1 answer

,(s): (24 points) 2. The open loop transfer Function of a certain plant is: G„( s(s+2)(s+10)...

Question:

,(s): (24 points) 2. The open loop transfer Function of a certain plant is: G„( s(s+2)(s+10) a. Create the root locus plot of
All parts are to be done in Matlab.
,(s): (24 points) 2. The open loop transfer Function of a certain plant is: G„( s(s+2)(s+10) a. Create the root locus plot of the plant (5 points) b. Use PD control to improve the response so that the dominant time constant is 0.625 or less and the damping ratio is 0.25 or greater. i. Put the transfer function in terms of Kp and Td (2 points) ii. Show a root locus plots for each Td and which Td meets the conditions. You may assume Kp = 1 (17 points)
All parts are to be done in Matlab.

Answers

MATLAB code is given below in bold letters.

clc;
close all;
clear all;

% define the Laplace variable s
s = tf('s');

% define the plant
G = 1/(s*(s+2)*(s+10));

% plot the rootlocus plot
figure;
rlocus(G);

Root Locus Imaginary Axis (seconds) -30 -20 -10 Real Axis (seconds)

Question b

% PD compensator design using SISOTOOL
sisotool(G);

- 0 x Control and Estimation Tools Manager Edit Help File Architecture Compensator Editor Graphical Tuning Analysis Plots Aut

SISO Design for SISO Design Task File Edit View Designs Analysis Tools Window Help Root Locus Editor for Open Loop 1 (OL1) Op

Now add the design requirements by right clocking on the rootlocus plot.

- O X SISO Design for SISO Design Task Edit View Designs Analysis File Tools Window Help Root Locus Editor for Open Loop 1 (O

settling time = 4* time constant = 4 * 0.625 = 2.5

SISO Design for SISO Design Task Edit View Designs Analysis File Tools Window Help Root Locus Editor for Open Loop 1 (OL1) Op

damping ratio = 0.25

Then the probable region of closed loop dominant poles is shown below.

- O X SISO Design for SISO Design Task File Edit View Designs Analysis Tools Window Help Root Locus Editor for Open Loop 1 (O

Now add a zero at at s = -3.(this comes through trial and error).

X - 0 SISO Design for SISO Design Task File Edit View Designs Analysis Control and Estimation Tools Manager File Edit Help To

from the above figure it is observed that the closed loop dominant poles travel through the desired region in s plane.

Therefore the PD controller gain is given by s = -3.

For a loop gain Kd > 46, the closed loop system meets the desired specifications.

Therefore Controller C(s) = 46(s+3)

design verification:

closed loop step response is plotted below to verify the validity of the design.

code:

C = 46*(s+3);

% Plot the step response
figure;
step(feedback(C*G,1));

Step Response System: untitled1 Peak amplitude: 1.14 Overshoot (%): 14.3 At time (seconds): 0.625 1.2 System: untitled1 Settl

from the above figure it is observed that the closed loop system meets the design requirements.

.

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