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4.3.5  Plant tolerances

The plant gain typically decreases with frequency, and the tolerances of the plant transmission function increase with frequency as illustrated by the limiting curves in Fig. 4.30. In plants whose transfer functions have only real poles and zeros, the plant responses and their variations are commonly smooth and monotonic. Such responses are typical for temperature control and rigid-body position control. Since the minimum necessary stability margins must be satisfied for the worst case, which is typically the case of the largest plant gain, the feedback will be smaller in the case of the minimum plant gain. This way the plant response tolerances reduce the minimum guaranteed feedback.

Fig. 4.30  Boundaries of monotonic
plant transfer functions

Fig. 4.31  Plant gain frequency

For plants with monotonic responses, it is convenient to consider some nominal plant response, as shown in Fig. 4.30. The feedback loop design is then performed for the nominal plant. Larger plant tolerances - larger deviations from the nominal - require increased stability margins for the nominal response, and thereby limit the nominal available feedback.

For computer design and simulation, the plant uncertainty is most often modeled as multiplicative uncertainty, that is multiplication of the loop transfer function by some error response (i.e., addition of some uncertainty to the gain and phase responses). The multiplicative uncertainty is typically either a constant as in Example 1 in Section 4.2.3 (see Fig. 4.11) or a function of frequency, as those shown in Fig. 4.30.

In general, the dependence of the plant transfer function on its varying parameters can be complicated. For some plants, parameter uncertainty causes deviations from the nominal plant responses which are neither symmetrical nor monotonic, like those shown in Fig. 4.31. Uncertainties are also sometimes modeled by vector addition of some error response to the transfer function (additive uncertainty).

Fig. 4.32  Plant structural resonance
on a Bode diagram

Fig. 4.33  Plant structural resonance
on an L-plane Nyquist diagram

Flexible plants, i.e., plants composed of rigid bodies connected with springs and dampers, have structural resonances corresponding to specific modes of vibration. Stiffness and mass variations in flexible plants change the pole and zero frequencies as shown in Fig. 4.32. Similar responses are obtained in low-loss electrical systems such as transformers and filters. The resonances typically produce loops nearly 180° wide on the L-plane Nyquist diagram as shown in Fig. 4.33. Neither multiplicative nor additive uncertainty conveniently characterizes these effects. It is better to describe such plants by specified uncertainties in the transfer function poles and zeros.

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