On multi zone tracking and non gaussian noise filtering for model predictive control

In this thesis, we proposed a method called Uniformity Model PredictiveControl UMPC to achieve output uniformity.. At each time instant, MPC uses a current measurement of the process out

ON MULTIZONE TRACKING AND NON-GAUSSIAN NOISE FILTERING FOR THE MODEL PREDICTIVE CONTROL

WANG XIAOQIONG

(B.Eng.(Hons.),NUS)

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF ELECTRICAL & COMPUTER

ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2014

ON MULTIZONE TRACKING AND NON-GAUSSIAN NOISE FILTERING FOR THE MODEL PREDICTIVE CONTROL

Copyright 2014byWANG XIAOQIONG

supervi-My sincere thanks also goes to Prof Tan Kok Kiong and Prof ArthurTay Ee Beng, for their time and efforts in assessing my research work, thevaluable suggestions and critical questions during my qualification exami-nation.

I would like to thank my colleagues and lab mates, Jose Vu, Qu Yifan,

Yu Chao, and Vathi for the stimulating discussions, for the accompanywhen we were working together, and all the fun we have had in the lastfour years Many thanks also goes to my dearest friends, Xie Yanxi, SunWen, and Li suchun, accompanied me through the happiness and sadness

1.1 An Overview of Model Predictive Control 1

1.2 Motivation of the Thesis 3

1.3 Contribution of the Thesis 7

1.4 Scope of the Thesis 10

2 Model Predictive Control for Uniform Output 12 2.1 Introduction 13

2.2 Algorithm of UMPC 15

2.2.1 Formulation of UMPC 18

2.2.2 Control Law of UMPC 20

2.2.3 Cost Function Comparison of UMPC and SMPC 24

2.3 Bake Plate Thermal Modeling 25

2.4 Experimental Results 28

2.4.1 UMPC Uniformity Validation Experiments 33

2.4.2 UMPC Robustness Experiments 40

2.5 Conclusion 54

3 Filtering of the ARMAX process with Generalized t-Distribution Noise: The Influence Function Approach 57 3.1 Introduction 57

3.2 Maximum Likelihood Estimation of the ARMAX Process with GT Noise 60

3.2.1 The ARMAX Process 60

3.2.2 The Diophantine Equation 61

3.2.3 Maximum Likelihood Estimation 64

3.3 Influence Function Approximation 65

3.3.1 The Recursive Algorithm 66

3.3.2 Mean, Variance and Outlier 68

3.4 Examples 69

3.4.1 Example 1: The Kalman Filter Connection 70

3.4.2 Example 2: Variance 80

3.4.3 Example 3: Outlier 84

3.4.4 Example 4: Liquid Level Estimation Experiment 87

3.5 Conclusion 95

4 MPC Closed-Loop Control with ARMAX and Kalman Filter 97 4.1 Introduction 97

4.2 MPC Examples 98

4.2.1 Outlier 100

4.2.2 Variance 104

4.3 Conclusion 107

5 Computational Load Comparison of Multiplexed MPC and Standard MPC 108 5.1 Introduction 109

5.2 The Experimental Setup 111

5.3 Controller Design 114

5.4 Experimental Results 116

5.5 Conclusion 122

List of Figures

Control 2

1.2 Model Predictive Control structure 3

distribution (dotted-line, µ = 0, σ = 28.5nm) and GT bution (solid-line, p = q = 2, σ = 29.5nm) to the thickness

distri-measurement distribution 6

1.4 Thickness measurements on 24 semiconductor wafers after ical Mechanical Polishing 6

temperature wafer placed on 15

temperature wafer placed on 16

2.3 A photograph of the multizone bake plate 26

2.4 UMPC Uniformity ISE trend with q1 = 1 35

2.5 Output and Control singals of SMPC (left) with q1 = q2 = q3 =

1 and UMPC (right) with q1 = 1, q2 = 3, q3 = 3 36

2.6 Zone 1 block diagram of bake-plate 39

upper from left to right are with q2 = q3 = 1, q2 = q3 = 5

respectively; lower from left to right are with q2 = q3 = 10,

q2 = q3 = 20 respectively 42

2.8 Output performance and input Signals of SMPC (left) with q1 =

q2 = q3 = 1 and UMPC (right) with q1 = 1, q2 = 20, q3 = 20 44

identified plant model is artificially increased by 2 times: SMPC

with q1 = q2 = q3 = 1 (left), and UMPC with q1 = 1, q2 = 20,

2.11 Output performance and input Signals when the gain of the identified plant model is artificially decreased by 2 times: SMPC

with q1 = q2 = q3 = 1 (left), and UMPC with q1 = 1, q2 = 20,

q3 = 20(right) 47

2.12 Output performance and input Signals when the gain of the identified plant model is artificially decreased by 5 times: SMPC with q1 = q2 = q3 = 1 (left), and UMPC with q1 = 1, q2 = 20, q3 = 20(right) 48

