- Email: [email protected]
Approximate ARX Model Estimation for Jacketing Adaptive Systems
Copyright © IFAC Adaptive Systems in Control and Signal Processing. Budapest. Hungary, 1995
APPROXIMATE ARX MODEL ESTIMATION FOR JACKETING ADAPTIVE SYSTEMS
Miroslav Karny, Petr Nedoma, Josef Bohm, Alena Halouskova Inltitute of Information Theory and Automation Academy of Sciencu of the Czedl Republic Pod voddrend:ou vUi -I, 18108, Prague 8, Czedl Republic Fa:J:: +-11.1.66-11-1903, tel: +-11.1.66,413,411, e·mail: ,chool§utia.au.cz
Abstract. Recently, a. substa.ntial progress hu been a.chieved in a.ttempts to merge systematically va.nous .pieces of. informa.tion of different precision, compa.tibility a.nd origin. The developed techmque prOVIdes a. new tool needed for jacketing a.da.ptive predictors/controllers bued on LQG the pa.ra.digm. ~en a.pplied to th~ a.utoregressive model with exogenous va.na.bles (ARX) the method contributes to the solutIon of the following tub: - incorpora.~on of prior knowledge into initial conditions of the recursive leut squa.res; - ~nstruction of a. reference for a.n a.dva.nced forgetting technique; - unprovement of Ba.yesia.n structure estima.tion algorithm.
Here, a. pra.ctical experience with the procedure will be reported using realistic simula.tion results.
Keywords. ARX model, Ba.yes rule, Forgetting, Structure estima.tion, Ada.ptive control.
1. INTRODUCTION
The input-output rela.tionship is modelled by the conditional proba.bility density functions (pdf) f(YrII(r),a) 98;r, where a E a· is a.n unknown pa.ra.meter a.bout which informa.tion should be collected a.nd which is recursively estima.ted during the a.da.ptive system functioning. The meuured inform&ti.on I(r) built in this model is a. (by mismodelling) distorted pa.rt of the full informa.tion D(t) meuured. V~ue prior informa.tion on a is expressed in the Ba.yesia.n ma.nner through a. (fia.t) prior pdf pea).
=
Ada.ptive predictors/controllers a.re mostly used when discrepa.ncies between the model a.nd system a.re expected to va.ry with time. Mismodelling u well u cha.nges of the inspected system ma.y be responsible for such va.na.tions. Consequently, the prelimina.ry estima.tion, which is highly desira.ble for a. prior design of the constructed a.da.ptive system (predictor/controller) (Bohm a.nd Ka.rny 1992, Ka.rny a.nd Halouskova. 1992), hu to fa.ce incomplete relia.bility of the informa.tion sources. At the sa.me time, a.ny piece of informa.tion should be used in order to decreue uncerta.inty of the results ga.ined in this wa.y. Thus, a. proper bala.nce hu to be sea.rched for between this need a.nd the informa.tion-sources unrelia.bility.
In (Ka.rnY et al. 1994a), a. generally a.pplica.ble hypermodel - a. noisy rela.tion 98;r, rE t·,- f(aID(t)) a.nd its Ba.yesia.n estima.tion ha.ve been proposed. This model of a. £actor a.nalysis type ca.n be estima.ted recursively u shown in (Ka.rnY et al. 1994c). The Ba.yesia.n estima.te of f(aID(t)) (obta.ined a.t the hyperlevel) merges inexa.ct a.nd not fully compa.tible informa.tion pieces a.nd hu been la.belled u (Ba.yesia.n) pooling algorithm. The a.va.ila.ble restricted experience with it hu been more tha.n promising. The construction of the hypermodel depends na.turally on modelling heuristics. It is thus necessa.ry to complement it by a. thorough a.nalysis in order to specify the potential a.nd restrictions of the resulting technique. Extensive simula.tions, summarized here, represent a. first step in this direction. All of them concentra.te on ARX model which is undoubtedly one ofthe most frequent models
The technique for a.chieving such a. bala.nce, we rely on a.nd we inspect here experimentally, is bued on identmca.tion of the model to be used in on-line phue by rela.ting it to the ideal posterior pdf with a. hypermodel which respects restricted compa.tibility of the model with da.ta..
=
Specifically, let the sequence D(t), t E t· {1, 2, ... }, be observed (A· denotes a. set of possible A-values). The da.ta. consist of the input-output pa.irs dr (Ur,Yr), r Er· {1, ... ,t}. The single output cue is considered.
=
=
119
of the Ba.yesian estima.te, ie. the evalua.tion of the posterior pdf f(eID(t)), is not a.pplica.ble and some a.pproxima.te trea.tment is una.voida.ble. The following proposition summarizes the pooling method used.
used in ada.ptive systems.
2. THEORY OVERVIEW The theory is supprelllled to minimum in order to spa.re the space for discussion of the experimental results. At the sa.me time, all details needed for progra.mming of the final algorithm a.re given.
Proposition 1 [Edimate of f(eID(t))] Let ge;" be positive on a. common support eo, sufficiently smooth fln2[ge;,,]p(e) de < 00 for in e and t p[ln2(ge.,,)] all rE rO.
!.1 Gau.uian ARX model
Then, the following pdf j(eID(t)) can be interpreted u the ma.ximum a posteriori proba.bility estima.te of f(eID(t)) bued on a fa.ctor ana.lysis type hypermodel
=
The Ga.ussian ARX model is specified by the pdf g[l/,r);r
=
=
j(eID(t)) ex n~.1 [ge;,,]"" p(e) where v = [V1, ... , ve]' is the mixing vector, ie. the
NII .. (8'~r,r) (1) (211'r)-0.5 exp[-0.5r- 1 (Yr - 6'~"
rt
eigenvector corresponding to the ma.ximum eigenva.lue c of the mixing symmetric positive semidefinite (t, t)ma.trix Q Q(t), with entries
=
NII(y, r) is the Ga.ussian pdf of y given by the expected value y and dispersion r > 0; , denotes transposition; 6 is an i.;-vector of unknown real regre.uion coeffi· cient.t; ~" is the regre.uor, a. known i.;-dimensional function of D(r - I), u,,; r > 0 is an unknown di,per,ion; e = [8, r] is the i. = (i.; + I)-dimensional unknown pa.ra.meter defining the model.
The length a of v = av, v'v = 1, is given by
=
where l' [1, ... ,1] and "'( > 0 is a. user specified degree of confidence into model compatibility ("'( increues with the confidence).
The self-reproducing Ga.uss-inverse-Wisha.rt (GiW) prior pdf (Anderson 1958) is selected
If the pdfs 9 belong to the exponentia.l fa.mily then the estima.tes j can be upda.ted recursively when the data sequence length t is growing.
'UT (L'L) -0.5[,,+,.+2] . GIHI/,r ,11 exr [-1, 61L'L[-I,8']' } exp { 2r ' (2)
=
(4)
where covp denotes second centra.l moment (cova.riance) ta.ken with respect to p(e).
