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-