2.13 Output performance and input Signals when the time constant of the identified plant model is artificially increased by 2 times: SMPC with q1 = q2 = q3 = 1 (left), and UMPC with q1 = 1, q2 = 20, q3 = 20(right) 50

2.14 Output performance and input Signals when the time constant of the identified plant model is artificially increased by 5 times: SMPC with q1 = q2 = q3 = 1 (left), and UMPC with q1 = 1, q2 = 20, q3 = 20(right) 51

2.15 Output performance and input Signals when the time constant of the identified plant model is artificially decreased by 2 times: SMPC with q1 = q2 = q3 = 1 (left), and UMPC with q1 = 1, q2 = 20, q3 = 20(right) 52

2.16 Output performance and input Signals when the time constant of the identified plant model is artificially decreased by 5 times: SMPC with q1 = q2 = q3 = 1 (left), and UMPC with q1 = 1, q2 = 20, q3 = 20(right) 53

2.17 Comparison of Uniformity ISE of UMPC(left) and SMPC(right) when modeling error is presence 54

3.1 Different choices of the GT distribution shape parameters p and q can give different well-known distributions. 61

3.2 Simulation results of Example 2 81

3.3 ARMAX filter output ˆy(N ). 85

3.4 Kalman filter output ˆy(N ). 85

3.5 Photo of the coupled-tank 87

3.6 Measurement y(N ) for the liquid level estimation experiment. 88

3.7 The maximum likelihood criterion was used to fit a GT distribu-tion (solid-line) and Gaussian distribudistribu-tion (dashed-line) to the noise distribution 89

3.8 ARMAX filter estimate ˆy(N ). 91

3.9 Kalman filter estimate ˆy(N ). 91

4.2 Silumation results of MPC Outlier Example 101

4.3 Simulation results of MPC Variance Example 106

5.1 Patterns of input moves for Standard MPC (solid), and for theMultiplexed MPC (dashed) 109

esti-mated model respectively 113

control horizon by experiment 117

SMPC (solid) with similar performance, N u = 5 118

SMPC (solid) with similar performance, N u = 20 119

SMPC (solid) with similar performance, N u = 25 120

3.2 Mean and Variance of ˆy(N ) in Figure 3.2 81

3.3 Variance (×10 −3) of ˆy(N ) in Figures 3.8 and 3.9. . 90

Exam-ples 4.2.1 and 4.2.2 99

4.2 Mean and Variance of the simulation output in Figure 4.2 102

4.3 Mean and Variance of the simulation output in Figure 4.3 105

Model Predictive Control (MPC) has been widely studied and adopted inindustrial applications because the actual control objectives and operatingconstraints can be represented explicitly in the optimization problem that

is solved at each control instant [1 3] Some attempts have been done fortemperature uniformity control [4 6] But these studies on temperatureuniformity usually focused on the set-point tracking uniformity from batch

to batch, not the uniformity of the zone-to-zone temperature trajectories

In this thesis, we proposed a method called Uniformity Model PredictiveControl (UMPC) to achieve output uniformity The idea of UMPC is toreconstruct the cost function of the Standard MPC Simulations and bake-plate experiments were carried out to show that UMPC gives better out-puts uniformity than SMPC Moreover, most of MPC control designs useKalman filter to filter the measurement noise which is assumed to be Gaus-sian distributed This might be a limitation in the case of non-Gaussiannoise as Kalman filter is well-known to be sensitive to outliers [7] We pro-posed a filter called ARMAX filter for MPC by modeling noise with the

GT distribution, as it can model other distributions (e.g t-distribution),

instead of the usual Gaussian distribution Moreover, the computationalload is also a problem when applying the MPC designs to the industrial

applications We provide one of the first experimental verification of thecomputational load reduction property of MMPC.

