Adaptive Control of Silicon Crystal Growth with Suboptimal Reference Model

Copyright© IFAC Evaluation of Adaptive Control Strategies, Tbilisi, USSR, 1989

ADAPTIVE CONTROL OF SILICON CRYSTAL GROWTH WITH SUBOPTIMAL REFERENCE MODEL A. I. Zalyaev*, I. B. Yadykin*, D. N. Lanin** and V. S. Leibovich** *IllStitllte of Control Sciences, Moscow, USSR **Central Institule of IndllStry Automation, AlosCOll', USSR

Abstract. The problem of optimal reference model construction for silicon crystal growth is discussed. The consistency of estimates of linearized model of crystal growth is proved. Keywords. Silicon crystal growth; model reference adaptive control; optimal control; estimation.

SUBOPTIMAL REFERENCE MODEL (RM) OF BASIC CONTROL LOOP

INTRODUCTION Adaptive control systems with reference model (ACSRM) have found wide application in most diverse areas. Their use is particularly effective in control of multidimensional plants, since theoretical analysis of spatial behaviour of ACSRM's for such plants is easier than for other types of adaptive control Systems (Tsypkin, 1985) • Of special application interest are adaptive multidimensional PID controllers which have proved to be sufficiently robust in a number of applications. Tuning of adaptive multidimensional PID controllers optimal in the sense of a certain adaptivity criterion has been described in (Barabanov, 1983). The seme paper suggests an adaptivity criterion for the basic control loop. If a control plant and a multidimensional PID controller are described by the equations x

= Ax

+ Bu, y

=

Let the control plant be presented in the canonical form by its phase variable (Kwakernaak, 1972), i.e. let ~t sta(~)vec­ tor be of the form Y = col(y,y, •• ,y ), where y is the scalar output of the plant. 1: Add the (n+1 )-st coordinate fJ :::f 0 d~dr.: _ Y == <.De (y ~H ) and re-numerate!:J; (j=1,l1 d ) so that ~ILH be number three. As a result, we obtain y =. coe ( If, ~, Sot: ~.Jt , .. , :r("-+l)). Then a q~+1

PD (PID) controller over output y may be treated line a bounded-structure controller (i.e. the one that shapes its control with the use of just a fragment of its state vector). In its general statement, the problem of bounded synthesis of controllers is verbalized as follows (Lanin, 1989) •

Cx,

Consider a control plant described by a system of linear differential equations:

x=-Ax-tB(.(., then the controller tuning algorithm is

T] G_ 1

=

~+

L

(f..i)

where x is an n-dimensional vector of state, u is an m-dimensional vector of control, and A and B is a Kalman-controlled pair of constant matrices of appropriate dimension.

~

. N,

where matrices Land N are obtained from matrices A, Band C of the control plant, and from matrices Am' Bm' Cm' G_ 1m , GQm' G1m of the reference control loop (Yadynin, 1985).

An optimal control must quarantee minimum of the following performance functional along trajectories of system (1):

J=S

There are two points that have remained outside the scope of paper (Yadykin, 1985): (1) optimality of the basic loop reference model itself and (2) consistency and convergence of estimates of the parameters under identification.

00

+

(XTQ.X

(.(T«.IA).,it ,

(,Ll)

o

where R > 0 and Q are constant symmetric weight matrices of appropriate dimension. Let only the first

275

t

< n

components of

276

A. l. Zalyaev et al.

state vector x, be observable. Find the control optimal in the class of feedback controls linear in this fragment of the state vector: (,t.3) u = SMx where S is the (m x 1) feedback matrix and M is the (1 x n) matrix of the form

Thus, operator P~ must be both symmetric snd skew-symmetric. These two conditions are true only of a zero matrix, so

I01·

M=[ E t

Substitute () x

Matrix P is symmetric with any argument of operator p(s) since the expression in the parentheses is always symmetric.

=

into (1), then

f Differentiate ( 6 ) with respect to s with the account of (~.i(); as a result, we obtain the following (index "*", for simplicity, will hereinafter be omitted):

(A + BSM)x

Denote G(s) T(s)

A + BSM Q + r,rTST RSM

Then f unctional (2) may be written as

r~ G- \ = Xo [~e o T

J

Te

(it

elt

JKo

(1.5)

where Xo E Rn is the vector of initial conditions for system (1). Denote

P

= S~e Grir e G "Cdt

• I f the

eigenva lues of ma trix G(S) have negative real parts (i.e. if our c losed-loop control system is steady-st a te), then P is a converging integral. In this case, ma trix P satisfies the following Lyapunovtype equation (Tsypkin, 1985)

