Optimal observation trajectory of an active maneuvering radar

This article deals with the path optimization problem of aircrafts observing ships, using Kalman estimation. Observing that the continuous version of the Kalman filter equation is a Riccati differential equation, we formalize the problem as a minimization of the integral of the covariance trace. This integral is constrained by the Riccati equation coupled with aircrafts trajectory equations, giving an optimal control problem. Because our problem contains multiple optima and because maximum principle cannot solve optimization problems with multiple optima, we propose a graph theory approach using the Bellman dynamic programming principle. First we show that if we look at a lower bound of the criterion, we can separate the state space of the estimation (covariance) and of the trajectory (space time coordinates).This allows the use of integer programming like Bellman shortest path algorithm with state space given by the trajectory and criterion based on the estimation evolution versus trajectory. Shortest path algorithm is computed in a graph whose nodes are given by a space time coordinates grid, and whose edges are given by the path authorized by the maneuvre constraints. Once the lower bound is obtained and the Branch and Bound procedure leads to the global optimum conditionally to the grid. The algorithm gives an initial condition to local optimization procedures like variation descent search. We found that in certain cases non-trivial trajectories, like circles, U-turns and broken lines appear with this method. These trajectories are actually inaccessible with the maximum principle, unless the initial trajectory happens to be close to the optimum. When applying these methods with noise to the filtering process, we obtained encouraging results. Comparing with minimal time trajectory; we gained up to 40 % on the quadratic error mean.