. Fo r

example, the stability of the adapt ive loop in the STR is not easy to ensure

a

pr or

because of the close integration between the recursive algorithm and t he

control law, and the highly non-linear nature of the resulting closed loop system.

On the other han d, the separation of estimation .and synthesis in t he S AI S

system means t ha t the question of convergence and stability is largely one of

ensuring th e identifiability of t he system under closed loop control. Thi s will

always be possible provided an external command input is present which is both

sufficiently exciting and statistically independent of the noise in the closed

loop. The requirement for such an inp ut can be problematical, however, in th e

pure regulatory situation, where the STR clearly comes into its own.

In cases where the SAIS procedure seems advantageous, the refined

V

algorith r provides the best, currently available recursive estimation str ategy

:

it is robust, can be applied in continuous or discrete-time and its results can

be used for either deterministic or stochastic control system design. Th e

efficiency of such an SAIS strategy is demonstrated in the self adapti ve

autostabilization system described by Young (1979 b) : here the recursive

estimation is used to synthesize a deterministically designed control system

based on closed loop pole assignment using sta te variable feedback. This

system achieved tight control of the simulated missile over the

whoIe of the

mission, which included a difficult boost phase where parameters changed

rapidly by factors ,of up to 30 in sec.

I n the case of adaptive, stochastic control by SAIS, the present paper has

shown th at the refined V approach provides an added bonus : the single IV

AML

algorithm yields not only the estimates of the model parameters b ut also

estimates of the sta te variables, which can then be used in st ate variable feed-

back control. And in the discrete-time, linear case, such an SAIS system could

be considered optima lly adaptive, since the st ate variable estimates would

then, as we have seen, be optimal in a Kalman sense.

9

Conclusions

This is th e third of three papers which have described and comprehensively

evaluated the refined V approach to time-series analysis.

In the present

paper, we have seen how this approach can be extended in various important

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Refined instrumental variable methods of recursive t ime-se ries an aly sis 763

directions a nd can also provide a conceptual basis for t he synthesis of refined

c

I V

algorithms for a wide class of s tochas tic dyn amic systems.

This conceptual base, which we have termed generalized equation-error

GE E) minimization, has some similarities with th e alte rnat ive prediction error

PE ) minimization concept but tends to yield algorithms which a re more ro bust

both in a computational sense and to mis-specification of t he noise charac-

teristics.

We

feel that this robustness, which arises primarily because of the

errors-in-variables formulation Johnst on 1963) and conse quent IV mechaniza-

tion, is an impo rta nt featur e of the refined I V algorithms and more detailed

discussion on this topic will appear in a forthcoming paper Young an d Jak em an

1979 c).

This paper was completed while the authors were visitors in the Control

an d Management Systems Division of the Engineering Dep art ment , University

of Cambridge.

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