AN INTRODUCTION

TO

ROBUST CONTROL SYSTEM DESIGN

by

Zhiqiang Gao, Associate Professor and

Automatic Control Resean:~h Laboratory Electrical and Computer Engineenng Department

, Cleveland State University

© April 1996

reserved

ru1 U!;;Jll . .'j It;.'SelVtxl

OUTLINE

• MOTIVATION

• BACKGROUND

FREQUENCY RESPONSE STABILITY DESIGN METHODS

• LIMITATIONS OF MATHEMATICAL MODELS

• ROBUST STABILITY

• Loop Shaping Design Technique

• A DESIGN EXAMPLE

• SUMMARY

• FU.EQUENCY RESPONSE

r(f) =A sil16J! ~--'j v(!)

-"- 'l_V~~r-----'

R (u)) == I vV (s ) I ! 5 _ jeu =-- I \V ( jeu ) I

¢(w) == phase angle of H/(S)!\=iCU == arg ~'V(Jw)

y (t) == ARCev) cos wt + epC(v) - 90° - AR(co) sin [cot + ¢(w)] ....

., .

ll(r )

I

y (s)

U(S)

) == Ie

q+

r y(f)

1+

1

1- - __ _

I-~t (

()

()

-

-

4

so

o

-so .~. ~.~ ... . ,.,;

10-2 10- 1 100 10 1 102

w (rad)

(0)

0

-100 --. :r. Il.t

~ C4J -200 2 --'3 '-..... -300 S CtJ ~

::::

-400

-500 I () - :' 10- 1 IOn 1 () 1 102

W (fad/sec)

(lJ)

;".,' = 10

c..' = 1

-I

C<.' = 0.1

I~--

I) ,

Stability of Closed-loop Control Systems:

~ H(s) .. G(s) -~ -.. ~ -

Closed-Loop Transfer Function:

C(s) = G(s)H(s)(l+G(s)H(s)yl

Stability Tests:

• Pole Locations The closed-loop control system is stable if C(s) has all the poles in the left half plane

• Nyquist Plot:

The closed-loop control system is stable if the Nyquist 1J6e cfoSea~loop co~1~o(syst~m is stahle·If th~Nyq·uist plot of G(s)H(s) satjsfy the Nyquist Stability Criterion

Nyquist Stability Criterion

If a contour, A, that encircles the right half-pI

is mapped through (5)H (5), then the number of clos loop

poles" in the right haif-plane equals the nUl?lher oj' open loop

po 1 P, [hat a in [he nght half-plane 171inus the nU!11ber

of un I e I 10 C A.'1It,) i s ere v 0 I uti s, tli! a r 0 U n d 1 of the

!17applngc: tha[ is, rnapp1l7g; {lzar IS,

'- P N

5) K(5 5) ~.~~ .. ~ .. ·~··~~·-T-··~~· ~ .. '~

(5 2)(5 4)

J CD

s plane

1 K

-pla

A' IS ,-l ()

"'" Re

• Gain Margin and Phase Margin

GH-plane

1

I I

di

! 1 I

IQ' I I '

"ElV \ instability

ist

. - G M - 20 log a

Unit .

nee ?iffase .

lns

Ph margin - <DNf -

Control Design Methods Review:

Reference Inpu t ,---_----,

Disturbance d(t)

+

u(t) ret) }----IIiI'-f.. ontrollell-----{ Prefilter

Design considerations:

• command following steady state and transient response

• stability (gain/phase) margin

• disturbance rejection

• sensor noise reduction "'-

• unmodeled dynamics/nonlinearity

Plant Output

yet)

+ Sensor Noise

CLASSICAL DESIGN METHODS

-PID

• Root Locus

• State Space

• methods ( Bode/Nyquist plots)

l\10DERN DESIGN METHODS

- Adaptive Control ystem

• Multi-Input and Multi-Output Systen1 Design

• Intelligent Control Methodology

Logic

Artificial Neural Network

System '--

• Optim.al Control

a cost function and use mathematical programming technique to minimize it

• PID

u(t) == Kpe(t) + KJ e(t)dt

• Root Locus Method:

K deCt) o dt

A P signment ll1ethod using constant gain

•

•

simple procedure easy to understand

can

quick solution

be directly

disturbance nOIse

uncertainty

to deal

R

+

Yes) K

R(s) K

s --

K 3A2

16

2

A 2 -

2

LIMITATIONS OF MODELS

• Parameter Inaccuracies

i 10% variations in R f and

Solution: feedback control

• Nonlinearity T

Field current {r J

Solution: Ii

• Unmodeled

No matter how there will always

a system' nominally modeled, unmodeled dynamics

-

Accounting for Uncertainty in Modeling

p (s) is u

nominal odel, i.e., the b Ie lant ode-l for the ntrol slgn

G p (s) is the rturbed model

a (s) is an additive urbation to t model

fa (jw)' a bound the agni tude of the addi ti ve perturbati

I m (jw) I

(s) LmCs) == --

u:

(0)

(b)

a(jw) p(jw) f3(jw)

!CjJ(jw)!- a(jw) + f3(jw) 2

IC ( . \) ')1 - f3(jw.) - a(jw) p JUJ JW - -----

a ( (.)' /) j ) - a

-<1

( G /) (j ) ( I

(G1J jw)

III (j )))

I) ( . ) LIII ( . ))

Unmod led Dynamics

1m is the InOlnent of in ia of the Inotor Pm is dampi in t motor if} is rnon1ent inertia of the platform Pp is the d pI of the platfonn

is the spring cons t

Lm 1m 8m (8m - ) I \

.. r. r. - - ~

(8m - 8p ) == l pep pp8p

· . ~ . . ..

Tm == JmGm JpGp f3m Gm + f3,/J p

1-.1 +fJ "

[/1/ - .I 8diff f3 8diff K.Jiciifl

f3s + Ks

C0difl(S) == [II.!

s(s + f3IJ) ----- Tm(s) 1 1.1 ] s2 + fJI.!s KsI.l

Time-Delay Systems

(s) -sT e (1

lO,-----------------------------------------------~

of (jev) for a time perturbation. s 2 s

ROBUST STABILITY

yes) R(s) <=> ---.1

e pertllrbed conflgurCltion.

~

G fJ (S) - {J (5 ) La (s )

I ( iev) I ILa(jw)\

La (jw) fa (j )

(;(5) Y (5)

G ) - G ) ( I) ) L(1 (s)) - G ) + G (s) Lu (s)

1m

, ) / (I

Ulst pto of (s) and (s).

At h w dist (jw) and the 1 point is iven by h ,the distance between (jw) and -1 point is given by

I ( jw ) ( 1 ) I == \1 G ( jw ) I

ROBUST STATBILITY Condition

\ G ( j w ) II a ( j LV ) 11 ( j w ) I

G(jw)1 I. I r I' ,! == j c (.I (u )

-G(s) - C7c (s)G()(S) (1 Lm(s)) - G(s) (J LmCs)} I

ROBUST STATBILITY Condition

!lm(jw)C(jw)! /1 C(jw)1

( jw)1

(j(j) ) 1

1 G (jel)) 1m (jw)

C(jw) tm (jw) I G (jw) 1

I I . m (

0.01

c (s) = a, a constant

a G )

s

G(s) a

G (s) s a

dB

40

0] 10 100 -----~---r~·---+--~---'-,~------·--,------- G

a = 100

-40 a = 10

{J

n n J

Loop SHAPING DESIGN TECHNIQUE

Objective:

Ge(s)

controller

Output

yet)

+ Sensor NQise

Manipulate loop gain frequency response, Gp(jco)Ge(jco), to closed-loop specifications

. n Procedure:

• D . ,Specifications Constraints on G(jco)H(jco)

• Find a Ge(s) to constrain ts

• Use pes), if necessary, to adjust transient response

From Des' Specifications Constraints on GpG CO )GcG co)

Specs:

• command following

Yes) _ Gp(s)Gc(s) ~: 1 1 := IGp(jco) co) I

R(s) 1 Gp(s) (s)

• disturbance rej on

Yes) s)

D(s) 1 Gp(s)Gc(s)

• nsor noi reduction

Yes) ,

N(s) Gp(s)Gc(s)

1 Gp )Gc(s)

• robust stability

all

1 Gp(jOJ) (jOJ) 1< 1

1 Gp(jOJ)Gc(jOJ) lm(jOJ)

• stability . n/phase)

pole roll-off at cra ver frequency

1

1

1

Constraints on GGco)HGco):

Low Frequency: I GGco)H(jco)1 1

High Frequency: I GGco)HGco)1 1

Crossover Frequency: good and phase margin

IGp(jeu)Gc(jeu)l

dB

low

constraints

-20db/dec o

Find suitable controller GeCs)

1 at low frequency:

1. Use constant and worry about high frequency constraints later

2. Us an integrator, lis

compensator

with abO

10

100 10 1

Frequency

o r---r--r-"--'-rTTr"""-~TTTl r<~ I I I I rTT~-r-==-r-r-TTTl

I -45

--90 L--~-'--L..Ll.J...J..l-___ L _L..L..LLLJ.~~ __ (-----'_---'--'-'-'-LLL--1 --'---'----'-~'_'_'_UI ()1 10-2 10- 1 10 1 10 -

(r;lci/s<:c)

• Adjusting Bandwidth, Gain and Phase Margin

1. Gain adjustment

. Use

20

-;::;

10

;.;:

40

u

~ 20 0::

lead compensator

)-s+a

with b s+b

IOu

a 0

10 1 10"

1 at high frequency:

1. gain adjustment

. Add poles high frequency (one decade from U\)

A Design Example:

ATS-6 (Applications Technology Satellite 6), 1974

Objective of the control system:

Point the antenna to target with .01 to .1 degree

,

.hlXis'

, , , , ,

,

, .' , , ,

30 fL diameter antenna

t, y-axis [0 orbit

((j )

e:mil

F

/ solar

pressure

I

I x

I

T

T-KTi-ce • UJ

e=K8 v w e - I\. tli /Jw 'I dl' y e.e- dt

Ri

_ reaction wheel

reference

line to earth

fJ{J

(3 P (5) 1

V(s) 5(5 0.01)

) 5 --V(S) 5 0.01

Gm(s) 10

Gp(S) 5 10

Gm(s) 10

V(s) 1035(5 0.01)(5 10)

10 10

(0 III (s ) - (0.00 1 ) ( 1 0)

D/(s) S2(S 10)

-0.01 + 10)

G (S)

0.01 + 0.01 )(5 + 10)

8 p

5 + 10 10

8 p

lm (jW) -(

JW

-0.03 1 0.000

30 30

~-0.01

s ~ (.\' + 10) ) D'(s) C~) III (S - 1 ( )

r S

I (-) Ii! ( .i ) I ma\

0,01 10 5 1 /

100) -

I ~ / Iii \.1 (' I i ma\

0.01 (10 5) 10 5)2 10 (1.7 10 )

First try: gain adjustment too little phase margin

40

o

90

- 135

Using a lead compensator Jor more phase margin:

o

5 (5) --

5

0.019

0.19

10(5 0.019)

5 + 0.19

IOO~·~~~-····-L~~·~~~~~-~~~-~~~~~'~~w

10-:1

180

, , ,

, , , ,

, ,

Using. lag compensator for higher gain at low frequency:

5 -: 0.005 lao (5) -~-

;:::-

5

10 0.005) (5 + 0.019) -------

5 (5

~ Ii! I III I I I , I III I IIII1I r-"

o

-40

~LL.LLL11L-i-l.-LLLlll-~-LlllJl I I I I I I II I I I I I III I I I I I I II

10-2 IO~1 t()O 10 1

ucncy )

((j )

-I

Transient Response Analysis

1.5 ~------------------,

o 0...

0... Q)

Vi 0.5

100 200 Time (SeC)

JOO 400

0.1 0 ~----,------.--.------.-------~

0.05

Magnified behavior

Magnified beh<lvior near the origin

~ I

O~----------~--~---T----~rr~

-o.OS

(two poles at origin)

I

--I

- 0.1 ~(L).-2-()---.. -._ (...Ll.-1 ~----(-Ll.-l (....L) ____ ()L-.(-)~-------'()

I · \

8 c

()

. " . ~

Prefilter and the Final Design

0.019 s + 0.019

10 (s + 0.005)(s + 0.019) s (s + 0.19)

10 s+10

t D'

10-:1

v 1 0-

3 r---~0-3-S(S + 0.01 )

1.5 ~----------------------------------~--------~

~ 0.5

o ~--------------------------------~----------~ o 100 200

Time (sec)

300 400

E _I G\(s) 1 Tn< ~I G2(s) I (0)

E Till ~0 (~/) G,(s) G 2(s) -

L(s) -

(IJ)

-Gp(s) == Gl(S) [G2(S) - L(s)]

== Gj (S)G2(S) - G1(s)L(s) == Gj(S)G2(S) - G1(s)L(s)

1/.1

(0)

1 / J

1 / J

(l.. \ L ___ ~~ __ ~ ______ ~ ________ _

(h)

L(s) = 2 S + (f3 / J ) s + Ks / J

l/J

( 7 2

lIes) == 4 s + l~~(s + 3s + 9) (s + 3)_1(s2 + 0.6s + 9)

IL(jw)1 < \I'(jw)\

20~--------------------------------------------~

o

~ -20~----------~~

~ Ks=9 ~

......:)

Z -Lin , - .. .. . ... . ·· ·· ·· · '0- · ·· ' · _ ... ,," ..... ".

