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Extended Self-Tuning - Practical Aspects
Copyright © IFAC Identification and System Parameter Estimation 1982. Washington D.C .. USA 1982
EXTENDED SELF-TUNING
PRACTICAL ASPECTS
S. P. Sanoff and P . E. Wellstead Control Systems Centre, University of Manchester Institute of Science and Technology, P. O. Box 88, Manchester M60 1 QD, UK
Abstract. The paper discusses an extended self-tuninR property which generalises previous self - tuning results by employing filtered data during parameter estimation. It is shown that the choice of filter has reper cussions on the computational efficiency, convergence and set-point tracking characteristics of the resulting algorithms. The argument is developed in terms of pole-assignment and is supported by computer demonstrations and discussion of practical issues. Keywords. Self-tuning regulators; stochastic control; digital control; parameter estimation; models; time-lag systems. form a modified system model .
1. INTRODUCTION In their early papers, Peterka (1970), and Astrom and Wittenmark (1973) discussed the on - line combination of a minimum variance control rule and a recursive least squares algorithm. It was shown that the combination of these algorithms could converge to the closed-loop configuration that would have been obtained by off - line design: This asymptotic feature of an adaptive algorithm was termed the 'self-tuning property ' and led to significant research effort into ways whereby other parameter estimators and control synthesis methods could be combined. Inparticular, other forms of the linear quadratic regulator algorithm were shown to 'self-tune' (Clarke and Gawthrop,1975,1979). By the same token, complementary classical control methods based around pole - zero assignment were shown to have the self-tuning property, (IJellstead and co-workers,1979 .; Prager and Well stead ,1981)
The object of this paper is to consider the apparen t ambigui t ies in the modellin5 of noise in self - tuning sy s tems and investigate the connec t ion with data reticulation (or so called impl i ci t ) model s in var ious self tuning schemes. Specifically, consider the conventional difference equation model of a single input/ single output sampled system given by Ay(t)
=
Bu(t)+Ce(t)
(1)
where A, B,C, are polynomials in z-l(the backward shift operator) , y(t), u(t) are respectively the output and input sequences of the sampled system and e(t) is a white noise sequence . In the references (Astrom and Wittenmark~973; Clarke and Gawthrop,1975) the system is rearranged into an implicit model of a form which allows direct estimation of the controller parameters. However , this is not possible with the self-tuner in references (lvellstead and Colleagues,1979; Prager and Wellstead,1981) indicating a distinctive operational difference between techniques .
Now, in terms of the convergence theory for adaptive algorithms, the self-tuning property is a relatively weak resul t. Nonetheless, it has the great merit of being easily under stood and readily applied. Indeed, numerous industrial applications of self-tuning have been made, only a fraction of which have appeared in the literature. In fact, selftuning methods (despite their formal limitations) are in the process of making the difficult transition from theory to industrial practice with remarkable speed.
Corresponding differences occur in the modcll ing of the noise process . In references (Peterka,1970; Wellstead and others,1979)the noise model is set arbitrarily equal to unity, with self - tuning to the correct configuration still applying . On the other hand, in (Cla~ and Gawthrop , 1975) C (the noise model in equation (1» is estimated directly, again with a self - tuning result .
With this transition in mind, we reflect upon two important practical aspects of selftuning of stochastically disturbed systems, namely: • the modelling of process noise, • the reticulation of the available data to
This last point suggests that the assumption that C = 1 in the self - tuning principle (Astrom and Wittenmark,1973; Wellstead and 1175
S. P. Sanoff and P. E. We llstead
1176
co-workers,l 979) is ac tuall y a spe c ial case of a more ge neral self-tunine prope rt y. This turns out t o be so, and the generalisation (termed in (Wellstead and Sanoff ,198l) the ' extended self-tunin g property') thr ows li gh t upon a number of important practical i ssues which include re~
•
combined servo con tr o l and stochastic ulation by po le-assi gnment
•
the uses of additional filterin g in selftunin g lo op s
•
implicit f orms o f pole-assignment re gu lation.
The current paper explores the practi ca l con sequen ce s o f extended self-tuning accordinp, t o the foll owin g plan. Se c tion 2 dis cusse s the basis o f the extended self-tunin g pr operty from a filtering viewpo int. Se c ti on 3 developes the combined servo and st oc hasti c self-tuning contro ller from the extended property, while Se c tion 4 indi ca tes how extended self -tuning fa c ilitates implicit po le-assi gn ment self -tunin g regulation. Se c ti on 5 iliustrates these and other matters in terms of computer simulati ons. 2 . EXTENDED SELF-TUNING PROPERTY
~(t) = Xu(t)
(2)
where X is a rational discrete time filter. The incorporation of filters in this manner ties in with techniques used in recursive estimation (Young,1976) and is a self-tunin g analogue of the auxiliary signals procedure of adaptive control(Lozano and Landau,198l) . Thus, if we wished to use estimators o ther than recursive least square(RLS), or t o build a theoretical link with adaptive control, then the schema of Fig.l is a reasonable starting point. The interesting point is, however,that by working thr ough the self-tuning lemma (e.g. appendix to(Wellstead,Prager and Zanker,1979» it is easily shown that only filters X of a highly restrictive form may be used to selftune stochastically disturbed systems defined by equation (1). In fact, the filter X must be a moving average process with order limited by the inequality, deg(T)+deg(X) < deg(A)+deg(B)-deg(C)-l
Cons id e r first the usual pole - assignment self tuning proof applied to the system described by equation (1). Here it is assumed that any time delay i n the underlying system ismodell ed by leading zeros in the B polynomial so that unknown and time varying transport dewys may be accommodated . Partial time delays are implicitly catered for in the pole - assignment c rit erion since open - loop zeros are not cance ll ed. I t is then assumed that the system of equat i on (1) is modelled at each sample interval by Ay(t) = Bu(t) +
ml)
where deg(T) = desired number of closed loop poles. Thus, we have the somewhat negative result that data prefiltering can only be used under limited circumstances. By the same token it implies that stochastic (RLS based) versions of adaptive controllers (e.g. Lozano and Landau,1981) would not ~elf-tune'. This disappointing conclusion is mitigated if one
~( t)
(3)
where A, B are biased recursive least squares estimates of the underlying systempolynom ia l ~ and the biasing arises because we ignore the no is e colorat i on polymon i al C. ;( t) is a l east squa r es fitting residual with the usua l orth ogonality conditions between it and the dat a y(t) , u(t). The r egulator polynomials F,G in Fu ( t) = Gy ( t)
Consider first the self-tuning pole-assi gnment schema shown in Fig.l. This is the convent ional self-tuning arrangement except that data filters have been included to modify the system input/output data, thus yet) = Xy(t)
conside r s a self - tuning scheme whereby the filters are qanqed through to the pole - assign ment synthesizer (PAS) . This conf iguration is shown in Fig.2 and leads to an extended self-tuning property whe r eby filtering of a gene r al and very useful form is al lowed.
