Optimal sampling for state estimation of stochastic dynamical systems

Aspeel, Antoine [UCL]

The problem of estimating a quantity that evolves with time from noisy measurements can be found in many fields, such as engineering (e.g., the position of a drone), epidemiology (e.g., the number of infected people), or medicine (e.g., the position of a lung tumor moving due to breathing). In some cases, the number of measurements must be reduced because the acquisition of measurements is expensive, energy-consuming, time-consuming or dangerous. In this thesis, we assume that the amount of measurements is limited, and we study the problem of the optimal choice of measurement times to estimate the state of a stochastic dynamical system over a finite time horizon. We first focus on the case of linear and Gaussian dynamical systems. We propose the optimal intermittent Kalman filter for which the measurement times are obtained by solving a combinatorial optimization problem. Then, we generalize this approach to non-linear and non-Gaussian dynamical systems. This leads to the formulation of three variants of optimal intermittent particle filters. The first one, for which the measurements are time-triggered, is based on combinatorial optimization. The other two are self-triggered. One requires a combinatorial program to be solved for each measurement, and the other requires to solve a stochastic program to choose when to perform a measurement. Robust estimation of the state of a linear dynamical system subject to bounded disturbances is also studied. Using the polytope containment method and Youla-parameterization, we show that the measurement times that minimize the worst-case estimation error are obtained by solving a mixed integer linear program. Numerical and empirical experiments are presented, with a particular focus on the problem of tumor tracking and its use in radiation therapy.