Extended Horizon Control of an Epitaxy Furnace

Copyright © I FA C Dmamics ami Control of Chemical ReaclOrs (DYCO RD+ '89). \Iaastricht. The ~etherland s. 1989

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PROCESSES

EXTENDED HORIZON CONTROL OF AN EPITAXY FURNACE

J.

Bordeneuve*,

J.

P. Babary*,

J.

L. Abatut* and K. Najim**

*L.A.A.S./C.X.R.S., 7, At'eIlue du Colollel R oche, 31077 Toulouse Cedex, Frallce **EN.S.l.G.C., c..V.R.S. UA 192, GRECO SARTA, Chemin de la Loge, 31078 T oulouse Cedex, France

Abstract. This paper is concerned with the multi variable adaptive long-range predictive control of an epitaxy furnace which operates in batch manner. The epitaxy furnace is used for the fabrication of electronic devices. The thermal behaviour of the furnace is described by a set of three non-linear partial differential equations. The output vector is composed of the temperature in the heart of the furnace, in the tube, and in the heating elements. The heating powers are considered as control actions. A finite difference discretization scheme is used for the furnace model simulation. A CARIMA plant model is used to represent the dynamics of the considered furnace . The parameters of this model are estimated on-line using the constant trace recursive least squares method. The temperature-time evolution consists of three steps: a rising period, stabilization, and a falling period. So, we are interested in the tracking and regulation problem . Simulation results show that the furnace operates well under this adaptive control algorithm, which is easy to implement. Keywords . Adaptive control; liquid phase epitaxy; long range prediction; partial state model reference; robust parameter estimator.

INTRODUCTION Liquid phase epitaxy used in the production of electronic components is a technique which has many advantages over, for example, gaseous phase epitaxy. It requires simple equipment, layers are quickly deposited, it allows a wider choice of dopants and easier growth of III-V compounds containing aluminium, thus avoiding the presence of corrosive gases. In order to use this technique, the operation of the reactor has to be completely and closely controlled, in particular so as to achieve a high level of reproducibility of phenomena, as well as fairly homogeneous layers. The process can be briefly decomposed into five phases : - Preheating up to 750 0 C, necessary for the purification of the inside atmosphere. - Heating up to 800 0 C without overshooting this temperature . - Regulation of a thermal profile along the reactor. - Slow cooling with controlled space-time thermal gradients. - Natural cooling . Some studies have been carried out for the reactor control. Firstly, PID regulators were used, then optimal and suboptimal control strategies were applied (Benhammou, 1979), (Dahhou, 1980). On the other hand, there has been a growing awareness over the last decade that adaptive control can produce improved performances in real applications, mainly as a result of considerable advances in the understanding of adaptive control theory as well as in the computer technology (M'Saad, 1987). Adaptive controllers can provide good performances for thermal processes which are highly non-linear and distributed in nature.

An adaptive controller will consist of the combination of a control law that minimizes a given criterion, and of a parameter estimator. Of fundamental interest is the ability of the controller to handle non-minimum phase behaviour, load disturbances, time varying dynamics and wrong assumptions about the time delay and model order. The aim of this paper is to show that an epitaxial reactor can be controlled by an adaptive multivariable long range predictive controller that allows easy performance specifications and which is robust with respect to bounded disturbances. Simulation studies are given to highlight the applicability of the involved adaptive controller.

REACTOR MODELlSATION Physical Description of the Epitaxial Reactor The reactor consists of two concentric quartz cylinders, heated by five identical coils distributed along the total length of the reactor, (see fig . 1). The main charge, being a partitioned graphite crucible with a sliding lid for multi layer deposits, is in the middle of the central zone. In this crucible, a Gallium solution is saturated with a polycrystalline Ga-As solution at constant temperature, generally around 800 0 C. This equilibrium having been obtained, the mixture is progressively cooled linearly in time. During this phase, the substrate is introduced. A gas circulates at a low velocity inside the quartz tube. The substrate and the heating element temperatures are available by means of thermocouples.

J.

