Self-Tuning Long-Range Predictive Controllers

Copyright © IFAC 10th Triennial World Congress, Munich, FRG, 198 7

SELF-TUNING LONG-RANGE PREDICTIVE CONTROLLERS G. Favier CNR SILASSY, 41 bd Napoleon Ill, F-0604 1 Nice Cedex, France

~.

A 00..., cless of control metOOds. called long-range predictive control metOOds. h8s been recently introdOO3d. Among these metOOds. Geoora1ized Predictive Control (GPC) appears very promising for applications, In this paper some improvements of this method are presented and experimental results obtained ...,ith a real tracking system are reported. Keltw'Ords. Mlptive control. Predictive control. Prediction. Tracki ng systems. Applications of Mlptive Control. Equation ( 1 ) can be re...,ritten as ()«2-') y(t) = B(2-') 6u(t-d) + C(2-')e(t) ...,ith 6u(t) 4 u(t) - u(t-1)

INTRODUCTION Long-range predictive control (LRPC) metOOds. i nt rod I£ed by Richalet and co-lJOrkers (1978). are b8sed on the tlJO follo...,ing steps : * Use a predictive model of the process to be controlled to forecast the process output over a long-range hari zon. This prediction is composed of tlJO terms : the fi rst 000 is function only of post input-output measurements and the second 000 is function of the future control input. * Determioo future inputs in order to satisfy the control objactives in terms of a desi red future reference trajectory. This step is function of e future control scenario. that means constraints on future controls.

()«2-')

4

A(2-') 1(2-') • nQC= n A+1


()(j

A

=a j -aH

iE [1.n l A

1.

...,ith

(10)

(11 )

n G. = J j=O

...,here d ) 0 is the time-delay. 2-' is the bocklo'8rd-shift operator. I( 2 -, )=1-2 -, is the bocklo'8rd difference. {e( t) }is a ...,hite noise sequence and the polynomials A( 2 -'). B (2 -') and C( 2-') are defi ned as :

1,gj z-j

2.

(12)

j

Computation of ij(t+j/t) by use of equation cij(t+j/t) = ~ y(t) + ej 6u(t+j-d)

(2)

...,ith

eJ =BF j • ne(nB+J-1

(13) (14)

(3)

When a long-range predictive control method is used \018 have to compute J-step-ahead predictions for j =1 .....J. Each prediction can be obtai ned by applyi ng the procedure of Table 1. Ho\ol8ver this solution is very time-consuming because it needs the sol ution of J diophanti 00 equatiOns (9) .

n 2- e rlc ~

(8)

Determination of polynomials FJ and Gj solution of the Diophanti 00 equation C = ()( FJ+ 2- j ~ J~ 1 (9)

(1)

Subsequently \018 shall essume that

(7)

The computation of the optimal J-step-ahead predictor ij(t+J/t) ...,hich minimizes the variance of the J-step-ahead prediction error y(t+J/t) ~ y(t+j) - ij(t+J/t). and ...,hich is b8sed on all available i nput/output measurements up to and incllldi ng time t {y( t) .6u( t-1 ). y( t-1 ) ,6u( t-2) ..... 6u( 0), y( O)} and the future control increments {6u( t) ,6u( t+ 1) .... . 6u(t+J-d»), is described in table 1 (Astrom.1970) . TABLE 1 Compytation of the i -step-ahead predictor

MODEL OF THE PROCESS The process is essumed to be described by a CARIMA model

C( 2-1 ) = 1 + Cl 2-1 + .... + C ne

(6)

OPTIMAL J-STEP-AHEAD PREDICTOR

Different LRPC metOOds \oI8re developed dependi ng on the type of predictive model. the control objectives and the future control scenario. De Kel)Ser. Von de Velde and Dumortier (1985) and Di Marti no (1985) compared different self adaptive LRPC metOOds.Recently. Clarke. Tuffs and Mohtadi (1984) proposed a 00..., method called "Geoora1ized Predictive Control" (GPC) . In this paper \018 give some improvements of this method and real experiments on a tracking system are described.

