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In this chapter we present an approach for solving the problem of finding the set of all PID controllers that stabilize an arbitrary -order plant with time delay. The results presented in Chapters 6 through 8 do not readily extend to the case of higher-order plants with time delay, and an alternative procedure is presented here. This procedure is based on a connection linking Pontryagin’s results on quasi-polynomials to the Nyquist criterion.

This final chapter presents a summary of algorithms that can be used to generate the entire set of stabilizing PID controllers for single-input single-output (1) continuous-time rational plants of arbitrary order, (2) discrete-time rational plants of arbitrary order, and (3) continuous-time first-order plants with time delay. These algorithms follow from the material presented throughout the book. They display the rich mathematical structure underlying the topology of PID stabilizing sets. By presenting these algorithms without the highly technical details of the underlying theory, we seek to make the results accessible to as many engineers as possible. We have incorporated the bare minimum mathematical background required to make it self-contained.