Transient Response

EES42042

Fundamental of Control Systems

Transient Response

Electrical Engineering Department University of Indonesia

Lecturer:

Dr. Wahidin Wahab M.Sc.

Aries Subiantoro, ST. MSc.

Transient Response - First 2

Order System

‹ Consider simple closed loop system with an integrator in feed-forward path

+ -

R(s) C(s)

Ts 1

1 1

+ Ts

R(s) C(s)

Transient Response - First 3

Order System

‹ Unit Step Response

1 1

+ Ts

R(s) C(s)

s T s

Ts T s s

C

s s Ts

s C s

R

1 1 1

1 ) 1

(

1 1 ) 1

( 1 ,

) (

− + + =

−

=

→

+ ⋅

=

=

Transient Response - First 4

Order System

‹ Unit Step Response

1 1

+ Ts

R(s) C(s)

T t

e t

c

s T s s

C

−

−

=

→

− +

=

1 )

(

1 1 ) 1

(

Transient Response - First 5

Order System

‹ Unit Step Response

1 1

+ Ts

R(s) C(s)

T t

e t

c ( ) = 1 −

⁻

1.0

Initial slope = 1/T

0.632

Transient Response - First 6

Order System

‹ Unit Step Response

1 1

+ Ts

R(s) C(s)

Time c(t)

T 0.632

2T 0.865

3T 0.95

4T 0.982

5T 0.993

Transient Response - First 7

Order System

‹ Unit Ramp Response

1 1

+ Ts

R(s) C(s)

1.0 t r(t)

1.0

2

) 1

( s s

R =

Transient Response - First 8

Order System

‹ Unit Ramp Response

1 1

+ Ts

R(s) C(s)

2 2

1 1

) 1 ( ) 1 (

s s Ts

C s s R

+ ⋅

=

→

=

Transient Response - First 9

Order System

‹ Unit Ramp Response

1 1

+ Ts

R(s) C(s)

T t

Te T

t t

c

Ts T s

T s s

C

s s Ts

C

+

−

−

=

∴

+ +

−

=

→

+ ⋅

=

) (

1 ) 1

(

1 1

) 1 (

2 2

2

Transient Response - First 10

Order System

‹ Unit Ramp Response

1 1

+ Ts

R(s) C(s)

{ } ^{e t T}

e T

t c t

t c t

r t

e

T t

=

=

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛ −

=

−

=

−

=

∞

→

−

) ( lim

Error Tracking

State Steady

1 )

(

) ( )

( )

( as error Define

input ramp

g in trackin Error

t

Transient Response - First 11

Order System

‹ Unit Ramp Response

1 1

+ Ts

R(s) C(s)

{ }

system.

the of

constant time

by the determined

offset small

a be always will

There ) ( lim

Error Tracking

State Steady

t

→

=

=

→∞

e t T

Transient Response - First 12

Order System

‹ Unit Ramp Response

1 1

+ Ts

R(s) C(s)

Steady State error = T

r(t)

c(t)

Transient Response - First 13

Order System

‹ Unit Impulse Response

1 1

+ Ts

R(s) C(s)

Note that the unit impulse response of an LTI system is

simply the inverse Laplace Transform of the transfer

function

Transient Response - First 14

Order System

‹ Unit Impulse Response

1 1

+ Ts

R(s) C(s)

T t

T e L Ts

t

c

⁻

=

⁻

⎭ ⎬

⎫

⎩ ⎨

⎧

= + 1

1 ) 1

(

¹

Transient Response - First 15

Order System

‹ Unit step response = derivative of unit ramp response

‹ Unit impulse response = derivative of unit

step response

16

Example from Ogata

‹ Liquid level controller - p140 Ogata

+ -

R(s) H(s)

K

_p

K

_v

+ 1 RCs

R

K

_b

17

Example from Ogata

‹ Upon block reduction

+ -

R(s) H(s)

+ 1 Ts 1/K

_b

K

X(s)

RC T

RK K

K

K

_{p v b}

=

=

18

Example from Ogata

‹ Consider response to step change in reference input r(t)

+ -

R(s) H(s)

+ 1 Ts 1/K

_b

K

X(s)

Note that this is equivalent to step change in x(t)

