Question 10.12: Proportional–Derivative Control of a DC Motor–Driven Cart Co...
Proportional-Derivative Control of a DC Motor-Driven Cart
Consider the feedback control system shown in Figure 10.33, in which the plant is the DC motordriven cart given in Example 10.2. The input to the plant is the voltage applied to the DC motor, and the output is the position of the cart. Design a proportional-derivative (PD) controller such that the maximum overshoot in the response to a unit-step reference input is less than 10 \%, and the rise time is less than 0.15 \mathrm{~s}.
The closed-loop transfer function is
\frac{Y(s)}{R(s)}=\frac{C(s) G(s)}{1+C(s) G(s)}=\frac{\left(k_{\mathrm{p}}+k_{\mathrm{D}} s\right)\left(3.778 /\left(s^{2}+16.883 s\right)\right)}{1+\left(k_{\mathrm{p}}+k_{\mathrm{D}} s\right)\left(3.778 /\left(s^{2}+16.883 s\right)\right)}=\frac{3.778 k_{\mathrm{D}} s+3.778 k_{\mathrm{p}}}{s^{2}+\left(16.883+3.778 k_{\mathrm{D}}\right) s+3.778 k_{\mathrm{p}}},
which is a second-order system. The closed-loop characteristic equation is
s^{2}+\left(16.883+3.778 k_{\mathrm{D}}\right) s+3.778 k_{\mathrm{p}}=0
where the coefficients are related to the natural frequency and the damping ratio of the closedloop system via
\begin{aligned} & 16.883+3.778 k_{\mathrm{D}}=2 \zeta \omega_{\mathrm{n}} \\ & 3.778 k_{\mathrm{p}}=\omega_{\mathrm{n}}^{2} . \end{aligned}
Note that there are two requirements for the transient response of the closed-loop system, that is, M_{\mathrm{p}}<10 \% and t_{\mathrm{r}}<0.15 \mathrm{~s}. To satisfy these two requirements, a set of reasonable values of \omega_{\mathrm{n}} and \zeta can be approximated using the relationships given in Section 10.2.
\begin{gathered} M_{\mathrm{p}}=\mathrm{e}^{-\pi \zeta / \sqrt{1-\zeta^{2}}}<10 \% \\ t_{\mathrm{r}} \approx \frac{1.12-0.078 \zeta+2.230 \zeta^{2}}{\omega_{\mathrm{n}}}<0.15 \mathrm{~s} \end{gathered}
The requirement for overshoot yields
\zeta>0.59.
Letting \zeta=0.59 and substituting it into the requirement for rise time gives
\omega_{\mathrm{n}}>12.33 \mathrm{rad} / \mathrm{s}.
This region is shown in Figure 10.34, which can be used as a starting point for control design. Note that damping slows the motion of the system. Thus, for a damping ratio higher than 0.59 , the critical value of natural frequency should be higher than 12.33 \mathrm{rad} / \mathrm{s} to speed up the motion of the system. If the closed-loop poles are located to the left of the gray boundary in Figure 10.34, then the closed-loop response to a unit-step reference input will very likely meet the desired requirements.
Choosing \zeta=0.65 and \omega_{\mathrm{n}}=13.5 \mathrm{rad} / \mathrm{s} yields k_{\mathrm{p}}=48.24 and k_{\mathrm{D}}=0.18. Figure 10.35 shows the response of the closed-loop system to a unit-step reference input.