!.! Prior pdf and canonical form of ARX model
p(8,r)
covp[ln(ge;,,) 1n(ge;,)], r, T E rO
Q",
Proof. See (Kaxny et al. 1994b).
0
where ex means proportionality. The normalizing constant in (2) is finite, ie. the pdf p(8, r) is well defined if the (i., i. )-ma.trix L is regula.r and the scala.r 11 is positive. The lower triangula.r ma.trix L with a. positive dia.gonal can a.lwa.ys be chosen.
This section describes fully the a.lgorithm &rising from the pooling method applied to the ARX model; its recursive version is presented.
The ARX model can be re-written into the following form which is used in the pooling a.lgorithm
3.1 The mixing matrix
gx."
.p'
+ ([p,x']~ .. )2]}
ex
exp
=
exp { -0.5.p~e } where
X' -
~~
{-~
= =
[-p, x']
[-2ln(P)
= r-
0 5 •
[_I,8']L'
With the vectors .p~, rE rO ta.ken u data dependent rows of the (t, i,,)-ma.trix 'It(t), 'It'(t) [.p1"'" .pe], the mixing (t,t)-ma.trix Q(t) given in (4) becomes Q(t) 'It(t)A'lt'(t).
=
=
(3)
[y(r),~~]L-1 [ 1,
-2
-2
-
.,.
-
The fixed symmetric positive definite (i.. , i,p)-matrix covp[e] becomes
-
~1"'" ~'.' ~1tP ,tP1[~2, ... , tP,.],
... '~'r1~'.]'
e' =
3. POOLING FOR ARX MODEL
i oP
A
= dim(.p)
[-2ln(p), p2, x~,
,xt., 2px',
2X1 [X2, . .. ,x,.],
,2X'._1X'.] .
= 8%!0;5:l
2(11: 2)
~
~
o
0 21 0 0 0 4(11 + 2)1 0 0 0 where unit matrices multiplied by 2 and 4(11 + 2) a.re ie-dimensional and 41 completes dimensions of A to (i,p, i,,). The sca.la.rs involved a.re defined
[
£.3 Pooling method Due to the &8sumed incomplete compa.tibility of the involved model and da.ta., the sta.nda.rd determina.tion
Po
120
o o
=(v + 2)[S(0.511) -
S(0.511
+ 1)]
S«()
= :, In(r«()),
of j(8ID(t)) is given a. special name {3
(> 0,
where r(.) is the Euler gamma. function. The function S(-) ca.n be well a.pproxima.ted by, (Abramowitz and Stegun 1972),
S(() ::::: In«() -
i'(t)BA(t)A- 1 = [P(t),H'(t)).
For the same reasons the remaining subvector H(t) is orga.nized into a. symmetric ma.trix form VH(t) (an inverse ma.pping corresponding to the transforma.tions X .... 8, ~(t) .... t/l(t), (3)), and the lower triangula.r ma.trix L (2) divided into blocks
i< - ~ + 12~'4'
In the implementa.tion it is useful to work with triangula.r Choleski squa.re root A of
(5)
A=AA'.
L -_ For its evalua.tion, just the Choleski squa.re root of the top-left (2,2)-sub-ma.trix of A is needed.
= T(t) [BA(t)oB~(t) ~] T'(t)
BA~;~ 1)
] .... [
B~(t)
L' 1/ 18 sc al a.r.
(9)
The evalua.tion of the estimate j(8ID(t)) summa.rizes into the following algorithm:
] .
S.... l Off-line ph/lle. 1. Select prior ,tat"tiCl L, v, respecting moments
of (2)
r:: lp[r) =
8 = lp(8)
The mixing vector can be found to ha.ve the form
= [i(t), O)T'(t)
0] ,
L~
S... Algorithm ,ummarll
S.! Mizing vector
v'(t)
onI
j(r, 8ID(t)) = (10) aiWr .9 (L'(a(t)VH(t) + I)L, v + a(t){3(t))
where T(t) is an orthogonal ma.trix and BA (t) hu dimension less than i.=dim( t/l) (3). The matrix T(t) u well u BA(t) ca.n be constructed recursively by "orthogonally" zeroing the new row
[
[LL~.
With elements introduced, it ca.n be found (Karny et al. 1994 b) tha.t the estima.te of the posterior pdf is also Ga.uss-inverse-Wisha.rt form
Using the introduced quantities, the mixing ma.trix ca.n be written in the form
Q(t)
(8)
cov p (8) fill
where T(t) is the orthogonal matrix constructed a.bove and i(t) is the unit-length eigenvector corresponding to the ma.ximum eigenvalue c(t) of the ma.trix BA(t)B~(t). Length a(t) of the mixing vector v(t) a(t)v(t), v'(t)v(t) = 1, ca.n be expressed
a
=
lC(t) (7) + IC-y(t)a b'(t)(W(t)-y-1 + I)- l b(t) + n}.(t)
lC(t)
=
c(t) -'(t)b(t) (t) _ 1 c(t) + or z ,IC-y - c(t) + or
!Po
(1).
S .... ! On-line pha,e. ... Set t = t + 1. S. Me/llure new da.ta. Ue, lie and construct the vector ~e and then t/le (3). 6. Update the ma.trix BA(t - 1), vector b(t - 1) and scala.r n~(t -1) to B(t), b(t) and n~(t) by
=
=
= rL;l(L;l)'.
2. Compute the matriz A a.ccording to (5). S. Set the initial condition BA(O) = 0, t = 0 and
(6)
a(t)
L~V-1 ,
= _L;l L"
1C 2 (t)
"orthogonal" zeroing of [t/l~+lA,I) with respect to
b(t - 1) ] t · [B(t 0- 1) n~(t-l)' th ema.nx 7. Compute the mixing vector v(t), ie. the scala.r {3(t), and the matrix VH(t) (6). 8. Find the appropria.te norm of the mixing vector a(t) (7). 9. Evaluate required cha.ra.ctemtics of the a.pproxima.te pdf (10) and go to step 4.
where the new is-vector b(t) and the scala.r n~ ca.n be obta.ined recursively by a.pplying the a.bove orthogonal rota.tion T(t) to the vector of t + 1 units: T(t)l =
[b'(t), n~)'.
S.S Edimate
4.
For the a.ssumed member of the exponential family, the estima.te j(8ID(t)) of f(8ID(t)) described in Proposition 1 can simply be given the form
EXPERIMENTS
...1 Lead 'quare, initialization ...1.1 Problem addre"ed. Upda.ting of the GiW sta.-
j(8ID(t)) ex p(8) exp[-0.58'1IT'(t)v(t)) = =p(8) exp[ -0.5a(t)8'(BA (t)A -1)'i(t)),
tistics is known to coincide formally with recursive leut squa.res (RLS). From this view point the adopted Ba.yesian pa.ra.digm adds non-trivial initial conditions to them (Peterka. 1981). It is known that transient beha.viour of RLS is substantially influenced by the initial conditions chosen. This fa.ct is especially urgent when the outcomes of RLS a.re used u a. pa.rt of an adaptive controller. Thus, a. ca.reful but automa.-
ie. it is determined by the sta.tistics BA(t), a(t), i(t) evalua.ted in previous steps.
