The relationship between terminal state constraints and penalties for the discrete-time LQP problem associated with the adjustment of accelerometer data

Abstract

One of the methods of adjusting and integrating acceleration data is to cast it within the discrete-time optimal control framework of minimizing the sum of squares of adjustments, here regarded as controls, subject to the linear dynamics of double integration. Excessive deviations in terminal velocity and displacement, the states, could be avoided by adding to the sum of squares a terminal penalty weighting such deviations. In this paper a procedure, using terminal state constraints, is developed as an alternative to terminal penalties to prevent drift in the terminal states. The explicit relationship between the controls for these two alternatives are derived. In particular, expressions for the dependence of the controls on terminal weights are obtained and used to account for the insensitivity of the state and control trajectories to certain weights. While the focus of this paper is on the dynamics associated with the double integration of acceleration, it is shown that the approach has wider application to other LQP problems.