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Lec 6. Second Order Systems
• 2nd order systems
• Step response of standard 2nd order systems
• Performance specifications
• Reading: 5.3, 5.4
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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.
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Motivating Example
Poles:
Response of H(s) is the sum of the two 1st order system responses
Zeros:
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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)
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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:
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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:
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Example of 2nd Order Systems
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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:
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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
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Some Typical
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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
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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)
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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?
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Critically Damped Case (=1)
Transfer function
has two identical real poles
Step response is
steady state response transient response
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Overdamped Case (>1)
Transfer function has two distinct real poles
Step response is
steady state response transient response
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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
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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.
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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)
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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
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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
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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:
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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
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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”
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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
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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”
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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-