MESA Lab

Interesting TOPICS in ICFDA2014

xiaodong sunMESA (Mechatronics, Embedded Systems and Automation)Lab

School of Engineering,University of California, Merced

E: xsun7@ucmerced.edu Phone:209 201 1947Lab: CAS Eng 820 (T: 228-4398)

June 2, 2014. Applied Fractional Calculus Workshop Series @ MESA Lab @ UCMerced

MESA LabWhat attract my attention about icfda2014?

topic 1: Improved Fractional Kalman Filter for Variable Order Systems

main idea:This 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

MESA Lab

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 integrators[3] .

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 &FDA’10, 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. 1111–1118, 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. 459–466, 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. 17–20.

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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 scheme[2].

＊ improve measurement results from MEMS sensors [3].

[1]A. Dzielinski and D. Sierociuk, “Ultracapacitor modelling and control using discrete fractional order state-space model,” Acta Montanistica Slovaca, vol. 13, no. 1, pp. 136–145, 2008.

[2]A. 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. 863–879, 2009.

[3]M. 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. 199–209, 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:

Unscented Kalman Filter

Kalman Filter

Extended Kalman Filter

Linear system

Unscented Kalman Filter

Kalman Filter

Extended Kalman Filter

Integer order

fractional order

nonlinear system

nonlinear system

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Improved Fractional Kalman Filter for Variable Order SystemsExtented 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 SystemsExtented 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 noise[1]

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) 863–879[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

MESA Lab

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 linearizi

ng the nonlinear function by using Taylor series expansions, UKF uses the true nonlinear models and approximates a Gaussian distribution of the state randomvariable, 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

UKF+Fractional calculus =future work?

A LOT OF WORK TO be doned(constant order +ukf ,variable order +ukf)

MESA LabA physical approach to the connection between fractal geometry and frac

tional calculusHeighlight:

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.

MESA LabA physical approach to the connection between fractal geometry and frac

tional calculusHeighlight:

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.

MESA LabA physical approach to the connection between fractal geometry and frac

tional calculus

THE MODELConsider 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

𝑁( ) = is the number of -th order pipes, ( ) is their surface, and ( , ) is 𝑛 𝑁𝑛 𝑛 𝑆 𝑛 𝑣 𝑛 𝑡the 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:

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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 scaleThe results are very close to what following eq.

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Thank you !