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

ISBN-10: 144195158X

ISBN-13: 9781441951588

ISBN-10: 1475724934

ISBN-13: 9781475724936

Artificial neural networks own numerous houses that lead them to fairly beautiful for functions to modelling and keep watch over of complicated non-linear structures. between those houses are their common approximation skill, their parallel community constitution and the provision of on- and off-line studying equipment for the interconnection weights. in spite of the fact that, dynamic versions that comprise neural community architectures can be hugely non-linear and tough to examine therefore. *Artificial Neural Networks for Modelling and**Control of Non-Linear Systems* investigates the topic from a procedure theoretical viewpoint. but the mathematical concept that's required from the reader is proscribed to matrix calculus, simple research, differential equations and simple linear procedure thought. No initial wisdom of neural networks is explicitly required.

The booklet offers either classical and novel community architectures and studying algorithms for modelling and keep an eye on. subject matters comprise non-linear process identity, neural optimum keep watch over, top-down version dependent neural keep an eye on layout and balance research of neural keep watch over platforms. an incredible contribution of this booklet is to introduce *NLq**Theory* as an extension in the direction of sleek regulate idea, with the intention to examine and synthesize non-linear platforms that include linear including static non-linear operators that fulfill a quarter : neural country area keep an eye on platforms are an instance. additionally, it seems that *NLq Theory* is unifying with appreciate to many difficulties bobbing up in neural networks, structures and keep an eye on. Examples exhibit that complicated non-linear platforms may be modelled and regulated inside of NLq concept, together with learning chaos.

The didactic taste of this e-book makes it compatible to be used as a textual content for a direction on Neural Networks. additionally, researchers and architects will locate many vital new concepts, specifically *NLq Theory, that experience purposes up to speed idea, process concept, circuit thought and Time sequence Analysis.*

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**Additional info for Artificial Neural Networks for Modelling and Control of Non-Linear Systems**

**Example text**

Now let Jl be a probability measure defined on (IRr, B r ) with B r C IRr a Borel O'-field and M r the set of aIl Borel measurable functions from IRr to IR. Functions f, 9 E M r are caIled Jl-equivalent if Jl{ x E IRr : f(x) g(x)} l. Then the metric Pp. : M r X M r --+ IR + is defined by Pp. (f, g) = inf {f > 0 : Jl{ x : If( x) - g( x) I > f} < ('}. Hence fand gare elose in the metric Pp. if and only if there is only a smaIl probability that they differ significantly and fand gare Jl-equivalent if Pp.

O This means that regardless of the dimension r of the input space and for any squashing function '1/), a feedforward neural network with one hidden layer can approximate any continuous function arbitrarily weIl in the Pp. metric. In the proof of Hornik's Theorems a central role is played by the Stone-Weierstrass theorem. 26 Chapter 2 Artificial neural networks In addition to the previous Theorems, more refined Theorems were formulated by Hornik (1991). More recently Leshno et al. (1993) showed that a standard multilayer feedforward network with a locally bounded piecewise continuous activation function can approximate any continuous function to any degree of accuracy if and only if the network's activation function is not a polynomial.

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. Remark that for linear state space models this is only up to a similarity transformation.

### Artificial Neural Networks for Modelling and Control of Non-Linear Systems by Johan A.K. Suykens, Joos P.L. Vandewalle, B.L. de Moor

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