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Order Reduction of Nonlinear Dynamic Models for Distributed Reacting Systems
Copyright © IFAC Dynamics and Control of Process Systems, Corfu, Greece, 1998
ORDER REDUCTION OF NONLINEAR DYNAMIC MODELS FOR DISTRIBUTED REACTING SYSTEMS S.Y. Shvartsman.· R. Rico-Martinez," E. S. Titi .•••• C. Theodoropoulos, ••• T. J. Mountziaris .••• and I.G. Kevrekidis'
• Department of Chemical Engineering, Princeton UnIVersity •• Depto. de Ingenzeria Quimlca , Instltuto Tecnol6glco de Celaya and Fritz-Haber-Institut der Max-Planck-Gesellschaft ••• Department of Chemical Engmeermg, SL'NYat Buffalo •••• Departments of Mathematics and of Mechalllcal and Aerospace Engineering, University of Califomza, Irvmt
Abstract: Detailed first-principles models of transport and react.ion (based on part.ial differential equations) lead , after discretization , to dynamical systems of very high order. Systematic methodologies for model order reduction are vital in exploiting such fundamental models in the analysis , design and real-time control of distributed reacting systems. We briefly review some approaches to model order reduction we have successfully used in recent years, and illustrate their capabilities through (a) the design of an observer and stabilization of a reaction-diffusion problem and (b) twodimensional simulations of the transient behavior of a horizontal MOVPE reactor. Copyright © 1998 IFAC
Keywords: Distributed Systems, Model Reduction , Estimation, Control , MOVPE.
1. INTRODUCTION
tion from infinite-dimensional dynamical systems to finite (albeit large) ones; the study of such truncations and their performance in controller design has been studied extensively by Balas (e.g. (Balas, 1982; Balas, 1991 ; Balas, 1998)) and others. In some cases reduced order models are formulated on the basis of extensive knowledge about the dynamic behavior of the system based on many years of modeling and experimentation; examples of using such models for dynamic analysis and design of controllers for distillation columns and packed bed reactors can be found in the series of papers by F. Doy le et aI., (Doy le et al.. 1996; Balasubramhanya and Doyle, 1997).
A (nonlinear) vectorfield constitutes the starting point for the analysis of the dynamic behavior of an open loop system and its dependence on operating parameters. It also constitutes the starting point for controller design and evaluation of closed-loop performance. Three-dimensional, time dependent PDEs modeling reaction and transport lead, through accurate , resolved discretizations to models with predictive capabilities; these models , however , contain O( 10 4 ) (and often many more) degrees of freedom. While current computational hardware and software allow for the routine simulation of such system sizes , non linear dynamic analysis and controller design can not (and probably should not) be based on such full scale models.
We are interested in systematic methodologies for reducing large vectorfields, obtained through PDE discretizations, to "small" , accurate, non linear dynamics models: obviously these reduced models will be predictive in a more narrow region of
It should be noted that truncation of the PDE discretization is . in itself, a model order reduc-
637
parallleter space than t.he full model. Th ey will . 011 t.h e other hand. [)(' better conditioned and more amenable to real-time simulation and integration in control algorithms. Furthermore , if the ope rating regime changes , the reduction procedures can be used again to provide new reduced models , adapted to the new region of phase and parameter space. Two m ethodologies. partly motivated by analysis of dissipative PDEs , are beginning to enjoy extensive use in control applicat.ions. The first is th e so-called method of empirical orthogonal eigenfunctions (EOFs), also termed Proper Orthogonal Decomposition (POD) and KarhunenLoeve expansion (KL) . Statistical analysis of exte nsive simulation databases leads (through , esse ntially, Principal Compon ent Analysis) to global basis functions that are then used in a G a lerkin weighted residual projection of the model PDEs (see the monograph (Holmes et al. , 1996) for a recent revie w and (Aling et al. , 1997 : Theodoropoulou et al. , 1998 ; Chen and C hang , 1992: Shvart.sman and Kevrekidis , 19980) for a couple of representative wnt.rol applications) . Th e second , the so call ed Inertial Manifold and Approximate Inertial Manifold approach (Fo ias et al. , 1988b: Constantin et 01 .. 1988 ; Temam. 1988) . can be thought of as an infinite-dimensional extension of singular perturbation methods based on the separation of time scales in finite sets of ODEs (e.g. (Kokotovic et al., 1986)). Center Manifold Theory provides such reductions for nonlinear dynamics syst.ems close to specific , low-codimension bifurcation points (Guckenheimer and Holmes , 1983; Chen and Chang , 1992). Examples ofthe application of this reduction technique to controller design can be found in (Brunovsky, 1991 ; Sano and Kunimatsu , 1994 ; Christofides and Daoutidis, 1996; Christofides and Daoutidis, 1998; Shvartsman and Kevrekidis . 1998b) .
(resolved)
Estimator (Low
Order)~----...J
Fig. 1. Closed loop system with a low-dim ension a l controller. K ey elements : dist ributed syst em (resolved PDE discretization or experiment). spatially resol ved sensing and low dilllf'lI sional controller and estimator . quite efficient in "detec ting" such slaving fun ctions (i. e. , nonlinear correlations between st.ate variables in POD-space ) and in yi elding vectorfi elds that are simply faster to evaluate (Aling et al .. 1997: Hangia et al., 1997). In this contribution we illustrate the effectiV('ness of POD- based model redu ction in th p desigll of low-di,nens ional observers and controllers fo r a particular. o ne-dimensio na l reaction-diffusion probl em : we also d emonstrat(' the combinat.ioIl of POD with further , black-box based slaving techIliques . W", condud (' wit.h an exalllpl e of PODbased reduction for a 2-D "complex geo met.ry " microelectronics fabrication reactor (a horizontal MOVPE reactor with differentially heated lower wall).
2. EXAMPLES 2.1 A NonlmeaT' Reaction-Diffuswn System
Consider the one-dimensional (in space ) reaction diffusion system with no-flux boundary conditions which we have used as an illustrative example in prevIOus work (Shvartsman and Kevrekidis , 1998(a,b)):
PODs have been especially successful in formulating low-order dynamical systems in complexgeometry problems, where no simple global basis function sets are available for (pseudo)spectral discretizations (e .g. (Theodoropoulou et al. , 1998 ; Banerjee et al., 1998)). Clearly, they can be combined with separation of time scales methods built on top of a (linear , "flat") POD-Galerkin reduction (Aling et al., 1997 ; Sirovich et al .. 1990) . Even without an explicit hierarchy of time scales, it is our experience (Bangia et aL., 1997 ; Krischer et al., 1993) that slaving can be quite effective in further reducing "flat" POD-Galerkin truncations to "nonlinear-POD-Galerkin" ones , in analogy to non linear Galerkin methods (implementations of Approximate Inertial Manifolds in conventional spectral discretizations , see (Foias et at.. 1988a; Jolly et al. , 1990; Titi, 1990; Jones and Titi, 1994 ; Foias et al .. 1989) . Remarkably, black box models (e. g . ANN-based architectures) can be
av a2 v at
aw
fit
= az2
+ I(v , w) + u(t)q(z)
aw
(1)
2
= J a z 2 + g(v , w)
v z lo ,L = w z lo ,L = O.
