Question 10.16: Proportional Control Design Using Root Locus Design a propor...
Proportional Control Design Using Root Locus
Design a proportional controller for the cart system in Example 10.13 using the root locus technique.
For proportional feedback control of the \text{DC} motor–driven cart, the loop gain L(s) is equal to the transfer function of the plant G(s),
L(s) = G(s) = \frac{3.778}{s^2 + 16.883s}
The root locus is plotted in Figure 10.56 using the MATLAB command rlocus. Note that the closed-loop system is a second-order system. Two closed-loop poles that are real appear for small values of K, specifically K ≤ 18.9. When K > 18.9, the closed-loop system has a pair of complex conjugate poles, which move along the complex portion of the root locus as K varies between 18.9 and ∞.
To select the value of the proportional control gain that will meet the performance specifications, we can turn on the grid lines to the root locus using the following command.
>> grid on
As discussed in subsection 10.2.2, the semicircles in Figure 10.56 indicate lines of constant natural frequencies ω_n and the diagonal lines indicate constant damping ratios ζ. In this example, we need an overshoot that is less than 10\% (which implies ζ > 0.59) and a rise time that is less than 0.15 \text{s} (which implies ω_n > 12.33 \text{rad/s}). In Figure 10.56, the solid diagonal lines indicate pole locations with a damping ratio of about 0.59. In between these lines, the poles have ζ > 0.59 and outside of the lines ζ < 0.59. The solid semicircle is the locus of all poles with ω_n = 12.33 \text{rad/s}, whereas those inside the semicircle have ω_n < 12.33 \text{rad/s}, and those outside correspond to ω_n > 12.33 \text{rad/s}. Thus, only the part of the root locus between the two diagonal lines and outside of the semicircle is acceptable. Figure 10.57 zooms in on the desired region, in which the vertical line is a part of the root locus. Left-clicking the root locus, you will see the values of the pole and the gain that is required to place one of the closed-loop poles at that particular location. Holding down the left mouse button and moving the mouse along the root locus, you can see the values of the pole and the gain varying correspondingly.
Let us select K = 50. Figure 10.58 is the corresponding unit-step response of the closedloop system with proportional feedback control. The closed-loop system meets the given specifications.
Note that the closed-loop poles cannot be placed arbitrarily in the s-plane with only a static proportional controller since the shape of the root locus is fixed for a given plant. A more useful design can be obtained by adding a pole or zero to the controller and making it a dynamic controller. This results in so-called lead or lag compensators, C(s) = (s + z)/(s + p). The reader can refer to the control systems books for more details.