NDLI: Galerkin Approximation of the Generalized Hamilton-jacobi Equation

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Galerkin Approximation of the Generalized Hamilton-jacobi Equation

Content Provider	Semantic Scholar
Author	Saridis, George N.
Copyright Year	1996
Abstract	If u is a stabilizing control for a nonlinear system that is aane in the control variable, then the solution to the Generalized Hamilton-Jacobi-Bellman (GHJB) equation associated with u is a Lyapunov function for the system and equals the cost associated with u. If an explicit solution to the GHJB equation can be found then it can be used to construct a feedback control law that improves the performance of u. Repeating this process leads to a successive approximation algorithm that uniformly approximates the Hamilton-Jacobi-Bellman equation. The diiculty is that it is very diicult to construct solutions to the GHJB equation such that the control derived from its solution is in feedback form. This paper shows that Galerkin's approximation method can be used to construct arbitrarily close approximations to the GHJB equation while generating stable feedback control laws. We state suucient conditions for the convergence of Galerkin approximations to the GHJB equation. The suucient conditions derived in this paper include standard completeness assumptions and the asymptotic stability of the associated vector eld. The method is demonstrated on a simple nonlinear system and is compared to a result obtained by using exact feedback linearization in conjunction with the LQR design method.
File Format	PDF HTM / HTML
Alternate Webpage(s)	http://www.ee.byu.edu/~beard/papers/preprints/BeardSaridisWen96a.ps
Language	English
Access Restriction	Open
Subject Keyword	ARID1A wt Allele Approximation algorithm Bellman equation Convergence (action) Epilepsy, Generalized Feedback Galerkin method Hamilton–Jacobi–Bellman equation Jacobi eigenvalue algorithm Jacobi method Lyapunov fractal Nonlinear system Optimal control Solutions
Content Type	Text
Resource Type	Article

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