Borde ne U\'e

1'1 (1 /,

The initial condition is defined as a space temperature profile(i=1 ,2,3) : (4) Numerical simulation, The numerical simulation of (1), associated with the boundary conditions (2) and (3), has been obtained using a finite difference method according to the following discretization scheme (Richtmyer et aI., 1967) :

:;~jl:;(I:;~:; ::;~'l~i:!;~:;I'l':;tij'I(~A:;I

1¥rtf~,li'*ft~~*iJ~w,Uj~Wjt, u,

ay

_' (xl)= at'

IJ

IJ

t. t

iVj (xt)= a(Yrr.i - 2v;'j1 + vi;:.i) + (1 -a)(V(i+1- 2V (i+ vi'H)

U1 U~ Us Fig . 1. The epitaxy furnace . U2

ax2

Mathematical Modelling The reactor's dynamic thermal behaviour is modelled via a balance of the heat exchanges between charge, gas , tubes and heating elements, under a certain number of simplifying hypotheses. This model, a system of three non-linear partial differential equations , enables the evolution of the thermal profile in the different constituents to be determined , given the evolution of the heating power profile . It will be assumed that (Benhammou, 1979): - the process has a cylindrical symmetry; - the gas flow is laminar and one-dimensional ; - the thermal exchanges between the gas and the charge, between the gas and the quartz , are by convection only; - the conduction phenomenon in the gas is negligeable ; - the temperature of the gas in a section of the tube is taken to be the mean value; - the charge has a cylindrical symmetry and is concentric with other tubes. Distributed model Taking into account the above assumptions, the modelisation leads to the following system of partial derivative equations (Benhammou, 1979): aY 1 i Y1 4 -a-=A1 - 2 - +A2 (~-Y1)+A3(Yr Y1) t x

a

aY 2 i Y2 4 4 • .4 • .4 = A4 - - + As (y 2 - Y 1) + As (Y 3 - y 2) + A7 (Y2 - Y1) at al aY 3 i Y3 4 =Aa - - + ~ ~ - Y2) + A10 (Y3 - To) + An P(x,t) at al (1)

The variables Yi(x,t) , i=1 ,2,3 represent the thermal profile of the charge, quartz tubes and heating elements respectively . The Ai coefficients are functions depending on thermal and physical characteristics of the reactor elements, and can be found in Souza Leao and Babary (1983) . To is the ambient temperature, and P(x,t) is the heating power profile . The tube length has been normalized so that x E [0. 1]. The boundary conditions are of the Dirichlet type (i=1,2 ,3): (2) Yi(O,t) = Yi-O Yi(1 ,t) = Yi-1

vf':'"1_y"

(3)

'

t. x2

(5) i=1,2,3.; t=n,';t n=O,1, ... (time discretization) x=j,';x j=O,1•... ,J J=1 /,';x (space discretization) The parameter a is such that ~ a ~ 1 . The particular

°

choice a=O .5 gives the Crank-Nicholson scheme. Using this discretization scheme, an algebraic tridiagonal system is obtained, which has been solved with an adaptation of the Gauss elimination procedure. The values of ,';x, ,';t, a have a direct influence on the algorithm stability. So, the optimal values will be experimentally determined. Sensitivity studies have been carri ed out in (Souza Leao and Babary, 1983).

THE ADAPTIVE CONTROL ALGORITHM An adaptive control algorithm consists of two distinct procedures : on-line identification of a linear process model using a parameter estimation algorithm, and a control strategy . that uses this model as the basis for closed-loop control. Extended horizon control is the basis of many recent studies which use this concept to build the control law. From a historical point of view. this is the result of a logical evolution : one of the earliest algorithms was the Generalized Minimum Variance (GMV) algorithm (Clarke et al. .1975), which was based upon a k-step prediction of the process output, where k is the assumed plant time delay. Two limitations appeared in this algorithm : when the time delay is unknown or time-varying, and when the plant is a non-minimum phase time-varying system . Recently, it was seen that long-range prediction can overcome these problems. Long-range predictive control is a more general approach that considers a range of future predictions of the process output. This extended horizon prediction allows one to deal with a time-varying time delay and a non-minimum phase plant. Of particular interest, from the point of view of both broad applicability and implementation simplicity, is the Generalized Predictive Control (GPC) design , suggested by Clarke et al. (1987) . The latter incorporates three basic des ign features : a Controlled Auto-Regressive Integrated Moving Average (CARIMA) model is used to deal with the underlying prediction process, the involved control objective is based on a general linear quadratic cost function minimization and the assumption made about the future control actions is the one we have seen before . There are . however, two performance limitations that should be outlined : - the control objective involved prevents from straightforward specification of the closed loop performances.

