08 models ap

1DYNAMICS and CONTROL

Presented by

Pedro AlbertosProfessor of Systems Engineering and Control - UPV

MODULE 1I (App)

Models of

Systems & Signals

Math review

2DYNAMICS & CONTROL

Y(z)

3DYNAMICS & CONTROL

G(s)

4DYNAMICS & CONTROL

ABSTRACTION!!!

Model

Magnitude: Yo

Frequency: w

Phase: j

)()( 0 j wtsenYty

Dealing with Signals: Sinusoidal

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DYNAMICS & CONTROL

Operations:

Sum

Linear combination

Delay

Derivative

Integral

Parameters

( )y t Laplace Transform:Dealing with Signals

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DYNAMICS & CONTROL

0( ) ( )stY s e y t dt

Laplace transform properties

( ) ( )y t Y sL

Unicity

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

Linearity

( ) ( )sy t e Y s L

Delay

( )( ) (0)

dy tsY s y

dt L

Derivative0

( )( )

tY s

y ds

L

Integral

Delay

Laplace Transform of typical signals

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DYNAMICS & CONTROL

( )f t ( )F s

Impulsounitario 1

Escalnunitario 1

s

Rampaunitaria 2

1

s

a te 1

s a

a t sen te 2 2s a

a t cos te 2 2s a

s a

Unitary Impulse

Unitary Step

Unitary Ramp

( )y t T ( )sTe Y s

Delay

Laplace Transform of typical signals

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DYNAMICS & CONTROL

( )f t ( )F s

Impulsounitario 1

Escalnunitario 1

s

Rampaunitaria 2

1

s

a te 1

s a

a t sen te 2 2s a

a t cos te 2 2s a

s a

Unitary Impulse

Unitary Step

Unitary Ramp

( )y t T ( )sTe Y s

se

s

Inverse Laplace Transform

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DYNAMICS & CONTROL

2

2( )

3 2Y s

s s

2( ) 2 2t ty t e e

2 2 2

( 1)( 2) 1 2s s s s

1( )

Laty t e

s a

325.25.10ky

0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

tiempo

Amplitud

Discrete-time Signals

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DYNAMICS & CONTROL

Time

Mag

nit

ude

1 2 3( ) 3 3 3 ...y z z z z

325.25.10ky

0 1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

tiempo

Amplitud

Discrete-time Signals

9

DYNAMICS & CONTROL

Time

Mag

nit

ude

Correspondence between CT and DT

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DYNAMICS & CONTROL

1( ) k ky ydy t

dt T

Correspondence between CT and DT

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DYNAMICS & CONTROL

. ( )sY s1

. ( )z

Y zT

1zs

T

But, the delay: 1 Tsz e Tsz e

1z sT

ZL

1( ) k ky ydy t

dt T

Linear / nonlinear

Estatic / Dynamic

Time variant / Invariant

Concentrated / Distributed

Continuous / DiscreteLogic/Binary Deterministic / estocasticApproximated / concreteMonovariable / multivariable

By the attached signals By the operator

Systems to be modeled

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DYNAMICS & CONTROL

Model as an operator

VC

DYNAMICS & CONTROL

( ) 1

( ) ( )c cdV t

V t V tdt RC

Voltage balance

Laplace transform

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

RC

1( ) ( )

1cV s V s

RCs

Set of differential equations

Set of algebraic equations

11 12 1

21 22 2

( )( ) ( ) ( )

( ) ( )2 ( ) ( ) ( )

dx ta x t a y t b u t

dt

dx t dy ta x t a y t b u t

dt dt

11 12 1

21 22 2

( ) ( ) ( ) ( )

2 ( ) ( ) ( ) ( ) ( )

sx s a x s a y s bu s

sx s sy s a x s a y s b u s

Compact System representation:

By means of an operator between transformed signals

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

)(95.0

1)( zu

zzy

Transfer Function :

95.0

1)(

z

zG

SYSTEMy(k) u(k)

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DYNAMICS & CONTROL

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

Series composition:

Block Diagrams

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DYNAMICS & CONTROL

X(z)

2

( )( )

( )

Y zG z

X z

1

( )( )

( )

X zG z

U z

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

2 1( ) ( ) ( )G z G z G z

Paralell composition:

Block Diagrams

16

DYNAMICS & CONTROL

22

( )( )

( )

Y zG z

U z

11

( )( )

( )

Y zG z

U z

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

1 2( ) ( ) ( )G z G z G z

1( )G z

2 ( )G z

+

2 ( )Y z

1( )Y z

( )U z ( )Y z

1 2( ) ( ) ( )Y z Y z Y z

Loop arrangement:Block Diagrams

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DYNAMICS & CONTROL

( )G z

( )H z

+

-

( )Y z( )U z ( )E z

( )M z

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

Y z G z E z

M z H z Y z

E z U z M z

( )( )

1 ( ) ( )

G zY z

G z H z

MATLAB

10( )

( 2)( 5)

sG s

s s

Simulation tools

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DYNAMICS & CONTROL

>> s=zpk('s');

>> G=(s+10)/(s+2)/(s+5)

Zero/pole/gain:

(s+10)

-----------

(s+2) (s+5)

>> step(G)

Programming Exploiting

A bit of Maths around modeling

What have we seen today?

Parameterizing the signals informationLaplace and Z- TransformationsTypical transformed signalsModels of systems as operatorsSystems connection and structureModeling and simulation tools

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DYNAMICS & CONTROL

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!

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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

08 models ap

Documents

s s s s1 laty t es

s rampaunitaria

y s bu ssx s sy s

s llinearity sy t e

t sen te

t cos te

properties y t y slunicity1

y t b u tdtdx t dy