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Recursive Analytic Solution of Nonlinear Optimal Regulators

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Document pages: 14 pages

Abstract: The paper develops an optimal regulator for a general class of multi-inputaffine nonlinear systems minimizing a nonlinear cost functional with infinitehorizon. The cost functional is general enough to enforce saturation limits onthe control input if desired. An efficient algorithm utilizing tensor algebrais employed to compute the tensor coefficients of the Taylor series expansionof the value function (i.e., optimal cost-to-go). The tensor coefficients arefound by solving a set of nonlinear matrix equations recursively generalizingthe well-known linear quadratic solution. The resulting solution generates theoptimal controller as a nonlinear function of the state vector up to aprescribed truncation order. Moreover, a complete convergence of the computedsolution together with an estimation of its applicability domain are providedto further guide the user. The algorithm s computational complexity is shown togrow only polynomially with respect to the series order. Finally, severalnonlinear examples including some with input saturation are presented todemonstrate the efficacy of the algorithm to generate high order Taylor seriessolution of the optimal controller.

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