Extellded H o ri zo ll COlltrol of a ll E)Jitax\' Fumace - the control increment assumption, though acceptable for regulation purposes , is not reasonable for tracking objectives. To overcome these problems, we will introduce an optimal partial state model reference control, which has been proposed by M'Saad, et al. (1987) . It consists of reformulating an appropriate partial state reference model control objective in terms of a linear quadratic cost function minimization. The resulting control objective is achieved using the GPC design , together with a suitable plant model reparametrization . The control law is combined with a robust parameter estimator with respect to external disturbances and unmodelled dynamics so as to provide an applicable adaptive controller. The estimator robustness is achieved by regularized least-quares algorithm with a dead-zone for parameter-estimates running on a good data model. The latter involves data filtering and signal normalization (M'Saad et aI. , 1986).

gain , say : ~=

if S(l)~ ; ~

1/S(1)

=

1

otherwise

The main interest of the above control objective is the possibility of separating the desired tracking behaviour and the desired regulation behaviour. Indeed, the tracking and regulation objectives are specified by the model reference partial state (13) and by the P polynomial (12), respectively . With the equations (10) , (11), (12) and (13) , the desired input-output behaviour of the closed loop system is given by :

, (14) (15)

DPAx (t) = DPl!.(t-k-l) Py(t) = PSx (t)

On the other hand , if we introduce the conceptual variables : €u(t) = p(q-l)[u(t) - A(q- l)x*(t+k+l)) (16)

The Control Law Calculation Ey(t) = p(q-l)[y(t) - S(q-l)x*(t)] The plant to be controlled is represented by the following model : A(q-l )y(t) = q-k S(q-l )u(t-l) + w(t) w(t) = e(t)/D(q-l) + v(t)

the involved control objective becomes :

(6)

(18)

(7)

where y(t) , u(t) and w(t) denote the plant output, input and disturbance respectively , {e(t)) is a bounded zero mean sequence , {v(t)) represents the unmodelled plant dynam ics , k the minimal expected value of the time delay . A,S ,D are polynomials in the backward shift operator q-l . A(q-l)=l +alq-l + ... +anaq-na

(8)

S(q-l) = bO + b l q-l + ... + bnbq-nb

(9)

The term e(t)lD(q-l) denotes the noise and load disturbance effects , the 0 polynomial being the load disturbance internal model. It is interesting to note that the introduct ion of such a model allows the incorporation of a natural offset-free disturbance rejection . The case D(q-l) = 1 - q-l has been proposed by Clarke (1987), and is of particular interest as it introduces an integral action in the control law.

More specifically , the {€u(t)) and the {€y(t)} sequences may be referred to as the tracking input error and tracking output error respectively, the input and output reference trajectories being defined by A(q-l)x * (t+k+ 1) and S(q - l)x'(t) respectively . So , the desired performances are completely defined by the EU(t) and

€y(t) variables. Note that these tracking errors can therefore be interpreted as suitable performance quantifiers for non-minimum phase systems. The control law will consist of the minimization of the quadratic cost function :

J=E

[

L

. ph (cyCt+j)

2+ ACj) (DCq- ' ) cu Ct+j- k-1) 2] I t J

J=sh (19) under the following basic assumption : j~ch

The control objective . Let us define the following partial state controllable representation : DAx(t) = Du(t-k-l) y(t) = Sx(t)

(17)

(10) (11 )

where x(t) denotes the partial state. Then, the control objective may be defined as : (12) where P is an arbitrarily chosen Hurwitz monic polynomial and {x'(t)} is the desired reference partial state sequence. This sequence is defined as the output of an asymptotically stable reference model : (13) where r*(t) is the desired set point sequence. A* and S* denote the involved partial state model reference. B is a scalar gain which is used to get a unit closed loop static

