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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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slide1
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 SAVACI

Izmir Institute of Technology

Dept. of Electrical Electronics Engineering

Urla 35430, Izmir

[email protected]

Serkan GÜNEL

Dokuz Eylül University

Dept. of Electrical Electronics Engineering

Buca, 35160, Izmir

[email protected]

contents
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
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.
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

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.

Reduced Fokker—Planck—Kolmogorov Equ.

frobenius perron operator

X

S(n-2)

S

x1

xn-1

xn

S

x0

fn

D(X)

P

f1

fn-1

P

Pn-2

f0

Frobenius—Perron Operator
estimating densities from observed data

i=1,...,n

d

Observation vector :

Estimating Densities from Observed Data
  • Parzen’s Estimator

}

 = 1

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

p(x)

Limit Cycles

a

x

Double Scroll

Period-2 Cycles

Details

Scrolls

the 2 nd law of thermodynamics information

Q : 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 in control systems i

x(t)p(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
  • Mutual Information
  • Theorem
optimal control with uncertain controller ii
Optimal Control with Uncertain Controller II

Select p(u) to maximize

subject to

summary i
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
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.
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