Process Parameter Estimation and Self Adaptive Control

PROCESS PARAMETER ESTDJATION AND SELF AlJAPTIVE CON1'ROL by Peter C. Young Research StudentLoughborough College of Advancea Technology Loughborough, Leicestershire. England ABSTRACT The paper describes a method of automatic process parameter estimation which has been mechanised with the aid of hybrid (analogue-digital) equipment. The technique is characterised by it's simplicity, and differs from earlier scheme s of this type in not requiring direct measurement of the input and output time derivatives of the process. The performance of the system is discussed, with particular reference to the effect of uncertainty on the sampled data caused by spurious noise contamination. Finally, a simple identification-adaptive control system for a nonstationary process is described, which utilises the hybrid parameter estimation techniques. INTRODUCTION One type of automatic process identification scheme which has received a great deal of attention in recent years is the 'Process Parameter Estimation' system. The basis of 3Uch systems is an a priori assumption on the form of the mathematical relationships which provide a reasonable description of the process dynamic performance characteristics (Ref. 1). The paper aescribes a technique for estimating the parameters of a liifferential equation or Laplace transform transfer function description of the process, using hybrid (analogue-digital) equipment. These techniques are important not only in their straight forward role in the experimental evaluation of dynamic processes (The Analysis of J)ynamic Experiments), but also because of their possible application to the problem of self adaptive control. This latter application is discussed and a simple identification-adaptive control system for a nonstationary process is described, which utilises the hybrid parameter estimation techniques. 1.1 THE TECHNIQUES OF PARAMETER ESTIMATION

(B) "Using an Explicit Mathematical Relation", in which the process is characterised by certain numerical values, which are chosen in order to satis~ a pre-determined and applicable mathematical relationship. This is possibly, a rather larger category than class (A), because the term 'numerical values', which appears in the definition, is capable of quite wide interpretation. The parameter estimation schemes dealt with in this paper come within this class (B) category. Fundamentally, they are concerned with the description of a physical process in terms of a general mathematical expression of the form, r:=k

~.>7r =

r:=o

•••••••••••• (1)

f

f(t)

can be considered an arbitrary input or forcing function to the process y = Y (t) are time dependent variables r r occurring in the process a = a (t) are slowly variable or fixed r r coefficients or parameters

where, f

Equation

(1)

yT A =

can be written in matrix form, •••••••••••• (2)

f

•••••••••••• (3)

or yT A - f where, yT AT

AT

.

let A [aOc ' ale' a2c •••• ~c] be the estimated value of the matrix AT. In general, it is possible to define an error signal E, where,

E yTA - f ••••••••••••• (4) The principal object of process parameter estimation is to obtain a realistic mathematical or, description of the process from a knowledge of E yT(A_A) (from Eqn.(2) •••••••••••• (5) it's normal operating characteri stics, and making use of ;my a priori information which is availah1e Here, E = E(t), the error in satisfaction of the on the nature of the process. Under this definiprocess equation (or more simply the 'Sati~tion tion it is possible to divide the various Error') is seen to be a function of the error approaches to the problem into two broad classes between the actual and estimated values ofA. (Ref.l). (A) ''Using a Physical Model", in which the It is possible to arrive at an estimate of characteristics of a physical model of some which minimises E or some function of E, in a form (usually an electrioal network) are maninumber of ways. pulated in 3Uch a way that they converge (A) By systematic adjustment of the parameters towards the characteristics of the process itself, in some pre-defined sense. *now with Dept. of Eng., University of Cambridge. "Superior numbers refer to similarly-numbered references at the end of this paper" 118

Src using the methods of steapest desoent, it is possible to oontinuously minimise oertain funotions of the satisfaction error by means of either ana.logue or iterative digital techniques. This approaoh has been investi[ated and disoussed in (Ref. 2.)

vl =

[[(Yo)h,",

and fL

(2) By samplin~ either the variables Yr and f or oertain funotions of these variables at S instants of time (where S ~ k+l) and so generatine k+l linear simultaneous algebraic equations in k+l unknowns, ~c (r=o -r=k). When S>k+l the samples oan be used to perform a 'least squares'eotimation. (3) Alternatively, it can be shown that by special treatment of the variables Yr and f it is possible to p'enerate the re~lired aleebraic equations from samples taken at only one instant of time. These latter two appraohes to the problem have been dealt with briefly in(Refs.3,4 and 5.) These investif,ations have been concerned with a special case in which the process can be described by the equation,

:L>

.Lx -

n=o nC dt l1

Lb

d f

m=o mC dtm

where, without loss of

1'1

•••..•. ,. •• •• (7)

~enerality

, ••

,[(f11 )]

LJ

[(fol] L dM = ~

M

n=o

ano l(Yn)] L J

L bmo

m=o

.[(fm)]

J

L

••••• (ll)

where j = 1,2, •••• S and boo = 1.0 Now, let the funotion

b oc = 1.0

ioan[(~)JL iobm[(::~)JL

r be defined, where

S

?= Emj 2

r

•.......... . (12)

J=l

and S

~1,:+N+l.

The necessary oondition for a minimum of the function r is the equality to zero of all i t.8 first orner partial derivatives with respeot to Snc (n=o+N) and bmo (m=l-M)o Thus,

(8)

£.[

aa nc

where the brackets [( )] indicate that the funotion enclosed has been physically filtered by L low pass filters eaoh havinB the transfer funotion o/s+o (aee Fig.14). The modified satisfaotion error, Em oan now be defined, where

2tEm .. a~j aa nc j=l

J

0 (n=o-N) ...... (13)

ar abmo These M+N+I linear simultaneous algebraic equations can be solved by normal methods to supply the required 'least squares' estimates of the M+N+l parameters (aoo ' a lo "'" ~o' blo'

tano[(~)] L n:o dt n

mt tbmo[(d )] ... (9) m=o dtID L This is a particnlar oase of equation (4) of the form,

b20'·'"

~o)·

In deoiding on the quantity of data to be used for eaoh separate oomputation, it is neoessary to strike a oompromise between two rather oon.tradiotory requirements. On the one hand thera is the need to aooumulate and prooess enough information to provide a reasonable statistioal estimate of the unknown parameters. On the other hand there is the need to restriot sampling time so that any non-stationary effects whioh the

•••••••..•. . (10) in whioh, AT

f)4

L

In effeotively removing the need for direot sampline; of the hieher order input and output darivatives, which oan be diffioult or even impossible in praotice and oan intensify the problem of spurious noise oontamination, this 'N-ethod of Multiple Filters' beoomes an essential prerequisite to the practioal implementation of the type of parameter estimation soheme desoribed in this paper.

