ADAPTIVE CONTROL FOR A CLASS OF STOCHASTIC PASSIVE HOPFIELD

Dedicated to Dr. Constantin VARSAN

on the occasion of his 70th birthday

ADAPTIVE CONTROL FOR A CLASS OFSTOCHASTIC PASSIVE HOPFIELD NETWORKS

ADRIAN-MIHAIL STOICA and ISAAC YAESH

The paper presents passivity conditions for a class of stochastic Hopfield neuralnetworks with state-dependent noise and with Markovian jumps. Our contribu-tions are mainly based on the stability analysis of the class of stochastic neu-ral networks considered, using infinitesimal generators of appropriate stochasticLyapunov-type functions. The passivity conditions derived are expressed in termsof the solutions of some specific systems of linear matrix inequalities. The theore-tical results are illustrated by a simplified adaptive control problem for a dynamicsystem with chaotic behaviour.

AMS 2000 Subject Classification: 93E15, 93E35.

Key words: stochastic stability control, Hopfield neural network, recurrent neuralnetwork, S-procedure, linear matrix inequality, adaptive control.

1. INTRODUCTION

Hopfield networks are symmetric recurrent neural networks which exhibitmotions in the state space which converge to minima of energy.

Symmetric Hopfield networks can be used to solve practical complexproblems such as implement associative memory, linear programming solversand optimal guidance problems. Recurrent networks which are non symme-tric versions of Hopfield networks play an important role in understandinghuman motor tasks involving visual feedback (see [3] and [2] and the referencestherein). Such networks are subject to effects of state-multiplicative noise,pure time delay (see [7], [10] and [15]) and even multiple attractors which canbe caused by Markov jumps. Even without Markov jumps, a non symmetricclass of Hopfield networks is able to generate chaos [9]. Therefore, Hopfieldnetworks can be used [14] as identifiers of unknown chaotic dynamic systems.The resulting identifier neural networks have been used in [14] to derive alocally optimal robust controller to remove the chaos in the system.

MATH. REPORTS 11(61), 4 (2009), 369–381

370 Adrian-Mihail Stoica and Isaac Yaesh 2

In this paper, the robust controller of [14] is replaced by a direct adap-tive controller. More specifically, we consider the so called Simplified Adap-tive Control (SAC) method [8] which applies a simple proportional controllerwhose gain is adapted according to the squared tracking error. Since suchcontrollers’ stability proof involves a passivity condition, a passivity result forthe Hopfield network is derived. The results presented in this paper are, infact, developed for a generalized version of non symmetric Hopfield networksincluding Markov jumps of the parameters and state multiplicative noise, thusallowing a wider stochastic class of chaos generating systems to be conside-red. They also represent an extension of the problem presented in [17] tothe case where the stochastic Hopfield network is passivied via unconstantstate-feedback gains.

The paper is organized as follows. In Section 2 the problem is formulatedwhile in Section 3 we derive Linear Matrix Inequalities (LMIs) based conditionsfor passivity analysis. Section 4 deals with an adaptive control application,Section 5 includes a chaos control example and Section 6 the conclusions.

Throughout the paper, Rn denotes the n dimensional Euclidean space,Rn×m is the set of all n × m real matrices, and the notation P > 0 (respec-tively, P ≥ 0) for P ∈ Rn×n means that P is symmetric and positive definite(respectively, semi-definite). Throughout the paper (Ω,F , P ) is a given prob-ability space; the argument θ ∈ Ω will be suppressed. Expectation is denotedby E· and conditional expectation of x on the event θ(t) = i is denoted byE[x|θ(t) = i].

2. PROBLEM FORMULATION

The neural network proposed by Hopfield can be described by a systemof ordinary differential equations of the form

(2.1) vi(t) = aivi(t) +n∑

j=1

bijgj(vj(t)) + ci = κi(v), 1 ≤ i ≤ n,

where vi is the voltage on the input of the ith neuron, ai < 0, 1 ≤ i ≤ n,bij = bji and the activations gi(·), i = 1, . . . , n are C1-bounded and strictlyincreasing functions.

This network is usually analyzed by defining the network energy func-tional

(2.2) E(v) = −n∑

i=1

ai

∫ vi

0u

dgi(u)du

du− 12

n∑i,j=1

bijgi(vi)gj(vj)−n∑

i=1

cigi(vi).

3 Adaptive control for a class of stochastic passive hopfield networks 371

It can be seen that dEdt = −

∑ dgi(vi)dvi

κi(v)2 ≤ 0, where the zero rate of theenergy is only obtained at the equilibrium points, also referred to as attractors,where

(2.3) κi(v0) = 0, 1 ≤ i ≤ n.

