0806ThermalSystems_webinar_2015

Modeling Feedback Control of Thermal Systems in COMSOL

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Lexi Carver Technical Marketing

Engineer COMSOL

Jon Ebert, PhD Director

SC SOLUTIONS

http://www.comsol.com/trademarks

Agenda• Introduction to COMSOL

Multiphysics® software• Simulating Feedback Control

of Thermal Systems• Demo: Rapid Thermal

Processing (RTP) System• Q&A• How To– Try COMSOL Multiphysics®– Contact Us

Prestressed plated metal layers in a model of a micromirror

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Displacement induced by thermal strains and applied pressure in a model of a capacitive pressure sensor

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Copyright © 2015, SC Solutions, Inc. All Rights Reserved 9August 2015

Feedback Control of Thermal Systemsin COMSOL

Jon EbertSC Solutions, Inc.

Email: [email protected]


Overview

Feedback control of temperature is important in many manufacturing processes.

At SC we have developed methods for the model-based control of systems where we use physics-based models to develop high-performance temperature control.

This presentation will give an example of one method for designing a simple PID feedback controller (Q-Design).

We will present a simple example of feedback control within a COMSOL model.

A demo with 5-zone temperature control of a Rapid Thermal Processing (RTP) system will be shown.


Closed-Loop Transfer Function

The Closed-Loop System

PC

Closed-Loop Response:

For a scalar system


Proportional + Integral + Derivative Control (PID)

This is the most common type of feedback controller

The “gains”, Kp, Ki, and Kd (and td) must be selected by the engineer

There are systematic methods

Here we’ll outline ‘Q’-design

To limit high-frequency noise amplification, we usually low-pass filter the error driving the derivative control:


Model-Based Control (Q-Design)

Incorporate a mathematical model of the system directly into the controller.

Often referred to as Q-parameterization or Youla parameterization.

References for Q-parameterization Control Design

For stable P, ALL stable controllers can be expressed in this form!

Control design becomes choice of transfer function Q


A Model-Based Design (‘Q-Design’)

Closed-Loop Response:

If we select T and we know P, we can calculate C

For scalar (single input, single output)


Selecting T (Closed-Loop Transfer Function)

Or, in time domain

Or, in time domain


Solving for Model-Based Controller Gains

where and select

Resulting controller:

Our PID form:

Tuning PID

It turns out that if you pick T as we have done for the second order P, the controller is in the form of a PID controller.

Assume the system (P) is a second order system


PID Example (w=1, z=1)


PID Example (w=2, z=1)


Implementing PID in COMSOL (An Example)

One dimensional heat transfer in a plate

Governing Equations:

Measure y on this surface

Heat surface at x=0 and measure temperature at surface x=L (y=T2)


Build the Model in COMSOL


Model Parameters


Apply Boundary Conditions

Control input


Create Mesh and Run the Simulation

This is not a second order system, but a much higher order PDE.

When we discretize the PDE using finite elements, we create a system of ODE’s.

For the mesh used here, there are 101 differential equations, or degrees of freedom (DOF).

Still, we will look for a PID controller based on a second order system.


Open-Loop Response

We step the input flux from 0 to qmax*uctrl


Measure Model Response Properties from OL Step

Time where y crosses 63% of final value


Finding Approximate Time Constants

Measure the rate of change in y.Maximum is approximately 18[K/s]


Approximate 2nd Order Model Parameters

We can use these to compute controller parameters.

We also need to pick the closed-loop speed, w, and damping, z.

Approximate 2nd order model parameters for plate model:


Implementing PID in COMSOL

Proportional control:

Integral control:

Derivative control:

We solve these differential equations in COMSOL.


Implementing PID in COMSOLAdd “Global ODEs and DAEs” to Model

Enter the Equations

State variableEquation


Implementing PID in COMSOLDefine T2 as the plate temperature at x=L

Define control variables

The control: limit to 0<= uctrl <=1


Define the Reference

Define the reference using an interpolation table.


Controller Parameters

Tuning PID


Closed-Loop Response

Works pretty well.

The open-loop response took almost 100s to reach the setpoint, the closed loop gets there in 12s.

Sensor

ref

Actuator face


More Advanced Model-based Control

(MIMO) Multi-input (u)/Multi-output (y) require more complicated methods.

Each operation is a matrix operation…

The design principle is the same, but the difficulty of control design is balancing the tradeoffs between bandwidth, noise, and actuator saturation.

PC


Summary

A methodology for tuning a PID controller has been presented.

A model for the system is built and an approximate second order model is assumed for choosing the PID parameters.

A desired closed-loop transfer function (T) is selected. Here we selected it so the resulting controller has the same form as the PID.

The PID gains were derived that give the desired closed-loop response.

An example of implementing PID in COMSOL was presented.

Poll QuestionHow do you model your feedback control law in your application?• In the same software that I use for studying

thermal systems.• In a different software that I interface.• In COMSOL using the Global ODEs and DAEs

interface.• In COMSOL with LiveLink™ for MATLAB®.• I haven't done it yet and want to learn how.

DemoClosed-Loop Temperature Control of a Rapid

Thermal Processing (RTP) System

Wafer (200mm dia.)

Lamps (5)

FeedbackControllerS

Rapid Thermal Processor (RTP)

-+ Lamp

power commands

Wafer temperature sensors

Temperature reference

Cold walls600°C

1000°C

Conclusion• A model-based method of tuning a PID

controller has been presented (Q-Design).

• For many thermal systems an open-loop step response can be used to characterize parameters (time-constants and gain) needed to tune the PID controller.

• COMSOL provides a simple ODE interface that allows quick implementation of PID control.

Q&A Session

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0806ThermalSystems_webinar_2015

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