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

1. Frequency Domain Design Demo IEE 362K (Buckman) Fall 03

2. 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

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

4. 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

5. 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!

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

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

8. 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

9. 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

10. 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

11. 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

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

13. 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

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

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

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

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

18. 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.