In this chapter, we developed the Bode and Nyquist frequency response techniques for control-system design. Gain and phase margins indicate the amount of uncertainty a feedback system can tolerate before becoming unstable. The gain margin is related to the change in gain (process or controller) that can be tolerated, while the phase margin is related to the change in time delay that can be tolerated. From a practical perspective, we recommend a gain margin of at least 2.5 and phase margin of at least 60° to assure robustness.
Amplitude ratio (AR):
Ratio of the output to input amplitudes
Phase angle (f):
Peak-to-peak shift between input and output curves
All-pass:
A transfer function with constant amplitude at all frequencies
Nonminimum phase:
Generally, transfer functions with time delays and RHP zeros
Bode stability criterion:
Amplitude ratio of gcgp < 1 at –180°.
Nyquist stability criterion:
The critical point (–1,0) is not encircled by gcgp.
Gain margin:
Multiplicative factor tolerated by gcgp before losing stability
Phase margin:
Phase-angle error tolerated by gcgp before losing stability
