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27.09.2019
RTI Vorlesung 2
Quelle: www.dreager.ch
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Signals and Systems
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inputs(cause)
outputs(effect)
system: operator on functions
signals: functions of time
Systems: Key Assumption
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1 1 2 2
Systems: Key Assumption
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?
1 1 2 2=
!
Standard Control System
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Main Tasks of Control Systems
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Disturbance Rejection
Stabilization
Reference Tracking
Water Clock of KtesibiosOldest known engineered feedback control system
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The Fly-Ball “Governor”
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20.09. Lektion 1 – Einführung
27.09. Lektion 2 – Modellbildung4.10. Lektion 3 – Systemdarstellung, Normierung, Linearisierung
11.10. Lektion 4 – Analyse I, allg. Lösung, Systeme erster Ordnung, Stabilität18.10. Lektion 5 – Analyse II, Zustandsraum, Steuerbarkeit/Beobachtbarkeit
25.10. Lektion 6 – Laplace I, Übertragungsfunktionen1.11. Lektion 7 – Laplace II, Lösung, Pole/Nullstellen, BIBO-Stabilität8.11. Lektion 8 – Frequenzgänge (RH hält VL)
15.11. Lektion 9 – Systemidentifikation, Modellunsicherheiten 22.11. Lektion 10 – Analyse geschlossener Regelkreise 29.11. Lektion 11 – Randbedingungen
6.12. Lektion 12 – Spezifikationen geregelter Systeme13.12. Lektion 13 – Reglerentwurf I, PID (RH hält VL)20.12. Lektion 14 – Reglerentwurf II, „loop shaping“
Modellierung
Systemanalyse im Zeitbereich
Systemanalyse im Frequenzbereich
Reglerauslegung
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Model
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real system:
model of :
is a mathematically tractable representations of
In RT I+II: • ordinary differential equations (ODE, today) or• transfer functions obtained by Laplace transformation (later)
expressed by
Purpose of the Model
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Find a (mathematical) model of the plant P, which predictshow the true plant’s output y reacts (approximately) to the input u.
This (mathematical) model is later used to determine the limits of performance of any closed-loop system and then to synthesize a suitable controller C.
General Modeling Guidelines1. Identify the system boundaries.
2. Identify the relevant reservoirs and level variables.
3. Formulate the conservation laws for the relevant reservoirs
4. Formulate the algebraic relations for the flows between the
reservoirs.
5. Identify the system parameters using experiments.
6. Validate the model with experiments other than those used
for the identification.
dd" (reservoir content) = / in0lows − /out0lows.
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Conservation Laws
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dd" (reservoir content) =/in0lows − /out0lows.
t
t
t
content
in0low
out0low
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!"($)
!&($)
'"(… )
')(… )
*(… )
+($)
,($)!)($)
'&(… )
'-(… )
,($)+($)
Reservoir (energy, mass, charge, …)
Level (state) variable
Flow (power, mass flow, current, …)
!.($)
Relevant Dynamics
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Example: Water Tank
Assumptions• The water temperature is assumed to change very slowly.• The actuator (inlet valve) and the sensor are very fast.
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Example: Water Tank
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Bernoulli
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Along a streamline, 12 #$
% + ' = const.holds for incompressible and frictionless flows.
Numerical Simulation
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dd" ℎ(") =
1() * " − , " ( 2.ℎ "
• F = 100 m2
• ρ = 1000 kg/m2
• g = 9.81 m/s2
• A = 0.1 -> 0.12 m2
• v = 600 kg/s
2626
speed v(t)
speed v(t)
Example: Cruise Control
Example: Cruise Control
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Example: Cruise Control
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Example: Cruise Control
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Example: Cruise Control
Example: Stirred Reactor
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Example: Stirred Reactor
Example: Loudspeaker
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Example: Loudspeaker
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Example: Loudspeaker
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! " # $ = &($)
)ind(t) = κ " //0 p(t)
Example: Loudspeaker
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ResultMathematical models of the plant P to be controlled can often be expressed as a set of usually nonlinear Ordinary Differential Equations (ODE).
Pv w z
… but there is more …
Example: Conveyor Belt
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