Thus, minimization of f unctional (2) becomes minimization over s of the quadratic form: ( .t . ~)

which manifests a transformation to the f inite-dimensional parametric (s)-optimiza t ion problem. Consider operator p(s) directed from the space of (m x I)-matrices s (i.e. spa c e Rml) into the sp~ce of (n x n)-matrices P (1. e. space Rn ). To satisfy (7) with any ne c essary that - PsI

\

5

XOE Rn, it is

"s·

s

=

G:I"'\. 1. (p ( s+ h ~ S) - p (.s) )

~-.. 0

h

Algebrai c ma trix equations ( 6 ) and (10) make a system with two unknown ma trices P and S : PB + J,lTST R = 0 (1. .(1) ( AT+MTSTB)P + P( A+BSM )+ Q+MTSTRSM o Note tha t with M = En' system (11) turns into e well known result obtained for the deterministic problem of state controller design: the exclusion of S fro m (11) gives us a Pic c ati-type algebraic equation for matrix P (Tsypkin, 1985). This a c tually le a ds to one more way to obtain feedback equations for a st a te controller which, so far as these authors know, has not been yet described in literature. Let us dis cuss the fo rm of ma trix P in the case when M = [ E1 I 0 ] , f < n. In order to make (11) sol vable in this case, matrix P must look a s follows:

p =- [

! ~ _: _~ _] o :

e

0

e. l1.-e

II-f

(the upper righths nd squa r e of ma trix P, due to its symmetry, is zero). Decompose matri c es A, B, C and Q into the following blocks:

(.e .f)

where pI - is the Fre chet variable (KolmogSrov, Fomin, 1967) of operator P(s), and s is the optimal feedback matrix. Relationship (8) is simplified if we take into account that p~ must at the same time be a symmetric operator, since it is a limit of the difference between two symmetric operators:

p'

wgere A S is sn element fro m ~l. Necessary and suf fi c ient f or any S ~l to satisfy (9) is the f ollowing e qua li t :l' : PB + MTSTR = 0 (2.. .10)

A :: [

All: A 12) e -A-2.~rA-z.:- l1.-t e

I'I.-e

e

l7-t

Ad a ptive Contro l o f Silicon Cr ystal Grow th

'2 77

For consistency of (11). it is necessary that Q22 = O. Expression (11) then may be brought to the form:

Let us say that of matrices T1 , the elements of of the elements

G11 = A11 + B11 S T G11P11+P11G11+SRS + Q11 , - T S R B11 P 11

In this sense, matrix Q is e function of matrices A and B (expressed via matrix P). If matrix Q is sign-definite. the necessary and sufficient condition for the existence of an optimal control and for the finiteness of functional (2) is not Kalman-controllability of pair (A.B) but. r a ther. satisfaction of the Yakubovich frequency condition expressed in the form ( Yakubovich. 1987)

= Q12

-P 11 A12

o}

(H")

(;..a)

( 11')

Subsystem (12') determines a st a te c ontroller for a subspace stretched over the observa ble components of the state vector. This sUbsystem is consistent And its solution is a (1 x I)-ma trix P 11 > 0 (i f Q11 ~ 0) when the pair (A 11 • B11 ) is c ontrollable (Tsypkin. 198 5). Howe ver. the condition (1 2 ') must be satisfied too. Since Q12 is not included in (12'~. the consistency of (12) re quires that Q12 ha ve the form of (12") where P11 is found from (12'). This requirement is not particularly stringent. since weight ma trix Q may be assigned in a rather arbitrary way. Even though. aa shown in Yakubovich (1973). Q becomes sign-definite. this does not imply (even with an infinite time of control used as a performance fun c tiona l) that there is no minimum for functional (2). For the existence of this minimum and for obtaining a steady-state c losed-loop control system. it is necessary and su ff icient that the Yakubovich fre~uency condition be met for all w Co R: «iW En - A)-1 B)T Q «iW E - A)-1 B)+R > 0 n

u . ("')

P 11

0 ) is. most evidently. Matrix P = [ 0 a solution to the Ricc a ti equation f or system (1) with weight matrix Q in the performance fun c tional ha ving the form:

Q11 .... I1I-P11 A12]

Q _ -

[

()T ~I A 11.