(2) (3)
Briefly here, v is the "activator" , w the "inhibitor", L the length of the one-dimensional domain, I(v, w) = v - v3 - W , g(v , w) = f(V PI w- Po), u(t) is an input (it will later become our control action) and q( z) the shape of the (slightly ad hoc) actuator influence function (see FigAb). The open loop system (u(t) = 0) is known , for L = 20 , J = 4 and PI = 2, Po = -0 .03 to exhibit, in a one-parameter bifurcat.ion diagram with respect to the parameter f (a ratio of reaction time scales) both a supercritical Hopf bifurcation (( ;::: 0.017) and a turning point (saddle node) bifurcation (f ;::: 0.945) of a sharp, front-like solution .
638
!
1.0
1.3
(a)
--~~-
0.3
a,
a,
-0.2
t
-1.7
0
z
- 1.0
L
r-
(C)
b1 "
~_
-0.2 -1.0
0.0
250
0
time
250
I
a1
-1.0 1.0
-0.2
0
z
L Fig .
=
Fig . 2. Open- loop system ( U.OI .,) ): (a) Spaeet illle plot of t he osc illat ory fro nt m o tio n ill a rbitrary graysca lf' units : black (white ) co rresponds to high (Iow ) levels of the l ' fwld : (b) a nd (d) shapes of th e first two emp iri ca l eige nfunctio ns for [' and u fi elds: (c) proj ection of thp limit cyr k in (a) onto th e plane of t he first two POD codti c iPllts . This soluti on. wh pll stable. coe xis ts with a num be r of ot.h<>r (d i:;t.a nt 111 ph ase space) attractors. Our (pseudospect ra l) reso l ved cl isc retization (t.he "full systl' lO ") is a 52-dimen siona l Fouril'r-space Illo del. Csing this IlI odel we coll ec ted a dat.abase of system transient responses for ( values d ose t.o t.h p Hopf bifurcat ion at various constant settings of u. This dat.abase was used to arrive at a set of empirical eigenfunctions (we chose the option of having separate eigenfunctions for each of the two concentration fields) . Galerkin projection of the original PDE on these empirical eigenfunctions m
0
250
time
~ . Low-dim ensio na l o bsp rvpr . Ti rrlf' serI P" of ttlP first t.wo POD compoll ents fo r bot h fi eld s a lo ng tll P spat iote mpo ra l limit cyc le . So lid lin e. reso lved discre ti za tio n : dash f'd lilw : "Hat'· two- m ode POD Galerkin trull cat.ion: lon g-dash lin e: Lu eIll w rger o!Jsp rv pr des ig ned OIl t1lf' basis of t.h e "fl at" 2-lJl odl' (4 deg ree of freedom) trull cat io n .
Xk+l
= (Xk . Uk)
= F(x. u)
=Xk +
J
(F(X(T))
+ ukb)
dT
k6t
Th e open loop dynami cs at this nominal param eter value (see Fig.2) exhibit a stable oscillation (a back-and-forth movement of the fro nt locat ion) . shown in a grayscale space-tim e plot in Fig . 2a . Th e first few empirical eigenfunct io ns are shown in Fig.2 b and 2d . whil e phase-space projection (on th e first. two eigenfunctions) of the full m ode l attractor is compared to the same projection of the red uced model one in Fig.2c.
(5)
where a and b are vectors of (POD) modal coeffi cients fo r the two concentration fi elds. In prev io us wo rk we assumed that th e full state x was accessible to measurement and thus available for feedhack . C urrent developments in spatially resolved sensing t echniques (e.g. microscopies and spectroscopies. (Rotermund et af .. 1997)) justify th e assumption of essentially continuous (in space) m easurem ents . Here we will consid er the case when only a (the l' fi eld) can be measured . The o utput y of the system is then given by
= [fmxm . Orn xrn]x == ex = a .
(7)
(k+l)6t
m
j.
-0.03
of th f' u'- profile and s ubsequently a co ntro ll er fo r the origin a l PDE (in our ca'ie. its co nverged discretization, the "fu 11 system" ); see th e schem at ic Fig . I. The contro ller will be impl e mented in discrete time with samplin g interval 1l1 = 2. We convert the POD-Gal erkin discret izat io n into disc rete form :
results in a dynami cal syst em fo r t he time evolu tion of state variable .r = [a ; b]
y
time
b,
b,
1
0
0.00
(d)
- - 31 Cos - - 2 POD
Q:J
-0.2
250
0.2
1.0 0.2
time
0
Both estimator and controller were d esigned based on the discrete time linear system (Ad . Bd . C)
= Adfk + BdUk Yk = Ce k
fk+1
(8)
(9)
obtained (with sampling tim e Ilt) from the linearization of red uced order (2 POD , four degree of freedom) nonlinear model. This linearizat.i on was perfo rmed around the steady state of the 0.0565. a 2 -0 .1476. bl reduced model (al -0 .0081 , b2 -0.0069) at the nominal paramet er value ({ 0.015) .
(6)
We a re going to use a low-dimensional POD(;alerkin discretization to d esign first an estimator
=
639
=
=
=
=
(b) - -
(a)
' POD
"CS
ffiO
,.
0.1
b,
"" ,
\
"",
t
~. 8
0.0 ' - - - - - - - - - ' z o L
0.2
I
0.0
0.1
\U
-0. 1
\
~. 4
50
-0. 1 - 1.0
a,
1.0
I
\ / ' ..
I,
2 POD 2 ANNIPOD
L..-_ _~_ _---'
0.0
....... .. 2
k
\1
- -
...... .
(c)
(\.,< ,',
....-:.0
3
1.2
a,
r---~----,
0.0
~
u{k)
L -_ _~_ _---'
~. 2
tIme
200.0
Fig .."l. (a) Architecture of the ANN used to slave higher POD mode components. (b) Longtt'rnl predictio n capabilities of tht' 4 A;\i]\; fitted ODEs based on tim e s!:'ries of th!:' first two PO 0 modal co!:'ffici ents. (c) Performance of t.h e observer based on t Iw AN:-'; / PO D vec torfield.
