# All of these problems should be done by hand. You may use a computer or calculator...

###### Question:

All of these problems should be done by hand. You may use a computer or calculator to find roots of polynomials with constant coefficients. All hand calculations can of course be verified with a computer. Show calculations for asymptote angles, center of mass, angles of departure, etc. for full credit.
3. Consider the proportional control system with measurement dynamics in Figure 2. (a) (5 points) - Determine the limiting value of gain K for a stable system. (b) (5 points) . For the gain that results in marginal stability, determine the magnitude of the imagi- nary roots. (c) (5 points) and/or = Reduce the gain to K = 400 and determine the damping ratio and natural frequency of the conjugate pair of closed loop poles. At this value of K, does the system qualify for the dominant pole approximation (software tools may be useful). If this system does qualify for the dominant pole approximation, write the second order transfer function that approximates it. If this system does not qualify, explain what aspects of the step response the third pole effects so much that it cannot be ignored. R(S) + Y(8) K 8+60 s(s+20) s+20 Figure 2: Block diagram for Question 3.

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