L

The analysis shown in Ap~endix 1 illustrates how, by performin~ a TAplace transformation on equation (6) and by then operating on the resultr ant r~ationship by a modnlatinc function of the form ~s+ct(where s is the TAplace operator and c is a constant), it is possible to express the relationshi!J between an and bm in the form,

Em

j

[(fl)]

Appendix 1 shows that the variables [(Yn)]r. (where n=l+N) and [(fm)] L (Where m=l-M) maYlle generated ~ the simple algebraio summation of the variables appearing at various stages in the chain of low pass filters shown in Fie.14 (i.e. suoh signals as [(ff)J 1'.... ' [(fo )] Land [(Yo)] 1'•••• , [(Yo) L j L~M+N+l).

m

11

~

here, YN

so that an error function, Es defined by n

({f)] L dN

'

If the required samples are taken at S discrete instants of time, it is possible to generate S equations of the form,

•••••••••••• (6)

N

[(YN)] L

[a oo , •••• , ~o,-blo' •••• ,-byc]

119

f

Y

PROCESS

1

1

high and

high and

low pass filters

low pass fi Iters

~ track store

-r--

-

~

circuits

-

-1----

INTER-FACE [a-d, d-a conversion eqpt.]

------t-----DIG ITAL

COtv1PUTATION

estimation of process parameters [see fig 3]

fig.l.

C

mechanisation of hybrid P.D.C. using general purpose hyb rid equipment

-----r---...

I in ea r second

Yo

or d e r 1---"';";;"'--,-

process

fig. 2.

multiple filters applied to a second order process [C.I, high pass filters not shown] 120

process may haTe do not result in the use of obsolete data and consequent error in the computation of the parameters.

feasibilit)· of the identification scheme, &\d yet provides a reasonably realistic description of a number of physical processes.

1.2 Tllli

Secondly, practicel limitations inherent in the eeneral purpo3e hybrid equipment used in the simulation prevented more sophisticated mechanisation.

~:ECHANISATION

ESTI~~TION

OF THE PARAMETER

SCHEME

An auto~atic Parameter Determination Computer (P.D.C.) utilising the techniques of parameter estimation discussed in the previous section can be mechanised with the aid of analolue and ~tal elements. Fig.l illustrates how such a 'HYbrid P.D.C.' has been simulated on the general purpose hybrid computational facility of Loughborough College of Technology (Refs.4 and 5).

Ideally, the sampling frequency and sample size per conputation should be capable of wide variation in order that well conditioned simultaneous equations may be generated and well defined parameter estimates obtained. At the time the investigations rlescr"i bed in this paper Vlere carried out, it was only possible to sample a set of four variables every two seconds. In additio~ the number of samples taken per computation was maintained at a fixed value to ease the problem of digital progra~ning. These restrictions imposed a practical Ij~it on the type of process which could be investigated; only relatively slow variation in the prooess parameters could be tolerated, undamped natu)'al periods needed to be less than 15 seconds and noise levels had to be reasonably loVl in order that the variance in the estimates was not to great.

It will be noticed that, in addition to the low pass filters which are necessary for the Method of Multiple Filters, high pass filters are incorporated in the system. By 'd.c. blooking' the signals in this manner, the detrimental effect of incorrect reference levels, or bias errors, is avoided and the need for soaling oircuitry in the hybrid system is red.uoed (Ref .4) and section 1.3). Consider the application of this system to a simple seoond order prooess which can be reasonably described by the differential equation,

aaY +

dy ~-

dt

d 2y + ~2

f

dt

Similarly, in a truly practicalsystem, it should be possible to modify the time constants of the hieh and 10Vl pass filters, in order that the P.D.C. can be 'matched' to the nrocess under investigation (see section 1.3). I~ the simple case de~cribed here, they were maintained at a fixed value of one second in order to further simplify the mechanisation and the digital programming.

•••••••••.•• (l4.)

.......... •• (15) Choosing 1=3 and c=l ra~sec the al~ebraic relationships derived in Appendix 1 can be used to obtain the following equations for the filtered state variables, [(Yz )] = [(yci]

3

3

- 2 [(yJ]

2

+ [(yciJ

In addition to it's use in the identification of second order analogue processes the equipment described here has been applied to the investigation of' a d.c. position oontrol servo system. The servo had a ntunber of mildly non-linear characteristics and presented a reasonably realistic problem in process identification. Fig.4 shows a comparison of the actual frequency response of' the servo and that calculated from the estimated parameters. The fact that this estimation was achieved with a sample size maintained at 6 sets per computation (8=6 equation (12)h illustrates the potentialities of the system even in it's present elementary form.

1

•••.•.• (16)

Now, if S is chosen to be 6 the signals (Yo)] l' [(Yo)] 2' [(Yo)] 3 and f(f o )] 3 (Fig.2) can be fed to the track store circuits and se.mpled at six discrete instants of time. They are then passed via the conversion equipment to the ~bU computer. The digital computer is programmed to perform the algebraic summations detailed above and then insert the results into equation (13), (Fi~.3). These three linear algebraic simultaneous equations in three unknowns (M+N+l=3) are then so17ed by normal methods to provide a least equares esti.mate of the prooess parameters. These estimates can then be either printed out, or processed and sent back to the analogue side of the equipment in the form of adaptive controller parameters (see section 2.1).

