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INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS, VOL. 24.25-36 (1996)
DISCRETE TIME INTERCONNECTED CELLULAR NEURAL
NETWORKS WITHIN NL, THEORY
JOHAN A. K. SUYKENS AND JOOS VANDEWALLE
Department of Electrical Engineering, ESAT-SISTA, Katholieke Universiteir Leuven. Kardinaal Mercierlaan 94,
8-3001 Leuven (Heverlee), Belgium
SUMMARY
Feedforward, cascade and feedback interconnections of CNNs were recently studied by Guzelis and Chua ( b i t . j . cir.
rheor. uppl., 21, 1-33 (1993)). Their framework was in continuous time with sufficient conditions for global
asymptotic and 1 / 0 stability and the relation with classical non-linear control theory such as the Lur’e problem was
revealed. In this paper such interconnected CNNs are considered in a discrete time setting. The original system
description is brought into a so-called NL, system form using state augmentation. NL,s are a general class of nonlinear systems in state space form with a typical feature of having q ‘layers’ with alternating linear and static nonlinear operators that satisfy a sector condition. Within NL, theory, sufficient conditions for global asymptotic stability
and 1/0 stability are available. The results are closely related to modem control theory ( H , theory and ,u theory).
Stability criteria are formulated as linear matrix inequalities (LMIs). Checking stability involves the solution to a
convex optimization problem. Furthermore, it is shown by examples that existing interconnected CNN configurations
result in q = 1 values. Hence, if one considers the q-value of the NL, as a measure of the complexity of the overall
system, such interconnected CNNs still have a low complexity from the analysis point of view. In addition, more
complex neural network architectures with 9 > 1 are discussed.
1. INTRODUCTION
After the introduction of the cellular neural network (CNN) paradigm by Chua and Yang’ it became clear
that CNNs are successful in solving image-processing tasks and partial differential equations (see e.g.
References 2 and 3). CNNs consist of simple non-linear dynamical processing elements (called cells or
neurons) that are only locally interconnected to their nearest neighbours and are usually arranged in a twodimensional array, which makes them attractive from the viewpoint of implementation. Besides continuous
time also discrete time versions, algorithms and technology are a ~ a i l a b l e Nevertheless,
.~~~
there are also
limitations on the use and applications of a single CNN. Therefore there is increasing interest in creating
more complex systems that are built up of existing less complex ones (CNNs in this case). Recently
Guzelis and Chua6 studied feedforward, cascade and feedback interconnections between CNNs. Also, at
the cell level, generalizations were made by considering ‘Chua circuits’ instead of ‘neurons’, leading to
cellular ‘non-linear’ networks instead of cellular ‘neural’ networks. Because even simple non-linear
dynamical systems such as the Chua circuit can already exhibit complex behaviour, letting such circuits
interact within an array results in new rich phenomena (see Reference 8 for an overview).
On the other hand, stability analysis of interconnected systems becomes more complicated. This
motivates the need for general frameworks for the analysis and design of such systems. The aim of this
paper is to show precisely how NL, theory may contribute towards this purpose in a discrete time setting.
NL, theory originated from the stability analysis of neural control systems (which consist of recurrent
neural network models and controllers) and was introduced by Suykens er al.’ Examples of NL,s
include neural control systems, the Lur’e problem, linear fractional transformations (LFTs), several
kinds of recurrent neural networks (including multilayer) and digital filters with overflow characteristic.
Using state augmentation,the original system descriptions are put into some standard non-linear state
space form that is called the NL, system. The value of q might be considered as a measure of the overall
CCC 0098-9886/96/010025- 12
0 1996 by John Wiley & Sons, Ltd.