The network is then described in matrix form as

(2.4) v(t) = Av(t) + Bg(v) + C, 1 ≤ i ≤ n,

where

A := diag(a1, . . . , an), B := [bij ]i,j=1,...,n,

C :=[c1 c2 . . . cn

]T, v :=

[v1 v2 . . . vn

]T

and

g(v) :=[g1(v1) g2(v2) . . . gn(vn)

]T.

The stochastic version of this network driven by white noise has been consid-ered in [7] where the stochastic stability of (2.1) has been analyzed and it hasbeen shown that the network is almost surely stable when the condition dE

dt ≤ 0is replaced by LE ≤ 0, where L is the infinitesimal generator associated withthe Ito type stochastic differential equation (2.4). This condition has beenshown in [7] to be only satisfied in cases where the driving noise in (2.1) is notpersistent. This non persistent white noise can be interpreted as a white-noisetype uncertainty in A and B, namely, state-multiplicative noise. In [16] and[15] Hopfield networks with Markov jump parameters have been considered torepresent also nonzero mean uncertainties in these matrices. Encouraged bythe insight gained in [3] and [2] regarding the role of state-multiplicative noiseand time delay (see also [13]) in visuo-motor control loops, we generalize theresults of [16] to include this effect. The Lur’e-Postnikov systems approach([12], [1]) is invoked to analyze stability and disturbance attenuation (in theH∞ norm sense), and the results are given in terms of Linear Matrix Inequa-lities (LMI).

To analyze input output properties we first define the error of the Hop-field network output with respect to its equilibrium points by

(2.5) x(t) = v(t)− v0.

and assume that the errors vector x(t) satisfy

dx(t) = [A0 (θ(t))x(t) + B0 (θ(t)) f(y(t)) + D (θ (t))u(t)] dt(2.6)

+A1 (θ (t))x(t)dη(t) + B1 (θ(t)) f(y(t))dξ(t),

where

(2.7) y(t) = C (θ(t))x(t)


and the system measured output is

(2.8) z(t) = L (θ(t))x(t) + N (θ(t))u(t).

Note that (2.6) was obtained from (2.4) by replacing Adt by A0dt + A1dη,Bdt by B0dt + B1gdξ and f(x) = g(x + v0) − g(v0). The control input u(t)is introduced to provide a stochastic version of [14] allowing the consideredHopfield network to also serve a chaotic system identifier. In [14] the controlsignal is of the form φ(r)u rather than just u, where φ(r) is a diagonal matrixhaving fi(ri) on its diagonal, with r = H (θ(t))x. For simplicity, here, φ = I,which also is motivated by the example presented in Section 4.

Note also that the matrices A0 (θ(t)), A1 (θ(t)), B0 (θ(t)), B1 (θ(t)),D (θ(t)), C (θ(t)) and L (θ(t)) are piecewise constant matrices of appropri-ate dimensions whose entries are dependent upon the mode θ(t) ∈ 1, . . . , r,where r is a positive integer denoting the number of possible modes betweenwhich the Hopfield network parameters can jump. Namely, A0(θ(t)) attainsthe values of A01, A02, . . . , A0r, etc. It is assumed that θ(t), t ≥ 0, is a rightcontinuous homogeneous Markov chain on D = 1, . . . , r with a probabilitytransition matrix

(2.9) P (t) = eQt, Q = [qij ], qij ≥ 0, i 6= j,r∑

j=1

qij = 0, i = 1, 2, . . . , r.

Given the initial condition θ(0) = i, at each time instant t, the mode maymaintain its current state or jump to another mode i 6= j. The transitionsbetween the r possible states i ∈ D may be the result of random fluctuationsof the actual network components (i.e., resistors, capacitors) characteristicsor can used to artificially model deliberate jumps which are the result of pa-rameter changes in an optimization problem the network is used to solve. Invisuo-motor tasks, one may conjecture that proportional and derivative feed-backs are applied on the basis of time sharing, where transition probabilitiesdefine the statistics of switching between the two modes. Although there isno evidence for this conjecture, the model analyzed in the present paper canbe used to check its stability and L2 gain.

In the forthcoming analysis, it is assumed that the components fi, i =1, . . . , n, of f(ξ) satisfy the sector conditions

(2.10) 0 ≤ ζifi(ζi) ≤ ζ2i σi,

which are equivalent to

(2.11) −Fi(ζi, fi) := fi(ζi)(fi(ζi)− σiζi) ≤ 0.