19

Example from Ogata

T s K

K K s

K K

s K Ts

s K H

K Ts

K s

X s H

+ + + ⋅

− + ⋅

=

+ ⋅

= +

+

= +

1 1 1

1 1

fractions partial

into Splitting

1 ) 1

(

response Step

1 )

(

)

(

20

Example from Ogata

K T T

t K e

t K h

T s K

K K s

K s K

H

T t

= +

⎟ ≥

⎟

⎠

⎞

⎜ ⎜

⎝

⎛ −

= +

→

+ + + ⋅

− + ⋅

=

−

1

0 ,

1 1 )

(

sides both

of LT inverse

the taking

1 1 1

1 ) 1

(

1

1

21

Example from Ogata

( ) K

e K K

h K

t K e

t K h

T t

t

T t

= +

⎪⎭

⎪ ⎬

⎫

⎪⎩

⎪ ⎨

⎧

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛ −

= +

∞

⎟ ≥

⎟

⎠

⎞

⎜ ⎜

⎝

⎛ −

= +

−

∞

→

−

1 1 lim 1

value state

Steady

0 ,

1 1 )

(

1 1

Note that this could also be found from the Final Value Theorem

of Laplace Transforms

22

Example from Ogata

( ) { }

K K

s K

Ts s K

s H s

h

s

s

= +

⎭ ⎬

⎫

⎩ ⎨

⎧ ⋅

+

⋅ +

=

⋅

=

∞

→

→

1

1 lim 1

) ( lim

theorem value

final the

using

0

0

23

Example from Ogata

( )

K K

K x

= +

− +

=

=

∞

1 1 1 1

offset or

error state

Steady

0 . 1 Since

1.0

Note that this offset could be eliminated by placing an

integrator (K/s) in the feed-forward path

24

Second Order Systems

‹ General form of 2nd order transfer function

C s

R s s s

n

n n

( ) ( ) =

+ +

=

=

ω

ςω ω ς

ω

2

2 2

2

damping ratio of system

undamped natural frequency

n

Time domain performance of a 2nd order system

linked to these two quantities

25

Effect of Damping Ratio

0 1

1

2 1

1 2

2 2

2

< < −

→

= ⋅

+ +

⎧ ⎨

⎩

⎫ ⎬

⎭

= − =

−

ς

ω

ςω ω

ω ω ς

UNDERDAMPED CASE poles are complex conjugate

damped oscillatory response Response to Unit Step input

Define damped natural frequency c t L

s s s

n

n n

d n

( )

26

Effect of Damping Ratio

→ = − +

−

⎛

⎝ ⎜⎜ ⎞

⎠ ⎟⎟

= − − ⎛ + −

⎝ ⎜⎜ ⎞

⎠ ⎟⎟

= +

−

⎛

⎝ ⎜⎜ ⎞

⎠ ⎟⎟

−

− −

−

c t e t t

e t

c t

e t t

n

n

n

t

d d

t

d

t

d d

( ) cos sin

sin tan ,

( )

cos sin

1

1 1

1

1

1

2

2

1 2

2 ςω

ςω

ςω

ω ς

ς ω

ς ω ς

ς

ω ς

ς ω

Error signal = 1 -

27

Effect of Damping Ratio

‹ Note that if the damping ratio is zero system will undergo sustained undamped

oscillations

ς = → 0 c t ( ) = − 1 cos ω

_n

t

Note also that if the damping ratio >1 then

response is OVERDAMPED and system will not

oscillate

28

Critical Damping

( )

( )

Critical damping = 1

System poles are equal and real L.T. of unit step response is

→

→

→

= +

→ = −

⁻

+ ≥

ς

ω ω

ω

ω C s

s s

c t e t t

n n

t

n n

( )

( ) ,

2 2

1 1 0

29

Overdamped Case

( )( )

( ) ( )

OVERDAMPED > 1 poles are real and distinct L.T. of unit step response

Poles:

→

→

→

= + + − + − −

= + − = − −

ς

ω

ςω ω ς ςω ω ς

ς ς ω ς ς ω

C s

s s s

s s

n

n n n n

n n

( )

,

2

2 2

1

2

2

2

1 1

1 1

30

Overdamped Case

( )