(4 )
ar e then determined at each sample interval by solving FA + BG
=
T
(5)
where deg( F)=d eg(B) - l, deg(G) =d eg(A) -l and degeT) < deg(A)+deg(B)-deg(C) - l, and T is the desi r ed clos ed l oop characteristic polynomial. I f the system converges , then it can easily be shown,by substituting the regulator (equation (4» i n t o the model (equation (3» and usin g the identity (equation (5» , that TyC t) = Fqt)
(6)
That is to say, the model has its poles correctly placed at the zeros of T , which repr esen ts the desired closed -l oop character isti c polynomial . A projection argument (Wellstead and co-workers ,1 979; Pr ager and Wellst ead ,1 981) is then deployed to show that ~( t)
=
e( t )
(7)
Hence Ty( t) = Fe(t) and s o the desired closed -l oop pole set is attained for the system (equation 1). The point being that"even th ough the incorrect assumpti on that C = 1 is made i n the model (equation 3) the co rre ct closed - loop conf i gurati on can be attained. This is the class ical self-tuning property wh i ch complements and is derivative of the optimal version i n (Astrom and Wittenmark,1 973) . Howeve r, as mentioned earlier , Clarke and Gaw thr op( 1975) implicitly estimate C and ye t still attain a 'self-tuning' state . This, t oge the r with the prev i ous filterin g discussion, t empts one to seek an extended self-tuner in whi ch C mayor may not be estimated. To see how this
1177
Extended Self-Tuning - Practical Aspects might function, consider the following argument based on Fig . 2.
In this instance A and C will share a common factor at z = 1.)
In the figure the familiar self-tuning schema is shown, rearranged to accommodate an addit ional filtering action on the observed data u(t), y(t), so that the filtered signals u(t). yet) are used in the RLS estimator, where the filtering action is now associated with a surrogate noise polynomial C , thus
In such cases the use of surrogate nOise filtering may be useful, since it allows . in the regulation case a de facto filtering to be affected. Thus, for example , a fixed filter C can be selected with zeros near the point 1,0 and used in the extended selftuning algorithm to mitigate possible conver gence problems. The simulations section indicates the use of this idea.
Cy(t)
=
Cu(t)
yet)
=
u(t)
(R)
The choice and naming of this filter becomes clear if one considers the resulting recursive least squares model Ay(t)
=
Bu(t) + Ut)
(9)
Clearly, the regulator rule (equation (4)) can be rewritten in terms of filtered variables, Fu (t)
(l0)
Gy(t)
and combined with equation (9) to give the closed loop model description (AF+BG)y(t)
=
F~(t)
(11)
Now if the modified identity AF + BG
=
TC
(12)
is used then the surrogate filter appears as a common factor and the relation (equation (6)) occurs as required . Thus, the extended self-tuning property arises which tells us that we can model the noise process C by any surrogate (denoted here as C)and still have the desired closed-loop con figuration. From a philosophical viewpoint the extended self-tuning property resolves the ambiguity bet\veen Clarke and Gawthrop IS self-tuner (in which C is estimated) and others which assume C = 1. By the same toke~ because of the general structure of the filter C, the original requirement of building data prefilters into the loop (as in Fig.l) is fulfilled. However, there are other deeply practical implications of extended self-tuni~ which we now explore. In particular. we con sider : (a)
composite servo and regulator self-tuning with a pole-assignment objective,
(b)
surrogate noise models,
(c)
implicit self-tuning regulators.
