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2006 The MathWorks, Inc.

MATLAB and Simulink forControl Design Acceleration

Control Design

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6 - 2Control Design


Section Outline

Uses of feedback

Control design methods

Time domain

Frequency domain Control design tools

SISO tool Simulink Response Optimization

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6 - 3Control Design


Uses of Feedback

Make things happen as closely as possible tosome specification e.g. rise time.

Solve potential problems

Overshoot Oscillation Positioning errors Plant instability

Place limits on parameters Reduce effect of plant uncertainty

Make efficient use of control energy

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6 - 4Control Design


Control Design Tools

The Control System Toolbox offers a variety oftools for designing and evaluating controllers

Root locus and Bode design (SISOTOOL) Pole placement

LQR (Kalman filtering) techniques

The advanced control toolboxes offer alternativedesign techniques

Robust, Mu-Analysis, and Synthesis

Fuzzy Logic, Neural Networks Model Predictive Control

>> doc control

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6 - 5Control Design


Loop Shaping

Control system design essentially consists ofshaping the nominal system in the frequencydomain.

This is typically done by adding transfer functionterms to ensure: Enough roll-off at high frequency Enough gain at low frequency Good stability margins

PID control is loop shaping with a constrainedcontroller structure and is sufficient for manyapplications

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6 - 6Control Design


Root Locus Diagram

Shows the effect ofincreasing gain k

Diagram plotted from openloop plant, rlocus(G)

Poles travel towards zerosor infinity as k increases

Determination ofdestabilising gain

>> ms_transferfcn

>> rlocus(ms_tf)

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6 - 7Control Design


Root Locus Loop Shaping

Place dominant closed-loop poles in the desired region bychanging the gain and by introducing additional poles and

zeros.

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6 - 8Control Design


Stability Margins

The amount by which gain and phase can changewhile still maintaining stability

Gain margin (GM) Phase margin (PM)

Phase Margin

Gain Margin

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6 - 9Control Design


Frequency Domain Design

Phase-Lead Compensator

PD is an ideal Phase-Lead Compensator Phase-Lag Compensator

PI is an ideal Phase-Lag Compensator Lag-Lead and Lead-Lag

PID is an ideal lead-lag or lag-lead compensator These techniques can be used to change the

phase and gain margins.

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6 - 10Control Design


Phase-Lead

Phase-Lead

z=1 p=5

z=1p=5

peak=2.23

peak=6.98

ps

zsksH

)(

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Phase-Lag

Phase-lag

p=1 z=5

z=5

p=1

peak=2.23

peak=-6.98

ps

zsksH

)(

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Lead-Lag and Lag-Lead

Lead-lag compensator

Phase-lead and phase-lag cascade

1> 2peak1< peak2

Increases the magnitude of the system frequencyresponse by 20log(a) in the range peak1-peak2

Lag-lead compensator

Phase-lag and phase-lead cascade

1> 2peak1< peak2

Increases the magnitude of the system frequency

response by -20log(a) in the range peak1-peak2

)1)(1(

)1)(1()(

21

21

sas

ssacsH

)1)(1()1)(1()(

21

21

ssasascsH

>> leadlaglaglead

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Lead-Lag and Lag-Lead Plots

>> leadlaglaglead

Lead-Lag Peak magnitude is20*log10(5)

Peak magnitude is-20*log10(5)

peak1=1 peak2=100

Lag-Lead

peak1=1 peak2=100

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SISO Design Tool

Feedbackstructure

Currentcontrol law

Tools for addingcontroller polesand zeros

Graphical displayof open-loop,closed-loop, andcontroller polesand zeros

Zoom controls

Root locus

Bode

response

Response Characteristics>> ms_transferfcn, sisotool(ms_tf)

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Adding Design Constraints

Right-click and select NewDesign Constraint

Specify Settling Time

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Adding a Controller

Controllers may be added and changed in several ways

1. Click here, and then drag

the poles, zeros or gain in

the Root Locus window.

2. Click in the CurrentCompensator box or on thered colored C block to see acontroller editor GUI.

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Choose theAnalysis menuto view timeresponses

Viewing Responses (LTI Viewer)

Opens the LTIViewer to display

responses

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Draw Simulink Diagram

LTI System

block

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Tuning the PID Gains

After the initial design there are severalways to further tune the controller Adjust the gains by trial and error

Write a custom optimization script Use the Optimization Toolbox Use Simulink Response Optimization

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Simulink Response Optimization

Simulink Response Optimization offers designersthe ability to: Optimize block parameters Place constraints on any signal

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Using Simulink Response

Optimization

>> mass_spring_respopt

Insert and connect the SignalConstraint block just like anyother Simulink block

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Specifying Constraints

There are two ways to specify constraints: Using the mouse to drag from the center point, end point ortime boundary

Right-clicking on a boundary and use the Constraint Editor

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Specifying Tunable Parameters

Specify the tunableparameters by name

The tunable parameters must be specified. This is doneusing the Optimization->Tuned Parameters pull down menu

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Running the Optimization

The constraint values that give this response can beloaded frommass_spring_constrains.mat.

If left to run, the finalresponse will be similar this

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Review: the Design Steps

We completed all steps in the design process:

1. Performance/control specifications2. Model development

3. System analysis

trim and linearize model4. Controller design

a.Import system into Control System Toolbox

b.Design a controller, inspecting time and

frequency domain response

c. Use Simulink Response Optimization to further

tune controller5. Performance specification tests simulatecomplete system to see if specifications are met

6. Implementation

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Additional Considerations

Real designs must take account of many ofthe following components

Signal conditioning and prefilters Noise filters

Notch filters Characteristics of the A/D and D/A

converters

Actuator dynamics

Sensor dynamics Discretization of the controller Implementation of the controller

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Implementation

Use Real-Time Workshop Rapid simulations: non-real time Rapid prototyping applications: hardware-in-the-loop Embedded system design: code generated for embeddedsystems

Rapid

Simulation

Hardware-in-

the-Loop

Simulation

Embedded

System

Code

GenerationPrototyping

Code

EmbeddedCode

Deployment

Test

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Summary Outline

Uses of feedback

Control design methods

Time domain

Frequency domain Control design tools

SISO tool Simulink Response Optimization

ct01 06 design

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