1.3 THE US.c: OF l!'ILTi!:RS IN Tllli PAMM}~ER ~_~lot; .§CHErr.E The statistical least squares approach to parameter estimation of the Hybrid P.D.C. system is further aided ~ the low pass filters inherent in the Method of Multiple Filters. In effeot, the filters introduoe a degree of memory into the system by weighting each sample with past data (Ref.6). The fact that different low pass filters are able to weight the samples in different ways, also sur,gests that a more sophisticated sampling techni~ue may be evolved and so form the basis of future equipment (preliminary work indicates that this should be possible, see Appendix 1 and section 1.5)

Up to the present time, a large amount of the work carried out on the Hybrid P.D.C. system has been ooncerned with second order processes of this type. This restriction was initially imposed for two prinoipal reasons. Firstly, the second order mathematical model provides a simple example to test the

Both the hieh and low pass filter time 121

filtered


obtained

from

set

process

A-D

adaptive controller

D-A CONVERTER read out estimated pararreters

read in samples for one equation

change to fixed point solve equations form coeffici from data

fig. 3.

compute leasl squares cefficients

schematic block dlqgram of hybrid p.d.c. digital programme

122

+10

- actual freq resp. - - estimated +5

0 -50

U1

-100

0

:0

~

er.

l>l 'J

l»

-150

c

·0

-200

-5

100 frequency rad/ sec.

100

1ig4 comparison of actual and estimated frequency response of the d.e. servo

v f

y

+

- - - - - + I process J=====~)C

z

fig.5 noise contamination of signals

123

f ilte rs

process

0

Cl-

phase

0-

U1

.c

-~~

Signal amp. .msv.of nOise

00

M7

10

nominal parameter values

3·57

5

2 1«-0'74.10' ..·0-56.10·< <1'00'77.10. "'00'13.10·' "'0'27.10'

---..

1·0

LJ

nn~

I

Ila >< '1·0

U~ U l

~"·0'76xI0· "·0'10.10·' ""0'19.10·' ..·0-20.10·' ""o-36xI0~

:::J

0

>

OS

'"

~ o E

~ ~·l

-45

Eo ",0-

n

n

L

"0", "'~

0

_·U

L~

lJ

·95

Vl

'"

~

c"::LJ1

~n-

04

·lftrbmLlf1r ;

Ha,;.0463!

Vl Vl

'"u

20-

02

"'0·14.10·' ""0'11,10' ""013.10·' ",0'24.10·' "'. 032.10'

·15

n

~

0'1

..L1Jl p~~

U

L

S

.flr[

~a,~0'1l41

0 0

10 20 computation Intervals

I _. ,.-- mean value I standard deviation

30

f40

!:JO

nOise. on OUlPUt. signal. wnl. e nOise. Wltn gauss lan amplltud e d IS lrl bu tl on ,obtained from Servomex Ltd. nOise aererator

flg.6. typical example ShOWing the effect of nOise contamination of the process olp Signal on parameter estimation accuracy s2(ond order process activated by±25volt amplitude step stgnals]

124

I

S

oOlUltants oan be cOlUlidered as design parameters in aystems of this type:

i.e.

~jYL.jYLTA- jYL.lL

j=l This may be written,

The time oonstant of the low pass filtera determines how muoh importance is attached to older data (Ref.6). If the time oonstant is ohosen to be large then the samples are quite heavily' 'weighted' with past data. Consequently, even if the process comes to rest the parameter estimation is not affeoted for some time because the P.D.C. 'remembers' the effect of previous activation. Alternatively, if the process is nonstationary, and the parameters are changing fairly rapidly, then the time constant should be chosen much smaller in order that obsolete data is not utilised. The time constant of the high pass filters controls the attenuation of the low frequency components of the signals. If possible it should be selected to 'block' ~ effects such as bias errors or slow drifts about the equilibrium point whilst at the same time passing a1\1 useful low frequency components of the signals.

= 0

S

....••.•••••. (19) ~jYL. jYL'rt,A-A) = 0 J=l Now if S is large and a common factor l/s is introduced this equation can be expressed in the form,

=

0

•...•......•• (20)

where the angular braokets denote the expectation operator. In practice the process signals are filtered by a high pass filter in addition to the low pass filters inherent in the Method of Multiple FilterL In effect, these high pass (or d.c. blooking) filters extract the mean value of each of the elements of Y (as defined by (10». Thus L T
In addition to these points, the low and high pass filters have considerable effect on the transient response characteristics of the sampled signals, a point whioh must be considered when deciding what sampling frequency is desirable.

L

L

>

matrix of YL , cov. [Y , Y ]. L t :. cov. [Y L, YL] (A-A) = 0

••••••••••••• (21)

One factor of great importance to the practical implementation of ~ parameter estimation scheme is it's sensitivity to uncertainty on the sampled signals caused by suoh effects as measurement noise. Consider then the matrix block diagram Fig.5, in which the vector, Y=Y(t), is contaminated by additive noise V=V(t), where

The ohoice of filter oharaoteristics for &l\Y partioular practioal applioation of the Hybrid P.D.C. has to be guided by oonsideration

of all the above points. With the present fixed oomponent aystem, the designer must take into account the possible range of parameter variation, the maximum rate of variation which might be expeoted, and the types of input exitation which are most likely to occur. He should then choose those filter characteristics which provide the best compromise between ~ conflicting requirements, and so produce the most satisfactory oveZ'all system performance.

T

V

= [v o ' v l ' •••• , vM+N+ l ]

The observed vector is now ZL and, ZL = (Y+V)L = Yt+VL

(by the principle

?f.

superpos:lt:lon) The elements ofV are dependent upon the nature of the noise andL on the characteristios of the filters (high and low pass) used in the meohanisation.

1.4 THE EFFECT OF NOISE CONTAMINATION

OF THE PROCESS SIGNALS Equation (11) oan be written in the matrix form,

Now, in this noisy case, Emj will be defined as,

Emj

•....•....•. (17)

=

[ j

Emj jz:.A - /L (j=l, 2, ••• , 5) and the equation equivalent to (21) becomes, oov. [ZL'ZL]A

[(Yo» L'···' j [(YN)] L' j [(f1 )] L'· •• ••• , j [

(fW)] L ] •••• (18)

COV.[ZL9VL] A

=

0

..... (22)

This equation can be expanded by use of the relationship, ZL- YL+VL' However, in the case where the input to the process contains no components of the noise V , cOV.[VL,yL ]

and j = 1, 2, ••••• S

~O

Thus equation (22) bec0mes, oonditions for a minimum of rare,

ar ar

=

{cov.[yL,YL1+ cov.[ VL'VL ] ~A - cov. [YL , YL]A so that,

0

0

-1

A-A = - cov. [yL,y ] L 125

.cov.[VL,V ] A L

(23)

'prtdicted paramd~r variotion

, ,

_______-.•..,15 2

~

" ..

