Time Response*, ME451...First order system response System transfer function : Impulse response :...
Transcript of Time Response*, ME451...First order system response System transfer function : Impulse response :...
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
Re(s)
Im(s)
Theorems
• Initial Value Theorem
• Final Value Theorem– If all poles of sX(s) are in the left half plane (LHP), then
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
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
Pure integrator
• ODE :
• Impulse response :
• Step response :
• If the initial condition is not zero, then :
Physical meaning of the impulse response
First order system
• ODE :
• Impulse response :
• Step response :
• DC gain: (Use the final value theorem)
R
C
First order system• If the initial condition was not zero, then
Physical meaning of the impulse response
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
First order system response
System transfer function :
First order system response
System transfer function :
Impulse response :
First order system response
System transfer function :
Impulse response :
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
First order system response
Re(s)
Im(s)
First order system response
Unstable
Re(s)
Im(s)
First order system response
Unstable
Re(s)
Im(s)
-1
First order system response
Unstable
Re(s)
Im(s)
-2
First order system response
Unstable
Re(s)
Im(s)
faster response slower response
constant
First order system – Time specifications.
First order system – Time specifications.
Time specs:
Steady state value :
Time constant :
Rise time :
Settling time :
Time to go from to
First order system – Simple behavior.
No overshoot
No oscillations
Second order system (mass-spring-damper system)
• ODE :
• Transfer function :
Polar vs. Cartesian representations.
Cartesian representation :
… Imaginary part (frequency)
… Real part (rate of decay)
Polar vs. Cartesian representations.
Cartesian representation :
… Imaginary part (frequency)
… Real part (rate of decay)
Polar representation :
… damping ratio… natural frequency
Polar vs. Cartesian representations.
Cartesian representation :
… Imaginary part (frequency)
… Real part (rate of decay)
Polar representation :
… damping ratio… natural frequency
Polar vs. Cartesian representations.
Cartesian representation :
… Imaginary part (frequency)
… Real part (rate of decay)
Polar representation :
… damping ratio… natural frequency
Unless overdamped
Polar vs. Cartesian representations.
… Overdamped
… Critically damped… Underdamped
… Undamped
Significance of the damping ratio :
System transfer function :
Polar vs. Cartesian representations.
System transfer function :
Significance of the damping ratio : … Overdamped
… Critically damped… Underdamped
… Undamped
Polar vs. Cartesian representations.
System transfer function :
Significance of the damping ratio : … Overdamped
… Critically damped… Underdamped
… Undamped
Polar vs. Cartesian representations.
System transfer function :
Significance of the damping ratio : … Overdamped
… Critically damped… Underdamped
… Undamped
All 4 cases Unless overdamped
Underdamped second order system
• Underdamped
• Two complex poles:
Underdamped second order system
Impulse response of the second order system
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
Unit step response of undamped systems
• Unit step response :
• DC gain :
Unit step response of undamped system
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
Second order system response.
Stable 2nd order system:
2 distinct real polesA pair of repeated real polesA pair of complex poles
Unstable
Re(s)
Im(s)
Second order system response.
2 distinct real polesA pair of repeated real polesA pair of complex poles
Unstable
Re(s)
Im(s)
Stable 2nd order system:
Second order system response.
2 distinct real polesA pair of repeated real polesA pair of complex poles
Unstable
Re(s)
Im(s)
Stable 2nd order system:
Second order system response.
2 distinct real polesA pair of repeated real polesA pair of complex poles
Unstable
Re(s)
Im(s)
Stable 2nd order system:
negative real partzero real part
Second order system response.
2 distinct real polesA pair of repeated real polesA pair of complex poles
Unstable
Re(s)
Im(s)
Stable 2nd order system:
negative real partzero real part
Second order system response.
2 distinct real polesA pair of repeated real polesA pair of complex poles
Unstable
Re(s)
Im(s)
Stable 2nd order system:
negative real partzero real part
Second order system response.
Unstable
Re(s)
Im(s)
2 distinct real poles = Overdamped
Second order system response.
Unstable
Re(s)
Im(s)
Repeated real poles = Critically damped
Second order system response.
Unstable
Re(s)
Im(s)
Complex poles negative real part = Underdamped
Second order system response.
Unstable
Re(s)
Im(s)
Complex poles zero real part = Undamped
Second order system response.
Unstable
Re(s)
Im(s)
Und
ampe
d
Overdamped or Critically damped
Underdamped
Underdamped
Overdamped system response
System transfer function :
Impulse response :
Step response :
Overdamped and critically damped system response.
Overdamped and critically damped system response.
Overdamped
Overdamped and critically damped system response.
Overdamped
Overdamped and critically damped system response.
Critically damped
Polar vs. Cartesian representations.
Polar vs. Cartesian representations.
System transfer function :
Significance of the damping ratio : … Overdamped
… Critically damped… Underdamped
… Undamped
All 4 cases Unless overdamped
… Cartesian overdamped
Polar vs. Cartesian representations.
System transfer function :
Significance of the damping ratio : … Overdamped
… Critically damped… Underdamped
… Undamped
All 4 cases Unless overdamped
… Cartesian overdamped
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
Second order impulse response – Underdamped and Undamped
Impulse response :
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
0 0.5 1 1.5 2 2.5 3-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Impulse Response
Time (sec)
Ampl
itude
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
Impulse Response
Time (sec)
Ampl
itude
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
Impulse Response
Time (sec)
Ampl
itude
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
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
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
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
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
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
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
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
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
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
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
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
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
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
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
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
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
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
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
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
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
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
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
Second order impulse response – Underdamped and Undamped
Increasing / Fixed
Impulse Response
Time (sec)
Ampl
itude
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
Second order step response – Underdamped and Undamped
0 5−2
−1
0
1
2
3
0 5−2
−1
0
1
2
3
0 5−2
−1
0
1
2
3
0 5−2
−1
0
1
2
3
0 5−2
−1
0
1
2
3
0 5−2
−1
0
1
2
3
0 10 200
0.5
1
1.5
0 10 20−10
0
10
20
30
40
−2+j0.5zeta=0.97
−10+j0.5zeta=0.998 +j0.5
zeta=0
0.2+j0.5
0.2+j5+j5zeta=0
−2+j5zeta=0.3714
−10+j5zeta=0.8944
time sec.
output
time sec. time sec. time sec.
time sec. time sec. time sec. time sec.
output
Second order impulse response – Underdamped and Undamped
Unstable
Faster response Slower response
Higher frequency oscillations
Lower frequency oscillations
Second order impulse response – Underdamped and Undamped
Unstable
Less damping
More damping
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
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 .
Second order step response – Time specifications.
… Steady state value.
More generally, if the numerator is not , but some :
Second order step response – Time specifications.
… Peak time.
Therefore,
is the time of the occurrence of the first peak :
Second order step response – Time specifications.
… Percent overshoot.
Evaluating at ,
is defined as:
Substituting our expressions for and :
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
Typical specifications for second order systems.
How many independent parameters can we specify?
3