B( 2- 1) = bo+ b z-1 + .... + b% 2-% 1

(5)

nA '

AC-O

83

84

G. Favier

T'w'O alternative solutions can be developed. The first one consists in computi ng the J-step-ahead predictor by use of a recursive formulation ..,ith respect to step j. The secooo one is based on the use of a multistep formulation of the j -step-ahead predictor. These solutions are presented belD'*'.

the j -step-ahead predictor. This metOOd is deseri bed in Table 2. TABLE 2 Recursive Computation of the j -step-ahead Predjctor

2.

Initialization (J=1) ( 15)

For j = 1..... J-1 •

Recursion of FJ j+l

fj

j

=go

(28)

J

Clarke. Tuffs aoo Mohtadi (1984.1985) proposed a recursive metOOd to compute the polynomials (Fj+1. ~+1) of the (j +1) -step-ahead predictor from the polynomials (Fj .Gj) of

F1= 1 • G1 = z(C-(X)

(27)

C. =

Recursive Formulation of the j-step -ahead predjctor

1.

..,ith

(16)

(X(Z-I) y(t+J/t) = B (z-1)~u(t+j-d) + Cj (Z-1)[y(t)- y(tlH)] (29 ..,here the operator (z- ') is only 'w'Orking on the first part of the time argument in Ul t+)/t) ..,h1Ch means that n(X (X(Z-I) y(t+j/t) = I (Xi y(t+j-i/t) i=o

(30)

..,ith y(t+J-1It) = y(t+j-1)

(31)

..,hile 'w'8 have n(X

(17)

•

Recursion of GJ j+l

gi-l j+l

gn

A

j

=gi- go (Xi

=-go (Xn A+ 1

dra..,bacl<: is that all the predictiOns y(t+1It) for i E [1.J-1] (19)

•

Computation of !?>J+1 = BFJ+1

•

Computation of y(t+J+1/t) byuseof

(20)

equation (13) in replacing j by j +1. We no.., give a recursive computation for the polynomial!?>j ' From equations (16). (17) aoo (20) is is easy to deduce the foll 0.., i ng relation !?>j+1 = !?>j + z-j g~ B (21) aoo usi ng (15) ..,e get the recursive procedure of Table 3 for computing !?>j. TABLE 3 1. 2.

Recursive Computation of !?>J

(22)

For j=1 .....J-1 ' 1 ' . 0.)+ + gO' b·1-). i E [O.nB+j] ~i -- 0.) ~i

(23)

Remark:

i (J

(24)

From equatiOns (23) -( 24). 'w'8 have

!?>~=!?>~

'v'(J.€»i~O

have to be previously calculated. Ho'w'8ver ..,hen a long-range predictive control is considered the determination of these predictions is necessary over e prediction hari zon Np. THE CI:NERALlZED PREDICTIVE CONTROL The GPCstrategy is based on the minimization of the follo..,lng Quadratic cost function. d+!!e-1 V = ,2 J:d

[e (t+»+ >- ~ u (t+J-(n] 2

2

(33)

..,ith respect to the future control increments ~u(t) .... .... ~u(t+Np-1). >- is a '<.'Bighting ord e( t+j) is the difference bet'<.lB8n the J-step-ahead prediction of the prooess output ard the future output YM(t+j) of a reference model. e(t+j) ~ YM(t+j) - ij(t+j/t)

Initialization !?>1 = B

..,1th bi-j= 0

(32)

Remark: An advantage of this multistep formulation is that the calculation of the polynomials (I?>j. ~) is not needed. A

j

j

(X (Z-1) ij(t+J/t) =I (Xi y(t+J-1It-i) i=o

(25)

(34)

Defining the vectors of dimension Np equal to the prediction horizon e,(t+d) = [e(t+d) e(t+d+1) .... e(t+d+Np-1)]T (35)

~(t+d)= [YM( t+d) Yt1( t+d+1) .... YM( t+d+Np-1)]T

(36)

i(t+d) = [ij(t+d/t) ij(t+d+1/t) ...ij(t+d+Np-1/t)]T Clll(t+d) = [6u(t) 6u(t+1) .... 6u(t+Np-1)]T

(37) (38)

the cost function (33) can be rewitten in vector form V = e,T(t+d) e,(t+d) + >- ClllT(t+d) Clll(t+d)

(39)

....ith

e,(t+d) ~Y.t1 (t+d)-i(t+d).

