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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.