Question 13.11: For the system of Figure 13.21, find the value of gain, K, t...
For the system of Figure 13.21, find the value of gain, K, to yield a damping ratio of 0.7.
Figure 13.23 shows the constant damping ratio curves superimposed over the root locus for the system as determined from the last example. Draw a radial line from the origin to the intersection of the root locus with the 0.7 damping ratio curve (a 16.62° line). The root locus program discussed in Appendix H.2 at www.wiley.com/go/Nise/ControlSystemsEngineering8e can now be used to obtain the gain by searching along a 16.62° line for 180°, the intersection with the root locus. The results of the program show that the gain, K, is 0.0627 at 0.719 + j 0.215, the point where the 0.7 damping ratio curve intersects the root locus.
We can now check our design by finding the unit sampled step response of the system of Figure 13.21. Using our design, K = 0.0627, along with R(z) = z/(z − 1), a sampled step input, we find the sampled output to be
C (z) = \frac{R (z) G (z)}{1 + G (z)} = \frac{0.0627z² + 0.0627z}{z³ − 2.4373z² + 2z − 0.5627} (13.87)
Performing the indicated division, we obtain the output valid at the sampling instants, as shown in Figure 13.24. Since the overshoot is approximately 5%, the requirement of a 0.7 damping ratio has been met. You should remember, however, that the plot is valid only at integer values of the sampling instants.
Students who are using MATLAB should now run ch13apB8 in Appendix B. You will learn how to use MATLAB to plot a root locus on the z-plane as well as superimpose a grid of damping ratio curves. You will learn how to obtain the gain and a closed-loop step response of a digital system after interactively selecting the operating point on the root locus. This exercise solves Example 13.11 using MATLAB.