Fig. 4. C1o~!:'d - loop system. (a) Spac!:'- tint(' pl ot of th e transi ent leading to the stabilized set point . (b) Shape of th e actuator influe nce fun ction. (c) TinH'-history of the (discret!:'time) co ntrol action. (d) Phase-plane projec tions of closed- loop transients from initial rond itions (points I . 2. 3) on the open- loop limit cycle leading to th e set-point. TIt(' estimator equat ion (a non linear Lu enberger obs!:'rve r) is obtained in a st.andard way (Muskf' and Edgar . 1991) ) from the red llced order model by adding an inj !:'ct.ion t.e rm 1.0 t.ht' vec torfi!:'ld :
where ,'; is the (positive definite) so luti on of the algebraic Ri ccati equation
Fig. 3 shows the performance (along the fullsystem open-loop stable limit cycl e) of the lowo rder model based observer: the observer is clearly capable of tracking (from a wide range of initial conditions) the state variables. This should be contrasted to the (also showil) performance of the straight "open-loop" reduced model: clearly it is not as accurate in capturing either the period or the shape of the "full system" limit cycle (a preview of this was contained in Fig. 2b). Figure 4 shows the successful stabilization of the system 's unstable steady state based on the controller/observer pair described above; any initial condition on the open-loop stable limit cycle surrounding the unstable steady state gives rise to closed-loop transients attracted to the set point. Higher accu racy (and better observer performan ce ) can be obtained using successively higher order POD models . These become increasingly more complicated to evaluate in real time. and time-scale separation methods become useful in accurately reducing them . As we discussed in the introduction . black-box models can also be efficient in further reducing "flat" POD-based Galerkin models , and in yielding easy to evaluate vectorfields for real-time applications. Here we used such an ANN to effectively "slave" the dynamics of POD modes 3 , 4 , 5,6 and 7 for each field to the first two pairs of modes. thus reducing a 14dimensional (7+7 POD mode) dynamical system to a 4-dimensional one.
The observer gain L was chosen to place the eigenvalues of A d - Le insid e the unit circle (norm below 0.8) : the tuning procedure involved testing its performance operating on a "medium size", 12dimensional (6 + 6) POD-Galerkin discretization. C learly, closed loop performance of observers and controllers bas!:'d on such low-dimensional models will bf' affect!:'d by int.eract ion with the (unm odelled) dynamics of higher-order spatial modes. Insight in the robustness of low-dimensional design in the face of such modelling error can be gained through performance checks with a hierarchy of increasing (but still comparatively low) dimension models (see (Erickson and Laub. 1994) for theoretical analysis of such a procedure for spectral systems}. In our particular example we designed a discretetime Linear Quadratic Regulator minimizing th e objective functi onal ( 12)
The selection of the (positive definite) weighting matrices Q and R was done following the rules of thumb in (Moore and Anderson . 1990) . The gain matrix !\' determining the control law Uk = - A' Ek is given by
640
and for developing control strategies. To exploit model-based control techniques, the ability to prpdict the spatiotemporal behavior of the process in a real time frame is required . Thus. there is a clear need for developing reduced order process models that can accurately capture the spatiotemporal behavior of MOVPE.
The A\N we used (see Fig. 5a) is a standard four-Iaver feed forward neural network with four neuron~ in the input layer (the amplitudes of the first t wo PO 0 modes for each field). and four neurons in the output layer that are trained to predict. upon convergence. the time derivatives of the four amplitudes. There are eight neurons in pach of the two hidden layers (nonlinear. with sigmoidal activation function). Training is achieved using back propagation and a conjugate gradient (CC) algorithm. The t.raining set consisted of .,)00 t.raining pxamples taken from transients st.art.ing close to the unstable fixed point and evolving towards the stable limit cycle. Convergence was declared after 50 complete CG iterations. We used standard cross-validation techniques to prevf'nt over-training. This four dimensional model does a much better job in approximating the open-loop attractors (see Fig. Sb) than the 2-POD based one . and the (excellent) performance of t.he corresponding observer on the full system limit cycle is shown in Fig . Se (in contrast. with the 2-POD based observer): the figu re shows f2 == a',! - a2.
:L.2 Transport
In
In this work. 2-D simulations are performpd to obtain t.he dynamics of the limiting reactant ('01\centration profiles in a MOVPE reactor during growth of GaAsI AlAs st.ruct.ures from trimethylgallium (TMG), trim ethyl-aluminum and an excess of arsine (Theodoropoulos et al.. 1997). The POD method is applied t.o t he resulting data. The IlIOSt. energetic empirical global eigenfunctiolls an' subsequently used to reconstruct the co ncent.ration profiles. The MOVPE reactor is a horizont.al quart.z duct with rectangular cross-section and different.ially heated lower wall (Theodoropoulos d al .. 1997). The heated susceptor. where the substrat.e is placed , has a uniform temperature. T, = 9S0J\·. The upper wall is wat.er-cooled (at. :WO /{). Th e operating pressure is 1 atm. Due to the high dilution of reactants ill H 2, the flow and heat transfer problem can be decoupled from t.he mass transfer and solved at steady state to obtain velocity and temperature profiles of H 2 .
a MOVPE reaciOl'
:v1etal-Organic Vapor Phase Epitaxy (MOVPE) is the most versatile and cost-effective technique for growing thin films of compound semiconductors (Kuech, 1992) . During MOVPE. vapors of organometallic compounds containing the chemical elements of t.he film to be deposited are diluted in a carrier gas (e.g. H 2 , He , Ar , etc.) and flow over a heated substrate, resulting in the deposition of a single-crystalline film on the surface of the substrate. Multi-layer structures of such materials. called heterostructures, form the basis of advanced electronic and opto-electronic devices. For example, quantum-well lasers emitting in the redlIR part of the spectrum consist of alternating layers of GaAs and AlAs.
