time responce analysis

Closed Loop Control System

C(s)= G(s).E(s)

B(s)= C(s).H(s)

E(s) = R(s)-B(s)

B(s)=G(s).E(s).H(s)

B(s)/E(s) = G(s).H(s) Open loop T.F.

C(s)= G(s).E(s)

=G(s)[R(s)-B(s)]

=G(s)R(s)-G(s)B(s)

= G(s)R(s)-G(s) .C(s).H(s)

C(s)[1+G(s)H(s)]=G(s)R(s)

C(s)/R(s)=G(s)/[1+G(s)H(s)] Closed Loop T.F.(with –ve feedback)

C(s)/R(s)=G(s)/[1-G(s)H(s)] Closed Loop T.F.(with +ve feedback)

Step Response of underdamped System

222222 2

21

nnnn

n

ss

s

ssC

)(

• The partial fraction expansion of above equation is given as

22 2

21

nn

n

ss

s

ssC

)(

2ns

22 1 n

2221

21

nn

n

s

s

ssC )(

22

2

2 nn

n

sssR

sC

)(

)(

22

2

2 nn

n

ssssC

)(

Step Response


• Above equation can be written as

2221

21

nn

n

s

s

ssC )(

22

21

dn

n

s

s

ssC

)(

21 nd• Where , is the frequency of transient oscillations and is called damped natural frequency.

• The inverse Laplace transform of above equation can be obtained easily if C(s) is written in the following form:

2222

1

dn

n

dn

n

ss

s

ssC

)(


2222

1

dn

n

dn

n

ss

s

ssC

)(

22

2

2

22

111

dn

n

dn

n

ss

s

ssC

)(

222221

1

dn

d

dn

n

ss

s

ssC

)(

tetetc dt

dt nn

sincos)(

21

1


tetetc dt

dt nn

sincos)(

21

1

ttetc ddtn

sincos)(21

1

n

nd

21

• When 0

ttc ncos)( 1


ttetc ddtn

sincos)(21

1

sec/. radn and if 310

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


ttetc ddtn

sincos)(21

1


0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4


0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

b=0

b=0.2

b=0.4

b=0.6

b=0.9


0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

wn=0.5

wn=1

wn=1.5

wn=2

wn=2.5

Time-Domain Specification

30

For 0< <1 and ωn > 0, the 2nd order system’s response due to a unit step input looks like


31

• The delay (td) time is the time required for the response to reach half the final value the very first time.


32

• The rise time is the time required for the response to rise from 10% to 90%, 5% to 95%, or 0% to 100% of its final value.

• For underdamped second order systems, the 0% to 100% rise time is normally used. For overdamped systems, the 10% to 90% rise time is commonly used.

32


33

• The peak time is the time required for the response to reach the first peak of the overshoot.

33 33


34

The maximum overshoot is the maximum peak value of the response curve measured from unity. If the final steady-state value of the response differs from unity, then it is common to use the maximum percent overshoot. It is defined by The amount of the maximum (percent) overshoot directly indicates the relative stability of the system.


35

• The settling time is the time required for the response curve to reach and stay within a range about the final value of size specified by absolute percentage of the final value (usually 2% or 5%).

Time Domain Specifications of Underdamped system

Time Domain Specifications (Rise Time)

ttetc ddtn

sincos)(21

1

equation above in Put rtt

rdrdt

r ttetc rn

sincos)(21

1

1)c(tr Where

rdrdt

tte rn

sincos21

0

0 rnte

rdrd tt

sincos

21

0


as writen-re be can equation above

0

1 2

rdrd tt

sincos

rdrd tt

cossin

21

21rd ttan

21 1

tanrd t


21 1

tanrd t

n

n

drt

21 11

tan

drt

b

a1 tan

Time Domain Specifications (Peak Time)

ttetc ddtn

sincos)(21

1

• In order to find peak time let us differentiate above equation w.r.t t.

ttettedt

tdcd

ddd

tdd

tn

nn

cossinsincos)(

22 11

tttte dd

dddn

dntn

cossinsincos22

2

11

0

tttte dn

dddn

dntn

cossinsincos2

2

2

2

1

1

1

0


tttte dn

dddn

dntn

cossinsincos2

2

2

2

1

1

1

0

0

1 2

2

tte ddd

ntn

sinsin

0 tne 0

1 2

2

tt dddn

sinsin

0

1 2

2

d

nd t

sin


0

1 2

2

d

nd t

sin

0

1 2

2

d

n

0tdsin

01 sintd

d

t

,,, 20

• Since for underdamped stable systems first peak is maximum peak therefore,

dpt

Time Domain Specifications (Maximum Overshoot)

pdpd

t

p ttetc pn

sincos)(

21

1

1)(c

1001

1

12

pdpd

t

p tteM pn

sincos

equation above in Putd

pt

100

1 2

dd

ddp

dn

eM

sincos

Time Domain Specifications (Maximum Overshoot)

100

1 2

dd

ddp

dn

eM

sincos

100

1 2

1 2

sincosn

n

eM p

1000121

eM p

10021

eM p

equation above in Put 21-ζωω nd

Time Domain Specifications (Settling Time)

ttetc ddtn

sincos)(21

1

12 nn

n

T

1

Real Part Imaginary Part

Time Domain Specifications (Settling Time)

n

T

1

• Settling time (2%) criterion • Time consumed in exponential decay up to 98% of the input.

ns Tt

44

• Settling time (5%) criterion • Time consumed in exponential decay up to 95% of the input.

ns Tt

33

Summary of Time Domain Specifications

ns Tt

44

ns Tt

33 100

21

eM p

21

nd

pt21

nd

rt

Rise Time Peak Time

Settling Time (2%)

Settling Time (4%)

Maximum Overshoot

Example#5 • Consider the system shown in following figure, where

damping ratio is 0.6 and natural undamped frequency is 5 rad/sec. Obtain the rise time tr, peak time tp, maximum overshoot Mp, and settling time 2% and 5% criterion ts when the system is subjected to a unit-step input.

Example#5

ns Tt

44

10021

eM p

dpt

drt

Rise Time Peak Time

Settling Time (2%) Maximum Overshoot

ns Tt

33

Settling Time (4%)

Example#5

drt

Rise Time

21

1413

n

rt.

rad 9301 2

1 .)(tan

n

n

str 550

6015

9301413

2.

.

..

Example#5

nst

4

dpt

Peak Time Settling Time (2%)

nst

3

Settling Time (4%)

st p 78504

1413.

.

sts 331560

4.

.

sts 1560

3

.

Example#5

10021

eM p

Maximum Overshoot

1002601

601413

.

..

eM p

1000950 .pM

%.59pM

Example#5 Step Response

Time (sec)

Am

plit

ude

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.2

0.4

0.6

0.8

1

1.2

1.4

Mp

Rise Time

time responce analysis

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