Self-Tuning Control: Towards Industrial Viability

Copyright © IFAC Low Cost Automation 1986 Valencia, Spain, 19R6

SURVEY PAPERS

SELF-TUNING CONTROL: TOWARDS INDUSTRIAL VIABILI1Y R. W. Jones and R. P. D. Baker* Industrial Control Unit, University of Strathclyde , Glasgow , United Kingdom 'Department of Mathematics, University of Strathclyde, Glasgow, United Kingdom Abstract. In this paper, a new goal is set for the topic of self-tuning control: the progress made to date, towards widespread industrial us age is reviewed critically and a speCification , for an, Industrial self-tuning control system IS proposed, with the objec tive of Indicating the direction in which self-tuning should move If it is to approach the realisation of its potential industrial value. To illustrate the Ideas being ad vocated, the generalised minimum variance self-tuning controller IS taken as an example: It is assessed against the proposed speCification and some developments put forward to move this particular algorithm closer to It; the points made are exemplified via simulations. Keywords. Self-tuning control, Industrial control, Practical issues in adaptive control, Microprocessor-based control, Generalised minimum variance self-tuning controller INTRODUCTION Clarke ( 1984a ) and M'saad and Colleagues ( 1985). So far as compensation of dead limes IS concerned such compensation is In fact achieved naturally with some control strategies In adaptive settings (pole placement is an example ) , and for workers who have adopted these strategies this has been a major motivation. Other approaches to the problem for alternative Situations havealso been considered, for example in Kurz ( 1979 ) , Sc humann et al ( 1981 ) , Wong and Bayoumi ( 1982 ) , Vogel and Edgar ' ( 1982 ) , Gurubasavaraj and Brogan ( 1983 ) , Cheln et al ( 1985), Hunl and Grimble ( 1985), Lammers and Verbr uggen ( 1985 ) and De Keyser ( 1986). Techniques for disturbance rejection in self-tuning are considered in Astrom ( 1980b ) , Clarke ( 1981), Isermann ( 198D), Moden and Nybrant (1980 ) , Wittenmark and Astrom (1980), Harris et al ( 1980), Allidina and Hughes ( 1982 ) , Gawthrop ( 1982 ) , Latawiec and Chyra ( 1983 ) , Clarke et al ( 1983 ) , McDermott and Mellichamp ( 1984 ) , Xianya and Evans ( 1984), Tuffs and Clarke (1985 ) , Berger ( 1986 ) , Warwick (1986 ) and Grimble (1986a). The important issue of acheiving tracking of changing process dynamics whilst preser ving satisfactor y stationary estimation performance is addressed in Cegrell ( 1976 ) , Astrom ( 1980b ) , Latawiec and Chyra ( 1983), Clarke (1980), Irvlng (1979), Karny (1982 ) , Fortescue et al ( 1981 ), Wellstead and Sanoff ( 1981 ) , Dexter ( 1983 ) , Vogel and Edgar ( 1982 ) , De Keyser ( 1983 ) , Tham ( 1985 ) , Tuffs ( 1984 ) , Hagglund ( 1985 ) , Kulhavy ( 1985), Bertin et al ( 1986 ) , Lammers ( 1985 ) , Goodwin and Teoh ( 1983 ) , Shah ( 1986 ) , Witt enmark(1979a), Hagglund ( 1984 ) , Andersson ( 1985 ) and Holst and Poulsen ( 1985 ) . Practical implementation aspects in self-tuning have attracted incidental study in many workers invesligations, but more in-depth coverage can be found in Clarke (1981 ) , Wittenmark and Astrom ( 1984 ) , Clarke and Gawthrop ( 1981 ) , Isermann nad Lachmann ( 1985 ) , Tuffs ( 1984 ) , Hang et al ( 1985 ) aAd M'saad et al ( 1986a ) . Proposals for the synthesis of self-tuning algorithms with PID type structures have involved a miscellany of approaches, including those presented in Witlenmark ( 1979a ) , Gawthrop ( 1982 ) , Corripio and Tompkins ( 1981 ) , Halme et al ( 1982 ) , Banyasz and Keviczky ( 1982 ) , Banyasz et al (1985), Cameron and Seborg ( 1983 ), Yuan ( 1985 ) , Song et al (1986 ) , Makila ( 1984), Radke and Isermann ( 1984 ) , Hunt and Grimble ( 1985) and Tjokro and Shah ( 1985 ) . This brief survey is of couse very far from being complete. Its selectivity has excluded numerous important papers and many

