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Midterm

This Midterm covers Weeks 1-4.  Show all required calculations, MATLAB code and MATLAB plots for full credit.

1. 1. Determine the partial fraction expansion for V(s) and compute the inverse Laplace transform.  The transfer function V(s) is given by:

V(s) = 400 / (s2 + 8s +400)

1. 1. A second-order system is Y(s)/R(s) = T(s) = (10/z)(s + z) / ((s + 1)(s + 8))

Consider the case where 1 < z < 8. Obtain the partial fraction expansion, and plot y(t) for a step input r(t) for z = 2, 4, and 6.

1. 1. Determine whether the systems with the following characteristic equations are stable or unstable:
      1. s3 + 4s2 + 6s + 100 = 0
      2. s4 + 6s3 + 10s2 + 17s + 6 = 0
      3. s2 + 6s + 3 = 0
   2. A single-loop negative feedback system has the loop transfer equation:

L(s) = Gc(s)G(s) = K(s + 2)2 / (s(s2 + 1)(s + 8))

1. 1. Sketch the root locus for 0 ≤ K ≤ infinity to indicate the significant features of the locus.
   2. Determine the range of the gain K for which the system is stable.
   3. For what value of K in the range of K ≥ 0 do purely imaginary roots exist?  What are the values of these roots?
2. Would the use of the dominant roots approximation for an estimate of settling time be justified in this case for a

   Assignment 6

   Stability in the Frequency Domain

   1. Watch video “EE495 – Week 6 – Lecture”
   2. Read Chapter 9 in the text Modern Control Systems, 12th Edition.
   3. Work the following problems:
      1. Sketch the Nyquist plots of the following loop transfer functions and determine whether the system is stable by applying the Nyquist criterion:
         • • L(s) = Gc(s)G(s) = K/(s(s2 + s + 6)
           • L(s) = Gc(s)G(s) = K(s + 1) / (s2(s + 6))

         If they system is stable, find the maximum value for K by determining the point where the Nyquist plot crosses the u-axis.

      2. A closed-loop system with unity feedback has a loop transfer function L(s) = Gc(s)G(s) = K(s +20) / s2
         • Determine the gain K so that the phase margin is 45 degrees.
         • For the gain K selected in part (a), determine the gain margin.
         • Predict the bandwidth of the closed-loop system.
   4. Save work in a file with the title: “HW6_StudentID”, with your student id substituted in the file name.  Show all work for full credit.
   5. Upload file “HW6_StudentID”

   Lab 6

   Stability in the Frequency Domain

   1. Consider a closed-loop system that has the loop transfer function L(s) = Gc(s)G(s) = Ke-TS/ s
      1. Determine the gain K so that the phase margin is 60 degrees when T = 0.2.
      2. Plot the phase margin versus the time delay T for K as in part (a).
   2. Include all MATLAB code, calculations and screenshots in a Word entitled “Lab6_StudentID”.
   3. Upload file “Lab6_StudentID”