Classical Feedback Control, Section 4.3.7: Unstable plants

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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 B_int 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 B_int = 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.

When the loop is disconnected at the input to the link P, the loop transfer function is T = (B_int + BCA)P. After the desired frequency response for T is specified, the required transfer function for the compensator is C = (T/P - B_int)/(AB). The method is especially convenient when B_int is small compared with BCA.

The compensator can be directly designed for the unstable plant. In this case, the main-loop Nyquist diagram must encompass the critical point in the counterclockwise direction, as required by the Nyquist-Bode multiloop stability criterion.

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