Chapter 1

Introduction

1.1 An Overview of Model Predictive

Control

The general design objective of Model Predictive Control (MPC) is to

com-pute a trajectory of a future manipulated input u to optimize the future behavior of the plant output y Figure1.1 shows the receding-horizon con-trol implementation of MPC The control horizon represents the number

of parameters used to capture the future control trajectory The predictivehorizon represents the number of samples we want to predict Although theoptimal trajectory of future control signal is completely described withinthe length of control horizon, the actual control input to the plant onlytakes the first sample of the control signal, while neglecting the rest of the

Figure 1.1: Receding-horizon control implementation of Model PredictiveControl

Figure1.2shows the structure of MPC At each time instant, MPC uses

a current measurement of the process output, and an internal model of theprocess, to compute and implement a new control input, which minimisessome cost function, while guaranteeing that constraints are satisfied In de-termining the MPC control, one needs a process model to predict the futureplant outputs, and an optimization criterion which is the cost function.MPC could well handle the highly complex, non-linear, uncertain, and

Figure 1.2: Model Predictive Control structure.

these advantages MPC has, therefore, been widely applied in practical dustries However, MPC also has disadvantages The derivation of thecontrol law is more complex than that of classical PID controllers Theamount of computation required is high when constraints are considered.And an appropriate model of the process needs to be available to implementMPC controller

in-1.2 Motivation of the Thesis

Model Predictive Control (MPC) has been well studied in the literature[8 10] MPC designs have the ability to yield high performance control sys-tems that is capable of operating without expert intervention for long peri-ods of time Hence, the MPC concept has been widely studied by academia[2, 11] and adopted in a wide range of practical applications [12–15], such

as ships [16], aerospace [17], road vehicles [18], Unmanned helicopter [19],and building cooling systems [20] However, MPC for multi-zone tracking

is not fully studied Some attempts have been done for temperature formity control [4 6] But these studies on temperature uniformity usuallyfocused on the set-point tracking uniformity from batch to batch, not theuniformity of the zone-to-zone temperature trajectories Moreover, most

uni-of MPC control designs use Kalman filter to filter the measurement noisewhich is assumed to be Gaussian distributed This might be a limitation

in the case of non-Gaussian noise as Kalman filter is well-known to besensitive to outliers [7]

For the convenience of differentiating the conventional MPC among

oth-er newly developed MPCs, the conventional MPC is refoth-erred to as the dard MPC (SMPC) in this thesis The objective of SMPC is to obtain theoptimal performance so that the output follow the pre-set reference [21].The general aim of SMPC cost function is that the future output with-

Stan-in the considered horizons should follow a pre-determStan-ined reference signaland, at the same time, the control effort necessary for doing so should

be penalized [11] However, in some practical applications, e.g ductor manufacturing baking processes, the uniformity of the outputs iscrucial [22–24] Disturbances added onto each output need not necessarily

semicon-to be the same The model of each output is different The trajecsemicon-tory ofhow each output reaches the reference could be very different compared to

guarantee the output uniformity Hence, an innovative MPC should bedeveloped to handle the uniformity cases.

The accuracy of the measurement is important when implementing theMPC system However, in practical system the measurement is usually con-taminated by the noise Hence, a good filter is necessary to enhance the ac-curacy of output measurement A commonly made assumption of Gaussiannoise is an approximation to reality The occurrence of outliers, transientdata in steady-state measurements, instrument failure, human error, pro-cess nonlinearity, etc can all induce non-Gaussian process data [7] Indeedwhenever the central limit theorem is invoked — the central limit theorembeing a limit theorem can at most suggest approximate normality for realdata [26] However, even high-quality process data may not fit the Gaus-sian distribution and the presence of a single outlier can spoil the statisticalanalysis completely Take the example of the chemical-mechanical polish-ing of semiconductor wafers [27–29] The histogram of the distribution

of 576 thickness measurements (see Figure 1.3) after chemical-mechanicalpolishing of twenty-four 200mm semiconductor wafers and after subtract-ing the mean are plotted in Figure 1.4 Using the maximum likelihoodcriterion, a Gaussian distribution was fitted to the histogram It is evident

in Figure 1.3 that the Gaussian curve does not give a good fit Hence, acapable observer is required to reduce the effect of the non-Gaussian noise.The computational load is also a problem when applying MPC to real-

Figure 1.3: The maximum likelihood criterion was used to fit a Gaussian

distribution (dotted-line, µ = 0, σ = 28.5nm) and GT distribution line, p = q = 2, σ = 29.5nm) to the thickness measurement distribution.

