Title: Modeling complex systems by simple mathematical models using selfsimilarity and fractionalorder syst
1Modeling complex systems by simple mathematical
models using self-similarity and fractional-order
system identification
2Outline
- Motivation
- Mathematical modeling
- Objectives
- Present Work
- Using self-similarity
- Using fractional-order system identification
- Future Work
3Mathematical Modeling
- Mathematical modeling finding a compromise
between an intractable problem and a model that
sufficiently describes the system - Intractable models can always be made more
complex - Sufficiently do not always need a complex model
- Compromise
- Complexity of our model depends on what we need
- Dont need great detail/accuracy
- ? dont need detailed or complex model
- a good model is a model that is as simple as
possible but still adequately describes the
system
4Mathematical Modeling
xo
- Electro-mechanical model
- Equations of motion
- Add complexity
- Experiments
5The philosophy of analysis
- Many ways to find a mathematical model
6Methodological 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
7Holism
- 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
8A 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
9Complex 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 or
solution - Properties
- Size/complexity
- Non-linear
- Emergent phenomena
- Memory
10Objectives
- Reduce complex systems to simple mathematical
models - Using both reductionist and holistic approaches
- Using self-similarity
- Can reduce complex mathematical models to simple
models - Both physical and mathematical similarity
- Using fractional-order system identification
- Modeling complex systems as a black-box
- Better results than traditional integer-order
models
11Mathematical self-similarity
- Current Work using a reductionist approach and
taking advantage of mathematical similarity to
reduce 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
12Mathematical Similarity Reducing infinite-order
ODEs
- High-order ODEs can be reduced in the Laplace
domain
Geometric Series!!
n ? 8
13Mathematical Similarity 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
14Mathematical Similarity Reducing infinite sets
of ODEs
- High-order (infinite) ODEs result from large
(infinite) equations sets
15Mathematical Similarity Reducing infinite sets
of ODEs
- Consider a very large system of springs and masses
16Mathematical Similarity Reducing PDEs
- Infinite sets of ODEs also result from the
reduction of PDEs - Finite-volume
- Spectral methods
- Finite difference
- Example heat equation
17Mathematical Similarity Reducing PDEs
- Finite-volume formulation
18Mathematical Similarity Reducing PDEs
- An infinite set of differential equations!
19Mathematical Similarity Reducing PDEs
- Can now reduce the continued fraction
- Now take the inverse Laplace transform
20Mathematical Similarity 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
21Mathematical Similarity Applications of PDE
reduction
- Only need a local solution
- Change of boundary conditions
- Laser or cryogenic surgery
22Physical 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
23Physical Similarity A self-similar model
- The bifurcating tree geometry
- Geometry seen in a wide variety of applications
- Potential-driven flow or transfer
- ex. heat, fluid, energy, ect.
- Conservation at bifurcation points
24Physical Similarity A self-similar model
- Assumptions
- Transfer governed by a linear operator
- i.e., for each branch
- Large system of DAEs
- 2n1 -2 differential branch equations
- 2n-1 continuity equations
25Physical Similarity Reduction
- System of DAEs can be reduced as before, even
without knowing the branch operators - Regular coupling from physical self-similarity
allows for reduction - For integro-differential operators
26Physical 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
27Physical Similarity Generation independent
operators
28Physical Similarity Generation dependent
operators
29Physical Similarity Applications
- Viscoelasticity
- Asymmetric, generation independent
- Fractional-order viscoelastic models
- Springs (k) and dampers (µ)
30Physical Similarity Applications
- Laminar flow through bifurcating trees
- Symmetric, generation dependent
- Using laminar pipe flow model for each branch
31Fractional-order system ID
- Current work using fractional-order models to
better fit nearly exponential data - System identification building a dynamic
mathematical model of a system using
measured/observed data - Holistic or black-box approach to reduction
- Grey- or black-box modeling
- Typically assumes integer-order models
- Fractional-order models can
- often do a better job describing
- real physical systems
32Fractional-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
33Fractional-order system ID Linear systems a toy
problem
- Consider the system
- System representation
- System identification
34Fractional-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 - N 1
- N 8
35Fractional-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
36Fractional-order system ID Nonlinear systems a
toy problem
- Consider the system
- System representation
- System identification
37Fractional-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
38Fractional-order system ID Nonlinear systems
experimental results
39Current Work Summary
- Reducing complex systems to simple 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 offer further
reduction - From a holistic perpective
- Fractional-order system identification
- Fractional-order models often better descriptors
of complex - systems than traditional integer-order models
40Future Work
- Work to proceed in two directions
- Reduction using similarity
- Reduction using fractional-order models
- Using Similarity
- Reducing PDEs
- Approximate boundary condition transformations
- for Pennes equation (and other PDEs)
- 2-D PDEs
- Reducing physical systems
- Grid-like transfer networks
41Future Work
- 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
42Questions?