Question 10.7.1: PD Control of a Neutrally Stable Second-Order Plant PD contr...

The output equation is
\Theta(s) = \frac{K_{P}  +  K_{D} s}{I s^{2}  +  (c  +  K_{D})s  +  K_{P}} \Theta_{r}(s)  −  \frac{1}{I s^{2}  +  (c  +  K_{D})s  +  K_{P}} T_{d} (s)
The characteristic polynomial is I s^{2} + (c + K_{D})s + K_{P} , and the system is stable if c + K_{D} > 0 and if K_{P} > 0. For unit-step inputs, the steady-state command response is K_{P} /K_{P} = 1, which
is perfect, and the disturbance response is −1/K_{P} . The damping ratio is
ζ = \frac{c  +  K_{D}}{2 \sqrt{I K_{P}}}
If ζ ≤ 1, the time constant is given by
\tau = \frac{2I}{c  +  K_{D}}
For P control (with K_{D} = 0), ζ = c/2 \sqrt{I K_{P}} . Thus, introducing rate action allows the proportional gain K_{P} to be selected large to reduce the steady-state  disturbance response, while K_{D} can be used to achieve an acceptable damping ratio. The rate action also helps to stabilize the system by adding damping (if c = 0, the system with P control is not stable).