By George A. Rovithakis PhD, Manolis A. Christodoulou PhD (auth.)
The sequence Advances in commercial keep watch over goals to file and inspire expertise move up to speed engineering. The speedy improvement of keep an eye on expertise has an impression on all components of the keep an eye on self-discipline. New conception, new controllers, actuators, sensors, new business strategies, computing device equipment, new functions, new philosophies ... , new demanding situations. a lot of this improvement paintings is living in commercial experiences, feasibility research papers and the studies of complicated collaborative initiatives. The sequence deals a chance for researchers to give a longer exposition of such new paintings in all facets of commercial keep watch over for wider and speedy 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 simply within the final decade that major theoretical paintings on balance, convergence and robustness for using neural networks up to the mark platforms has been tackled. George Rovithakis and Manolis Christodoulou were drawn to those theoretical difficulties and within the sensible elements of neural community purposes to business difficulties. This very welcome boost to the Advances in commercial keep an eye on 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 an entire theoretical and simulation improvement is gifted. varied readers will locate diversified features of the advance of curiosity. The final bankruptcy of the monograph discusses the matter of producing or creation procedure scheduling.
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Additional info for Adaptive Control with Recurrent High-order Neural Networks: Theory and Industrial Applications
5) z=h(x,TJ)+'f/, where h(x, TJ) is defined as the quasi-steady-state of z and 'f/ as its fast transient. 6) J-try where we define . - 8h . h(e, W, Wl,TJ,u) = J'}e ve oh':' 8h,:. 8h . + -W + ---Wl + J'}u. oW OWl vU Notice, however, that in the control case, u is a function of e, W, making h(e, W, Wb 'f/, u) to be equal to . - h(e, W, Wl,TJ,u) 8h 8h,:. 1. 3) can be viewed as correction terms in the input vectorfields and in the drift term of x = Ax + BW*S(x) + BW[S'(x)u, in the sense that the unknown system can now be described by a neural network plus the correction terms.
42) The function Fi(X, u) denotes the i-th component of the vector field F(X, u), while the unknown optimal weight vector wi is defined as the value of the weight vector Wi that minimizes the Loo-norm difference between Fi(X, u) + ajX and z(X, u) for all (X, u) EYe ~n+m, subject to the constraint that IWil ~ Mi, where Mi is a large design constant. , (X(t), u(t)) E Y for all t ~ O. Since by assumption u(t) is uniformly bounded and the dynamical system to be identified is BIBO stable, the existence of such Y is assured.
Since e E L2nLoo,17 E L2nLoo, using Barbalat's Lemma we conclude that limt ..... oo e(t) 0, limt ..... oo 17(t) O. Now using the bo~nde~ness ofu, S(x), S'(x) and the convergence of e(t) to zero, we have that W , WI also converges to zero. 2. Again, we cannot conclude anything about the convergence of the weights to their optimal values from the above analysis. 2 provides some details concerning the problem. 2 Indirect Control In this section we investigate the regulation problem. The unknown nonlinear dynamical system which, by now, has already been identified around an operational point, is regulated to zero adaptively.
Adaptive Control with Recurrent High-order Neural Networks: Theory and Industrial Applications by George A. Rovithakis PhD, Manolis A. Christodoulou PhD (auth.)