00-1Automatizácia a riadenie v teórii a praxi,ARTEP 2012
workshop odborníkov z univerzít, vysokých škôl a praxe 
22. 2. – 24. 2. 2012
Stará Lesná, SR


Prospects of Gradient Methods for Nonlinear Control
Ivo BUKOVSKÝ1, Jiří BÍLA1, Homma NORIASU2, 
1Czech Technical University in Prague,  2Tohoku University, Japan

Abstract: The paper introduces a novel approach to evaluation of stability of nonlinear adaptive systems via monitoring of spectral radius of the matrix of dynamics for a class of polynomial nonlinear systems and with the use of flattening operation 
Key words: control, neural networks, gradient descent, spectral radius, weight update system, stability
Introduction
Linear modelling and control approaches are superior regarding their clear analytical solvability and thus possibility of evaluating the stability of the control loop. T
Flattenning of HONU 
HONU are nonconventional neural units with polynomial aggregation of neural inputs with a polynomial of order r. 
	
where x0 is already substituted as x0=1, w0,0 is analogy to neural bias, xi stands for ith neural .

Sketch of optimization error surfaces Linear  x  MLP Networks x HONU
Stability of Gradient Descent AdaptatioN of HONU
This section shows the derivation of the stability of the weight update systems of static and dynamical adaptive models that are linear in parameters such as HONU; recall that training 
Acknowledgement
This research has been supported by SGS grant No SGS10/252/OHK2/3T/12
References
A. G. IVAKHNENKO, Polynomial Theory of Complex Systems, IEEE Tran. on Systems. Man, and Cybernetics. Vol. SMC-1 (4). pp. 364-378, 1971.
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