Observability-Based Optimal Sensor Placement for Flapping Airfoil Wake Estimation

Abstract

This work considers the optimal sensor placement problem for a general nonlinear system using the eigenvalues of the observability Gramian in the cost function. The problem is formulated as a mixed-integer convex optimization problem. Using the empirical observability Gramian, the input data to this optimization problem are computed from a simulation of the nonlinear system with no analytical model required. A piecewise linear approximation to the observability Gramian is proposed using special ordered sets of type two, allowing a coarser sensor location mesh and thus fewer binary variables and shorter solution times compared with standard gridded approaches. The solution methodology is applied to vortex estimation in the wake of a flapping airfoil using velocity sensors on the surface of the airfoil, which is modeled using unsteady potential flow and a Joukowski conformal mapping. Resulting optimal sensor sets are found near the trailing edge in pairs on the upper and lower surface. Although the observability Gramian is computed from the linearized system, the optimal sensor sets yield improved performance of an unscented Kalman filter estimating wake vortex structure on the nonlinear dynamics; the optimal sets outperform more than 99% of the sampled feasible sensor sets, thus validating the observability eigenvalues as a measure of nonlinear estimator performance.