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 / (s²Â + 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. s³Â + 4s²Â + 6s + 100 = 0
      2. s⁴ + 6s³Â + 10s²Â + 17s + 6 = 0
      3. s²Â + 6s + 3 = 0
   2. A single-loop negative feedback system has the loop transfer equation:

L(s) = G_c(s)G(s) = K(s + 2)²Â / (s(s²Â + 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?
   4. Would the use of the dominant roots approximation for an estimate of settling time be justified in this case for a large magnitude of gain (K ≥ 50)?