For proper controlling action, a feedback system must be stable. Previous chapters have indicated that feedback systems have the serious disadvantage that they may inadvertently act as oscillators. A feedback control system must maintain stability when the system is subjected to commands at its input, extraneous inputs anywhere within the feedback loop, power-supply variations, and changes in the parameters of the elements comprising the feedback loop.

In the ensuing discussion, if a control system has zero initial conditions, then for every bounded input, the output is bounded and the system is stable. This is popularly referred to by control engineers as bounded input-bounded output (BIBO) stability. In this chapter, analysis is limited to linear time-invariant systems, that is, systems for which the principle of superposition is valid and which may be described by an ordinary linear differential equation with constant coefficients. The analysis of nonlinear systems is presented in Chapter 5, and digital control system stability is presented in Chapter 4 of the accompanying volume.

We showed in Section 4.4 that a control system’s total response was the sum of the forced response due to the external forcing function and the natural or homogeneous response due to the initial conditions. The analysis was performed for the second-order control system illustrated in Figure 4.7, and the resulting response to a unit-step input for this control system was given by ...

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