This results in demanding on-line computational load and can be a limitingfactor when applying MPC to complex systems with fast dynamics or toembedded applications where computational resources are limited [34] As

a strategy to reduce computational complexity of MPC, Multiplexed ModelPredictive Control (MMPC) has been proposed [35] Results for MMPChave also been established [36] A robust version of MMPC which ensuresatisfaction of hard constraints in the presence of unknown but boundeddisturbances is also available [37] However, the computational advantage

of MMPC has only been proven by MATLAB simulation This thesisprovides the first experimental verification

1.3 Contribution of the Thesis

In this thesis, first we proposed a method called Uniformity Model dictive Control (UMPC) as a strategy to ensure good output uniformity.The algorithm of UMPC could be implemented by using the already exist-ing MPC software Simulations and experiments were carried out to showthat UMPC gives better output uniformity than SMPC SMPC solves theoptimization problem with the cost function which minimizes the errorsbetween the outputs and the references With SMPC method applied, theperformances of each outputs could reach the reference, but the trajectoriescould be very different SMPC method could not ensure good uniformity

Pre-of the outputs However, UMPC solves the optimization problem with adifferent cost function The cost function of UMPC minimizes one errorbetween one output and its reference, and errors between this output andthe other outputs In order to have good output uniformity, only one out-put follows the setpoint, while this output becomes the references of theother outputs.

Work on applications of MPC as a feedback controller for bake-platetemperature control can be found in [38], and feed-forward control wasreported in [39] In addition, a Linear Quadratic Gaussian (LQG) con-troller has been applied to a state-of-the-art 49-zone bake-plate [40] LQGand SMPC are optimal control strategies In this thesis, we derived thealgorithm of UMPC The detailed derivation will be discussed in chapter

2 Bake-plate experiments were conducted We compared the load bance performance of UMPC and SMPC The experimental results showthat when the set-points are the same, UMPC has better output unifor-mity compared to SMPC We also show that when the plant modellingerror exists, the UMPC maintains the uniformity performance whereas theSMPC does not

distur-We then proposed a filter called ARMAX filter for MPC by modellingnoise with the GT distribution instead of the usual Gaussian distribution

bility to characterize data with non-Gaussian statistical properties [41–43].

It is evident in Figure 1.3 that the GT distribution fit the experimentaldata better than the Gaussian curve In this work, we use the InfluenceFunction (IF) to analyse the state estimation problem with GT noise Theanalysis is further generalized to the case where the estimator designed with

probability density function f (ε) is applied to noise with different bility density function g k (ε) at different sampling instance, k, to provide

proba-a frproba-amework for the proba-anproba-alysis of outliers Influence Function (IF) is proba-alsoused to formulate a recursive algorithm that gives an approximate solutionmaking it suitable for real-time and on-line implementation Specificallythe problem is formulated as the filtering of the ARMAX process with GTnoise We also show how the IF can be used to analyse the filter, specifi-cally how it can predict the filter output due to outliers and the variance

of the output To put things in perspective, it will be shown through anexample that if the noise is Gaussian then the proposed ARMAX filter isequivalent to the Kalman filter [44] Otherwise the ARMAX filter has theextra degrees of freedom to model the noise

In Chapter 5, we provided one of the first experimental verification ofthe computational load reduction property of MMPC SMPC updates al-

l the control signals simultaneously However, MMPC only updates onecontrol signal at a time (see Fig 5.1) The main idea of MMPC is topartition the entire system into smaller subsystems, solve each subsystem

sequentially, and update subsystem controls as soon as the solution comes available [45] This is in contrast to SMPC, which solves the entireoptimisation problem in one go An estimate of the computational com-

be-plexity of SMPC (or quadratic programming) is O((m × N u)3), where m

is the number of control inputs and N u the control horizon, i.e., the putational complexity increases as a cubic function of the total number of

com-decision variables (m × N u) In MMPC, only one control is updated at atime and the process is repeated sequentially, the computational complexi-

ty of MMPC is roughly m × O(N3

u) In other words, for each control move,MMPC solves a smaller optimisation problem, and resulting in reducedcomputational complexity and hence computational load Simulation workhas been done to show the MMPC computational advantage [33] However,

in the real practical application, real measurement includes the necessaryoverhead Hence, experiments need to be conducted to consolidate thetheory

