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1
DYNAMICS and CONTROL
Presented by
Pedro AlbertosProfessor of Systems Engineering and Control - UPV
MODULE 1I (III)
Models of
Systems & Signals
Formalism
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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
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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
3
DYNAMICS & CONTROL
( ) ( )y t Y sL
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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
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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Signals: TRANSFORMED
Systems: OPERATOR
)(
)()(
za
zbzG
)(
)()(
za
zbzY )(ty
Transfer Function
Representations
14
DYNAMICS & CONTROL
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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
15
DYNAMICS & CONTROL
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The steem generatorA historical curiosity
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DYNAMICS & CONTROL
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The steem generatorA historical curiosity
16
DYNAMICS & CONTROL
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The steem generatorA historical curiosity
16
DYNAMICS & CONTROL
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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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18
DYNAMICS & CONTROL
Lets explore the
control systems,
their structure,
their goals, and
the benefits.
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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
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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