( )

c t e

e

n

n

t

t

( ) = +

− + −

− − − −

− +⎛ −

⎝⎜ ⎞

⎠⎟

− −⎛ −

⎝⎜ ⎞

⎠⎟

1 1

2 1 1

1

2 1 1

2 2

1

2 2

1

2

2

ς ς ς

ς ς ς

ς ς ω

ς ς ω

Step response = sum of two decaying exponential functions

31

Overdamped Case

( )

( )

c t e

e

n

n

t

t

( ) = +

− + −

− − − −

− +⎛ −

⎝⎜ ⎞

⎠⎟

− −⎛ −

⎝⎜ ⎞

⎠⎟

1 1

2 1 1

1

2 1 1

2 2

1

2 2

1

2

2

ς ς ς

ς ς ς

ς ς ω

ς ς ω

When faster decaying exponential has died out response

is similar to 1st order system

Performance of Second Order 32

Systems

0 2 4 6 8 10 12 14

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Unit Step Response Curves of Second Order System ω_n=1.0 rad/sec

c(t)

Underdamped Critically damped Overdamped

ζ = 2.0 ζ = 1.0 ζ = 0.9 ζ = 0

ζ = 0.2 ζ = 0.4

ζ = 0.7

ζ = 0.8

Performance of Second Order 33

Systems

‹ Underdamped systems respond fastest but will oscillate about steady state value

‹ Overdamped systems tend to be sluggish but lack oscillation.

‹ Critically damped systems give the quickest

rise time without overshoot.

Transient Response 34

Specifications

‹ Means of specifying the desired performance of a control system

‹ Frequently specified in terms of response to

unit step input

Transient Response 35

Specifications

‹ Terms

– Delay Time, t

_d

– Rise Time, t

_r

– Peak time, t

_p

– Maximum Overshoot, M

_p

– Settling Time, t

_s

Transient Response 36

Specifications

‹ Delay Time

– Time to reach 50% of final value for the first time

1.0 0.5

Example - Unit Step Response

Transient Response 37

Specifications

‹ Rise Time

– Time for system to go from 0-100% of final value or 10-90%, or 5-95%

1.0

Example - Unit Step Response

Transient Response 38

Specifications

‹ Peak Time

– Time for system to reach first peak

1.0

Example - Unit Step Response

Transient Response 39

Specifications

‹ Maximum Percent Overshoot

– Maximum percent value of response from steady state value

1.0

Example - Unit Step Response

M

_p

Transient Response 40

Specifications

‹ Maximum Percent Overshoot

1.0

Example - Unit Step Response M

_p

( ) ^{( )}

( ) ^{− ∞ ∞ × 100}

= c

c t

M

_p

c

^p

Transient Response 41

Specifications

‹ Settling Time

– Time for response to reach and stay within a certain range of the steady state value -

typically 5% or 2%

1.0

Example - Unit Step Response

Allowable

tolerance

Transient Response 42

Specifications

‹ Desirable Response

– fast and well damped

– damping ratio should be between 0.4 and 0.8 – too small a damping ratio leads to a poorly

damped response – too much overshoot

– too large a damping ratio leads to a sluggish response

Conflict between max. rise time and allowable overshoot

Second Order Systems and 43

Transient Response Specs.

‹ Second Order system pole locations

2 2

1

1 ,

,

Poles

ς ω

ω ςω

σ

ω σ

−

=

−

=

±

=

n d

n

j

d

s

s

Second Order Systems and 44

Transient Response Specs.

‹ Second Order system pole locations

2 2

1

1 ,

, Poles

ς ω

ω ςω

σ

ω σ

−

=

−

=

±

=

n d

n

j

d

s s j ω

σ x

x -σ=−ζω

_n

ω

_n

ω

_d

β

β ς cos

Note =

s

₁

s

₂

=s

^*₁

Second Order Systems and 45

Transient Response Specs.

‹ Rise Time

large.

be

must time

rise small

a for that

Note

1 tan

₁

d d

d d

t

r

ω ω

β π

σ ω ω

= −

⎟ ⎠

⎜ ⎞

⎝

⎛

=

⁻

−

Second Order Systems and 46

Transient Response Specs.

‹ Peak Time

proof.

of details for

153 -

152 pp.

Ogata See

d

t

p

ω

= π

Second Order Systems and 47

Transient Response Specs.

‹ Maximum Overshoot

proof.

of details for

153 p.

