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Optimal control problems are generally nonlinear and therefore, generally do not have analytic solutions (e.g., like the linear-quadratic optimal control problem). As a result, it is necessary to employ numerical methods to solve optimal control problems.
Nonlinear control theory is the area of control theory which deals with systems that are nonlinear, time-variant, or both. Control theory is an interdisciplinary branch of engineering and mathematics that is concerned with the behavior of dynamical systems with inputs, and how to modify the output by changes in the input using feedback ...
This solution is optimal, although possibly not unique. The algorithm may also be stopped early, with the assurance that the best possible solution is within a tolerance from the best point found; such points are called ε-optimal. Terminating to ε-optimal points is typically necessary to ensure finite termination.
The Gauss pseudospectral method (GPM), one of many topics named after Carl Friedrich Gauss, is a direct transcription method for discretizing a continuous optimal control problem into a nonlinear program (NLP).
Its solution is the value function of the optimal control problem which, once known, can be used to obtain the optimal control by taking the maximizer (or minimizer) of the Hamiltonian involved in the HJB equation. [2] [3] The equation is a result of the theory of dynamic programming which was pioneered in the 1950s by Richard Bellman and ...
The problem of optimal non-linear filtering (even for the non-stationary case) was solved by Ruslan L. Stratonovich (1959, [1] 1960 [2]), see also Harold J. Kushner's work [3] and Moshe Zakai's, who introduced a simplified dynamics for the unnormalized conditional law of the filter [4] known as the Zakai equation.
Inspired by—but distinct from—the Hamiltonian of classical mechanics, the Hamiltonian of optimal control theory was developed by Lev Pontryagin as part of his maximum principle. [2] Pontryagin proved that a necessary condition for solving the optimal control problem is that the control should be chosen so as to optimize the Hamiltonian. [3]
The phrase H ∞ control comes from the name of the mathematical space over which the optimization takes place: H ∞ is the Hardy space of matrix-valued functions that are analytic and bounded in the open right-half of the complex plane defined by Re(s) > 0; the H ∞ norm is the supremum singular value of the matrix over that space.