A 2-D steady state model based on fundam ental equations of continuity, momentum and energy balances was used (Ingle and Mountziaris, 1994). The equations were discretized using the Galerkin Finite Element (FEM) method on a mesh consisting of 16 x 72 quadrilateral elements. The mesh was denser near the reactor inlet and the susceptor in order to capture steep gradients. A low inlet velocity (Semis) led to buoyancy-driven recirculations (Fig. 6a) (Ingle and Mountziaris , 1994). The corresponding temperature profile is shown in Fig. 6b. 2-D dynamic simulations of TMG transport were performed to obtain instantaneous concentration profiles. Reactor filling wi th the precursor switched on at time t = 0 is demonstrated here . The same rectangular FEM mesh is used for the mass transfer calculations and the solution is advanced in time via an implicit Euler scheme. A small time step of O.OSs is used to capture the initial shock from the introduction of the precursor and 2S0 timesteps are taken until the reactor almost reaches steady state. Four snapshots of the TMG mole fraction profile (x - TMC) are shown in Fig 6c . Note that at t= 12 s the highest TMG concentration is near the cooled upper wall (and not at the inlet) due to thermal diffusion (Mountziaris et al., 1993). The eigenvalues of the two-point correlation matrix show that the first S modes capture 99.S% of the energy. The spatial
To grow heterostructures by MOVPE, reactants are switched on and off periodically into a steady flow of the carrier gas. The resulting transient behavior in the reactant supply to the substrate controls t.he interface abruptness in the film . To form heterostructures with abrupt interfaces. transients must ideally last less than the time needed to grow one atomic layer (Theodoropoulos et al., 1997). Increasing the flow rate of the carrier gas leads to shorter transients, but also minimizes the conversion of the expensive precursors into film. Optimization of such operations requires the ability to monitor and control film quality across large substrates in real time. A recent.ly developed technique based on spect.ral reflectance (Breiland and Killeen , 1995) can be used for on line monit.oring of MOVPE processes
641
(a)
rr--------~--~
(b)
~
It--~ (c)
t=ls
(d)
<1> 1 max
t=4s
t .. 8s
t .. 12s
min (e)
(f)
t=4s
t=8s
Fig. 6. a). Pathlin es of hydrogen. The bar indicates the heated susceptor. Flow is from left to right. (b) Temperature profil e . (c) Snapshots of x - TMG from the FEM model. (d) Spatial structure of the 4 most energeti c eigenmodes (e) Snapshot of x - TMG from the FEM model at t = 45 and 85. (f) Reconstruction of :r - T M G snapshot from the 5 most energetic eigenmodes for t = 45 and t = 85.
A POD-based 5-mode Galerkin projection of the dynamic model is currently under development .
structure of the 4 most energetic eigenmodes 0; is shown in Fig . Gd Th e eigenmod es capture well the shape of the advancing concentration front . The accuracy of the technique is evident by comparing the results from the FE~I model at t= 4s and t=8s (Fig. 6e) with their reconstructions shown in Fig. 6f. Because of the ensemble used , the reconstruction is slight Iy biased towards higher concentrations . This is obvious in the reconstruction of the snapshot at t =45. On the other hand. the reconstruction at t= 85 is extremely accurate .
3. SUMMARY
Model order reduction techniques can be effectively used in the extraction of low-dimensional , control-relevant dynamic models of distributed reacting processes with nontrivial dynamics and / or in complex geometries. While the methods are systematic and often successful , theory does not provide practical a priOr! guarantees of the sizes of accurate reduced models, and therefore issues
642
normal incidence reflectance . .J. .4.1']11. Ph .liS. 78.6726. Brunovsky. P. (1991). Controlling t he dynam ics of scalar reaction diffusion equat ions by finite dimensional controllers. In: ."-'fodelmg and Inl'erse problems of control for D'.5tnbutcd paramfier systems. (A. Kurzhanski and I. Lasiecka. Eds.). Springer- Verlag. Chen . c.-C and H-C. Chang (1992). Accpl f' rated disturbance damping of an unkllowlI distributed system by non linear feedback . AIChE Journal 38(9). 1461 - 1476. Christofides. P.D. and P Daoutidis (1996). :\011linear control of diffusion - convection reactioll processes . Camp. Chem. Engllg . 20. SlO71 I07f). Christofidcs . P.D. alld P Daoutidis (1998). Finitedim ensional control of parabolic pde systems using approximate inertial manifolds. J M AA . in pnss. Constantin. P.C .. C. Foias. H. i\icolaenko and R. Temam (1988). Integral A1(/nifold~ mid InFrtzal Mamfolds fm' f)zsslpatn'( Partzal Dlff el'entwl EquatIOns. Springer- Verlag. NY Doyle , F.J , H. M. Hudfllall and M. Morari (19Y6). Linearizing controller design for a packed-bed reactor using a low-order wave propagatioll model. Ind. Eng. Chem . Res. 36. :~567 - ~5tiO. Erickson, M.A. and A.J . Laub (1994). An algorithmic test for checking stability of feedback spectral systems. Automatica 31, 125-13·). Foias, c., D.Manley and R. Tcmam (19880). Modelling of the interaction of small and large eddies in two dimensional turbulent flows. Math. Mod. Num. Anal. 22, 93- 114. Foias, C., G .R . Sell and E.S. Tfti (1989). Exponential tracking and approximation of inertial manifolds for dissipative non linear equations. Journal of DynamiCs and Dlff. Eq. I , 199 254. Foias, c. , G.R. Sell and R. Tcmam (l988b). Inertial manifolds for nonlinear evolutionary equations. 1. Differential Equations 73. 309353. Guckenheimer, J. and P. Holmes (1983) . Nonlmear Oscillations. dynamical systems and blfur'cations of vectorfields. Springer- Verlag. Holmes. P., J.L. Lumley and G. Berkooz (1996). Turbulence, coherent structures. dynamICal systems and symmetry. Cambridge University Press. Ingle, N.K. and T.J . Mountziaris (1994). The onset of transverse recirculations during flow of gases in horizontal duct.s with differentially heated lower walls . 1. Fluid Mech. 277. 2-19. Jolly, MS. I.G. Kevrekidis and E.S. Titi (1990). Approximate inertial methods for the Kuramoto- Sivashinsky equation: analysis and computations. Physica D 44, 36- 60.