Self-tuning control is widely regarded as a method that can potentially become a major industrial control technique. The industrial pOSSibilities of the method, following the publication of the seminal paper by Astrom and Wittenmark ( 1973 ) , have been the main stimulus In the development of the self-tuning research area into one of considerable size and importance. The Industrial use of self-tuning controllers is, however, as yet still limited, with their rmployment for the most part being confined to Isolated examples of successful applications. Since such examples have been appearing the literature for a decade or more ( a case of a selftune r being in regular operation on a full-scale industrial plant was reported ten years ago in Cegrell ( 1976 ) ; the same application having previously provided in Cegrell and Hedquist ( 1974) one of the fir s t published accounts of an Industrial trial of self-tuning control ) , It might have been expected that rather greater progress towards more general usage would have been made by now. The absence of thi s has to be viewed as disappointing. One r eas on for the failure of se lf-tu ning to make a more s ubstantial impact In Industry is that In lhe research area, although the emphasis has been slrohgly praclical, the tendency has been to concentrate on one or two specific areas of difficult y rather than pursuing the development of self-t uning control systems that meet Industrial requirements In all respects. Thus researchers have been pre-occupied with particular issues like the problem of stabilising ( so-called) nonminimum phase s ystems, the question of handlin g variable and/ or unknown dead times, rejection of unmeasurable deterministic disturbances, the provision of effective tracking of process variations without the performance of the estimator under static conditions being compromised, the practical implementation of se lf-tuners, and the options for constructing self-tuning algorithms that have PID type control structures. The problem of nonminimum phase systems has been a key concern In much work, but Instances where this is especially so are in Astrom and Wittenmark ( 19 74 ) , Wellstead et al ( 1979b ) , Astrom ( 1980a ) , Gawthrop(1980 ) , Samson ( 1982 ) , Makila(1982 ) , Grimble ( 1982 ) , Grimble et al ( 1982 ) , Elliot ( 1982 ) , Lozano Leal and Landau ( 1982 ) , Lammers and Verbruggen ( 1982 ) , Kumar and Moore ( 1982 ) , Grimble ( 1984 ) , Bars and Keviczky ( 1985 ) and Goodwin et al ( 1986 ) - see also the reviews by

CIT- 8

5

6

R. W. Jones and R . P. D. Baker

well known ones. The literature on self-tuning is in fact vast : but the bulk of t hi s considera bl e effort has been dire c t ed along a basical l y circular path, with progress towards methods that are globally suitable for industrial use being retarded as a consequence. A possible expalnation for what would appear to be a lack of will amongst researchers to examine the broader requir ements of industrial self-tuning systems is that in many cases they have misnterpreted the role that se lf-tuners will pla y If the y do find application in industry. The popular view is that since the ma jo rit y of industrial plants are contemporarily controlled by PlO controlers, this must mean that se lf-tuning IS for the "difficult" pl ant s, l e. for those on which PID does badl y of fails comp l etely. Coupled with the supposition that difficulty necessarily indicates a high degree of s pecial it y this can be in turn taken to impl y that it will be j us tifiable to ha ve for each installation of a self-tuning controller a major commissioning phase, in which detailed plant trials would be carried out to aid in the selection of the most appropriat e algorithm, facilitate the development of va riou s application specific measures necessary to resolve outstanding weaknesses of the algorithm se l ected, and provide a priori information for the sett ing up of a range of key design parameters . From this sta ndpoint, the status of the self-tuning r esearc h area, with such a ri c h di ve r s it y of techniques bei ng on offer, appear excellent. Now, although sel f-tuning control ma y find application through this philosophy ( indeed, It basically has underl ai n the instances of successful applications that were observed ear lier to ha ve been occurring for a numbe r of yea r s now), it is improbable that It will ever lead to the pene tration of self-tuning techniques into industry becoming substantially greater than at pr ese nt. The problem with th e notion of basing the application of se lf-tuning on a large commissioning effort lies in the naivety of pr esuming that there will be an automatic willingn ess t o undertake thi s effort. Apart from the qu est ion of the necessary expertise, a r ange of other fa ctors will mak e it unlikel y that the Idea of a major commissioning excercise will be v i ewed favourably without there being a ve ry high degree of confidence about the outcome in terms of both its ecomomic and lt S operational Impluc at ions (a useful di sc ussion In this area IS gi ve n by Rij nsdo r p and Seborg ( 1976 ) , in which the importance of, to use the term they coin, brainware costs IS emphasised). Whilst there are situations in whi c h there will be a willingness to undertake a s ubstantial commissioning effort for the application of se lf-t uning, t hese are likel y to be small In number compared to the situations in which self-tuning methods might be useful. The addition of commercial packaging to the philosophy, pro v ided for example by ASEA's Nova tune ( Syding and Einarsso n 1984, Moden 1984 , 8engtsson and Egardt 1984 ) , or the introduction of Implementation aids employing expert systems concepts, as suggested by Sanoff and Well stead ( 198 5 ) and Astrom and colleagues ( 1986 ) , will not alter this basic conclusion, although an extenSion of limited magnitude in the usage of delf-tuning can be expected to occur with th ese de vel opments. Furthermore, recollecting that the rationale for apprroaching the application of self-tuning via a large commissioning effort r es t s on the idea that the function, in industrial usage, of this kind of ad vanced technique is to provide satisfactory control in what is viewed as the essentially specialised situation of a plant where PlO control pro ves to be inadequate , per mi ts another per t inent point to made, which IS that the preferabilit y of self-tuning as the advanced contro l st rateg y utilised in a co ntext s uc h as this i s by no mean s obvious: the effort which i t is assumed would be required for the commi ss ioning o f a self-tuner could

