Modeling complex systems by simple mathematical models using selfsimilarity and fractionalorder syst

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Modeling complex systems by simple mathematical
models using self-similarity and fractional-order
system identification

• April 19, 2007
• Mechanical Systems GS
• Jason Mayes

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Outline

• Present Outline
• Talk
• Answer Questions

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Outline

• Mathematical modeling
• Philosophy
• Complex systems
• Self-similarity
• Mathematical
• Physical
• Fractional-order system identification

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Mathematical Modeling
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• Electro-mechanical model
• Equations of motion
• Add complexity
• Experiments

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Mathematical Modeling

• Finding a mathematical model is finding a
  compromise between an intractable problem and
  finding a model that sufficiently describes the
  system
• Complexity of our model depends on what we need
• Dont need great detail ? dont need detailed
  model

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The philosophy of analysis

• Many ways to find a mathematical model

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Methodological Reductionism

• Decartes 1637 Discourse on Method
• The world is like a machine
• Everything can be reduced to many smaller,
  simpler things
• The best way to understand a system is to first
  gain a clear understanding of its smallest
  subsystems
• Models are based on first principles
• Problems
• Size and complexity

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Holism

• Behavior must be studied on the level of the
  system as a whole
• Aristotles Metaphysics the whole is more than
  the sum of its parts
• Examples
• Neural nets
• On/off, PID, fuzzy logic
• Expert systems
• Are useful for describing or controlling systems,
  but dont explain observed behavior

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A model-based compromise

• Mix of holistic and reductionist approaches
• Model-based system identification
• Can form a model containing a few free parameters
• Very common in heat transfer
• Nusselt number correlations
• Conductivity, convection
• Heat exchangers
• Trend in science Holistic?Reductionist analysis
• Example chemical reactions

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Complex systems

• What is a complex system?
• Physically any system composed of a large number
  of components and interactions that creates
  difficulties in both understanding and modeling
• Mathematically a large system of coupled
  equations which are either too complex or too
  large to admit a sufficiently useful model of
  solution
• Properties
• Size/complexity
• Non-linear
• Emergent phenomena
• Memory

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New approaches

• Using self-similarity
• Can reduce complex mathematical models to simple
  models
• IF full solutions arent needed
• Fractional-order system identification

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Mathematical self-similarity

• Reducible mathematical models
• PDEs ? system of ODEs
• System of ODEs ? scalar ODE
• High-order scalar ODE ? lower order scalar ODE
• Mathematical similarity
• Equations of the same form
• Repeating pattern or coupling in equations

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Reducing infinite-order ODEs

• High-order ODEs can be reduced in the Laplace
  domain

Geometric Series!!
n ? 8
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Reducing infinite-order ODEs

• Can reduce an infinite-order ODE to a simple,
  finite-order ODE
• Only useful in complex mechanical systems if
  infinite-order ODEs occur in modeling

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Reducing infinite sets of ODEs

• High-order (infinite) ODEs result from large
  (infinite) equations sets

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Reducing infinite sets of ODEs

• Consider a very large system of springs and masses

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Reducing PDEs

• Infinite sets of ODEs also result from the
  reduction of PDEs
• Finite-volume
• Spectral methods
• Finite difference
• Example heat equation

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Reducing PDEs

• Finite-volume formulation

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Reducing PDEs

• An infinite set of differential equations!

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Reducing PDEs

• Can now reduce the continued fraction
• Now take the inverse Laplace transform

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Reducing PDEs

• PDE has been reduced to a single fractional-order
  ODE
• Reduction process
• Alternative
• Solve PDE numerically
• Differentiate at the boundary
• Get a global solution

PDE ? System of ODEs ? Single high-order ODE ?
Single low-order ODE
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Applications

• Only need a local solution
• Change of boundary conditions
• Laser or cryogenic surgery

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Using physical self-similarity

• Self-similar equations sets also result from
  physically self-similar systems

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Additional forms of similarity

• Similarity between operators can allow for
  further reduction
• Symmetric/Asymmetric
• Generation dependence
• Four cases
• Symmetric and generation independent
• Asymmetric and generation independent
• Symmetric and generation dependent
• Asymmetric and generation dependent
• All cases allow for reduction

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Fractional-order system identification

• Traditional system identification usually assumes
  integer-order models
• Fractional-order models can often provide a
  better fit
• For nearly-exponential systems

• Need models for control
• PID vs. PI?Dµ
• Parameter tuning for fractional-order controllers

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The end.
Questions?