of model validation. ensemble construction etc. are an important. and largely empirical. part of the task. On t he other hand. properties of these models , such as the incorporation of explicit parametric/control dependence, good use of the fundamental conservation equations, better conditioning and appropriateness for real-time use . make the effort , in our opinion, worthwhile. Acknowledgements: This work was partially supported by NSF (SYS ,EST.lGK). ARPA/ONR and liTRe (SYS. IGK). One of the authors (RRM) was supported by an Alexander von Humboldt Foundation Fellowship, and wishes to thank Prof. Or. G. Ertl and the FHI in Berlin for their hospitality. During this work EST was the Orson Anderson Scholar at. the Institute of Geophysics and Planetary Physics (IGPP) at the Los Alamos National Laborat.ory. The work of TJM and CT has been supported by t.he National Science Foundation (CTS-9622204) and by the Pittsburgh Supercomputing Center. 4. REFERENCES Aling . H. , S. Banerjee, A.K. Bangia, V. Cole. J.L. Ebert , A. Emami-Naeini. K.F. Jensen , LG. Kevrekidis and S. Shvartsman (1997). Nonlinear model reduction for simulation and control of rapid thermal processing. Proc. Amer. Cont . ConL Albuquerque, NM. Balas, M.J. (1982). Toward a more practical control theory for distributed parameter systems. Control and Dynamic Systems pp. 361 - 420. Balas, M.J. (1991). Nonlinear finite-dimensional control of a class of nonlinear distributed parameter systems using residual-mode filters: a proof of local exponential stability. lMAA 162,63-70. Balas, M.J. (1998). Exponentially stable feedback control of discrete-time nonlinear distributed parameter systems on Banach spaces. lMAA, to appeaL Balasubramhanya, L.S. and F.J. Doyle (1997). Nonlinear control of a high purity distillation column using a traveling wave model. AIChE 10uma143, 703- 714 . Banerjee, S., J .V. Cole and K.F. Jensen (1998). Nonlinear model reduction for rapid thermal processing systems. IEEE Trans . Semicon. Man . , in press. Bangia, A.K., P.F. Batcho, I.G. Kevrekidis and G.Em. Karniadakis (1997) . Unsteady 2-D flows in complex geometries: comparative bifurcation studies with global eigenfunction expansions. SIAM, 1.Sci.Comp 18(3), 775805. Breiland. W .G. and K .P. Killeen (1995). A virtual interface method for extracting growth rates and high t.emperature optical const.ants from thin semiconductor films using in situ 643
./olles. D. ami E.S. Titi (1994). A remark on qua..';;i st.at.ionary approximate inertial manifolds for the :\avier- Stokes equat.ions SIAA/. J. Math Anal. 25.894 - 914. Kokotovic. P.V. , H.K. Khalil and J. O'Reilly (1986). Singular perturbation methods lTI control: analysIs and design. Academic Press. Krischer. K .. R. Rico-Martinez. I.G.Kevrekidis . H.H.Rot.ermund. G.Ertl and J .L.Hudson ( 1993). Model identification of a spatiotemporally varying catalytic reaction. AIChE Journal 39 . 89-98. Kuech . T.F. (1992) . Recent advances in metalorganic vapor phase epitaxy. Proc. IEEE 80. 1609. Moore . B .O. and J .B. Anderson (1990). Optimal Control: Linear Quadratic methods. Prentice Hall. Mountziaris , T.J. , S. Kalyanasundaram and N.K. Ingle (1993) . A reaction-transport model of GaAs growth by metalorganic chemical vapor deposition using trimethyl-galliurn and t.ertiary-butyl-arsine. J. Crystal Growth 131. 289- 299. M uske , K. R. and 1'. F. Edgar (1996). Nonlinear stat.e estimation. In: Nonlinear Process Control (M.A. Henson and D .E. Seborg , Eds.). pp. 311 - 371. Prentice Hall. Rot.ermund , H.-H .. K. Krisher and B. Pettinger (1997). Imaging of reaction fronts at surfaces. In : Frontiers of Elechtrochemistry. vol. I V: Imaglng of surfaces and mterfaces. (J. Lipkowski and P .N . Ross, Eds.). VCH Publishers. Sano , H. and N. Kunimatsu (1994) . Feedback control of semilinear diffusion systems: inertial manifolds for closed-loop systems. IMA Journal of Math. Control f3 InformatIOn 11 . 75- 92. Shvartsman , S.Y. and LG. Kevrekidis (1998a). Low-dimensional approximation and feedback control of periodic solutions in extended syst.ems. Phys. Rev. E , in press. Shvartsman, S.Y. and LG. Kevrekidis (1998b). Nonlinear model reduction for control of distributed parameter systems: A computer assisted study. AIChE Journal, in press. Sirovich, L. , W.B. Knight and J .D. Rodriguez (1990) . Optimal low-dimensional dynamical approximations. Quart. Appl. Math . 3 , 535548. Temam , R. (1988). Infinite dimensional dynamical systems in mechanics and physics. Springer. Theodoropoulos, C. N.K . Ingle and T.J . Mountziaris (1997) . Computational studies of the transient behavior of horizontal MOVPE reactors. J . Crystal Growth 170, 722. Theodoropoulou. A.. R.A . Adomaitis and E. Zafiriou (1998). Model reduction for
optimization of rapid thermal chemical vapor deposition systems. IEEE Trarl.~. Semlcon . Man. 11.85- 98. Titi . E.S. (1990). On approximate inertial manifolds to the Navier- Stokes equations. Journ. of Math. Anal. [" Appl. 149.540- .')57.
644
ORDER REDUCTION OF NONLINEAR DYNAMIC MODELS FOR DISTRIBUTED REACTING SYSTEMS S.Y. Shvartsman.· R. Rico-Martinez," E. S. Titi .•••• C. Theodoropoulos, ••• T. J. Mountziaris .••• and I.G. Kevrekidis'
• Department of Chemical Engineering, Princeton UnIVersity •• Depto. de Ingenzeria Quimlca , Instltuto Tecnol6glco de Celaya and Fritz-Haber-Institut der Max-Planck-Gesellschaft ••• Department of Chemical Engmeermg, SL'NYat Buffalo •••• Departments of Mathematics and of Mechalllcal and Aerospace Engineering, University of Califomza, Irvmt
Abstract: Detailed first-principles models of transport and react.ion (based on part.ial differential equations) lead , after discretization , to dynamical systems of very high order. Systematic methodologies for model order reduction are vital in exploiting such fundamental models in the analysis , design and real-time control of distributed reacting systems. We briefly review some approaches to model order reduction we have successfully used in recent years, and illustrate their capabilities through (a) the design of an observer and stabilization of a reaction-diffusion problem and (b) twodimensional simulations of the transient behavior of a horizontal MOVPE reactor. Copyright © 1998 IFAC
Keywords: Distributed Systems, Model Reduction , Estimation, Control , MOVPE.
1. INTRODUCTION
tion from infinite-dimensional dynamical systems to finite (albeit large) ones; the study of such truncations and their performance in controller design has been studied extensively by Balas (e.g. (Balas, 1982; Balas, 1991 ; Balas, 1998)) and others. In some cases reduced order models are formulated on the basis of extensive knowledge about the dynamic behavior of the system based on many years of modeling and experimentation; examples of using such models for dynamic analysis and design of controllers for distillation columns and packed bed reactors can be found in the series of papers by F. Doy le et aI., (Doy le et al.. 1996; Balasubramhanya and Doyle, 1997).
A (nonlinear) vectorfield constitutes the starting point for the analysis of the dynamic behavior of an open loop system and its dependence on operating parameters. It also constitutes the starting point for controller design and evaluation of closed-loop performance. Three-dimensional, time dependent PDEs modeling reaction and transport lead, through accurate , resolved discretizations to models with predictive capabilities; these models , however , contain O( 10 4 ) (and often many more) degrees of freedom. While current computational hardware and software allow for the routine simulation of such system sizes , non linear dynamic analysis and controller design can not (and probably should not) be based on such full scale models.