(40)

Multistep Formulation of the i -step-ahead predictor Akai ke (1975). (De Kelll9r aoo Van Cau'w'8nberghe ,1981) ard (Good..,i n aoo Si n .1984) proposed a multistep formulation of the j -step-ahead predictor. See also (Favier-Cresp.1 981) aoo (Favier.1 987) for a comparative study aoo a revie.., of various formulations of the j -step-ahead predictor . Writing (26)

Clarke. Tuffs and Mohtadi (1984.1985) proJX)Sed 0 GPC ol(}lrithm in the case .... hen nc=O. This al(}lrfthm uses the recursive computotion of the j-step-ahead predictor .... hich is described in teble 2. The control la.... minimizing the cost function (39) can be obtained in separating the predicted output vector into t'JO terms : ore is function of the future control increments vector ~(t+d) to be opti mi zed and the other t+d) is

Yt(

only depending on available {y(t) .6u(t-1) .y(H) .6u(t-2) .... }.

input/output

data

85

Self-tuning Long-range Predictive Controllers i(t+d) = M ~(t+d) + ~(t+d) ....Ith

~O

(41)

0

0

litter .... hlch ell control I ncrements ere taken to be zero. The M matrIx Is then of dImension (NP.Nu) arc! the matrIx O=(M TM+},I) to be Inverted is of dimension(Nu.Nu), ....Ith In general Nu « Np. To decrease the computatIon time for the inversIon of 0 'J8 prolQe to use a recursive al(J)rlthm ....1th respect to the dImensIon Nu,lrQed this matrix has the follO'JIr¥,) partItioned form

~~ ~ (42)

M=

Np-1

~N

~N Np-1

Np-2

I

I(mY+~:

0

i:o__ ~ __ J

I

I m.. m. 1 : j~

1

1+: I I

Np-2

1~

%+1( 2-' ).y(t)+

+22[~+1(2-')-~~-~~(2-') ].~u(t-1) ~(t+d)= :

I~

1

I

( 3) Nq;:.Nu NII,.:.Nu Nq;:.Nu l..m.ml +Nu -1 l..m . ml Nu_2·····-·l..(ml+},

GN( Z-, )y(t)+ZNP[~N( Z-')- ~~~Z-I].~U(H) i=o

N~ d + Np-1

I~ I

I~ 1

I~

+

(50)

(44)

Or In 11 concise form

Minimization of (39) IEms to the 'J811-koo.... n leest-squares solution ~(t+d)=(M TM+), 0- 1 MT(Yr1(t+d)-~(t+d» (45)

I

°t: -----I

I I I I

I I I I I I I I I

O:

_________ 2 J

In practice only the first control input u(t)=u(t-1)+6U(t) is appl1ed to the process. arc! at time (t+1) the ....hole procedure Is repeated to compute ~u(t+1). etc. We 00 .... describe some improvements of this el(J)rlthm. First. 'J8 give a generali zation to the case .... hen nC>O by use of the multistep

0=

(51)

°Nu-t:

---- -------------~

formulation (29) of the j-step-aheed predictor. The complete procedure Is descri bed in TABLE 4. TABLE 4

:

I ( mj)2+},:

-----------------j

0=

Hp-t

arc!