Item (b) is in the manner of an informal suggestion and runs as follows. It has been noted in the stochastic convergence literature that the nature of the noise dynamics as embodied in C influence the convergence or otherwise of an adaptive algorithm. Likewise. application tells us that a noise process with a zero near the point 1 . 0 in the z-plane causes practical problems. (This is especially true when integral action in the form of incremental control is built into the loop. ISPE- 2-L*
3. SELF-TUNING REGULATION AND SERVO CONTROL It has become clear (Clarke and Gawthrop ,1 97~ that , without prior knowledge or implied est imation of C in equation (1), it is not possible to obtain a self -tuning algorithm for combined servo tracking and stochastic regulation with a pole-assignment objective . As a result, compl ementar y pole-assignment self-tuners have evolved for stochastic regulation (Wellstead and co - workers,1979;Prager and \.Jellstead,198l) and deterministic servo tracking(Astrom and Wittenmark,1980).Although in practice eithe r of these dual approaches works acceptably on the composite problem, we are led to seek a clean, theoretically justified solution whi ch in co rporates explicit estimation of C (Allidina and Hughes,1980) . The extended self-tuner shows how this may be done by employ in g a recursive extended least squares (RELS) (Soderstrom and others ,1974) algorithm t o self-tune the coefficients of controller polynomials F, G, H in Fig.3. The self-tuning of this system runs as follows . •
Estimate A, B, C, using RELS in the model A yet)
•
=
Bu(t) +
(13)
C~(t)
Synthesize the controller polynomial coefficients from the identity FA + GB H
= CT
(l4a)
CT( 1) (l4b)
B(l)
•
Apply control Fu (t)
(l5)
= -Gy (t) + HYr (t)
The nume r ator factor C in equation (14b) cancels the noise dynamics from tte closedloop servo -r esponse and scalars B(l), T(I) ensure zero steady state tracking error for step inputs l where B(l)
T(1) = TI
z=1
BI Iz=1
(J 6)
Known (or "factorizable") zeros of B which are inverse stable can be cancelled and new desirable zeros introduced as z~ros of H (cf . Astrom and Wittenmark,19RO). However, lUith more dynamics in H, steady state errors fc>r ramp and parabol i c inputs can be zeroed.
S. P. Sanoff and P. E . Wellstead
1178
our practical experience is that discrete time zeros of sampled data systems cannot usually be cancelled with confidence . In particular, sampled data zeros are mobile, being functions of partial time delays in the underlying system (see the discussion in (Wellstead, Prager and Zanker,1979;). The algorithm of equations (13) through (16) uses the extended self-·tuning principle in the sense that the surrogate noise filter c is replaced by an estimate ~ of the true fikeG where C is used principally in the servo-, response shaping. Even if the estimates C are poor~ the regulation results will be good because C appears as a common factor . Likewise, experience has spown that the influence of poor estimates of C in the servo-response is not significant. It is also interesting to note that whe~ the underlying system is noise-free then C -+ 1 and a deterministic servo controller arises. The use of extended least squares, or other estimators which give unbiased estimates of system parameters, means that it is also straight-forward to incorporate feedforward compensators. However, to be effective these need an accurate knowledge of the system time delay. Estimators of system delays are the subject of current research.
where an implicit solution to equation (19) is FT-aB G
The coefficient scaling factor. •
(20)
o.A
1S an arbitrary polynomial
0.
Apply control Fy ( t ) ; Gu ( t )
(21)
If the system converges then the closed loop equation is Ty (t)
(T-CtB)e(t)
(22)
and is achieved without solving the identity (equation (5». The compromise made in return for this practical benefit is possible aggravated convergence problems (Hersh and Zarrop,198l). Nevertheless, in the selftuning spirit, the desired closed loop configuration (as measured by the desired pole set T) remains a possible convergence point. A potentially attractive feature of the implicit algorithm is that it offers control, albeit of a limited kind, over the closedloop zeros . EXAMPLES
5.
4. IMPLICIT SELF-TUNING The pole-assignment approach to self-tuning is robust and insensitive to certain system properties which are unique to sampled data systems (e.g. Wellstead, Prager and Zanker, 1979). However, the approach has been criticised for computational problems associated with solving the polynomial identity FA + BG ; T
(5)(bis)
By equating coefficients of like powers of z-l in equation (5) a consistent set of simultaneous equations arise which are solved by standard methods. However, a significant computational effort is required and the equations can be poorly conditioned (Clarke, . 1981). An implicit pole-assignment regulator algorithm which avoids the identity and concommitant numerical problems runs as follows, •
Select a surrogate noise model
c •
or
(17)
A
Estimate the coefficients of the model Ay (t)
Bu (t) + A£, (t)
Ay( t)
Bu (t)
(18)
where ~(t) ; yet) - E;(t) is the one step prediction of yet). •
Synthesize the regulator polynomial coefficients from ,.
,
FA + EG ; TA or
A(F-T) ; -BG
(19)
The simulated examples which follow demonstrate the three forms of the extended selftuning algorithm. Fixed Prefilter
(1)
This example shows the improvemeRt in convergence possible when the filter C is chosen appropriately. To make convergence difficult, the system -2 -3 -1 -2 (1-1.2 z +0.35 z )y ; (0 . 5 z -0.8 z )u +(1-0.99 z-l)e was simulated, C(z-l) having a root almost on the unit circle in the z-plane. Figure 4 compares the regulator parameters obtained with Cl; 1 and C 2 ; 1-0.95 z-l Figure 5 compares the difference E;(t)-e(t), whlch sho'lld tend to zero if the self-tuner converges . Note that C corresponds to the algorithm given in (I.,retlstead, Prager and Zanker, 1979) and that C was chosen because of its 2 closeness to C. In practice C is not known and techniques for the selection of C 1n such cases are to be investigated. Lastly, it should be noted that when computer storage is limited, the effect of filtering can be achieved without additional coding. The usual RELS algorithm is employed with the initial conditions of C being the desired C and the autocovariance of C set to zero to prevent its updating. Implicit Algorithm
(2)
The aim here was to design an implicit regulator for the non-minimum phase system
Extended Self-Tuning - Practical Aspects (1-0.2 z -l)y = (0.5 z-2 + z-3)u + O - O.B z-l)e Fig.6 shows the self - tuned parameters, with a , as defined in equation (20), equal to unity. It has been found that the convergence of the algorithm is greatly affected by the openloop poles. This leads us to believe that the convergence properties normally associated with the ratio C/C now hinge in some way on the positive realness of A by virtue of the model structure (lB) and is being further investigated. Simulations have sh.own that the algorithm will fail when A has unstable or marginally stable zeros . Consequently , an integrator cannot be introduced into the loop to handle offsets or servo-command following , as in the explicit regulator (Wells t ead and co - workers,1979). Deterministic Controller
(1-1.5z
+ 0 .7 z
-2
- 1 -2 )y = (1.2z +1.4z )u + 1.0
Note that it is subject to an offset and no disturbance. The open-loop poles a t z = 0.75 0 . 37j give rise to an oscillatory response but by choosing to shift these to z = 0.72 ± 0.2j the damping is improved (see Fig.7a). Figures 7b and 7c show that using RELS identification in the presence of no disturbances produces rapid convergence to unbiased est imates and consequently good control is achieved quickly.
