Model-Based Controller Design

1. Introduction

2. Direct synthesis method

3. Internal model control (IMC)

4. IMC derived PID tuning rules

5. Simulink example

Model-Based Control

PID controller tuning

» Restrict controller to PID form

» Seek “best” tuning parameters

» Can perform with FOPTD model if available

Model-based controller design

» Controller is not restricted to PID form

» Requires a process model that is used to

determine the controller form as well as the

tuning parameters

» Not restricted to FOPTD models

» Makes full use of available model

» Generates PID controllers for many model types

Direct Synthesis Method

Closed-loop transfer function for setpoint changes

Simplification of CLTF

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pvcm

sp GGGG

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1

c

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sp

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Control Objective

Rearrange CLTF

Desired setpoint response

» Gd is the desired CLTF

» The controller Gc depends explicitly on the

inverse of the process model G

» The equation for Gc is known as the control law

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sp

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Desired Closed-Loop Transfer Function

The desired CLTF Gd is specified such that:

» The resulting Gc has a single tuning parameter with an easily understood effect on closed-loop stability and performance

» Gc is implementable – does not require prediction and has the appropriate properness

Properness

» If n >= m, the controller is proper no derivative control

» If n = m-1, the controller is improper derivative control

» If n = m-2, the controller is improper requires second derivative of measured output (not desirable)

» Seek controllers that are proper or improper with n = m-1

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)()(

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Selecting the Desired CLTF

Common choice

» tc > 0 is the desired closed-loop time constant

» Gd is stable for all tc > 0

» Gd has a steady-state gain of unity ensuring

offset-free performance due to integral action

in Gc

» Closed-loop speed of response is determined

by tc; typical choice is tc = 0.5t

Other choices of Gd may be required to

ensure that Gc is implementable

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Simple Examples

First-order system

» This is a PI controller

Second-order system

» This is a PID controller

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Systems with Time Delays

Model representation:

Desired CLTF must include time delay

FOPTD model

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d

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Non-Minimum Phase Systems

Process Model

Zeros: N(s) = 0 » Systems with right-half plane zeros can exhibit

inverse response

» Such systems are said to be non-minimum phase

Direct synthesis controller

» Zeros of model become poles of controller

» Controller is unstable if model is non-minimum phase not acceptable

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Internal Model Control

Applicable to both minimum-phase and non-minimum phase systems

Does not invert non-invertible elements: time delays and right-half plane zeros

IMC approach

» Factor model into invertible and non-invertible parts

» Design IMC controller using the IMC control structure

» Convert IMC controller into standard feedback controller

» Implement standard feedback controller as usual

IMC Structure

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1 *

*

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)~

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)~

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IMC Design

Factor the process model

» contains any time delays and right-half plane

zeros, has unity gain and is an all-pass element

Construct the IMC controller

» f is the IMC filter, tc is the desired closed-loop

time constant and r is chosen to make G*c proper

Resulting closed-loop relation

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1~1*

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~1*

First-Order System

This is a PI controller

Same result as direct synthesis method

Two methods always yield same result when G+ = 1

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Non-Minimum Phase Examples

Right-half plane zero

Time delay

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)3)(2(

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1

1

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)3)(2(

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PID Tuning Rules

Example: IMC Design

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)3)(2(

)1(4)4)(1(

)3)(2(

)3)(2(

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)3)(2(

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)3)(2(

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)3)(2(

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*

*

*

41

Example: Simulink Implementation

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)65(

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)3)(2(

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)3)(2(

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2

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2

2

Example: Setpoint Tracking

0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

1

1.5

Outp

ut

Time0 2 4 6 8 10 12 14 16 18 20

-2

-1.5

-1

-0.5

0

0.5

Input

Time

Example: Disturbance Rejection

0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

1

1.5

Outp

ut

Time

0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

1

1.5

2

Input

Time