Representing complex engineering systems by simple mathematical models

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Representing complex engineering systems by
simple mathematical models

• May 23, 2007
• Jason Mayes

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Outline

• Motivation
• Mathematical modeling
• Complex systems
• Objectives
• Present Work
• Using self-similarity
• Using fractional-order system identification
• Future Work

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Mathematical Modeling

• A mathematical model is a mathematical
  description of a physical system or process
• Design
• Prediction
• Control
• Mathematical modeling finding a compromise
  between an intractable problem and a model that
  sufficiently describes the system
• Characteristics of a desirable model
• Simple
• Adequately describes the system

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Mathematical Modeling

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

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The philosophy of analysis

• Many ways to find a mathematical model

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Methodological Reductionism

• Descartes 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
• Black-box analysis is useful for describing or
  controlling systems, but it doesnt explain
  observed behavior

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A model-based compromise

• Mix of holistic and reductionist approaches
• Model-based parameter identification
• Can form a model containing a few free parameters
• Very common in heat transfer
• Nusselt number correlations, convection
• Contact resistance, fouling coefficients
• Heat exchangers
• Friction factor (Moody diagram)
• Trend in science Holistic?Reductionist analysis
• Example chemical reactions

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Complex systems

• Q What is a complex system?
• A ???
• Properties
• Size/complexity
• Non-linear
• Emergent phenomena
• Memory
• Our working definition complex engineering
  systems
• 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 solution

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Objectives

• To find simple mathematical models of complex
  engineering systems
• Using both reductionist and holistic approaches
  to modeling
• Systems with similarity
• Both physical and mathematical similarity
• Can reduce complex mathematical models to simpler
  models
• Using fractional-order system identification
• Modeling complex systems as a black-box
• Better results than traditional integer-order
  models

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Mathematical self-similarity

• Current Work using a reductionist approach and
  taking advantage of mathematical similarity to
  simplify complex models
• Mathematical similarity
• Equations of the same form
• Repeating patterns or coupling in equations
• Reducible mathematical models
• Complex system large system of ODEs
• System of first order ODEs ? scalar ODE
• High-order scalar ODE ? lower order scalar ODE

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Mathematical Similarity Reducing infinite-order
ODEs

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

Geometric Series!!
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Mathematical Similarity Reducing infinite-order
ODEs

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

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Mathematical Similarity Reducing infinite sets
of ODEs

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

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Mathematical Similarity Reducing infinite sets
of ODEs

• Consider a very large system of springs and masses

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Mathematical Similarity 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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Mathematical Similarity Reducing PDEs

• Finite-volume formulation

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Mathematical Similarity Reducing PDEs

• An infinite set of differential equations

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Mathematical Similarity Reducing PDEs

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

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Mathematical Similarity Reducing PDEs

• Reduced the PDE in a way that we have extracted
  the heat flux at the boundary
• Reduction process
• Alternative
• Solve PDE
• Get a global solution
• Differentiate at the boundary

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

• Only need a local solution
• Change of boundary conditions
• Blast furnace monitoring Oldham and Spanier,
  1974
• Laser or cryogenic surgery
• Heat equation Taler, 1996
• Need only one thermocouple

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Physical self-similarity

• Current work using physical self-similarity to
  reduce complex models
• Potential driven flows through bifurcating trees
• Self-similar equations sets can also result from
  physically self-similar systems

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Physical Similarity A self-similar model

• The bifurcating tree geometry
• Geometry seen in a wide variety of applications
  Bejan, 2000
• Potential-driven flow or transfer
• ex. heat, fluid, energy, etc.
• Conservation at bifurcation points

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Physical Similarity A self-similar model

• Assumptions
• Conservation at nodes
• Transfer governed by a linear operator
• i.e., for each branch
• Large system of DAEs
• 2n1 -2 differential branch equations
• 2n-1 continuity equations

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Physical Similarity Reduction

• System of DAEs can be reduced as before
• Regular coupling from physical similarity allows
  for reduction
• For N1 generation network

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Physical Similarity Reduction

• For N-generation network

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Physical Similarity Additional forms of
similarity

• Similarity in operators can be used to further
  simplify
• Two forms of similarity
• Similarity within a generation
• Symmetric networks the operators within a
  generation are identical
• Asymmetric networks the operators within a
  generation are not identical
• Similarity between generations
• Generation dependent operators depend on the
  generation in which the operator occurs and
  change between successive generations
• Generation independent operators do not change
  between generations
• Four possible combinations
• Symmetric with generation independent operators
• Asymmetric with generation independent operators
• Symmetric with generation dependent operators
• Asymmetric with generation dependent operators