~ - 40 -- " ... ... . .. , . .. ....- ..

~ Ks=.LOO ...

-60 . s·-· · ..... . ,,'

-80L---------------------------------------------~ 10- 1 100 10 1 102

W (fad/sec)

Bending modes <lnd bounding IU jeo)l.

II' (jw) I I 2(jW) I

1

0.1 )

40

The bound lm(jw)

SUMMARY

• Basics of control system design

• Limitations of modeling

• The concept of robust stability

• Loop shaping design

• A design example

Further Reading:

and Schultz, 1993

lfff:{) I.>,. ~M

~r'i 2..or91

DISCRETE TIME SYSTEMS AND

DIGITAL CONTROL

Reference

Digital computer

u Plant

oomputer-controlled feedback system.

ADVANTAGE OF DIGITAL CONTROL ." Flxibility

• Accuracy • Reliability • Drift -free intergration • l-l1 dUVll

• Remote Control

y

• scale interconnected system

DISADVANTAGE OF DIGITAL CONTROL

• Cost (sometimes)

• Time Delay

IMPLEMENTATION OF DIGITAL CONTROLLER

• with Data Acquisition Card

• Stand alone DSP board

411 PLC

• Microcontroller based embedded control system

Intel: 805 1 , 8096, 80196 Motorola: 68 11 . (E9, , N)

zzy/Neural Chips: AL220 etc.

DISCRETE SIGNALS:

In discrete system, observation and actio are only taken at

Input

u(t)

Order Hold

The Sampling of the Output

Output

yet)

yit)

o delay could destabili~e the system e on Gain margin

to yjl) Output

(s) -::: (;)

10'

....... : .: . : . : .: : . : -: : :-: :.: :.: :-: .: :: : . : : . .. : ..

* • • ~ • • • : ~. ~." -.- .~ .. ~. ~~. " ...

10-1 ' ___ ~_~ ... ~~~~~~.L~ __ ~_~ .. _.~ .. ~~~~~~

10-' 10° 10' , I

10° 10'

Unsainple.d .

-150

. : : : . : :Samp\ed '. • . .' ..• '. ~ ~. *'. " .~. : • . . . ....... . . . .

-250

-300L---~---~~~~~~L_--_~ __ -~.~.~~.~~~~ 10-' 10° 10'

DIGITIZING ANALOG CONT.ROLLER

Digitizing an analog controller.

L Numerical approximation of differential equation

'1 l\ If '" trh; n a <:tf'n and other responses. 2. Matching step and other respo

3. Pole-zero matching.

Numerical Approximation

dJ dt

(s) 1

(s) s

e(t) .

J(t) - J(tO) jl e(l)dl, 1 >10 10

(k ) - I(kY) lkT

kT

T

e(t)dr.

Trapezoidal Method (Two Samples)

A

f(kT A

T) - f(kT) T 2 {e(kT) e(kT + T)}

A

(z - l)F(z) -2

1) (z),

H(z) i~~~ = ~ G-~) .

H(z) == Gc(s)L= .• z+

PULSE WIDTH MODULATION (PWM)

• High powered linear actuator is expensive

• Easier/cheaper implementation by turning power on and off

• Achievable in digital controller by turning on the power during part of the sampling interval

• The control variable, U, becomes "duty cycle" ( a variable fraction of sampling interval)

• Many rnicrocontrollers have ADC and PWM ports

A Design Example

• zero steady state

disturbances

• settling time 20

s

in the presence of input

• implementable in hardware using DSP

Solution:

abs 1 s +1

s

PI =1 +aT/2,

2z-1

Tz+l

2z-1 2z-1 1 ab---

1 T 1

=k PiP3 P2P3)Z-i Ps (P6 - Ps )z-! - P6 -2

- 1 ,

Programmable Difference

-2

Z

u(k)=(l/ps)[ (Ps-P6)u(k-l )+P6u(k-2)+kpIP3e(k) +k(PIP4+P2P3)e(k-l )+kp2p4e(k-2)]

Implementation Issues:

• Sampling rate selection

• Anti-Aliasing Fitler

• ADC and DAC resolution: number of bits

• Signal Conditioning

• Sensor Noises

• Averaging and Low pass filter

• Processor speed consideration

• Isolati-on of analog and digital circuitry

• Spectrum analysis of noise/disturbance characteristics