Sensor Placement for Optimal Control of Infinite-Dimensional Systems

An important challenge in controlling distributed parameter systems is implementing feedback control laws over an infinite-dimensional space. One widely studied approach is to place sensors at a set of discrete locations and then approximate the state feedback using the sensor outputs. This approach naturally raises the question of where the sensors should be located. In this paper, we investigate the problem of placing a set of sensors on the unit interval in order to minimize the mean-square deviation between a desired infinite-dimensional control law, and an approximate finite-dimensional controller obtained by applying state feedback at the chosen sensor positions. We propose a greedy algorithm and derive optimality bounds on the selected set of sensors. We also present a simplified greedy algorithm in which the incremental improvements from adding each sensor are not updated at each step. We analyze the performance of the approaches under two scenarios. In the case where the sensor placements are constrained such that each of the detection radii of each pair of sensors do not overlap, we show that the two algorithms are equivalent and achieve a 1/2 + ε optimality guarantee, where ε can be made arbitrarily small at the cost of increased computational overhead. When overlaps exist between sensor detection areas, we derive an optimality bound for the simplified algorithm. The value of the optimality bound is determined by the sensor model, cardinality of sensor set, and the allowed minimum sensor distance. Our approach is illustrated through numerical study, in which we compare our proposed greedy algorithm, the current state-of-the-art approach, and the true optimum obtained from exhaustive search.