By George A. Rovithakis PhD, Manolis A. Christodoulou PhD (auth.)

The sequence Advances in business keep watch over goals to file and inspire know-how move on top of things engineering. The speedy improvement of keep an eye on expertise has an impression on all components of the keep watch over self-discipline. New conception, new controllers, actuators, sensors, new business strategies, computing device tools, new purposes, new philosophies ... , new demanding situations. a lot of this improvement paintings is living in commercial studies, feasibility research papers and the stories of complicated collaborative tasks. The sequence deals a chance for researchers to offer a longer exposition of such new paintings in all facets of commercial keep an eye on for wider and fast dissemination. Neural networks is a kind of components the place an preliminary burst of enthusiasm and optimism ends up in an explosion of papers within the journals and lots of shows at meetings however it is barely within the final decade that major theoretical paintings on balance, convergence and robustness for using neural networks on top of things structures has been tackled. George Rovithakis and Manolis Christodoulou were drawn to those theoretical difficulties and within the sensible facets of neural community functions to business difficulties. This very welcome boost to the Advances in business keep watch over sequence offers a succinct record in their study. The neural community version on the middle in their paintings is the Recurrent excessive Order Neural community (RHONN) and a whole theoretical and simulation improvement is gifted. diversified readers will locate diverse points of the advance of curiosity. The final bankruptcy of the monograph discusses the matter of producing or construction procedure scheduling.

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**Extra resources for Adaptive Control with Recurrent High-order Neural Networks: Theory and Industrial Applications**

**Example text**

MaX~~i-1) ; i = 1,2, ... ) TI¢;j2 :::; form (TiM? 50) can be written in the V:::; -aV +K, where K := 2::7=1 ((TiM? 2ai) and Vi is an upper bound for Vi; therefore, for V ;::: Vo = Kia, we have V :::; 0, which implies that V E 'coo. Hence ei, ¢i E 'coo· 24 2. J < ~ (-a·e. - e·v-) < ~ --e· 'I I. J 2' i=l i=l 2) v· +-' 2. 51) yields V(t) - V(O) ~ t (. e(rW dr ~ - . e(rW dr + _1_ where amin:= min{ai ; i = 1, .. 1. 46) guarantees that ei and 1>i remain bounded for all i = 1, ... n, and furthermore, the "energy" of the state error e(t) is proportional to the "energy" of the modeling error v(t).

1. 46). Then for i = 1, ... l it Iv(rW dr. J ei + ¢i r i ¢i . 48) where I~. is the indicator function defined as I~ i = 1 if 1Wi 1> Mi and I~ i = 0 if IWi 1 :::; Mi. Since ¢i = Wi - wi, we have that T 1 T 1( T T *) ¢i Wi = "2¢i ¢i +"2 ¢i ¢i + 2¢i Wi , 1 2 1 2 1 *2 = "21¢il + "2l wd - "2IWi 1 . maX~~i-1) ; i = 1,2, ... ) TI¢;j2 :::; form (TiM? 50) can be written in the V:::; -aV +K, where K := 2::7=1 ((TiM? 2ai) and Vi is an upper bound for Vi; therefore, for V ;::: Vo = Kia, we have V :::; 0, which implies that V E 'coo.

This signal cannot be measured since h is unknown. To circumvent this problem, we use the following filtered version of II e+lI":e=lI, ah T = ax WS(x) . 3) a strictly positive constant. 3), we take t:;. e=(-h. 5) with the state ( E lR. This method is referred to as error filtering. Furthermore, we choose h(x) to be h(x) = 1 2 21xl . 5) becomes (+ 11":( = II":h - x T Ax + xTWS(x) + XTWiS'(x)u. 7) with respect to time we obtain £'=e~+tr{WTW}+tr{W[Wd. 10) Hence, if we choose tr{WTW} = -exTWS(x), .