All the systems to be assessed are closed-loop feedback systems.

A simple dynamic system (Fig. 2.28a), which has a response R to an input signal I, can be represented by a transfer function G(s) = R/I, where s is the Laplace variable, which is generally a complex variable. The response of this system to a steady sinusoidal input of frequency со of unit amplitude is R — G(ico), which is called the frequency response (7).

A simple feedback loop (Fig. 2.28b) has a forward function G(j), a feedback transfer function H(s), and a feedback signal F, which is the output R modified by the feedback function

F = RH(s) (2.9)

The input to the forward transfer function G(s) is now the difference be­tween the input signal I and the feedback F. It is clear that with the loop closed the response R is given by

R = G(j)[7— ВД] (2.10)

R/I=G(s)/[l +G(s)H(s)} (2.11)

and it should be noticed that the positive sign in the denominator is indica­tive of a negative feedback.

This is called the closed-loop transfer function.

The function G(s)H(s), the feedback response to unit amplitude F/I, lis called the open-loop transfer function.

Instability in the system is exhibited when the signals D, R, and F in the oop become self-sustaining without an input 7. Instability is indicated by the poles (where the function becomes infinite) of the closed-loop transfer function in the right half of the complex plane s, where the poles indicate exponentially increasing time functions in the time domain.

The number of poles is equal to the number of unstable modes in the system while the position of the pole gives information about the type of instability shown; the real s coordinate is the divergence rate while the imaginary s coordinate is the divergence frequency. There are several methods by which this information can be used to analyze the stability of the reactor system.