1. If f(t)=est then u(t)= H(s)est
Transfer Function:
f(t): Input,
u(t): Output
Example 7.1 (continued)
If f(t)=est then u(t)= H(s)est
Transfer Function:
s3H(s)est+4s2H(s)est+14sH(s)est+20H(s)est=3est+sest
a=[1,4,14,20];roots(a)
(s3+4s2+14s+20)H(s)=3+s
Eigenvalues: -1±3i, -2
Exponential/Harmonic Input:
s= i;
hs=(s+3)/(s^3+4*s^2+14*s+20);
abs(hs), angle(hs)
RESONANCE

2. H(s) Input-Output relationship in s domain: x(t) y(t) Y(s)=X(s) H(s)
Impulse Response:
y(t)=h(t)
Impulse function
Δ(s)=1
Example 7.1 (Continued):
p1=[1,3]; p2=[1,4,14,20]; [r,p,k]=residue(p1,p2)

3. z=-0.05+0.1833i; 2*abs(z), phase(z)
Impulse response of the system
ξ= (s=-1±3i), for the system Δt=0.099, t∞=6.283
clc;clear;
t=0:0.099:6.283;
yt=0.3801*exp(-t).*cos(3*t-1.837)+0.1*exp(-2*t)
plot(t,yt)
Steady-state response is:
For stable systems, response returns to zero or reaches a finite value determined by the input amplitude.
STABILITY

4. p1=[1,3]; p2=[1,4,14,20,0]; [r,p,k]=residue(p1,p2)
Step Input Response:
Example 7.1 (Continued):
u(t): Step function 1
p1=[1,3]; p2=[1,4,14,20,0]; [r,p,k]=residue(p1,p2)

5. clc;clear;
t=0:0.099:6.283;
yt=0.1202*exp(-t).*cos(3*t )-0.05*exp(-2*t)+0.15;
plot(t,yt)

6. Final value theorem:
Example 7.1 (continued) Step input response