rti vorlesung 4 - eth z
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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
24
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
27
Stability Conditions I
Stability Conditions II
28
Result: If is diagonizable, then
and
Stability Conditions III
30
à need further analysis (not treated in RT1)
Recipe to Determine Stability
31
Calculate eigenvalues of :
Then:
What About the Original Nonlinear System?
32
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