=
Combined Regulation and Control
(4)
In this example, a controller was designed for the system (1 - 1.3z- 1 + 0.45z- 2 )y = z- K(z - 1+2z-2)u + (l - O.Bz - l)e + 3. with K = 0 up to t=1500 and K=l thereafter. Fig.B shows the parameter estimates and the system output. Note how the algorithm copes with the unstable roots of B and how insensitive it is to a change in time delay. Effect of \.Jrong Model Order
(5)
This last example concerns the system (1 - 1.2z
-1
+0.6 z +0.56z- 2 )e
-2
)y =(z
-1
+O.Bz
-2
)u +(1-1.5z
-1
Fig.9 illustrates the e~fect of modellin~this with deg(A) = 2, deg(B) =2, but deg(C) =1 " deg(C). Notice that despite the fact that C cannot cancel C in the servo transfer function, good control is still possible . Interestingly, the estimator used in this case was not P~LS but Recursive Maximum Likelihood (Soderstrom and co - workers , 1974) . This uses C as a data filter to improve convergence in certain pathologi~al systems, but again the wrong order of C has not prevented its successful application in this context.
6.
This paper has considered an extended selftuning property whereby a simple least squarffi model may be estimated on-line and combined with a control law to yield desired closed loop poles. The enhancement over previously reported work (Wellstead and co-workers,1979) arises through prefiltering the data used in the estimator. By employing the same filter in the synthesis of the regulator, the bias in the estimates has no effect on the final design. The filter may be chosen to improve convergence or, in s ome cases , to simplify the calculation of the regulator parameters. Moreover , it has been shown how control and regulation may be combined with only a marginal increase in computation.
(3)
We now turn our attention to the system gov erned by -1
1179
CONCLUSIONS
ACKNOWLEDGEMENT This work was supported by Science Research Council Grants GR/A/B035B and GR/B/799Bl. REFERENCES Allidina, A.Y., and Hughes,F.M. (19BO). Generalised Self-Tuning Controller with PoleAssignment. Proc.IEE, 127 . 1, pp . 13 - lB . Astrom, K.J., and Wittenmark, B. (1973). On Self-Tuning Regulators. Automatica , 9,pp. IB5-199. Astrom, K.J., and Wittenmark, B. (19BO). Self-Tuning Controllers Based on PoleZero Placement. Proc.IEE,127.3, pp.120-DU Clarke, D.W. (19Bl). Model-Following and Pole-Placement Self-Tuners. Proc . 3rd Conf . on Organisation & Automation of Experimental Research, Rousse, Bulgaria. Clarke, D.W., and Gawthrop , P.J . (1975).Self Tuning Controller. Proc .IEE,122 . 9,pp.929934. -Clarke, D.W., and Gawthrop, P.J. (1979) . Self Tuning Control. Proc.IEE,126.6,pp.6 33~O Hersh, M.A., and Zarrop, M.B.--cI9Bl). Stochastic Adaptive Control of Nonminimum Phase Systems . Control Systems Centre Report 524, UMIST. Lozano. R., and Landau, I. D. (19Bl). Redesign of Explicit and Implicit Discrete Time Model Reference Adaptive Control Schemes. Int. J. Control,33.2, pp.247-26B. Peterka, V. (1970). Adaptive Regulation of Noisy Systems. Proc.IFAC Symposium on Identification and Parameter Estimation, Prague. Prager, D.L., and Wellstead, P.E. (19Bl). Multivariable Pole-Assignment Self-Tuning Regulators. Proc.IEE,12B.l,pp.9-lB. Soderstrom , T., Ljung, L., and Gustavsson,I. (1974). A Comparative Study of p.ecursive Identificati on Methods. Report 7427,Lund Inst. of Technology, Dept.of Automatic Control. Wellstead, P.E., Edmunds, J.M., Prager, D. L., and Zanker, P.M. (1979) . Self-Tuning Pole/Zero Assignment Regulators. Int. J. Control, 30.1, pp.I-26. Well~teadi P.E' I P,ager, D.L., and Zanker,P.M . t1979)Pole asslgnment Selr-Tuning Regul-
S. P. Sanoff and P. E. Wellstead
1180
ator . Proc.IEE,126 . R, pp,781-787. Wellstead, P.E., and Sanoff, S.P. (1981). Extended Self-Tuning Algorithm . Int. J. Control, 34.3, pp . 433 - 455. Young, P.C. (1976). Some Observations on Instrumental Variable Methods of TimeSeries Analysis. Int. J . Control , 23.5, pp.593-6l2 .
