LQR performance index distribution with uncertain boundary conditions

The track assignment problem in applications with large gaps in tracking measurements and uncertain boundary conditions is addressed as a Two Point Boundary Value Problem (TPBVP) using Hamiltonian formalisms. An L₂-norm analog Linear Quadratic Regulator (LQR) performance function metric is used to measure the trajectory cost, which may be interpreted as a control distance metric. Distributions of the performance function are determined by linearizing about the deterministic optimal nonlinear trajectory solution to the TPBVP and accounting for statistical variations in the boundary conditions. The performance function random variable under this treatment is found to have a quadratic form, and Pearson´s Approximation is used to model it as a chi-squared random variable. Stochastic dominance is borrowed from mathematical finance and is used to rank statistical distributions in a metric sense. Analytical results and approximations are validated and an example of the approach utility is given. Finally conclusions and future work are discussed.