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
Serkan GÜNEL
Izmir Institute of Technology
Dept. of Electrical Electronics Engineering
Urla 35430, Izmir
[email protected]
Dokuz Eylül University
Dept. of Electrical Electronics Engineering
Buca, 35160, Izmir
[email protected]
Contents

Deterministic and indeterministic systems under influence of uncertainty...



Estimating state probability densities using kernel density estimators



Evolution of state probability densities
Transformations on probability densities
Markov Operators & Frobenius—Perron Operators
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
X
S(n-2)
x1
xn-1
S
S
xn
x0
fn
D(X)
P
f1
fn-1
P
Pn-2
f0
Calcutating FPO

S differentiable & invertible
Logistic Map


α=4
Estimating Densities from Observed Data

Parzen’s Estimator
Observation vector :
d
}
=1
i=1,...,n
Logistic Map — Parzen’s Estimation
Logistic Map a =4
Chua’s Circuit
E
-E
Chua’s Circuit — Dynamics
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


Entropy = Disorder of the system = Information gained by observing the system
Classius
Q
H 
0
T
Boltzman
Q : Energy transfered to the system
T
Shannon
n: number of events
pi: probability of event “i”
: Temprature (Average Kinetic Energy)
Thermodynamics
Information Theory
Entropy


Estimated Entropy – Logistic Map
Estimated Entropy — Chua’s Circuit
Estimated Entropy — Chua’s Circuit II
Entropy in Control Systems I

External Effects
e(t)
p(e)
x(t)
p(x)
Change in entropy :
If State transition transformation is measure preserving, then


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.