It is further assumed that

(2.12)dfi

dζi≤ σi, i = 1, . . . , n,


which is not restrictive since it is fulfilled by the usual nonlinearities as satu-ration, sigmoid, etc., used in the neural networks.

As mentioned above, the passivity (in stochastic sense) condition forsystems (2.6)–(2.8) which is expressed as

(2.13) J = E

∫ ∞

0

(zT(t)u(t)

)dt

> 0, x(0) = 0,

will be further analyzed.

3. PASSIVITY ANALYSIS

Introduce the Lyapunov-type function

(3.14) V (x(t), θ(t))xT(t) P (θ(t))x(t) + 2n∑

k=1

λk

∫ C(k)(θ(t))x

0fk(s)ds

depending on the nonlinearities fk(yk) = fk(C(k)i x) via the constants λk, k =

1, . . . , n, where C(k)i is the kth row in Ci, i = 1, . . . , r. As it was mentioned

in [1], V of (3.14) defines a parameter-dependent Lypaunov function. Indeed,in the special case of fi(xi) = σixi one gets V (x, σ1, σ2, . . . , σn) = xT(P +S

12 ΛS

12 )x, where the notation

(3.15) S := diag (σ1, σ2, . . . , σn) and Λ := diag (λ1, λ2, . . . , λn)

has been introduced.Using the Ito-type formula (see [4], [5] and [6]) for V (x, θ) we get

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

∫ t

0LV (x, θ(s)) ds

with the infinitesimal operator L given by

LV (x, θ) := (A0 (θ) x + B0(θ)f (y) + D (θ) u)T∂V

∂x+(3.16)

+xTAT1 (θ)PA1 (θ) x + fTBT

1 (θ)PB1 (θ) f +r∑

j=1

qijxTPjx,

where

P (θ, λ1, λ2, . . . , λn) : P (θ) + diag(

λ1∂f1

∂x1, . . . , λn

∂fn

∂xn

)and f := f(y(t).

Here, the dependence on the argument t was omitted for simplicity. Thencondition (2.13) is fulfilled if

(3.17) LV < 2zTu,


which becomes (xTAT

0i + fTBT0i + uTDT

i

) (Pix + CT

i Λf)+(3.18)

+(xTPi + fTΛCi

)(A0ix + B0if + Diu) +

+xTAT1iPiA1ix + fTBT

1iPiB1if +r∑

j=1

qijxTPjx−

−uTLix− xTLTi u− uT(Ni + NT

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

where Pi denotes P (θ = i, λ1, λ2, . . . , λn).In order to explicitly express (3.18), assumption (2.12) will be used.

Indeed, using inequalities (2.12) we see that conditions (3.18) are fulfilled if

−Fi0 (x, f) := xT

[AT

0iPi + PiA0i + AT1i

(Pi + CT

i S12 ΛS

12 Ci

)A1i +

r∑j=1

qijPj

]x+

(3.19)

+uTDTi CT

i Λf + fTΛCiDiu + fT[BT

0iCTi Λ + ΛCiB0i+

+BT1i

(Pi + CT

i S12 ΛS

12 Ci

)B1i

]f + fT

(BT

0iPi + ΛCiA0i

)x+

+xT(PiB0i + AT

0iCTi Λ

)f − uT(Li −DT

i Pi)x−−xT(LT

i − PiDi)u− uT(Ni + NTi )u < 0.

Using the S-procedure (e.g., [1]), we deduce that (3.17) subject to (2.11) issatisfied if there exist τi ≥ 0, i = 1, 2, . . . , n, such that

(3.20) Fi0(x, f)−n∑

k=1

τkFk(x, f) ≥ 0.

Denoting

(3.21) T := diag(τ1, τ2, . . . , τn)

and noticing that

−n∑

k=1

τkFk(x, f) =n∑

k=1

τkf2k − τkσkfkyk = fTTf − 1

2fTTCiSx− 1

2xTSCT

i Tf,

one gets from (3.20) that

xTZi11x + fTZTi12x + xTZi12f + fTZi22f − uT(Li −DT

i Pi)x + uTDTi CT

i Λf

+fTΛCiDiu− xT(LTi − PiDi)u− uT(Ni + NT

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


where

Zi11 := AT0iPi + PiA0i + AT

1iPiA1i +r∑

j=1

qijPj ,

Zi12 := PiB0i + AT0iC

Ti Λ +

12SCT

i T,

Zi22 := BT0iC

Ti Λ + ΛCiB0i + BT

1iPiB1i − T,

(3.22)

with

(3.23) Pi := Pi + CTi S

12 ΛS

12 Ci .

These conditions are fulfilled if

(3.24)

Zi11 Zi12 PiDi − LTi

ZTi12 Zi22 ΛCiDi

DTi Pi − Li DT

i CTi Λ −(NT

i + Ni)

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

with the unknown variables Pi, Λ and T .