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Physical Similarity Symmetric and generation
independent

• Can further reduce the continued fraction

For infinite networks
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Physical Similarity Asymmetric and generation
independent

• Further reduction

For infinite networks
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Physical Similarity Symmetric and generation
dependent

• Further reduction

• For infinite networks
• Can further reduce for certain forms of

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Physical Similarity Asymmetric and generation
dependent

• Further reduction
• No general reduction
• Can reduce further for specific cases

For infinite networks
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Physical Similarity Applications

• Viscoelasticity
• Asymmetric, generation independent
• Fractional-order viscoelastic models
• Springs (k) and dampers (µ)
• Heymans and Bauwens, 1994

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Physical Similarity Applications

• Laminar flow through bifurcating trees
• Symmetric, generation dependent
• Using laminar pipe flow model for each branch

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Using Similarity Summary of current work

• Representing complex engineering systems by
• simple mathematical models
• From a reductionist perspective
• Mathematical systems with similarity
• PDEs ? Systems of ODEs ? High-order scaler ODE ?
  Lower-order scaler ODE
• Physical systems with similarity
• Use physical structure to reduce model
• Additional forms of similarity can offer further
  reduction

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Fractional-order system ID

• Current work using fractional-order models to
  better fit nearly exponential experimental data
• System identification building a dynamic
  mathematical model of a system using
  measured/observed data
• Holistic approach to modeling
• Grey- or black-box modeling
• Typically assumes integer-order models
• Fractional-order models can
• often do a better job describing
• real physical systems
• Podlubny, 1999

?
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Fractional-order system ID Nearly exponential
transitions

• Focus on nearly exponential transitions
• A transition from one steady-state to
• another, usually the result of step change
• in input
• Typically assumed to be some combination
• of exponentials
• Commonly modeled as first or second
• order systems
• Fractional-order models can often do a better job

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Fractional-order system ID Linear systems a toy
problem

• Consider the system
• System representation
• System identification

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Fractional-order system ID Linear systems a
fractance device

• Fractance device an electrical circuit having
  properties which lie between resistance and
  capacitance or resistance and inductance.
• Has an impedance
• Construction infinite bifurcating network of
  resistors and capacitors
• Nakagawa and Sorimachi, 1994
• N 1
• N 8

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Fractional-order system ID Linear systems a
fractance device

• For N1, 3, 6, and 9 generation fractance devices

N 1
N 9
Fractional-order model is a special case of the
integer-order model
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Fractional-order system ID Nonlinear systems a
toy problem

• Consider the system
• System representation
• System identification

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Fractional-order system ID Nonlinear systems
experimental results

• Shell-and-tube heat exchanger
• Steady state correlations are
• readily available
• Interested in a dynamic correlation
• for the response to a step input
• Complex systems reductionist approach
• leads a complex model
• System of PDEs
• Turbulence, recirculation
• Very detailed
• Holistic or black-box experimentally determine
  response
• Measured hot-side outlet temperature
• Step change in cold-side inlet flow rate

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Fractional-order system ID Nonlinear systems
experimental results

• Experimental results

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Fractional-order system ID Summary of current
work

• Representing complex engineering systems by
  simple mathematical models
• From a holistic perspective
• Fractional-order models often better descriptors
  of complex systems than traditional integer-order
  models
• Integer-order models are special cases of
  fractional-order models
• Useful for modeling linear and non-linear systems
• Heat exchanger shown to be modeled well using a
  fractional-order model Aoki, 2006

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Future Work

• Work to proceed in two directions
• Reductionism
• Finding simple mathematical models of complex
  engineering systems that exhibit similarity
  properties
• Holism
• Using simple fractional-order models

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Future Work Reduction using similarity

• Reducing PDEs
• Approximate boundary condition transformations
• Pennes equation
• 2-D PDEs

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Future Work Reduction using similarity

• Reducing physical systems
• Grid-like transfer networks

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Future Work Reduction using fractional-order
models

• Using fractional-order models
• Continue heat exchanger experiments to better
  develop dynamic correlation
• Fractional-order control
• Fractional-order systems can be controlled more
  efficiently using fractional-order control
  techniques
• Using fractional-order system identification and
  experimental results to choose proper controller
  gains
• Similar to Ziegler-Nichols method of parameter
  tuning
• Based on offline time response characteristics

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Questions?