ME 330 Control Systems

SP 2011

Lecture 2

Complex Numbers Complex numbers contain both

real and imaginary parts Complex functions “ ”

Example

1, jjs

IR jFFsF )(

2222 11

1

1

11

1)(

j

j

ssF

2222 dc

adbci

dc

bdac

idc

idc

idc

iba

idc

iba

Laplace Transform The Laplace transform is given by

0

)()()( dtetfsFtf stL

0,)(2

1)()(

tdsesFj

tfsFjc

jc

st

1-L

The inverse Laplace transform is given by

c > 0 and c > singular values of f(t) is an integration constant

The transformation is no loss of “information” Can convert between the two (not like averaging)

Common Functions 1 Exponential

0,

0,0)(

tAe

ttft

s

A

es

A

dteA

dteAeAe

ts

ts

sttt

0

)(

0

)(

0

L

Step

0,1

0,0)(

t

ttu

s

A

es

A

dteA

dtAetAu

st

st

st

0

0

0

)(L

not defined at t = 0

Common Functions 2

stAedvtu

vduuvdvu

,

Sinusoid

0),sin(

0,0)(

ttA

ttf

22

00

0

1

2

1

2

22

2)sin(

s

A

jsj

A

jsj

A

dteej

Adtee

j

A

dteeej

AtA

sttjsttj

sttjtjL

2

0

00

0

s

A

dtes

A

dts

Ae

s

eAt

dtAteAt

st

stst

st

L

Ramp

0,

0,0)(

tAt

ttf tjtj ee

jtA

2

1)sin(

Euler Formula

General Functions Function Derivatives

)0()()(

)(1)0(

)()(

)()(

00

0

fssFtfdt

d

tfdt

d

ss

f

dts

etf

dt

d

s

etf

dtetftf

stst

st

L

L

L

)0()0()()( 22

2

fsfsFstfdt

d

L

stedvtfu

vduuvdvu

),(

Properties of Laplace Transform Linearity

Frequency Shift

Convolution

)()()()()()( sBGsAFtgBtfAtBgtAf LLL

)()()()(0

sGsFdgtft

L

)()( sFtfe tL

Laplace Tables

In general, don’t solve Laplace integrals. Common functions

and corresponding Laplace transforms are well documented

Usefulness of Laplace Transform Basic mathematical framework for control systems

analysis and design differential equations algebraic equations

)()()()( tkxtxctxmtf f(t)

)()(

)()()(

)()()()(

2

2

sXkcsmssF

skXscsXsXms

txktxctxmtf

LLLL

Inputs Outputs

Example Process Solving differential equations to get the time-domain

response of the dynamical system.

)(1

)(2

sFkcsms

sX

)()()()( tkxtxctxmtf f(t)

Step 2) Transform f(t) to F(s)

Step 1) Represent the system differential equation in Laplace domain “transfer function”

Step 3) Multiply “transfer function” with Laplace domain input F(s)

Step 4) Solve for time-domain response with inverse Laplace transform

Laplace Domain Mathematical Framework Solutions for time-domain response to specific

input excitation Analysis of system stability

given any bounded input, the output will be bounded Performance of system

speed of time-domain response, oscillatory behavior, steady-state (persistent) errors

Next Lecture

Techniques for solving inverse Laplace Solutions for time-domain response to specific

input excitation Derivation of simple dynamical system models