MECH466 L19 FreqDomSpecs

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2008/09 MECH466 : Automatic Control 1

MECH466: Automatic Control

Dr. Ryozo Nagamune Dr. Ryozo Nagamune

Department of Mechanical Engineering Department of Mechanical Engineering

University of British Columbia University of British Columbia

Lecture 19 Lecture 19

Examples of gain margin & phase margin Examples of gain margin & phase margin

Frequency domain specifications Frequency domain specifications

Course roadmap

Laplace transform

Transfer function

Models for systems

•

•electricalelectrical •

•mechanicalmechanical •

•electromechanicalelectromechanical Linearization

Linearization Modeling

Modeling AnalysisAnalysis DesignDesign

Time response

•

•TransientTransient •

•Steady stateSteady state Frequency response

Frequency response

•

•Bode plotBode plot Stability

Stability

•

•RouthRouth--HurwitzHurwitz

•

•NyquistNyquist

Design specs

Root locus

Frequency domain

PID & Lead

PID & Lead--laglag Design examples

Design examples

Matlab

Matlabsimulations & laboratoriessimulations & laboratories

Gain margin (GM) (review)

Phase crossover Phase crossover

frequency

frequency ωωpp::

Gain marginGain margin(in dB)(in dB)

Indicates how much Indicates how much OL gain can be OL gain can be multiplied without multiplied without violating CL stability.

violating CL stability. NyquistNyquistplot of L(splot of L(s))

Phase margin:

CORRECTION

Gain crossover Gain crossover frequency

frequency ωωgg::

Phase marginPhase margin

Indicates how much Indicates how much OL phase lag can be OL phase lag can be added without added without violating CL stability. violating CL stability.

Nyquist

(2)

Relative stability on Bode plot (review)

ω

ωgg

ω

ωpp

GM GM

PM PM

Example of GM & PM

dB

deg

--2020

--4040

ω

ωpp: GM:: GM:

ω

ωgg: PM: : PM:

Example of GM & PM

Second order systemSecond order system

Bode plot

Bode plot

10-2 10-1 100 101 102 -100

-50 0

10-2 10-1 100 101 102 -150

-100 -50 0

Nyquist

Nyquistplotplot

-1 -0.5 0 0.5 1 -1

-0.5 0 0.5 1

ω

ωpp: GM:: GM:

ω

ωgg: PM: : PM:

Example of GM & PM

Third order systemThird order system

Bode plot

Bode plot

10-2 10-1 100 101 102 -150

-100 -50 0

10-2 10-1 100 101 102 -200

-100 0

Nyquist

Nyquistplotplot

-1 -0.5 0 0.5 1 -1

-0.5 0 0.5 1

ω

ωpp: GM:: GM:

ω

(3)

How to compute GM?

Frequency response functionFrequency response function

PM computation

often requires

computational tools.

Course roadmap

Laplace transform

Transfer function

Models for systems

•

Time response

•

Frequency response

•

Stability

•

•NyquistNyquist

Design specs

Root locus

Frequency domain

PID & Lead

Design examples

Matlab

Controller design comparison

Design specifications in time domain

(Rise time, settling time, overshoot, steady state error, etc.)

Desired closed

Desired closed--loop loop pole location

pole location

in s

in s--domaindomain

Desired open

Desired open--loop loop

frequency response

in s

in s--domaindomain

Root locus shaping

Root locus shaping Frequency response shapingFrequency response shaping (Loop shaping)

(Loop shaping)

Approximate translation

Feedback control system design

Given Given G(sG(s), design ), design C(sC(s) that satisfies CL stability ) that satisfies CL stability and time domain specs, i.e., transient and

and time domain specs, i.e., transient and

steady

steady--state responses.state responses.

We learn typical qualitative relationships We learn typical qualitative relationships between

between openopen--loop Bode plotloop Bode plot and timeand time--domain domain responses.

responses.

G(s G(s)) C(s

C(s))

Plant

Controller

OL:

CL:

(4)

Typical desired OL Bode plot

Steady

Steady--state accuracystate accuracy Sensitivity

Sensitivity

Disturbance rejection Disturbance rejection

Noise Noise reduction reduction Transient

Transient Response speed Response speed

Transient Transient Overshoot Overshoot Relative stability

Relative stability

Relative stability Relative stability

Noise

reduction

Steady

-

state accuracy

G(s G(s)) C(s

C(s))

Plant

Controller

For steady

For steady--state accuracy, state accuracy,

L should have high gain at low frequencies.

y(t

y(t) tracks ) tracks r(tr(t) composed of ) composed of low frequencies very well. low frequencies very well.

Steady

-

state accuracy (cont

’

d)

Step Step r(tr(t))

Increase Increase

Ramp Ramp r(tr(t)) Increase Increase

Parabolic Parabolic r(tr(t)) Increase Increase

For

For KvKvto be nonzero,to be nonzero,

L must contain

at least one integrator.

For Ka to be nonzero,

L must contain

at least two integrators.

<

<--2020 _<_<_-_-₄₀₄₀

Typical desired OL Bode plot

Noise Noise reduction reduction Transient

Relative stability Relative stability Noise

Noise reduction reduction Steady

Sensitivity

(5)

A second order example

For illustration, we use the feedback system: For illustration, we use the feedback system:

G(s G(s)) C(s

C(s))

Plant

Controller

Percent overshoot

0 5 10 15 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

For small percent overshoot,

L should have larger phase margin.