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EXTENDED SELF-TUNING
PRACTICAL ASPECTS
S. P. Sanoff and P . E. Wellstead Control Systems Centre, University of Manchester Institute of Science and Technology, P. O. Box 88, Manchester M60 1 QD, UK
Abstract. The paper discusses an extended self-tuninR property which generalises previous self - tuning results by employing filtered data during parameter estimation. It is shown that the choice of filter has reper cussions on the computational efficiency, convergence and set-point tracking characteristics of the resulting algorithms. The argument is developed in terms of pole-assignment and is supported by computer demonstrations and discussion of practical issues. Keywords. Self-tuning regulators; stochastic control; digital control; parameter estimation; models; time-lag systems. form a modified system model .
1. INTRODUCTION In their early papers, Peterka (1970), and Astrom and Wittenmark (1973) discussed the on - line combination of a minimum variance control rule and a recursive least squares algorithm. It was shown that the combination of these algorithms could converge to the closed-loop configuration that would have been obtained by off - line design: This asymptotic feature of an adaptive algorithm was termed the 'self-tuning property ' and led to significant research effort into ways whereby other parameter estimators and control synthesis methods could be combined. Inparticular, other forms of the linear quadratic regulator algorithm were shown to 'self-tune' (Clarke and Gawthrop,1975,1979). By the same token, complementary classical control methods based around pole - zero assignment were shown to have the self-tuning property, (IJellstead and co-workers,1979 .; Prager and Well stead ,1981)
The object of this paper is to consider the apparen t ambigui t ies in the modellin5 of noise in self - tuning sy s tems and investigate the connec t ion with data reticulation (or so called impl i ci t ) model s in var ious self tuning schemes. Specifically, consider the conventional difference equation model of a single input/ single output sampled system given by Ay(t)
=
Bu(t)+Ce(t)
(1)
where A, B,C, are polynomials in z-l(the backward shift operator) , y(t), u(t) are respectively the output and input sequences of the sampled system and e(t) is a white noise sequence . In the references (Astrom and Wittenmark~973; Clarke and Gawthrop,1975) the system is rearranged into an implicit model of a form which allows direct estimation of the controller parameters. However , this is not possible with the self-tuner in references (lvellstead and Colleagues,1979; Prager and Wellstead,1981) indicating a distinctive operational difference between techniques .
Now, in terms of the convergence theory for adaptive algorithms, the self-tuning property is a relatively weak resul t. Nonetheless, it has the great merit of being easily under stood and readily applied. Indeed, numerous industrial applications of self-tuning have been made, only a fraction of which have appeared in the literature. In fact, selftuning methods (despite their formal limitations) are in the process of making the difficult transition from theory to industrial practice with remarkable speed.
Corresponding differences occur in the modcll ing of the noise process . In references (Peterka,1970; Wellstead and others,1979)the noise model is set arbitrarily equal to unity, with self - tuning to the correct configuration still applying . On the other hand, in (Cla~ and Gawthrop , 1975) C (the noise model in equation (1» is estimated directly, again with a self - tuning result .
With this transition in mind, we reflect upon two important practical aspects of selftuning of stochastically disturbed systems, namely: • the modelling of process noise, • the reticulation of the available data to
This last point suggests that the assumption that C = 1 in the self - tuning principle (Astrom and Wittenmark,1973; Wellstead and 1175
S. P. Sanoff and P. E. We llstead
1176
co-workers,l 979) is ac tuall y a spe c ial case of a more ge neral self-tunine prope rt y. This turns out t o be so, and the generalisation (termed in (Wellstead and Sanoff ,198l) the ' extended self-tunin g property') thr ows li gh t upon a number of important practical i ssues which include re~
•
combined servo con tr o l and stochastic ulation by po le-assi gnment
•
the uses of additional filterin g in selftunin g lo op s
•
implicit f orms o f pole-assignment re gu lation.
The current paper explores the practi ca l con sequen ce s o f extended self-tuning accordinp, t o the foll owin g plan. Se c tion 2 dis cusse s the basis o f the extended self-tunin g pr operty from a filtering viewpo int. Se c ti on 3 developes the combined servo and st oc hasti c self-tuning contro ller from the extended property, while Se c tion 4 indi ca tes how extended self -tuning fa c ilitates implicit po le-assi gn ment self -tunin g regulation. Se c ti on 5 iliustrates these and other matters in terms of computer simulati ons. 2 . EXTENDED SELF-TUNING PROPERTY
~(t) = Xu(t)
(2)
where X is a rational discrete time filter. The incorporation of filters in this manner ties in with techniques used in recursive estimation (Young,1976) and is a self-tunin g analogue of the auxiliary signals procedure of adaptive control(Lozano and Landau,198l) . Thus, if we wished to use estimators o ther than recursive least square(RLS), or t o build a theoretical link with adaptive control, then the schema of Fig.l is a reasonable starting point. The interesting point is, however,that by working thr ough the self-tuning lemma (e.g. appendix to(Wellstead,Prager and Zanker,1979» it is easily shown that only filters X of a highly restrictive form may be used to selftune stochastically disturbed systems defined by equation (1). In fact, the filter X must be a moving average process with order limited by the inequality, deg(T)+deg(X) < deg(A)+deg(B)-deg(C)-l
Cons id e r first the usual pole - assignment self tuning proof applied to the system described by equation (1). Here it is assumed that any time delay i n the underlying system ismodell ed by leading zeros in the B polynomial so that unknown and time varying transport dewys may be accommodated . Partial time delays are implicitly catered for in the pole - assignment c rit erion since open - loop zeros are not cance ll ed. I t is then assumed that the system of equat i on (1) is modelled at each sample interval by Ay(t) = Bu(t) +
ml)
where deg(T) = desired number of closed loop poles. Thus, we have the somewhat negative result that data prefiltering can only be used under limited circumstances. By the same token it implies that stochastic (RLS based) versions of adaptive controllers (e.g. Lozano and Landau,1981) would not ~elf-tune'. This disappointing conclusion is mitigated if one
~( t)
(3)
where A, B are biased recursive least squares estimates of the underlying systempolynom ia l ~ and the biasing arises because we ignore the no is e colorat i on polymon i al C. ;( t) is a l east squa r es fitting residual with the usua l orth ogonality conditions between it and the dat a y(t) , u(t). The r egulator polynomials F,G in Fu ( t) = Gy ( t)
Consider first the self-tuning pole-assi gnment schema shown in Fig.l. This is the convent ional self-tuning arrangement except that data filters have been included to modify the system input/output data, thus yet) = Xy(t)
conside r s a self - tuning scheme whereby the filters are qanqed through to the pole - assign ment synthesizer (PAS) . This conf iguration is shown in Fig.2 and leads to an extended self-tuning property whe r eby filtering of a gene r al and very useful form is al lowed.