In order to describe simply the results, the first entry of the da.ta. dependent i.-vector used in the exponent 121
f(8ID(t)) ex fl'(8ID(t))f~->.(8ID(t))
tized choice of these initial conditions is an inevitable pa.rt of ja.c.keting of any a.daptive controller.
where the probability of time invariant model 0 < A < 1 acts u a fixed forgetting factor, f, denotes the posterior pdf obtained under time-inva.riance hypothesis and fG denotes an alterna.tive pdf.
.... 1.£ Solution. The a.dvocated algorithm can be used for the job: all pieces of information should be combined. The mixture of expert knowledge (eg. rough information on step response), of possibly obsolete meuured data and data gained from a simulation of a more complex model can be obtained by the algorithm. In this way an appropriate GiW pdf, ie. initia.li.zation of RLS, is obtained.
For ARX models, it is reuonable to select the alterna.tive of the GiW form. It means that the alternative statistics, the matrix L G and the scala.r 110., have to be selected. Then, this technique reduces to the so called stabilized forgetting described and analyzed in (Milek and Kraus 1991) .
....1.3 Re$ult$.
The influence of prior knowledge on the identification and control sta.rt wu tested extensively (Nedoma et al. 1994). Generally, prior informa..tion hu the awaited positive influence, especially under "difficult" conditions. Combining imprecise prior information sources does not necessa.rily lead to better system behaviour, but dangerous overshoots a.re substantially suppressed. The less uncertain the prior information is, the more pronounced the improvement.
The forgetting factor A and the alternative fG(')' for ARX model given by L 4 , 114 , a.re tuning knobs of this forgetting technique. Within a rea.sona.ble range of Avalues the forgetting results a.re not very sensitive to the choice of A. The choice of fG may influence the results ra.dically. Its construction requires a substantial skill and thus it is desira.ble to support the prospective user .
....1.... Rludrative e%ample. The typical system behaviour is documented graphically. In Fig.l, time normalized control error squa.red is given u function of time. AB the results a.re dependent of noise realization, the error is a.vera.ged over 50 simulation runs of length 80. Four experimental results a.re presented, relying upon different kinds of prior knowledge implemented: - approximate static gain (points); - a.pproxima.te transient response (da.shed); - data sample (duhdot); - no prior knowledge, default initialization (solid).
.... £.£ Solution. The ba.sic idea is obvious: the prior
pdf obtained through the inspected algorithm is ta.ken a.s a sa.fe compromise of prior knowledge elements. It is expected to be valid for the given system permanently. Thus, it is reuonable to ta.ke the GiW pdf (10) u the needed alternative. .... £.3 Re$ulb. Comparison of simple recursive leut squa.res with exponential forgetting to stabilized algorithms (Milek and Kraus 1991) shows, tha.t even their simplest form with diagonal alternative covaria.nce removes the wind-up da.nger (paying by a slight decrea.se in model quality). The full matrix alternative cova.ria.nce, computed by the algorithm presented here from prior knowledge of rough approximate transient response, had the stabilizing effect without any nega..tive consequences. Moreover, the mere knowledge of the approximate system gain proved to be helpful for substantial performance improvement. More extensive experience is summa.rized in (Halouskova. et al. 1994) .
. ~'<"~>:~.:~ ........-::.: ........
.......
.... £.... Rludrative e%ample.
00
10
20
::JiO
40
,
50
80
70
eo
5lQ
Model of a steam power plant super-heater wa.s simulated and then identified by a simplified model: due to mismodelling, forgetting wu important. To show the performance under insufficient excitation, the random input signal wa.s switched off in the observed interval; graph of inputs and outputs is given in Fig.2.
100
Fig.I. Adaptive control initialization.
The illustration compa.res the results of recursive lea.st squares identification with conventional exponential forgetting (RLS-EF) to recursive lea.st squares with stabilized forgetting (RLS-SI). The alternative full matrix cova.riance for the stabilized forgetting wa.s gained by the advocated algorithm, using prior knowledge of rough transient response.
.... £ Stabilized forgetting
.... £.1 Problem addreued. Adaptivity is mostly rea.ched by applying a sort of forgetting. Its simplest form, the exponential forgetting, bea.rs danger of cova.riance wind-up in ca.se of insufficiently exciting inputs. A general proba.bilistic approa.ch of removing this problem wu proposed in (Kulha.vY and Za.rrop n.d.) where a. suitable decision tuk is formula.ted balancing estima.tion under hypothesis of time-inva.riant pa.ra.meters with an alternative which admits para.meter changes according to pre-specified pdf. The best compromise found ha.s the form
AB the experiment outcomes depend heavily on system noise rea.li.zation, a series of 50 experiments were performed and average values plotted.
122
4.3 Structure e,timation 4.3.1 Problem addreued. General theory of Bayesia.n estimation was elaborated (Ka.rnY a.nd Kulhavy 1988) for estimating structure of the regressor ~ (1). It reduces to the 8euch for the maximum of the posterior probability of the structure conditioned on the obeerved data.
-eo
If we denote Ll k1,l/[k1 the statistics determining the GiW distribution (10) assuming that the regressor ~[kJ of a kth structure is appropriate for describing the system, then the posterior probability of this structure has the form
y . . . .u ~
-4
20
40
eo
eo
tOO
t20
t40
teo
teo
200
I
Fig.2. Data for the identification experiment. (11) The predictive ability of the pa.rticula.r estimation procedures was qua.ntified by average squa.red prediction errors in appropriate time insta.nts. As shown in Fig.3, the resulta a.re better for the new type of forgetting tha.n for the classical exponential one.
where the time index distinguishes the statistics before using data (prior statistics) (2) a.nd after using them (10). Splitting L into sub-blocks (9), the functions .J ca.n be expressed in the form
(12) 1.._-_~-~-~-_~-~-~-_--,
C i
The formula (11) has been repeatedly used with success but cases were encountered when data brought too little information a.nd procedure became sensitive to the choice of the prior statistics. Typically, it happened for weakly excited continuous time system with a short sampling period. Then, the absolute values of coefficients at inputs a.re much smaller tha.n those at delayed outputs a.nd a tendency to select autoregresHive structure has been observed.
t
•i i"0. ti
20
40
eo
~
tOO
t20
t40
tElO
teo
4.3.! Idea of the ,olution. Introduction of expert knowledge is the only possible remedy for the outlined problem. The algorithm tested allows us to introduce wide ra.nge of such information pieces into L(O), 1/(0). The typical global vague knowledge of the static gain (Karny 1984) is expected to help in solving the addressed problem by a priori disqualifying purely autoregression models.
200
t
Fig.3. Adaptive forgetting - prediction errors.