Np-2

~ mj . m'+1

Np-1 0,= I(m )2+~

....Ith

I~

A Ne", GPC Aloorlthm

0Nu =0

I

(52)

arc! 1. Constrl.Ctlon of the M matrix defl ned!lS • dimOt(I.O.le[2.Nu]

(53)

dim Bi =(1-1,1) • dim Ej =(1.1-1), dim Dj=( 1.1) (54) By USlr¥,) the matrix inversIOn lemma 'Je get the recursIve al(J)rlthm of Table 5 for invertir¥,) O. ....here the coefficients mj can be recurslvely calculated by use of the follo .... ir¥,) equatiOns mo =bo (47)

TABLE 5

ftl

1.

Hin(llotJ)

m. =b. )

)

I

1=1


.

calculate

(48)

2.

)-1

-1

(55)

For i =2 .....Nu • Form the Quantities BI' El. 01 !IS H 21 j ~I E{[el .... el el] ....Ith e{ (:, mk mk+j

3. calculation of the vector ~(t+d) by use of equation (29) for j=1 .... ,d+Np-1. ....hen repleclr¥,) ~u( t+i) by zero for hO. erc! of the vector Yr1( t+d) 4. calculatIon of the first control Increment u(t)=u(H)+ l(Yr1(t+d)-it(t+d»

0~1 = [ 2. m:+ ~ ] k~

2. calculation of the fl rst ro .... ':LT of the matrix (MT M +},W 1 MT

=>

Recursive Aloorlthm for jnvertloo 0

T

~I

2

1

1 k~

•

B.= E. • 0.= ~ m. + ~ 1

(49)

ThIs el(J)rlthm Is sI mpler then Clarke's one because It doesn't need the recursive calculetion of (~j.~) . Only the coefficients mj hIlve to be recurslvelydetermlned. A dra.... tx.:k Is caused by the InversIon of the matrix (MTM+ },I) .... hlch Is of dimension (NP.Np). Clarke.Tuffs arc! Mohtedi (1984,1985) prolXJSlld to simplify the solutIon by constrainir¥,) the vector ~(t+d) In Sl.Ch a 'JaY thet there Is a control horIzon Nu

(56)

(57)

• calculate oi' by use of the formula

oI

1~

0- B~ H I 1 J [ =[ ---~-+ ----- [0 -[ 0- B] E 0-1 0 10 I I H 1 I H .

-1

I

-1 : 0 OH I I

I

1

(58) ] -1

-1

3. Then 0-' = O-NU

(59)

I

G. Favier

86 ADAPTIVE IINERALlZED PREDICTIVE IllNTROL

I"

