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4.3.7  Unstable plants

Unstable plants are quite common. For example, the Saturn V launch vehicle and some airplanes are aerodynamically unstable; a slug formed in the combustion chamber or a turbulence in the chamber can make a rocket unstable; rotation of a prolate spacecraft is unstable; a large-gain electronic amplifier without external feedback circuitry is often unstable. For the purposes of analysis and design, an unstable plant can be equivalently presented as a combination of a stable forward path link P with internal feedback path Bint that makes the plant unstable, as shown in Fig. 4.38(a).

(a) (b) (c)

Fig. 4.38  (a) Plant with internal feedback, and the diagrams of (b) Bode and
(c) Nyquist for the internal feedback


Example 1. Consider the system diagrammed in Fig. 4.38(a). Assume that the plant is a double integrator with an internal feedback path having a low-pass transfer function Bint = b/(s + a). The Bode diagrams are shown in Fig. 4.38(b). The internal loop phase lag exceeds 180° at all frequencies, and the plant becomes unstable as seen from the Nyquist diagram in Fig. 4.38(c).

There are two convenient ways of analyzing and designing such systems.

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