Frequency Domain Design Demo I EE 362K (Buckman) Fall 03

Start with a nasty, complicated plant with not one but two strong resonant frequencies: more complicated than anything we’ve tackled so far. . . Showing Plant only Frequency-domain info even helps in describing the plant: note how 2 resonances not obvious in time-domain Two resonances

Turning on the unity feedback reveals another problem: Need more gain at low frequencies Bad steady state error

Problems identified: Steady-state error: need more gain near DC  try pole on real axis Two resonances: need less gain at: w=3.15 w=9.13 Try two complex-conjugate pairs of zeros

Start with a real pole in C(s), at s = -50 Pole at s = -48.45 introduced here This pole is too far left in the s-plane to have any observable effect on either the frequency or the time-domain response of the closed loop system: both are unchanged!

Bringing this new controller pole to the right starts boosting the gain at low frequencies at about s = -0.4 Low-frequency gain increased Steady-state error still bad

It might be tempting to just increase the overall gain….BUT The peaks come back. Although, SSE is a bit less.

We also backed down the DC gain Since the peak near w=3 is now the most obvious problem, attack it next: introduce a complex conjugate pair of zeros, starting with a large negative real part. . . We also backed down the DC gain New zeros No observable changes in frequency- or time-domain behavior yet: zeros are too far away from the w-axis to have any effect

You can suppress the low-frequency oscillations completely by bringing this pair of zeros closer to the w-axis and adjusting the w value slightly: Adjusted w slightly to make the zeros “cancel” the low-frequency pair of poles. This is most easily done looking at the pole-zero plot. No low-frequency peak No low-frequency oscillations

Cranking up the DC Gain reveals that the high-frequency resonance is now threatening to drive the closed-loop system unstable Poles about to cross w-axis Phase shift becoming discontinuous

So once again, back down the DC gain and introduce another pair of zeros near the high-frequency resonance New pair of zeros supresses high-frequency peak, cancels poles

Try increasing DC gain now: About 2% steady-state error +2% risetime

The only thing wrong with this picture is the unrealistic controller: More zeros than poles for C(s) 4 zeros, 1 pole indicates 4 more poles needed Gain increasing without bound at high frequencies

Put in two pole pairs at s=-40+j0.0, and move the real pole to –0.12: Stays within +3% in 0.55

You can do better on risetime and steady-state error, but it requires even more controller gain than this: Max gain = 41dB

Translated to digital, this design holds up well down to a sampling frequency of 30:

Shifting the two pole pairs from –40 to –100 lets you bring the sampling frequency down to 8.0 and still maintain performance:

Frequency-domain design summary Careful placement of poles and zeros in the controller C(s) lets you smooth out peaks and valleys in the closed loop transfer function H(s). First, identify problems with the frequency-domain shape of H(s): Too little gain at low frequency Peaks or dips to smooth out Increasing DC gain will accent the biggest problem areas: fix them first Fix frequency-domain of H(s) using more poles and zeros, keeping DC gain relatively low until the final steps Introduce extra poles if necessary to keep your C(s) realistic Translate your controller C(s) to a digital design D(z), lowering sampling frequency to realistic levels.