be put instead into, fo r example , the building up o f a gain schedul e , or into t he prosecution of a robust co ntr o l de s i gn methodology ( for instance an approach to robust control system design involving a range of simple proc~dures requiring only a modest amount of process information ha s been recentl y put forward by Owens and colleagues (eg. Owens and Chotai 198 3 , Chotai et a l 1984, Owens 1985 » , or perhaps into the se tting up o f a control technique like Dynamic Matrix Control ( Cut ler and Ramaker 1980 ). For several rea so ns c hoo s ing an alternative advanced strategy to se lf-tuning co uld be desirable: for one thing the Installed cont rol syste m with an alternative strategy will general l y be s ubstantiall y si mpler than with se l f-t uni ng and normally will not include s uch potentially p r oblemat i c feature s as online identification or on-line co ntr o ll e r design; another factor is that with a n alternative strategy there may be guarantees offered relating to stability and other c losed-loop properties that are sign ificantl y more mea ni ngful than any available with se lf-tuning. It is unfor tunat e that the pr ev alence of the commissioning phase approach ha s led to the focu s in the de ve lopment of the subject being directed away from the production of se lf-tuning sys tems satisfying the full spectrum of indu stria l requirement s f or which a major effo rt for commi ss i oning need not be necessa r y. Cl ea rly a shift towards consideration o f a compl ete range of requirements i s needed , and one major aobjective of this paper is to define a spec ifi ca tion that indust ri a l systems need to mee t if they are to mo ve the method towards t he kind of usage that is commonly envisaged. Although the main point here is to emphasise that any self-tuner mu st be globa ll y Industrially s u i table rather than to advocate a particular algorithm, the example of th e generalised minimum variance (GMV ) delf-tuning controller (e g. Gawthrop 1977 ) is taken to illustrate the kind of development the authors' belei ve are nee ded. This includes a discussion of this method' s s tr engt hs and weaknesses , some s ugge st ions as to how the weaknesses could be ove r come, and simulation exa mple s to exemplify the poi nt s made. A POS SIBLE SPE CIFICATION FOR AN I NDUSTRIAL SELF-TUNER The major requirements tha t a n industrial self-tuning system mu st satisy can be s ummarised thus ( some of these are ada pt ed fr om Edgar ( 1976 ) and Rijn s dorp and Se borg ( 1976) :(a ) The performance of the se lf-tuner should be largel y insensitive to incorrect choices of model st ruct ur e par~meters, es peciall y mode order and dead time. ( b ) Tailoring the pe rformance of the self-tuner to meet typical process control requirement s , s uch as overs hoot and rise time in servo r esponses, load disturbacne rejection characteristics, noise s uppres s ion nad avoi dance of excessive control s ignal s . s hou ld be si mpl y achei ved: it should preferably be automated but at least s hould be tr anspare nt l y obtainable via a minimum numher of tuning parameters.

( c ) The self-tuner's performance s hould not be affected by t he presence of (or even require awareness of ) diffi cu lt process chaacteristics like non-minimum phase dynamic s or variable dead time. (d ) Installation of the sel f-tuner ahould not necessitate th e availability of any expertise other than basic process control expe rtise - sett ing up design parameters

Self-tuning Control: Towards Industrial Viability should be trivial, including the tuning parameters referred to in ( b ) above.

7

The expression for the minimum variance prediction (4) is in fact obtained from the model ( 1) via the following relationships :-

( e ) The start-up pahse which is required with all self-tuners should be automated, and should also not compromise plat operation.

