NL q theory: checking and imposing stability of recurrent neural networks for nonlinear modelling J.A.K. Suykens, J. Vandewalle, B. De Moor Katholieke

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1 NL q theory: checing and imposing stability of recurrent neural networs for nonlinear modelling J.A.K. Suyens, J. Vandewalle, B. De Moor Katholiee Universiteit Leuven, Dept. of Electr. Eng., ESAT-SISTA Kardinaal Mercierlaan 94, B-3001 Leuven (Heverlee), Belgium Tel: 3/16/ Fax: 3/16/ johan.suyens,joos.vandewalle,bart.demoor@esat.uleuven.ac.be (Author for correspondence: J. Suyens) to appear in IEEE-SP (special issue: NNs for SP), SP EDICS Abstract It is nown that many discrete time recurrent neural networs, such as e.g. neural state space models, multilayer Hopeld networs and locally recurrent globally feedforward neural networs, can be represented as NL q systems. Sucient conditions for global asymptotic stability and input/output stability of NL q systems are available, including three types of criteria: diagonal scaling and criteria depending on diagonal dominance and condition number factors of certain matrices. In this paper, it is discussed how Narendra's dynamic bacpropagation procedure, used for identifying recurrent neural networ from I/O measurements, can be modied with an NL q stability constraint in order to ensure globally asymptotically stable identied models. An example illustrates how system identication of an internally stable model, corrupted by process noise, may lead to unwanted limit cycle behaviour and how this problem can be avoided by adding the stability constraint. Keywords. multilayer recurrent neural networs, NL q systems, global asymptotic stability, LMIs, dynamic bacpropagation This research wor was carried out at the ESAT laboratory and the Interdisciplinary Center of Neural Networs ICNN of the Katholiee Universiteit Leuven, in the framewor of the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister's Oce for Science, Technology and Culture (IUAP-17, IUAP-50) and in the framewor of a Concerted Action Project MIPS (Modelbased Information Processing Systems) of the Flemish Community, ICCoS (Identication and Control of Complex Systems), Human Capital and Mobility Networ (SIMONET) and SCIENCE-ERNSI (SC1-CT9-0779). 1

2 1 Introduction Recently, NL q theory has been introduced as a modelbased neural control framewor with global asymptotic stability criteria [0, 4]. It consists of recurrent neural networ models and controllers in state space form, for which the closed-loop system can be represented in so-called NL q system form. NL q systems are discrete time nonlinear state space models with q layers of alternating linear and static nonlinear operators that satisfy a sector condition. It has been shown then how Narendra's dynamic bacpropagation, classically used to learn a controller trac a set of specic reference inputs, can be modied with NL q stability constraints. Furthermore several types of nonlinear behaviour, including systems with a unique equilibrium, multiple equilibria, (quasi)-periodic behaviour and chaos have been stabilized and controlled using the stability criteria [0]. In this paper we focuss on nonlinear modelling applications of NL q theory, instead of control applications. Lie for tracing problems, where Narendra's dynamic bacpropagation [9, 10] has been modied with a closed-loop stability constraint [0], we will modify dynamic bacpropagation for system identication with a stability constraint in order to obtain identied recurrent neural networs that are globally asymptotically stable. Also for linear lters (e.g. IIR lters [16, 17]) this has been an important issue. We will consider the class of discrete time recurrent neural networs, which is representable as NL q systems. Examples are e.g. neural state space models and locally recurrent globally feedforward neural networs, that are models consisting of global and local feedbac respectively. In order to chec stability of identied models, sucient conditions for global asymptotic stability of NL q s are applied [0]. A rst condition is called diagonal scaling, which is closely related to diagonal scaling criteria in robust control theory [3, 13]. Checing stability can be formulated then as an LMI (Linear Matrix Inequality) problem, which corresponds to solving a convex optimization problem. A second condition is based on diagonal dominance of certain matrices. Certain results of digital lters with overow characteristic [8] can be considered as a special case for q = 1 (one layer NL q ). Finally, we demonstrate how global asymptotic stability can be imposed on the identied models. This is done by modifying dynamic bacpropagation with stability constraints. Besides the diagonal scaling condition, criteria based on condition numbers of certain matrices [0] are proposed for this purpose. In many applications one has indeed the a priori nowledge that the true system is globally asymptotically stable or one is interested in a stable approximator. It is illustrated with an example that process noise can indeed cause identi-

