MECH466 L15 FreqResponse

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2008/09 MECH466 : Automatic Control 1

MECH466: Automatic Control

Dr. Ryozo Nagamune

Department of Mechanical Engineering

University of British Columbia

University of British Columbia Lecture 15

Lecture 15

Root locus: Design with

Root locus: Design with MatlabMatlab

Frequency response

Course roadmap

Laplace transform Laplace transform Transfer function Transfer function

Models for systems Models for systems •

•electricalelectrical •

•mechanicalmechanical •

•electromechanicalelectromechanical

Linearization Linearization

Modeling

Modeling AnalysisAnalysis DesignDesign

Time response Time response •

•TransientTransient •

•Steady stateSteady state

Frequency response Frequency response •

•Bode plotBode plot

Stability Stability •

•RouthRouth--HurwitzHurwitz •

•NyquistNyquist

Design specs Design specs Root locus Root locus Frequency domain Frequency domain

PID & Lead PID & Lead--laglag

Design examples Design examples

Matlab

Matlabsimulations & laboratoriessimulations & laboratories

A design example (review)

Consider a systemConsider a system

Analysis of CL system for Analysis of CL system for C(sC(s)=1)=1

Damping ratio Damping ratio ζζ=0.5=0.5

UndampedUndampednatural freq. natural freq. ωωnn=2 =2 rad/srad/s

RampRamp--error constant error constant KvKv=2=2

Performance specificationPerformance specification

Damping ratio Damping ratio ζζ=0.5=0.5

UndampedUndampednatural freq. natural freq. ωωnn=4 =4 rad/srad/s

RampRamp--error constant error constant KvKv=50=50

G(s

G(s)) C(s

C(s))

Plant Plant Controller Controller Re Re Im Im Desired pole Desired pole

Lead & lag compensators (review)

Lead compensatorLead compensator

Improve transient responseImprove transient response

Improve stabilityImprove stability

Lag compensatorLag compensator

Reduce steady state errorReduce steady state error

Lead-Lead-lag compensatorlag compensator

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SISO Design Tool in

Matlab

GraphicalGraphical--user user interface (GUI) that interface (GUI) that allows you to design allows you to design compensators. compensators.

Type Type ““sisotoolsisotool””in in Matlab

Matlabprompt.prompt.

Select Select ““Root locusRoot locus”” from

from ViewÆViewÆDesign Design Plots configuration Plots configuration..

Root locus plot Root locus plot

SISO Design Tool (cont

’

d)

Input the plantInput the plant

Import the plant sysGImport the plant sysG from

from FileFileÆÆImportImport

’

d)

You will see the root You will see the root locus plot for

locus plot for

The default setting The default setting C(s

C(s)=1.)=1.

The corresponding The corresponding closed

closed--loop poles are loop poles are indicated by pink indicated by pink squares.

squares.

SISO Design Tool (cont

’

d)

Add a pole & a zero Add a pole & a zero of a compensator, of a compensator, and adjust its gain: and adjust its gain:

If necessary, move If necessary, move the pole and zero the pole and zero

by clickby click--andand--drag, ordrag, or

DesignDesignÆÆEdit Edit Compensator

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SISO Design Tool (cont

’

d)

See the step responseSee the step response

DesignÆDesignÆEdit CompensatorEdit Compensator……ÆÆAnalysis PlotsAnalysis Plots

’

d)

Modify the pole/zero for “Modify the pole/zero for “goodgood””step responsestep response

SISO Design Tool (cont

’

d)

Add a pole & a zero of Add a pole & a zero of

If necessary, adjust controller parameters.If necessary, adjust controller parameters.

Announcement

See the updated schedule for Lab 4 See the updated schedule for Lab 4

MECH466_LabSchedule_0809 (Mar 5).pdf. MECH466_LabSchedule_0809 (Mar 5).pdf.

If your group wants to perform Lab 4 on Tuesday If your group wants to perform Lab 4 on Tuesday March 24 (4

March 24 (4--6pm, or possibly 66pm, or possibly 6--8pm), please 8pm), please contact me ASAP.

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Course roadmap

Laplace transform Laplace transform

Transfer function Transfer function

Models for systems Models for systems •

•electricalelectrical •

•mechanicalmechanical •

•electromechanicalelectromechanical

Linearization Linearization

Modeling

Modeling AnalysisAnalysis DesignDesign

Time response Time response •

•TransientTransient •

•Steady stateSteady state

Frequency response Frequency response •

•Bode plotBode plot

Stability Stability •

•RouthRouth--HurwitzHurwitz •

•NyquistNyquist

Design specs Design specs

Root locus Root locus

Frequency domain Frequency domain

PID & Lead PID & Lead--laglag

Design examples Design examples

Matlab

Matlabsimulations & laboratoriessimulations & laboratories

What is frequency response?

We would like to analyze a system property by We would like to analyze a system property by applying a

applying a test sinusoidal inputtest sinusoidal inputr(t) and r(t) and observing a response

observing a response y(ty(t).).

