Control Systems · Chibum Lee -Seoultech Control Systems Linear vs. nonlinear system Linear system: the principle of superposition holds •Linearity in mathematics Let Vand Wbe vector

Control Systems

Mathematical Modeling

of Control Systems

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Control SystemsChibum Lee -Seoultech

Outline

Mathematical models and model types.

Transfer function models

System poles and zeros


Mathematical Models

Models are key elements in the design and analysis of

control systems qualitative mathematical model

We must make a compromise b/w the simplicity of the

model vs. the accuracy of the results of analysis

???)t(u )t(y


Linear vs. nonlinear system

Linear system: the principle of superposition holds

• Linearity in mathematics

Let V and W be vector spaces over the same field K.

A function f: V → W is said to be a linear map if for any 2 vectors x and y

in V and any scalar α in K, the following conditions are satisfied:

• Linearity in system

A general system can be described by operator H, that maps an input

x(t) as a function of t to an output y(t) a type of black box description.

Linear systems satisfy the properties of superposition and homogeneity.

additivity

homogeneity


Linear Time Invariant System

Linear Time Invariant = Linear & time invariant

A time-invariant (TIV) system is one whose output does not depend explicitly on time.• If the input signal x(t) produces an output y(t), then

any time shifted input, x(t+), results in a time-shifted output y(t+ )

• Time invariant means that the coefficients in the differential equations are constant and don’t change with respect to time.

We can apply impulse response & Laplace transform in LTI system


Time-Varying Models

A time-varying system is a system that is not

time invariant its output depend explicitly

upon time

• Eg. a spacecraft control system.

The mass of fuel consumption changes due to fuel

consumption

Dynamic system

Linear Nonlinear

LinearTime

Invariant

LinearTime

Varying

Our focus


Example

Transfer Functions

Fkyybym

ei

ymFybky

ymmaF

.

y = displacement from spring equilibrium


Transfer Functions

• Assuming zero initial conditions, take the Laplace

Transform of both sides

)()(

)()()()(

2

2

sFsYkbsms

sFskYsbsYsYms

)(1

)(

)(2

sGkbsmssF

sY

output

input

G(s)input output

TransferFunction


Transfer Functions

Transfer Function: the ratio of the Laplace

transform of the input and output of a linear

time-invariant system with zero initial conditions

and zero-point equilibrium.

Rational function in the complex variables

• Let x(t) : input , y(t) : output

xbxbxbxb

yayayaya

m

m

m

m

n

n

n

n

01

)1(

1

)(

01

)1(

1

)(

(n > m)


Transfer Functions

n-th order system since the highest power in the

denominator is n.

Note:

• limited to time-invariant, differential equation

• independent of the input magnitude. (homogeneity)

• no information on physically structure. (MKS and RLC)

01

1

1

01

1

1

ICs zero

)(

)(

][

][)( :TF

asasasa

bsbsbsb

sX

sY

inputL

outputLsG

n

n

n

n

m

m

m

m


System Poles and Zeros

• Roots of N(s)=0 : the system zeros z1, z2, …, zm

• Roots of D(s)=0 : the system poles p1, p2, …, pn

Note

• (System) Poles and zeros: real or either complex conjugate pairs

))(())((

))(())((

)(

)(

)(

121

121

01

1

1

01

1

1

nn

mm

n

n

n

n

m

m

m

m

pspspsps

zszszszsK

sD

sN

asasasa

bsbsbsbsG

numerator

denominator


System Poles and Zeros

Ex.

)2)(1(

)3()(

ss

sKsG


Outline

Convolution

Impulse Response


Convolution Integral

If a transfer function is

The output can be written as Y(s) = G(s)U(s)

The inverse Laplace transform is given by the convolution integral

)()(

)(sG

sU

sY

output

input

tt

dtugdtguty00

)()( )()()(

0for 0)()( where ttutg


Impulse Response Function

Impulse response of a dynamic system is its output

when its input is a unit impulse

• Laplace transform of the unit impulse 1][ L

)()]()([)(

)]([)(en wh)()()(

1 tgsUsGLty

tLsUsUsGsY

Impulse response function

G(s)

Unit impulse(t)

Impulse responseg(t)


Impulse Response Function

The transfer function and impulse response function

of a LTI system contain the same information

abut the system dynamics.

