Question

you can use matlab to solve

1. Given the plant model differential equation: y" + 6y'+ 12y 12u(t) Find: a) G(s) continuous transfer function he step response of the unity feedback system c) The appropriate sampling time d) G(z) pulse transfer function e) Continuous State Space, A, B, C, D f) Discrete State Space, A, B, C, D

Answer 1

MATLAB code:

clc;clear all;close all;
%G(s)
b=12;%coefficients of u
a=[1 6 12];%coefficients of y
t=0:0.1:20;
disp('Continuous transfer fucntion G(s)')
sys=tf(b,a)
xt=(t>=0);%step input
%step response
y=lsim(sys,xt,t);
subplot(321)
plot(t,xt,'b','linewidth',3)
xlabel('t');ylabel('x(t)')
title('step input')
subplot(322)
plot(t,y,'r','linewidth',3)
xlabel('t');ylabel('y(t)')
title('step response of G(s)')

Ts=1;
sysd1 = c2d(sys,Ts);
n=0:1:20;
x=(n>=0);
y=lsim(sysd1,x,n);

subplot(323)
stem(n,y,'r','linewidth',3)
xlabel('n');ylabel('y(n)')
title('step response for Ts=1s')

Ts=0.5;sysd2 = c2d(sys,Ts)
y=lsim(sysd2,x,n);

subplot(324)
stem(n,y,'r','linewidth',3)
xlabel('n');ylabel('y(n)')
title('step response for Ts=0.5s')

Ts=2;sysd3 = c2d(sys,Ts);
y=lsim(sysd3,x,n);

subplot(325)
stem(n,y,'r','linewidth',3)
xlabel('n');ylabel('y(n)')
title('step response for Ts=2s')

disp('contunous state space model')
[a,b,c,d] = ssdata(sys)
disp('Discrete state space model')
[aa,bb,cc,dd] = ssdata(sysd2)

%step response of unty feedback system
%Gf(s)=G/(1+(G*H)) where H=1
%Gf=G/(1+G)=12/(s^2 +6s +24)
bb=12;
aa=[1 6 24];
syss=tf(bb,aa)
%step response
t=0:0.01:5;
xt=(t>=0);
y=lsim(syss,xt,t);
subplot(326)
plot(t,y,'m','linewidth',3)
xlabel('t');ylabel('y(t)')
title('Step response of unity feedback system')

Command window:

Continuous transfer fucntion G(s)

Transfer function 'sys' from input 'u1' to output ...

12
y1: --------------
s^2 + 6 s + 12

Continuous-time model.

Transfer function 'sysd2' from input 'u1' to output ...

0.561 z + 0.1996
y1: ------------------------
z^2 - 0.2891 z + 0.04979

Sampling time: 0.5 s
Discrete-time model.
contunous state space model
a =

0.00000 -1.20000
10.00000 -6.00000

b =

-1.20000
0.00000

c =

0 -1

d = 0
Discrete state space model
aa =

0.00000 0.04979
-1.00000 0.28911

bb =

-0.19963
0.56104

cc =

0 1

dd = 0

Transfer function 'syss' from input 'u1' to output ...

12
y1: --------------
s^2 + 6 s + 24

Continuous-time model.

step response of G(s) step input 1.15 1.1 1.05 0.8 0.6 0.4 0.2 0.95 0.9 0.85 15 20 15 20 step response for Ts=1s step response for Ts=0.5s 1.4 E 0.8 0.6 0.4 0.2 E 0.8 0.6 0.4 0.2 15 15 10 20 20 step response for Ts2s Step response of unity feedback system 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.8 0.6 0.4 0.2 20 2 3 4 15

_________________________________________________________________________________________

Observation:

The continouos transfer function is

12
G(s)= --------------
s^2 + 6 s + 12

From the graphs (2) (3) (4), the appropriate sampling time is Ts=0.5s.

The pulse transfer function is

G(z) = 0.561 z + 0.1996
   ------------------------
z^2 - 0.2891 z + 0.04979

The unity feedback system is

Gf(s)= 12
--------------
s^2 + 6 s + 24

The continuous state space model is

a =

0.00000 -1.20000
10.00000 -6.00000

b =

-1.20000
0.00000

c =

0 -1

d = 0

The discrete state space model is

aa =

0.00000 0.04979
-1.00000 0.28911

bb =

-0.19963
0.56104

cc =

0 1

dd = 0