07_Models_3

1

DYNAMICS and CONTROL

Presented by

Pedro AlbertosProfessor of Systems Engineering and Control - UPV

MODULE 1I (III)

Models of

Systems & Signals

Formalism

2

DYNAMICS & CONTROL

Modules: Examples of systems and signals

Models of systems and signals

Representations

Analogies

Formalism

Controlled systems: properties

Control systems design

Control benefits

Topics to study

Or by means of transformed

representations.

)()( 0 wtsenYty

220)(

ws

wsYsy

Laplace Transform:

0 0

1( ) ; ( )aty t Y e Y s Y

s a

Compact representation of CT Signals

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DYNAMICS & CONTROL

( ) ( )y t Y sL

Compact representation of DT Signals

Or using transformed representations

)5.01(3)( kky

)5.01)(1(

5.1)(

11

1

zz

zzy

1z Delay operator

325.25.10ky

0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

tiempo

Am

plit

ud

...25.25.1)( 21 zzzy

Discrete-time Signals

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DYNAMICS & CONTROL

Time

Mag

nit

ude

It is a sequence

ALGORITHMInputConst.

y(0) = 25;

for k=1 to k=100

y(k+1)=0.95*y(k)+10;

Represent y(k)

Computer sequence

y(k+1) = 0.95*y(k) + u(k);

u(k) = 10; y(0) = 25MODEL:

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DYNAMICS & CONTROL

Outp

ut:

y(k)

Compact System representation:

Or by means of an operator between transformed signals

u(k) y(k)*0.95 1)y(k

)(95.0

1)( zu

zzy

T.F. :

95.0

1)(

zzG

SYSTEMy(k) u(k)

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DYNAMICS & CONTROL

T.F. : Transfer Function

1z Delay operator

zy( ) 0.95*y(z) u(z)z

Tank system

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DYNAMICS & CONTROL

( ) ( ) ( )i r oq t q t q t ( ) ( )oq t F h t

( )r

Adh tq

dt

( ) 1

( ) ( )odh t

q t F h tdt A

Water balance Outlet flow

Retained flow

Electric Circuit

( ) . ( ) ( )cV t R I t V t

VC

C

tQtVc

)()(

DYNAMICS & CONTROL

( ) ( )t

Q t I d

( ) cdV

I t Cdt

( )( ) ( ) cc

dV tV t V t RC

dt

( ) 1

( ) ( )c cdV t

V t V tdt RC

Voltage balance Capacitor

Accumulated charge

Model as an operator

VC

DYNAMICS & CONTROL

( ) 1

( ) ( )c cdV t

V t V tdt RC

Voltage balance Capacitor

Laplace transform properties (see Appendix)

( ) ( )y t Y sL

Unicity

1 2 1 2( ) ( ) ( ) ( )ay t by t aY s bY s L

Linearity

( )( )

dy tsY s

dtL

Derivative

1( ) ( ) ( )c csV s V s V s

RC

1( )

1G s

RCs

1( ) ( )

1cV s V s

RCs

0 1 2 3 4 5 6 7 8 9 1020

40

60

80

100

120

140

160

180

200

11

10)(

101010

zzu

uk 0 1 2 3 4 5 6 7 8 9 10

9

9.2

9.4

9.6

9.8

10

10.2

10.4

10.6

10.8

11

)95.0)(1(

10)(

1

zzzy

( ) 1( )

( ) 0.95

y zG z

u z z

SYSTEMy(k) u(k)

Experimental Modeling

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DYNAMICS & CONTROL

G1(z) G2(z)U(z) Y(z)

G2(z)G1(z)U(z) Y(z)

G1(z)+G2(z)U(z) Y(z)

G1(z)

G2(z)

U(z) Y(z)

+

+

Series composition:

Paralell composition:

Loop arrangement:

G(z)U(z) Y(z)+

H(z))(

G(z)H(z)1

G(z)Y(z) zU

Block Diagrams

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DYNAMICS & CONTROL

Linear / nonlinear

Static / Dynamic

Time variant / Invariant

Concentrated / Distributed

Continuous / Discrete

Logic/Binary

Deterministic / stochastic

Approximated/concrete

Monovariable / multivariable

By the attached signals By the operator

Kind of systems

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DYNAMICS & CONTROL

Kind of Models

Internal and external relationships

Behavioral description: first principles

Stated as equations

Initial conditions are considered

Input / Output relationship

Experimental modeling

Use of analogies

Null initial conditions (in

equilibrium)

Black box White box

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DYNAMICS & CONTROL

Signals: TRANSFORMED

Systems: OPERATOR

)(

)()(

za

zbzG

)(

)()(

za

zbzY )(ty

Transfer Function

Representations

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DYNAMICS & CONTROL

Formalism in modeling systems and signals

What have we seen today?

Parameterizing the signals information

Expressing in equivalent representations

Transformations

Models of systems as operators

Models of systems as set of equations

Systems connection and structure

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DYNAMICS & CONTROL

The steem generatorA historical curiosity

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DYNAMICS & CONTROL

The steem generatorA historical curiosity

and the Watts regulator (17361819)

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DYNAMICS & CONTROL

How it works?

On Governors J.

Clerk Maxwell (1868)

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DYNAMICS & CONTROL

Lets explore the

control systems,

their structure,

their goals, and

the benefits.

What is next?

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DYNAMICS & CONTROL

Modules: Examples of systems and signals

Models of systems and signals

Controlled systems: properties

Dynamic and static behavior

Sensitivity and Robustness

Control systems design

Control benefits

Topics to study

Thank you!

20

The sources of some of these figures are:

Slide 12-1 http://upload.wikimedia.org/wikipedia/commons/0/01/Newcomen_atmospheric_engine_%28Heat_Engines%2C_1913%29.jpg. Author: Andy Dingley Public Domain.

Slide 12-2 http://upload.wikimedia.org/wikipedia/commons/1/16/Newcomen_atmospheric_engine_animation.gif. Author: Emoscopes. GNU Free Documentation License

Slide 13-1 http://commons.wikimedia.org/wiki/File:Boulton_and_Watt_centrifugal_governor-MJ.jpg By Dr. Mirko Junge (Own work) [CC-BY-3.0 (http://creativecommons.org/licenses/by/3.0)], via Wikimedia

Commons

Slide 14. http://upload.wikimedia.org/wikipedia/commons/thumb/5/55/Catalonia_Terrassa_mNATEC_MaquinaDeVapor_ReguladorDeWatt.jpg/800px-

Catalonia_Terrassa_mNATEC_MaquinaDeVapor_ReguladorDeWatt.jpg. Author Friviere GNU

DYNAMICS & CONTROL
http://commons.wikimedia.org/wiki/File:Boulton_and_Watt_centrifugal_governor-MJ.jpg

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07_Models_3