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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

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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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