1. Eigenvalues: p1=-4.526, p2,3=-0.4993±2.7883i, p4=-0.4753
2.1 Step Input Response:
Roots([ ])
Eigenvalues: p1=-4.526, p2,3= ±2.7883i, p4=
Steady-state response
Laplace transform of unit step input
Final value theorem:
Stability: If the step response reaches to a constant value, the system is said to be stable.
Steady State Error: ess=1-css ess=0
[r,p,k]=residue(nh,[dh,0])
r = i i
1.0000
z=r(3);a2=2*abs(z),fi2=angle(z)

2. If (Damping ratio) Different two roots (No oscillation)
Repeated roots
Complex roots
(Oscillatory response)
(Characteristic polynomial)
The form of the system response is determined by the value of
Critical damping coefficient
(Damping ratio)

3. For F(s)=
For a control system whose output is 1 to a unit step reference input, the prototype transfer function is defined as

4. Step Response of a Second Order System:
ωn : Undamped natural frequency
ξ: Damping Ratio OR
clc,clear
wn=1;ksi=0.2;
tp=2*pi/wn;dt=tp/20;ts=tp/ksi;
t=0:dt:ts;w=wn*sqrt(1-ksi^2);
a=wn/w;sigma=ksi*wn;fi=-acos(ksi)-pi/2;
c=a*exp(-sigma*t).*cos(w*t-fi)+1;
plot(t,c)

5. tr : Rise Time
td : Delay Time
ts : Settling time
cmax : Peak value, tmax: Peak time, cmax-css:Maksimum overshoot
css : Steady State Response, 1-css : Steady State Error Re
- σ  Im 

6. Design Criteria of Control Systems:
Stability (Control system must be stable)
Steady State Error ess=1-css → 0 (Steady-state error should be zero)
Sensitivity to a disturbance [css]d → 0 (Output from uncontrolled input should be zero)
Overshoot , typical value % 5, damping ratio ξ=0.7
Settling time : ts (Depends on the application)
Transfer Function of a PID Controller
Sensitivity to disturbance
At the first stage apply P control: observe the stability ess, [css]d
If necessary apply PI control, it eliminates/reduces the steady state error
If necessary apply PD kontrol, it reduces the overshoot
Apply PID control, it is useful for all criteria