"

\

"

t --+

~t. f-' t\:)

m

h,

c

correct ed ~ prediction

p

......

'4t~ \'''et•• ~.

~'"

'i. ""'\

,

t--.

t-

digital to analogue? converter

dlglta I computer

ke - - actual p'~rameter varlO lion

nb.

t.~-t.,

c

A

th~

S :: co~ulation

tolal computation tIme

d~lay

-

p.d.c. output

...... predicted variation

1ig .7

first

order

prediction

circuit

equipment would be based on nethod 3 mentioned in Appendix 1 and Ref.3. By utilising at least M+N+l different sets of low pass filters (Fig.15) it appears that all the information required to determine the M+N+l parameters could be extracted by takinB samples at only one instant of time. In any practioal system of this type it would probably be best to reach a compromise, incorporating only the minimum M+N+l sets of filters, but at the same time sampling at more than one instant of time in order that sufficient data is available for use in the stati stica.l least squares proced\lr~

The effect of the noise or 'uncertainty' on the observed signals is apparent if this equation is campared with equation (21). In the long term the parametric estimate error, A-A, is biased hy a factor dependent upon the statistical properties of the noise and noise free process signals. Fig.6 demonstrates the effect of various levels of additive white noise on the output signal y(t) of a simple second order Erocess. As might be expected from the theoretical consideration just discussed quite high levels of uncertainty can be tolerated. In the case of white noise, the low pass filters used in the Method of Multiple Filters can help to reduce the magnitude of the elements of the covariance matrix, cov.[vL,v ] (equation (23»,as compared L with the elements of the straight noise covariance matrix,cov.[V,V]. 2onsequently, the parametric estimate error, A-A , is correspondingly less in aI\Y system usinB the Method of Multiple Filters, as compared with a similar system based on straight observation of the state variables.

In the present system all of the S samples required for each computation are taken lit one sampling interval. A possible im)rovement could be ohtained by takint; onJ,y Sl samples, where S 1
The limitations of the present mechanisation prevented larBe sets of observation being taken. As a result, it has only been possible to compute short teno statistical properties, and equation (23) has not been verified experimentally. However, V.S. Levadi (Ref.7) has derived a similar expression for a continuous analogue system usin~ direct observation of the state variables, and has shown that excellent correlation exists between theory and experiment.

~

1.5 THE PROB:r..Md OF 'l'HE TO'l'AL COMPUTATION TDili DELAY In common with many other parameter estimation Bchemes, the Hybrid P.D.C. is limited to the investigation of processes having fixed or slowly variable parameters. Although in some measure this is the result of restrictions imposed by the mathematical assumptions which are the basis of the techniques, it arises principally from more practical conaiderations. The computer requires a finite time to sample data and to perform its computations, and as a result parameter estimates can only be supplied at discrete intervals of time. During the sampling and computation period the variation of the parameters must be within hounds otherwise outdated estimates are obtained which do not provide a satisfactory description of the aotual process characteristics. It is possible that the total computation time delay (i.e. including sampling, A-D and ~A transfer times) could be reduced ~J the design of a special purpose Qybrid computer. Thiscould also reduce pqysical size and make the equipment a practical proposition for use in on-line adaptive control applications and as a tool in the analysis of dynamio experiments. In addition, an initial feasibility study has indicated that a further stage of the hybrid equipment may be developed. The operation of this

1.6 FUTURE DEVELOPMENT OF THE PA.tW>iE'fER ESTU1ATION TSCHNIQlm3 Uncertainty caused by additive noise contamin&tion of the sampled signals can be a very real problem with any system relyine on reeasured data. The 'bias' effect of additive noise has been

127

jProcess l.KlO:!r IrNestl9Jtd y 1

If

1-0

=

~t·a,s

+

1

d

, ~ .~/ ~

t"I

1. actual

variation 0-

0-5

;!.p.dc. output 3 predicted /vartatlon

~~

lA

'E"

0

8-

=

"t :;.

~ ,,~

.... ti

Iprocess activated by repe
20 .." 3 sccs.

~

4.cor.rec~ predICtion i

0 0

80

40

120

1

160

200

time (5IZCS) f--

i fig.B.

I~

hybrid pdc IT with flrs;t order analogu e prediction elements Investigating a non stationary second order process;

MODEL OR

~

SELF ADAPTIVE CONTROLLER synthesIs

---.L-

PRE-FILTER

of

PROCESS

f l--o~

Y output

KG(s)

s KG(s)

PARAMETER

P::TERMNtIT1Ot1 COMPUTER P.DC estimcted

fig. 9.

schematic diagram

showing

parameters

1

prtnClpal components of the SESAC system

128

I

discussed in section 1.4. One possible approach to the problem is menti3ned in Appendix 2. It entails the LlSe of an auxiliary m:>del constructed from the parameter estimates. The technique would be a straight development of work carried out on a similar analogue parameter estimation scheme, by Victor S. Levadi (Ref.7). An ~~tension of the hybrid techniques to higher order and non-linear processes needs to be investigated. Although no extensive difficulties are enVisaged as far as higher order linear processes are concerned (except for the necessary increase in sampling and computation time), the problem of identifying certain types of nonlinear process should pose a number of difficult questions. Recent investigations have been concerned with the estimation of the second order equivalent parameters of higher order processes. Results suggest that the estimated parameters provide good agreement in the time domain (a second order model constructed with the aid of the estimated parameters and subjected to the same input as the higher order process under i~vestigati0~. produces an output response which agrees very well with the output response of the process itself).

