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Continuous System Modeling

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newer domains such as information and bio-technologies ... Fifth Edition (Hardcover) by Dennis G. Zill, Brooks Cole; 5 edition (December 8, 2000) ... – PowerPoint PPT presentation

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Title: Continuous System Modeling


1
Continuous System Modeling
2
Need various types models
  • Advances in system development ultimately rely on
    well-constructed predictive models
  • Applications
  • traditional fields such as electrical and
    mechanical engineering
  • newer domains such as information and
    bio-technologies
  • Using appropriate simulation software, we can
    derive solutions to difficult problems using such
    models
  • Success often depends on having a variety of
    modeling approaches available to formulate the
    right model for the particular issue at hand
  • Therefore, a broad familiarity with different
    types of models is desirable

3
Continuous System Models
  • Continuous system models were the first widely
    employed models and are traditionally described
    by ordinary and partial differential equations.
  • Such models originated in such areas as physics
    and chemistry, electrical circuits, mechanics,
    and aeronautics.
  • They have been extended to many new areas such as
    bio-informatics, homeland security, and social
    systems.
  • Continuous differential equation models remain an
    essential component in multi-formalism
    compositions.

4
Multi-formalism Compositions
  • A host of formalisms have emerged in the last few
    decades that greatly increase our ability to
    express features of the real world and employ
    them in engineering systems.
  • Such formalisms include Neural Networks, Fuzzy
    Logic Systems, Cellular Automata, Evolutionary
    and Genetic Algorithms, among others.
  • Hybrid models combine two or more formalisms,
    e.g., fuzzy logic control of continuous
    manufacturing process.
  • Most often, applications will require such
    hybrids to address the problem domain of
    interest.

5
Fundamental Systems Problems
6
MS Entities and Relations
Device for executing model
Real World
Simulator
Data Input/output relation pairs
modeling relation
simulation relation
Each entity is represented as a dynamic
system Each relation is represented by a
homomorphism or other equivalence
Model
structure for generating behavior claimed to
represent real world
7
MS Framework Continuous case
Real World
Simulator
modeling relation
simulation relation
  • Numerical Integration
  • Accuracy
  • Error effects

Model
  • Validity
  • Accuracy of
  • -retro-diction
  • -prediction

8
Specification Levels for Differential Equation
Systems
9
Canonical Ordinary Differential Equation Model
d q1(t)/dt f1(q1(t), q2(t), ..., qn(t), x1(t),
x2(t),..., xm(t)) d q2(t)/dt f2(q1(t), q2(t),
..., qn(t), x1(t), x2(t),..., xm(t)) ... d
qn(t)/dt fn(q1(t), q2(t), ..., qn(t), x1(t),
x2(t),..., xm(t))
10
Numerical Integration
Euler or rectangular method.
q((n1)h)q(nh)hf(q(nh),x(nh))
11
Feedback Coupling
12
Feedback Qualitative Analysis
13
2nd Order Linear System (undamped)
14
Continuous system simulation languages and
systems
  • state-space description languages
  • Continuous System Simulation Language (CSSL)
    standard, e.g., ACSL
  • block oriented simulation systems, e.g.,
    Simulink

Van der Pol Oscillator
CSSL PROGRAM Van der Pol INITIAL constant k
-1, x0 1, v0 0, tf 20 END DYNAMIC D
ERIVATIVE x integ(v, x0) v integ((1
x2)v kx, v0) END termt
(t.ge.tf) END END
15
Simulink building blocks
16
Van der Pol Oscillator in Simulink Block Diagram
17
1st Order Linear System exponential growth/decay
18
1st Order constant input toexponential decay
19
Damped Linear oscillator of second order
20
Van der Pol Oscillator Dynamic Behavior
21
Lotka Volterra Model and Behavior
  • Exercise
  • write the ODE for the model
  • find the equilibrium point of the Lotka-Volterra
    model
  • investigate the oscillations around this
    equilibrium.
  • where do the maximum and minimum populations
    occur?
  • show that small oscillations around the
    equilibrium are approximated by the 2nd order
    linear oscillator.

