time response*, me451
DESCRIPTION
Time Response*, ME451. Instructor: Jongeun Choi. * This presentation is created by Jongeun Choi and Gabrial Gomes. Zeros and poles of a transfer function. Let G(s)=N(s)/D(s), then Zeros of G(s) are the roots of N(s)=0 Poles of G(s) are the roots of D(s)=0. Im(s). Re(s). Theorems. - PowerPoint PPT PresentationTRANSCRIPT
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Time Response*, ME451
Instructor: Jongeun Choi
* This presentation is created by Jongeun Choi and Gabrial Gomes
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Zeros and poles of a transfer function
• Let G(s)=N(s)/D(s), then– Zeros of G(s) are the roots of N(s)=0– Poles of G(s) are the roots of D(s)=0
Re(s)
Im(s)
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Theorems
• Initial Value Theorem
• Final Value Theorem– If all poles of sX(s) are in the left half plane
(LHP), then
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DC gain or static gain of a stable system
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
1.2
1.4
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DC Gain of a stable transfer function
• DC gain (static gain) : the ratio of the steady state output of a system to its constant input, i.e., steady state of the unit step response
• Use final value theorem to compute the steady state of the unit step response
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Pure integrator
• ODE :
• Impulse response :
• Step response :
• If the initial condition is not zero, then :
Physical meaning of the impulse response
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First order system
• ODE :
• Impulse response :
• Step response :
• DC gain: (Use the final value theorem)
R
C
+ K
u(t) -
+ y(t) -
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First order system• If the initial condition was not zero, then
Physical meaning of the impulse response
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Matlab Simulation
• G=tf([0 5],[1 2]); • impulse(G)
• step(G)
• Time constant
0 0.5 1 1.5 2 2.5 30
0.51
1.52
2.53
3.54
4.55
Impulse Response
Time (sec)
Am
plitu
de
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5Step Response
Time (sec)
Am
plitu
de
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First order system response
System transfer function :
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First order system response
System transfer function :
Impulse response :
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First order system response
System transfer function :
Impulse response :
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First order system response
System transfer function :
Impulse response :
Step response :
0 100 200 300 400 500 6000
10
20
30
40
50
60
70
80
90
100Step Response
Time (sec)
Am
plitu
de
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First order system response
Re(s)
Im(s)
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First order system response
Unstable
Re(s)
Im(s)
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First order system response
Unstable
Re(s)
Im(s)
-1
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First order system response
Unstable
Re(s)
Im(s)
-2
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First order system response
Unstable
Re(s)
Im(s)
faster response slower response
constant
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First order system – Time specifications.
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First order system – Time specifications.
Time specs:
Steady state value :
Time constant :
Rise time :
Settling time :
Time to go from to
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First order system – Simple behavior.
No overshoot
No oscillations
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Second order system (mass-spring-damper system)
• ODE :
• Transfer function :
k b
y(t)
F(t)=ku(t)
m
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Polar vs. Cartesian representations.
Cartesian representation :
… Imaginary part (frequency) … Real part (rate of decay)
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Polar vs. Cartesian representations.
Cartesian representation :
… Imaginary part (frequency) … Real part (rate of decay)
Polar representation :
… damping ratio … natural frequency
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Polar vs. Cartesian representations.
Cartesian representation :
… Imaginary part (frequency) … Real part (rate of decay)
Polar representation :
… damping ratio … natural frequency
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Polar vs. Cartesian representations.
Cartesian representation :
… Imaginary part (frequency) … Real part (rate of decay)
Polar representation :
… damping ratio … natural frequency
Unless overdamped
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Polar vs. Cartesian representations.
… Overdamped … Critically damped
… Underdamped
… Undamped
Significance of the damping ratio :
System transfer function :
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Polar vs. Cartesian representations.
System transfer function :
Significance of the damping ratio :
… Overdamped … Critically damped
… Underdamped
… Undamped
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Polar vs. Cartesian representations.
System transfer function :
Significance of the damping ratio :
… Overdamped … Critically damped
… Underdamped
… Undamped
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Polar vs. Cartesian representations.
System transfer function :
Significance of the damping ratio :
… Overdamped … Critically damped
… Underdamped
… Undamped
All 4 cases Unless overdamped
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Underdamped second order system
• Underdamped
• Two complex poles:
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Underdamped second order system
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Impulse response of the second order system
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Matlab Simulation
• zeta = 0.3; wn=1;• G=tf([wn],[1 2*zeta*wn wn^2]);• impulse(G)
0 2 4 6 8 10 12 14 16 18 20-0.3-0.2-0.1
00.10.20.30.40.50.60.7
Impulse Response
Time (sec)
Am
plitu
de
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Unit step response of undamped systems
• Unit step response :
• DC gain :
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Unit step response of undamped system
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Matlab Simulation
• zeta = 0.3; wn=1; G=tf([wn],[1 2*zeta*wn wn^2]);
• step(G)
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
1.2
1.4Step Response
Time (sec)
Am
plitu
de
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Second order system response.
Stable 2nd order system:
2 distinct real poles
A pair of repeated real poles
A pair of complex poles
Unstable
Re(s)
Im(s)
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Second order system response.
2 distinct real poles
A pair of repeated real poles
A pair of complex poles
Unstable
Re(s)
Im(s)
Stable 2nd order system:
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Second order system response.
2 distinct real poles
A pair of repeated real poles
A pair of complex poles
Unstable
Re(s)
Im(s)
Stable 2nd order system:
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Second order system response.
