ETEC 415 – Control Systems – HW 4

Module 4 Homework Assignment

1. The general form of a first-order system is
   𝜏d𝑦dt+𝑦=𝑘𝑥
   Or as a transfer function
   𝐺(𝑠)= 𝑌(𝑠)/𝑋(𝑠)=𝑘/(𝜏𝑠+1)
   Describe the following in your own words.
   a. Time constant when subjected to an impulse input
   b. Time constant when subjected to a step input
   c. Delay time when subjected to a step input
   d. Rise time when subjected to a step input
2. The general form of a second-order system is
   d2𝑦/dt2+2𝜁𝜔𝑛d𝑦/dt+𝜔𝑛2𝑦=𝑘𝜔𝑛2𝑥
   Or as a transfer function 𝐺(𝑠)= 𝑌(𝑠)𝑋(𝑠)=𝑘𝜔𝑛2𝑠2+2𝜁𝜔𝑛𝑠+𝜔𝑛2
   If subjected to a step input, describe the following in your own words:
   a. The difference in output behavior if over damped (𝜁>1), critically damped (𝜁=1), and under damped (𝜁<1)
   b. Rise time for an under-damped system
   c. Overshoot for an under-damped system
   d. Settling time for an under-damped system
3. Use MATLAB to find the partial fraction expansion of the following. Write the commands you used. Also write the final partial fraction expansion using an equation editor. If there are imaginary roots, write them as a quadratic like Example 4 in the MATLAB tutorial.
   (13𝑠2+21𝑠−4)/(𝑠3+2𝑠2−5𝑠−6)
4. Use MATLAB to find the partial fraction expansion of the following. Write the commands you used. Also write the final partial fraction expansion using an equation editor. If there are imaginary roots, write them as a quadratic like Example 4 in the MATLAB tutorial.
   (3𝑠2+12𝑠+5)/(𝑠4+8𝑠3+26𝑠2+48𝑠+45)
5. Use MATLAB to find the time response equation (hint: inverse Laplace) of the following transfer function when the input is a step function with a magnitude of 2. Plot the time response up to a time of 10 seconds (include names for the y and x-axes). Include the plot, MATLAB commands used, and the time response equation written with an equation editor.
   𝑌(𝑠)/𝑋(𝑠)=1/(𝑠2+6𝑠+8)
6. Use MATLAB to plot the time response of the following transfer function when the input is a unit step function. Make sure the response is plotted up to 5 seconds. What is the rise time, settling time, overshoot, peak, and peak time? Include the plot and MATLAB commands used.
   𝑌(𝑠)/𝑋(𝑠)=1/(𝑠2+6𝑠+109)
7. Use Simulink to simulate a step input with the transfer function in problem 6. Take screen captures of your Simulink model and the resulting plot. The screen captures will be the answer for this question.
8. Describe in your own words how knowing the locations of the poles in the s-domain of a transfer function lets you know whether a system will be stable or unstable. For the transfer functions in problems 3-6, are the systems stable or unstable? Why?