Lec 6. Second Order Systems
• 2nd order systems
• Step response of standard 2nd order systems
• Performance specifications
• Reading: 5.3, 5.4
2nd Order SystemsGeneral second order system transfer function:
Two poles p1,p2 of H(s) are the two roots of denominator polynomial:
The locations of p1 and p2 have important implication in system responses.
Motivating Example
Poles:
Response of H(s) is the sum of the two 1st order system responses
Zeros:
Motivating Example (cont.)
Step response:
Step Response
Time (sec)
Am
plit
ud
e
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Each pole p contributes a transient term
in the response
Transient will settle down if all poles are on the left half plane
Convergence to final value no longer monotone (overshoot)
Step Response
Time (sec)
Am
plit
ud
e
0 1 2 3 4 5 6 7 80
0.05
0.1
0.15
0.2
0.25
Another Motivating Example
Poles:
Step response of H(s) is:
Standard 2nd Order SystemsStandard form of second order systems:
Represents only a special family of second order systems• Numerator polynomial is a constant• Denominator polynomial is 2nd order with positive coefficients• H(0)=1 (unit DC gain)
Standard form is completely characterized by two parameters , n
• n: undamped natural frequency (n>0)• : damping ratio (>0)
Ex:
Example of 2nd Order Systems
Poles of Standard 2nd Order Systems
has two poles
Underdamped case (0<<1):
Two complex conjugate poles:
Critically damped case (=1):
Two identical real poles:
Overdamped case (>1):
Two distinct real poles:
Underdamped Case (0<<1)
has two complex poles
Damped natural frequency
Damping ratio determines the angle As increases from 0 to 1, changes from 0 to 90 degree
Undamped natural frequency n is the distance of poles to 0
Some Typical
Step Response: Underdampled Case
Step response of
steady state response transient response
Pole p=-+jd contributes the term ept in the transient response
Transient responses are damped oscillations with frequency d, whose amplitude decay (or grow) exponentially according to e- t
Step Response
Time (sec)
Am
plit
ud
e
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Step Responses: Underdamped Case(n is constant)
Summary of Underdamped Case• Overshoot and oscillation in the step response
• (Negative of) real part =n of the poles determines the transient amplitude decaying rate
• Imaginary part d of the poles determines the transient oscillation frequency
• For a given undamped natural frequency n, as damping ratio increases– larger, poles more to the left, hence transient
dies off faster– Transient oscillation frequency d decreases– Overshoot decreases
• What if we fix and increase n?
Critically Damped Case (=1)
Transfer function
has two identical real poles
Step response is
steady state response transient response
Overdamped Case (>1)
Transfer function has two distinct real poles
Step response is
steady state response transient response
Step Responses for Different Step Response
Time (sec)
Am
plit
ud
e
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Remarks• An overdamped system is sluggish in responding to inputs.
• Among the systems responding without oscillation, a critically damped system exhibits the fastest response.
• Underdamped systems with between 0.5 and 0.8 get close to the final value more rapidly than critically dampled or overdampled system, without incurring too large an overshoot
• Impulse response and ramp response of 2nd order systems can be obtained from the step responses by differentiation or integration.
Step Response
Time (sec)
Am
plit
ud
e
0 5 10 150
0.2
0.4
0.6
0.8
1
1.2
1.4
Time Specifications
• td: delay time, time for s(t) to reach half of s(1)
• tr: rise time, time for s(t) to first reach s(1)
• tp: peak time, time for s(t) to reach first peak
• Mp: maximum overshoot
• ts: settling time, time for s(t) to settle within a range (2% or 5%) of s(1)
A typical step response s(t)
Remarks
Step Response
Time (sec)
Am
plit
ud
e
0 10 20 30 40 50 600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Not all quantities are defined for certain step responses
Step response of a 1st order systemStep response of a 2nd order critically damped or overdamped system
Step Response
Time (sec)
Am
plit
ud
e
0 5 10 150
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Not defined: tr, tp, Mp
Time Specifications of 2nd Order Systems
• A standard 2nd order system
is completely specified by the parameters and n
• What are the time specifications in terms of and n?
• Focus on the underdamped case (0<<1) as tr, tp, Mp are not defined for critically damped or overdamped systems
Time Specifications of Underdamped Systems
Delay time td: smallest positive solution of equation
Rise time tr: smallest positive solution of equation
Recall that . Hence tr is smaller (faster rise) for larger n)
where
Underdamped Systems (0<<1)
Step response:
Step Response
Time (sec)
Am
plit
ud
e
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Step Responses for Fixed n and Different
For fixed n, rise time tr is smallest when =0, and approaches 1 as approaches 1
Peak time tp and Maximum Overshoot Mp
Peak time tp: smallest positive solution of equation
Maximum overshoot Mp:
(tp decreases with n)
“The smaller the damping ratio, the larger the maximum overshoot”
Step Response
Time (sec)
Am
plit
ud
e
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Step Responses for Fixed n and Different
For fixed n, peak time tp increases to 1 as increases from 0 to 1
Settling Time ts
Settling time ts: the smallest time ts such that |s(t)-1|< for all t>ts
Idea: approximate s(t) by its envelope:
Settling time ts when =5%:
Settling time ts when =2%:
Analytic expression of ts is difficult to obtain.
“The more to the left the poles are, the smaller the settling time”
Effect of Pole Locations on Responses of 2nd Order SystemsStep Response
Time (sec)
Am
plit
ud
e
0 5 10 150
0.2
0.4
0.6
0.8
1
1.2
1.4
Step Response
Time (sec)
Am
plit
ud
e
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Step Response
Time (sec)
Am
plit
ud
e
0 1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
1.2
1.4
Step Response
Time (sec)
Am
plitu
de
0 0.5 1 1.5 2 2.50
2
4
6
8
10
12
14
16
18
20
unstablestable
Step Response
Time (sec)
Am
plitu
de
0 5 10 150
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
equa-n
equa-