By Johan A.K. Suykens, Joos P.L. Vandewalle, B.L. de Moor

Artificial neural networks own numerous homes that lead them to fairly beautiful for functions to modelling and keep an eye on of complicated non-linear structures. between those homes are their common approximation skill, their parallel community constitution and the supply of on- and off-line studying tools for the interconnection weights. although, dynamic versions that comprise neural community architectures may be hugely non-linear and hard to examine therefore. *Artificial Neural Networks for Modelling and**Control of Non-Linear Systems* investigates the topic from a procedure theoretical perspective. but the mathematical idea that's required from the reader is restricted to matrix calculus, simple research, differential equations and simple linear procedure conception. No initial wisdom of neural networks is explicitly required.

The publication provides either classical and novel community architectures and studying algorithms for modelling and regulate. subject matters contain non-linear approach id, neural optimum keep an eye on, top-down version established neural keep watch over layout and balance research of neural keep watch over platforms. a massive contribution of this publication is to introduce *NLq**Theory* as an extension in the direction of glossy keep watch over thought, with a view to study and synthesize non-linear platforms that comprise linear including static non-linear operators that fulfill a quarter situation: neural kingdom house keep watch over platforms are an instance. additionally, it seems that *NLq Theory* is unifying with appreciate to many difficulties coming up in neural networks, structures and keep watch over. Examples exhibit that complicated non-linear structures might be modelled and regulated inside NLq concept, together with learning chaos.

The didactic style of this ebook makes it appropriate to be used as a textual content for a direction on Neural Networks. moreover, researchers and architects will locate many vital new options, specifically *NLq Theory, that experience functions on top of things concept, procedure conception, circuit concept and Time sequence Analysis.*

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The bias vectors are ß2 E jR n h 2 , ß1 E jR nhl for the second and first hidden layer respectively. 5) j=l = where 1 1, ... , L is the layer index, NI denotes the number of neurons in layer 1 and x~ is the output of the neurons at layer I. The thresholds are considered here to be part of the interconnection matrix, by defining additional constant inputs. ) may depend on the application area. 2. For applications in modelling and control the hyperbolic tangent function tanh(x) = (1- exp( -2x))j(1 +exp( -2x)) is normally used.

G. 7) and Yk is the estimated output. 7), this corresponds to a feedforward neural network. One has a static nonlinear mapping from the input to the output space of the neural network. 2. 8) Because of the recurrence of recurrent neural network. 2) by a feedforward neural network may either lead to an overall feedforward or a recurrent neural network architecture. The difference between series-parallel models and parallel models is quite essential because the learning algorithms for recurrent neural networks are more complicated than for feedforward neural networks, as we already explained in Chapter 2.

G. Ljung (1987)). 2. 1. 17) = P J, with Papermutation matrix and J diag{ ± I}. 2. 2. 2. 22) WCD tanh(Vczk + VDUk + ßCD). 23) Pi a permutation matrix and Ji = diag{±l} and Then putting zk = SXk with 5' E ~nxn and of ~nhyXnhy. 24) wtth WAB = 5' WABT1 , VA = Tl VA 5', VB = Tl VB, ßAB = Tl ßAB, , -1' ' - 1 ' -1' -1 J{ = 5' J{, WCD = WCDT2 , Vc = T 2 Vc5', VD = T 2 VD, ßCD = T 2 ßCD. Although we do not have a formal proof, this suggests that the representation of the neural state space model is unique up to a similarity transformation and sign reversals.