1.4 Scope of the Thesis

This thesis is organised as follows In chapter 2, we propose the algorithm

of UMPC which can deal with the output tracking problem

Simulation-MAX filter in chapter 3 The derived ARMAX filter can be used as aneffective observer as well In chapter 4, simulations of the closed-loop M-

PC control system with ARMAX filter as an observer are given to showthe advantage of the ARMAX filter when the output measurement is con-taminated by non-Gaussian noise In chapter 5, practical experiments areconducted on wafer bake-plate to support the computational load advan-tage of MMPC compared to SMPC The conclusion chapter summarisesthe work of this thesis

chap-PC We show that when the plant modeling error exists, the UMPC tains the uniformity performance whereas the SMPC does not We formu-late the UMPC such that the weighting parameter tuning is related to theuniformity performance whereas the SMPC does not have this property.

main-2.1 Introduction

In manufacturing, maintaining product uniformity is important For ample, in semiconductor manufacturing, current photoresist processes inadvanced lithography system are especially sensitive to temperature Thekey output in photolithography is the linewidth of the photoresist pattern

ex-or critical dimension (CD) and the CD is significantly impacted by severalvariables that must also be monitored to ensure quality [46, 47] Thermalprocessing of semiconductor substrate is common and critical in the pho-tolithography sequence Temperature uniformity control is an importantissue with stringent specifications and has a significant impact on the CD[38, 48, 49] The most temperature sensitive step in the photolithographysequence is the post-exposure bake step As the photolithography industrymoves to bigger substrate and smaller CD, the stringent requirements forpost-exposure bake processing still persist [50, 51] Beyond year 2013, thepost-exposure bake resist sensitivity is expected to be less than 1 nm/◦C,making temperature control even more critical [52] A number of recentinvestigations also showed the importance of proper bake-plate operation

on CD control [53–55]

Thermal processing of semiconductor wafers is commonly performed byplacement of the wafer on a heated plate for a given period of time Theheated plate is of large thermal mass relative to the wafer and is held at

a constant temperature by a feedback controller that adjusts the resistiveheater power in response to a temperature sensor embedded in the platenear the surface The plate is designed with multiple radial zone configu-rations The wafer may be placed in direct contact or on proximity pins.Processes that utilize this thermal process include photoresist processing,chemical vapor deposition and rapid thermal annealing, and span a largetemperature range [38, 46].

A general requirement for these systems is the ability to reject theload disturbance induced by placement of a cold wafer on the bake-plate.Fig.2.1 shows the SMPC closed-loop temperature response of a bake-plateused for photoresist processing when a 200 mm wafer at room temperaturewas placed on the bake-plate Initially the temperature dropped and thenrecovered because of closed-loop control However, the 3-zone responsetrajectories are very different from each other In some practical applica-tions, e.g semiconductor manufacturing baking processes, the non-uniformtemperature trajectories will affect critical dimension of the wafer [36] andcould result wafer warping Warped wafers can affect device performance,reliability, and linewidth control in various processing steps [56–58] Hence,

a suitable MPC should be developed to make the response trajectories ofdifferent zones on the wafer as similar as possible, as shown in Fig 2.2,

Figure 2.1: SMPC Temperature response of 3-zone bake-plate with roomtemperature wafer placed on.

2.2 Algorithm of UMPC

UMPC is proposed by reformulating the cost function of the SMPC Thereformulated cost function is aimed to minimise the difference between eachoutput, while SMPC cost function focuses on tracking the pre-set reference.Model Predictive Control (MPC) operates by solving a constrained op-timization problem on-line, in real-time, in order to decide how to updatethe control inputs (manipulated variables) at the next update instant The

0 200 400 600 800 1000 88

Figure 2.2: UMPC Temperature response of 3-zone bake-plate with roomtemperature wafer placed on

objective of the Standard MPC (SMPC) is to control the inputs so that theoutputs could reach the references as fast as possible With SMPC methodapplied, the performances of each outputs could reach the references, butthe trajectories could be very different SMPC method could not ensuregood uniformity of the outputs, as shown in Fig 2.1 The general aim ofSMPC cost function is that the future outputs within the considered hori-zons should follow the pre-determined reference signals and, at the same

objective can be expressed mathematically as

where p is the output number, N is the control horizon, w1, w2, , w p are

the references for the p outputs respectively, and F (x k+N) is a suitablechosen terminal cost