(4 )
ar e then determined at each sample interval by solving FA + BG
=
T
(5)
where deg( F)=d eg(B) - l, deg(G) =d eg(A) -l and degeT) < deg(A)+deg(B)-deg(C) - l, and T is the desi r ed clos ed l oop characteristic polynomial. I f the system converges , then it can easily be shown,by substituting the regulator (equation (4» i n t o the model (equation (3» and usin g the identity (equation (5» , that TyC t) = Fqt)
(6)
That is to say, the model has its poles correctly placed at the zeros of T , which repr esen ts the desired closed -l oop character isti c polynomial . A projection argument (Wellstead and co-workers ,1 979; Pr ager and Wellst ead ,1 981) is then deployed to show that ~( t)
=
e( t )
(7)
Hence Ty( t) = Fe(t) and s o the desired closed -l oop pole set is attained for the system (equation 1). The point being that"even th ough the incorrect assumpti on that C = 1 is made i n the model (equation 3) the co rre ct closed - loop conf i gurati on can be attained. This is the class ical self-tuning property wh i ch complements and is derivative of the optimal version i n (Astrom and Wittenmark,1 973) . Howeve r, as mentioned earlier , Clarke and Gaw thr op( 1975) implicitly estimate C and ye t still attain a 'self-tuning' state . This, t oge the r with the prev i ous filterin g discussion, t empts one to seek an extended self-tuner in whi ch C mayor may not be estimated. To see how this
1177
Extended Self-Tuning - Practical Aspects might function, consider the following argument based on Fig . 2.
In this instance A and C will share a common factor at z = 1.)
In the figure the familiar self-tuning schema is shown, rearranged to accommodate an addit ional filtering action on the observed data u(t), y(t), so that the filtered signals u(t). yet) are used in the RLS estimator, where the filtering action is now associated with a surrogate noise polynomial C , thus
In such cases the use of surrogate nOise filtering may be useful, since it allows . in the regulation case a de facto filtering to be affected. Thus, for example , a fixed filter C can be selected with zeros near the point 1,0 and used in the extended selftuning algorithm to mitigate possible conver gence problems. The simulations section indicates the use of this idea.
Cy(t)
=
Cu(t)
yet)
=
u(t)
(R)
The choice and naming of this filter becomes clear if one considers the resulting recursive least squares model Ay(t)
=
Bu(t) + Ut)
(9)
Clearly, the regulator rule (equation (4)) can be rewritten in terms of filtered variables, Fu (t)
(l0)
Gy(t)
and combined with equation (9) to give the closed loop model description (AF+BG)y(t)
=
F~(t)
(11)
Now if the modified identity AF + BG
=
TC
(12)
is used then the surrogate filter appears as a common factor and the relation (equation (6)) occurs as required . Thus, the extended self-tuning property arises which tells us that we can model the noise process C by any surrogate (denoted here as C)and still have the desired closed-loop con figuration. From a philosophical viewpoint the extended self-tuning property resolves the ambiguity bet\veen Clarke and Gawthrop IS self-tuner (in which C is estimated) and others which assume C = 1. By the same toke~ because of the general structure of the filter C, the original requirement of building data prefilters into the loop (as in Fig.l) is fulfilled. However, there are other deeply practical implications of extended self-tuni~ which we now explore. In particular. we con sider : (a)
composite servo and regulator self-tuning with a pole-assignment objective,
(b)
surrogate noise models,
(c)
implicit self-tuning regulators.