• Transfer function in s-domain

• Impulse response function in time domain

Exciting a system with an impulse input and measuring

the response

can obtain the dynamic characteristics of the systems

(impact hammer test)

http://www.youtube.com/watch?v=VT1ndotq3dE


Impulse Response Function Example

You can find the system response to an arbitrary input

Example:

Find repose to the following input

)()()()( tftytyty

Impulse response function (HW)



Approximation with impulsesA sample of 0.8 s discrete function



By superposition:

t 0 , summation convolution integration.

)1)()(()()(

)()()()()()()()()(

1

112211

ttututtg

tuttgtuttgtuttgtuttgty

n

i

ii

nnnn

t

dutgty0

)()()(



Summation of the responsesEach impulse response(scaled and delayed)

6

1

6

1

66552211

)()()(

)()()()()()()()()(

i

i

i

ii ttgtuttg

tuttgtuttgtuttgtuttgty



Summation of the responsesSmaller time step


Outline

Block diagrams


Block Diagrams

Block diagram: a pictorial representation of the functions

performed by each component and of the flow of

signals

G(s))(sU )(sX

Y(s)


Block Diagrams

Blocks in cascade

Blocks in parallel

G1(s))(sR )(sY

G2(s) G2(s) G1(s))(sR

G1(s)

G2(s)

+

+G1(s)+G2(s)

)(sY

)(sY)(sR)(sR )(sY


Block Diagrams

Feedback system

Ex.

G(s))(sR

H(s)

)(sY)(sE

-+

)()(1

)(

)(

)(

)()()()()(1

)()()()()()()(

)()()()(

sHsG

sG

sR

sY

sRsGsYsHsG

sYsHsRsGsEsGsY

sYsHsRsE

56

1

)6(

51

)6(

1

)(

2

ss

ss

sssT

1/S)(sR

5

)(sY1/(S+6)

-+


Block Diagrams


Block Diagrams


Block Diagrams

Example



Outline

Mathematical model of mechanical system

Mathematical model of electrical system


Mechanical System

Mass-spring-damper system

y

m

ky

m

b

m

Fy

Fkyybym

ei

ymFybky

ymmaF

.

y = displacement from spring equilibrium


Mechanical System

Block Diagram representation

1/s 1/s

b/m

k/m

1/m +

+

+

-

y yyF

)(1

)(

)(2

sGkbsmssF

sY


RC Circuit

ioo

o

CRi

vvdt

dvRC

idtC

v

idtC

Rivvv

1

1R

iv i C ov

Capacitance

)(1

21 vvidtC

1v 2vC

t

)(tv o

RC

1

RC: Time constant63.2% of final value)1()(

)/1/(1/1(1

)/1(

/1

)1(

1)(

)()()(

/ RCt

o

o

ioo

etv

RCss

RCss

RC

RCsssV

sVsVsRCsV

For step Vi(t),

Electrical Systems


RL circuit

i

RLi

vR

idt

di

R

L

Ridt

diLvvv

1

L

iv i R

1/

/1

)(

)(

)(1

)()(

sRL

R

sV

sI

sVR

sIssIR

L

i

i

t

)(ti

RL /

R/1

)1(/1)(

)//(1/1(/1

)/(

/1)(

)(/1)()(/

/ LtR

i

eRti

LRssR

LRss

LsI

sRVsIsRsIL

Electrical Systems

For step Vi(t),

L/R: Time constant63.2% of final value

L

vv

dt

di )( 21

1v 2vL

Inductor


Electrical Systems

LRC circuit

o

i

VidtC

VidtC

Ridt

diL

1

1

iv ov

1

12

RCsLCsV

V

i

o

)()(1

)()(1

)()(

sVsICs

sVsICs

sRIsLsI

o

i


Recall Impedance method

Resistor Capacitor Inductor

T-domain

S-domain

ImpedanceZ

Riv idtC

v1

dt

diLv

v R

i

v L

i

v C

i

)()( sRIsV )(1

)( sICS

sV )()( sLsIsV

RCs

1Ls

R

i

L

Civ ov

LCsLRs

Ls

RCsLCs

Cs

CsLsRsV

sI

sICsLsRsV

i

i

/1)/(

/

1

/1

1

)(

)(

)()/1()(

22

LCsLRs

LC

sV

sI

CssV

sV

sCsVsI

sICs

sV

ii

o

o

o

/1)/(

/1

)(

)(1

)(

)(

)()(

)(1

)(

2

Electrical Systems

Documents

Control Systems · Chibum Lee -Seoultech Control Systems Linear vs. nonlinear system Linear system: the principle of superposition holds •Linearity in mathematics Let Vand Wbe vector