The cov&ria.nce behaviour, for control robustness even more importa.nt, was cha.r&Ctenzed by average cov&ria.nce matrix tr&ee. The wind-up in the classical appro&ch is clea.rly visible in Fig.4, while for the stabilized forgetting the cov&ria.nce converges to a reasonable robust value.
4.3.3 Open problem". When prepa.ring ea.rly version of this paper we believed that at least prelimina.ry experimental results will be presented. Two problems were, however, encountered:
•.. _-_~-~-~--~-~-~----,
R~
20
40
eo
eo
'00
'20
Prior statistic is not nested. The formula (11) requires evaluation of statistics Llk1, 1/1k1 over the entire eet of hypotheses. For the prior a.nalysis assumed, all regressors obtained by relaxing some entries from a. "maximum" regressor ~ a.re to be considered. It mea.ns that the number of hypotheses 2d,m(~) is UBUally extremely la.rge for full evaluation. For this rea.son, maximum of the posterior probability (11) is eea.rched for using a.n adapted algorithm (Ka.rnY a.nd Kulhavy 1988) which strongly relies on the nesting
. RLS-EF
'40
teo
teo
~[k1 = [~[IJ,.]' ~ L~J = [L.~J
200
1
.].
This property is, however, lost for the constructed prior a.nd consequently posterior statistics. It makes the available algorithms almost useless a.nd gives no
Fig.4. Adaptive forgetting - cov&ria.nce norms.
123
prior informa.tion for sta.bilized forgetting. Technical Report 1821. UTIA AV CR. POBox 18, 182 08 Pr~ue 8, Czech Republic. Ka.rnY, M. (1984). Qua.ntifica.tion of prior knowledge a.bout global cha.ra.cteristics of linea.r normal model. Kllbernetika 20(5), 376-385. Ka.rnY, M., A. Halouskova. a.nd L. ZOrnigova. (1994a). On pooling expert opinions. In: Preprinu SYS/D'9-4 (M. Bla.nke T. SOderstrom, Ed.). Vol. 2. pp. 477-482. Da.nish Automa.tion Society. Kopenh~en. Ka.rny, M., A. Halouskova. a.nd P. Nedoma. (1994b). Recursive a.pproxima.tion by ARX model: a. tool for grey box modelling; submitted. International Journal of Adaptave Control and Signal Proce,,· ang. Ka.rnY, M. a.nd A. Halouskova. (1992). User supplied informa.tion in the design of linea.r qua.dra.tic Ga.ussia.n controllers. In: Preprinu of -4th IFAC SlImpo,aum Adaptave Control and Signal Procelling ACASP'9£ (LD. Landa.u, L. Duga.rd a.nd M. M'Sa.a.d, Eds.). Vol. 2. pp. 451-456. Aca.demic Press. Grenoble. Ka.rny, M. a.nd R. Kulha.vY (1988). Structure determina.tion of regression-type models for a.da.ptive prediction a.nd control. In: Balle,ian Analll,a, of Time Serie, and Dllnamic Model& (J.C. Spa.ll, Ed.). Ma.rcel Dekker. New York. cha.pter 12. Karny, M., D. Ha.jma.n a.nd A. Halouskova. (1994c). On conserva.tive recursive lea.rning. In: Preprinu of the European IEEE Work,hop CMP'9-4 (L. Kulha.va., M. Ka.rny and K. Wa.rwick, Eds.). pp. 77-82. UTIA AV CR. Pr~ue. Kulha.vY, R. a.nd M. B. Za.rrop (n.d.). On general concept of forgetting. International Journal of Control fiS(4), 905-924. Milek, J.J. a.nd F.J. Kra.us (1991). Sta.bilized leut squares estima.tors. Technical Report 91-02. ETH Zurich, Automa.tic Control La.bora.tory. ACL ETH Zurich, Switzerla.nd. Nedoma., P., M. Ka.rny a.nd A. Halouskova. (1994). Recursive a.pproxima.tion by ARX model: Simula.tion verifica.tion of a.da.ptive controller initializa.tion. Technical Report 1816. UTIA AV CR. POBox 18, 182 08 Pr~ue 8, Czech Republic. Peterka., V. (1981). Ba.yesia.n system identifica.tion. In: TrendJ and Progrell in SlIdem Identification (P. Eykhoff, Ed.). pp. 239-304. Perga.mon Press. Oxford.
cha.nce for a.ny efficient solution. The P value is scaling sensitive. This qua.ntity (8) determining the number of degrees of freedom in the derived GiW pdf (10) depends too much on a.ppropria.te selection of the prior sta.tistic (2). This property becomes observa.ble just in structure estima.tion tuk due to the explicit dependence of the result on JI + Otp. When studying this problem a. possible remedy wu found: instead of rela.ting loga.rithms of ge;r a.nd f(8ID(t)), their gradients Ve with respect to 8 a.re used in the hypermodel. This -
ca.n be shown to resolve both the discuaed prob-
lems; - modifies the presented algorithm very slightly: just the mixing ma.trix covp[ln(g.;.)] is repla.ced by its Fisher-informa.tion-ma.trix like counterpa.rt covp{Ve [In(g.;.)]} a.re used. Detailed trea.tment of this line will, however, be given elsewhere.
5. CONCLUSIONS
Simula.tion bued verifica.tion of the potential a.nd restrictions of the general algorithm for combining va.rious informa.tion pieces form the core of the pa.per. It concentra.tes on use of this algorithm for ARX models - the key building block for a. substa.ntial set of a.da.ptive systems. The pa.rticula.r subtuks like RLS initializa.tion, construction of a.n alterna.tive pdf for sta.bilized forgetting technique a.nd improvements of the structure estima,tion algorithm a.re inspected in detail. Their solution is necessa.ry pa.rt of a.n extensive project which is to provide sue a.nd well justified ja.cketing of esta.blished LQG a.da.ptive controllers. For this reuon, the potential use of the technique u a.n a.pproxima.tion and/or model reduction tool hu been left uide. It ca.n be concluded tha.t the proposed technique brings expected a.dva.nt~es for RLS initializa.tion a.nd tuning of the a.dva.nced forgetting technique. For structure estima.tion specific problems ha.ve been encountered a.nd possible remedy proposed but promises in this respect should be verified.
Acknowledgment This work has been partially supported by GA tR, Grant No. 102/93/0228 and by GA AV tR, Grants No. 27554, 275109, 275110.