C P

EAPd + z

n

G

Another important consideration, which underlies several o f the above points, is the issue of operator acceptabilit y . This is strongly a function of the pres entation of the control policy as well though (eg. Umbers 1984 ) so it is given less emphasis here. The algorithm on which the self-tuner is based is not, In the authors' view a paramount issue: in the sequel, the GMV self-tuning controller is comsidered, but seve ral other techniques could also be employed, such as pole placement ( eg. Wellstead et al 1979a , Well stead and Sanoff 1981, Clarke 1982) or LQG ( eg. Astrom 1983, Jones et al 1986, M'saad et al 1986b. Shieh et al 1983 ) . The alternative algorithms share so me of the problems of GMV and as well have some distinct ones of there own. The key point here, though. is that the self-tuner should yield the kind of performance that the above specification defines in a practical situation - less importance is attached to theoretically possessed or idealised proper ties.

,

EB

C

l-z

-1

-k

F

,

nf=na-l, ne=k-l

(6)

ng=k-l+nb

(7)

H

(8)

The conventional implicit approach uses extendedleast-squares (ELS) identification to eliminate G,F and H in the following pseudo-linear regression prediction model:y* (t) = Gu(t-k) + FYpd(t-k) + HY* (t-l;1t-k-l) + P p

The control signal u(t)

(10)

which upon substitution of y*(t+kl't) the implemented control sign~l as :

pd

(9)

is calculated from

u(t) = ( Rw(t)-y*(t+k/t)) /Q p

U(t)=(Rw(t)-G'u(t)-Fy

~(t)

from (4) gives

(t)+Hy*(t+k-l/,t-l))/(Q+go) p (11)

where ng

GENERAL MINIMUM VARIANCE CONTROL

G'

A general discrete time model of the process being controlled is assumed to take the form: A(z

-1

) y(t) = z

-k

B(z

-1

) u(t) + C("

-1

)

~(t)

(1)

where z is the back shift operator, Z-iy(t)= y(t-k) The time delay , k, of this controlled autofegressive moving average process (CARMA) is an integer number of sample intervals, k)/ 1. This definition implies b 0 , so that polynomials A?B and C have the form: o -nx + ... + X z (2) nx

I

wi th a = 1 and Co =1. The degrees of the system O polynomials A;B and C ( denoted na,nb,nc) are assumed known and C is further assumed to be a satble polynomial with a zero-mean unco rrelated random input sequence ~(t) .

The cost function for the GMV controller can be written as :-

*

= v.p (t+k /t ) -Rw(t))

J

where y /

p

p

* ( t+k/t ) ,

2 +).l (Qu(t) 2 )

(3)

given by:-

( t+k t ) = Gu ( t) + F y(t) + H /

p

(t+k-l/t-l )

p

== p1d

(5)

y(t)

and P,Q and R are transfer functions defined as ratios of polynomials, for example, :P

=

P

n

(Z-I)

P d (z

-1 )

gi z

i=O Using the above equations the closed-loop behaviour is given by RB (PB + QA)

w(t-k) +

(EB+QC) e(t)

( 12)

(PB+QA)

If the transfer function Q(z-l) is chosen to be zero the closed-loo~lsystem behaves as a prespecified model I/P(z ), (for minimumphase systems), with respect to the delayed set point w(t-k), A non-zero Q is used to provide 'detuned' model-following control. Both P and Q can be used to stabilise the closed loop poles of the characteristic equation, PB + QA =0.

Recent work (Hodgson (1982)) has extended the predictor to become incremental over k steps (k incremental). The advantages gained from the use of this include the rejection of steadystate errors due to parameter bias in the positional predictor (4) and improved disturbance rejection properies. The k-incremental prediction is defined as :

(4)

is the minimum variance prediciton of y (t) k steps ahead, and encapsulates the stochastic Rspects ef the problem. In these formulae, G,F and Hare polynomials in the backward shift operator, ).l is a constant that disappears in the minimisation of the cost function, ypd(t) is defined as :Y d (t )

-i

2:

y;(t+k/t)=y(t)-~(t t-k)+F6 y(t)+G6 u(t) k k

+ H6 y*(t+k-ll't-l) k P

(13)

where 6 ~(I_z-k) and y(t/t-k) is the prediction error a~ time (t). A by-product of the predictor is that the estimation becomes better conditioned numerically as it works on zero-mean data. Moreover the design is robust in the sense that the freezing of the estimator does not detract from its offset elimination property.

All the GMV self-tuning results presented in the paper were generated using the k-incremental predictor in the self-tuning control law.

8

R. W. lones and R. P. D. Baker

The implementation of the implicit algorithm requires knowledge about the process, which might be extremely time consuming to obtain, to set up the control and output the filters and time delay, k. By treating the implicit G~W controller from an explicit point of view a considerable simplification in the setting-up procedure can be achieved. This explicit approach apart from simplifying the setting-up procedure also leads to

wholly a function of the model order. The sample time can thus be chosen purely on dynamic considerations. The result is faster initial tuning and

better adaptability. A disadvantage is the increased computational complexity and calculation time due to the need to solve a Diophantine equation. Solving

this recursively (Lee and Lee 1983) minimises the extra calculation time.

increased performance characteristics. The

following advantages can be listed:

(1) smaller number of parameters are identified for the time-delay case, (2) variable time-delays can be handled (3)

the adaptive characteristics of the controller are improved,

(4) each of these points will be discussed briefly in the next section.

