11.10.2019

RTI Vorlesung 4

Control System

2

Remember: we want to control the plant

àWe need “information” about the plant, i.e.,a mathematical model (ODE, …)

Steps1. Develop a mathe-

mathical modelof the system

2. Choose an equilibrium point for control

3. Normalize usingexpected values(often equilibrium)

4. Linearize aroundthe equilibrium point

C

y0

0

v

δx(t) = A δx(t) + b δu(t)δy(t) = c δx(t) + d δu(t)

. . .

. .

0u0

.z(t)=f(z(t),v(t))w(t)=g(z(t),v(t))

w0-1

δyy

δu

u v w

approximated by:

P

ueye

4

Real plant approximated by:

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

5

Analysis of Linear Systems

7

Given:

Goal:

Linear, time-invariant description of a system from input ! to output "

• Compute # $ and "($) for given !($) and # 0• Understand the qualitative behavior of #($)

Exponential Function

8

Scalar case (! is a scalar)

The derivative of an exponential function is a linear function of the same exponential

Multidimensional case (" is a matrix)

As for the scalar case!

Remarks on Matrix Exponentials

9

In general:

Only if !" and !# commute:(Proof: Quick Check 4.2.1)

and

(Proof: Quick Check 4.2.2)

$ ⋅ & and $ ⋅ * obviously commute ( are scalars)

Therefore:

therefore

How to compute x(t)?

10

Key idea: The derivative of an exponential function is a linear function of the same exponential function; this is interesting!

Pro memoria: Partial integration

∫"# $(&) ( )*(+))+ ,& = $ . ( / . − $ 0 ( / 0 - ∫"

# )2(+))+ ( / & ,&

Known: u(t), x(0)

To be computed: x(t) and y(t)

11

System Output

Transition matrix:

Convolution:

with

13

Remarks on the General Solution

14

In general (n>1, arbitrary u(t)), the solution to the linear ODE must be calculated numerically

A closed-form solution is possible only for low-order systems and simple input signals (”test signals”)

Test Signals

!

!

!15

Strictly-Proper First-Order Systems

16

Impulse Response

17

Step Response

18

Ramp Response

19

Harmonic Response

20

Higher-Order Systems

21

Quantitative solutions to test signals not easy to compute

Qualitative «solutions» easy à stability

Stability

22

(later in lecture 7)

(today)

Lyapunov Stability

23

Then, system is said to be

Consider autonomous system ( ) given by

Pro Memoria I

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Linear transformation:

Eigenvalues and Eigenvectors:

!" = $" + & ' (" $", (" ∈ ℝ & = −1

25

Form a coordinate transformation with:

Not always the case! If Eigenvalues distinct then yes.

Definition

Then and therefore

Pro Memoria II

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Stability Conditions I

Stability Conditions II

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Result: If is diagonizable, then

and

Stability Conditions III

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à need further analysis (not treated in RT1)

Recipe to Determine Stability

31

Calculate eigenvalues of :

Then:

What About the Original Nonlinear System?

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