Page 673 - Mechanical Engineers' Handbook (Volume 2)

664 Controller Design

Figure 39 PI controlled added to the system of Fig. 6a:(a) open-loop frequency response.

idealize the nonlinearity as a simple gain when no saturation takes place and as zero gain
when fully in the saturation region. A similar rationale can be applied to Fig. 41b. Generally
speaking, linear techniques can be used to ensure system stability by analyzing the system
with a range of gains determined by the minimum and maximum slopes of the nonlinear
amplitude curve. Small-perturbation step response and frequency response of the system
around an operating point can also be determined by linear analysis.
Describing-function analysis can provide useful insight into the behavior of the com-
bined deadzone–saturation nonlinearity shown in Fig. 42. Its describing function is a gain
that is zero in the deadzone region, increases to a maximum as the input amplitude ap-
proaches saturation, then decreases again as the input pushes well into the saturation region.
If linear analysis predicts instability at the maximum value of the nonlinear gain, this type
of describing function will result in a sustained oscillation at an amplitude corresponding to
maximum gain. This behavior is called a stable limit cycle because any tendency of the
oscillation to diverge will result in lower gain, which will reduce the tendency to oscillate.
Stable limit cycles can also result from deadzone in a system that is marginally stable
at low gains. For example, Section 6.3 explains that a PI compensator used in a system
having an inherent integration can exhibit 180 of phase lag at low gains, become stable at
intermediate gains, then become unstable at high gains. In this case, a low-frequency oscil-
lation can develop whose amplitude will grow until the describing-function gain is high
enough to produce a stable limit cycle.
The effects of saturation and deadzone on system stability are generally straightforward
to analyze by operating-point analysis or describing functions, as long as the system consists
of single control loops. However, when multiple feedback and feedforward loops are present,
the linearized analysis must be performed very carefully. For example, when an inner feed-