THE TIME VARYING DELAY PROBLEII Controllers providing dead time compensation tend to be more sensitive to errors in the dead time estimate than errors in the model of the dynamics.

In the GIN self-tuner the process dead time is assumed to be known apriori. However in most

industrial applications, the process time delay is either unknown or, if known,

time-varying. In its

explicit form the algorithm can easily be extended to cope with the variable time delay problem with an apriori value for the time-delay not to be

needed. PROBLEMS OF THE IMPLICIT FORMULATION The loss of information when a continuous

process is subjected to the sampling operation is not of practical significance as long as the sampling time is about one-tenth of the dominant time (Clarke 1984b ) or IllS t "' h~1/4 t ,where t 5 is 95%of the settling tim~5(Isermann,19g1). The ttme delay parameter k depends on the ratio of the time delay(T ) to the sampling period, h. Clearly the choices gf k and h interact with the number of G polynomial parameters required being a linear function of k. Choosing a sample time on the above guidelines could lead to h being very small and thus produce a correspondingly large k.

The consequences of a large k are twofold. With the large number of parameters the initial tuning-in period is necessarily longer to obtain

Recently several groups (that is, Kurz and Goedecke, 1981, Wong and Bayoumi,1981) have developed explicit self-tuning control techniques using the polynomial approach where the identified B polynomial in the plant model is extended to account for varying process dead time. There are

two related meth~~s for doing this. The first rewrites the B(z ) polynomial in (1) with a sufficient number of terms to include the delay, that is,

B' (z

-1

) =B (z

-1

)z

-kmax

( 14)

where

satisfactory estimates. Secondly the adaptive

( 15)

capability is undermined since it takes a long time for a large number of parameters to retune in a

dynamically time variant system. These imply long periods of potentially unsatisfactory control and

and r is determined from

are particularly of significance to industrial processes which may be grossly nonlinear and contain substantial deadtime.

r

This drawback of the GITV self-tuning controller has received little attention previously. Clarke (1984b) merely recommends increasing the sample interval to reduce k to a value 2-3. A large number of parameters are avoided but at the expense of inadequate consideration bein~ given by the controller to the dynamic behaviour of the system. The use of the self-tuner in this way to a large extent cancels out any initially conceived advantage over conventional control.

THE EXPLICIT

FOR~ruLATION

( 16)

= nb + kmax

with kmax being the maximum expected deadtime. Ideally for a dead-time of k sampling interval.s the estimated value for the first k B' polynomial coefficients, b

the time delay

, b

,

...

, b

will be zero. Thus

~ will be equal-to the number of

leading coefficients which are zero, or in the case of noisy systems, close to zero.

A drawback of this method is the possibility of a dimensionality problem. If the sampling time is small relative to the dead-time the number of sampling intervals in the discrete dead-time , k is large. Thus r is large resulting in a large number of B' polynomial coefficients to be estimated. In this case Dumont (1982) suggested that a range for the dead-time be specified. The B polynomial can now be written as

By identifying the process model(l) explicitly the Diophantine equation (6) can then be solved to obtain the control parameters. Thus the number of eliminated parameters become independent of the process time-delay and sample time and are

(b

oz

-1

+ b z 1

-2

+ ... + b s-l z

-s

)

z

-lanin

where kmin is the minimum expected deadtime,

(s -n)

Self-tuning Control: Towards Industrial Viability is the number of intervals over whi ch the dead time may vary,that is, k and the maximum expected dead time is kmin + Ys - n). Using this method with an appropriate choice of sampling interval means that any si gn ifica nt dead-time variation will likely be over only a few sampling intervals a llowing s to be small. The inclusion of the above time - varying delay facility i n the contrlller increases the number o.f" es . . im - t e' pa --am ters over the purely explicit form of the G~N . More parameters st il l need to be estimated in the i mpl icit G~ as against the extended explicit GI,N except for k~l when the numbers will be equal.

characteris tic equation which is much easier to solve . Jurys' stability cr iteria (1964) will be utilised to auto -tune A to maintain stabi li ty of the characteristic equation. Consider

The characteristic equation becomes (b + A )Z3 + ( b +}\ (a -1 »z2 1 O l

/l (a2 -al )z -Aa2~ 0

+ Let s3

TABLE 1. Numbers of Estimated Parameters Form of GIN

Paramet ers to be estimated

where

N~

na + nb + nc,

k

v

~

b b

s2

So

O

+/1.)