3 ed models that show limit cycle behaviour, instead of global asymptotic stability as for the true system. We show how this problem can be avoided by applying the modied dynamic bacpropagation algorithm. This paper is organized as follows. In Section we present two examples of discrete time recurrent neural networs, that are representable as NL q systems: neural state space models and locally recurrent globally feedforward neural networs. In Section 3 we review the classical dynamic bacpropagation paradigm. In Section 4 we discuss sucient conditions for global asymptotic stability of identied models. In Section 5 Narendra's dynamic bacpropagation is modied with NL q stability constraints. In Section 6 an example is given for a system corrupted by process noise. NL q systems The following discrete time nonlinear state space model is called an NL q system [0]: 8 < : p +1 =? 1 ( V 1? ( V :::? q ( V q p + B q w )::: + B w ) + B 1 w ) e = 1 ( W 1 ( W ::: q ( W q p + D q w )::: + D w ) + D 1 w ) (1) with state vector p R np, input vector w R nw and output vector e R ne. The matrices V i, B i, W i, D i (i = 1; :::; q) are constant with compatible dimensions, the matrices? i = diagf i g, i = diagf i g (i = 1; :::; q) are diagonal with diagonal elements i (p ; w ), i (p ; w ) [0; 1] for all p ; w. The term 'NL q ' stands for the alternating sequence of linear and nonlinear operations in the q layered system description. In this Section we rst explain the lin between NL q s and multilayer Hopeld networs and then discuss two examples: neural state space models and locally recurrent globally feedforward networs, which are models with global feedbac and local feedbac respectively. Other examples of NL q s are e.g. (generalized) cellular neural networs, the discrete time Lur'e problem and linear fractional transformations with real diagonal uncertainty bloc [0, 3]..1 Multilayer Hopeld networs NL q systems are closely related to multilayer recurrent neural networs of the form: 8 < : p +1 = 1 ( V 1 ( V ::: q ( V q p + B q w )::: + B w ) + B 1 w ) e = 1 ( W 1 ( W ::: q ( W q p + D q w )::: + D w ) + D 1 w ) () with i (:), i (:) vector valued nonlinear functions that belong to sector [0; 1] [8]. 3

4 Let us illustrate the lin between (1) and () for the autonomous NL q system (zero external input) p +1 = ( qy i=1? i (p )V i ) p by means of the following autonomous Hopeld networ with synchronous updating: x +1 = tanh(w x ): (3) This can be written as x +1 =?(x )W x (4) with? = diagf i g and i = tanh(w T i x )=(w T i x ), which follows from the elementwise notation x i := tanh( P j wi j xj ) := tanh( P j wi j xj ) Pj wi j xj : P j wi j xj := i i P j wi j xj : (5) The time index is omitted here because of the assignment operator 0 := 0. The notation i i means that this corresponds to the diagonal matrix?(x ). In case wi T x = 0 de l' Hospital's rule can be applied or a Taylor expansion of tanh(:) can be taen, leading to i = 1. In a similar way the multilayer Hopeld neural networ x +1 = tanh(v tanh(w x )); (6) can be written as x +1 =? 1 (x )V? (x )W x (7) because x i := tanh( P j vi j tanh(p l wj l xl ) := tanh( P j vi j j i := 1i Pj vi j j j P j l wj l xl ) P l wj l xl : (8). Neural state space models Neural state space models (Fig.1) for nonlinear system identication have been introduced in [19] and are of the form: 8 < : ^x +1 = W AB tanh(v A^x + V B u + AB ) + K y = W CD tanh(v C ^x + V D u + CD ) + (9) 4

