Consider the closed-loop control system shown in Figure 4.74.

(a) Use proportional control \(D(s)=K_{P}\). Does this controller provide additional damping?

(b) Use PD control, \(D(s)=K_{P}+K_{D}\) s. Determine the tracking and disturbance-rejection properties for step inputs on \(\theta_{r}(t)\) and \(w(t)\).

(c) Use PI control, \(D(s)=K_{P}+\frac{K_{I}}{s}\). Discuss the effect of this controller on the stability of the system.

(d) Use PID control, \(D(s)=K_{P}+\frac{K_{I}}{s}+K_{D}\) s. Discuss the effect of this controller on the stability and steady-state errors of the system. What are the two system types when the PID is employed?

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w(t) e,(t) 1 D(s) Js S Figure 4.74 PID control system problem. 1 e (t)