Received 15 January 1995
Revised 6 April 1995
26
J. A. K. SUYKENS AND J. VANDEWALLE
coniplexity of the system. Whereas up to now the Lur'e problem (which consists of a linear dynamical
system feedback interconnected to a static non-linearity that satisfies a sector condition) has been
extensively studied in non-linear control theory, in NL, theory higher complex systems can be analysed and
synthesized. The Lur'e problem corresponds to an NL, system. In Reference 6 it was shown that stability
analysis of interconnected CNNs can be made in the context of the Lur'e problem. Hence it is not very
surprising that the discrete time interconnected CNNs considered in this paper will reduce to NL,s.
Sufficient conditions for global asymptotic stability and 1 / 0 stability (dissipativity with finite L,-gain) of
NL,s were derived in Reference 9. The conditions are very much related to conditions arising in modem
control theory ( H , and p theory; see e.g. Reference 10) and can be expressed as linear matrix inequalities
(LMIs) (see Reference 11 for LMIs in systems and control). LMI problems lead to non-differentiable but
convex optimization problems, which means there is a unique minimum and moreover this minimum can be
found in polynomia1 time."
This paper is organized as follows. In Section 2 we discuss discrete time interconnected CNNs. Then
NL,s are defined in Section 3 and CNNs are written as NL,s in Section 4.In Section 5 sufficient conditions
for global asymptotic and 1/0stability of NL,s are presented, which are formulated as LMI problems in
Section 6.
2. DISCRETE TIME INTERCONNECTED CNNs
Feedforward, feedback and cascade interconnected CNNs were studied in Reference 6. Such systems have
a certain interlayer (between the CNNs) and intralayer (within the CNN) connectivity. Let us consider an
interconnected CNN that consists of L CNNs with layer index I = I , ...,L. The dynamics of the fth CNN
are assumed to take the discrete time form
/
/ I
I '
x k + != A x , + B u p
z /p = c x/ I,
with internal state x, E R"1, input to the linear dynamical subcircuit u, E Rm/,output y , E IW"' and external
input v kE RY1.The upper index of the matrices and vectors is the layer index. The notation Eli, F:. means
that the Ith CNN is connected to the output of the ith CNN and the jth external input respectively. The
discrete time index is k. The non-linearity f(.)is taken elementwise when applied to a vector. If f(-)is a
static non-linearity belonging to sector [0,1], we will use the notation a(.) instead of f(.) in the sequel.
Alternatively the dynamics (1) are written as
y: = f(c::)
The dynamics (1) are formulated at the CNN level and not at the cell level as was done in Reference 6. In
this paper we make abstraction of the intralayer connectivity. The local interconnection between the cells is
then revealed by the sparseness of certain matrices in (1).
3. NL, SYSTEMS
NL, systems are non-linear discrete time systems of the state space form'
DISCRETE TIME INTERCONNECTED CNNs
27
with state vector p k E R input vector W , E R and output vector e , E R ‘c. Here r,,A, ( i = 1, ..., 9 ) are
w , ) € [0,11 for all values of p , , W , and
diagonal matrices with diagonal elements y,(p,, w L ) , AJ(pA,
depending continuously on the variables p , , w,. The matrices V,, W,, B,, D, for i = 1, ...,q are constant
with compatible dimensions.
An 1/0 equivalent representation to ( 3 ) is
‘$1,
‘$1
with Q, = diag( I‘,.e , A,,e , 0 ) (i = 1, ..., q ) and R, = blockdiag{ M,, N,, O ) , R, = [M,; Nq;01 (i = 1, ..., 9 - l),
where rl,e=l-l,
r , . e = d i a g { r , , I ) , M I = [ V , B , l , M I = [V,B,;O 11, I I , , ~ = A , ,A,,e=diag{A,,II,
corresponds to e, augmented with a
N , = ( W , D,],N,= [W, D,;O I] ( i = 2 , ...,q). Furthermore
1 and
number of zero elements in order to make n;=,R , square. Note that I)R,I)a1 because )(r,()a
1) A, 11 Q 1 (1,2- or -norm). A typical aspect of NL,s are the 9 ‘layers’ in the state equation and the output
equation (Figure 1).
