Intro to Robust Control System Design
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Transcript of Intro to Robust Control System Design
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