This paper proposes a symbolic-numeric Bayesian filtering method for a class of discrete-time nonlinear stochastic systems to achieve high accuracy with a relatively small online computational cost. The proposed method is based on the holonomic gradient method (HGM), which is a symbolicnumeric method to evaluate integrals efficiently depending on several parameters. By approximating the posterior probability density function (PDF) of the state as a Gaussian PDF, the update process of its mean and variance can be formulated as evaluations of several integrals that exactly take into account the nonlinearity of the system dynamics. An integral transform is used to evaluate these integrals more efficiently using the HGM compared to our previous method. Further, a numerical example is provided to demonstrate the efficiency of the proposed method over other existing methods.
This paper studies nonlinear finite-horizon optimal control problems with terminal constraints, where all nonlinear functions are rational or algebraic functions. We first extend a recursive elimination method, which decouples the Euler-Lagrange equations into sets of algebraic equations, where each set contains only the variables at the same time instant. Therefore, a candidate of an optimal feedback control law at each time instant is obtained by solving each set of algebraic equations. Next, we provide a sufficient condition such that each set of algebraic equations gives a unique local optimal feedback control law at each time instant. Illustrative and practical examples are provided to illustrate the proposed method and sufficient condition.
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