We are interested in systematic methodologies for reducing large vectorfields, obtained through PDE discretizations, to "small" , accurate, non linear dynamics models: obviously these reduced models will be predictive in a more narrow region of
It should be noted that truncation of the PDE discretization is . in itself, a model order reduc-
637
parallleter space than t.he full model. Th ey will . 011 t.h e other hand. [)(' better conditioned and more amenable to real-time simulation and integration in control algorithms. Furthermore , if the ope rating regime changes , the reduction procedures can be used again to provide new reduced models , adapted to the new region of phase and parameter space. Two m ethodologies. partly motivated by analysis of dissipative PDEs , are beginning to enjoy extensive use in control applicat.ions. The first is th e so-called method of empirical orthogonal eigenfunctions (EOFs), also termed Proper Orthogonal Decomposition (POD) and KarhunenLoeve expansion (KL) . Statistical analysis of exte nsive simulation databases leads (through , esse ntially, Principal Compon ent Analysis) to global basis functions that are then used in a G a lerkin weighted residual projection of the model PDEs (see the monograph (Holmes et al. , 1996) for a recent revie w and (Aling et al. , 1997 : Theodoropoulou et al. , 1998 ; Chen and C hang , 1992: Shvart.sman and Kevrekidis , 19980) for a couple of representative wnt.rol applications) . Th e second , the so call ed Inertial Manifold and Approximate Inertial Manifold approach (Fo ias et al. , 1988b: Constantin et 01 .. 1988 ; Temam. 1988) . can be thought of as an infinite-dimensional extension of singular perturbation methods based on the separation of time scales in finite sets of ODEs (e.g. (Kokotovic et al., 1986)). Center Manifold Theory provides such reductions for nonlinear dynamics syst.ems close to specific , low-codimension bifurcation points (Guckenheimer and Holmes , 1983; Chen and Chang , 1992). Examples ofthe application of this reduction technique to controller design can be found in (Brunovsky, 1991 ; Sano and Kunimatsu , 1994 ; Christofides and Daoutidis, 1996; Christofides and Daoutidis, 1998; Shvartsman and Kevrekidis . 1998b) .
(resolved)
Estimator (Low
Order)~----...J
Fig. 1. Closed loop system with a low-dim ension a l controller. K ey elements : dist ributed syst em (resolved PDE discretization or experiment). spatially resol ved sensing and low dilllf'lI sional controller and estimator . quite efficient in "detec ting" such slaving fun ctions (i. e. , nonlinear correlations between st.ate variables in POD-space ) and in yi elding vectorfi elds that are simply faster to evaluate (Aling et al .. 1997: Hangia et al., 1997). In this contribution we illustrate the effectiV('ness of POD- based model redu ction in th p desigll of low-di,nens ional observers and controllers fo r a particular. o ne-dimensio na l reaction-diffusion probl em : we also d emonstrat(' the combinat.ioIl of POD with further , black-box based slaving techIliques . W", condud (' wit.h an exalllpl e of PODbased reduction for a 2-D "complex geo met.ry " microelectronics fabrication reactor (a horizontal MOVPE reactor with differentially heated lower wall).
2. EXAMPLES 2.1 A NonlmeaT' Reaction-Diffuswn System
Consider the one-dimensional (in space ) reaction diffusion system with no-flux boundary conditions which we have used as an illustrative example in prevIOus work (Shvartsman and Kevrekidis , 1998(a,b)):
PODs have been especially successful in formulating low-order dynamical systems in complexgeometry problems, where no simple global basis function sets are available for (pseudo)spectral discretizations (e .g. (Theodoropoulou et al. , 1998 ; Banerjee et al., 1998)). Clearly, they can be combined with separation of time scales methods built on top of a (linear , "flat") POD-Galerkin reduction (Aling et al., 1997 ; Sirovich et al .. 1990) . Even without an explicit hierarchy of time scales, it is our experience (Bangia et aL., 1997 ; Krischer et al., 1993) that slaving can be quite effective in further reducing "flat" POD-Galerkin truncations to "nonlinear-POD-Galerkin" ones , in analogy to non linear Galerkin methods (implementations of Approximate Inertial Manifolds in conventional spectral discretizations , see (Foias et at.. 1988a; Jolly et al. , 1990; Titi, 1990; Jones and Titi, 1994 ; Foias et al .. 1989) . Remarkably, black box models (e. g . ANN-based architectures) can be
av a2 v at
aw
fit
= az2
+ I(v , w) + u(t)q(z)
aw
(1)
2
= J a z 2 + g(v , w)
v z lo ,L = w z lo ,L = O.
(2) (3)
Briefly here, v is the "activator" , w the "inhibitor", L the length of the one-dimensional domain, I(v, w) = v - v3 - W , g(v , w) = f(V PI w- Po), u(t) is an input (it will later become our control action) and q( z) the shape of the (slightly ad hoc) actuator influence function (see FigAb). The open loop system (u(t) = 0) is known , for L = 20 , J = 4 and PI = 2, Po = -0 .03 to exhibit, in a one-parameter bifurcat.ion diagram with respect to the parameter f (a ratio of reaction time scales) both a supercritical Hopf bifurcation (( ;::: 0.017) and a turning point (saddle node) bifurcation (f ;::: 0.945) of a sharp, front-like solution .
638
!
1.0
1.3
(a)
--~~-
0.3
a,
a,
-0.2
t
-1.7
0
z
- 1.0
L
r-
(C)
b1 "
~_
-0.2 -1.0
0.0
250
0
time
250
I
a1
-1.0 1.0
-0.2
0
z
L Fig .
=
Fig . 2. Open- loop system ( U.OI .,) ): (a) Spaeet illle plot of t he osc illat ory fro nt m o tio n ill a rbitrary graysca lf' units : black (white ) co rresponds to high (Iow ) levels of the l ' fwld : (b) a nd (d) shapes of th e first two emp iri ca l eige nfunctio ns for [' and u fi elds: (c) proj ection of thp limit cyr k in (a) onto th e plane of t he first two POD codti c iPllts . This soluti on. wh pll stable. coe xis ts with a num be r of ot.h<>r (d i:;t.a nt 111 ph ase space) attractors. Our (pseudospect ra l) reso l ved cl isc retization (t.he "full systl' lO ") is a 52-dimen siona l Fouril'r-space Illo del. Csing this IlI odel we coll ec ted a dat.abase of system transient responses for ( values d ose t.o t.h p Hopf bifurcat ion at various constant settings of u. This dat.abase was used to arrive at a set of empirical eigenfunctions (we chose the option of having separate eigenfunctions for each of the two concentration fields) . Galerkin projection of the original PDE on these empirical eigenfunctions m
0
250
time
~ . Low-dim ensio na l o bsp rvpr . Ti rrlf' serI P" of ttlP first t.wo POD compoll ents fo r bot h fi eld s a lo ng tll P spat iote mpo ra l limit cyc le . So lid lin e. reso lved discre ti za tio n : dash f'd lilw : "Hat'· two- m ode POD Galerkin trull cat.ion: lon g-dash lin e: Lu eIll w rger o!Jsp rv pr des ig ned OIl t1lf' basis of t.h e "fl at" 2-lJl odl' (4 deg ree of freedom) trull cat io n .
