PASSIVITY OF A CLASS OF HOPFIELD NETWORKS

Application to Chaos Control

Adrian–Mihail Stoica

Faculty of Aerospace Engineering, University ”Politehnica” of Bucharest, Str. Polizu, No. 1, Ro-011061, Bucharest, Romania

Isaac Yaesh

Control Department, IMI Advanced Systems Div., P.O.B. 1044/77, Ramat–Hasharon, 47100, Israel

Keywords:

Chaos, Stochastic systems, Hopﬁeld Neural Networks, Recurrent Neural Networks, Sprocedure, Linear matrix

inequalities, Direct Adaptive Control.

Abstract:

The paper presents passivity conditions for a class of stochastic Hopﬁeld neural networks with state–dependent

noise and with Markovian jumps. The contributions are mainly based on the stability analysis of the considered

class of stochastic neural networks using inﬁnitesimal generators of appropriate stochastic Lyapunov–type

functions. The derived passivity conditions are expressed in terms of the solutions of some speciﬁc systems of

linear matrix inequalities. The theoretical results are illustrated by a simpliﬁed adaptive control problem for a

dynamic system with chaotic behavior.

1 INTRODUCTION

Hopﬁeld networks are symmetric recurrent neural

networks which exhibit motions in the state space

which converge to minima of energy.

Symmetric Hopﬁeld networks can be used to solve

practical complex problems such as implement asso-

ciative memory, linear programming solvers and op-

timal guidance problems. Recurrent networks which

are non symmetric versions of Hopﬁeld networks play

an important role in understanding human motor tasks

involving visual feedback (see (Cabrera and Milton,

2004) - (Cabrera et al., 2001) and the references

therein). Such networks seem to be subject to effects

of state-multiplicative noise, pure time delay (see (Hu

et al., 2003), (X. Liao and Sanchez, 2002) and (Sto-

ica and Yaesh, 2006)) and even multiple attractors

which can be caused by Markov jumps. Even with-

out Markov jumps, a non symmetric class of Hop-

ﬁeld networks is able to generate chaos (Kwok et al.,

2003). Therefore, Hopﬁeld networks can be used

(Poznyak and Sanchez, 1999) as identiﬁers of un-

known chaotic dynamic systems. The resulting iden-

tiﬁer neural networks have been used in (Poznyak and

Sanchez, 1999) to derive a locally optimal robust con-

troller to remove the chaos in the system.

In this paper, we consider to replace the robust

controller of (Poznyak and Sanchez, 1999) by a direct

adaptive controller. More speciﬁcally we consider the

so called Simpliﬁed Adaptive Control (SAC) method

(Kaufman et al., 1998) which applies a simple propor-

tional controller whose gain is adapted according the

squared tracking error. Since such controllers’ stabil-

ity proof involves a passivity condition, we derive a

passivity result for the Hopﬁeld network. Our results

are, in fact, developed for a generalized version of

non symmetric Hopﬁeld networks including Markov

jumps of the parameters and state multiplicative noise

thus allowing a wider stochastic class of chaos gener-

ating systems to be considered.

The paper is organized as follows. In Section 2,

the problem is formulated and in Section 3 Linear Ma-

trix Inequalities (LMIs) based conditions are derived

for passivity analysis. In Section 4 a chaos control

example is given and ﬁnally Section 5 includes con-

cluding remarks.

Throughout the paper R

n

denotes the n dimen-

sional Euclidean space, R

n×m

is the set of all n ×

m real matrices, and the notation P > 0, (respec-

tively, P≥ 0) for P ∈ R

n×n

means that P is symmet-

ric and positive deﬁnite (respectively, semi-deﬁnite).

Throughout the paper (Ω, F , P ) is a given probability

space; the argumentθ ∈ Ω will be suppressed. Expec-

tation is denoted by E{.} and conditional expectation

of x on the event θ(t) = i is denoted by E[x|θ(t) = i].

84

Stoica A. and Yaesh I. (2008).

PASSIVITY OF A CLASS OF HOPFIELD NETWORKS - Application to Chaos Control.