(20)

where ph , sh and ch are the maximum prediction horizon , the minimum prediction horizon and the control horizon . {Ad)} is the input tracking error weighting sequence . Control law derivation . Consider the case where v(t) = O. Substituting expression (16) and (17) into (6) and (7) leads to the following input-output error and tracking error relationship (M'Saad et aI. , 1986). A(q-l )D(q-l )Ey(t)=S(q-l )D(q-l )€u(t-l-k)+P(q-l )e(t) (21) This input-output model for the performance quantifiers E u (t)

and

E y (t)

will

be

referred

to

as

a

performance-oriented model. In fact , it is possible to say that the partial state tracking problem has been turned into a regulation one , with respect to the performance oriented model. Finally , there are three

J. Bo rd e neu H' Ft

26

steps in the control calculation : - Computation of a j-step ahead predictor of the output tracking error, this using (17) . - Calculation of the sequence (EU(t+j)} that minimizes the cost function (19), under the assumption (20) . - Deduce the control law u(t) , in receding horizon sense , using expression (16) . To elaborate the j-step ahead predictor of Ey(t) , consider the following identity : p(q-1 )F(q-1 )=A(q-1 )D(q-1 )G(q-1 )Sj(q-1 )+q-jRj(q-1) (22) with aOSj = j -1 and aORj = max (aoA+ aOD + aOG -1 ,aoF + aop - H. F and G are arbitrarily chosen Hurwitz monic polynomials . Multiplying each member of (22) by ey(t+j) , then using the performance oriented model (21) : ey' (t+Vt) = {BGSj I PF} D Eu (t+j-k-1)} + {RrF} ey(t) (23) Ey ' (t+j/t) is the expectation value of the error Ey(.) at time t+j, conditional on the data available up to time 1. In order to minimize (19) , one needs an expression of ey' (t+j/t) as a function of the sequence (D(q-1) Eu(t+i) } for i=O,ch-1 (these components have to appear explicitly in the prediction equation (23)) . To do so, we will consider the polynomial identity : B(q-1 )G(q-1 )Sj(q-1 )=p(q-1)F(q-1 )Lj_k(q-1) +qk- JKj_k(q-1 ) (24) with aOLj_k= j-k-1 aOK j _k + 1 =maxWp+aoF-1 , aOB+aoG+k-1)

(25)

Then we can rewrite equation (23) as : (26) The control sequence has been separated into past and future control increments. The term p(t,j) is equal to : (27)

al.

DEu ° = - [ MT M+ AI

r 1[ MT N ]

(30)

The long range predictive control law is then given by :

Multivariable adaptive control. Firstly, we consider that the number of inputs and the number of outputs are equal , say m. Instead of a full multivariable controller, which have required m inputs and m outputs, we have chosen a distributed algorithm , which is composed of m independant SISO controllers, each controlling the temperature in the corresponding zone . Distributed control has got many advantages ( Lambert, 1987): -it requires the estimation of less parameters than the corresponding full control. -it's possible to specify different cost functions for each loop. -the robustness of the overall control system is improved. In principle, its decoupling properties should be weaker to that of a multivariable algorithm . A solution to that problem could be a feed-forward compensation , but for typically slow process controls, this is not expected to make a great difference. So, we will not use any compensation . Conclusion We have developed the partial state model reference control law , which has been performed using a reparametrization of a suitable model, and using the Generalized Predictive Control Strategy . The use of the input and output errors allows the specifications of both regulation and tracking behaviours. The unknown parameters involved in the control law are being identified recursively and, in self-tuning manner, the control design computations are performed with these unknown parameters being replaced by their estimates . In next section an advisable parameter estimation algorithm is described. Parameter Estimation The plant model can be written in the following form : (32)

The objective function (19) to be minimized can be rewritten as :

where e and are the parameter and regressor vectors respectively : eT= [a1 ... ana bO ... bnb ]

where :

