System Response Characteristics

ISAT 412 -Dynamic Control of Energy Systems(Fall 2005)

Review

We have overed several O.D.E. solution techniques Direct integration Exponential solutions (classical) Laplace transforms

Such techniques allow us to find the time response of systems described by differential equations

Generic 1st order model

Solution in Laplace domain

Solution comprised of Free Response (homogeneous solution) Forced Response (non-homogeneous solution)

tfcxdt

dxm

cms

sF

cms

mxsX

0

Free response of 1st order model

Free response means: Converting back to the time domain:

0 sFtf

m

ct

h

h

ex

mc

sxtx

mc

s

x

cms

mxsX

01

0

00

1L

Time constant

Define the system time constant as

Rewriting the free responsec

m

m

c

1

or

t

m

ct

h exextx

00

Free response behavior

0

0

0

Unstable

Stable

Unstable

tx

00 x

00 x

Meaning of the time constant

When t =

When t = 2, t = 3, and t = 2,

0368.000 1 xexexxh

0018.04

0045.03

0135.02

xx

xx

xx

h

h

h

Transfer Functions and Common Forcing Functions

ISAT 412 -Dynamic Control of Energy Systems(Fall 2005)

Forced response of 1st order system

The forced response corresponds to the case where x(0) = 0

In the Laplace domain, the forced response of a 1st order system is

cms

sFsX

Transfer functions

Solve for the ratio X(s)/F(s)

T(s) is the transfer function Can be used as a multiplier in the

Laplace domain to obtain the forced response to any input

cmssF

sXsT

1

sXsFsT

Using the transfer function

Now that we know the transfer function for a 1st order system, we can obtain the forced response to any input if we can express that input in the Laplace domain

Step input

Used to model an abrupt change in input from one constant level to another constant level Example: turning on a light switch

tf

b

Dt

Heaviside (unit) step function

Used to model step inputs

s

tu

t

ttu

s

s

1

0 1

0 0

L

Time shifted unit step function

For a unit step shifted in time,

Using the shifting property of the Laplace transform (property 6)

Dt

DtDtus 1

0

s

eDtu

sD

s

L

Step input model

For a step of magnitude b at time D

s

betf

DtbutfsD

s

L

Pulse input

tf

M

At Bt

Pulse input model

Use two step functions

tf

M

At Bt

0

M

Pulse input model

s

Me

s

Metf

BtMuAtMutfsBsA

ss

L

For a pulse input of magnitude M, starting at time A and ending at time B

Impulse input

tf

M

At

Examples: explosion, camera flash, hammer blow

Impulse input model

Unit impulse function

For an impulse input of magnitude M at time A

At

AtAt

0

1

sAMetf

AtMtf

L

Ramp input

tfm

At

Ramp input model

For a ramp input beginning at time A with a slope of m

2s

me

AtuAtmtf

AtmAuAtmtu

AtuAtmtf

sA

s

ss

s

LL

Other input functions

Sinusoidal inputs Combinations of step, pulse,

impulse, and ramp functions

Modeling periodic inputs

tf

A

B B2 B3 B4 B5 B6

t

Square wave input model

Addition of an infinite number of step functions with amplitudes A and -A

1

11

...3

2

ns

n

ss

ss

nBtuA

BtAuBtAu

BtAuBtAutf

Laplace transform of square wave

1

1

1

1

1

1

n

snBn

ns

n

s

e-A

nBtu-Atf LL