In Proceedings of the Fifth International Conference on Informatics in Control, Automation and Robotics - SPSMC, pages 84-89

DOI: 10.5220/0001478500840089

Copyright

c

SciTePress

2 PROBLEM FORMULATION

The neural network proposed by Hopﬁeld, can be de-

scribed by an ordinary differential equation of the

form

˙v

i

(t) = a

i

v

i

(t)+

n

∑

j=1

b

ij

g

j

(v

j

(t))+ ¯c

i

= κ

i

(v), 1 ≤ i ≤ n

(1)

where v

i

represents the voltage on the input of the ith

neuron, a

i

< 0, 1 ≤ i ≤ n, b

ij

= b

ji

and the activations

g

i

(·), i = 1, ..., n are C

1

–bounded and strictly increas-

ing functions.

This network is usually analyzed by deﬁning the

network energy functional:

E(v) = −

n

∑

i=1

a

i

Z

v

i

0

u

dg

i

(u)

du

du (2)

−

1

2

n

∑

i, j=1

b

ij

g

i

(v

i

)g

j

(v

j

) −

n

∑

i=1

¯c

i

g

i

(v

i

)

where it can be seen that

dE

dt

= −

∑

dg

i

(v

i

)

dv

i

κ

i

(v)

2

≤ 0

where the zero rate of the energy is obtained only in

the equilibrium points, also referred to as attractors,

where

κ

i

(v

0

) = 0, 1 ≤ i ≤ n (3)

The network is then described in matrix form as:

˙v(t) = Av(t) + Bg(v) +

¯

C,1 ≤ i ≤ n (4)

where

A := diag(a

1

, ..., a

n

), B := [b

ij

]

i, j=1,...,n

,

¯

C :=

¯c

1

¯c

2

... ¯c

n

T

, v :=

v

1

v

2

... v

n

T

and where

g(v) :=

g

1

(v

1

) g

2

(v

2

) ... g

n

(v

n

)

T

The stochastic version of this network driven by white

noise, has been considered in (Hu et al., 2003) where

the stochastic stability of (1) has been analyzed and

where it has been shown that the network is almost

surely stable when the condition

dE

dt

≤ 0 is replaced

by L E ≤ 0 where L is the inﬁnitesimal generator as-

sociated with the Itˆo type stochastic differential equa-

tion (4). This condition has been shown in (Hu et al.,

2003) to be satisﬁed only in cases where the driv-

ing noise in (1) is not persistent. This non persistent

white noise can be interpreted as a white-noise type

uncertainty in A and B, namely state-multiplicative

noise. In (Stoica and Yaesh, 2005)-(Stoica and Yaesh,

2006) Hopﬁeld networks with Markov jump parame-

ters have been considered to represent also non zero

mean uncertainties in these matrices. Encouraged

by the insight gained in (Cabrera and Milton, 2004)

and (Cabrera et al., 2001) regarding the role of state-

multiplicative noise and time delay (see also (Mazenc

and Niculescu, 2001)) in visuo-motor control loops,

we generalize the results of (Stoica and Yaesh, 2005)

to include this effect. The Lur’e - Postnikov systems

approach ((Lure and Postnikov, 1944), (Boyd et al.,

1994)) is invoked to analyze stability and disturbance

attenuation (in the H

∞

norm sense) and the results are

given in terms of Linear Matrix Inequalities (LMI).

To analyze input output properties we ﬁrst deﬁne

the error of the Hopﬁeld network output with respect

to its equilibrium points by

x(t) = v(t) − v

0

. (5)

and assume that the errors vector x(t) satisfy

dx = (A

0

(θ(t)) x + B

0

(θ(t)) f(y) + D(θ(t))u(t))dt

+A

1

(θ(t)) xdη + B

1

(θ(t)) f(y)dξ

(6)

where the system measured output is

z = L(θ(t)) x+ N (θ(t))u (7)

and where

y = C(θ(t))x (8)

Note that (6) was obtained from (4) by replacing Adt

by A

0

dt + A

1

dξ, Bdt by B

0

dt + B

1

gdξ and f(x) =

g(x + v

0

) − g(v

0

). The control input u(t) as intro-

duced to provide a stochastic version of (Poznyak and

Sanchez, 1999) allowing the considered Hopﬁeld net-

work to serve also a chaotic system identiﬁer. We

note that (Poznyak and Sanchez, 1999) the control

signal is u = φ(r)u rather than just u where φ(r) is

a diagonal matrix having f

i

(r

i

) on its diagonal, where

r = H (θ(t))x. We have taken for simplicity φ = I

which is also motivated by our example in Section IV.

