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In this chapter we give a quick overview of control theory, explaining why-integral feedback control works, describing PID controllers, and summarizing some of the currently available techniques for PID controller design. This background will serve to motivate our results on PID control, presented in the subsequent chapters.

In this chapter we introduce the classical Hermite-Biehler Theorem for Hurwitz polynomials. We also present several generalizations of this theorem that are useful for solving the problem of finding the set of proportional (P), PI, and PID controllers that stabilize a given finite-dimensional linear time-invariant system.

In this chapter we utilize the Generalized Hermite-Biehler Theorem presented in the previous chapter to give a solution to the problem of feedback stabilization of a given finite-dimensional linear-time invariant plant by a constant gain controller and by a PI controller. In each case the complete set of stabilizing solutions is found.

In this chapter we first consider the problem of stabilizing a continuous-time system described by a rational transfer function using a PID controller. A solution based on the Generalized Hermite-Biehler Theorem of Chapter 2 is presented. The solution provided here characterizes the entire family of stabilizing controllers in terms of a family of linear programming problems. The discrete-time counterpart is then solved.

In this chapter we present an important generalization of the Hermite-Biehler Theorem due to Pontryagin for characteristic equations or systems that contain time-delay terms. Some other results for analyzing systems with time delay are also introduced and will be the basis for the work presented in the following chapters. Proofs of all results are omitted and the reader is referred to the extensive literature on the subject.

In this chapter we present a solution to the problem of stabilizing a first-or second-order plant with time delay using a proportional (P) controller. This solution is built upon the results presented in Chapter 5 for quasi-polynomials and computes the complete set of stabilizing gains. Examples are included to illustrate the application of these results.

In this chapter, we continue with the line of work presented in the last chapter and solve the problem of stabilizing a first-order plant with time delay using a PI controller. As before, the results in Chapter 5 will play a crucial role. Examples are included to clarify the detailed steps associated with the solution.

In this chapter we present a complete solution to the problem of characterizing all PID controllers that stabilize a given first-order system with time delay. As will be seen shortly, the PID stabilization problem is considerably more complicated than the P and PI cases considered in previous chapters. The solution presented here makes use of the results introduced in Chapter 5. The range of admissible proportional gains is first determined in closed form. Then for each proportional gain in this range the stabilizing set in the space of the integral and derivative gains is shown to be either a trapezoid, a triangle, or a quadrilateral.

In this chapter we present some tools that are useful when designing a PI or a PID controller for a first-order system with time delay. These tools show the importance of knowing the set of controller parameter values that stabilize the closed-loop system derived in the previous chapters. The chapter also provides a solution to the problem of robustly stabilizing a given delay-free interval plant family using the PID controller.

In this chapter we present an analysis of some PID tuning techniques that are based on first-order models with time delays. Using the characterization of all stabilizing PID controllers derived in Chapter 8, each tuning rule is analyzed to first determine if the proportional gain value dictated by that rule lies inside the range of admissible proportional gains. Then the integral and derivative gain values are examined to determine conditions under which the tuning rule exhibits robustness with respect to controller parameter perturbations.