lecture7 design

7/31/2019 Lecture7 Design

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EE392m - Winter 2003 Control Engineering 7-1

Lecture 7 - SISO Loop Design

Design approaches, given specs

Loopshaping: in-band and out-of-band specs

Design example

Fundamental design limitations for the loop

Frequency domain limitations

Structural design limitations

Engineering design limitations


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Modern control design Observable and controllable system

Can put poles anywhere

Can drive state anywhere

Can design optimal control

Issues Large control

Error peaking in the transient

Noise amplification

Poor robustness, margins

Engineering trade off vs. a single optimality index


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Feedback controller design Conflicting requirements

Engineers look for areasonable trade-off

Educated guess, trial and

error controller parameter

choice Optimization, if the

performance is really

important

optimality parameters areused as tuning handles

Analysis and simulation

es

kk

s

sku IP

D

D

++

+=

1

Plant model

Design process

D

I

P

kk

kStability

Performance

Robustness

Indexes

Constraints

Specs


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Loopshape requirementsPerformance

Disturbance rejection and reference tracking

|S(i)|


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Loopshape requirementsRobustness

Multiplicative uncertainty

|T(i)| < 1/(), where () is the uncertainty magnitude

at high frequencies, relative uncertainty can be large, hence, |T(i)|

must be kept small must have |L(i)|


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Gain and phase margins

Are less informative

than the noisesensitivity

ImL(s)

ReL(s)

1/gm

m-1

[ ] 1)()(1)()( += sCsPsCsSu

Additive uncertainty

|(i)| radius

sensitivity

peak margin

Can use uncertainty

characterization and

the sensitivity instead

Margins are useful fordeciding upon the loop

shape modifications


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Loop Shape Requirements Low frequency:

high gainL

= small S

High frequency:

small gainL

= small T large

Bandwidth

performance can be

only achieved in a

limited frequency

band: B B is the bandwidth

Fundamental tradeoff: performance vs. robustness

0 dB

gc

|L(i )|

B

Performance

Robustness

Bandwidth

slope


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Example - disk drive servo The problem from HW Assignment 2

data in diskPID.m, diskdata.mat

Design model: is an uncertainty

Analysis model: description for Design approach: PID control based on

the simplified model

)()(2

0 sPs

gsP +=

1)(

+++=

s

sk

s

kksC

D

DIP

Disk servo control

EDISTURBANCVCM TTJ +=&&

Voice

Coil

Motor

)(sP

)(sP


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Disk drive servo controller Start from designing a PD controller

poles, characteristic equation

( )

0

010)()(1

00

2

2

0

=++

=++=+

PD

DP

kgksgs

s

gskksPsC

0

2

000 /;/2 gwkgwk PD ==

Critically damped system

where frequency w0 is the closed-loop bandwidth

In the derivative term make dynamics fasterthan w0. Select 0/25.0 wD = 1+s

skD

D


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Disk drive servo Step up from PD to PID control

0

0

1

1

000

23

2

0

=+++

=

+++

IPD

IDP

kgksgkgss

s

g

ksskk

00

3

0000

2

0 //;/;/ wcgbwkgawkgwk DIDP ====

Keep the system close to the critically damped, add integrator

term to correct the steady state error, keep the scaling

where a, b, and c are the tuning parameters

Initial guess: w0 =2000; a=2; b=0.1; c=0.25

Tune a, b, c and w0 by watching performance and robustness


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Disk drive - controller tuning Tune a, b, w0 , and D by trial and error

Find a trade off taking into the account Closed loop step response

Loop gain - performance

Robustness - sensitivity

Gain and phase margins

Try to match the characteristics of C2 controller (demo)

The final tuned values:w0 =1700; a=1.5; b=0.5; c=0.2


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Disk servo - controller comparison PID is compared

against a reference

design

Reference design: 4-th

order controller: lead-lag + notch filter

Matlab diskdemo

Data in diskPID.m,

diskdata.mat

4th-order compens a tor C2 (blue, das hed), PID (red)

Time (sec)

Am

plitude

0 0.005 0.01 0.015-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3


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Loop shape, marginsLOOP GAIN - C2 (blue, dashe d), P ID (red)

Frequency (rad/sec)

Phase(deg)

Magnitude

(dB)

-100

-50

0

50

100

150

200

100

101

102

103

104

-900

-720

-540

-360

-180

0

Crossover

area of

interest


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Disk drive servo - robustness

104

-30

-25

-20

-15

ROBUSTNES S TO PLANT UNCER TAINTY (dB) - C2 (blue , da s hed), PID (red)104

-50

-40

-30

-20

-10

TRANSFER FUNCTION AND ACCEP TABLE UNCERTAINTY - C2 (blue , da s he d), PID (red, dotted)

[m2,ph2]=bode(feedback(C2,Gd),w))

[mP,phP]=bode(feedback(PIDd,Gd),w))

Full model

Simple model

Robust stability

bounds

PID

C2 - Matlab demo

plot(w,1./mP,w,1/m2)

1/|Su(i)|


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Fundamental design limitations If we do not have a reference design - how do we know if

we are doing well. May be there is a much better

controller?

Cannot get around the fundamental design limitations

frequency domain limitations on the loop shape

system structure limitations

engineering design limitations


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Frequency domain limitationS(i) + T(i) = 1

Bodes integral constraint - waterbed effect

Robustness: |T(i)|


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Structural design limitations Delays and non-minimum phase (r.h.s. zeros)

cannot make the response faster than delay, set bandwidth smaller

Unstable dynamics

makes Bodes integral constraint worse

re-design system to make it stable or use advanced control design

Flexible dynamics cannot go faster than the oscillation frequency

practical approach:

filter out and use low-bandwidth control (wait till it settles)

use input shaping feedforward


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Unstable dynamics Very advanced applications

need advanced feedback control design


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Flexible dynamics Very advanced

applications

really need control of 1-3flexible modes


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lecture7 design

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