10-1 100 101 -20

0 20

10-1 100 101 -180

-160 -140 -120 -100

CL step response CL step response OL Bode plot

OL Bode plot

PM

Typical desired OL Bode plot

Noise Noise reduction reduction Transient

Sensitivity

0 5 10 15 0

0.2 0.4 0.6 0.8 1 1.2 1.4

100 -20

0 20

10-1 100 101 -180

-160 -140 -120 -100

Response speed

For fast response,

L should have larger gain crossover frequency.

CL step response CL step response OL Bode plot

(6)

Typical desired OL Bode plot

Noise Noise reduction reduction Transient

Sensitivity

Relative stability

We require adequate GM and PM for:We require adequate GM and PM for:

safety against inaccuracies in modelingsafety against inaccuracies in modeling

reasonable transient response (overshoot)reasonable transient response (overshoot)

It is difficult to give reasonable numbers of GM It is difficult to give reasonable numbers of GM and PM for general cases, but usually,

and PM for general cases, but usually,

GM should be at least 6dB GM should be at least 6dB

PM should be at least 45deg PM should be at least 45deg

(These values are not absolute but approximate!) (These values are not absolute but approximate!)

In controller design, we are especially interested In controller design, we are especially interested in PM (which typically leads to good GM).

in PM (which typically leads to good GM).

Typical desired OL Bode plot

Noise Noise reduction reduction Transient

Sensitivity

Noise reduction

y(t

y(t) is not affected by ) is not affected by n(tn(t) ) composed of high frequencies. composed of high frequencies. G(s

G(s)) C(s

C(s))

Plant

Controller

n(t

n(t): noise): noise y(t

y(t))

For noise reduction,

L should have small gain at high frequencies.

(7)

Noise Noise reduction reduction Transient

Sensitivity

Sensitivity reduction

SensitivitySensitivityindicates the influence of plant indicates the influence of plant variations (due to temperature, humidity, age.)

variations (due to temperature, humidity, age.)

on closed

on closed--loop performance. loop performance.

Sensitivity functionSensitivity function

For sensitivity reduction,

L should have large gain

at low frequencies.

Disturbance

Unwanted signalUnwanted signal

ExamplesExamples

Wind turbulence in airplane altitude controlWind turbulence in airplane altitude control

Wave in ship direction controlWave in ship direction control

Sudden temperature change outside the temperatureSudden temperature change outside the temperature- -controlled room

controlled room

Air pressure brake to DC motorAir pressure brake to DC motor

Bumpy road in cruise controlBumpy road in cruise control

Often, disturbance is neither measurable nor Often, disturbance is neither measurable nor

predictable. (Use feedback to compensate it!)

Disturbance rejection

y(t

y(t) is not affected by ) is not affected by d(td(t) ) composed of low frequencies. composed of low frequencies.

For disturbance rejection,

L should have large gain at low frequencies.

L should have large gain at low frequencies. G(s

G(s)) C(s

C(s))

Plant

Controller

d(t

d(t): disturbance): disturbance

y(t

(8)

Typical shaping goal (Summary)

Steady

Steady--state accuracystate accuracy

Sensitivity

Disturbance rejection

Noise Noise reduction reduction Transient

Relative stability Relative stability Noise reduction

Noise reduction

Next, Next, frequency shaping (loop shaping) designfrequency shaping (loop shaping) design

Course roadmap

Laplace transform

Transfer function

Models for systems

•

Time response

•

Frequency response

•

Stability

•

•NyquistNyquist

Design specs

Root locus

Frequency domain

PID & Lead

Design examples

Matlab

Frequency shaping (Loop shaping)

Reshape Reshape Bode plot of G(jBode plot of G(jωω)) into a

into a ““desireddesired””shape ofshape of

by a series connection of

appropriate

appropriate C(sC(s).).

G(s G(s)) C(s

C(s))

Stable plant

Controller

An advantage of Bode plot (review)

Bode plot of a series connection GBode plot of a series connection G11(s)G(s)G22(s) is (s) is

the addition of each Bode plot of G

the addition of each Bode plot of G11and Gand G22..

GainGain

PhasePhase

We use this property to design We use this property to design C(sC(s) so that ) so that G(s)C(s

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Simple controllers

We use simple controllers for shaping.We use simple controllers for shaping.

GainGain

Lead and lag compensatorsLead and lag compensators G(s G(s)) C(s

C(s))

Stable plant

Controller

Bode plot of a gain (review)

dB

deg

Bode plots of lead and lag

C(s

)

10-2 10-1 100 101 102 103 -20

-15 -10 -5 0

10-2 10-1 100 101 102 103 -60

-40 -20 0

10-2 10-1 100 101 102 103 0

5 10 15 20

10-2 10-1 100 101 102 103 0

20 40 60

Lead

Leadcompensatorcompensator LagLagcompensatorcompensator

PHASE LEAD

PHASE LEAD _{PHASE LAG}_{PHASE LAG} MEMORIZE THESE

MEMORIZE THESE SHAPES!!! SHAPES!!!

Summary and exercises

Examples of gain margin and phase marginExamples of gain margin and phase margin

Frequency domain specificationsFrequency domain specifications

Frequency shaping (Loop shaping) on Bode plotFrequency shaping (Loop shaping) on Bode plot

GainGain

Lead, lag, and leadLead, lag, and lead--lag compensatorslag compensators

Next, more detail about frequency shaping.Next, more detail about frequency shaping.

Read Sections 10.6, 10.7 and 10.10.Read Sections 10.6, 10.7 and 10.10.