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 :



Impulse response :



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


Re(s)

Im(s)


Unstable

Re(s)

Im(s)


Unstable

Re(s)

Im(s)

-1


Unstable

Re(s)

Im(s)

-2


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 representation :

… damping ratio… natural frequency





Polar representation :

… damping ratio… natural frequency

Unless overdamped


… Overdamped

… Critically damped… Underdamped

… Undamped

Significance of the damping ratio :




Significance of the damping ratio : … Overdamped


… Undamped





… 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)



Unstable

Re(s)

Im(s)




Unstable

Re(s)

Im(s)


negative real partzero real part


Unstable

Re(s)

Im(s)

2 distinct real poles = Overdamped


Unstable

Re(s)

Im(s)

Repeated real poles = Critically damped


Unstable

Re(s)

Im(s)

Complex poles negative real part = Underdamped


Unstable

Re(s)

Im(s)

Complex poles zero real part = Undamped


Unstable

Re(s)

Im(s)

Und

ampe

d

Overdamped or Critically damped

Underdamped

Underdamped

Overdamped system response


Impulse response :

Step response :

Overdamped and critically damped system response.


Overdamped


Critically damped





… Undamped


… Cartesian overdamped





… Undamped


Overdamped case:

… Cartesian overdamped

Second order impulse response – Underdamped and Undamped

Impulse response :


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


Increasing / Fixed


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


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


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


Unstable

Faster response Slower response

Higher frequency oscillations

Lower frequency oscillations


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


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 .


… Steady state value.

More generally, if the numerator is not , but some :


… Peak time.

Therefore,

is the time of the occurrence of the first peak :


… Percent overshoot.

Evaluating at ,

is defined as:

Substituting our expressions for and :


… 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

Time Response*, ME451...First order system response System transfer function : Impulse response :...

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