Interesting TOPICS in ICFDA2014

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Interesting TOPICS in ICFDA2014

• xiaodong sun
• MESA (Mechatronics, Embedded Systems and
  Automation)Lab
• School of Engineering,
• University of California, Merced
• E xsun7_at_ucmerced.edu Phone[phone]
• Lab CAS Eng 820 (T 228-4398)

June 2, 2014. Applied Fractional Calculus
Workshop Series _at_ MESA Lab _at_ UCMerced
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What attract my attention about icfda2014?

• topic 1 Improved Fractional Kalman Filter for
  Variable Order Systems
• main ideaThis paper presents generalization of
  the Improved
• Fractional Kalman Filter (ExFKF) for variable
  order discrete
• state-space systems.
• topic 2 A physical approach to the connection
  between fractal geometry and fractional calculus
• main idea its goal is to prove the existence of
  a connection between fractal geometries and
  fractional calculus

3
Improved Fractional Kalman Filter for Variable
Order Systems

• Background Fractional variable order is the case
  in fractional order is vary in time.
• it is more complicated than constant order
  case.It can be found in many areas
• Fractional order modeling of physical
  electrochemical system 1.
• Description history of drag expression using
  Fractional variable order model 2.
• The variable order interpretation of the
  analog realization of fractional orders
  integrators3 .
• The applications of variable order
  derivatives and integrals in signal processing
  4.
• Numerical simulations algorithm of variable
  order systems 5.
• 1 H. Sheng, H. Sun, Y. Chen, and G. W.
  Bohannan,Physical experimental study of
  variable-order fractional integrator and
  differentiator,4th IFAC FDA10, 2010.
• 2 L. Ramirez and C. Coimbra, On the variable
  order dynamics of the nonlinear wake caused by a
  sedimenting particle, Physica D-Nonlinear
  Phenomena, vol. 240, no. 13, pp. 11111118, 2011.
• 3 D. Sierociuk, I. Podlubny, Experimental
  evidence of variable-order behavior of ladders
  and nested ladders, Control Systems Technology,
  IEEE Transactions on, vol. 21, no. 2, pp.
  459466, 2013.
• 4 H. Sheng, Y. Chen, and T. Qiu, Signal
  Processing Fractional Processes and
  Fractional-Order Signal Processing. Springer,
  London, 2012.
• 5 C.-C. Tseng and S.-L. Lee, Design of
  variable fractional order differentiator using
  infinite product expansion, ECCTD, 2011, pp.
  1720.

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Improved Fractional Kalman Filter for Variable
Order Systems

• Application of FKF algorithm
• parameters and fractional order estimation for
  fractional order systems. eg, statevariables in
  the system with ultracapacitor 1.
• obtain a new chaotic secure communication
  scheme2.
• improve measurement results from MEMS sensors
  3.
• 1A. Dzielinski and D. Sierociuk,
  Ultracapacitor modelling and control using
  discrete fractional order state-space model,
  Acta Montanistica Slovaca, vol. 13, no. 1, pp.
  136145, 2008.
• 2A. Kiani-B, K. Fallahi, N. Pariz, and H.
  Leung, A chaotic secure communication scheme
  using fractional chaotic systems based on an
  extended fractional kalman filter,
  Communications in Nonlinear Science
• and Numerical Simulation, vol. 14, no. 3, pp.
  863879, 2009.
• 3M. Romanovas, L. Klingbeil, M. Traechtler, and
  Y. Manoli, Applicationof fractional sensor
  fusion algorithms for inertial mems sensing,
  Mathematical Modelling and Analysis, vol. 14, no.
  2, pp. 199209, 2009.

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Improved Fractional Kalman Filter for Variable
Order Systems

• what is Kalman filter a set of mathematical
  equations that provides an efficient
  computational (recursive) solution of the
  least-squares method.
• It can estimate past, present, and future states
  even when the precise nature of the modeled
  system is unknown.
• Applications areas signal processing, control,
  and communications.
• Develop history

Integer order
Linear system
nonlinear system
nonlinear system
fractional order
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Improved Fractional Kalman Filter for Variable
Order Systems

• Extented Kalman Filter fractional (a few papers
  )
• The EKF has been applied extensively to the field
  of nonlinear estimation. General application
  areas may be divided into state-estimation and
  machine learning. it propagated analytically
  through the first-order linearization of the
  nonlinear system.
• Drawbacks
• This can introduce large errors in the true
  posterior mean and covariance of the transformed
  Gaussian random variable, which may lead to
  sub-optimal performance and sometimes divergence
  of the filter.
• In addition, the state distribution only for
  Gaussian random variable.