Xk+l
= (Xk . Uk)
= F(x. u)
=Xk +
J
(F(X(T))
+ ukb)
dT
k6t
Th e open loop dynami cs at this nominal param eter value (see Fig.2) exhibit a stable oscillation (a back-and-forth movement of the fro nt locat ion) . shown in a grayscale space-tim e plot in Fig . 2a . Th e first few empirical eigenfunct io ns are shown in Fig.2 b and 2d . whil e phase-space projection (on th e first. two eigenfunctions) of the full m ode l attractor is compared to the same projection of the red uced model one in Fig.2c.
(5)
where a and b are vectors of (POD) modal coeffi cients fo r the two concentration fi elds. In prev io us wo rk we assumed that th e full state x was accessible to measurement and thus available for feedhack . C urrent developments in spatially resolved sensing t echniques (e.g. microscopies and spectroscopies. (Rotermund et af .. 1997)) justify th e assumption of essentially continuous (in space) m easurem ents . Here we will consid er the case when only a (the l' fi eld) can be measured . The o utput y of the system is then given by
= [fmxm . Orn xrn]x == ex = a .
(7)
(k+l)6t
m
j.
-0.03
of th f' u'- profile and s ubsequently a co ntro ll er fo r the origin a l PDE (in our ca'ie. its co nverged discretization, the "fu 11 system" ); see th e schem at ic Fig . I. The contro ller will be impl e mented in discrete time with samplin g interval 1l1 = 2. We convert the POD-Gal erkin discret izat io n into disc rete form :
results in a dynami cal syst em fo r t he time evolu tion of state variable .r = [a ; b]
y
time
b,
b,
1
0
0.00
(d)
- - 31 Cos - - 2 POD
Q:J
-0.2
250
0.2
1.0 0.2
time
0
Both estimator and controller were d esigned based on the discrete time linear system (Ad . Bd . C)
= Adfk + BdUk Yk = Ce k
fk+1
(8)
(9)
obtained (with sampling tim e Ilt) from the linearization of red uced order (2 POD , four degree of freedom) nonlinear model. This linearizat.i on was perfo rmed around the steady state of the 0.0565. a 2 -0 .1476. bl reduced model (al -0 .0081 , b2 -0.0069) at the nominal paramet er value ({ 0.015) .
(6)
We a re going to use a low-dimensional POD(;alerkin discretization to d esign first an estimator
=
639
=
=
=
=
(b) - -
(a)
' POD
"CS
ffiO
,.
0.1
b,
"" ,
\
"",
t
~. 8
0.0 ' - - - - - - - - - ' z o L
0.2
I
0.0
0.1
\U
-0. 1
\
~. 4
50
-0. 1 - 1.0
a,
1.0
I
\ / ' ..
I,
2 POD 2 ANNIPOD
L..-_ _~_ _---'
0.0
....... .. 2
k
\1
- -
...... .
(c)
(\.,< ,',
....-:.0
3
1.2
a,
r---~----,
0.0
~
u{k)
L -_ _~_ _---'
~. 2
tIme
200.0
Fig .."l. (a) Architecture of the ANN used to slave higher POD mode components. (b) Longtt'rnl predictio n capabilities of tht' 4 A;\i]\; fitted ODEs based on tim e s!:'ries of th!:' first two PO 0 modal co!:'ffici ents. (c) Performance of t.h e observer based on t Iw AN:-'; / PO D vec torfield.
Fig. 4. C1o~!:'d - loop system. (a) Spac!:'- tint(' pl ot of th e transi ent leading to the stabilized set point . (b) Shape of th e actuator influe nce fun ction. (c) TinH'-history of the (discret!:'time) co ntrol action. (d) Phase-plane projec tions of closed- loop transients from initial rond itions (points I . 2. 3) on the open- loop limit cycle leading to th e set-point. TIt(' estimator equat ion (a non linear Lu enberger obs!:'rve r) is obtained in a st.andard way (Muskf' and Edgar . 1991) ) from the red llced order model by adding an inj !:'ct.ion t.e rm 1.0 t.ht' vec torfi!:'ld :
where ,'; is the (positive definite) so luti on of the algebraic Ri ccati equation
Fig. 3 shows the performance (along the fullsystem open-loop stable limit cycl e) of the lowo rder model based observer: the observer is clearly capable of tracking (from a wide range of initial conditions) the state variables. This should be contrasted to the (also showil) performance of the straight "open-loop" reduced model: clearly it is not as accurate in capturing either the period or the shape of the "full system" limit cycle (a preview of this was contained in Fig. 2b). Figure 4 shows the successful stabilization of the system 's unstable steady state based on the controller/observer pair described above; any initial condition on the open-loop stable limit cycle surrounding the unstable steady state gives rise to closed-loop transients attracted to the set point. Higher accu racy (and better observer performan ce ) can be obtained using successively higher order POD models . These become increasingly more complicated to evaluate in real time. and time-scale separation methods become useful in accurately reducing them . As we discussed in the introduction . black-box models can also be efficient in further reducing "flat" POD-based Galerkin models , and in yielding easy to evaluate vectorfields for real-time applications. Here we used such an ANN to effectively "slave" the dynamics of POD modes 3 , 4 , 5,6 and 7 for each field to the first two pairs of modes. thus reducing a 14dimensional (7+7 POD mode) dynamical system to a 4-dimensional one.
The observer gain L was chosen to place the eigenvalues of A d - Le insid e the unit circle (norm below 0.8) : the tuning procedure involved testing its performance operating on a "medium size", 12dimensional (6 + 6) POD-Galerkin discretization. C learly, closed loop performance of observers and controllers bas!:'d on such low-dimensional models will bf' affect!:'d by int.eract ion with the (unm odelled) dynamics of higher-order spatial modes. Insight in the robustness of low-dimensional design in the face of such modelling error can be gained through performance checks with a hierarchy of increasing (but still comparatively low) dimension models (see (Erickson and Laub. 1994) for theoretical analysis of such a procedure for spectral systems}. In our particular example we designed a discretetime Linear Quadratic Regulator minimizing th e objective functi onal ( 12)
The selection of the (positive definite) weighting matrices Q and R was done following the rules of thumb in (Moore and Anderson . 1990) . The gain matrix !\' determining the control law Uk = - A' Ek is given by
640
and for developing control strategies. To exploit model-based control techniques, the ability to prpdict the spatiotemporal behavior of the process in a real time frame is required . Thus. there is a clear need for developing reduced order process models that can accurately capture the spatiotemporal behavior of MOVPE.