Disturbance added onto each output need not necessarily to be thesame In general, the model of each output could be different The trajec-tory of how each output reaches reference could be very different compared

to each other, even when the references for each output are the same Thus,SMPC in general, does not ensure good output uniformity

In contrast, as a strategy to ensure the good output uniformity, formity Model Predictive Control (UMPC) is proposed UMPC solves theoptimization problem with a different cost function The cost function ofUMPC minimizes the error between one output and the pre-set reference,and also the errors between this output and the other outputs In order tohave good output uniformity, only one output follows the setpoint, whilethis output becomes the references of the other ouputs Hence, the outputtrajectories could be very similar to each other The output uniformity issecured, as shown in Fig 2.2 The cost function of UMPC is expressed as

Uni-J umpc =

N∑−1 i=0

is an effective way of achieving uniformity amongst the different outputs

We compared the load disturbance performance of UMPC and SMPC perimental results showed that in the presence of modelling errors, e.g gainand time constant, UMPC maintains the uniformity performance whereasthe SMPC did not

with an incremental input, ∆u, and one possibility is given as

∆u k = u k − u k −1 is the incremental control move applied to input at time

step k We assume that at time step k the complete state vector x k isknown exactly from measurements

Let

N∑−1 i=0

∥ ∆u k+i ∥2

r +F (x k+N) (2.3)

w.r.t ∆u k+i s.t x k+i+1 = Ax k+i + B∆u k+i

where N is the control horizon, a design parameter which denotes the

number of control moves to be optimized per input channel of the originalsystem (2.1); p is the output number; F (x k+N) is the stabilising terminal

cost, here we set it to be F (x k+N ) = x T

k+N Q n x k+N Q n is obtained bysolving the discrete LQR problem of the augmented system (Equation2.2)

The different constructions of J x,k is the only different part of UMPC andSMPC cost functions SMPC focuses on minimising the error between theoutput and its reference While UMPC focuses on minimising the errorbetween one output and another output

For a plant with p outputs,

For simplicity, the setpoint here is assumed to be zero.

N∑−1

i=0

(

y 1,k+i+1 T q1y 1,k+i+1 + (y 1,k+i+1 − y2,k+i+1)T q2(y 1,k+i+1 − y2,k+i+1)

+ + (y 1,k+i+1 − y p,k+i+1)T q p (y 1,k+i+1 − y p,k+i+1))

If no constraints are active, or the set of constraints which will be active isconstant and known, then the UMPC control law is a linear state feedback:

and the state feedback gain K can be computed similar to the standard

MPC design procedure The state feedback gain can be computed as

K = −H −1 E

where

2.2.3 Cost Function Comparison of UMPC and

SMPC

The setpoint is assumed to be zero for simplicity, let

N∑−1 i=0

∥ ∆u k+i ∥2

r +F (x k+N ).

N

)

The difference of the cost functions of UMPC and SMPC is the

differen-t consdifferen-trucdifferen-tions of Q umpc and Q smpc This difference shows that UMPCfocuses on the output uniformity while SMPC focuses on tracking presetreferences Thus, UMPC has the output uniformity advantage over SMPC

by design

We can easily find out that the format of the UMPC state-feedbackcontrol law is the same as the one of SMPC The only difference is theway of calculating the control law parameters Thus, UMPC could beimplemented by the existing MPC software

2.3 Bake Plate Thermal Modeling

The physical model of an m-zone bake-plate as shown in Figure 2.3 hasbeen derived in [12] According to this paper, a distributed lumped modelcan satisfactorily describe the bake-plate characteristics, and with energy

balance and heat transfer laws, an m-zone bake-plate can be modeled as

Figure 2.3: A photograph of the multizone bake plate.

where

i = 1, 2, · · · , m denotes zone i

C i = heat capacity of the ith zone (J/K)

T i (t) = the ith zone temperature above ambient (K)

r i = thermal resistance between zone i and surrounding air (K/W)

r (i −1)i = thermal resistance between zone

i − 1 and zone i; r(i −1)i =∞ for i = 1 (K/W)

i and zone i + 1; r i(i+1)=∞ for i = m (K/W)

p i (t) = heater power to zone i (W)

At steady state, ˙T i(∞) = 0 and Eq (2.7) becomes

Defining new variables θ i (t) = T i (t) − T i(∞), u i (t) = p i (t) − p i(∞) and

substituting Equation (2.8) into Equation (2.7) gives