5 with estimated state vector ^x R n, input vector u R m, output vector y R l, zero mean white Gaussian noise input (corresponding to the prediction error y? ^y ). W, V are the interconnection matrices with compatible dimensions, are bias vectors and K is a steady Kalman gain. If the multilayer perceptrons in the state equation and output equation are replaced by linear mappings, the neural state space model corresponds to a Kalman lter. Dening p = x, w = [u ; ; 1] and e = y in (1), the neural state space model is an NL system with? 1 = I, V 1 = W AB, V = V A, B = [V B 0 AB ], B 1 = [0 K 0], 1 = I, W 1 = W CD, W = V C, D = [V D 0 CD ], D 1 = [0 I 0]. For the autonomous case and zero bias terms, the neural state space model becomes: ^x +1 = W AB tanh(v A^x ): (10) By introducing a new state variable = tanh(v A^x ), this can be written as the NL 1 system: 8 < : ^x +1 = W AB +1 = tanh(v A W AB ):.3 Locally recurrent globally feedforward networs In [6] the LRGF networ (Locally Recurrent Globally Feedforward networ), has been proposed, which aims at unifying several existing recurrent neural networ models. Starting from the McCulloch-Pitts model, many other architectures have been proposed in the past with local synapse feedbac, local activation feedbac and local output feedbac, time delayed neural networs etc., e.g. by Frasconi-Gori-Soda, De Vries-Principe and Poddar-Unnirishnan. The architecture of Tsoi & Bac (Fig.) includes most of the latter architectures. Assuming on Fig. that the transfer functions G i (z) (i = 1; ::; n) (which may have both poles and zeros) have a state space realization (A i ; B i ; C i ) an LRGF networ can be described in state space form as 8 >< >: (i) +1 = A (i) (i) z (i) = C (i) (i) + B(i) u (i) ; i = 1; :::; n? 1 (n) = A (n) (n) +1 + B (n) f( P n j=1 z (j) ) z (n) = C (n) (n) y = f( P n j=1 z (j) ): Here u (i) ; z(i) R (i = 1; :::; n? 1) are the inputs and local outputs of the networ, y R is the output of the networ and z (n) R the ltered output of the networ. f(:) is a static 5 (11) (1)

6 nonlinearity belonging to sector [0; 1]. Applying the state augmentation = f( P n j=1 z (j) ) an NL 1 system is obtained with state vector p = [ (1) w = [u (1) ; :::;u(n?1) ] and matrices V 1 = 6 4 ; () ; :::; (n?1) A (1) 0 0 A ()... A (n?1) ::: 0 A (n) B (n) C (1) A (1) C () A () ::: C (n?1) A (n?1) C (n) A (n) C (n) B (n) B 1 = 6 4 B (1) B ()... B (n?1) 0 0 ::: 0 C (1) B (1) C () B () ::: C (n?1) B (n?1) 3 Classical dynamic bacpropagation ; (n) ; ], input vector Dynamic bacpropagation, according to Narendra & Parthasarathy [9, 10], is a well-nown method for training recurrent neural networs. In this Section we shortly review this method for models in state space form, because this ts into the framewor of NL q representations. Let us consider discrete time recurrent neural networs that can be written in the form: 8 < : ^x +1 = (^x ; u ; ; ) ; ^x 0 = x 0 given ^y = (^x ; u ; ) where (:), (:) are twice continuously dierentiable nonlinear mappings and the weights ; are elements of the parameter vector, to be identied from a number of N input/output data Z N = fu ; y g =N: =1 min J N (; Z N ) = 1 N NX =1 : (13) l( ()): (14) A typical choice for l( ) in this prediction error algorithm is 1 T with prediction error = y? ^y. For gradient based optimization algorithms one computes the = 1 N NX =1 (? ): 6

7 Dynamic bacpropagation [9, 10] maes use then of a sensitivity model 8 >< ^x @ ^x @ ^x ^x @ in order to generate the gradient of the cost function. The sensitivity model is a dynamical system with state and at the Rl, Rn, driven by the input vector @ ^x R ln are evaluated around the nominal trajectory. Rn, Rl are generated. ^x R nn and Examples and applications of dynamic bacpropagation, applied to neural state space models, are discussed in [19, 0, 1]. pruning of neural networ models see e.g. [1, 15, 18]. For aspects of model validation, regularization and 4 Checing global asymptotic stability of identied models In many applications, recurrent neural networs have been used in order to model systems with a unique or multiple equilibria, (quasi)-periodic behaviour or chaos. On the other hand one is often interested in obtaining models that are globally asymptotically stable, e.g. in case one has this a priori nowledge about the true system or one is interested in such an approximator. In this Section, we present two criteria that are sucient in order to chec global asymptotic stability of the identied model, represented as NL q system. and let Theorem 1 [Diagonal scaling] [0]. Consider the autonomous NL q system V tot = 6 4 p +1 = ( 0 V 0 0 V V q V 1 0 qy i= ? i (p )V i ) p (17) ; V i R n h i n hi+1 ; nh1 = n h q+1 = n p: A sucient condition for global asymptotic stability of (17) is to nd a diagonal matrix D tot such that D tot V tot D?1 tot q = D < 1; (18) 7