The above developments can be stated as

Theorem 1. System (2.6)–(2.8) is stochastically stable and strictly pas-sive if there exist symmetric matrices Pi > 0, i = 1, . . . , r, and diagonal ma-trices Λ > 0 and T > 0 satisfying system (3.24) of LMIs.

4. SIMPLIFIED ADAPTIVE CONTROL

In this section it is shown how a stochastic system of the form (2.6)–(2.7)can be regulated using a direct adaptive controller of the type

(4.25) u = −Kz,

where

(4.26) K = zzT.

This type of adaptive control is well-known in the deterministic case (see,e.g., [8]) and in the present section we analyze the particularities arising inthe stochastic framework (see also [18]). Throughout this section it is assumedthat Ci = 0, which corresponds to the situation when the nonlinearity f ismissing, or CiDi = 0, i = 1, . . . , r. Under this assumption, system (2.6)–(2.7)with Ni = εI for ε tending to zero satisfies the strictly passivity condition(3.24) if there exist symmetric matrices Pi > 0, i = 1, 2, . . . , r, satisfyingthe conditions [

Zi11 Zi12

ZTi12 Zi22

]< 0(4.27)


and

(4.28) PiDi = LTi , i = 1, 2, . . . , r,

(see e.g. [18] and [8]). The stochastic closed-loop system obtained from (2.6),and (2.7) with u = Kz can be written as:

dx = [(A0 (θ)−D (θ) KeL (θ))x + B0 (θ) f(y) + D (θ) u] dt(4.29)

+A1 (θ) xdη + B1 (θ) f(y)dξ, z = L (θ) x,

where u := − (K −Ke) z. The above equations hold for any Ke of appropriatedimensions but it is assumed that Ke is such that system (4.29) is stochas-tically passive (in this case some authors call the open-loop system almostpassive-AP). Note that the existence of Ke is needed just for stability analy-sis, but its value is not used in the implementation. In the present context,the stochastic stability of this direct adaptive controller (4.25), (4.26) (whichis usually referred to as simplified adaptive control -SAC) will be guaranteedby the stochastic version of the AP property. Two cases will be emphasizedin what follows.

Case a): Ke is constant. As in [8], to prove the closed-loop stability, wewill choose a generalization of the Lyapunov function of (3.14), namely

(4.30) V (x(t),K(t), θ(t)) = V (x(t), θ(t)) + tr (K(t)−Ke)T (K(t)−Ke) ,

where tr denotes the trace and V is given by (3.14) with Pi, i = 1, . . . , r,satisfying conditions of the form (4.27) and (4.28) written for the passivesystem (4.29) relating u and z. Then, denoting A0i := A0i − DiKeLi, theinfinitesimal generator of V of the form (4.30) along the trajectory (4.29) hasthe expression

LV (x(t),K(t), θ(t) = i) = 2[A0ix + B0if + Diu

]T (Pix + CT

i Λf)

(4.31)

+xTAT1iPiA1ix +

r∑j=1

qijxTPjx + fTBT

1iB1if + 2 tr K(K −Ke)T

= xT

(AT

0iPi + PiA0i + AT1iPiA1i +

r∑j=1

qijPj

)x + fT

(BT

0iPi + ΛCiA0i

)x

+xT(PiB0i + AT

0iCTi Λ

)f + uTDT

i Pix + xTPiDiu + uTDTi CT

i Λf

+fTΛCiDiu + fT(BT

0iCTi Λ + ΛCiB0i + BT

1iPiB1i

)f + 2 tr K(K −Ke)T

=[

xT fT] [

Zi11Zi12

ZTi12Zi22

] [xf

]+ uTDT

i Pix + xTPiDiu

+uTDTi CT

i Λf + fTΛCiDiu + 2 tr K(K −Ke)T,


where

Zi11 := AT0iPi + PiA0i + AT

1iPiA1i +r∑

j=1

qijPj ,

Zi12 := PiB0i + AT0iC

Ti Λ,

Zi22 := BT0iC

Ti Λ + ΛCiB0i + BT

1iPiB1i,

(4.32)

with Pi defined by (3.23). Taking into account that PiDi = LTi , u = −(K −

Ke)z and the assumption CiDi = 0, it follows from (4.31) that

LV (x(t),K(t), θ(t) = i) =[

xT fT] [

Zi11 Zi12

ZTi12 Zi22

] [xf

]+(4.33)

2 tr(K − zzT

)(K −Ke)T.