~

~~~~~~~~~r;,j~Tr.Cklna window

When the mOOel perameters (A,B ,C) are unkro..,n aM/or slOlo'ly time-varylrg, it is possible to use a self-tunlrg approach to IJ)t the OOeptlve GPC algJrlthm described In Table 6. TABLE 6 Fmptjve GPC Aloorithm

I

I I. I I

IC

IS

I

I

1. Choose the tunirg perameters Np, Nu, },. 2. Estimate the coefficients of the polyromials (A,B,C) of the process mode I fly(t) = !pT(t) e + e(t) ~ 6(t)

a

Q

: I

-------1

I I

I

I

Llne-

I I

I I I

Be.ri"" .)(1.

I"

-I" :

(60)

T e = [a, ... anA bo ... b nB c, ... cnC 1 opT(t) fl [-fly(t-1) ... -fly(t-nA) flu(t-d) ... flu(t-d-nB) e(t-1 ) ... e(t-nc) 1 e(t) =fly(t)_opT(t) e(H) 3. calculate = (1-2-') A. 4. llilntical to step 1 of Table 4. 5. Use the al(Jlrlthm of Table 5 for computirg 0-' . 6. calculate the first ro.., ~T of 0-' MT. 7.8 llilntical to steps 3-4 of Table 4.

C

I I

I Target I

I I I

I

I I I I

(61)

I I

(62)

I

J

(63)

-I"

(64)

APPLICATION OF GPC FOR IllNTROLLlNG ATRACKING SYSTEM The GPC method 'J8S applioo to control a trackl rg system ..,hich h8s to be able to point as Quickly as possible In a preselectoo direction (tolJ8rd a tariJ)t lDC8too at (YM ,YM » aM to tracle automatically s g movlrg taflJ)ts (helicopters, alrplanes aM missiles). This system ..,hich is equi pad ..,ith an optical sensor (a troolel rg-TV camera), h8s tlo'O degrees of rotational fraaoom correspoMlrg to motions arouM tlo'O axes (vertical aM horizontel motions) . The position of the system Is lilflnOO In terms of tlo'O argles : elevation (vertical position: Ys) aM bearirg (horizontal position : ~). The polntlrg control system ..,as Implementoo In Assembler on a microprocessor MOTOROLA 68000, aM the block di~rom of the system is sho..,n in Fig.1.

Fig. 2.

TV im8lJl aM argular dlfferarces

To simplify the control calculation the trooleirg system 'J8S consllilroo as tlo'O moro-I nput/moro-output systems, one for eooh axis. Various control lalo'S Io'ere implementoo on the mIcroprocessor aM experlmentoo for controllirg the troolelrg system : Pole Plooement methods (Oustri, 1984), Generalized Minimum Variance Control (Favler aM DI Martlro, 1984-1986), (calaiS, 1985), Generelized Predictive Control (Di Mortiro, 1985) eM Mlptive PlO controllers (Dirraca, 1986). Here balo.., Io'e report some results obtai nOO ..,ith the GPC method. The control system blocle dl~ram for a si rgle axis Is presentoo In Flg.3. TW'O situotions are possible. In case 1 (see Fig. 1 am 3) the traclelng system has to be pointed in a preselected direction am the desired future reference trejectory Is then exactly leno..,n. In cese 2, correspomlng to eutometlc traclelng of e movi rg target, the future refererce trajectory is unkno..,n am it is calculated by use of a Self-Tuning Predictor (STP). This STP Is based on the Identification of a polynomial model of the trajectory (Favler am Smolders, 1984), (SmolderS,1986) .

~

Position ______~Comm.nd

Electra Mech.nlc.1 System + Sensors

Potentiometers

• Filt. 1

SlIltem constral nts t'elght : 0.82m ROOlus (for a 3600 bearlrg rotation) : 0.62m Weight (..,ith the peyload) : up to 360 leg Inertia (maximum value for eooh axis) : 35m 2 leg. TIo'O types of sensors are evailable : Potentlometers deliver position meesurements (Ys'~) of the TV camere optical exis (line-m-sight) aM the .TV camera IJ)nerates an Im8lJl ..,hlch Is processOO (I ffi8IJl processlrg system = trooleer) to deliver video signals (cs' fog) corres~Mlrg to argular dlffererces betW'OOn the llne-m-slght aM the terlJ)t position es is sho..,n In Fig. 2.

Three types of contro 1 are COJ)Sidered : - NAGPC : Non lldapti ve GPC. - AGPC : lldaptlve GPC ..,Ith a preprogrammed control gain. - STGPC : Self-Tunl ng GPC ..,Ith recursive Identification. G( p) am Bo( p) represent the transfer functions of the tracking sl"lStem am of 8 zero order hold (Dlgit81 to Analog Convertor) respectively. The non linearity is slX:h that _