REFERENCES Abra.mowitz, M. a.nd LA. Stegun (1972). Handbook of mathematacalfunction&. Dover Publica.tions, Inc.. New York. Anderson, T.W. (1958). An Introductaon to Multivari. ate Statadacal Analll&a&. John Wiley. Bohm, J. a.nd M. Ka.rnY (1992). Merging of user's knowledge into self-tuning controllers. In: Preprinu of -4 th IFA C SlImpO&aUm Adaptive Control and Sagnal Proceuang ACASP'9! (LD. La.nda.u, L. Duga.rd a.nd M. M'Sa.a.d , Eds.). Vol. 2. pp. 427-432. Aca.demic Press. Grenoble. Halouskova., A., M. Ka.rny a.nd P. Nedoma. (1994). Recursive a.pproxima.tion by ARX model: Use of 124
APPROXIMATE ARX MODEL ESTIMATION FOR JACKETING ADAPTIVE SYSTEMS
Miroslav Karny, Petr Nedoma, Josef Bohm, Alena Halouskova Inltitute of Information Theory and Automation Academy of Sciencu of the Czedl Republic Pod voddrend:ou vUi -I, 18108, Prague 8, Czedl Republic Fa:J:: +-11.1.66-11-1903, tel: +-11.1.66,413,411, e·mail: ,chool§utia.au.cz
Abstract. Recently, a. substa.ntial progress hu been a.chieved in a.ttempts to merge systematically va.nous .pieces of. informa.tion of different precision, compa.tibility a.nd origin. The developed techmque prOVIdes a. new tool needed for jacketing a.da.ptive predictors/controllers bued on LQG the pa.ra.digm. ~en a.pplied to th~ a.utoregressive model with exogenous va.na.bles (ARX) the method contributes to the solutIon of the following tub: - incorpora.~on of prior knowledge into initial conditions of the recursive leut squa.res; - ~nstruction of a. reference for a.n a.dva.nced forgetting technique; - unprovement of Ba.yesia.n structure estima.tion algorithm.
Here, a. pra.ctical experience with the procedure will be reported using realistic simula.tion results.
Keywords. ARX model, Ba.yes rule, Forgetting, Structure estima.tion, Ada.ptive control.
1. INTRODUCTION
The input-output rela.tionship is modelled by the conditional proba.bility density functions (pdf) f(YrII(r),a) 98;r, where a E a· is a.n unknown pa.ra.meter a.bout which informa.tion should be collected a.nd which is recursively estima.ted during the a.da.ptive system functioning. The meuured inform&ti.on I(r) built in this model is a. (by mismodelling) distorted pa.rt of the full informa.tion D(t) meuured. V~ue prior informa.tion on a is expressed in the Ba.yesia.n ma.nner through a. (fia.t) prior pdf pea).
=
Ada.ptive predictors/controllers a.re mostly used when discrepa.ncies between the model a.nd system a.re expected to va.ry with time. Mismodelling u well u cha.nges of the inspected system ma.y be responsible for such va.na.tions. Consequently, the prelimina.ry estima.tion, which is highly desira.ble for a. prior design of the constructed a.da.ptive system (predictor/controller) (Bohm a.nd Ka.rny 1992, Ka.rny a.nd Halouskova. 1992), hu to fa.ce incomplete relia.bility of the informa.tion sources. At the sa.me time, a.ny piece of informa.tion should be used in order to decreue uncerta.inty of the results ga.ined in this wa.y. Thus, a. proper bala.nce hu to be sea.rched for between this need a.nd the informa.tion-sources unrelia.bility.
In (Ka.rnY et al. 1994a), a. generally a.pplica.ble hypermodel - a. noisy rela.tion 98;r, rE t·,- f(aID(t)) a.nd its Ba.yesia.n estima.tion ha.ve been proposed. This model of a. £actor a.nalysis type ca.n be estima.ted recursively u shown in (Ka.rnY et al. 1994c). The Ba.yesia.n estima.te of f(aID(t)) (obta.ined a.t the hyperlevel) merges inexa.ct a.nd not fully compa.tible informa.tion pieces a.nd hu been la.belled u (Ba.yesia.n) pooling algorithm. The a.va.ila.ble restricted experience with it hu been more tha.n promising. The construction of the hypermodel depends na.turally on modelling heuristics. It is thus necessa.ry to complement it by a. thorough a.nalysis in order to specify the potential a.nd restrictions of the resulting technique. Extensive simula.tions, summarized here, represent a. first step in this direction. All of them concentra.te on ARX model which is undoubtedly one ofthe most frequent models
The technique for a.chieving such a. bala.nce, we rely on a.nd we inspect here experimentally, is bued on identmca.tion of the model to be used in on-line phue by rela.ting it to the ideal posterior pdf with a. hypermodel which respects restricted compa.tibility of the model with da.ta..
=
Specifically, let the sequence D(t), t E t· {1, 2, ... }, be observed (A· denotes a. set of possible A-values). The da.ta. consist of the input-output pa.irs dr (Ur,Yr), r Er· {1, ... ,t}. The single output cue is considered.
=
=
119
of the Ba.yesian estima.te, ie. the evalua.tion of the posterior pdf f(eID(t)), is not a.pplica.ble and some a.pproxima.te trea.tment is una.voida.ble. The following proposition summarizes the pooling method used.
used in ada.ptive systems.
2. THEORY OVERVIEW The theory is supprelllled to minimum in order to spa.re the space for discussion of the experimental results. At the sa.me time, all details needed for progra.mming of the final algorithm a.re given.
Proposition 1 [Edimate of f(eID(t))] Let ge;" be positive on a. common support eo, sufficiently smooth fln2[ge;,,]p(e) de < 00 for in e and t p[ln2(ge.,,)] all rE rO.
!.1 Gau.uian ARX model
Then, the following pdf j(eID(t)) can be interpreted u the ma.ximum a posteriori proba.bility estima.te of f(eID(t)) bued on a fa.ctor ana.lysis type hypermodel
=
The Ga.ussian ARX model is specified by the pdf g[l/,r);r
=
=
j(eID(t)) ex n~.1 [ge;,,]"" p(e) where v = [V1, ... , ve]' is the mixing vector, ie. the
NII .. (8'~r,r) (1) (211'r)-0.5 exp[-0.5r- 1 (Yr - 6'~"
rt
eigenvector corresponding to the ma.ximum eigenva.lue c of the mixing symmetric positive semidefinite (t, t)ma.trix Q Q(t), with entries
=
NII(y, r) is the Ga.ussian pdf of y given by the expected value y and dispersion r > 0; , denotes transposition; 6 is an i.;-vector of unknown real regre.uion coeffi· cient.t; ~" is the regre.uor, a. known i.;-dimensional function of D(r - I), u,,; r > 0 is an unknown di,per,ion; e = [8, r] is the i. = (i.; + I)-dimensional unknown pa.ra.meter defining the model.
The length a of v = av, v'v = 1, is given by
=
where l' [1, ... ,1] and "'( > 0 is a. user specified degree of confidence into model compatibility ("'( increues with the confidence).
The self-reproducing Ga.uss-inverse-Wisha.rt (GiW) prior pdf (Anderson 1958) is selected
If the pdfs 9 belong to the exponentia.l fa.mily then the estima.tes j can be upda.ted recursively when the data sequence length t is growing.
'UT (L'L) -0.5[,,+,.+2] . GIHI/,r ,11 exr [-1, 61L'L[-I,8']' } exp { 2r ' (2)
=
(4)
where covp denotes second centra.l moment (cova.riance) ta.ken with respect to p(e).
!.! Prior pdf and canonical form of ARX model
p(8,r)
covp[ln(ge;,,) 1n(ge;,)], r, T E rO
Q",
Proof. See (Kaxny et al. 1994b).