2 distinct real poles
A pair of repeated real poles
A pair of complex poles
Unstable
Re(s)
Im(s)
Stable 2nd order system:
negative real part
zero real part
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Second order system response.
2 distinct real poles
A pair of repeated real poles
A pair of complex poles
Unstable
Re(s)
Im(s)
Stable 2nd order system:
negative real part
zero real part
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Second order system response.
2 distinct real poles
A pair of repeated real poles
A pair of complex poles
Unstable
Re(s)
Im(s)
Stable 2nd order system:
negative real part
zero real part
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Second order system response.
Unstable
Re(s)
Im(s)
2 distinct real poles = Overdamped
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Second order system response.
Unstable
Re(s)
Im(s)
Repeated real poles = Critically damped
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Second order system response.
Unstable
Re(s)
Im(s)
Complex poles negative real part = Underdamped
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Second order system response.
Unstable
Re(s)
Im(s)
Complex poles zero real part = Undamped
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Second order system response.
Unstable
Re(s)
Im(s)
Un
dam
ped
Overdamped or Critically damped
Underdamped
Underdamped
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Overdamped system response
System transfer function :
Impulse response :
Step response :
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Overdamped and critically damped system response.
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Overdamped and critically damped system response.
Overdamped
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Overdamped and critically damped system response.
Overdamped
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Overdamped and critically damped system response.
Critically damped
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Polar vs. Cartesian representations.
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Polar vs. Cartesian representations.
System transfer function :
Significance of the damping ratio :
… Overdamped … Critically damped
… Underdamped
… Undamped
All 4 cases Unless overdamped
… Cartesian overdamped
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Polar vs. Cartesian representations.
System transfer function :
Significance of the damping ratio :
… Overdamped … Critically damped
… Underdamped
… Undamped
All 4 cases Unless overdamped
… Cartesian overdamped
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Polar vs. Cartesian representations.
System transfer function :
Significance of the damping ratio :
… Overdamped … Critically damped
… Underdamped
… Undamped
All 4 cases Unless overdamped
Overdamped case:
… Cartesian overdamped
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Second order impulse response – Underdamped and Undamped
Impulse response :
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Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Am
plitu
de
0 0.5 1 1.5 2 2.5 3-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
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Impulse Response
Time (sec)
Am
plitu
de
0 0.5 1 1.5 2 2.5 3-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
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Impulse Response
Time (sec)
Am
plitu
de
0 0.5 1 1.5 2 2.5 3-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
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Impulse Response
Time (sec)
Am
plitu
de
0 0.5 1 1.5 2 2.5 3-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
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Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Am
plitu
de
0 0.5 1 1.5 2 2.5 3-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
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Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Am
plitu
de
0 0.5 1 1.5 2 2.5 3-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
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Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Am
plitu
de
0 0.5 1 1.5 2 2.5 3-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
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Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Am
plitu
de
0 0.5 1 1.5 2 2.5 3-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
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Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Am
plitu
de
0 0.5 1 1.5 2 2.5 3 3.5 4-1
0
1
2
3
4
5
6
-6 -4 -2 0 2-10
-5
0
5
10
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Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Am
plitu
de
0 0.5 1 1.5 2 2.5 3 3.5 4-1
0
1
2
3
4
5
6
-6 -4 -2 0 2-10
-5
0
5
10
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Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Am
plitu
de
0 0.5 1 1.5 2 2.5 3 3.5 4-1
0
1
2
3
4
5
6
-6 -4 -2 0 2-10
-5
0
5
10
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Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Am
plitu
de
0 0.5 1 1.5 2 2.5 3 3.5 4-1
0
1
2
3
4
5
6
-6 -4 -2 0 2-10
-5
0
5
10
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Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Am
plitu
de
0 2 4 6 8 10 12-4
-3
-2
-1
0
1
2
3
4
5
-5 0 5-6
-4
-2
0
2
4
6
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Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Am
plitu
de
0 2 4 6 8 10 12-4
-3
-2
-1
0
1
2
3
4
5
-5 0 5-6
-4
-2
0
2
4
6
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Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Am
plitu
de
0 2 4 6 8 10 12-4
-3
-2
-1
0
1
2
3
4
5
-5 0 5-6
-4
-2
0
2
4
6
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Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Am
plitu
de
0 2 4 6 8 10 12-4
-3
-2
-1
0
1
2
3
4
5
-5 0 5-6
-4
-2
0
2
4
6
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Second order step response – Underdamped and Undamped
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Second order impulse response – Underdamped and Undamped
Unstable
Faster response Slower response
Higher frequency oscillations
Lower frequency oscillations
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Second order impulse response – Underdamped and Undamped
Unstable
Less damping
More damping
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Second order step response – Time specifications.
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
1.2
1.4
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Second order step response – Time specifications.
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
1.2
1.4
… Steady state value.
… Time to reach first peak (undamped or underdamped only).
… % of in excess of .
… Time to reach and stay within 2% of .
… Time to rise from 10% to 90% of .
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Second order step response – Time specifications.
… Steady state value.
More generally, if the numerator is not , but some :
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Second order step response – Time specifications.
… Peak time.
Therefore,
is the time of the occurrence of the first peak :
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Second order step response – Time specifications.
… Percent overshoot.
Evaluating at ,
is defined as:
Substituting our expressions for and :
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Second order step response – Time specifications.
… Settling time.
Defining with , the previous expression for can be re-written as:
As an approximation, we find the time it takes for the exponential envelope to reach 2% of .
when
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Typical specifications for second order systems.
How many independent parameters can we specify?
3