Note also that the matrices

A

0

(θ(t)) , A

1

(θ(t)) , B

0

(θ(t)) , B

1

(θ(t)) , D(θ(t)),

C

1

(θ(t)) ,C

2

(θ(t)) and L(θ(t)) are piecewise

constant matrices of appropriate dimensions whose

entries are dependent upon the mode θ(t) ∈ {1, ..., r}

where r is a positive integer denoting the number of

possible models between which the Hopﬁeld network

parameters can jump. Namely, A

0

(θ(t)) attains the

values of A

0,1

, A

0,2

, ..., A

0,r

, etc. It is assumed that

θ(t),t ≥ 0 is a right continuous homogeneous Markov

chain on D = {1, ..., r} with a probability transition

matrix

P(t) = e

Qt

; Q = [q

ij

]; q

ii

< 0;

∑

r

j=1

q

ij

= 0;

i = 1, 2, .., r.

(9)

Given the initial condition θ(0) = i, at each time

instant t, the mode may maintain its current state or

jump to another mode i 6= j. The transitions between

PASSIVITY OF A CLASS OF HOPFIELD NETWORKS - Application to Chaos Control

85

the r possible states, i ∈ D , may be the result of ran-

dom ﬂuctuations of the actual network components

(i.e. resistors, capacitors) characteristics or can used

to artiﬁcially model deliberate jumps which are the re-

sult of parameter changes in an optimization problem

the network is used to solve. In visuo-motor tasks one

may conjecture that proportional and derivative feed-

backs are applied on the basis of time sharing, where

transition probabilities deﬁne the statistics of switch-

ing between the two modes. Although there is no ev-

idence for this conjecture, the model analyzed in the

present paper can be used to check its stability and L

2

gain.

In the forthcoming analysis, we will assume that

the components f

i

, i = 1, ..., n of f(ξ) are assumed to

satisfy the sector conditions

0 ≤ ζ

i

f

i

(ζ

i

) ≤ ζ

2

i

σ

i

(10)

which are equivalent to

−F

i

(ζ

i

, f

i

) := f

i

(ζ

i

)( f

i

(ζ

i

) − σ

i

ζ

i

) ≤ 0 (11)

We shall further assume that

∂f

i

∂ζ

i

≤ σ

i

, i = 1, ..., n. (12)

Although the latter assumption of (12) further re-

stricts the sector–type one class of (11), the applica-

bility of our results remains since it is fulﬁlled by the

usual nonlinearities as saturation, sigmoid, etc., used

in the neural networks.

Some additional notations are now in place. We

deﬁne

S = diag

σ

1

, σ

2

, ... , σ

n

where σ

i

are the nonlinearity gains of (11).

As mentioned above we shall analyze passivity (in

stochastic sense) conditions for the systems (6)-(8b)

which is expressed as:

J = E

Z

∞

0

z

T

(t)u(t)

dt

> 0, x(0) = 0. (13)

3 PASSIVITY ANALYSIS

Introduce the Lyapunov–type function:

V (x(t), θ(t)) = x

T

(t)P(θ(t))x(t)

+2

∑

n

k=1

λ

k

R

C

k

x

0

f

k

(s)ds.

