6) A continuous time plant with transfer function is to be included in

the control system of Figure 1, with D(z) = K and T = 0.1s: system

a) Find the open loop discrete transfer function of the

b) Find the closed loop discrete transfer function of the system for K = 1

c) Sketch the root locus of the system in the z-plane and find the marginal value of K for stability. Show this value on the root locus

d) Find the values of K (K, and K 2 ) fc zero steady state error for a step input and 5% steady state error for a ramp input.

e) For this plant, using only the proportional control, can we have zero steady

state error for a unit step and a damping ratio of 0.7? Answer by yes or no and explain

Root Locus Design

7) A system is given by the following transfer function G(s) = K * (z + 1)/((z - 1) ^ 2)

a) Sketch the root locus as function of K 20 assuming unity negative feedback. b) Design a compensator to achieve a damping ratio 0.7 and settling time 4 seconds assuming a sampling interval 0.1 sec.

8) A DC servo-systemis represented by the transfer function 1 * v/v = 0/(x ^ 2) +0.84 +0.001 where

Bis the angular position in rad and I is the input voltage in volts. It is required to

design a compensator such that the closed loop system will have a settling time of 2 seconds and a damping ratio of 0.5. Assume the sampling interval is 0.1 second. a) Sketch a diagram (showing the physical components) to illustrate how a

digital control system can be implemented. Then, draw the corresponding block diagram.

b) Design the required compensator using the root locus.

c) Find the steady state value of the angular position if the set-point is 0.3 rad.

d) Modify the controller to compensate a constant input disturbance with unknown magnitude. Simulate the system to verify your design.

e) Modify the controller to compensate any constant bias in the sensor. Simulate

the system to verify your design.

9) For G(s) = 1/((s + 0.1)(s + 3))

being controlled with a digital controller using a sample period of T=0.1 sec a) Design compensation using z-plane root locus that will respond to a step with a settling time of ≤1.5 sec and an overshoot ≤ 5%.

b) Obtain the response of the system with the controller to a unit step input. Plot the first few samples of the output in this case. What is the steady state error? c) Update your compensator to achieve half the steady state error calculated in (b).

10 ) G(s) = 1/(s(s + 1)) 10) A system, given by the transfer function is controlled by a proportional controller with D(z) = K as shown in Figure 1, given that the sampling time is T = 1

1. Find the open loop discrete transfer function of the system b) Find the value of K that results in critical damping for the system and find the
2. time constants of the roots in this case.
3.  c) Design a lag compensator such that the system has twice the value of error steady state constant and results in critical damping, with roots having approximately the same time constant found above.

11) The transfer function of an antenna tracking controller is given by

G_{p}(s) = 1/(s(10s + 1))

a) Using s-domain root locus design a compensator for the system to obtain an overshoot less than or equal to 16%, with settling time of less than or equal to 10s.

b) Calculate the steady state error of the controlled system for unit ramp input. c) Design a controller to reduce the steady state error computed to half its value.

d) Assuming a sampling time f * 0.2s, Dig the obtained controller using the following transformation techniques:

Bilinear

iv. Zero-Pole mapping e) Compute the ZOH equivalent of the plant. Find the closed loop poles of the

overall system with each of the obtained digitized controllers. Find the overshoot and the settling time for each design

f) Comment on the obtained results

12) Consider the system

G(s)= 0.2083 s( s + 1.71

Let the system be controlled using a cascade controller in a unity feedback configuration, a) Design a continuous time compensator to achieve a settling time of 2 seconds

and a damping ratio of 0.5 b) Assume T = 0.02 sec. Descretize the controller using the bilinear transformation

c) Repeat (b) gT =0.5 sec .

d) Repeat (b) and (c) using the forward and backward methods of approximating continuous systems. e) Use simulation to compare the performance of continuous-time and discrete-time controllers.