EG1110 SIGNALS AND SYSTEMS
Behaviour of Second order systems
• If second order system has no numerator dynamics (no zeros), standard form is
Second order systems
G(s) =
Overview
• “Second order” systems so called because equivalently
Y (s)
ωn2
= 2
U (s) s + 2ζωns + ωn2
• Behaviour of system dependent on two parameters:
1. ζ - damping ratio: stability, transient response
– Differential equation describing the system is of second order e.g.
2. ωn - undamped natural frequency - speed of response
ÿ(t) + 2ζωnẏ(t) + ωn2 y(t) = ωn2 u(t)
– System’s transfer function is of second order, e.g.
• The qualitative ‘shape’ of the response (in time and frequency domains) is dependent on the
damping ratio ζ.
Y (s)
ωn2
G(s) =
=
U (s) s2 + 2ζωns + ωn2
where ζ is damping ratio and ωn is undamped natural frequency
1
Time response - effect of damping ratio ζ
2
Response to initial conditions - free response
• Three main categories:
• For second order systems in standard form, can determine free response using formulae (in data
1. ζ > 1 - “Over damped system” - poles are real and unequal.
2. ζ = 1 - “Critically damped system” - poles are real and equal.
3. ζ < 1 - “Under damped system” - poles are both complex
• With these categories a particular type of behaviour is associated:
1. ζ > 1 - System behaves as product of two first order systems - one fast and one slow pole
components
2. ζ = 1 Fastest “no oscillation” response a 2nd order system can have.
3. ζ < 1 Some oscillatory behaviour present. Smaller ζ, greater oscillation.
3
book!)
• Assuming initial conditions y(0) = y0 and ẏ(0) = y00 , response can be calculates as
1. ζ < 1.
y(t) = A exp(−ζωnt) cos(ωt − Φ)
r
ω = ωn 1 − ζ 2


y00 + ζωny0 


Φ = tan−1 
ωy0
Y0
A =
cos Φ
4
2. ζ = 1
Qualitative behaviour - time domain
y(t) = (y0 + Bt) exp(−ωnt)
• Effect of damping ratio demonstrated by observing response of second order system to unit step
B = ωny0 + y00
(1)
input:
1.6
3. ζ > 1
1.4
s2 = −ωn[ζ + ζ 2 − 1]
s2y0 − y00
A1 =
s2 − s1
s1y0 − y00
A2 =
s1 − s2
• Note that in all cases the initial conditions
(y0, y00 ),
0.8
0.8
0.7
1
0.7
0.8
0.6
Response − normalised units
− 1]
r
Response − normalised units
Response − normalised units
s1 = −ωn[ζ −
ζ2
1
0.9
1.2
y(t) = A1 exp(s1t) + A2 exp(s2 t)
r
1
0.9
0.6
0.5
0.4
0.3
0.2
0
• Bode plots of second order system are similar except for the decades above and below the undamped natural frequency.
Bode Diagram
−20
0
−20
−20
−40
−40
−60
−40
0
−80
0
−80
0
−45
−45
−45
Phase (deg)
−90
−135
−180
−1
10
Phase (deg)
−60
−30
Phase (deg)
Bode Diagram
0
Magnitude (dB)
Magnitude (dB)
Magnitude (dB)
0
−90
−135
0
10
Frequency (rad/sec)
1
10
−180
−2
10
−90
−135
0
10
Frequency (rad/sec)
7
2
10
0.3
0
5
10
15
Time − seconds (underdamped)
20
0.2
0.1
0.1
0
0
5
10
15
Time − seconds (critically damped)
6
Qualitative behaviour - frequency domain
−10
0.4
0.2
the damping ratio (ζ) and undamped natural
5
Bode Diagram
0.5
0.4
frequency (ωn) completely determine the free response of the system.
10
0.6
−180
−2
10
0
10
Frequency (rad/sec)
2
10
20
0
0
5
10
15
Time − seconds (over damped)
20