2.1 APPLICATION OJ.;' AUTOMATIC PARAMETER ESTIMATION TO THE ffiOBLb.'M OF SELF ADAPTIVE CONTROL In 1958 R.E. Kalman (Ref.9) discussed a generalised concert of control in which he compared the role of the self adaptive controller with that of the control systems designer. In accordance with this concept, it is SUF,fested that the controller should perform three major functions. These are : 1. 5stimation of the dynamic characteristics of the pr9cess during normal operation. 2. Determination of the characteristics of the control element of reference t~ this inrorm-

Theoretically the self adaptive controller in the S.E.S.A.• C. system is intended to conti.nuously synthesise a function of the inverse transfer function of the process to be controlled. However, in practical terms the st~lcture of the controller will often be limited to a second order-form. As a result, it is more realistic to regard it as a variable parameter rroportional-integral-deriv&ive element (three term controller) which shapes the input demand to the process in order to produce a desired output response. The system achieves adaptive control by using the information obtained from the P.D.C. in a certain organised mannaI'. However, this same information might well be used in the design of systems having completely di~erent adaption al£.ori1:hms to that of S.F..S.A.C. So far, no attempt has been made to iDIprove, or modify in any way the basic system. However, it is intended to investigate the S.E.S.,p..C. prinoiple in more detail subsequent to the realisation of a completely acceptable method of parameter estiDIation. Despite it's 1lllitations, the present S.E.S.A.~ system does provide an excelJent framework in which to assess the ori-line operation of the Parameter Determination Computer. Fig.lO illustrates the control of a non-stationary second order process. In this case step chanees in the process parameters ere deteoted quiokly and accurately by the P.D.C. and it is possible to maintain tight control over the transient response characteristics. This can be compared with the more realistic case shown in Fig.ll. Here the system has been appliea to the d.c. servo system mentioned earlier (section 1.2). Although the control action is not as good as in the previous example it is quite satisfactory considering the elementary nature of the present P.D.C. The effect of the control system on the step response of the d.c. servo is shown in Fig.12. Finally, the improvement in control obtained by incorporatinr analogue prediction circuits into the P.D.C. (Appendix 2) is demonstrated in Fig.13. Here the system is controlline an analogue second order process having a ramp variation in one of it's parameters. The reduction in response error, ~, obtained by the use of a single prediction element on the variable parameter channel is apparent from this diagram. 2.2. DPROVING THE PERFORl,:ANCE OF THE S.E.S.A.C. SYSThM The S.B.S.A.C. approach to control is attractive because of i~s simplicity. However, it is just this simplici.ty which provides some of the major limitations and di.sadvantages of the system.

lltj.ort.

3. Construction or Ir.odificll.tion of the control ele@mt in accol'danne with the details computed in 2. above. A sim~le self adaptive control system, designed in accordance with these basic principles, and using the teohniq\les of rarameter estimation d.iscussed in the previous sections, has been described in(Ref.4.l The fundamental idea behind this Hybrid Satisfaction 3rror Self Adal't:ive Control, S.B.S.A.C., syster'l is illustrated in Fig.9. In a more adve.nced form this system might be usea in the self adaptive autostabilisation of aircraft.

The basic control principle must be further developed so that it may be applied in any practical situation. At the moment, it would appear to be limited to low order linear processes. Further investigation must be aimed at finding out whether the system can be used with non-linear or higher order processes. As mentioned earlier, initial attempts at applying the present system

129

~Il,i~i~i~,:~i~I;~~~~~~~~~f0i~~;~"~~":'O;;',:\=,L~,~~,i,~~~~;i'~:;;:::=~';O","~;-\~~t;I1~~;!{_~~i;~:;3l~~~i~~~~~~~Ji~t~ PAPER SPEED"! rill'" Ine

..... CV

o

nnlc:-,;,'s

"',,,; ",. ';7,,<':"':- - '~-.,""":-:- ~;-- - ,,_c:' ;-

-)

- ••• , - _. - - -

flg.IO. hybrid SESAC

--

-[(yjJa-- _..--

system applied to a

DESIRED RESPONSE CHARACTERISTICS

2nd order analogue process

~

\-In!)- lOAAbJs((,)0'<>7

... ,c.ooc,"''lCO''

..... .....

cv

tRUSH

DES,Qeo RfSPONSE .

f I g.11.

hybrid SESAC system

applied

to a

CH~RJo.C.TE~ISiICS ;

d.c. position contro l

S ervo-m

Wno • 3-0 RAO/CIG..

echan Ism

;0& 0-7

l!'\o

uncontrolled response --- desired response ---- controlled response

1·5

tlI

~:;

"0

,1" ...... -

"0

::l ~

;1

--:..-------------...:::::

/I"

/1

0.. E

0

0'5

t'

/

,f

I'

0

0

2·0

10

time (secs)

fig.12. hybrid SE.5AC. system applied to a dc position control servo. : step response for one particular servo condition

132

to a d.c. position control servo have provided encouraging results, and it is possible that even in it's basic form the system may have wider application than at first envisaged. As a next step in the development of this approach to adaptive oontrol, the information obtained from the P.D.C. could be used to further improve the overall performance of the system. A knowledge of the process parameters could provide the basis for optimisation calculations (to be performed in the digital side of the equipment), whioh would be aimed at minimising or maximising oertain performance criteria. An example which immediately springs to mind is the minimisation of gust effeots in an aircraft oontrol system. Dynamic Programming and/or variational techniques (including Pontryagins Maximum Principle) would provide the basis for such computations. Another possibility is the incorporation of extra feedback loops around the process to provide some improvement in performance and so make more simple the task of the adaptive portion of the controller. Circulatory noise within the closed loop puts a limit on the final aoouracy of the present parameter estimation scheme. It's e~eot will have to be evaluated in more detail and investigations should proceed to discover whether any method of easing the problem can be found. The methods suggested in AppendiX 2 would be most effective when no circulatory noise exists in the system. However, if they are successful they should help to aleviate the difficulties to some extent. The open loop nature of the parameter estimation methods means that the stability of the adaptive loop itself is not in question. Nevertheless, the stability of the whole S.E.S.A.C. system is dependent upon the speed of identifioation (Ref.lO). The use of more sophisticated special purpose hybrid equipment and simple prediction elements provides a reasonable approach to this problem.