(14)

depending on the nonlinearities f

i

(y

i

) = f

i

(C

i

x) via

the constants λ

i

where C

i

is the i’th row in C. As it

was mentioned in (Boyd et al., 1994), V of (14) de-

ﬁnes a parameter-dependent Lypaunov function. To

see this, consider the simple case of f

i

(x

i

) = x

i

σ

i

and

get V(x, σ

1

, σ

2

, ..., σ

n

) = x

T

(P+ S

1

2

ΛS

1

2

)x which de-

pends on the parameters σ

i

, i = 1, 2, .., n and on the

constants λ

i

, i = 1, 2, .., n via (18) in the sequel. Ap-

plying the Itˆo–type formula (see (Dragan and Mo-

rozan, 1999), (Dragan and Morozan, 2004) and (Fen

et al., 1992)) for V (x, θ) it follows that:

E {V (x, θ(t)|θ(0))} − E{V (0, θ(0)|θ(0))} =

E

R

t

0

L V (x, θ(s))ds

where

L V (x, θ) := (A

0

(θ)x+ B

0

(θ) f (y) + D(θ)u)

T

∂V

∂x

+x

T

A

T

1

(θ)

¯

PA

1

(θ)x+ f

T

B

T

1

(θ)

¯

PB

1

(θ) f

+

∑

r

j=1

q

ij

x

T

P

j

x.

(15)

where

¯

P(θ, λ

1

, λ

2

, ..., λ

n

) := P(θ)+ diag

λ

1

∂f

1

∂x

1

, ..., λ

n

∂f

n

∂x

n

,

with the dependence on its arguments being omitted

and where for simplicity we have used the notation

f := f(y(t)). Then the condition (13) is fulﬁlled if

L V < 2z

T

u (16)

which becomes:

x

T

A

T

0i

+ f

T

B

T

0i

+ u

T

D

T

i

P

i

x+C

T

Λf

+

x

T

P

i

+ f

T

ΛC

(A

0i

x+ B

0i

f + D

i

u)

+x

T

A

T

1i

¯

P

i

A

1i

x+ f

T

B

T

1i

¯

P

i

B

1i

f

+

∑

r

j=1

q

ij

x

T

P

j

x− u

T

L

i

x− x

T

L

T

i

u

−u

T

(N

i

+ N

T

i

)u < 0, i = 1, ..., r,

(17)

where

¯

P

i

denotes

¯

P(θ = i, λ

1

, λ

2

, ..., λ

n

) and where

Λ := diag

λ

1

, λ

2

, ... , λ

n

. (18)

In order to explicitly express (17), the assumption

(12) will be used. Indeed, using the inequalities (12)

it follows that conditions (17) are satisﬁed if the fol-

lowing inequalities are satisﬁed:

−F

i0

(x, f) :=

x

T

h

A

T

0i

P

i

+ P

i

A

0i

+ A

T

1i

P

i

+C

T

S

1

2

ΛS

1

2

C

A

1i

+L

T

i

L

i

+

∑

r

j=1

q

ij

P

j

i

x+ f

T

B

T

0i

C

T

Λ+ ΛCB

0i

+B

T

1i

P

i

+C

T

S

1

2

ΛS

1

2

C

B

1i

i

f

+ f

T

B

T

0i

P

i

+ ΛCA

0i

x+ x

T

P

i

B

0i

+ A

T

0i

C

T

Λ

f

−u

T

(L

i

− D

T

i

P

i

)x− x

T

(L

T

i

− P

i

D

i

)u

−u

T

(N

i

+ N

T

i

)u < 0.

(19)

Using the S -procedure (e.g. (Boyd et al., 1994))

one, therefore, obtains that (16) subject to (11) is sat-

isﬁed if there exist τ

i

≥ 0, i = 1, 2, ..., n so that

F

i0

(x, f) −

n

∑

k=1

τ

k

F

k

(x, f) ≥ 0. (20)

ICINCO 2008 - International Conference on Informatics in Control, Automation and Robotics

86

Denoting

T := diag

τ

1

, τ

2

, ... , τ

n

(21)

and noticing that

−

n

∑

k=1

τ

k

F

k

(x, f) =

n

∑

k=1

τ

k

f

2

k

− τ

k

σ

k

f

k

y

k

= f

T

T f −

1

2

f

T

TCSx

−

1

2

x

T

SC

T

T f,

we get from (20) that:

x

T

Z

i11

x+ f

T

Z

T

i12

x+ x

T

Z

i12

f + f

T

Z

i22

f

−u

T

(L

i

− D

T

i

P

i

)x− x

T

(L

T

i

− P

i

D

i

)u

−u

T

(N

i

+ N

T

i

)u < 0, i = 1, ..., r.