T(t-1) = [-y(t-1) ... -y(t-na) u(t-1-k) ... u(t-1-k-nb)]

Eu = [EU(t) ,EU(t+1) , .. . ,EU(t+ ph-1)]T N = [p(t,k) , p(t,k+ 1) , .. . ,p(t,ph)]T

10

0

I,

10 0

0 (29)

M=

0 Iph-k- ,

Iph -{;

The li parameters are the Lj_k polynomials coefficients. The solution of the minimization problem is given by :

(33) (34)

"f" denotes signal filtering by G'/F' , G' and F' being two asymptotically stable polynomials. Low-pass filtering of the measurements may be used in the parameter-estimation procedure in order to reduce the high frequency modes due to both no ise and unmodelled dynamics. This ensures smooth parameter-estimates behaviour and reduces their variations. The latter property is quite coherent with stability requirements in adaptive control (M'Saad et al.,1987) . Normalization of input-ouput data is useful to prevent the effects of unbounded mode lling errors and to ensure the boundedness of the Signals before being

EXl end ed Hurizon CUl1lrol u f an Epitaxy Furnace

2i

processed by the algorithm (Praly , 1986). Then, the output, the reg ressor and the noise are divided by the special norm vTI(t), given by :

The reference model is slow enough to prevent a partial state overshoot . The set-point sequence r* (t) is composed of step changes and of a linear decrease.

TI(t) =

-For the parameter estimation: dO =0.1 tr(P(t)) = 30 (P(t) is the covariance matrix)

j..l

TI(t-1) + (1-

where 0

~ j..l ~

1

j..l)

max{o
and

(3S)

na = 2 G' = O.S

TlO > 0

To ensure numerical efficiency , it is advisable to use the U-O factorization to update the estimator gain, i.e., P(t) = U(t)O(t)U T(t)

(36)

where the factor U is an unitary upper triangular matrix and the factor 0 is a positive diagonal matrix (Sierman, 1977). It is straightforward to ensure a lower bound on the estimator gain matrix by monitoring the elements of O(t) = diag{di(t)). as suggested in Ljung and Soderstrom (1983), i.e.,

nb = 1 k=3 F' = 1 - 0.Sq-1

Generally , a second order model is sufficient to describe the temperature evolution of a thermal process . -For the furnace simulation: T = 18 seconds (sampling period) J = 20 (number of discretization points) 0.1Kw < Ui(t) < 0.3Kw i = 2,4 and O.OSKw < U3(t) < 0.1SKw (during the 200 first samples) OKw < Ui(t) < 2Kw U1 (t) = 0.3Kw

i = 2,3,4

(after 200 samples)

US(t) = 0.3Kw

(37) where d0 is a regularizing constant . It is therefore advisable to compute an information measure at every sampling interval using the input-output data, and to implement a decision ru le to take into consideration how the current information can improve the estimation process . The following function can be used to measure how much the incoming information differs from the past one : s(t) = 0
(38)

If s(t) is below a certain value , one can freeze the parameters at their previous values. These improvements are used together with the recursive least-squares algorithm, in which the trace of the covariance matri x is kept constant. Robustness of this parameter adaptation algorithm has been provided because of good data treatment. The control law implementation is relatively simple , and some simulation results will be shown in the next section .

SIMULATION RESULTS In this section, we present some simu lation results . Three of the five heating powers are used for inputs. Indeed , it's reasonable to consider that the extremal powers U1 (t) and US (t) are used to compensate the thermal losses at each side of the furnace . So , they are considered constant. The output vector is composed of the temperatures in the heart of the furnace and at the points x = 0.4 , x = O.S and x = 0.6, say Y 12(t), Y 13(t) and Y 14(t) ( the temperature in the central zone must be strongly controlled) . The boundary conditions are: Yi-O = Yi-1 for all t> O. For a sake of simplicity , we have chosen the same control and est imat ion specifications for the three loops, because their dynamics are qu ite similar. -For the control algorithm : ph = 24 ch = 3 sh = 3 A. =0 .01 D = 1 _q-1 A* = (1 -0 . 9~-1)2 S* = 0.01 P = 1 -0.1qG=1 F= 1

Figures 2a and 2b represent the partial state and the corresponding output evo lut ion for the zones 2 and 3. The corresponding inputs are dep icted on figures 3a and 3b . The results are exact ly symmetric between zones 2 and 4, so we have not depicted the results for the fourth zone. It may be seen that the partial state following in each loop is quite tight and that there is no cvershoot. Figure 4 show the thermal profile in the furnace at differents sampling times (before and after the temperature decrease) . In particular during the decrease, the temperature of the charge is constant in the central zone , this being very important for the layers deposition.

