Neighboring Optimal Feedback Law for Higher-Order Dynamic Systems

In recent years, using tools from linear and nonlinear systems theory, it has been shown that classes of dynamic systems in first-order forms can be alternatively written in higher-order forms, i.e., as sets of higher-order differential equations. Input-state linearization is one of the most popular tools to achieve such a representation. The equations of motion of mechanical systems naturally have a second-order form, arising from the application of Newton’s laws. In the last five years, effective computational tools have been developed by the authors to compute optimal trajectories of such systems, while exploiting the inherent structure of the dynamic equations. In this paper, we address the question of computing the neighboring optimal for systems in higher-order forms. It must be pointed out that the classical solution of the neighboring optimal problem is well known only for systems in the first-order form. The main contributions of this paper are: (i) derivation of the optimal feedback law for higher-order linear quadratic terminal controller using extended Hamilton-Jacobi equations; (ii) application of the feedback law to compute the neighboring optimal solution.