Unscented optimal control

In mathematics, unscented optimal control combines the notion of the unscented transform with deterministic optimal control to address a class of uncertain optimal control problems.^[1][2][3][4] It is a specific application of tychastic optimal control theory,^[1][5][6][7] which is a generalization of Riemmann-Stieltjes optimal control theory,^[8][9] a concept introduced by Ross and his coworkers.

Mathematical description

Suppose that the initial state ${\displaystyle x^{0}}$ of a dynamical system,

${\displaystyle {\dot {x}}=f(x,u,t)}$

is an uncertain quantity. Let ${\displaystyle \mathrm {X} ^{i}}$ be the sigma points. Then sigma-copies of the dynamical system are given by,

${\displaystyle {\dot {\mathrm {X} }}^{i}=f(\mathrm {X} ^{i},u,t)}$

Applying standard deterministic optimal control principles to this ensemble generates an unscented optimal control.^[10][11][12] Unscented optimal control is a special case of tychastic optimal control theory.^[1][5][13] According to Aubin^[13] and Ross,^[1] tychastic processes differ from stochastic processes in that a tychastic process is conditionally deterministic.

Applications

Unscented optimal control theory has been applied to UAV guidance,^[12][14] spacecraft attitude control,^[6] air-traffic control^[15] and low-thrust trajectory optimization^[2][10]