Transcript continuous

SiSy 2012 Dqtm Chapter 2, SysMod, 1
Signale und Systeme
SiSy
Chapter 2
System Description and Modelling
References
[1] P.D. Cha, J.I. Molinder, „Fundamentals of Signals and Systems – A Building Block Approach“
[2] G.Lekkas, J.Wild, „Signale und System“, ZHW-Vorlesung, 2007.
Modelling Approaches
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Example: DC-Motor to be used in a robot arm
How to drive it in order
to follow a desired
trajectory?
Is there an output
proportional to input
voltage?
Is there a latency in the
response? How big?
Model
as simple as possible
for required simulation
Analytical Approach
transparent box +
physical laws
Empirical Approach
black box +
measurements
stimuli
Differential
Equations
+
BlockDiagram
Response
to test
signals
measurement
Types of Systems
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x(t)
Continuous System
y(t)
LTI
(linear time invariant)
Discrete System
x[n]
LTD
(linear time invariant and
discrete)
y[n]
System Description with Differential Equations
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Examples: Models for Mechanical Systems with Lumped Elements
• Elements:
– spring (Feder) & torsional/rotational spring (Drehfeder)
– damper (Dämpfer) & rotational damper (Drehdämper)
– mass (Masse) & inertia (Trägheit)
mass moment of inertia (Massenträgheitsmoment)
• Physical Laws:
– relationships between force and displacement (Gesetze)
• Input & Ouput:
–
–
–
–
force (Kraft) & torque (Drehmoment)
position (Position) & angle (Winkel)
velocity (Geschwindigkeit) & angular velocity (Dreh-)
acceleration (Beschleunigung) & angular acceleration (Dreh-)
System Description with Differential Equations
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Mechanical Systems Physical Laws
Element
Spring
Translational
Rotational
k [N/m]
kt [N.m/rad]
Fs  k  x
Damper
c [N.s/m]
dx
Fd  c 
dt
Mass/Inertia
m [kg]
d 2x
Fm  m  2
dt
Ts  kt  
ct [N.m.s/rad]
d
Td  ct 
dt
I [N.m.s2/rad]
d 2
TI  I 
dt 2
System Description with Differential Equations
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Mechanical Systems Examples
• Example 1: spring-mass (horizontal)
• Example 2: spring-mass suspended
• Example 3: base excitation system
• Example 4: propeller of a ship
• Example 5: suspended pendulum
• Exercise List 2 : mechanical and electrical systems
System Description with Differential Equations
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Electrical Systems Physical Laws:
Kirchhoff node and mesh rules,
plus basic equation for passive elements R, L and C.
Element
Governing Equation
Inductor
L [H =V.s/A]
(Spule)
Resistor
(Widerstand)
Capacitor
(Condensator)
di
vL  L  L
dt
Nature (Energy)
stores energy –
magnetic field
R [Ω= V/A]
v R  iR  R
C [F = A.s/V]
dv
iC  C  C
dt
Dissipates energy
stores energy –
electrical charges
Discrete Systems
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Properties
•
Linearity
•
Time
Invariance
•
Causality
Discrete Systems
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Description with difference equations, examples:
moving average filter
Discrete Systems
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Examples: Response of discrete systems a, b, c
to a unit impulse (Kronecker Delta)
Discrete Systems
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Examples: Response of discrete systems a, b, c
to a unit step
Discrete Systems
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Example exercises:
System Modelling with Test Signals
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Empirical or «black-box» approach
for continuous systems
System Modelling with Test Signals
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Based on step response,
identify: LPF, HPF, BPF
System Modelling with Test Signals
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Compare step to impulse response:
which relationship? Why?
System Modelling with Test Signals
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Corresponding Frequency Responses :
LPF, HPF, BPF
Modelling of Discrete Systems
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Empirical or «black-box» approach for discrete systems
Classification of discrete systems based on the impulse response
Which kind of system in question 2-3?
Non-recursive Discrete System (FIR)
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Determine Filter coefficients based on impulse response:
•
•
Measure continuous LTI system step response and derivate to get the corresponding
impulse response;
Sample desired impulse response to get gs[n]
•
Non-zero values corresponds to filter coefficients (check delays!)
Convolution (Faltung)
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«Sum of weighted and shifted responses»
Discrete Convolution
Discrete Convolution
•
•
Operator Notation (star)
Commutative Property
Continuous Convolution
Convolution (Faltung)
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Comparing discrete and continuous convolution
Complex Number Notations
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Polar and Cartesian Notation
Frequency Response
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Calculation method of frequency response
from system’s differential equation
Frequency Response
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Example
Polar Notation
Frequency Response
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Bode Diagram
Frequency Response
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Bode Diagram: calculation of asymptotes
and reference points
Frequency Response
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Bode Diagram: dB scale
•
Powers of 2
•
Powers of 10
•
Slopes
•
Basic Terms
Reference Systems
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First Order
Reference Systems
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Second
Order
Reference Systems
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Second
Order
Example