3.1

COI'lCLlBIOI'B

A method of automatic process parameter esti_ IIIB.tion has been described which is easily mechanised from hybrid (analogue-digital) eqUipment. The estimation is based on the normal operation input signals to the process and does !lOt require the L~jection of special test disturbances. A technique is introduced which eliminates the need for direct measurement of the derivatives of the process input a.~d output sienals which has been the major disadvantage of earlier sJ~tems of this type. It is shown that uncertainty on the sanpled signals, caused by such effects as measure~ent noise, tends to bias the parameter estimates. One possible approach to this problem is suggested. One application of the automatic parBffieter estimation scheme is in the field of self adaptive c~ntrol systems. This is discussed and the simulation of a simple identification-adaptive system is described. In addition to it's use with linear second Qrder analogue processes this system has been used to control a d.e. position control servomechanism. IJlPENDJX 1. The Uethod of Lultiple Filters Consider the general differential

equati~n,

.

(1.1)

Performing a Laplace transformation on this equation and then multiplLing by a modulating function of the fnrm _c__ one obtains, (s+c)L

The basic S.E.S.A.C. system desoribed in this paper has no inherent check on whether the adaptive system is maintaining satisfactory oontrol. This will have to be remedied,possibly by the use of 'model reference' trimming adjustments. Finally the memory store inherent in the digital side of the hybrid equipment provides the future possibility of 'second generation' self adaptive control. It is envisaged that the store would have the secondary purpose of imparting a rudimentary 'learning' capacity to the oontrol system by supplying a long term memory for the calculated process parameters. In the case of a self adaptive airoraft autostabilisation system, these solutions could be used again should similar environmental conditions oocur in the future. Control parameters oould be set up on the basis of this information, so reducing adaption time; the P.D.C. would merely act as a ch~ on these values and 'update' them if necessary.

133

••..•.•..• (1.2) where the brackets ( )0+ indicates the initial conditions of the enclosed variables. Considering initially the left hand side of equation (1.2), it is possible to expand binomially in terms of (s+c) those terms which do not depend upon the intial conditions of y( t) and its derivatives. Thus,

~

\

~ ~

u

a

-
2

:

~

error

~

u

~

'"'~

...u 0

er '"'

~Oj 1 0

~

.~o

2

.it ;:)

(no predict ion)

between

d~sired

process

and actuol response

E:.

Il~

y

output,

~;p

computed

parameter

F+-!-!-r.' / l ! !"! . / / f !

f

variation, ale

!f! r!f/

r

iT

!. J

(with first ord er

! rr! r/"7 ! ! rFiT !

-,OOj -: -0

~

0

>tloo

!- [

prediction) !

! ! r/ : ! :i-:' ! !! ;: : "! ;' / / ;' I-H-f--f-f-f--f :' ! +-~. r!-. i

\t\ .t'.I~~~

~~j~~~:, ." \';,", ;~:\ ;0~'~,', ;j':\,\ ~.~~; \';m', I~,";l'~~f\ ;J;~\rtiliti I

0.

0

varKJtion, ale

!!.~:':~~;SS'\~~JJB:~~~~~~f~~~~:~~~t~:i:~~~~!~lt~~~~¥

1= u

parameter

~ ~~~\l~(((t\\(I:.~ :~:j~ controlled

...... w

\\\\\t

1.

comput ed

,.....",~....--.....,..-r"......,......-r.....,e_r-r-ror

fi

"l

between desired and actual

response

E;.

,(

controlled process output , y

fig.13.

P'1>~ ~cd

Improvement in ad~tive control by the introduction of first order prediction element rnto the p.d.c CA) ramp variation In parameter a. no predIctIOn (9)

similar to

(t\)

but

using

predictive

Circuit

llMl/s«c

cL

L ___c L s n y(s) = (-1)nn{ c ~ Y( s) ( s +c )L ( s-+c )

----L

J ...... (1.3)

where the brackets (

= (_l)ncn {(Yo)] L

and,

n =

...... (1.4)

~

(nao -N; m.o-M) to be obtained by the summation of signals originating from the outputs of the var ious low pass l' ilters, Fig.14.

dtn

(YO)]L;

(YO)]L-l; (Yo )]L-2 etc.,

This simple procedure, prOVides a method of the relationship shown in equation (1.6) which does not require direct measurement of' the derivatives of the input and output signals. This fact is important because these derivatives can be difficult or even impossible to obtain directly in practice. ~eneratL'lg

may be considE:L'ed physically as the outputs of a series of identical low pass filters as shoval in Fig.14. Inspection of equation (1.2) will show that the inverse Laplace transformation of the remaining terms on the left hand side of the equ~tion, that is the terlilS involving initial cond~tJ.ons of the dependent variable yet) and it's derivatives, will produce functions which die o~t very. rapicD..y as time progresses, provided the filter t~e constant l/c is reasonably small.

A similar, although more complex, analysis might be used to indicate how non-identical filters could be used for the same purpose.

Equations of the form of equation (1.6) may be solved for the parameters 8n and bm in a number of ways.

The ri~ht hand side of equation (1.2) may be treated ~ a similar mB..\1!ler to the left hand side and the various filtered input derivatives obtained by an expression of the form,

l.}'uncti"ns of the modified satL"Jfacti-:>n error, Fnt, may be continuously minimised. 'iVhere,

(f)] = (_l)mcm {(f)] - m(f )] o 1.-1 m L 0 L + m(m-l) (f)] _ .....} 2~ () 1.-2

E

m

(fO)]L;

=

~a

~nc

~b

(y)] (1'») n L ~mc m L

•.•••••.• (1.'7)

this ;nethod is similar to that discussed in Ref .2. The !Il:i.nimisation could be accomplished for fairly low order systems by using the steepest descent of the particular error parameter hypersurface selected.