8 where D tot = diagfd ; D 3 ; :::; D q ; D 1 g and D i R n h i n hi are diagonal matrices with nonzero diagonal elements. Proof: See [0]. The condition is based on the Lyapunov function V (p) = D 1 p, which is radially unbounded in order to prove that the origin is a unique equilibrium point. Finding such a diagonal matrix D tot for a given matrix V tot can be formulated as the LMI (Linear Matrix Inequality) in D tot: V T totd totv tot < D tot: (19) It is well-nown that this corresponds to solving a convex optimization problem [3, 1, 7]. Similar criteria are nown in the eld of control theory as 'diagonal scaling' [3, 7, 13]. However, it is also nown that such criteria can be conservative. Sharper criteria are obtained by considering a Lyapunov function V (p) = P 1 p a non-diagonal matrix P 1 with instead of D 1. The next Theorem is expressed then in terms of diagonal dominance of certain matrices. Let us recall that a matrix Q R nn is called diagonally dominant of level Q 1 if the following property holds [0]: q ii > Q The following Theorem holds then. n X j=1 (j6=i) jq ij j ; 8i = 1; :::; n: (0) Theorem [Diagonal dominance] [0]. A sucient condition for global asymptotic stability of the autonomous NL q system (17) is to nd matrices P i, N i such that qy i=1 Qi with P tot = blocdiagfp ; P 3 ; :::; P q ; P 1 g, and P i R n h i n hi ( Qi? 1 )1= P tot V tot P tot?1 q < 1 (1) full ran matrices. The matrices Q i = P T i P in i are diagonally dominant with Qi > 1 and N i are diagonal matrices with positive diagonal elements. Proof: See [0]. 8

9 In order to chec stability of an identied model, i.e. for a given matrix V tot, one might formulate an optimization problem in P i and N i such that (1) is satised. LMI conditions that correspond to (1) are derived in [0]. For the case of neural networs with a sat(:) activation function (linear characteristic with saturation) Theorem can be formulated in a sharper way [0]. In that case it is sucient to nd matrices P i, N i with: P tot V tot P?1 tot < 1 such that Qi = 1; (i = 1; :::; q): () The latter follows from a result on stability of digital lters with overow characteristic by Liu & Michel [8], which corresponds to the NL 1 system: x +1 = sat(v x ): (3) A sucient condition for global asymptotic stability is then to nd a matrix Q = P T P with: Also remar that for a linear system P V P?1 < 1 such that Q = 1: (4) x +1 = Ax (5) one obtains the condition P AP?1 < 1; (6) from the Lyapunov function V (x) = P x with P a full ran matrix. The spectral radius of A corresponds to (A) = min P R P AP?1 : (7) nn 5 Modied dynamic bacpropagation: imposing stability In this Section we discuss modied dynamic bacpropagation algorithms, by adding an NL q stability constraint to the cost function (14). In this way it will be possible to identify recurrent neural networ models that are guaranteed to be globally asymptotically stable. Based on the condition of Theorem 1, one may solve the following constrained nonlinear optimization problem: min J N (; Z N ) = 1 ;Dtot N NX =1 l( ()) such that V tot () T D totv tot () < D tot: (8) 9