0

U . ff)

Since checking of frequenc y c ondition (13) is too difficult (for this purpose. one should analyse the behaviour of the resolvent of ma trix A on the ima gina ry axis). it suffies to simply check the st a bility of matrix G. If G appears to be a st a ble matrix. then conditi on (13) is evidently satisfied. In this case. brings minimum to fun c tional ( 2 ) in the f orm (7) with weight matrix Q of the form (14) •

Let us consider a situation which occurs owing to the sign-definite and non-arbitrary nature of matrix Q. This matrix takes a special form:

~11

-P 11 A12 ] -A 12 P 11 0

Q =[

matrix U is a function T2 , •••• Tn (n=1.2 •••• ) if matrix U are functions of matrices T1 , T2 ••••• Tn •

(for the functional of type (2) not c ontaining any terms with cross-produc t s of the c omponents of control and state vectors U and X. respectively). Then.

Denote the characteristic polynomial of the (n x n) matrix A:

and then expression (*) may be transformed into

04 where

(i 4J) hA I

~

(-iw)

*" cL

dd. &i(A,B) (* * )

=det BR-1 BT (is independent of

A) •

Th us. expression (**) must be true with any wc-R1. Note that in differentiating the Lyapunov equat ion. ma trix Q was re ga rded cons ta nt And its deriva tive was t a ken equal to zero. Howev er. Q in (14) depends on S but it turn s out t hat Q's = O. sinc e Q11 = const, A12 = c onst a nd Pi1s = 0 by virtue of optimality of S aa a state co ntroller f or pair A11 • B11 • there f ore none of the blocks of matrix Q in the form (14) i s independent of s in point S*. Consequently, t~e e~tire matrix Q.iS independent of S ~n S. ~.e. Q I s=s = O. PARAMETER ESTDlIATION FOR ONE CLASS OF NONLINEAR PLANTS Ref. (Linnik, 1962) overviews extensive literature on appli cation of the least squares technique (LST) to estimation of the constant parameter vector linearly incorporated in a noisy observation model. Presently. the consistency of estimates is generally determined. as shown in (Barabanov. 1983; Chen. 1986; Lai. 1982) by some properties of the information

2i8

A. 1. Zalyaev et al.

matrix (to be more exact. by its minimal and maximal eigenvalues). This section describes a certain class of nonlinear control plants linear with respect to the parameters such that the conditions of theorem from (Barabanov. 1983) are satisfied and a strong consistency of the LST estimates is present. Some efficient estimates of the rate of convergence of the estimated parameters to their true values may be suggested. Consider the problem of identification of unknown vector Q in noisy observations over a scalar signal. Y: T

U

>It

+ i = e~... cp~~

+

Z.

t; +t

(

t = 1,.t.,,) (~ .

i.)

Process
6r

T

Pt .pt: ('PH

( -

T

Pt.-f

where

<:f>-t; (1+
e; 1't)

7"

~-1 ~ )1i ~., (?, . .tJ

Pt is the sequence of matrix:

where

;( min and

d max

mal and maximal eigenvalues of information matrix R1 • It is then evident. that the above limit is 0 if (5) is satisfied. This makes it possible to obtain efficient eSf timates for the rate of convergence of

(2).

Using (5). it is easy to estimate 2 min and ~ from above, as well as their max * relations A. Win"" k t, Amax"" k*t. where k* and k are respectively, the minimal and maximal eigenvalues of the constant matrix whose every element is a majorant of the respective element of information matrix R1 (namely. elements r ij • and r ji of matrix Rt are majorated by the product : j ). It may be possible that (5) is violated for some arbitrary values of X. Let. for instance, ~K be multiplicatively dependent on X:


fl'"

Xj

«.. 1'-;

(oLKj

J~i

The major results of this section are: A. A new proof based on the results of (Barabanov. 1983; Chen. 1986; Lai. 1982) is obtained for a strong consistency of LST estimates in the case of the regressors bounded in space L2 • B. A class of the so-called positively dissipative system is determined. in which the regressors multiplicative-dependent upon the stave vector components are bounded and positive in L2 • and consequently. a strong consistency of LST estimates linearly incorporated in the system of parameters is guaranteed. The LST estimates are strongly consistent (Fomin. 1980). if information matrix (3) satisfies the inequality:

o

~

Et a ,.

~

Rt

~

,.

atE; P ( a,. ). 0 )

= 1.

M(a

*2 )

(,00

•

It is evident that if each of regressors ~ k (k=1 ••••• p) satisfies inequality ,. (5) 0 <4>k < Cf>k' then (4) is true. However. in this case no efficient estimates exist for the rate of convergence*of the sequence to the true values of e . At the same time. as shown in (Barabanov. 1983). there exist conditions of strong consistency of LST estimates weaker than (4). These conditions sre as follows:

sre the mini-

E f(.1,

f(,=.

f,,,,f)

(3 . ")

in which case (5) is certainly violated for Vx E ({It • There exists, however, a class of control plants for which x Eo X C. Rn. On the tot ality X. assertion (5) for to

o <

*

k <; Xk = const Theorem. Let (1) an object whose parameters are to be identified feature the property of positive dissipativeness. (2) be linear in the parameters to be identified and (3) its regressors are multiplicatively dependent on the components of the state vector, then the LSM estimates for the unknown parameters are strongly consistent. X

The proof follows from the fact that wi thin the region {X Eo Rn : V x E: X, o " Xi < X • regressors (6) satisfy inequality (5).

