Control Design with Frequency Response

Reading: FPE, 6.7

Heuristic for controller/compensator design:

Goals: - stability - good damping - bandwidth frequency near 0.5

Strategy: - if PM is near 90 degrees, good damping and - to have PM near 90, need gain slope near -1

Work with the double integrator plant, for simplicity.

Note that proportional feedback is not going to work: won’t shift Phase plot away from -180 degrees…

Y -- ¥9,12

PM = god} we = .5⇐ wow )

Gcs) = 162.

PD controllerOpen loop transfer function:

We add a type 2 factor (in the numerator, equivalently adding a ramp to Gain):

For the Gain-Phase relationship to be valid, choose the break-point several times smaller than desired frequency:

Now, choose the scalar gain

gale from> K=kp Kfcsti)2. KI %2

Dstkp)G=K( Gcs) "P

g YIv

^

We >> YT >take e- to Zoe -5331

Tue 's> I Def ='

t.IQ← K ?

to have were-5IKDec.sjj-GC.si/=t--KFaf=tK--ufTzr.o5

What have we accomplished: PM≈90, good!Trade-offs: - want bandwidth freq to be large enough for fast response (larger =>larger =>faster rise), but - not too large to avoid noise amplification at high frequencies - usual complaint: D-gain is not physically realizable, so let’s try lead compensation

Moving to lead compensator

or, in Bode form:

K¥211 lost2052

.

WBW Wh .

KEEP Pat.

k SAI fslptl

Bye:÷ " KITT, - taModerates magnitude

Yp- at high w .

Lead Compensator

Note that gain levels off at high frequencies - not as susceptible to noise as PI controller…

Choices of z and p: the crossover frequency halfway between them, on log scale:

Larger p/z means larger PM shift (closer to 90), but also larger gain at high frequencies (noisier)

He = K SAI Highest lift ofHpt ' lead compensator at Ep

Yz Yps

a# Ep

Shape of Phase plotbecomes flatter as Hp grows

Eph → center of humpFp → width of bump Yp -40 Tp- to

Lag compensator

We see that lag compensators are very similar to the lead ones.

In particular, the same heuristics about the relation of p, z and the shape of the phase and magnitude plots apply.

KD = K SAI 2- XpSlp-l l

Still,renter at Fop

width or Hp

Consider plant

Design goals: - stability - bandwidth at - DC error for constant inputs <1%

What we are starting with:

Example

GG) -I

Y

Hoste)Cst)✓

WBw =L

DC ( r →E) = ¥€µ= FISK S ' 01

Purely scalar gain ⇒ Knew

⇒ we mores to the rights PMs zooNeed to do dyuaeuie compere selves .

Adding lead compensator

Choose

The phase margin improved, but not enough, - one obvious reason is that we increased magnitude at high frequencies, shifting too much to the right.

How to compensate for it? Obviously, by adding a counterbalancing term that would keep the magnitude where it was on the original Bode plot, shifts the phase down where we need it, and keeps the DC gain fixed…

KstztlslptI '

~1Z s weep plz - large .

2-=.8 p - 8 got 5=0 KD x 20

kD=2o%I KDG - too

What if we combine lag and lead compensators?

The sums would do more or less what we want: - push PM up near the desired frequencies (around 1); - keep magnitude gains where we want them at 0 (to satisfy the constant input error requirements), and - won’t increase magnitude at high frequencies (not amplifying the noise)

Note the alternation of the zeros and poles of the lead-lag compensator

X Ed

Die.is#f.I4Rag=El9fzZiu9is/pz4ppzpsz-ispz

D= @/Z,tc)(4zz#Glp , eiscslpzti)

-XX

Here’s the proposed design:

PE . 01 ZE. I 7=-8 p ,-8 ✓( keep zeros and poles apart)So KDG - Coo

(%eDcsl.IN)(

%eDGI.co#los+DCstDKDGCo)--looIggTogcl?k/PMnqo0✓Weil ✓

Design based on Bode plots is good for:

- easily visualizing the concepts;

- evaluating the design and seeing which way to change it using experimental data (frequency response of the uncontrolled system can be measured experimentally)

Design based on Bode plots is not good for: - exact closed-loop pole placement (root locus is more suitable for that) - deciding if a given K is stabilizing or not: we can only measure how far we are from instability (usingGM or PM), if we know that we are stable…however, we don’t have a way of checking whether a given K is stabilizing from frequency response data.

Summary