+

1

A.

sI

N+k N+1 N+k +1 v

Implic it Explicit Ex. Explicit

9

/\. (a -l ) l

(a - a ) 2 l -,\. a

2

Give n a polynomial

(s - n)

EXAMPLE 1.0 In this f i rst example the tuning-in behaviour of the explicit GHV contro l is compared against the implicit formulation . The discrete system is stable and minimum phase (1 + 1. 036z -

1

- 0.2636z ~

2

the necessary and sufficient conditions for the polynomial to hav e no r oots on and ou tside the unit circ l e in the zplane are: 5(1)

>0 ,

) y(t)

( 0 . 1387 + 0.0889z

-1

)u(t - 4) +e ( t )

which g ive the following cond itions for

A

Fig . l(a) is i mplicit while Fig l (b ) i s explicit. It can be seen that the initial tuning characteristics are better due to the fewer estimated parameters in the explicit contro ller.

AUTO TUN ING OF THE CONTROL WEIGHTING I-a To simplify bhe auto-tuning scheme the f o llowing weights were chosen P ~ 1, R ~ 1 and Q ~~ (l _Z-l) This leads to an interpretation of GIN s ome times referred to as detuned minimum variance contr o l. The equivalent closed -l oop character istic equation is

B(z

- 1

) + Q( z

-1

) A(z

-1

) - 0

( 17)

and the criterion which will be used to autotune the value of \ in Q is the maintenance of closed loop stability. Auto-tuning of the Q filter is possible for b o th formulations of G~ . For the implicit form we can define A from the Diophantine as :

A~ (C - Fz - k)/E and , B

G/ E ,

C

1- z

- 1 H

)..>

/\.>

-b

2

o 2

( - b +/ -

I

(b - 4ac»/2a

/

where

a

(a b

b

c

a a 2 1

~

~

and (b +b

O

b l

2o

-

c b

2 O

- a b

2 1

+ 2b )

O

O

):>

0

to be satisfied .

In the controller the )\ . value calculated from the model estimatesC~fid stability criteria is multipli ed by a factor to give better stab ili ty characteristics . Two examples are now presented which demonstrate this aato -tuning facility. EXI1J1PLE 2 . 0

giving ( 18) for the closed - loop characteristic equation. nq~l , nc~nh-l and nf~na-l thus it can be clearly seen that the o r der of the equation is a function of k , the time delay . For a va lue of k of 4 (a reasonab l e va lue ) the order of the equation becomes ( for nb~2) equal to 5. It thus becomes ve ry d ifficult to compute the value of 1\ t o stabili~e ( 18) . Fo r the explicit form for nb~na~2 and nq~ l this implies the solution of a third order ng~(k-1+nb) ,

Cons i de r the nonminimumphase system previously used by Clarke and Gawthrop ( 1975) (1 - 0 . 95 z

-1

)y(t)

z

- 2

(1 +2. 0z

- 1

+ ( 1- 0 . 7z

)u(t)

- 1

)~(t)

For detuned minimum variance control to stabilise the closed loop system A has to be ~ 0.26 in the control weig hting. The factor

10

R. W. lones and R. P. D. Baker

A.

used is 2.0 which gives a value of 0.52 at steady state. Figure 2(a) shows the response of the ystem to setpoint changes of magnitude 10 . Figure 2(b) shows the corresponding calculated ( and implemented ) value of ?l. It can be seen that the initial tuning (helped by the process noise and set point changes) is complete by the 40th time sample. Unfortunately the stability conditions make no distinction about the positions of the closed loop poles except that z ~ 1. Thus application of this method might result in the production of a control weighting that gives a very poor, but stable output response. For example in the simulation the response is very oscillatory with the closed loop poles at poSitions z = - 0.3347 +/jO.456. One alternative would be to use more complicated aperiodicity conditions to force the 0 roots of the clos ed loop characteristic equation to lie in the segment (0,1) in the zplane. (Sz araniec 1973) In this case a mo re complicated form of autotuned Q filter would be required to acheive the condition. Another alternative to improve the system response is to use the de tuned model-following interpretation of GMV control. f10re details can be found in Jones and Baker (1987). EXAMPLE 3.0 This example is intended to illustrate how the auto-tuning facility peI'forms when applied to a realistic system. The self-tuner was applied to the nonlinear simulation of a pH control loop (tham and Jones 1986). This loop is deterministic in nature and quite nonlinear with a process gain

variation of approximately -1.5 to 3.0 in the area of interest (pH 2.7 to pH 3.0). A first order model of the process at pH 2.7 is

