Synchronization of Chaotic and Stochastic Neurons Using Differential Flatness Theory - Advanced Models of Neural Networks - page 182

0 1 0 1 0 1 0 1

y 1
y 2
y 3

010
001
000

y 1
y 2
y 3

0
0
1

@ A D @ A @ A C @ A v

(9.6)

It is noted that differential flatness can be proven for several other types
of chaotic oscillators, such as Lorenz's system, Rössler's system, and Chua's
system. By expressing all state variables and the control input of such systems
as functions of the flat output and their derivatives it is possible to obtain again a
description in the linear canonical (Brunovsky) form.
Having written the chaotic oscillator in the linearized canonical form

y .n/ D v (9.7)

Then it suffices to apply a feedback control of the form

v D y .n/
d

y .n1/
d

v k 1 .y .n1/

/ k n1 . y y d / k n .y y d / (9.8)

to make the oscillator's output y.t/ convergence to the desirable setpoint y d .t/.
Considering that v D f.y; y; ; y .r/ /C u the control input that is finally applied
to the system is u D v f.y; y; ; y .r/ /. Using the control input u all state
vector elements x i of the chaotic oscillator will finally converge to their own
setpoints.

9.2 Stabilization of Interacting Particles Which

Are Modelled as Coupled Stochastic Oscillators

It will be shown that it is possible to succeed synchronizing control of models of
stochastic neural oscillators, which are equivalent to interacting diffusing particles
[ 35 , 57 , 124 ]. To this end one can consider either open-loop or closed-loop control
approaches. Open-loop control methods are suitable for controlling micro and
nano systems, since the control signal can be derived without need for on-line
measurements [ 12 , 28 , 41 , 102 ]. A different approach would be to apply closed-
loop control using real-time measurements, taken at micro or nano-scale. The next
sections of this chapter are concerned with open-loop control for particle systems.
The proposed control approach is flatness-based control [ 58 , 126 , 153 , 172 ].
A multi-particle system that consists of N particles is considered. It is assumed
that the particles perform diffusive motion, and interact to each other as the theory of
Brownian motion predicts. As explained, Brownian motion is the analogous of the
quantum harmonic oscillator (QHO), i.e. of Schrödingers equation under harmonic
(parabolic) potential [ 56 ]. Moreover, it has been shown that the diffusive motion
of the particles (kinematic model of the particles) can be described by Langevin's
equation which is a stochastic linear differential equation [ 56 , 66 ].