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. &le 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. 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