Ogata See

% 100

1 2

×

=

^⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

− −

ς π ς

e M

_p

Note that this ONLY depends on damping ratio ζ

Second Order Systems and 48

Transient Response Specs.

‹ Settling Time

s

n

t ςω

ςω

%) 3 5

(

value final

its of

5%

within be

will system

constants time

3 After

constant 1 time

of value

on depends

response of

decay of

Speed

=

∴

Second Order Systems and 49

Transient Response Specs.

‹ Settling Time

n

t

s

ςω

%) 4 2

(

value final

its of

2%

within be

will system

constants time

4 After

=

∴ These Settling Time formulae are very

important to remember as settling time

is a very common control system spec.

Second Order Systems and 50

Transient Response Specs.

‹ Settling Time

– inversely proportional to product of damping ratio with natural frequency

– damping ratio is usually set by max. overshoot – therefore settling time determined by choosing

ω

_n

once damping ratio has been set.

51

Example

Consider the closed loop system shown below. Determine the values of K

_h

& K

such that the max. overshoot to a step input is 0.2 and the peak time is 1 second. Also determine the rise time and settling time.

Assume J=1, B=1

52

Example

R(s) +

- +

- _{Js B}

K

+ s

1

C(s)

K

h

Need to reduce this to a single loop

53

Example

R(s) +

- ^s ( ^{Js + B K + KK}

_h

) _C(s)

Reduced system

( )

J K KJ

KK B

K s

KK B

Js

K R(s)

C(s)

n h

s

+ =

=

→

+ +

= +

ω

ς ,

2

T.F.

loop Closed

2

54

Example

456 .

0

61 . 1 1

2 . 0 Overshoot

Max

2 1 ²

=

→

⎟ =

⎟

⎠

⎞

⎜ ⎜

⎝

⎛

→ −

=

=

^⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

− −

ς

ς π ς

ς π ς

e

M

_p

55

Example

53 . 3 1

14 . 3

1 Time Peak

2

=

= −

→

=

→

=

=

ς ω ω

ω ω

π

d n

d d

t

p

56

Example

178 .

0 456

. 2 0

5 . 12 53

. 3

is frequency natural

Now

=

→ + =

=

=

→

=

=

h h

n

KJ K KK B

J K K

ς

ω

57

Example

65 . 0

10 . 1 tan

, Time Rise

1

=

→

⎟ =

⎠

⎜ ⎞

⎝

= ⎛

= −

⁻

r

d d

r

t

t σ

β ω ω

β

π

58

Example

sec 86

. 3 1

time settling

5%

sec 48

. 4 2

time settling

2%

Time Settling

=

=

=

=

σ

σ

Impulse Response of Second 59

Order System

‹ Simply inverse Laplace transform of transfer function

‹ or derivative of unit step response

‹ different expressions for max. overshoot

‹ see Ogata pp. 158-160

60

Significance of Pole Locations

‹ Real and negative

– impulse response is a stable exponential decay

‹ Real and positive

– impulse response is an unstable exponential rise

‹ Complex with negative real parts

– impulse response is oscillatory and stable

» damped oscillation

61

Significance of Pole Locations

‹ Complex with positive real parts

– impulse response is oscillatory and unstable

» undamped oscillation

‹ angle of complex pole w.r.t. negative real axis = cosine of damping ratio

‹ magnitude of complex pole = natural frequency

Note that complex poles ALWAYS occur in conjugate pairs

62

Significance of Pole Locations

63

Higher Order Systems

‹ Consider general transfer function of the form

( )

( ) ( )

∏ ∏

∏

= =

=

+ +

+

+

=

_q

j

r

k

k k

k j

M

i

i

s s

p s

z s

K s

G

1 1

2 2

1

2 )

(

ω ω

ς

M zeros, at least q real poles, & at most r pairs of complex

conjugate poles

64

Higher Order Systems

‹ The response to a unit step input is determined as

( )

( ) ( )

fractions.