-1. 59 e -76s

erised by large time lags, exhibit large and time varying delays , are nonlinear and are of

high order. In self-tuning low order models can usually be chosen to give a good approximation to the real system. The following simple exampl p demonstrates that models of lower order can be utilised. EXAMPLE 4.0 Consider the following slow second order minimumphase process model

(1-1.7z

-1

+ 0.72z

-2

)y(t)

(0.2

O.lz

-1

)u(t -1 )

was simulated and a first-order model

was assumed in the GMV algorithm. By comparing Figures 5 (a) (correct parameterisation) and 5 (b) (incorrect parameterisation) it can be seen that there is very little differnece in the performance of the controller. The initial overshoots can be attributed to tuning-in of the paramter estimates. It should be noted that this is not a tutorial example because neither of the poles of th~plant is negligible. EXAMPLE 5.0 This example demonstrates some genera l

properties of self-tuning control when the cho ice of model order is different. Linear quadratic gaussian (LQG) self-tuning control (Jones et al 1986) is applied to the nonlinear pH loop as used previously in example 3.0. As before a sample ime of 16 seconds and a discrete delay of k=5 samples was used and setpoint change tests (w ith converged values of eliminated paramters), carried

(1 + 112s with the time constant and time delay being in seconds. A sample time of 16 seconds was used (k=5) and a second order model estimated. Figure 3(a) shows the controlled response tuning-in via an initial open-loop variation of u(t) and then closed loop control to setpoint changes. Figure 3(b) shows the corresponding value of It can be seen that a fairly satisfactory response is finally acheived. This run was carried out with a multiplication factor of 2.0. Figures 4(a) and 4(b) show the equivalent characteristics when this was increased to 3.0.

1\.

out for 1st, 2nd a~? 3rd order estimated deterministic (C(z )=1.0) models. The control weighting in the LQG self-tuner was chosen to be unity. The results are shown in Figures 6(a), (b) and (c). It can be seen from these that the system damping is improved by increasing the order of the estimated model but also that the response time increases when more b- parameters are estimated. Also as the number of estimated parameters was increased the convergence rate of the estimates decreased.

This example has shown that very good behaviour can be obtained by estimating models which are considerably simpler than the process. This observation is of substantial practical significance as

INCORRECT MODEL PARAMETRISATION The model paramterisation problem can be broadly separated into two areas, choice of model order and choide of the system time delay. To all intents and purposes

it supports the use of simple

models. In some cases though it must be noted that if the model used in the self-tuner is too simple the closedloop will be unstable. It would thus be of interest to have theory which gives insight into the properties of slef-tuners based on model

Qverparameterisation in the GMV self-tuner is

structures that are simpler than on the realprocess. Unfortunately there are few results of

immaterial. This can be seen by studying (6)

this type available. Most theory of system

CP

n

EAP

d

+ z

-k

identification and adaptive control is based on

F

which shows that common factors in A and B (induced by overparameterisation) do not effect its solution. For pole placement self-tuning ( and LQG) the Diophantine that has to be solved is affected by common factors. In this case a numerically robust result can be obtained by calculating a minimum-norm solution (Lawson and Hanson 1974). In general industrial control systems are charact-

the assumption that the class of fitted models is sufficiently rich to include the real process. For insight into the problem see Astrom and Wittenmark (1973) ,Astrom (1980c) , Owens and Chotai( 1983) . With respect to an incorrect cho ice delay this can always b~ltaken care of by the method of extending the B(z ) polynomial (and estimating more B paramters) mentioned earlier. In this case

the incorrect choice of delay will be compensated. For cases where this method is not used and the delay is again set up incorrectly, not sGrprisingly there will be a degradation in control performance

depending upon the error in the delay. This can easily be compensated for by changing the Qfilter (Tahmassebi et al 1985 ) , Montague et al 1986)

11

Self-tuning Control: Towards Industrial Viability DISCUSSION The preceding sections have described some developments that contribute to one particular selftuning algorithm coming closer to the specification set out for an i ndustrially rele vant self-tuning control system. Clearly other measures are needed to complete the system: the provision of an effective estimation routine, the addition of a range of supervisory features, particularly for the start-up phase, and the inclus ion of a viable method of disturbance rejection are probably the most important. In all these areas a large range of ideas are already available, which should be clear from the introduction - the authors are not especially concerned again with advocating preferred techniques, but rather would want to emphasise strongly the need for practically successful methods ahead of supposed theoretical properties. Here practically successful implies not only that good performance is yielded under realistic conditions but also that the methods should be straightforward in operation as perceived by the user. A further development in the subject area that the authors feel could make a contribution to many algorithms is the use of the delta-operator in place of the backward shift operator commonly employed in formulating self-tuning algorithms. Numerical properties (which again have not been discussed here ) are significantly enhanced by the use of this device, as also are more pure control propertie~ of the algorithms obtained (e g. Goodwin et al 1986 ) . A second interesting area of possible future development is the introduction of non-linear methods into self-tuning. This should as it were take the presure off the estimator in the self-tuning algorithm in tracking non-linear effects. Initial moves in this direction ha ve been taken by Agarwhal and Seborg ( 1985 ) and Grimble (19 86b ). The packaging of the industrial self-tuner that is produced IS of course also of very grpat importance. Here the experience gained with the simple loop tuning de v ices now on the market could also be integrated into the controller. Ultimately, self-tuners utilising the kind of ideas described in this paper may be commonplace on industrial sites where they will be seen as hardwhere black boxes performing alongside (o r perh~ps instead of) the similar PlO devices currently so familiar.