) u(t)

u(tt.=/sign(u(t»uMax

if lu(t)1 ~ uMax if lu(t)1 > uMax

(65)

87

Self-tuning Long-range Predictive Controllers

-- -----.---- --- ---- ._. -._- --- ---_._-----------l--r··_··~~~~~··!.!:~~·C:!.. ··_··_··_··_··_··_··_·_ _ _ _ _ _ _ _ _ _ _ _:

Microprocessor

AGPC

_ _~

Preprolr.mmed . ._ - - - - I control a.ln

'-=-~':""::=--

____I Prediction

Model 14----1 Identifier

2

Control System Block Di.lr.m

A freque~y domain aneh.sis gave the follo .... ing approximated model .... hich is valid for a 10.... elevation angle.

~: u

s

Yo u

:.21i: z-'(0.016+0.0318z-'+0.0045zA

0

1-1,406z 1+ O,472z

2

2

)

#

-O,066z-'

#Z[Bp(P)G(p)]

(66)

P

We can notice that this model contains an integrator factor (1-z-1) and a zero out of the unit ci rcle (non mi nimum phase model). We no .... present experimentel results obteined .... ith the three GPC algorithms: NAGPC, STGPC and AGPC. The tuni ng parameters ....are cl'osen such that Nps= NPg= 3, Nus: NUg= 2, >'s=0.6 10- 3 , >'g= 1.2 10- 3 (67) We first present results obtained .... ith sinusoIdal trajectories, in using NAGPC and STGPC algorithms. Then,....a she .... real experiments of AGPC algorithm to track an helicopter. Use of NAOPC aloorithm

The reference trajectories (I.M ,I.M ) are si nusoids varyi ng from -80° to 80° for the beari ~ ang~e and from 0° to 80° for the elevation angle. Moreover these trajectories are in phase in that the maximum values (80°) are obtained Simultaneously for the bearing and the elevation engles.In Table 7 ....a summari zed some experiments .... hich ....are carried out .... ith NAGPC algorithm and the results of .... hich are 111 ustrated on FigA-9.

.

t-kl n tm gt'l~ GPC Sinusoidal Traiectories

TABlE 7 Fil

o·

YM s '+1g

~

<4

Ys

7

6

9

8

80·

80· 80·

-80·

YMs

I

!

Us

IYMo' ~ I Uo

-80·

-

YMo' ~

The reference trajectories (YM ,YM) and the tuni ng s 9 parameters (67) ....are cl'osen as preVIOusly. In Table 8 the figures 10-17 correspondi ng to some experi ments are described. The identification algorithm .... hich ....as used is a factori zed versIOn of the RecurSIve WeIghted least Squares algorithm ....ith a constant trace (Favier, 1984, 1987) . In Table 8 Tr represents the value of the constant trace. By comparing Fig.6-9 .... ith Fig.l0-17....a can co~lude that the behavIOur of the controlled bearIng IS Improved In USIng the STGPC. Moreover this improvement is more important .... ith Tr=0.01 than .... ith Tr:0.1. ltI....aver this improvement isn·t sufficient and an adaptive GPC .... 1th a preprogrammed control gain...-es Implemented. TABLE 8 Self-Tuning GPC - SinusoIdal Tralectorie$ Fill

10

Y~

o·

U o

By anah.sing FigA-7 ....e can co~lude that the elevation anglE< is very ....all controlled .... hile the beari ng angle presents osc1l1atlons .... hen elevation Is high ( ~ 450). This means that the model (66) IS satIsfactory for representIng the elevatIOn motion, .... hile it isn·t valid for the bearing motion .... hen elevation is high. This is Quite clear in Fig. 8-9 .... here elevation ....as fixed at 80° and the beari ng output is asci llati ng. ThIS behavIOur can be explal ned by the fact that model (66) ....as identified for a 10 .... elevation angle and the

11

12

13

1<4

US

0.1

. 80· 0.01

YMo 'Yol Uo YMO ' ~

J

16

17

80·

80·

YMg -80· Tr

80'

1

system inertia .... hich enters into the bearing model is depending on the elevation position. That involves a time-v!lrYlng model for the bearing motion .... hen elevatIOn IS varying. T....a adaptive solutions ....ere implemented to solve the control problem of the bearing angle. The experimental results obtained .... ith these t....a solutions (STGPC and AGPC) are no .... reported. t-klte that the elev!ltlOn control poses no problem and an adaptive sol ution ....as·nt needed for this motion. Use of STGpC aloorjthm