partial of

method by the

expanded be

can then This

2 ) 1

(

1 1

2 2

1

∏ ∏

∏

= =

=

+ +

+

+

⋅

=

_q

j

r

k

k k

k j

M

i

i

s s

p s

z s

K s s

C

ω ω

ς

65

Higher Order Systems

( )

responses order

second and

first of

set a

of

sum system

order higher

of Response

2

1 fractions

partial by

Expanding

1

2 2

2

1

=

→

+ +

− +

+ + + +

= ∑ ∑

=

=

r

k k k k

k k

k k

k k

q

j j

j

s s

c s

b p

s a s

C(s) a

ω ω

ς

ς ω

ω ς

Residues

66

Higher Order Systems

( ) ( )

( )

∑

∑

∑

=

−

=

−

=

−

− +

− +

+

=

r

k

k k

t k

r

k

k k

t k

q

j

t p j

t e

c

t e

b e

a a

t c

k k

k j k

1

2 1

2 1

1 sin

1 cos

then is

response domain

Time

ς ω

ς ω

ω ς

ω ς

Relative size of each component determined by system residues

which relate to system zeros.

67

Higher Order Systems

‹ Note that components of c(t)

corresponding to poles with large negative real parts will decay rapidly

– shorter settling times

‹ More dominant poles are those with longer settling times

– closer to the imaginary axis in the s-plane

Additional Pole in 2

Additional Pole in 2 ^{nd nd} Order Order Model

Model

Step Response Change Due to Step Response Change Due to

Extra Zero

Extra Zero

70

Dominant Poles

‹ Relative dominance of pole determined by ratio of real parts

‹ If ratio of real parts is greater than 5 & no

zeros are close by then the closed loop poles

closest to the imaginary axis will dominate

the system response.

71

Dominant Poles

‹ Useful for getting an approximate idea of system behaviour.

‹ Make sure of underlying assumptions

before using.

Transfer Functions in 72

MATLAB

‹ Transfer functions may be implemented in Matlab using the “tf” command

s polynomial r

denominato and

numerator the

of ts coefficien the

containing arrays

are and

where )

, (

syntax has

tf.m

den num

den num

tf

sys =

Transfer Functions in 73

MATLAB

);

, (

];

10 1

[ ];

1 [

Matlab in

write could

10 we ) 1

(

function transfer

the implement to

example For

den num

tf G

den num

s s G

=

=

=

= +

Transfer Functions in 74

MATLAB

);

, (

];

10 2

1 [

];

5 1 [

Matlab in

write could

10 we 2

) 5 (

function transfer

the implement Or to

2

den num

tf G

den num

s s

s s G

=

=

=

+ +

= +

In this way Matlab has the transfer function stored as variable G

which it can use in the control design tools you are going to use.

75

A Useful Trick

( ^{form 2} )( ^{for example 3} ) ^{5 6 but might be given as}

expanded in the

ly convenient be

not given will

are you

polynomial r

denominato or

numerator the

Sometimes

2

+ +

+ +

s s

s s

Although in this instance it is should be an easy matter to

expand it may be time consuming when you have many 1

^st

and

second order factors

76

A Useful Trick

( )( )

);

2 , 1 (

];

3 1 [ 2

];

2 1 [ 1

Matlab in

type out,

this expand

to

algebra he

through t going

to e alternativ quick

a As

3 2

given are

we Suppose

a a conv a

a a

s s

=

=

=

+ +

The resulting array “a” contains the expanded form of

(s+2)(s+3) – By the way the “conv” stands for convolution

and the operation is in fact a discrete convolution.

77

Parameter Sensitivity

‹ System parameters may be only measured approximately or may vary over time

‹ Sensitivity factors have been derived to

indicate how sensitive a system transfer

function is to changes in parameters

78

Pole-Zero Cancellation

‹ Sometimes it may be possible to cancel the effect of an unwanted pole by placing a zero over the top of it.

‹ Provided the location of the zero is known exactly this is not a problem.

‹ Generally a dangerous procedure if system parameters are uncertain or may change

with time

79

Pole-Zero Cancellation

‹ This is especially true if pole top be cancelled is unstable

‹ Example

1 1

− s Unstable pole

1 1

−

s 1

1 +

− s s

Unstable pole cancelled by zero in this block

1 1 1

1 1

) 1

( = +

+

⋅ −

= −

s s

s s s

G

_equiv

Homeworks Homeworks

Nise chapter 4: 26, 29, 54, 57, 58

81

Next Lecture

‹ System Stability

‹ Routh Hurwitz criterion

– Up to slide 38 of lecture 3 from last year