The A\N we used (see Fig. 5a) is a standard four-Iaver feed forward neural network with four neuron~ in the input layer (the amplitudes of the first t wo PO 0 modes for each field). and four neurons in the output layer that are trained to predict. upon convergence. the time derivatives of the four amplitudes. There are eight neurons in pach of the two hidden layers (nonlinear. with sigmoidal activation function). Training is achieved using back propagation and a conjugate gradient (CC) algorithm. The t.raining set consisted of .,)00 t.raining pxamples taken from transients st.art.ing close to the unstable fixed point and evolving towards the stable limit cycle. Convergence was declared after 50 complete CG iterations. We used standard cross-validation techniques to prevf'nt over-training. This four dimensional model does a much better job in approximating the open-loop attractors (see Fig. Sb) than the 2-POD based one . and the (excellent) performance of t.he corresponding observer on the full system limit cycle is shown in Fig . Se (in contrast. with the 2-POD based observer): the figu re shows f2 == a',! - a2.
:L.2 Transport
In
In this work. 2-D simulations are performpd to obtain t.he dynamics of the limiting reactant ('01\centration profiles in a MOVPE reactor during growth of GaAsI AlAs st.ruct.ures from trimethylgallium (TMG), trim ethyl-aluminum and an excess of arsine (Theodoropoulos et al.. 1997). The POD method is applied t.o t he resulting data. The IlIOSt. energetic empirical global eigenfunctiolls an' subsequently used to reconstruct the co ncent.ration profiles. The MOVPE reactor is a horizont.al quart.z duct with rectangular cross-section and different.ially heated lower wall (Theodoropoulos d al .. 1997). The heated susceptor. where the substrat.e is placed , has a uniform temperature. T, = 9S0J\·. The upper wall is wat.er-cooled (at. :WO /{). Th e operating pressure is 1 atm. Due to the high dilution of reactants ill H 2, the flow and heat transfer problem can be decoupled from t.he mass transfer and solved at steady state to obtain velocity and temperature profiles of H 2 .
a MOVPE reaciOl'
:v1etal-Organic Vapor Phase Epitaxy (MOVPE) is the most versatile and cost-effective technique for growing thin films of compound semiconductors (Kuech, 1992) . During MOVPE. vapors of organometallic compounds containing the chemical elements of t.he film to be deposited are diluted in a carrier gas (e.g. H 2 , He , Ar , etc.) and flow over a heated substrate, resulting in the deposition of a single-crystalline film on the surface of the substrate. Multi-layer structures of such materials. called heterostructures, form the basis of advanced electronic and opto-electronic devices. For example, quantum-well lasers emitting in the redlIR part of the spectrum consist of alternating layers of GaAs and AlAs.
A 2-D steady state model based on fundam ental equations of continuity, momentum and energy balances was used (Ingle and Mountziaris, 1994). The equations were discretized using the Galerkin Finite Element (FEM) method on a mesh consisting of 16 x 72 quadrilateral elements. The mesh was denser near the reactor inlet and the susceptor in order to capture steep gradients. A low inlet velocity (Semis) led to buoyancy-driven recirculations (Fig. 6a) (Ingle and Mountziaris , 1994). The corresponding temperature profile is shown in Fig. 6b. 2-D dynamic simulations of TMG transport were performed to obtain instantaneous concentration profiles. Reactor filling wi th the precursor switched on at time t = 0 is demonstrated here . The same rectangular FEM mesh is used for the mass transfer calculations and the solution is advanced in time via an implicit Euler scheme. A small time step of O.OSs is used to capture the initial shock from the introduction of the precursor and 2S0 timesteps are taken until the reactor almost reaches steady state. Four snapshots of the TMG mole fraction profile (x - TMC) are shown in Fig 6c . Note that at t= 12 s the highest TMG concentration is near the cooled upper wall (and not at the inlet) due to thermal diffusion (Mountziaris et al., 1993). The eigenvalues of the two-point correlation matrix show that the first S modes capture 99.S% of the energy. The spatial
To grow heterostructures by MOVPE, reactants are switched on and off periodically into a steady flow of the carrier gas. The resulting transient behavior in the reactant supply to the substrate controls t.he interface abruptness in the film . To form heterostructures with abrupt interfaces. transients must ideally last less than the time needed to grow one atomic layer (Theodoropoulos et al., 1997). Increasing the flow rate of the carrier gas leads to shorter transients, but also minimizes the conversion of the expensive precursors into film. Optimization of such operations requires the ability to monitor and control film quality across large substrates in real time. A recent.ly developed technique based on spect.ral reflectance (Breiland and Killeen , 1995) can be used for on line monit.oring of MOVPE processes
641
(a)
rr--------~--~
(b)
~
It--~ (c)
t=ls
(d)
<1> 1 max
t=4s
t .. 8s
t .. 12s
min (e)
(f)
t=4s
t=8s
Fig. 6. a). Pathlin es of hydrogen. The bar indicates the heated susceptor. Flow is from left to right. (b) Temperature profil e . (c) Snapshots of x - TMG from the FEM model. (d) Spatial structure of the 4 most energeti c eigenmodes (e) Snapshot of x - TMG from the FEM model at t = 45 and 85. (f) Reconstruction of :r - T M G snapshot from the 5 most energetic eigenmodes for t = 45 and t = 85.
A POD-based 5-mode Galerkin projection of the dynamic model is currently under development .
structure of the 4 most energetic eigenmodes 0; is shown in Fig . Gd Th e eigenmod es capture well the shape of the advancing concentration front . The accuracy of the technique is evident by comparing the results from the FE~I model at t= 4s and t=8s (Fig. 6e) with their reconstructions shown in Fig. 6f. Because of the ensemble used , the reconstruction is slight Iy biased towards higher concentrations . This is obvious in the reconstruction of the snapshot at t =45. On the other hand. the reconstruction at t= 85 is extremely accurate .
3. SUMMARY
Model order reduction techniques can be effectively used in the extraction of low-dimensional , control-relevant dynamic models of distributed reacting processes with nontrivial dynamics and / or in complex geometries. While the methods are systematic and often successful , theory does not provide practical a priOr! guarantees of the sizes of accurate reduced models, and therefore issues
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