Continuous System Modeling

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Continuous System Modeling
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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

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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.

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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.

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Fundamental Systems Problems
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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
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MS Framework Continuous case
Real World
Simulator
modeling relation
simulation relation

• Numerical Integration
• Accuracy
• Error effects

Model

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

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Specification Levels for Differential Equation
Systems
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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))
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Numerical Integration
Euler or rectangular method.
q((n1)h)q(nh)hf(q(nh),x(nh))
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Feedback Coupling
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Feedback Qualitative Analysis
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2nd Order Linear System (undamped)
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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
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Simulink building blocks
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Van der Pol Oscillator in Simulink Block Diagram
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1st Order Linear System exponential growth/decay
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1st Order constant input toexponential decay
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Damped Linear oscillator of second order
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Van der Pol Oscillator Dynamic Behavior
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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
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Locating Min/Max using Zero Crossings
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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.

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Rössler Model and Chaotic Behavior
state plane (v , z) to x
time behavior
interactive applet at http//www.geom.uiuc.edu/
worfolk/apps/Rossler/
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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/
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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/
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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.