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This research has been supported in part by European Commission FP6 IYTE-Wireless Project (Contract No: 017442). EVOLUTION OF THE STATE DENSITIES AND THE ENTROPIES OF DYNAMICAL SYSTEMS. Ferit Acar SAVACI Izmir Institute of Technology Dept. of Electrical Electronics Engineering

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Evolution of the state densities and the entropies of dynamical systems

This research has been supported in part by European Commission FP6 IYTE-Wireless Project (Contract No: 017442)


Evolution of the state densities and the entropies of dynamical systems

EVOLUTION OF THE STATE DENSITIES AND Commission FP6 IYTE-Wireless Project (Contract No: 017442)THE ENTROPIES OF DYNAMICAL SYSTEMS

Ferit Acar SAVACI

Izmir Institute of Technology

Dept. of Electrical Electronics Engineering

Urla 35430, Izmir

acarsavaci@iyte.edu.tr

Serkan GÜNEL

Dokuz Eylül University

Dept. of Electrical Electronics Engineering

Buca, 35160, Izmir

serkan.gunel@eee.deu.edu.tr


Contents
Contents Commission FP6 IYTE-Wireless Project (Contract No: 017442)

  • Deterministic and indeterministic systems under influence of uncertainty...

    • Evolution of state probability densities

    • Transformations on probability densities Markov Operators & Frobenius—Perron Operators

  • Estimating state probability densities using kernel density estimators

    • Parzen’s density estimator

    • Density estimates for Logistic Map and Chua’s Circuit

  • The 2nd Law of Thermodynamics and Entropy

    • Estimating Entropy of the system using kernel density estimations

      • Entropy Estimates for Logistic Map and Chua’s Circuit

    • Entropy in terms of Frobenius—Perron Operators

  • Entropy and Control

    • Maximum Entropy Principle

    • Effects of external disturbance and observation on the system entropy

    • Controller as a entropy changing device

    • Equivalence of Maximum Entropy minimization to Optimal Control


Motivation
Motivation Commission FP6 IYTE-Wireless Project (Contract No: 017442)

  • Thermal noise effects all dynamical systems,

  • Exciting the systems by noise can alter the dynamics radically causing interesting behavior such as stochastic resonances,

  • Problems in chaos control with bifurcation parameter perturbations,

  • Possibility of designing noise immune control systems

  • Densities arise whenever there is uncertainty in system parameters, initial conditions etc. even if the systems under study are deterministic.


Frobenius perron operators
Frobenius—Perron Operators Commission FP6 IYTE-Wireless Project (Contract No: 017442)

  • Definition


Evolution of the state densities of the stochastic dynamical systems
Evolution of The State Densities of The Stochastic Dynamical Systems

  • i’s are 1D Wiener Processes

Fokker—Planck—KolmogorovEqu.

  • p0(x) : Initial probability density of the states


Infinitesimal operator of frobenius perron operator

Systems

Infinitesimal Operator of Frobenius—Perron Operator

AFP : D(X)D(X)

D(X): Space of state probability densities

FPK equation in noiseless case


Stationary solutions of fpk eq
Stationary Solutions of FPK Eq. Systems

Reduced Fokker—Planck—Kolmogorov Equ.


Frobenius perron operator

X Systems

S(n-2)

S

x1

xn-1

xn

S

x0

fn

D(X)

P

f1

fn-1

P

Pn-2

f0

Frobenius—Perron Operator


Calcutating fpo

S Systemsdifferentiable & invertible

Calcutating FPO


Logistic map
Logistic Map Systems

  • α=4


Estimating densities from observed data

i=1,...,n Systems

d

Observation vector :

Estimating Densities from Observed Data

  • Parzen’s Estimator

}

 = 1



Logistic map a 4
Logistic Map Systemsa =4


Chua s circuit
Chua’s Circuit Systems

E

-E



Chua s circuit the state densities
Chua’s Circuit — The state densities Systems

p(x)

Limit Cycles

a

x

Double Scroll

Period-2 Cycles

Details

Scrolls


The 2 nd law of thermodynamics information

SystemsQ : Energy transfered to the systemT : Temprature (Average Kinetic Energy)

The 2nd Law of Thermodynamics & Information

Entropy = Disorder of the system = Information gained by observing the system

Classius

Shannon

Boltzman

n: number of events pi: probability of event “i”

Thermodynamics

Information Theory


Entropy
Entropy Systems





Entropy in control systems i

x(t) Systemsp(x)

e(t)p(e)

x(t)p(x)

y(t)p(y)

Entropy in Control Systems I

  • External Effects

Change in entropy :

If State transition transformation is measure preserving, then

  • Observer Entropy


Entropy of control systems ii
Entropy of Control Systems II Systems

  • Mutual Information

  • Theorem


Uncertain v s certain controller
Uncertain v.s Systems. Certain Controller

  • Theorem

  • Theorem



Optimal control with uncertain controller ii
Optimal Control with Uncertain Controller II Systems

Select p(u) to maximize

subject to





Summary i
Summary I Systems

  • The state densities of nonlinear dynamical systems can be estimated using kernel density estimators using the observed data which can be used to determine the evolution of the entropy.

  • Important observation : Topologically more complex the dynamics results in higher stationary entropy

  • The evolution of uncertainty is a trackable problem in terms of Fokker—Planck—Kolmogorov formalism.

    • The dynamics in the state space are converted to an infinite dimensional system given by a linear parabolic partial diff. equation (The FPK Equation),

    • The solution of the FPK can be reduced to finding solution of a set of nonlinear algebraic equations by means of weighted residual schemes,

  • The worst case entropy can be used as a performance criteria to be minimized(maximized) in order to force the system to a topologically simpler dynamics.


Summary ii
Summary II Systems

  • The (possibly stochastic) controller performance is determined by the information gather by the controller about the actual system state.

  • A controller that reduces the entropy of a dynamical system must increase its entropy at least by the reduction to be achieved.