Question

Consider the model of a spring-mass-damper system, where the following parameter values are assumed: m 1,b 2,k 2. 3. a. Write down the transfer function of the system b. Choose a sample time for the system c. Find the pulse transfer function (use MATLAB 'c2d' command) d. Find the range of K for stability for the closed-loop sampled-data system 4. Consider a series RLC circuit driven by a voltage source with capacitor voltage as output. Assume the following parameter values: R- 200,L 1mH, C SuF a. Write down the transfer function of the system b. Choose a sample time for the system c. Find the pulse transfer function (use MATLAB 'c2d' command) d. Find the range of K for stability for the dosed-loop sampled-data system
answer 3 and 4 please

Answer 1

I have used matlab to solve the problems 3 and 4. the code is given below along with the solutions for all problems. comments give the explanation part.

clc;
close all;
clear all;

% define m b k
m = 1;
b = 2;
k = 2;

% transfer function is defined below
G = tf(1,[1 2 2]);

% bandwidth of G
B = bandwidth(G)

% The bandwidth is 1.5 rad/sec approximately
% lets consider the sample rate 50 times the banwidth
ws = 50*B

% sample time is Ts = 2*pi/ws
Ts = 2*pi/ws

% pulse transfer function is given below
Gd = c2d(G,Ts,'tustin'); % bilinear approximation for converting continuous to discrete

% use rootlocus technique to determine the range of K for stability
figure;
rlocus(Gd);

RESULTS:

Transfer function: G(s)
1
-------------
s^2 + 2 s + 2

Bandwidth =

1.4124 rad/sec

ws (sampling frequency in rad/sec) =

70.6219

Ts ( sampling time ) =

0.0890

Root Locus System: God Gain: 1.82e+005 Pole: 0.9940.105i Damping: 8.13e-005 Overshoot(%): 100 requency (rad/s): 34.1 -0.5 -0.5 0.5 Real AXis

From the above figure, it is observed that the closed loop poles lie within the unit circle for all values of gain K from 0 to inifinity. Therefore the ROC contains teh unit circle for all K. Therefore the system is stable for all K.

% Question 4
% output is taken across the capacitor
% Transfer function is defined below
C = 5e-6;
L = 1e-3;
R = 20;
s = tf('s');
G = minreal(1/(C*s)/(R + 1/(C*s) + L*s))

% bandwidth of G
B = bandwidth(G)

% The bandwidth is 1.5 rad/sec approximately
% lets consider the sample rate 50 times the banwidth
ws = 50*B

% sample time is Ts = 2*pi/ws
Ts = 2*pi/ws

% pulse transfer function is given below
Gd = c2d(G,Ts,'tustin'); % bilinear approximation for converting continuous to discrete

% use rootlocus technique to determine the range of K for stability
figure;
rlocus(Gd);

RESULTS:

Transfer function: G(s)
2e008
---------------------
s^2 + 2e004 s + 2e008

B (bandwidth) =

1.4124e+004 rad/sec

ws, sampling frequency (radsec)=

7.0622e+005

Ts sampling time =

8.8969e-006

Root Locus System: Gd Gain: 9.08e+004 Pole: 0.9940.105i Damping: 8.13e-005 Overshoot(%): 100 Frequengy (radis) 3.41e+005 0.5 1.5 -0.5 0.5 Real Axis

From the above figure, it is observed that the closed loop poles lie within the unit circle for all values of gain K from 0 to inifinity. Therefore the ROC contains teh unit circle for all K. Therefore the system is stable for all K.