0
where ex means proportionality. The normalizing constant in (2) is finite, ie. the pdf p(8, r) is well defined if the (i., i. )-ma.trix L is regula.r and the scala.r 11 is positive. The lower triangula.r ma.trix L with a. positive dia.gonal can a.lwa.ys be chosen.
This section describes fully the a.lgorithm &rising from the pooling method applied to the ARX model; its recursive version is presented.
The ARX model can be re-written into the following form which is used in the pooling a.lgorithm
3.1 The mixing matrix
gx."
.p'
+ ([p,x']~ .. )2]}
ex
exp
=
exp { -0.5.p~e } where
X' -
~~
{-~
= =
[-p, x']
[-2ln(P)
= r-
0 5 •
[_I,8']L'
With the vectors .p~, rE rO ta.ken u data dependent rows of the (t, i,,)-ma.trix 'It(t), 'It'(t) [.p1"'" .pe], the mixing (t,t)-ma.trix Q(t) given in (4) becomes Q(t) 'It(t)A'lt'(t).
=
=
(3)
[y(r),~~]L-1 [ 1,
-2
-2
-
.,.
-
The fixed symmetric positive definite (i.. , i,p)-matrix covp[e] becomes
-
~1"'" ~'.' ~1tP ,tP1[~2, ... , tP,.],
... '~'r1~'.]'
e' =
3. POOLING FOR ARX MODEL
i oP
A
= dim(.p)
[-2ln(p), p2, x~,
,xt., 2px',
2X1 [X2, . .. ,x,.],
,2X'._1X'.] .
= 8%!0;5:l
2(11: 2)
~
~
o
0 21 0 0 0 4(11 + 2)1 0 0 0 where unit matrices multiplied by 2 and 4(11 + 2) a.re ie-dimensional and 41 completes dimensions of A to (i,p, i,,). The sca.la.rs involved a.re defined
[
£.3 Pooling method Due to the &8sumed incomplete compa.tibility of the involved model and da.ta., the sta.nda.rd determina.tion
Po
120
o o
=(v + 2)[S(0.511) -
S(0.511
+ 1)]
S«()
= :, In(r«()),
of j(8ID(t)) is given a. special name {3
(> 0,
where r(.) is the Euler gamma. function. The function S(-) ca.n be well a.pproxima.ted by, (Abramowitz and Stegun 1972),
S(() ::::: In«() -
i'(t)BA(t)A- 1 = [P(t),H'(t)).
For the same reasons the remaining subvector H(t) is orga.nized into a. symmetric ma.trix form VH(t) (an inverse ma.pping corresponding to the transforma.tions X .... 8, ~(t) .... t/l(t), (3)), and the lower triangula.r ma.trix L (2) divided into blocks
i< - ~ + 12~'4'
In the implementa.tion it is useful to work with triangula.r Choleski squa.re root A of
(5)
A=AA'.
L -_ For its evalua.tion, just the Choleski squa.re root of the top-left (2,2)-sub-ma.trix of A is needed.
= T(t) [BA(t)oB~(t) ~] T'(t)
BA~;~ 1)
] .... [
B~(t)
L' 1/ 18 sc al a.r.
(9)
The evalua.tion of the estimate j(8ID(t)) summa.rizes into the following algorithm:
] .
S.... l Off-line ph/lle. 1. Select prior ,tat"tiCl L, v, respecting moments
of (2)
r:: lp[r) =
8 = lp(8)
The mixing vector can be found to ha.ve the form
= [i(t), O)T'(t)
0] ,
L~
S... Algorithm ,ummarll
S.! Mizing vector
v'(t)
onI
j(r, 8ID(t)) = (10) aiWr .9 (L'(a(t)VH(t) + I)L, v + a(t){3(t))
where T(t) is an orthogonal ma.trix and BA (t) hu dimension less than i.=dim( t/l) (3). The matrix T(t) u well u BA(t) ca.n be constructed recursively by "orthogonally" zeroing the new row
[
[LL~.
With elements introduced, it ca.n be found (Karny et al. 1994 b) tha.t the estima.te of the posterior pdf is also Ga.uss-inverse-Wisha.rt form
Using the introduced quantities, the mixing ma.trix ca.n be written in the form
Q(t)
(8)
cov p (8) fill
where T(t) is the orthogonal matrix constructed a.bove and i(t) is the unit-length eigenvector corresponding to the ma.ximum eigenvalue c(t) of the ma.trix BA(t)B~(t). Length a(t) of the mixing vector v(t) a(t)v(t), v'(t)v(t) = 1, ca.n be expressed
a
=
lC(t) (7) + IC-y(t)a b'(t)(W(t)-y-1 + I)- l b(t) + n}.(t)
lC(t)
=
c(t) -'(t)b(t) (t) _ 1 c(t) + or z ,IC-y - c(t) + or
!Po
(1).
S .... ! On-line pha,e. ... Set t = t + 1. S. Me/llure new da.ta. Ue, lie and construct the vector ~e and then t/le (3). 6. Update the ma.trix BA(t - 1), vector b(t - 1) and scala.r n~(t -1) to B(t), b(t) and n~(t) by
=
=
= rL;l(L;l)'.
2. Compute the matriz A a.ccording to (5). S. Set the initial condition BA(O) = 0, t = 0 and
(6)
a(t)
L~V-1 ,
= _L;l L"
1C 2 (t)
"orthogonal" zeroing of [t/l~+lA,I) with respect to
b(t - 1) ] t · [B(t 0- 1) n~(t-l)' th ema.nx 7. Compute the mixing vector v(t), ie. the scala.r {3(t), and the matrix VH(t) (6). 8. Find the appropria.te norm of the mixing vector a(t) (7). 9. Evaluate required cha.ra.ctemtics of the a.pproxima.te pdf (10) and go to step 4.
where the new is-vector b(t) and the scala.r n~ ca.n be obta.ined recursively by a.pplying the a.bove orthogonal rota.tion T(t) to the vector of t + 1 units: T(t)l =
[b'(t), n~)'.
S.S Edimate
4.
For the a.ssumed member of the exponential family, the estima.te j(8ID(t)) of f(8ID(t)) described in Proposition 1 can simply be given the form
EXPERIMENTS
...1 Lead 'quare, initialization ...1.1 Problem addre"ed. Upda.ting of the GiW sta.-
j(8ID(t)) ex p(8) exp[-0.58'1IT'(t)v(t)) = =p(8) exp[ -0.5a(t)8'(BA (t)A -1)'i(t)),
tistics is known to coincide formally with recursive leut squa.res (RLS). From this view point the adopted Ba.yesian pa.ra.digm adds non-trivial initial conditions to them (Peterka. 1981). It is known that transient beha.viour of RLS is substantially influenced by the initial conditions chosen. This fa.ct is especially urgent when the outcomes of RLS a.re used u a. pa.rt of an adaptive controller. Thus, a. ca.reful but automa.-
ie. it is determined by the sta.tistics BA(t), a(t), i(t) evalua.ted in previous steps.