Item (b) is in the manner of an informal suggestion and runs as follows. It has been noted in the stochastic convergence literature that the nature of the noise dynamics as embodied in C influence the convergence or otherwise of an adaptive algorithm. Likewise. application tells us that a noise process with a zero near the point 1 . 0 in the z-plane causes practical problems. (This is especially true when integral action in the form of incremental control is built into the loop. ISPE- 2-L*
3. SELF-TUNING REGULATION AND SERVO CONTROL It has become clear (Clarke and Gawthrop ,1 97~ that , without prior knowledge or implied est imation of C in equation (1), it is not possible to obtain a self -tuning algorithm for combined servo tracking and stochastic regulation with a pole-assignment objective . As a result, compl ementar y pole-assignment self-tuners have evolved for stochastic regulation (Wellstead and co - workers,1979;Prager and \.Jellstead,198l) and deterministic servo tracking(Astrom and Wittenmark,1980).Although in practice eithe r of these dual approaches works acceptably on the composite problem, we are led to seek a clean, theoretically justified solution whi ch in co rporates explicit estimation of C (Allidina and Hughes,1980) . The extended self-tuner shows how this may be done by employ in g a recursive extended least squares (RELS) (Soderstrom and others ,1974) algorithm t o self-tune the coefficients of controller polynomials F, G, H in Fig.3. The self-tuning of this system runs as follows . •
Estimate A, B, C, using RELS in the model A yet)
•
=
Bu(t) +
(13)
C~(t)
Synthesize the controller polynomial coefficients from the identity FA + GB H
= CT
(l4a)
CT( 1) (l4b)
B(l)
•
Apply control Fu (t)
(l5)
= -Gy (t) + HYr (t)
The nume r ator factor C in equation (14b) cancels the noise dynamics from tte closedloop servo -r esponse and scalars B(l), T(I) ensure zero steady state tracking error for step inputs l where B(l)
T(1) = TI
z=1
BI Iz=1
(J 6)
Known (or "factorizable") zeros of B which are inverse stable can be cancelled and new desirable zeros introduced as z~ros of H (cf . Astrom and Wittenmark,19RO). However, lUith more dynamics in H, steady state errors fc>r ramp and parabol i c inputs can be zeroed.
S. P. Sanoff and P. E . Wellstead
1178
our practical experience is that discrete time zeros of sampled data systems cannot usually be cancelled with confidence . In particular, sampled data zeros are mobile, being functions of partial time delays in the underlying system (see the discussion in (Wellstead, Prager and Zanker,1979;). The algorithm of equations (13) through (16) uses the extended self-·tuning principle in the sense that the surrogate noise filter c is replaced by an estimate ~ of the true fikeG where C is used principally in the servo-, response shaping. Even if the estimates C are poor~ the regulation results will be good because C appears as a common factor . Likewise, experience has spown that the influence of poor estimates of C in the servo-response is not significant. It is also interesting to note that whe~ the underlying system is noise-free then C -+ 1 and a deterministic servo controller arises. The use of extended least squares, or other estimators which give unbiased estimates of system parameters, means that it is also straight-forward to incorporate feedforward compensators. However, to be effective these need an accurate knowledge of the system time delay. Estimators of system delays are the subject of current research.
where an implicit solution to equation (19) is FT-aB G
The coefficient scaling factor. •
(20)
o.A
1S an arbitrary polynomial
0.
Apply control Fy ( t ) ; Gu ( t )
(21)
If the system converges then the closed loop equation is Ty (t)
(T-CtB)e(t)
(22)
and is achieved without solving the identity (equation (5». The compromise made in return for this practical benefit is possible aggravated convergence problems (Hersh and Zarrop,198l). Nevertheless, in the selftuning spirit, the desired closed loop configuration (as measured by the desired pole set T) remains a possible convergence point. A potentially attractive feature of the implicit algorithm is that it offers control, albeit of a limited kind, over the closedloop zeros . EXAMPLES
5.
4. IMPLICIT SELF-TUNING The pole-assignment approach to self-tuning is robust and insensitive to certain system properties which are unique to sampled data systems (e.g. Wellstead, Prager and Zanker, 1979). However, the approach has been criticised for computational problems associated with solving the polynomial identity FA + BG ; T
(5)(bis)
By equating coefficients of like powers of z-l in equation (5) a consistent set of simultaneous equations arise which are solved by standard methods. However, a significant computational effort is required and the equations can be poorly conditioned (Clarke, . 1981). An implicit pole-assignment regulator algorithm which avoids the identity and concommitant numerical problems runs as follows, •
Select a surrogate noise model
c •
or
(17)
A
Estimate the coefficients of the model Ay (t)
Bu (t) + A£, (t)
Ay( t)
Bu (t)
(18)
where ~(t) ; yet) - E;(t) is the one step prediction of yet). •
Synthesize the regulator polynomial coefficients from ,.
,
FA + EG ; TA or
A(F-T) ; -BG
(19)
The simulated examples which follow demonstrate the three forms of the extended selftuning algorithm. Fixed Prefilter
(1)
This example shows the improvemeRt in convergence possible when the filter C is chosen appropriately. To make convergence difficult, the system -2 -3 -1 -2 (1-1.2 z +0.35 z )y ; (0 . 5 z -0.8 z )u +(1-0.99 z-l)e was simulated, C(z-l) having a root almost on the unit circle in the z-plane. Figure 4 compares the regulator parameters obtained with Cl; 1 and C 2 ; 1-0.95 z-l Figure 5 compares the difference E;(t)-e(t), whlch sho'lld tend to zero if the self-tuner converges . Note that C corresponds to the algorithm given in (I.,retlstead, Prager and Zanker, 1979) and that C was chosen because of its 2 closeness to C. In practice C is not known and techniques for the selection of C 1n such cases are to be investigated. Lastly, it should be noted that when computer storage is limited, the effect of filtering can be achieved without additional coding. The usual RELS algorithm is employed with the initial conditions of C being the desired C and the autocovariance of C set to zero to prevent its updating. Implicit Algorithm
(2)
The aim here was to design an implicit regulator for the non-minimum phase system
Extended Self-Tuning - Practical Aspects (1-0.2 z -l)y = (0.5 z-2 + z-3)u + O - O.B z-l)e Fig.6 shows the self - tuned parameters, with a , as defined in equation (20), equal to unity. It has been found that the convergence of the algorithm is greatly affected by the openloop poles. This leads us to believe that the convergence properties normally associated with the ratio C/C now hinge in some way on the positive realness of A by virtue of the model structure (lB) and is being further investigated. Simulations have sh.own that the algorithm will fail when A has unstable or marginally stable zeros . Consequently , an integrator cannot be introduced into the loop to handle offsets or servo-command following , as in the explicit regulator (Wells t ead and co - workers,1979). Deterministic Controller
(1-1.5z
+ 0 .7 z
-2
- 1 -2 )y = (1.2z +1.4z )u + 1.0
Note that it is subject to an offset and no disturbance. The open-loop poles a t z = 0.75 0 . 37j give rise to an oscillatory response but by choosing to shift these to z = 0.72 ± 0.2j the damping is improved (see Fig.7a). Figures 7b and 7c show that using RELS identification in the presence of no disturbances produces rapid convergence to unbiased est imates and consequently good control is achieved quickly.
