Question

Problem 1: The impulse response ht) for a particular LTI system is shown below hit) Be5e (4 cos(3t)+ 6 sin(3t) e. u(t) 1. Plot the impulse response for h(t) directly from the above equation by creating a time vector 2. Use the residue function to determine the transfer function H(s). Determine the locations of the poles and zeros of H(s) with the roots function, and plot them in the s-plane (x for poles, o for zeros). Use the freas function to plot the magnitude and phase of the transfer function. 3. Document your work, include plots to illustrate your work and your conclusions. Include your MATLAB code.

Answer 1

matlab code:

t=0:0.001:10;
f=3*exp(-3*t)+5*exp(-2*t)+exp(-t).*(4*cos(3*t)+6*sin(3*t))+exp(-4*t);
figure
plot(t,f),grid on
r=[3;5;2;2;-3*i;3*i;1];
p=[-3;-2;-1+3*i;-1-3*i;-1+3*i;-1-3*i;-4];
k=[];
[b a]=residue(r,p,k)
sys=tf(b,a)
figure
pzmap(sys)
p=pole(sys)   
z=zero(sys)
figure
bode(sys)
margin(sys)

output:

b =

13 142 834 3352 8536 13392 10824

a =

1 13 86 384 1180 2516 3560 2400

sys =

13 s^6 + 142 s^5 + 834 s^4 + 3352 s^3 + 8536 s^2 + 13392 s + 10824
---------------------------------------------------------------------
s^7 + 13 s^6 + 86 s^5 + 384 s^4 + 1180 s^3 + 2516 s^2 + 3560 s + 2400

Continuous-time transfer function.

p =

-1.0000 + 3.0000i
-1.0000 - 3.0000i
-1.0000 + 3.0000i
-1.0000 - 3.0000i
-4.0000 + 0.0000i
-3.0000 + 0.0000i
-2.0000 + 0.0000i

z =

-0.9147 + 3.7651i
-0.9147 - 3.7651i
-3.8609 + 0.0000i
-2.6326 + 0.0000i
-1.3001 + 1.9407i
-1.3001 - 1.9407i

impulse responce:

14 12 10 0 10

p-z map:

bode plot:

Pole-Zero Map 0 -0.5 1.5 -2 2.5 3.5 Real Axis (seconds)

Bode Diagram Gm Inf, Pm 99.1 deg (at 12.8 radis) 20 15 10 -5 10 15 -20 45 0. -90 101 10° 101 102 Frequency (rad/s)