The NL, system is related to systems of the form
p
, , = a (v,a ( v*... a (V,p, + B , wa)
+ B*w,) + B ,W e )
e , = a ( W , a ( W ,... a(W,p,+D,w,) ... + D 2 w k ) + D I w a )
+
1..
where a ( - )is a static non-linear operator that satisfies the sector condition [0, 1 1. Special cases for a(.) are
e.g. tanh(.) and sat(-). The fact that this NL, is related to non-linear operators that satisfy a sector condition
can be understood as follows. Consider a system related to the NL,
(6)
XA+I=a(wXA)
This can be written as
X,
+
, = r(X,)wX,
with r = diag{ P I ) and y, = a(w,‘x,)/ (w:~,). This is easily obtained by using an elementwise notation and
based on the fact that tanh(.) is a diagonal non-linearity. Equation (6) becomes
i
I
*
The time index is omitted here because of the assignment operator ‘ ’. The notation y,’ means that this
corresponds to the diagonal matrix T(x,). In the case W ; X , = 0, de 1’Hospital’s rule or a Taylor expansion of
a(.) leads to y, = 1.
Figure 1. 1/0 equivalent representation for NL,, system with typical q-‘layer’ feature. The matrices R,, are constant and the
matrices R,(y,, wA)are diagonal and satisfy the condition )I R, 1) G 1 for all values of p r , w 1
28
J. A. K. SUYKENS AND J. VANDEWALLE
For an additional layer
x l ; +I
= fY(Va(Wx,))
(7)
one obtains
r,( X , > ~ ? ( X k ) W X ,
where r,= diag( y , , E o ( v ~ a ( W x , ) ) / a ( v ~ a ( W xand
, ) ) r2=diag{y r , } with y2,= ~(w,Tx,)/(w,?x,).Indeed,
XI + I =
from an elementwise notation one has
4. INTERCONNECTED CNNs as NL,s
We will show now by examples how CNNs and interconnected CNNs can be written as NL, systems. This
is done by applying state augmentation.
Example 1
Consider a CNN of the form
X,
+
I
=AX,+ Ba(Cx,)+ F v ,
+B
Using the state augmentation
51 =
dCx,)
one obtains the state space description
+ Fv, +/I
I = a(CAx, + CB(, + CFV,+ C/I
x , + , = AX, + B(,
(,+
Defining p L= [x,;
5,] and w L= [v,;11, one has the NL, system
PL + I = rl(VIP, + B ,w,)
with matrices
v , = CA
[
CB
”].
4
and r,= diag{I, T(x,, Ca, vk)I with (1 r, I( 1.
&ample 2
Consider six CNNs, interconnected according to Figure 2, containing feedforward, feedback and
29
DISCRETE TIME INTERCONNECTED CNNs
-
c_L
CNN
C”l
C”
2
-
C”
4
c
-
CNN5
Figure 2. Interconnected CNNs containing feedforward, cascade and feedback interconnections. The system can be written as an
NL, with q = 1 (see Example 2 )
cascade interconnections, The system is described as
= A ’ x i + B’a(C’x’,)+E:a(C’xS,)+F’v,+B’
x:+ I =A2X?,+B2a(C2x:.)+E:a(C’x’,)+E&(C6$)+P2
x i + I = A3$+ B3a(C3.4)+ E3,a(C2x?,)
+ B3
x t + I = A4$+ B4a(C4xi)+ Ga(C26)+ E,4a(C3$) + /I4
x i + I = A 5 4 + B 5 a ( C 5 4+
) E:a(C4x1)+ B’
x:+ = A 6 $ + B6a(C6g)
+ g a ( C 4 x i )+ B6
with external input vk.Defining additional state variables
5’,-- ~ ( C ’ X ; ) , i = 1,2, ..., 6
and taking
4;
pic.= [2k; 4;xi; $; 4; ci; 5:;
c;;
ci; ti.;5 3
and w k= [v,; 11, one obtains again an NL, system with q = 1 and matrices
c
B’
E:
A’
A2
A3
A4
A5
C’A’
I
Ei
B3
E;
E:
B4
E:
A6
v, =
B2
E:
BS
B6
C’B’
C’E:
C2Ef C2B2
C2A2
C”:
C3A3
C3E: C 3 B 3
C4A4
C4Ei C4E: C4B4
C5A5
C’E; C S B 5
C6A6
C6E:
C6B6
30
.I.A. K. SUYKENS AND J . VANDEWALLE
B, =
(1 1)
Hence, although the interconnectivity of this system is rather complex, the overall system description is
still low complex ((I = 1) in the sense of NL, theory. This is due to the fact that the CNNs contain a linear
