University of Notre Dame
Aerospace and Mechanical Engineering

AME 437: Control Systems Engineering
Homework 5

B. Goodwine
Spring, 2000		 Issued: February 17, 2000
Due: February 23, 2000

The main objective of this homework is for you to be able to predict the nature of the transient response of a system as a function of pole and zero locations in the s-plane.

1.
Consider the second order system:

\begin{displaymath}\frac{Y(s)}{R(s)} = \frac{1}{s^2 + 3 s + 2}. \end{displaymath}

(a)
On the s-plane, plot the pole and zero locations for the transfer function.
(b)
Using the Matlab step() command, plot the response, y(t) to a step input in r.
(c)
Let R(s) be a step input, and, by using a partial fraction expansion and the inverse Laplace transform, compute the analytical expression for the time response.
(d)
On the same plot, plot each of the (three) terms appearing in the expression for the response, as well as their sum. Is the sum the same as what Matlab gave in part (a)?
2.
Consider the second order system with an additional pole at $s=-\alpha$:

\begin{displaymath}\frac{Y(s)}{R(s)} = \frac{\omega_n^2}{(s^2 + 2 \zeta \omega_n^2 s + \omega_n^2)(s/\alpha+1)}. \end{displaymath}

Using a partial fraction expansion, determine the time response of the system to a step input. Describe the response when
(a)
$\alpha$ is large and positive;
(b)
$\alpha$ is small and positive; and,
(c)
$\alpha$ is negative.
3.
4.5 This, and the following problems from the book illustrate the use of feedback to track input signals and to reject disturbances.
4.
4.8 (Note: 1/T_Is should be 1/(T_I s), i.e., the integral term is divided by s.)
5.
4.9 (wait until after Monday's class to try this one).
6.
4.36
7.
Match each of the pole zero plots with the corresponding step response in the following plots. For each match, write one or two sentences explaining your choice. Number the pole/zero plots 1 to 6, left to right and top to bottom, so that the lower left figure is 4. Label the step response plots a through f, left to right and top to bottom, so that the lower left figure is d. (Postscript versions of the figures are available here and here.

Last updated: February 17, 2000.
B. Goodwine (goodwine@controls.ame.nd.edu)