Question

PROBLEM 4 Suppose that a system is shown in Figure 4. There are three controllers that might be incorporated into this system. 1. Ge (s)-K (proportional (P) controller) 2. GS)K/s (integral (I) controller) 3. G (s)K(1+1/s) (proportional, integral (PI) controller) The system requirements are T, < 10 seconds and P0 10% for a unit step response. (a) For the (P) controller, write a piece of MATLAB code to plot root locus for 0<K<,and find the K value so that the system requirements are met. (b) Solve part (a) for the (l) controller. (C) Solve part (a) for the (PI) controller. (d) Create the subplot of the unit step responses for the closed loop systems with the controllers established in (a), (b), and (c) (e) Juxtapose all the controllers established in (a), (b), and (c) concerning the steady state error and transient response. Y(s) R(s) s2+ Figure 4: A unity feedback closed loop control system with a controller and process respectively

Answer 1

Matlab code is given below in bold letters.

clc;

close all;

clear all;

% define the lapalce variable s

s = tf('s');

% define the plant

Gp = 1/(s^2+5*s+6);

% plot the root locus with proportional controller

% define gain k

k = 0:0.1:100;

figure;rlocus(Gp,k);

Root Locus 15 System: Gp Gain: 11.9 Pole: -25 3.41i Damping: 0.591 Overshoot(%): 10 Frequency (radis: 4.23 0 0.5 43.5 -3-2.5 -21.5 -1 0.5 Real Axis (seconds)

From the above figure, it is observed that the %overshoot is 10 % for a gain of 11.9.

Time constant = 1/2.5 = 0.4 sec

Therefore the settling time = 4*T = 4*0.4 = 1.6 sec.

% Question 2 integral controller

% MATLAB code is in bold letter

% plot the root lcous with integral controller alone

% define gain k

k = 0:0.1:100;

figure;rlocus(Gp/s,k);

Root Locus System: untitled1 Gain: 4.4 Pole: -0.650.876i Damping: 0.596 Overshoot (% 9.7 Frequency (rad/s): 1.09 -3 -4 -6 -4 Real Axis (seconds)

From the above figure, it is observed that the %overshoot is 10 % for a gain of 4.4.

Time constant = 1/0.65 = 1.5385 sec

Therefore the settling time = 4*T = 4*1.5385 = 6.1538 sec.

% Question 3 Proportional + integral controller

% plot the root lcous with proportional integral controller

% define gain k

k = 0:0.1:100;

figure;rlocus(Gp*(s+1)/s,k);

Root Locus 15 System: untitled1 10 Gai: 10 Pole: -2.112.87i Damping: 0.591 Overshoot(%): 9.99 Frequency (radis): 3.56 153.52.52 -0.5 Real Axis (seconds)

From the above figure, it is observed that the % overshoot is 10 % for a gain of 10.

Time constant = 1/2.11= 0.4739sec

Therefore the settling time = 4*T = 4*0.4739 = 1.8957sec.

% Question d
figure;
subplot(311);
rlocus(Gp,k);
title('proportional controller');

subplot(312);
rlocus(Gp/s,k);
title('intgral controller controller');

subplot(313);
rlocus(Gp*(s+1)/s,k);
title('proportional + integral controller');

proportional controller 20 -10 -3.5 -2.5 0.5 0.5 Real Axis (seconds intgral controller controller Real Axis (seconds) proportional +integral controller 20 20 -3.5 -2.5 0.5 0.5 Real Axis (seconds-')