Steady state response ySteady state response yssss(t) (after transient dies (t) (after transient dies out) of a system to sinusoidal inputs is called out) of a system to sinusoidal inputs is called

frequency response frequency response..

System

A simple example

RC circuitRC circuit

Input a sinusoidal voltage Input a sinusoidal voltage r(tr(t))

What is the output voltage What is the output voltage y(ty(t)?)? R

R

C

r(t

r(t)) y(ty(t))

An example (cont

’

d)

TF (R=C=1)TF (R=C=1)

r(t)=r(t)=sin(tsin(t))

0 5 10 15 20 25 30 35 40 45 50 -1

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

r y

At steady

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An example (cont

’

d)

Derivation of Derivation of y(ty(t))

Inverse LaplaceInverse Laplace

0 as t goes to infinity.

Partial fraction expansion

Partial fraction expansion

(Derivation for general

(Derivation for general G(sG(s) is given at the end of lecture slide.)) is given at the end of lecture slide.)

Response to sinusoidal input

How is the steady state output of a linear system How is the steady state output of a linear system when the input is sinusoidal?

when the input is sinusoidal?

Steady stateSteady stateoutput output

FrequencyFrequencyis same as the input frequencyis same as the input frequency

AmplitudeAmplitudeis that of input (A) multiplied byis that of input (A) multiplied by

PhasePhaseshifts shifts GainGain

G(s

G(s))

y(t

y(t))

Frequency response function

For a stable system For a stable system G(sG(s), ), G(jG(jωω) () (ωωis positive) is is positive) is called

called frequency response function (FRF)frequency response function (FRF)..

FRF is a complex number, and thus, has an FRF is a complex number, and thus, has an amplitude

amplitudeand a phaseand a phase..

First order exampleFirst order example

Re

Im

First order example revisited

FRFFRF

Two graphs representing FRFTwo graphs representing FRF

Bode diagram (Bode plot) (Today and next lecture)Bode diagram (Bode plot) (Today and next lecture)

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Bode diagram (Bode plot) of

G(j

ω

)

Bode diagram consists of Bode diagram consists of gain plotgain plot& & phase plotphase plot

Log

Log--scalescale

Bode plot of a 1st order system

TF TF

10-2 10-1 100 101 102 -50

-40 -30 -20 -10 0

10-2 10-1 100 101 102 -100

-80 -60 -40 -20 0

Corner frequency Corner frequency

Exercises of sketching Bode plot

First order systemFirst order system

Remarks on Bode diagram

Bode diagram shows amplification and phase Bode diagram shows amplification and phase shift of a system output for sinusoidal inputs with shift of a system output for sinusoidal inputs with various frequencies.

various frequencies.

It is very useful and important in analysis and It is very useful and important in analysis and design of control systems.

design of control systems.

The shape of Bode plot contains information of The shape of Bode plot contains information of stability, time responses, and much more! stability, time responses, and much more!

It can also be used for system identification. It can also be used for system identification. (Given FRF experimental data, obtain a transfer (Given FRF experimental data, obtain a transfer function that matches the data.)

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System identification

Sweep frequencies of sinusoidal signals and Sweep frequencies of sinusoidal signals and obtain FRF data (i.e., gain and phase). obtain FRF data (i.e., gain and phase).

Select G(sSelect G(s) so that ) so that G(jG(jωω) fits the FRF data.) fits the FRF data. Agilent Technologies: FFT Dynamic Signal Analyzer

Agilent Technologies: FFT Dynamic Signal Analyzer

Unknown

system

Generate sin signals Generate sin signals Sweep frequencies Sweep frequencies

Collect FRF data Collect FRF data Select

Select G(sG(s))

Summary and exercises

Frequency response Frequency response

Steady state response to a sinusoidal inputSteady state response to a sinusoidal input

For a linear system, sinusoidal input generates For a linear system, sinusoidal input generates

sinusoidal output with

sinusoidal output with same frequencysame frequencybut but different different

amplitude and phase

amplitude and phase..

Bode plot is a graphical representation of Bode plot is a graphical representation of frequency response function. (

frequency response function. (““bode.mbode.m””))

ExercisesExercises

Read Section 10.1.Read Section 10.1.

Solve Problem 10.1.Solve Problem 10.1.

Another example of FRF

Second order systemSecond order system

Re

Im

Derivation of frequency response

Term having denominator of

Term having denominator of G(sG(s))

0 as t goes to infinity.

MECH466 L15 FreqResponse

MECH466: Automatic Control

Lecture 15

Course roadmap

A design example (review)

Lead & lag compensators (review)

SISO Design Tool in

Root locus plot Root locus plot

SISO Design Tool (cont

SISO Design Tool (cont

SISO Design Tool (cont

SISO Design Tool (cont

Course roadmap

What is frequency response?

A simple example

At steady

An example (cont

Partial fraction expansion

Response to sinusoidal input

Frequency response function

 FRFFRF

Bode diagram (Bode plot) of

Corner frequency Corner frequency

Exercises of sketching Bode plot

System identification

Generate sin signals Generate sin signals Sweep frequencies Sweep frequencies

Summary and exercises

Another example of FRF

FRFFRF