•.•... (1.5 )

where, and,

(similarly for fm)'

Equation (1.6) may be considered true for all time t ) E , where £ is a small interval of time immediately follOWing the initiation of the filtration process ~~ whose magnitude is dependent upon the filter time constant, l/c. Equations (1.4~ and (1.5) allow the functions (Yn)] L and (fm ] L

1.-1

+ n(n-l) (y )] _ .....} 2! 0 1.-2

where, y

y =~ n dtn

- n(yo)] L

)]L indicate that the

enclosed f~'1ction has been filtered L times by a filter of the above for~ and

Taking the inverse Laplace transformation of this equation, one can obtain an expression of the form, (Yn)]

and bm to be

•.......• (1.6)

1.-3 c y(s) + ... (s+c)1.-3 \

3~

8n

(s+c) related by equation (1.6),

L-1 ~(~,\ 1.-2 nc y(s) + ~ c 1.-2 y(s) (s+c )1.-1 2! (s-+c)

n(n-1)(n-2)

enables the paraIlieters

2.bo may be assumed unity without loss of generality. Thus samples of the various signals required may be taken at a minimum of ~+N+l discrete instants of time to prOVide M+N+l linear algebraic equ8tions in M·N+l unknowns (80- aN; bl-bl£). ~Ihen samples are taken at more than lbN+1 L'1stants of time they can be used in a least squares estimation (section 1.1 nf the paper).

(fo)] L-1; (1'0»)1.-2 etc.,

may be considered physically in a similar manner to the analo gous func tions in y( t ) ( see eqn. (1.4) and Fig.14). The foregoL'1g analysis suggests that the continuous filtration of the various terms appearing in equation (1.1) by a filter of the form 135

fo Yo _ _ _...,._ _~ p rocess ....--....,;.---4~

[( x,ll,

[(~l.

,

~ s+c

[(-x)l ~

fig.l4. multiple filters I

"-r'_~ process

_""'_ _"-r'_f_o

,((fJJ.

, I

,

.l(f~)l.

~~ S ... C,

S+C 1

,((f,,)]~

•[(f.)],

.[(fo)),

,[(foll. ,

•

I-_..,.._ _";';'Yo:",

,, •

.[(fo)]~

.,...~

J(yo)),

,,

.[(foll.

,[(Y.)].

I

1.[(~)J2

I

, .J(yJl. I

~~ ~

~

s+c,

s+c

s+c.· •

,((y,A

"W)l..

fig. IS. multiple filters II

136

[ note K~M·N·I

.[(Yo)]~

1

s+c

,J(Y.ll..

3. Alternatively. at least M+N+l sets of different. filters rNJ.y be used {see Fig. 15 } in order that ll+N+l equations in M+N+l unknowns may be generated by the use of samples taken at only one instant of time.

In other words, if the least squares equations are generated in the special way detailed above it S'ould be possible. in the long term. to remove the biae effect of the additive noise contamination. Levadi has used this a)proach in a continuous parameter estimation scheme and has shown that it achieves the desired effect.

APPENDIX. 2.

This appendix has dealt with one possible approach to the additive noise problem encountered in a parameter estimation scheme such as that described in the present paper. The problem is a classical one in statistical parameter estimation theory and the method suggested is certainly not the only one available. Other techniques might be based on Ka1rNJ.n - Bucy filter theory {Ref.ll} or 1Jaxi.1m.un Likelihood Estimates {Refs.12 and 13}.

Elimination of Noise Effects One disadvantage of the simple least squares approach to parameter estimation is the presence of a bias on the parametric estilliate if the samples signals are contaminated by noise. This appendix deals with a particular approach to the problem which has been discussed by V.S. Levadi (Ref.7).

A model of the process is constructed with the aid of the estimated parameters A. This model is subjected to the same input as the process and let it's output be denoted by Yoc (i.e. the estimated value of Yo).

If this signal

lDMENCIAT URE

N:>te: Boldface upper case letters denote a matrix quantity. A superscript T uDed with such a boldface letter indicates a tr&~posed ~ When shown in more detail the elements of the matrix are enclosed by square brackets.

is

filtered and sampled in a similar way to the actual process outp~t {Fig.16}. it is possible to estimate a vector Y • where, L T [ YL = [(YoC}]L'·· ···.[{yNc)]L' [{flc}]L'

t f

A

Yr

•••• [{fMc}]L] using the method of multiple filters.

Em.; Emj=

can be defined as in section 1.4 by. TA

J~A-lL

(j s l,2, ••••S)

ential equations of process. (mao-oM)

but let the least squares equations for the

n

~ • output dependent variable and dtn it's derivatives in general lL~ear differential equation of process. (nao-N) coefficients of input and it's derivatives in general linear differential equation of process. {moo-M} coefficients of dependent variable and it's derivatives in general linear differential equation of process. (nao-N)

S

function. r = 5

A

2 • LFmj jal

TA

L: {L· jZy. A j-l

be replaced by.

A

.YL• .fL

J

time ( f{t}) arbitrary input or forcing function to a process { Yr ( t» time dependent variables of the process. (rao-k) { a (t}) slowly variable or fixed coefficients or parameters which appear in the mathematical model of the process. {rao-.\{} estimated values of the parameters Sr. m 2...!m • arbitrary input and it's derivdt atives in general linear differ-

0

J

where, as before, the prefix j indicates that the function is sampled and is the jth sample of a set.

estimated values of bm and an

Thus, in place of equation (22) of section 1.4 one obtains. cov. [YL,Zy.]A - cov. [YL.YL]A "0

Laplace operator. integer denoting sample size. brackets enclosing a time variable function L~dicates that the function enclosed has been filtered L times by a low pass filter having a Laplace transform transfer function c/s+c (where c is a constant). If j is present it ll1dicates that the total filtered quantity is sampled and is the jth sample of a set. (j=l-S)

•••.•• {2.1}

If the input to the process and the model, f. is uncorrelated with the additive noise, v, then.

cov. (Y ,Z ]

L L

and. cov. (YL.Y ] {A-AI = 0 t

•••••• {2.2} 137

~r·

f.

Yo +

- - - - - -- - - - - - ----, r PARAMETER DETERMINATION COMPUTERI I I

Zo

I

'
(f)

a: f.

w

~

u..

-

MODEL

yo<.

I I ~ I I I

ZL

YL

YL by the

m~thod

mUltiple

I

of

estimation

of

~stimation

~

process parameters

filters

the auxiliary model approoch to the

138

noise problem

A

jA

rr r""" I

_ _ _ _ _ _ _ _ _ _ _ _I

A

flg.16.