10 The cost function is dierentiable but the constraint becomes non-dierentiable when the two largest eigenvalues of the matrix V tot () T D totv tot ()? D tot coincide [14]. Convergent algorithms for such non-convex non-dierentiable optimization problems have been described e.g. by Pola & Wardi [14]. The gradient based optimization method maes use of the concept of a generalized gradient [] for the constraint. An alternative formulation of the problem is: min J N (; Z N ) = 1 N NX =1 l( ()) such that min D tot V tot ()D tot?1 < 1: (9) Dtot The evaluation of the constraint corresponds then to solving a convex optimization problem. Though LMI conditions have been formulated for the condition of diagonal dominance of Theorem, the use of these LMI conditions is rather unpractical for a modied dynamic bacpropagation algorithm. Therefore we will mae use of another Theorem in order to impose stability on the identied models. Theorem 3 [Condition number factor] [0]. A sucient condition for global asymptotic stability of the autonomous NL q (17) is to nd matrices P i such that qy i=1 (P i ) P tot V tot P?1 tot q < 1 (30) where P tot = blocdiagfp ; P 3 ; :::; P q ; P 1 g and P i R n h i n hi are full ran matrices. The condition numbers (P i ) are by denition equal to P i P?1 i. Proof: See [0]. In practice, this Theorem is used as: min Ptot maxf(p i )g such that P tot V tot P tot?1 < 1: (31) i The constraint in (31) imposes local stability at the origin. The basin of attraction of the origin is enlarged by minimizing the condition numbers. Even when condition (30) is not satised, the basin of attraction can often be made very large (or probably innitely large as simulation results suggest). This principle has been demonstrated on stabilizing and controlling systems with one or multiple equilibria, periodic and quasi-periodic behaviour and chaos [0]. 10

11 Based on (31), dynamic bacpropagation is modied then as follows: min J N (; Z N ) = 1 ;Qtot N NX =1 l( ()) such that 8 >< >: V tot () T Q tot V tot () < Q tot I < Q i < i I: (3) The latter LMI corresponds to (P i ) < i and Q tot = P T tot P tot, Q i = P T i P i. An alternative formulation to (3) is: min J N (; Z N ) = 1 N NX =1 l( ()) such that min P tot V tot ()P tot?1 < 1 with maxf(p i )g < c Ptot (33) with c a user-dened upper bound on the condition numbers. For (9)-(33) the dierence in computational complexity between classical and modied dynamic bacpropagation is in the LMI constraint which has to be solved at each iteration step and is O(m 4 L 1:5 ) for L inequalities of size m [7]. One can avoid solving LMIs at each iteration step at the expense of introducing additional unnown parameters to the optimization problem as Q tot in (3). Finally, for the case of NL 1 systems, the problem can be formulated by solving a Lyapunov equation: min J N (; Z N ) = 1 ; 0<<1 N NX =1 l( ()) such that 8 >< >: V tot () T QV tot () + I = Q min 0 s:t: R = Q + I > 0 (P ) < c; R = P T P: 6 Example: a system corrupted by process noise Problem statement (34) In this example we consider system identication of the following neural state space model as true system: 8 < : x +1 = W AB tanh(v A x + V B u ) + ' y = W CD tanh(v C x + V D u ): It is corrupted by zero mean white Gaussian process noise '. I/O data were generated by (35) 11

12 taing?0:1986?0:635 0:408?1:033 3 W AB = 4 0:4157?0:006 0:160?0:037 1:171?0:0401?0:6084 0: VA = 4?0:315?0:839?0:33?0:5119?0:87?0:7385?0:4354 0:416 5 VB =?0:141 0:4840?0:966?0:007?0:1009?0:6593?0:5717?0:8109 W CD =?0:5546?0:603 1:3030?1:3587?0:8550?0:7005?0:0671 0:059 0:5938?0:655 0:5651?1: :156 1:334 1:0599?1:7554 V C = 4 0:3600 0:597 1:7870?1:4743 1:3145?0:945 0:0347 0:681 5 VD = 4 1:675?:3663?1:15?0:9360 0:355 1:7658?0:8700?0:7000 with zero initial state. The input u is zero mean white Gaussian noise with standard deviation 5. The standard deviation of the noise ' was chosen equal to 0.1. A set of 1000 data points was generated in this way, with the rst 500 data for the training set and the next 500 data for the test set. Some properties of the autonomous system are: (W AB V A ) = 0:98 < 1 and min D tot D totv tot D?1 tot = 1:66 > 1, which means that the origin is locally stable, but global asymptotic stability is not proven by the diagonal scaling condition. However, simulation of the autonomous system for several initial conditions suggests that the system is globally asymptotically stable (Fig.4). Because the state space representation of neural state space representation is only unique up to a similarity transformation and sign reversal of the hidden nodes [0], we are interested in identifying the true system, not in the sense of nding bac the original matrices, but in order to obtain the same qualitative behaviour for the autonomous system of the identied model as for the true system Application of classical versus modied dynamic bacpropagation System identication by using classical dynamic bacpropagation and starting from randomly chosen parameter vectors (deterministic neural neural state space model with the same number of hidden neurons as (35)), yields identied models with limit cycle behaviour for the autonomous system (Fig.5). This eect is due to the process noise '. No problems of this ind were met for the system (35) in the purely deterministic case or in the case of observation noise. In order to impose global asymptotic stability on the identied models, modied dynamic bacpropagation procedure was applied with diagonal scaling (8) and with condition number constraint (34). The autonomous system of (35) can be represented as the NL 1 system (11). Fig.3 shows a comparison between classical and modied dynamic bacpropagation for the error on the training set and test set, starting from 0 random parameter vectors for the 3 methods. Besides the fact that the identied models appear to be globally asymptotically stable for modied dynamic bacpropagation (Fig.6-7), the performance on 1