=
Combined Regulation and Control
(4)
In this example, a controller was designed for the system (1 - 1.3z- 1 + 0.45z- 2 )y = z- K(z - 1+2z-2)u + (l - O.Bz - l)e + 3. with K = 0 up to t=1500 and K=l thereafter. Fig.B shows the parameter estimates and the system output. Note how the algorithm copes with the unstable roots of B and how insensitive it is to a change in time delay. Effect of \.Jrong Model Order
(5)
This last example concerns the system (1 - 1.2z
-1
+0.6 z +0.56z- 2 )e
-2
)y =(z
-1
+O.Bz
-2
)u +(1-1.5z
-1
Fig.9 illustrates the e~fect of modellin~this with deg(A) = 2, deg(B) =2, but deg(C) =1 " deg(C). Notice that despite the fact that C cannot cancel C in the servo transfer function, good control is still possible . Interestingly, the estimator used in this case was not P~LS but Recursive Maximum Likelihood (Soderstrom and co - workers , 1974) . This uses C as a data filter to improve convergence in certain pathologi~al systems, but again the wrong order of C has not prevented its successful application in this context.
6.
This paper has considered an extended selftuning property whereby a simple least squarffi model may be estimated on-line and combined with a control law to yield desired closed loop poles. The enhancement over previously reported work (Wellstead and co-workers,1979) arises through prefiltering the data used in the estimator. By employing the same filter in the synthesis of the regulator, the bias in the estimates has no effect on the final design. The filter may be chosen to improve convergence or, in s ome cases , to simplify the calculation of the regulator parameters. Moreover , it has been shown how control and regulation may be combined with only a marginal increase in computation.
(3)
We now turn our attention to the system gov erned by -1
1179
CONCLUSIONS
ACKNOWLEDGEMENT This work was supported by Science Research Council Grants GR/A/B035B and GR/B/799Bl. REFERENCES Allidina, A.Y., and Hughes,F.M. (19BO). Generalised Self-Tuning Controller with PoleAssignment. Proc.IEE, 127 . 1, pp . 13 - lB . Astrom, K.J., and Wittenmark, B. (1973). On Self-Tuning Regulators. Automatica , 9,pp. IB5-199. Astrom, K.J., and Wittenmark, B. (19BO). Self-Tuning Controllers Based on PoleZero Placement. Proc.IEE,127.3, pp.120-DU Clarke, D.W. (19Bl). Model-Following and Pole-Placement Self-Tuners. Proc . 3rd Conf . on Organisation & Automation of Experimental Research, Rousse, Bulgaria. Clarke, D.W., and Gawthrop , P.J . (1975).Self Tuning Controller. Proc .IEE,122 . 9,pp.929934. -Clarke, D.W., and Gawthrop, P.J. (1979) . Self Tuning Control. Proc.IEE,126.6,pp.6 33~O Hersh, M.A., and Zarrop, M.B.--cI9Bl). Stochastic Adaptive Control of Nonminimum Phase Systems . Control Systems Centre Report 524, UMIST. Lozano. R., and Landau, I. D. (19Bl). Redesign of Explicit and Implicit Discrete Time Model Reference Adaptive Control Schemes. Int. J. Control,33.2, pp.247-26B. Peterka, V. (1970). Adaptive Regulation of Noisy Systems. Proc.IFAC Symposium on Identification and Parameter Estimation, Prague. Prager, D.L., and Wellstead, P.E. (19Bl). Multivariable Pole-Assignment Self-Tuning Regulators. Proc.IEE,12B.l,pp.9-lB. Soderstrom , T., Ljung, L., and Gustavsson,I. (1974). A Comparative Study of p.ecursive Identificati on Methods. Report 7427,Lund Inst. of Technology, Dept.of Automatic Control. Wellstead, P.E., Edmunds, J.M., Prager, D. L., and Zanker, P.M. (1979) . Self-Tuning Pole/Zero Assignment Regulators. Int. J. Control, 30.1, pp.I-26. Well~teadi P.E' I P,ager, D.L., and Zanker,P.M . t1979)Pole asslgnment Selr-Tuning Regul-
S. P. Sanoff and P. E. Wellstead
1180
ator . Proc.IEE,126 . R, pp,781-787. Wellstead, P.E., and Sanoff, S.P. (1981). Extended Self-Tuning Algorithm . Int. J. Control, 34.3, pp . 433 - 455. Young, P.C. (1976). Some Observations on Instrumental Variable Methods of TimeSeries Analysis. Int. J . Control , 23.5, pp.593-6l2 .
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