For K = zzT we have

LV (x(t),K(t), θ(t) = i) =[

xT fT] [

Zi11 Zi12

ZTi12 Zi22

] [xf

].(4.34)

Since system (4.29) satisfies the passivity condition (3.24), we get[Zi11 Zi12

ZTi12 Zi22

]< 0.(4.35)

By repeating the S-procedure arguments that follow (3.20), we deduce that if(4.35) is fulfilled then

[xT fT

] [Zi11 Zi12

ZTi12 Zi22

] [xf

]< 0

for x and f satisfying (2.11). Thus, LV (x(t),K(t), θ(t) = i) < 0, which provesthe stochastic stability of the resulting closed-loop system.

Case b): Ke is time-varying. In this case one can use a control law ofthe form (4.25) with the modified adaptive gain

K = zzT + Ke(4.36)

but in this case, instead of the constant positive definite matrices in the ex-pression (3.14) of the Lyapunov function, one must use the uniformly positivedefinite matrices Pi(t), t ≥ 0, i = 1, . . . , r, satisfying the system[

Pi + Zi11 Zi12

ZTi12 Zi22

]< 0, i = 1, . . . , r,(4.37)


where Z11i, Z12i and Z22i are given by (4.32). The existence of such positivedefinite matrices Pi(t), i = 1, . . . , r, together with the diagonal matrix Λ > 0,is guaranteed by the fact that Ke(t) was assumed to be such that the closedloop system (4.29) is passive. The implementation of the adaptive control lawwith K given by (4.36) requires in this case the knowledge of the functionKe(t), t ≥ 0.

We next apply this result to a chaos control problem.

5. EXAMPLE – CHAOS CONTROL

Consider a slightly modified version of the third order chaos generatormodel of [9] described by (2.6)–(2.7), where

A0 =

−ε 1 00 −ε 1a1 a2 a3

, B0 = D = LT =

001

,

A1 =

0 0 00 0 00 0 σ

, B1 =

0 0 00 0 00 0 0

, CT =

β00

,

(5.1)

with a1 = −2, a2 = −1.48, a3 = −1, σ = 0.1, ε = 0.01 and β = 10. Thenonlinearity is f(y) = α tanh(y), where α = 1.

To establish stability, we verify (4.27)–(4.28) with Ke =105 using YALMIP[11] where A0 is replaced by A0 − DKeL. Therefore, by the results of Sec-tion 4, the closed-loop system with controller (4.25), (4.26) is expected to bestochastically stable.

Next, we simulate the above system for 500 sec with an integration step of0.001 s with u = 0 for t ≤ 250 s and with the SAC controller u = −Kz, whereK = z2 in the rest of the time. The results are given in Figures 1–3: the phase-plane (i.e., x1 versus x2) trajectories are depicted in Figure 1, the componentsxi, i = 1, 2, 3, of the state-vector and the control input are depicted in Figure 2while the adaptive gain K is depicted in Figure 3. It is seen from these figuresthat the chaotic behavior characterizing the system in open-loop, is replacedby a stable trajectory at t ≥ 250 s, where the SAC is applied.

6. CONCLUSIONS

A class of stochastic Hopfield networks subject to state-multiplicativenoise where the network weights jump according a Markov chain process hasbeen considered. Stochastic passivity conditions for such systems have beenderived in terms of Linear Matrix Inequalities. The results have been illus-trated via simplified adaptive control of a dynamic system which exhibits a


chaotic behavior when it is not controlled. The control efficiency in stabilizingthe chaotic process has been demonstrated by simulations. The results of thispaper should encourage further study of attempts to control chaotic systemswith simplified adaptive controllers.

Fig. 1. Simulation results.




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Received 3 May 2009 “Politehnica” University of BucharestFaculty of Aerospace Engineering

Str. Polizu, No. 1, 011061, Bucharest, [email protected]

and

Control Department, IMI Advanced Systems Div.P.O.B. 1044/77

Ramat-Hasharon, 47100, [email protected]

ADAPTIVE CONTROL FOR A CLASS OF STOCHASTIC PASSIVE HOPFIELD

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