13 the test data is often better than with respect to classical dynamic bacpropagation. One might argue that by taing a Kalman gain into the predictor the limit cycle problem might be avoided as well. However, often the deterministic part is identied rst and secondly the Kalman gain, while eeping the other parameters constant. Moreover the stability constraint can also be taen into account for stochastic models. Software In the experiments, a quasi-newton optimization method with BFGS updating of the Hessian [4, 6] was used for classical dynamic bacpropagation (function fminu in Matlab [5]). For modied dynamic bacpropagation, sequential quadratic programming [4, 6] was used (function constr in Matlab). Numerical calculation of the gradients was done. In the experiments, 70 iteration steps were taen for the optimization (Fig.3). For the case of diagonal scaling the Matlab function psv was used in order to calculate (8). Other software for solving LMI problems is e.g. LMI lab [5]. For the case of condition numbers, c = 100 was chosen as upper bound in (34). 7 Conclusion In this paper we discussed checing and imposing stability of discrete time recurrent neural networ for nonlinear modelling applications. The class of recurrent neural networs that is representable as NL q systems has been considered. Sucient conditions for global asymptotic stability on diagonal scaling and diagonal dominance have been proposed for checing stability. Dynamic bacpropagation has been modied with NL q stability constraints in order to obtain models that are globally asymptotically stable. Therefore criteria of diagonal scaling and condition number factors have been used. It has been illustrated on an example how a globally asymptotically stable system, corrupted by process noise, may lead to unwanted limit cycle behaviour for the autonomous behaviour of the identied model, if one applies classical dynamic bacpropagation. The modied dynamic bacpropagation algorithm overcomes this problem. 13

14 References [1] Billings S., Jamaluddin H., Chen S., \Properties of neural networs with applications to modelling non-linear dynamical systems," International Journal of Control, Vol.55, No.1, pp.193-4, 199. [] Boyd S., Barratt C., Linear controller design, limits of performance, Prentice-Hall, [3] Boyd S., El Ghaoui L., Feron E., Balarishnan V., Linear matrix inequalities in system and control theory, SIAM (Studies in Applied Mathematics), Vol.15, [4] Fletcher R., Practical methods of optimization, second edition, Chichester and New Yor: John Wiley and Sons, [5] Gahinet, P., Nemirovsii A., \General-Purpose LMI Solvers with Benchmars," Proceedings Conference on Decision and Control, pp , [6] Gill P.E., Murray W., Wright M.H., Practical Optimization, London: Academic Press, [7] Kaszurewicz E., Bhaya A., \Robust stability and diagonal Liapunov functions," SIAM Journal on Matrix Analysis and Applications, Vol.14, No., pp , [8] Liu D., Michel A.N., \Asymptotic stability of discrete time systems with saturation nonlinearities with applications to digital lters," IEEE Transactions on Circuits and Systems-I, Vol.39, No.10, pp , 199. [9] Narendra K.S., Parthasarathy K., \Identication and control of dynamical systems using neural networs," IEEE Transactions on Neural Networs, Vol.1, No.1, pp. 4-7, [10] Narendra K.S., Parthasarathy K., \Gradient methods for the optimization of dynamical systems containing neural networs," IEEE Transactions on Neural Networs, Vol., No., pp.5-6, [11] Nesterov Y., Nemirovsii A., Interior point polynomial algorithms in convex programming, SIAM (Studies in Applied Mathematics), Vol.13, [1] Overton M.L., \On minimizing the maximum eigenvalue of a symmetric matrix," SIAM Journal on Matrix Analysis and Applications, Vol.9, No., pp.56-68,

