Suppose you are to design a unity feedback controller for a first-order plant depicted in Fig. 3.59. (As you will learn in Chapter 4, the configuration shown is referred to as a proportional-integral controller.) You are to design the controller so that the closed-loop poles lie within the shaded regions shown in Fig. 3.60.

(a) What values of ω_n and ξ correspond to the shaded regions in Fig. 3.59? (A simple estimate from the figure is sufficient.)

(b) Let K_α = α = 2. Find values for K and K_I so that the poles of the closed-loop system lie within the shaded regions.

Figure 3.59 Unity feedback system for Problem 3.28

Figure 3.60 Desired closed-loop pole locations for Problem 3.28

(c) Prove that no matter what the values of Kα and α are, the controller provides enough flexibility to place the poles anywhere in the complex (left-half) plane.