CONCLUSION Th is pape r was concerned with the study of the applicability of a multivariable partial state model reference controller to an epita xy furnace . This control strategy consists of combining a long range predictive control law with a robust parameter estimation . Its implementation is easy , as well as the desired performances specification . Thus , these advantages. together with simulation studies show that this very simple multivariable adaptive controller could be succesfully implemented to control industrial thermal processes . It's worth mentionning that this controller cou Id be improved by introducing feed-forward compensation or by using a full multi variable strategy.

REFERENCES -Senhammou A. (1979) . Identification et commande opt imale d'un reacteur d'epitaxie en phase liquide. These de 3eme cycle , Univ. Paul Sabatier, Toulouse. -Sierman G.J. (1977) . Factorization Methods for Discrete Sequential Estimation. Academic Press, New York. -Clarke DW, C. Mohtadi and P.S Tuffs (1987) . Generalised Predictive Control- Part I.The basic Algorithm and Part 11. Extensions and Interpretations. Automatica, 23, W. 2 , 137-160 . -Clarke OW , O. Ph il and P.J Tuffs (197S) . Self-tuning controller. Proc. lEE, 122, N°.9, 929-934.

J. Bo rd eneu ve

2H

-Dahhou B. (1980) . Commande sous-optimale des sequences de fonctionnement d'un reacteur d'epitaxie en phase liquide. These de 3eme cycle, Univ. Paul Sabatier, Toulouse . -Lambert E.P. (1987) . Process control applications of long-range prediction . Ph.D . thesis , Oxford. -Ljung L. and T. SOderstrom (1983) . Theory and MIT Press , Practice of Recursive Identification . Cambridge, MA .. -M'Saad M. (1987) . Sur I'applicabilite de la commande adaptative. These d'etat, Institut National Poly technique , Grenoble. -M'Saad M., M. Duque and 1.0. Landau (1986) . On the applicability of adaptive control. 2nd IFAC Workshop on Adaptive Systems in Control and Signal Processing, Lund , Sweden. -M'Saad M., M. Duque and E. Irving (1987) . Thermal process robust adaptive control : an experimental evaluation . 10th World Congress of IFAC, Munich. -Praly L. (1986) . Robustesse des Algorithmes de Commande Adaptative . In 1.0. Landau et L. Dugard Ed. , Commande Adaptative : Aspects theoriques et pratiques . Masson , Paris. -Richtmyer R.D ., K.W . Morton (1967). Difference methods for initial-value problems. Wiley Interscience, New-York. -Souza Leao J., J.P . Babary (1983) . Numerical simulation of the space-time thermal profile in an epitaxial reactor. Simulation in Engineering Sciences , IMACS , North Holland.

.... ...

DC

et al.

kW

•

..

Fig 3.a. Evolution of the control variable (zone 2) .

u u

I., u u

....

...

,. ,. +---------'

- -

..

..

lot

"'+----r--~----r---,----r--~----r_~

Fig 2.a. Evolution of the partial state and the output (zone 2).

•

time

-..

..

.+----r--~L---r_--4----r-~---r_--~

DC

-..

...

..

Fig 3.b. Evolution of the control variable (zone 3) .

DC

.

..

x·( t }

,. ,.

y 13 (t)

,. ne

- -

..

time

kW

....

,.

..

~----------------------------------,

...

..

- -

- -

..

Fig 2.b. Evolution of the partial state and the output (zone 3) .

..

time

7>11

,.

•

....

u

u

Fig 4. Thermal profile in the furnace .

time