py prey population pd predator population
22
Locating Min/Max using Zero Crossings
23
Limit Cycle and Chaos are Opposites
  • limit cycles initial state eventually winds up
    in a periodic loop or cycle
  • chaos trajectories are sensitive to initial
    states small difference in initial state
    results in large difference in trajectory
  • Note ODE models are deterministic if the
    input is zero, then if a trajectory returns to an
    earlier state, it will get into a cycle
  • If a chaotic model has a trajectory that comes
    close to an earlier state than it diverges from
    that earlier portion due to its sensitivity to
    initial states
  • BUT a chaotic model can have a strange
    attractor i.e., a subset to which always
    returns, though not with a fixed period.

24
Rössler Model and Chaotic Behavior
state plane (v , z) to x
time behavior
interactive applet at http//www.geom.uiuc.edu/
worfolk/apps/Rossler/
25
Rössler Behavior
a b 0.2, and c 8.0.
http//mathforum.org/advanced/robertd/rossler.html
http//astronomy.swin.edu.au/pbourke/fractals/ros
sler/
26
Lorenz Attractor Butterfly Effect
For a lt 1 the solution rapidly decays to the
origin XYZ0. This corresponds to no motion in
the fluid context. For a gt 1 (e.g. a5) the
orbit approaches one of two fixed points
(depending on the initial values) away from the
origin. The fixed points are at X 2 Y 2Za-1.
In the convection context this corresponds to
nonzero but steady fluid flow (in a circulating
"roll" configuration). At larger values of a,
for example a24.1, the long time dynamics may
either approach one of the fixed points or a
strange attractor (depending on the choice of
initial values), which coexist at these values of
a. (Choose nearby initial values to find
solutions that converge to the fixed points.)
For agt24.74 the strange attractor collides with
the fixed points, which become unstable so that
practically all initial values lead to the
familiar butterfly dynamics. a28 gives the
usual picture.
Java applet http//www.cmp.caltech.edu/mcc/chaos
_new/Lor_docs/intro.html
http//astronomy.swin.edu.au/pbourke/fractals/lor
enz/
27
References/Literature
  • Course Notes from B. P., H. Praehofer and T. G.
    Kim (2000). Theory of Modeling and Simulation
    Integrating Discrete Event and Continuous Complex
    Dynamic Systems, (2nd Ed.) Academic Press, NY.)
  • On reserve A First Course in Differential
    Equations The Classic Fifth Edition (Hardcover)
    by Dennis G. Zill, Brooks Cole 5 edition
    (December 8, 2000)
  • Others
  • The Nonlinear Workbook Chaos, Fractals, Celluar
    Automata, Neural Networks, Genetic Algorithms,
    Gene Expression Programming, Support Vector
    Machine, Wavelets, Hidden Markov M (Paperback) by
    Willi-Hans Steeb, 588 pages, Publisher World
    Scientific Publishing Company 3rd edition (July
    15, 2005)
  • Modeling and Analysis of Post-Conflict
    Reconstruction, Damon B. Richardson, Richard F.
    Deckro, and Victor D. Wiley, JDMS The Journal of
    Defense Modeling and Simulation,October 2004,
    Volume 1 Number 4
  • Fernando J. Barros, A Formal Representation of
    Hybrid Mobile Component, SIMULATION, May 2005
    81 381 - 393.
  • What is signal and what is noise in the brain?
    A.Knoblauch, G.Palm, Biosystems 79(1-3), pp
    83-90, 2005.
  • Discrete Event Multi-Level Models for Systems
    Biology, Uhrmacher, A.M. and Degenring, D. and
    Zeigler, B.P, LNCS Transactions on Computational
    Systems Biology, Vol. 1, 3380/2005, pp. 66-85.
  • Modifications of the Helbing-Molnár-Farkas-Vicsek
    Social Force Model for Pedestrian Evolution,
    Taras I. Lakoba, D. J. Kaup, and Neal M.
    Finkelstein, SIMULATION 2005 81 339-352.
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