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CDS 101/110a: Lecture 1.2
System Modeling
Douglas G. MacMynowski
29 September 2010
Goals:
Define a “model” and its use in answering questions about a system
Introduce the concepts of state, dynamics, inputs and outputs
Review modeling using ordinary differential equations (ODEs)
Reading:
Åström and Murray
Feedback Systems
Sections 2 1–23 31[40m
in
Åström and Murray,
Feedback Systems,
Sections 2.1 2.3, 3.1 [40 min]
Advanced: Lewis,
A Mathematical Approach to Classical Control
, Chapter 1
Monday: Introduction to Feedback and Control
Sense
Actuate
Control =
•
Sensing + Computation +
Actuation
Feedback Principles
Compute
•
Robustness to Uncertainty
•
Design of Dynamics
Many examples of feedback and control in natural & engineered systems:
BIO
Douglas G. MacMynowski, Caltech CDS
CDS 101/110, 29 Sep 10
2
BIO
ESE
ESE
CS
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ModelBased Analysis of Feedback Systems
Analysis and design based on
models
•
A model provides a prediction of how the system
will behave
•
Feedback can give counterintuitive behavior;
models help sort out what is going on
Weather Forecasting
•
For control design, models don’t have to be
exact: feedback provides robustness
The model you use depends on the questions
you want to answer
•
A single system may have many models
•
Time and spatial scale must be chosen to suit
the questions you want to answer
•
Question 1: how much will it rain
tomorrow?
•
Question 2: will it rain in the next
510 days?
•
Question 3: will we have a
Douglas G. MacMynowski, Caltech CDS
CDS 101/110, 29 Sep 10
3
•
Formulate questions
before
building a model
Controloriented models:
inputs
and
outputs
•
Capture input/output behaviour “sufficiently” well
Question 3: will we have a
drought next summer?
Different questions
⇒
different models
Example #1: Spring Mass System
Applications
•
Flexible structures (many apps)
•
Suspension systems (eg, “Bob”)
•
Molecular and quantum dynamics
m
1
m
2
q
1
u(t)
q
2
Questions we want to answer
•
How much do masses move as a
function of the forcing frequency?
•
What happens if I change the values
of the masses?
•
Will Bob fly into the air if I take that
speed bump at 25 mph?
Modeling assumptions
c
k
3
k
2
k
1
Douglas G. MacMynowski, Caltech CDS
CDS 101/110, 29 Sep 10
4
•
Mass, spring, and damper constants
are fixed and known
•
Springs satisfy Hooke’s law
•
Damper is (linear) viscous force,
proportional to
velocity
3
Modeling a Spring Mass System
m
1
m
2
q
1
u(t)
q
2
Model: rigid body physics (Ph 1)
•
Sum of forces = mass * acceleration
•
Hooke’s law:
F
=
k
(
x
–
x
rest
)
•
Viscous friction:
F = c v
c
k
3
k
2
k
1
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This note was uploaded on 01/04/2012 for the course CDS 101 taught by Professor Murray,r during the Fall '08 term at Caltech.
 Fall '08
 Murray,R

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