dynamical subcircuit. In Reference 9 systems with higher q-values were obtained in the context of neural
control systems. So-called neural state space models and controllers were considered, e.g. models of the
form
and controllers of the form
with x, the state of the plant, u, the control signal, y , the output of the plant, zk the state of the controller
and d, the reference input. W* and V* are interconnection matrices and B* are bias terms. These types of
recurrent neural networks are general in the sense that they correspond to non-linear models and controllers
of the form
x, + I = J’(x,, UL)
Y , = g(x,,
Un)
and
z, + I = N z , , Y,,
4)
u, = s(z,, ya, d,)
where the functions f(.), g ( . ) , h ( . ) and s(.) are parametrized by multilayer feedforward neural network
architectures. It is well known that these are universal approximators because any continuous non-linear
function can be approximated arbitrarily well on a compact interval by means of a multilayer feedforward
neural network with one or more hidden layers.l3-I6 This leads then to NL,s with q > 1 because of the
multilayer feature of the recurrent neural networks.
Based on this insight, interconnected CNNs that correspond to q > 1 can be obtained as follows. For
CNNs like (1) the non-linearity f(.) satisfies usually a sector condition [0, 11. However, it has been shown
DISCRETE TIME INTERCONNECTED CNNs
31
in Reference 6 that the Chua circuit can be written in a form like (1) (but in continuous time and on the cell
level). The non-linearity of the non-linear resistor plays then the role of the non-linearity f(.) in the
continuous time version to of (1). Hence, instead of taking y : = ~(C'Z;.)in (l), a useful extension is to
take
y ; = a(W'u(V'x:+ 6 ' ) )
where W'a(V'xi + 6') is a multilayer feedforward neural network with interconnection matrices W', V' and
bias vector 6', which is able to represent any continuous non-linear mapping (including the non-linearity
used in the Chua circuit) provided that there are 'enough' hidden neurons. This leads to the type of
recurrent neural network proposed in the following example.
Example 3
The use of a multilayer perceptron for f(.)in (1) gives
X,
+, = A x , + Bu(WU(VX,+ 6 ) )+ F v , + j3
Applying the state augmentation
<,= u ( W u ( V x , + 6 ) )
one obtains the state space description
,
x , + = A ,+ Bc,+ Fv,+
j3
~,+,=u(WU(VA~,+VB~,+VFV,+V~~+~))
Defining p , = [x,; C1] and w,= [v,; 11, this yields an NL, system with q = 2 and
t
o
VF Vj3
A similar interconnection between CNNs as in Example 2 can be considered which consists of CNNs of
the form (14). This would also lead to an NL1.
Hence from the viewpoint of NL, theory one has the following ordering in terms of the network
complexity q:
linear dynamical system d CNN or interconnected CNNs with u sector [0, 1] non-linearity
d CNN or interconnected CNNs with multilayer perceptron non-linearity
4 neural state space models and neural control systems
d NL, systems
The fact that interconnected CNNs are reducible to NL,s implies that such systems have still low
complexity with respect to analysis (at least from the viewpoint of NL,s).
5. STABILITY CRITERIA FOR NL,s
In this section we review some sufficient conditions for global asymptotic stability and 1 / 0 stability of
NL,s that were derived in Reference 9.
5.1. Autonomous NL,s
The following Theorem holds for autonomous NL,s.
32
J. A. K. SUYKENS AND J. VANDEWALLE
Theorem I (Diagonal scaling)
A sufficient condition for global asymptotic stability
?
I
-
0
0
VIOI
0
v2
v3
=
0
Vl
v
q
0J
n,,, = n,lq , = np, and D,, = diag{D2,D,, ...,D,,D l ] , D,E 08%
V ,E
non-zero diagonal elements.
aB'''1,
n'l,
+
1,
+
"'+diagonal matrices with
Proof. The proof given here is according to Reference 9. Propose a Lyapunov function
V ( p )= 11 D,p ]I2 (positive, radially unbounded and V ( 0 )= 0) with D ,E R"r "p a diagonal matrix with
nonzero diagonal elements. Hence V, = 11 Dip, and V , + I = 11 D,T,V,rzl V, ...TyV4p,112. The following
procedure is applied then: (a) insert D,-'D, after r,,where D, E Rnh, "h, (i = 1, ..., q - l), and insert 0;'
after V',; (b) the diagonal matrices I', and D,-' commute; ( c )use the properties of induced norms and the
fact that (1 r,11, G 1. Hence
v,+ = II ~ l ~ l D ; l D l ...
v l~ , D ; l ~ , v q ~ ; l ~ l P , l l *
I
~ll~,11211DIVID?'1I2l1~llD2V,D,'Il2
2112
* a *
Ilzv,
l l ~ , l 1 2 1 1 ~ , ~ , ~ ?
41DIVIW 112 II &v2D;' II, ...II D,V,D;I Ilzv,
Defining BD = nr= (motlulo o) 1) D,V, D,-l , a sufficient condition for global asymptotic stability of the NL,
system is bD < 1, because then AV, = V, , - V, < (BD- 1)V, < 0. This condition BD < 1 is satisfied if
+
max ( \ l D , V l D ~ ~ il ]= \ 21,: . . . , q (modq)) s
I
or if
or if
D,V~D;I
Bp < 1
DISCRETE TIME INTERCONNECTED CNNs
33
The condition on this permuted block diagonal matrix is then satisfied if
I1D,o,v,olD,: 114 6 B D < 1
Remarks
1. For (I = 1 the theorem reduces to the results in Reference 17 and is closely related to diagonal scaling
in ,u control theory.'"
2. Less conservative criteria were also derived in Reference 9. Instead of diagonal matrices D,, full
matrices P i are allowed provided that some condition of diagonal dominance on the matrices PTP,
holds.
5.2. Non-autonomous NL,s: I10 properties
The conditions of the following theorem are sufficient in order to guarantee 1/0 stability of the NL,.
Theorem 2 ( I 2 theory-Diagonal
scaling)
Given the representation (4), if there exist diagonal matrices D j such that
II D,o,Rto,D,: )I;=
B D
<1
then the following holds.
1. For a finite time horizon N
N- 1
N-l
with rn = II DIP" 112.
2 . For N + = there exist constants c I and c2 such that
0
R2
0 R,
0
-
Rto, =
Rl
0 R,
0-
34
J. A. K. SUYKENS AND J. VANDEWALLE
because 11 0,
IIzS1. Defining fiD = 11 Ds,R,D;' [I2 n:l=;l)I D,R,D,-+', ] I 2 (1 D & D ; , ' \I2, it follows from the proof
Defining r,= 11 D , p , [I2, the following holds
in Theorem 1 that fiD< 1 is satisfied if (1 D , o , R , , , D ~ ~ I l1.~ <
provided that PI, < 1:
2
rl+l
N-
N- 1
Furthermore, for N+=, (1 - 8);
one obtains the result.
11 r. ;1
+IIe,115 ~B:,r;+B;llwL115
N-l
I
11 e 1:
N-
N- I
1
N-l
N- I
N- I
N- I
4
ri+ ,!?; 11 w .;1
2
Defining constants cl = &,, and c2= a,,
,
0
Remarks
1. For q = 1 the theorem and the proof reduce to these given in Reference 10 for the state space upper
bound test in p control theory. Indeed, LFTs (linear fractional transformations) with real diagonal
uncertainty block are special cases of NL,s.
2. As for the autonomous case, sharper criteria based on diagonal dominance instead of diagonal scaling
were derived in Reference 9.
3. There exists a close relationship between the internal stability of the autonomous case and the
property of finite L,-gain of Theorem 2. This has already been stated in the work of Hill and
Moylan,'8,'YWillems20-22and van der Schaft" and becomes clear through the concept of
dissipativity. A dynamic system with input w k ,output e, and state vector p , is called dissipative if
there exists a non-negative function @ ( p ) : R "P+ R with @ (0) = 0, called the storage function, such
E R "* and Vk 2 0
that VJW
@(P,+t)- @ ( P , ) ~ w ( e , ? % )
where W(e,, w,) is called the supply rate. The NL, system is dissipative under the condition of
Theorem 2, with storage function @ ( p )= 11 D , p )I:, supply rate W(e,, w,) =
w L1 ; - 11 e, 1: and
finite &-gain fi < 1. Similar results hold for the case of diagonal dominance.
6. CHECKING STABILITY IS AN LMI PROBLEM
The conditions of Theorems 1 and 2 are of the form
with
V,,, global asymptotic stability
ZIOt = R,,, finite,!,,-gain < 1
and D,,, = diag( D 2 ,D,, ..., D , , D s ,1 with
global asymptotic stability
I), finite L2-gain < 1
DISCRETE TIME INTERCONNECTED CNNs
35
This condition (20) can be written as the linear matrix inequality (LMI)
~ L , D l z i<fD:,I
The notation A < O means A negative definite. A < B means A - B negative definite. For an LMI M ( x ) < O
with M = M T and unknown x E R “ the matrix M depends affinely on x: M ( x ) = M , + ~ x,M,
~ , with
M, = M:, M ,= M:. Finding a feasible x to such an LMI corresponds to solving a convex problem which has
a unique minimum. The book by Boyd ef ul.“ gives an overview of LMIs arising in systems and control.
Moreover, the convex problem can be solved in polynomial time.I2 Other related work has been done by
Nemirovskii and Gahinet,24Vandenberghe and BoydZ5and Overton.26Matlab software for solving (21) for
the case Z
,, = V,,, is the function psv of Matlab’s Robust Control To~lbox.’~
Software for solving generaltype LMIs is available in Matlab’s LMI lab.23
7. CONCLUSIONS
In this paper we investigated interconnected CNNs in a discrete time setting. Some sufficient conditions for
global asymptotic stability and 1/0 stability were presented by transforming these systems to NL, form.
Interconnected CNNs that consist of CNNs with non-linearities in sector [0,1] were reduced to NL,s,
which still have low complexity from the viewpoint of NL, theory. Existing sufficient stability criteria
within NL, theory reduce for q = 1 to well-known results in modern control theory. Checking stability
according to these criteria involves solving an LMI problem which is convex and computationally feasible.
Furthermore, it has been illustrated with examples how state augmentation can be applied in order to
transform given systems into an NL, form. This gives insight into the overall structure and complexity of
the interconnected system. We have to stress that the stability criteria derived in this paper are related to the
case of a unique equilibrium point. As a result the criteria are not suitable for image-processing tasks. On
the other hand the global asymptotic stability criteria might open up new possibilities and challenges for
applications to modelling and control of non-linear systems. Hence, when novel multilayer highly complex
and interconnected systems are created, NL, theory may contribute towards a unifying framework for the
analysis and design of such dynamical systems.
ACKNOWLEDGEMENTS
This research work was carried out at the ESAT laboratory and the Interdisciplinary Center of Neural
Networks (ICNN) of the Katholieke Universiteit Leuven in the framework of the Belgium Programme on
Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister’s Office for Science,
Technology and Culture (IUAP-17) and in the framework of a Concerted Action Project MIPS of the
Flemish Community. The scientific responsibility rests with its authors.
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