I

I

I I

I

IL

of

j (f)]L

[

permissi~n

of the Head of Department. Professor K•.L.C. Legg. The author wishes to thank Mr. C.D. Dwyer for his collab~ratio~ on the various problems of hybrid instrumentation and digital programming. Thanks are also due to Dr. J.D. Roberts of the Jniversity of Cambridge f0r his helpful discussion on the paper.

yO·yl·····yk] or [yO.yl ..... yN.fl.f2 ••

M]

•••• l'

Y

T

L

• [(f l )] •••• [ [(yo)] L ••••• [(YI,r)] . L L .••• (f})]L]

REFERENCES

[[(y)] ••••• j

L

0

[(Yr~)]'

j.

Lj

[(f )] •••• l L .••• [(f,J] ] j

Y

....

L

T

L

[ [ (y.oc )] L ••••• [ (yNc .)] L • [(flc )] L • ••• •••• (fYc)]L] estimated state variables obtained by Vlay Clf an auxiliary mdel constructed from the parsl.!.Le>ter estimates

[ao'~' .... ~] or [ao'~""

,aN;b l ,b2 ••

... ,\,;:J

.T

A E

r

T

as for A but with thp- estimated values of the parameters for elements. errClr in eatisfaction of the prClcess e~uation _ 'satisfaction error'. modified satisfaction error. N !od E'mJ' =~anc .[(Yn)]L - ) ' bmc .[(fm)]L ~ J ~ J

t~2 j-l

<>

denCltes ~€ctation operator (c~semble average). covariance matrix. additive noise conta.minBtion (k"o.1j-+N-+l) [VO.v1.v2.···.vll+N+l] column matrix of noise components. The elements depend upon the elements of and on the nature of the filters used in the estimation scheme. observable process variables (k-o+k+N+l)

[Y"'~L =

YtVL

The investi,ations described in this paper were carrif"d o~t under a research scholarship in the Department of Aeronautical a..rld Automobile E..'1gineering. Loughborough College of Advanced Technology. The paper is published with the 139

(1) Eykhoff. P.. IISome l''undaUlentcll Aspects of Process Parameter Estimation", Trans. I.E.E.E<. on Auto~tic Control, Vol.AC-8. No.4, October 1963. pp. 347-57. (2) Young, P.C •• liThe Determination of the Parameters I)f' a D~rnamic Process ll • The Radio s!1d Electronic Engineer. Vol.29. No.6. June 1965, pp. 345=61. 0) Young. P.C •• "In~light Dynamic Checkout:a discussion", TrailS. I.E.E.E. on Aerospace, Vol.!..5-2. No.3, July 1964, pp.llOb-ll. (4) Young, P.C., and Dwyer. C.D •• "A Hybrid Self Adaptive Control Systemll , recipient of P.L.C.E. Prize 1964. awarded by Electronic Associates LW• (5) DNyer. C.D •• llHybrid COIDputationll , A:asters Thesis sub~itted to J~iversity of Leicester. (6) Truxal. J.G.. IICO!1tX'?1 System Synthesis". l:cGraw-Hill Book Company Inc., ;'iew York, 1955. p.~5 et seg. (7) Levsdi, V.S •• llParametcr Estimation of Linear Systems in the Presence of ;~oise". Pape>r presented at the InternBti )£1;;.1 Conference on ~crowaves, Circuit Theory and ll~ormation Theory. Tokyo • 7th-11th September. 1964. (8) Young. P.C •• IIA Simple An.abzue Predictive l;ircuit". int~r!181 tectnical n;)te Jeoartment of Aeronautical and AJtomobile Er,gineering. LoughborouFh C.A.T. (9) Kalwan. R.E•• "Dpsign of a Self Optimising Gontro~ Systemll , Trans, A'S1.E•• Vol.BO. ~jo.2. rebruary 1953. pp.852 2. (10) YOIIDg. F.G., "Pcr<:>.meter Esti!llOtion and Self Adaptive Controlll J A.L. C. Thesis, Wur,hbo!OLlgJ1 C~llege of Advanced Tec~'1010gy! 1965. (11) Kalrnan. R.E. &'1d Bucy. It.S •• "New Results in Linear Filtering and Prediction Theory". TrAns, A.S.~.E•• Vol.S3. 1961, pp.95-108. (12) Li.r>_'1ik, Y.V •• "Liethod of Least Squares and Principles of the Theory of Observationll • ~rgamon Press. London, 1961. (13) ~tr5m. K. J. and Bohlin. T•• IlNumerical Identification of li1ear DyiJ.ami.c Systems from Normal Operating .l{ecords". paper presented at the I.F.A.C. (Teddington) Symposium 1965. on liThe Theory of Self Adaptive Control Systems". Teddington. EJ.1bland. 14th-17th September 1965.

Discussion The following remarks were based on Mr. Young's verbal contribution to the Symposium before the text of his paper was available. Mr. Eykhoff (Netherlands) interpreted Mr. Young as implying that each filter provided a coefficient of the differential operators of the differential equation describing the generalized model - a point in fact possible only with pure differentiators.

Mr. Rowe (U.K.) remarked on the similarity of the method and that described by Mishkin and Braun for generating Laguerre polynomial coefficients. Mr. Young stated in clarification that the "Method of Multiple Filters" generated certain functions of the derivative terms which could then be used, in the manner described in his paper, to arrive at a least squares estimate of the unknown parameters. Mr. Young considered the problem of removing noise induced bias and saw promise in a technique similar to that suggested by V. S. Levadi. By the incorporation of an auxiliary model of the process based on the estimated parameters, it should be possible to discriminate against any noise present on the sampled signals and so reduce the bias effect. Again, an adaptive filter of the Kalman-Bucy type might be utilised to obtain optimal unbiased estimates, whilst at the same time providing an answer to the optimal regulator problem. Mr. Young stated that the method described in the paper was not related to the method of Laguerre polynomials dealt with by Mishkin and Braun. However, if ~~. Rowe was interested in an approach to identification along those lines he should be referred to the paper, "The Synthesis of Dynamical Models of Plants and Processes" by W. D. T. Davies, which was available in the Proceedings of the U.K.A.C. Convention on Advances in Automatic Control, Nottingham, 1965. Mr. Duckenfield (U.K.) suggested that feedforward of observed noise might prove helpful.

140