controller design by r.l
Post on 02-Feb-2016
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Controller design by R.L.Typical setup:
sGsC
sGsC
1:TF c.l.
01:poly char. c.l. sGsC
21
21Let psps
zszsKsC
sd
snsG
01 :Then
1
1
sd
sn
ps
zsK
This is the R.L. eq.
• With no zi, pi, controller design means to pick good K for R.L.
• Those zi, pi means to pick additional poles /zeros to R.L.
Proportional control design
1. Draw R.L. with given
2. Pick a point on R.L. to be desired c.l. pole: PD
3. Compute
KKsC P
call
01sGK
01sd
snK
sd
snsG
DPGsGK
11
When to use:If R.L. of G(s) goes through
the region for desired c.l. poles
What is that region:– From design specs, get desired Mp, ts, tr, etc.– Use formulae for 2nd order system to get desired
ωn , ζ, σ, ωd
– Identify these in s-plane
Example:
When C(s) = 1, things are okay
But we want initial response speed as fast as possible; yet we can only tolerate 10% overshoot.
Sol: From the above, we need that means:
6
1
sssG
%10pM6.0
This is a cone around –Re axis with ±60° area
We also want tr to be as small as possible.i.e. : want ωn as large as possible
i.e. : want pd to be as far away from s = 0 as possible
1. Draw R.L.
2. Pick pd on R.L. in cone & | pd | max
3. 3661
ddd
pppG
K
dp pole
desired
forRegion
?333or
6
:max
jp
p
p
d
d
d
Example:
Want: , as fast as possible
Sol:
1. Draw R.L. for2.
Draw cone ±45° about –Re axis
3. Pick pd as the cross point of the ζ = 0.7 line & R.L.
4.
62
10
ssssG
%5pM
6.010631
dddd
P ppppG
K
7.0%5 pM
Controller tuning:
1. First design typically may not work
2. Identify trends of specs changes as K is increased.e.g.: as KP , pole
3. Perform closed-loop step response
4. Adjust K to improve specse.g. If MP too much, the 2. says reduce KP
Pd M&,,
PD controller design
•
• This is introducing an additional zero to the R.L. for G(s)
• Use this if the dominant pole pair branches of G(s) do not pass through the desired region
zsKsKKsC DDP
DP KKz
Design steps:
1. From specs, draw desired region for pole.Pick from region.
2. Compute
3. Select
4. Select:
dd jp dpG dd pGzpz s.t.
dd pGz tan i.e.
DP
ddD
KzKpGzp
K1
Example:
Want:
Sol:
(pd not on R.L.)
(Need a zero to attract R.L. to pd)
sec2%,5 sp tM
7.0%5 pM
24
sec2 s
s tt
22 Choose jpd 707.0,2,2 d
2.
3.
4.
4
tan dz
22222
1
jjdpG
222 jj
4
3
4
5
24
3
4122
8
222222422
11
DP
D
KzKjjj
K
ssC 28
44
3 dpG
Closed-loop step response simulation:
» ng = [1] ;
» dg = [1 2 0] ;
» nc = [Kp, Kd]
» n = conv(nc, ng) ;
» d = dg + n ;
» stepspec(n, d) ;
Tuning: for fixed z:
Q: What’s the effect of tuning z?
PD MKK
PD MK
gcgc
gc
ddnn
nn
here 1cd
Drawbacks of PD
• Not proper : deg of num > deg of den
• High frequency gain → ∞:
• High gain for noiseSaturated circuits
cannot be implemented physically
as jKK DP
Lead Controller
• Approximation to PD
• Same usefulness as PD
•
• It contributes a lead angle:
0
zpps
zsKsC
zppC dd
ppd
Lead Design:
1. Draw R.L. for G
2. From specs draw region for desired c.l. poles
3. Select pd from region
4. LetPick –z somewhere below pd on –Re axisLetSelect
dd jp dpG
121 ,zpd
2 s.t. ppp d
2tan i.e. dp
• There are many choices of z, p
• More neg. (–z) & (–p) → more close to PD & more sensitive to noise, and worse steady-state error
• But if –z is > Re(pd), pd may not dominate
dpdp
zdp pGK
1Let
ps
zsKsC :is controllerYour
Example: Lead Design
MP is fine,
but too slow.
Want: Don’t increase MP
but double the resp. speed
Sol: Original system: C(s) = 1
Since MP is a function of ζ, speed is proportional to ωn
5.022,2 nn
4224 TF c.l.
ss
Draw R.L. & desiredregion
Pick pd right at the
vertex:
(Could pick pd a little
inside the region
to allow “flex”)
5.0 new want weHence 4 new nω
322 jpd
Clearly, R.L. does not pass through pd, nor the desired region.need PD or Lead to “bend” the R.L. into region.(Note our choice may be the easiest to achieve)
Let’s do Lead:
2 ddd pppG
623
2
Pick –z to the left of pd
4,4Pick zz
3
41
dp
663then 12
2tanthen dp 8322
3
1
7
2
4
1
dpdppdp
zdpK
8
47
s
ssC
Suppose we select –z = -1 instead.
zpd 1
612
7.2tan 2 dp
1322 j
321 j
32tan 1
32tan6
5 1
The new R.L.
closed-loop
has pole pair
at pd & pd* but
the 3rd pole is
at somewhere
(-1, 0)
Which one is better?
Particular choice of z :
2221
ddddd
pABpApzpzOzp
ppd2
OpOAp dd
dp
OApBp dd bisect
s.t. B Choose
ABpOBp dd OApd
2
1
dp2
1
22
dp
1tan dz 2tan dp
322 :example prev.In jpd 32,2 d
get weprocedure, above Follow
46.5
93.273.4
s
ssC
359.0,%21 :step c.l. rp tM
repeat. , 5.2 to2 change σ
375.0,%1.16 :step c.l. rp tM
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