15 [13] Pacard A., Doyle J., \The complex structured singular value," Automatica, Vol.9, No.1, pp , [14] Pola E., Wardi Y., \Nondierentiable optimization algorithm for designing control systems having singular value inequalities," Automatica, Vol.18, No. 3, pp.67-83, 198. [15] Reed R., \Pruning algorithms - a survey," IEEE Transactions on Neural Networs, Vol.4, No.5, pp , [16] Regalia Ph., Adaptive IIR ltering in signal processing and control, New Yor: Marcel Deer Inc., [17] Shyn J., \Adaptive IIR ltering," IEEE ASSP Magazine, pp.4-1, [18] Sjoberg J., Zhang Q., Ljung L., Benveniste A., Delyon B., Glorennrc P., Hjalmarsson H., Juditsy A., \Nonlinear blac-box modeling in system identication: a unied overview," Automatica, Vol.31, No.1, pp , [19] Suyens J.A.K., De Moor B., Vandewalle J., \Nonlinear system identication using neural state space models, applicable to robust control design," International Journal of Control, Vol.6, No.1, pp.19-15, [0] Suyens J.A.K., Vandewalle J.P.L., De Moor B.L.R., Articial neural networs for modelling and control of non-linear systems, Kluwer Academic Publishers, Boston, [1] Suyens J.A.K., Vandewalle J., \Learning a simple recurrent neural state space model to behave lie Chua's double scroll," IEEE Transactions on Circuits and Systems-I, Vol.4, No.8, pp , [] Suyens J.A.K., Vandewalle J., \Control of a recurrent neural networ emulator for the double scroll," IEEE Transactions on Circuits and Systems-I, Vol.43, No.6, pp , [3] Suyens J.A.K., Vandewalle J., \Discrete time Interconnected Cellular Neural Networs within NL q theory," International Journal of Circuit Theory and Applications (Special Issue on Cellular Neural Networs), Vol.4, pp.5-36, [4] Suyens J.A.K., De Moor B., Vandewalle J., \NL q theory: a neural control framewor with global asymptotic stability criteria," to appear in Neural Networs. 15

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17 Captions of Figures Figure 1. Neural state space model, which is a discrete time recurrent neural networ with multilayer perceptrons for the state and output equation and a Kalman gain for taing into account process noise. Figure. Locally recurrent globally feedforward networ of Tsoi & Bac, consisting of linear lters G i (z) (i = 1; :::; n? 1) for local synapse feedbac and G n (z) for local output feedbac. Figure 3. Comparison between dynamic bacpropagation (-), modied dynamic bacpropagation with diagonal scaling (- -) and condition number constraint (..) for the error on the training data (*) and test data (o). The errors are plotted with respect to the experiment number. 0 random initial parameter vectors were chosen. The experiments were sorted with respect to the error on the training set. Figure 4. Behaviour of the true system to be identied. Shown are the state variables of the autonomous system for some randomly chosen initial state and zero noise. Figure 5. Unwanted limit cycle behaviour of the identied model, after applying classical dynamic bacpropagation. The I/O data were generated by a globally asymptotically stable system, corrupted by process noise. Shown are the state variables of the identied model for the autonomous case. Figure 6. Model identied by applying modied dynamic bacpropagation with diagonal scaling constraint. The model is guaranteed to be globally asymptotically stable as is illustrated for some randomly chosen initial state on this Figure. Shown are the state variables of the identied model for the autonomous case. Figure 7. Model identied by applying modied dynamic bacpropagation with condition number constraint. Local stability at the origin is imposed. Its region of attraction is determined by the condition number. Shown are the state variables of the identied model for the autonomous case. 17

18 -1 z f ^ x + ^ x +1 + Kalman Gain ε - + y g ^ y u Figure 1: 18

19 (1) u G 1 (z) (1) z () u G (z) () z a f(.) y (n-1) u G n-1 (z) (n-1) z z (n) G n (z) Figure : 19

20 Figure 3: 0

21 Figure 4: Figure 5: 1

22 Figure 6: Figure 7: