4. Consider a unity (negative) feedback control system whose open-loop transfer function is given by the following. 1 G(s): s³ (s + 2) What is the steady state error of the system for input u(t) = t³ 1(t)? Recall from the class lecture that steady-state error is given by the following formula. S ess = lim S-01 + G(s) U(s)

Feedback and Control Systems Write the general form of, and determine the exact response of the capacitor voltage for the electrical network shown below. R₁ = 10 kQ ww R₂ = 10 ks C= 10 pF v(t)=u(t) L=200 H 0000

Consider the feedback system. It is known that G(s) has one unstable zero and one unstable pole. On the G(s)-plane -1+j0 point is encircled one time in the clockwise direction as s moves in the clockwise direction on the Nyquist path. What can be said about the closed-loop system? a. Stable O b. C. e. Of. G(s) Has 1 unstable pole Od. Has 2 unstable poles and 1 stable zero Has 1 unstable pole and 1 unstable zero Has 2 unstable poles and 2 unstable zeros Has 2 unstable poles and 1 ustable zero None g.

The control system G(s) = Kp / s (s+2) (s+6) has a unity feedback a. Kp is variable, discuss stability by using Routh Hurwitz method. Mark the poles ofthe system on root-locus curve below.  b. When Kp = 26.7 , roots of the characteristic equation becomes s1= -0.6-1.89i, s2= -0.6+1.89i and s3= -6.8. Are you happy about this case? How do you evaluate,why ? c. How the root locus and step response would be effected by adding a controller Gc= Kp ( s + 0.5)2/ s . Give a comment about the addition of Gc on root locus andon step response.

- 6. The unity feedback system shown in Figure P9.1 with K(s+6) G(s) (s+2)(s + 3)(s + 5) is operating with a dominant-pole damping ratio of 0.707. Design a PD controller so that the settling time is reduced by a factor of 2. Compare the transient and steady-state performance of the uncompensated and compensated systems. Describe any problems with your design. [Section: 9.3]

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5. Consider following feedback system. R(s) + 100 S+4 +1 Find the steady-state error for (i) step input and (ii) ramp input.

The unity feedback system with open loop transfer function G(s) K S² is to be designed to have closed loop poles at -2.4 + j4.16. If the compensator's zero is placed at -1, do the following: a) Find the compensator pole b) Find the system gain c) Find the location of the third pole. d) Evaluate the steady-state error characteristic. e) Plot the step response of the compensated system

Control

For the open-loop system G(s) = 3/s^2 + 2s - 3 assume there is a feedback with a proportional gain, K. And sketch a locus of closed-loop rootsverse K. What is the minimum value of K to achievea stable system?