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 THE ENTROPIES OF DYNAMICAL SYSTEMS

Ferit Acar SAVACIIzmir Institute of Technology

Dept. of Electrical Electronics Engineering

Urla 35430, Izmir

acarsavaci@iyte.edu.tr

Serkan GÜNELDokuz Eylül University

Dept. of Electrical Electronics Engineering

Buca, 35160, Izmir

serkan.gunel@eee.deu.edu.tr

Contents 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

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

Definition

Evolution of The State Densities of The Stochastic Dynamical Systems

• i’s are 1D Wiener Processes

Fokker—Planck—Kolmogorov Equ.

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

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.

Reduced Fokker—Planck—Kolmogorov Equ.

Frobenius—Perron Operator

x0

x1 xn-1

xnS

S(n-2)S

X

D(X)

f0

f1 fn-1

fn

n-2

Calcutating FPO

S differentiable & invertible

Logistic Map

α=4

Estimating Densities from Observed Data Parzen’s Estimator

Observation vector : d i=1,...,n

} = 1

Logistic Map — Parzen’s Estimation

Logistic Map =4

Chua’s Circuit

-E

E

Chua’s Circuit — Dynamics

Chua’s Circuit — The state densities

x

p(x)

Limit Cycles

Period-2 Cycles

Scrolls

Double Scroll

Details

The 2nd Law of Thermodynamics & Information

0

T

QH

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

Classius Boltzman

Thermodynamics

Shannon

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

Information Theory

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

Entropy

Estimated Entropy – Logistic Map

Estimated Entropy — Chua’s Circuit

Estimated Entropy — Chua’s Circuit II

Entropy in Control Systems I External Effects

x(t)p(x)

e(t)p(e)

If State transition transformation is measure preserving, then

Change in entropy :

Observer Entropy

x(t)p(x)

y(t)p(y)

Entropy of Control Systems II Mutual Information

Theorem

Uncertain v.s. Certain Controller Theorem

Theorem

Principle of Maximum Entropy

Theorem

Optimal Control with Uncertain Controller II

Select p(u) to maximize

subject to

Optimal Control with Uncertain Controller III

Optimal Control with Uncertain Controller IV

Optimal Control with Uncertain Controller V

Theorem

Summary I 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

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.