In order to describe simply the results, the first entry of the da.ta. dependent i.-vector used in the exponent 121
f(8ID(t)) ex fl'(8ID(t))f~->.(8ID(t))
tized choice of these initial conditions is an inevitable pa.rt of ja.c.keting of any a.daptive controller.
where the probability of time invariant model 0 < A < 1 acts u a fixed forgetting factor, f, denotes the posterior pdf obtained under time-inva.riance hypothesis and fG denotes an alterna.tive pdf.
.... 1.£ Solution. The a.dvocated algorithm can be used for the job: all pieces of information should be combined. The mixture of expert knowledge (eg. rough information on step response), of possibly obsolete meuured data and data gained from a simulation of a more complex model can be obtained by the algorithm. In this way an appropriate GiW pdf, ie. initia.li.zation of RLS, is obtained.
For ARX models, it is reuonable to select the alterna.tive of the GiW form. It means that the alternative statistics, the matrix L G and the scala.r 110., have to be selected. Then, this technique reduces to the so called stabilized forgetting described and analyzed in (Milek and Kraus 1991) .
....1.3 Re$ult$.
The influence of prior knowledge on the identification and control sta.rt wu tested extensively (Nedoma et al. 1994). Generally, prior informa..tion hu the awaited positive influence, especially under "difficult" conditions. Combining imprecise prior information sources does not necessa.rily lead to better system behaviour, but dangerous overshoots a.re substantially suppressed. The less uncertain the prior information is, the more pronounced the improvement.
The forgetting factor A and the alternative fG(')' for ARX model given by L 4 , 114 , a.re tuning knobs of this forgetting technique. Within a rea.sona.ble range of Avalues the forgetting results a.re not very sensitive to the choice of A. The choice of fG may influence the results ra.dically. Its construction requires a substantial skill and thus it is desira.ble to support the prospective user .
....1.... Rludrative e%ample. The typical system behaviour is documented graphically. In Fig.l, time normalized control error squa.red is given u function of time. AB the results a.re dependent of noise realization, the error is a.vera.ged over 50 simulation runs of length 80. Four experimental results a.re presented, relying upon different kinds of prior knowledge implemented: - approximate static gain (points); - a.pproxima.te transient response (da.shed); - data sample (duhdot); - no prior knowledge, default initialization (solid).
.... £.£ Solution. The ba.sic idea is obvious: the prior
pdf obtained through the inspected algorithm is ta.ken a.s a sa.fe compromise of prior knowledge elements. It is expected to be valid for the given system permanently. Thus, it is reuonable to ta.ke the GiW pdf (10) u the needed alternative. .... £.3 Re$ulb. Comparison of simple recursive leut squa.res with exponential forgetting to stabilized algorithms (Milek and Kraus 1991) shows, tha.t even their simplest form with diagonal alternative covaria.nce removes the wind-up da.nger (paying by a slight decrea.se in model quality). The full matrix alternative cova.ria.nce, computed by the algorithm presented here from prior knowledge of rough approximate transient response, had the stabilizing effect without any nega..tive consequences. Moreover, the mere knowledge of the approximate system gain proved to be helpful for substantial performance improvement. More extensive experience is summa.rized in (Halouskova. et al. 1994) .
. ~'<"~>:~.:~ ........-::.: ........
.......
.... £.... Rludrative e%ample.
00
10
20
::JiO
40
,
50
80
70
eo
5lQ
Model of a steam power plant super-heater wa.s simulated and then identified by a simplified model: due to mismodelling, forgetting wu important. To show the performance under insufficient excitation, the random input signal wa.s switched off in the observed interval; graph of inputs and outputs is given in Fig.2.
100
Fig.I. Adaptive control initialization.
The illustration compa.res the results of recursive lea.st squares identification with conventional exponential forgetting (RLS-EF) to recursive lea.st squares with stabilized forgetting (RLS-SI). The alternative full matrix cova.riance for the stabilized forgetting wa.s gained by the advocated algorithm, using prior knowledge of rough transient response.
.... £ Stabilized forgetting
.... £.1 Problem addreued. Adaptivity is mostly rea.ched by applying a sort of forgetting. Its simplest form, the exponential forgetting, bea.rs danger of cova.riance wind-up in ca.se of insufficiently exciting inputs. A general proba.bilistic approa.ch of removing this problem wu proposed in (Kulha.vY and Za.rrop n.d.) where a. suitable decision tuk is formula.ted balancing estima.tion under hypothesis of time-inva.riant pa.ra.meters with an alternative which admits para.meter changes according to pre-specified pdf. The best compromise found ha.s the form
AB the experiment outcomes depend heavily on system noise rea.li.zation, a series of 50 experiments were performed and average values plotted.
122
4.3 Structure e,timation 4.3.1 Problem addreued. General theory of Bayesia.n estimation was elaborated (Ka.rnY a.nd Kulhavy 1988) for estimating structure of the regressor ~ (1). It reduces to the 8euch for the maximum of the posterior probability of the structure conditioned on the obeerved data.
-eo
If we denote Ll k1,l/[k1 the statistics determining the GiW distribution (10) assuming that the regressor ~[kJ of a kth structure is appropriate for describing the system, then the posterior probability of this structure has the form
y . . . .u ~
-4
20
40
eo
eo
tOO
t20
t40
teo
teo
200
I
Fig.2. Data for the identification experiment. (11) The predictive ability of the pa.rticula.r estimation procedures was qua.ntified by average squa.red prediction errors in appropriate time insta.nts. As shown in Fig.3, the resulta a.re better for the new type of forgetting tha.n for the classical exponential one.
where the time index distinguishes the statistics before using data (prior statistics) (2) a.nd after using them (10). Splitting L into sub-blocks (9), the functions .J ca.n be expressed in the form
(12) 1.._-_~-~-~-_~-~-~-_--,
C i
The formula (11) has been repeatedly used with success but cases were encountered when data brought too little information a.nd procedure became sensitive to the choice of the prior statistics. Typically, it happened for weakly excited continuous time system with a short sampling period. Then, the absolute values of coefficients at inputs a.re much smaller tha.n those at delayed outputs a.nd a tendency to select autoregresHive structure has been observed.
t
•i i"0. ti
20
40
eo
~
tOO
t20
t40
tElO
teo
4.3.! Idea of the ,olution. Introduction of expert knowledge is the only possible remedy for the outlined problem. The algorithm tested allows us to introduce wide ra.nge of such information pieces into L(O), 1/(0). The typical global vague knowledge of the static gain (Karny 1984) is expected to help in solving the addressed problem by a priori disqualifying purely autoregression models.
200
t
Fig.3. Adaptive forgetting - prediction errors.
The cov&ria.nce behaviour, for control robustness even more importa.nt, was cha.r&Ctenzed by average cov&ria.nce matrix tr&ee. The wind-up in the classical appro&ch is clea.rly visible in Fig.4, while for the stabilized forgetting the cov&ria.nce converges to a reasonable robust value.
