TABLE OF CONTENTS As discussed in the first chapter, systems of interest to us will generally be described by integro-differential equations. Consider the Laplace transforms of the input and output of such a system, X( s) and Y( s) respectively. Assume that we want to set some combination of derivatives and integrals of the output to some combination of derivatives and integrals of the input. We find that the relation between X( s) and Y( s) is:
where the polynomial P( s) is related to the initial conditions to which the system is subject, and R( s) and Q( s) are related to the form of the integro-differential equation.
In 1.5.2 we saw that for a system to be stable it is necessary that all of the zeros of Q( s) be in the left half-plane. It is crucial that any system that one uses be stable. The output of an unstable system as we have seen will almost invariably tend to "run away."
Assuming that we are dealing with a stable system, we find that the portion of the response that is related to the initial conditions to which the system is subject, P( s)/ Q( s), has all of its poles in the left half-plane. By considering the partial fraction expansion of this function, it is clear that the function to which it corresponds decays exponentially quickly. Thus, after a...