*J

In compliance with the above. a strong consistency of the RLS estimates for is guaranteed here, and there exist efficient estimates for the rate of con-

Adaptive Control of Silicon Crystal Growth

vergence of P

ef:

to

[1119." - e* It <. E ~ >

&*

(namely,

0).

Thus, a cless of nonlinear control plants is defined for which the method of guarantees strong consistensy of parameter estimates. These control plants are dissipative and their attractor must en. tirely lie in the positive N-tant of R (where n is the dimension of the state vector). CONCLUSION Consider a control system describing the process of cilicon crystal growth from the welt (Leibovich, 1986): At+1 =

e

= F(xt,Ut)+A( )f(x t ), where the (n x p) matrix A( B) has the form A (tJ ) = [

e1

&;... J fY P

Parameters e are included in the second control only. The components of vectorvalued function f(x t ) are multiplicatively dependent upon Xj (j=1,n) and serve as regressors with .(j=1,n). The obJ served output is the sequence of values X (t). The regressors are precisely de2 termined, because all components of the state vector are measured accurately. Thus, the described control plant satisfies the conditions of the theorem, and the estimates of ~j are strongly consistent. The control plant turns to be instable in Lyapunov's sense which, however, does not effect its identifiability via the use of the LST estimates (Lai, 1982) • After the unknown parameters are estimated in compliance with section 1, the tuning controls are found for the twochannel PID controller using the suboptimal reference model of the basic loop determined in compliance with section 2 for the model linearized with respect to the nominal (stationary) mode. In the above adaptive control system with reference model the accuracy of the crystal diameter control amounts to 0.01 percent which disproves the assertion (Astrom, 1987) that it is impossible to design an adaptive system controlling the process of growth of silicon crystals. REFERENCES Astrom, K.J. (1987). Adaptive Feedback Control. Proc. of the IEEE, 75, 185217 •

Barabanov, A.E. (1983). About a strong convergens of the LMS. Autom. & Rem. Control, 10, 35-38 (in Russian). Chen, H.-F., and Guo, L. (1986). Convergence rate of least-squares identification and adaptive control for stochastic systems, Int.Jour.Contr. 44, 5, 243-250. Kolmogorov, A.N. and Fomin, S.V. (1967). Elements of the Theory of Functions and Functional Anal)sis. Nauka Publ., Moscow. (in Russian •

279

Kwakernaak, H. and Sivan R. (1972). Linear optimal control systems. WileyInterscience, N.Y. Lai, T.L. and Wei, C. Z. (1982). Least squares estimates in stochastic regression model with application to identification and control in dynamic systems. Analls of statistics, 10, 1, 68-74. Lanin, D. and Yadykin, I. (1989). The Computer-Aided Design of Regulators with constrained structure. In CAD System Desi~n IFAC!IMAX Workshop, Moscow, p. 9 • Leibovich, V.S. (1986). Dynamics of crystalization process. In Crfstals growth, Moscow, Nauka Pub • (in Russian) • Linnik, Yu.V. (1962). LSM and mathematicstatistics theory foundations of observations processing. Fizmatgiz, Moscow. (in Russian). Rouche, N., Habets, P. and Laloy, M. (1977). Stability Theory by Lyapunov's Direct Method. Springer-verlag, New York. Fomin, V.K., Fradkov, A.L. and Yakubovich, V.A. (1980). Ada~tive Control of Dynamic Plants. Nau a Publ., Moscow. (in Russian). Tsypkin, Ya.Z. and Kel'mans, G.K. (1985). Discrete Adaptive Control Systems. In Itogi Naukie, Kibernetika.VINITI, Moscow. (in Russian) Yadykin, LB. (1985). Optimal tuning of linear regulators. Report's of Ac.Sc. USSR, 285, 3, 517-524. (In Russian). Yakubovich, V. A. (1973). Frequency theorem in Control Theory. Sib. Math.Jour., 14. 2, 118-125. (In Russian). Yakubovich, V.A. (1987). Frequency theorem for periodic system and analytical regulator construction theory. In Lyapunov's functions method in s'stem d namics anal siSt Nauka Publ., Novosibirsk. In Russian).