-80· 0.1

80· 0.Q1

IUo YM o 'Yo!uoj YMo 'Yol Uo

Use of AOPC Aloorithm The follo .... ing adaptive control la........as implemented 6u (t)= 6Ug(t) cos(Ys(t» (68) .... here 64J( t) is obtal ned by use of the GPC algorithm described in Table 4, and the adaptive gain ....as cl'osen equal to the cosi nus of the elevation angle. Fig.18-20 she .... some experimental results obtained .... ith this adaptIve control la .... for the trackIng of an helIcopter . These results are Quite satisfactory.

g

G. Favier

88

De Keyser ,R.M,C, Ph.G.A. Van de Velde,F.A,G.Oumortier( 1985),

CONCLUSION In this paper IJ8 have presented a ne", GPC method providi ng a generalizatIOn of Clarke's algorlthm to the case rc>G by USlng a multistep formulation of the j-step-ahead predictor. Moreover some numerlCal lmprovements have been proposed in order to redl..C8 the computation ti me (tables 3 and 5). Finally, real experiments on a tracking sl"lStem have been reported and an original adaptive controlla'w' based on the GPC method has been proposed to solve the tracklng problem. Some experi mental results obtained fi rst "'ith si nusoldal trajectories, then ",ith real helicopter trajectories,have been presented to she", that this type of adaptive predictive control is very 1J811 suited to control tracki ng sl"lStems. A::kno'w'ledQement. This 'w'Ork

\oI6S

supported by the Di rection

des Constructions et Armes Navales of Toulon . We are grateful

to M. Oi MartH'() 'w'he realized the experiments "'lth the tracking sl"lStem. REFERENCES Akai ke, H. (1975) . Markovian representation of stochastic processes and its application to the anall"lSis of Autoregressive moving average processes, Ann. Inst. Stat.Math., 26, 363-387 Astrtim (1970). Introduction to stochastic control theory, A::ademic Press, vc I. 70, Ne'w' Yori< Calais,F. (1985). Identification et commands d'un sl"lSteme de visOO, OEA, LASSY, Univ. of Nice, (July) Clarke, D.W., P.S. Tuffs, C. Mohtadi (1984,1985,1986), Self-tuni ng control of a difficult process, Colloque CNRS "Commands adaptotive. Aspects protlques et thEiorlques", St.Martin d't-t9res (1984), also In "Commands adaptative. Aspects pratiques et thEioriques" ,Ed.Masson, Paris ,( 1986) and in Proc. of IFAC Symp. on "Identification and Sl"IStem Parometer Estimotion", York, 1009-1014 (1985) De Keyser R.M .C.,A.R,Van Cau",enberghe( 1982), Aself-tuning multistep predictor application, Automatlca, 1 ,167-174

Acomparotive study of self-adoptive long-ronge predictive control methOds, Proc. of IFAC Symp. on Identification and Sl"IStem Parameter Esti matlon, York, 1317-1322 Oi Marti no, M. ( 1985), Commands opti male pr8dictlve et commands adoptative, Application a10 commonds d'un telepol nteur .These de 3° cyc le , LASSY, Unlv .NIce ,( Dec.) DIracca C.( 1986).Etude comparative de PlO numeriques auto-ajustables. Application ala commands d'un telepointeur. These de 3° cycle, LASSY, Univ. Nice, (Morch) Favier, G. ( 1984) , Regulateurs numeriques auto-aj ustables, DRET Final Report n081/548, LASSY, Univ. Nice (Jan.) Favler, G. (1987a), Computatlonally efficient adaptive identification algorithms, IEEE Int.Conf. on ASSP ,0011118 ,( Apr.) Favier, G. (1 987b), Acomparative study of self-tuning predictors, to be published Favler, G., M. Cresp (1983) , Etude comparative de predicteurs outo-ojustobles, 9th GRETSI Symp. on Signal Processing and its Applications, Nice, (May) Favier, G., M.Di Martino,(1984,1986), Commands adaptative d'un telepoi nteur. Colloque CNRS "Commands adaptative Aspects pratiques et thEioriques", St. Martin d'H9res, 1984, also in "Commands Adaptative - Aspects pratiques et thBoriques", Ed. Masson, Paris (1986) Favier, G., A. Smolders( 1984), Adaptive smoother-predictors for traci
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