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Improved Fractional Kalman Filter for Variable
Order Systems

• Extented Kalman Filter fractional (a few papers
  )
• Two relative papers
• The chaotic synchronization is implemented by
  the EFKF design in the presence of channel
  additive noise and noise1
• synchronization of chaotic systems is
  achieved by the EFKF algorithm in the presence of
  channel additive noise and processing noise.2

1.Arman Kiani-B , Kia Fallahi, Naser Pariz,
Henry Leung,A chaotic secure communication scheme
using fractional chaotic systems based on an
extended fractional Kalman filter. Communications
in Nonlinear Science and Numerical Simulation 14
(2009) 863879 2. Guanghui Sun, Mao Wang
.UNEXPECTED RESULTS OF EXTENDED FRACTIONAL KALMAN
FILTER FOR PARAMETER IDENTIFICATION IN FRACTIONAL
ORDER CHAOTIC SYSTEMS. ICIC , 2011 ISSN 1349-4198
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Improved Fractional Kalman Filter for Variable
Order Systems

• Unscented Kalman Filter (UKF) is a nonlinear
  filter, first proposed by Julier and
• Uhlmann (1997). Unlike Extended Kalman Filter
  (EKF) which is based on the linearizing the
  nonlinear function by using Taylor series
  expansions, UKF uses the
• true nonlinear models and approximates a Gaussian
  distribution of the state random
• variable, A central and vital operation performed
  in the Kalman filter is the propagation
• of a Gaussian random variable (GRV) through the
  system dynamics. the state
• distribution is represented by using a minimal
  set of carefully chosen sample points.
• These sample points completely capture the true
  mean and covariance of the GRV, and
• when propagated through the true nonlinear
  system, captures the posterior mean and
• covariance accurately to the 3rd order (Taylor
  series expansion) for any nonlinearity.
• Remarkably, the computational complexity of the
  UKF is the same order as that of the
• EKF.

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Improved Fractional Kalman Filter for Variable
Order Systems

• UKF and EKF

UKFFractional calculus future work? A LOT OF
WORK TO be doned (constant order ukf ,variable
order ukf)
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A physical approach to the connection between
fractal geometry and fractional calculus

• Heighlight
• the paper prove the existence of a
  connection between fractal geometries and
  fractional calculus.
• It show the connection exists in the
  physical origins of the power laws ruling the
  evolution of most of the natural phenomena, and
  that are the characteristic feature of fractional
  differential operators.
• the relevant example show that a power law
  comes up every time we deal with physical
  phenomena occurring on a underlying fractal
  geometry.
• The order of the power law depends on the
  anomalous dimension of the geometry, and on the
  mathematical model used to describe the physics.
• In the assumption of linear regime, by taking
  advantage of the Boltzmann superposition
  principle, a differential equation of not integer
  order is found, ruling the evolution of the
  phenomenon at hand.

11
A physical approach to the connection between
fractal geometry and fractional calculus

• Heighlight
• the paper prove the existence of a
  connection between fractal geometries and
  fractional calculus.
• It show the connection exists in the
  physical origins of the power laws ruling the
  evolution of most of the natural phenomena, and
  that are the characteristic feature of fractional
  differential operators.
• the relevant example show that a power law
  comes up every time we deal with physical
  phenomena occurring on a underlying fractal
  geometry.
• The order of the power law depends on the
  anomalous dimension of the geometry, and on the
  mathematical model used to describe the physics.
• In the assumption of linear regime, by taking
  advantage of the Boltzmann superposition
  principle, a differential equation of not integer
  order is found, ruling the evolution of the
  phenomenon at hand.

12
A physical approach to the connection between
fractal geometry and fractional calculus
THE MODEL
Consider a fractal medium, and we study the time
evolution of the flux of a viscous fluid seeping
through it. The expression for the flux of the
fluid flowing through this medium is
Fig. 2. Examples of Sierpinski carpets
??(??) ???? is the number of ??-th order pipes,
??(??) is their surface, and ??(??, ??) is the
velocity of the out flowing fluid.
rewrite the above formula
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A physical approach to the connection between
fractal geometry and fractional calculus

• In the paper, it firstly deal with the case of
  seeping of fractal medium, if the fluid through
  the fractal medium has appearance of
  fractional operators and power law character ,
  then we can got the connection between an
  expression for in modelling phenomena evolving on
  fractal geometries .
• For short , The time evolution of the flux
  through the medium is

14
A physical approach to the connection between
fractal geometry and fractional calculus
The numerical result for the time evolution of
the flux from fractal medium are shown in fig.4
in log-log scale The results are very close to
what following eq.
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• Thank you !