University of Notre Dame
Aerospace and Mechanical Engineering

AME 437: Control Systems Engineering
Homework 2

B. Goodwine
Spring, 2000		 Issued: January 26, 2000
Due: February 2, 2000

Unless otherwise indicated, all problems are from the course text.

  1. The following is a simple differential equation that describes the velocity of a car, v,

    \begin{displaymath}\dot v + \frac{b}{m}v = u(t), \end{displaymath}

    where u(t) is the force on the car due to the output of the engine, b is a drag coefficient and m is the mass of the car.

    The purpose of this problem is to solve the differential equation using the convolution integral. To do this:

    a.
    Determine the impulse response for the system, i.e., solve

    \begin{displaymath}\dot v + \frac{b}{m} v = \delta(t) \end{displaymath}

    where $v(0) = \dot v(0) = 0$.
    b.
    Use the convolution integral to determine the solution of the system for the input

    \begin{displaymath}u(t) = \sin t. \end{displaymath}

    c.
    Solve the differential equation

    \begin{displaymath}\dot v + \frac{b}{m} v = \sin t \end{displaymath}

    using another method (Laplace transform, perhaps?) to show that you get the same answer.

    Hint: Unless you have a great memory, the integral may be a bit tricky to solve. Feel free to find the answer in a table of integrals or use something like Mathematica or Maple.

  2. 3.2 (a - d)
  3. 3.6 (a - d)
  4. 3.7 (a,d,g, and j)
  5. 3.8
  6. Explain in terms that an Arts and Letters major could understand, i.e., no equations,
    1. why adding a derivative term to a proporational controller reduces the magnitude of an oscillatory response;
    2. why adding an intergral term to PD control can eliminate steady state error; and,
    3. why adding an integral term to PD control can potentially make the response more oscillatory and may even cause the system to be unstable.

Last updated: January 26, 2000.
B. Goodwine (goodwine@controls.ame.nd.edu)