## Description

1. A process Gp(s) is in feedback control with a P-controller using a measuring element Gsens(s).

Gp(s) = s

2 − 4s + 8

s(s + 1)(s + 3); Gsens(s) = 1

s + 10

(a) Sketch the root locus of a feedback compensated closed-loop system consisting of as the

proportional controller gain Kc varies from 0 to +∞. Compute the asymptotes angles,

centroid, angles of arrival, break-in and entry points.

(b) Generate the root locus on the computer and verify your sketch (do not reverse the order

of parts (b) and (a) for your own benefit!)

(c) Determine the ultimate gain by hand (show your calculations) and verify your answer

with the computer generated plot.

(d) Find the value of Kc such that the closed-loop response to a set-point change has the

minimum settling time.

(e) If a PI-controller Gc(s) =

Kc +

KI

s

was used instead, find the ultimate value of KI

with the value of Kc fixed to what you obtained in (1d).

2. A process with the transfer function G(s) = 2(s + 4)

10s

2 + 7s + 1

e

−2s

is placed in feedback with a

controller Gc(s)

(a) Suppose Gc is a P-controller. Design Kc s.t. the gain margin is 8.2 dB. Report the

corresponding PM.

(b) DelayNaJaane, who is in charge of the controller design, is uncertain about the delay but

would like to know the extent of delay for which the control system can robustly remain

stable. What is the maximum delay uncertainty with the value of Kc chosen in (2a)?

(c) Using the Kc value in part (2a), now design a PI controller of the form Gc(s) =

Kc

1 +

1

τIs

s.t. the phase margin is 60◦

. Report the corresponding GM.

(d) Evaluate the sensitivity function of the feedback system with the above settings. Verify

numerically that indeed Bode’s sensitivity integral holds (up to the numerical approximation).

3. A process has the transfer function G(s) = 2(s + 2)

s

2 + 2s − 3

e

−s

(a) Using Pade’s first-order approximation, design a P controller (call it Gc1) such that the

closed-loop system is stable and has the dominant pole located at p = −2.

(b) Design another P controller (call it Gc2) using the Nyquist diagram such that the gain

margin is 10.5 dB. Calculate the offset in output to a step-type set-point change for this

value of Kc.

(c) Using SIMULINK, compare the performances of above two controllers for step-type setpoint change and disturbance. Would the performance of first controller improve if we

had taken into account the delay using a Pad´e’s second-order approximation?