where

Z

i11

:= A

T

0i

P

i

+ P

i

A

0i

+ A

T

1i

ˆ

P

i

A

1i

+

r

∑

j=1

q

ij

P

j

Z

i12

:= P

i

B

0i

+ A

T

0i

C

T

Λ+

1

2

SC

T

T (22)

Z

i22

:= B

T

0i

C

T

Λ+ ΛCB

0i

+ B

T

1i

ˆ

P

i

B

1i

− T

where

ˆ

P

i

= P

i

+C

T

S

1

2

ΛS

1

2

C (23)

These conditions are fulﬁlled if:

Z

i11

Z

i12

P

i

D

i

− L

T

i

Z

T

i12

Z

i22

0

D

T

i

P− L

i

0 −(N

T

i

+ N

i

)

< 0, (24)

i = 1, ..., r, with the unknown variables P

i

, Λ and T.

The above developments are concluded in the fol-

lowing result:

Theorem 1

. The system (6)–(7) is stochastically

stable and strictly passive if there exist the symmetric

matrices P

i

> 0, i = 1, ..., r, and the diagonal matrices

Λ > 0 and T > 0 satisfying the system of LMIs (24)

with the notations (22)-(23).

4 SIMPLIFIED ADAPTIVE

CONTROL

In this section we show that the system of (6)-(8)

should be regulated using a direct adaptive controller

of the type:

u = −Kz (25)

where

˙

K = zz

T

. (26)

Since this type of adaptive control is well–known

in the deterministic case (see e.g. (Kaufman et al.,

1998)), we shall just give a sketch of the proof, em-

phasizing the particularities arising in the stochastic

framework (see also (Yaesh and Shaked, 2005)) ana-

lyzed in this paper. We ﬁrst note that the system (6)-

(8) is strictly passive when the passivity condition of

(24) is satisﬁed with N

i

= εI for ε that tends to zero.

The latter is satisﬁed (see e.g. (Yaesh and Shaked,

2005) and (Kaufman et al., 1998)) if there exist the

symmetric matrices P

i

> 0, i = 1, 2, ..., r such that

Z

i

< 0 and i = 1, 2, ..., r, (27)

and

P

i

D

i

= L

T

i

, i = 1, 2, ..., r, (28)

where Z

i

=

Z

i11

Z

i12

Z

T

i12

Z

i22

. The stochastic closed-

loop system obtained from (6), (8) with u = Kz can

be written as:

dx = [(A

0

(θ(t)) − D(θ(t))K

e

L(θ)x)

+B

0

(θ(t)) f(y) + D(θ(t)) ¯u]dt

+A

1

(θ(t))xdη + B

1

(θ(t)) f(y)dξ

z = L(θ(t))x. (29)

where ¯u = − (K − K

e

)z. The above equations hold

for any K

e

of appropriate dimensions but in the fol-

lowing it will be assumed that K

e

is a constant gain

for which the system (29) is stochastically passive

(some authors call in this case the open-loop sys-

tem almost passive-AP). Note that K

e

’s existence is

needed just for stability analysis and but its value is

not utilized in the implementation. In our stochas-

tic context the stochastic stability of this direct adap-

tive controller (25), (26) (which usually referred to as

simpliﬁed adaptive control-SAC) will be guaranteed

by the stochastic version of the AP property. To this

end, as in (Kaufman et al., 1998) we will choose the

following generalization of the Lyapunov function of

(14) to prove the closed-loop stability:

V (x(t), K(t), θ(t)) = V (x(t), θ(t))

+ tr(K(t) − K

e

)

T

(K(t) − K

e

)

(30)

where tr denotes the trace and V has the expression

(14) with P(i), i = 1, ..., r satisfying the conditions of

form (27) and (28) written for the passive system (29)

relating ¯u and z. Then, direct computations show that

the inﬁnitesimal generatorof V of the form (30) along

the trajectory (29) and subject to the conditions (28)

has the expression:

L V (x(t), K(t), θ(t) = i) = ¯x

T

Z

i

¯x+ 2tr

¯

K

T

˙

¯

K −

¯

K

T

zz

T

(31)

where

¯

K := K − K

e

and ¯x =

x

f

. Since the system

(29) was assumed passive (i.e. (27) is satisﬁed with

PASSIVITY OF A CLASS OF HOPFIELD NETWORKS - Application to Chaos Control

87

A

0i

−D

i

K

e

L

i

replacing A

0i

) it follows that L V < 0 and

then, choosing

˙

¯

K = zz

T

it results that L V < 0 which

proves the stochastic stability of the resulting closed-

loop system.

We next apply this result in a chaos control prob-

lem.

5 EXAMPLE - CHAOS CONTROL

Consider a slightly modiﬁed version of the third or-

der chaos generator model of (Kwok et al., 2003) de-

scribed by (6)-(8), where

A

0

=

−ε 1 0

0 −ε 1

a

1

a

2

a

3

, B

0

= D = L

T

=

0

0

1

,

A

1

=

0 0 0

0 0 0

0 0 σ

, B

1

=

0 0 0

0 0 0

0 0 0

,

C

T

=

β

0

0

,

(32)

where a

1

= −2, a

2

= −1.48, a

3

= −1, σ = 0.1, ε =

0.01 and β = 10. The nonlinearity is f(y) = αtanh(y)

where α = 1.

To establish stability we verify (27)-(28) with

K

e

= 10

5

and ﬁnd using YALMIP (L¨ofberg, 2004a)-

(L¨ofberg, 2004b) where A

0

is replaced by A

0

− DK

e

L.

Therefore, by the results of Section 4 above, the

closed-loop system with the controller (25), (26) is

expected to be stochastically stable.

Next we simulate the above system for 500sec

with an integration step of 0.001sec with u = 0 for

t ≤ 250sec and with the SAC controller u = −Kz

where

˙

K = z

2

in the rest of the time. The results are

given in Fig. 1 - 3 : the phase-plane (i.e. x

1

versus

x

2

) trajectories are depicted in Fig. 1, the components

x

i

, i = 1, 2, 3 of the state-vector and the control input

are depicted in Fig. 2, and the adaptive gain K is de-

picted in Fig. 3. It is seen from these ﬁgures that the

chaotic behavior characterizing the system in open-

loop, is replaced by a stable trajectory at t ≥ 250sec

where the SAC is applied.

6 CONCLUSIONS

A class of stochastic Hopﬁeld networks subject to

state-multiplicative noise where the network weights

jump according a Markov chain process have been

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

x

1

x

2

Figure 1: Simulation Results.

0 100 200 300 400 500 600 700 800 900 1000

−1

0

1

x

1

0 100 200 300 400 500 600 700 800 900 1000

−1

0

1

x

2

0 100 200 300 400 500 600 700 800 900 1000

−1

0

1

x

3

Time [sec]

0 100 200 300 400 500 600 700 800 900 1000

−1

0

1

u

c

Time [sec]

Figure 2: Simulation Results.

considered. Stochastic passivity conditions for such

systems have been derived in terms of Linear Matrix

Inequalities. The results have been illustrated via sim-

pliﬁed adaptive control of a dynamic system which

exhibits a chaotic behavior when its is not controlled.

ICINCO 2008 - International Conference on Informatics in Control, Automation and Robotics

88

0 100 200 300 400 500 600 700 800 900 1000

−0.5

0

0.5

1

1.5

λ

1

Time [sec]

0 100 200 300 400 500 600 700 800 900 1000

−1

0

1

2

λ

2

Time [sec]

0 100 200 300 400 500 600 700 800 900 1000

−2

0

2

4

6

λ

3

Time [sec]

Figure 3: Simulation Results.

The control efﬁciency in stabilizing the chaotic pro-

cess has been demonstrated with simulations. The

results of this paper should encourage further study

of attempts to control chaotic systems with simpliﬁed

adaptive controllers.

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