FIGURES

ZII.III

IQ. III Q.III -IUQ

Fig. l (a) Impli cit Identification

ZII.III IQ.III Q.III -IQ.III

11.111

-4Q .1II

ZII.III

611.111

811.111

tlll.1II

Fig. l(b) Explicit Identification

AUTO-TUNING FACTOR-2

I~

..

~.

10

JI'

IV' .....

6

10

12

14

rr

11

XlOI

Fig. 2(a) Output and Setpoint XIO- 1 10.,-~~

AUTO-TUNING OF LAMBDA ____________________________________ __

CONCL US IONS In this paper,the authors have called for a shift towards the development of globally industrially suitable techniques in the self-tuning field. Their belief is that the full potential of 'elf-tuning cannot be realised unless the method is made more available as a tool for low cost automation and ceases to be just a specialised method for irregular use. The y have suggested that the selftuning sustem obtainDd as an end product is more important than the choice of algorithm. To Illustrate the approach advocated the GMV self-tuning controller has been considered, and some steps that would take this particular algorithm closer the specification put forward in the paper are proposed.

10

12

rr Fig. 2(b) The Control Weight XIO-'

AUTO-TUNING GMV Q- (1-2 ~',-------------------------------------~

II

ACKNOWLEDGEMENTS AND NOTE The authors would like to thank the Industrial Control Unit and the Department of Mathematics of the University of Strathclyde for the use of their facilities; they would also like to thank Miss J. Wilkie for her help with the preparation of the paper. Correspondence should be sent to the first named author at Industrial Control Unit, University of Strathclyde, Royal College, 204 George Street, Glasgow, Gl lXW, United Kingdom.

10

rr

12

XI02

Fig. 3(a) Non-Linear system, Factor 2.0

R. W. lanes and R . P. D. Baker

12 XIO- I 16.~

AUTO-TUNING OF LAMBDA ____________________________________- ,

GMV Q-0.1( 1-z- l ) KINC NA z 1

2t

U 15

12

•

10 8

S

--..

•

ft

~

ftr ".

..n

2;

1/\

-

~

A. cV'

I

V T}

.

.

./\

[/\

D-

O

10

0

Fig . 3(b) The Control Weight lUO-'

J2

12

I.

2t

15

Xl02

a,

Fig. 5(b) Under Parame teri sati on

Factor 2 . 0 XIO- I

AUTO-TUNING GMV Q- (1-% -1)

LG1G CONTROL

31

JI

R -1.9. 1st. ORDER MODEL

JO

JO ~

~

28 28 27

27 28

28

12

10

10

15

20

XI0 2

0-

a-

Fig. 6(a)

Fig. 4 (a) No n-Linear system , Factor 3 . 0

40

JII

a-Y

XIO I

1st Order Model

XIO- 1

XIO- 1

AUTO-TUNING OF LAMBDA 167-____________________________________ -,

LOG CONTROL .9.2nd ORDER MODEL JI:r---------__________ :.....-__________

R-'

-=__---.

u JO

12 10 28 27

28·~~~~TnTnnT~~~~TnTnTn~nT~~·~~~~~~S~ 10

10

12

15

XIO I

Fig. 6(b)

Fig. 4 (b) The Control ',eight, Factor 3.0 20

2nd Order Model

XIO- 1

GMV Q~kl.' ( 1-z -\) KINC NA=2

LOG CONTROL R -1 • kl. 3rd ORDER MODEL

31.~----------------------------------~

15

10

40

JII

Xl02

JO

la

.L rvor

1\

rv

V

~

27

V

V

1/ 28,~nT~~TnTnTn~nTnT~TMTnTn~~~~·~~~~~~~~rrl

10

15

20

10

15

25

JO

0-lU

Fig . 5(a) Correct Parameterisation

Fig. 6(c)

3rd Order r10del

40

JII XIO I

Se lf-tuning Co ntro l: Tow ards Industri a l Viabilit y

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14

R . W . la nes and R . P . D. Baker

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