4.3.3 Open problem". When prepa.ring ea.rly version of this paper we believed that at least prelimina.ry experimental results will be presented. Two problems were, however, encountered:
•.. _-_~-~-~--~-~-~----,
R~
20
40
eo
eo
'00
'20
Prior statistic is not nested. The formula (11) requires evaluation of statistics Llk1, 1/1k1 over the entire eet of hypotheses. For the prior a.nalysis assumed, all regressors obtained by relaxing some entries from a. "maximum" regressor ~ a.re to be considered. It mea.ns that the number of hypotheses 2d,m(~) is UBUally extremely la.rge for full evaluation. For this rea.son, maximum of the posterior probability (11) is eea.rched for using a.n adapted algorithm (Ka.rnY a.nd Kulhavy 1988) which strongly relies on the nesting
. RLS-EF
'40
teo
teo
~[k1 = [~[IJ,.]' ~ L~J = [L.~J
200
1
.].
This property is, however, lost for the constructed prior a.nd consequently posterior statistics. It makes the available algorithms almost useless a.nd gives no
Fig.4. Adaptive forgetting - cov&ria.nce norms.
123
prior informa.tion for sta.bilized forgetting. Technical Report 1821. UTIA AV CR. POBox 18, 182 08 Pr~ue 8, Czech Republic. Ka.rnY, M. (1984). Qua.ntifica.tion of prior knowledge a.bout global cha.ra.cteristics of linea.r normal model. Kllbernetika 20(5), 376-385. Ka.rnY, M., A. Halouskova. a.nd L. ZOrnigova. (1994a). On pooling expert opinions. In: Preprinu SYS/D'9-4 (M. Bla.nke T. SOderstrom, Ed.). Vol. 2. pp. 477-482. Da.nish Automa.tion Society. Kopenh~en. Ka.rny, M., A. Halouskova. a.nd P. Nedoma. (1994b). Recursive a.pproxima.tion by ARX model: a. tool for grey box modelling; submitted. International Journal of Adaptave Control and Signal Proce,,· ang. Ka.rnY, M. a.nd A. Halouskova. (1992). User supplied informa.tion in the design of linea.r qua.dra.tic Ga.ussia.n controllers. In: Preprinu of -4th IFAC SlImpo,aum Adaptave Control and Signal Procelling ACASP'9£ (LD. Landa.u, L. Duga.rd a.nd M. M'Sa.a.d, Eds.). Vol. 2. pp. 451-456. Aca.demic Press. Grenoble. Ka.rny, M. a.nd R. Kulha.vY (1988). Structure determina.tion of regression-type models for a.da.ptive prediction a.nd control. In: Balle,ian Analll,a, of Time Serie, and Dllnamic Model& (J.C. Spa.ll, Ed.). Ma.rcel Dekker. New York. cha.pter 12. Karny, M., D. Ha.jma.n a.nd A. Halouskova. (1994c). On conserva.tive recursive lea.rning. In: Preprinu of the European IEEE Work,hop CMP'9-4 (L. Kulha.va., M. Ka.rny and K. Wa.rwick, Eds.). pp. 77-82. UTIA AV CR. Pr~ue. Kulha.vY, R. a.nd M. B. Za.rrop (n.d.). On general concept of forgetting. International Journal of Control fiS(4), 905-924. Milek, J.J. a.nd F.J. Kra.us (1991). Sta.bilized leut squares estima.tors. Technical Report 91-02. ETH Zurich, Automa.tic Control La.bora.tory. ACL ETH Zurich, Switzerla.nd. Nedoma., P., M. Ka.rny a.nd A. Halouskova. (1994). Recursive a.pproxima.tion by ARX model: Simula.tion verifica.tion of a.da.ptive controller initializa.tion. Technical Report 1816. UTIA AV CR. POBox 18, 182 08 Pr~ue 8, Czech Republic. Peterka., V. (1981). Ba.yesia.n system identifica.tion. In: TrendJ and Progrell in SlIdem Identification (P. Eykhoff, Ed.). pp. 239-304. Perga.mon Press. Oxford.
cha.nce for a.ny efficient solution. The P value is scaling sensitive. This qua.ntity (8) determining the number of degrees of freedom in the derived GiW pdf (10) depends too much on a.ppropria.te selection of the prior sta.tistic (2). This property becomes observa.ble just in structure estima.tion tuk due to the explicit dependence of the result on JI + Otp. When studying this problem a. possible remedy wu found: instead of rela.ting loga.rithms of ge;r a.nd f(8ID(t)), their gradients Ve with respect to 8 a.re used in the hypermodel. This -
ca.n be shown to resolve both the discuaed prob-
lems; - modifies the presented algorithm very slightly: just the mixing ma.trix covp[ln(g.;.)] is repla.ced by its Fisher-informa.tion-ma.trix like counterpa.rt covp{Ve [In(g.;.)]} a.re used. Detailed trea.tment of this line will, however, be given elsewhere.
5. CONCLUSIONS
Simula.tion bued verifica.tion of the potential a.nd restrictions of the general algorithm for combining va.rious informa.tion pieces form the core of the pa.per. It concentra.tes on use of this algorithm for ARX models - the key building block for a. substa.ntial set of a.da.ptive systems. The pa.rticula.r subtuks like RLS initializa.tion, construction of a.n alterna.tive pdf for sta.bilized forgetting technique a.nd improvements of the structure estima,tion algorithm a.re inspected in detail. Their solution is necessa.ry pa.rt of a.n extensive project which is to provide sue a.nd well justified ja.cketing of esta.blished LQG a.da.ptive controllers. For this reuon, the potential use of the technique u a.n a.pproxima.tion and/or model reduction tool hu been left uide. It ca.n be concluded tha.t the proposed technique brings expected a.dva.nt~es for RLS initializa.tion a.nd tuning of the a.dva.nced forgetting technique. For structure estima.tion specific problems ha.ve been encountered a.nd possible remedy proposed but promises in this respect should be verified.
Acknowledgment This work has been partially supported by GA tR, Grant No. 102/93/0228 and by GA AV tR, Grants No. 27554, 275109, 275110.
REFERENCES Abra.mowitz, M. a.nd LA. Stegun (1972). Handbook of mathematacalfunction&. Dover Publica.tions, Inc.. New York. Anderson, T.W. (1958). An Introductaon to Multivari. ate Statadacal Analll&a&. John Wiley. Bohm, J. a.nd M. Ka.rnY (1992). Merging of user's knowledge into self-tuning controllers. In: Preprinu of -4 th IFA C SlImpO&aUm Adaptive Control and Sagnal Proceuang ACASP'9! (LD. La.nda.u, L. Duga.rd a.nd M. M'Sa.a.d , Eds.). Vol. 2. pp. 427-432. Aca.demic Press. Grenoble. Halouskova., A., M. Ka.rny a.nd P. Nedoma. (1994). Recursive a.pproxima.tion by ARX model: Use of 124