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. Nomenclature A, B, C = linearized system dynamics matrices a = Joukowski cylinder radius E = expectation operator F = Fisher information matrix f = nonlinear dynamics model H = linear measurement model h = nonlinear measurement model k = discrete time step k r = reduced frequency P = covariance matrix p = number of desired sensors r = number of candidate sensors s i = sensor location for sensor i u = control input u = control linearization deviation u 0 = control linearization trajectory v = measurement noise W = observability Gramiañ W = empirical observability Gramian W = interpolated observability Gramian W d = discrete-time observability Gramian w j = special ordered set weight vector for sensor j x = state vector x = state linearization deviation x 0 = state linearization trajectory

Optimal linear combination of sensors and actuators to enforce an interior conic open‐loop system

“…Method 1 ( 2 -Optimal Sensor Combination). Solve for N s , P = P T > 0, 12 , Z = Z T > 0, and ∈ ℝ >0 that minimize subject to (13), (14), (15), (16), and tr(Z) < 1.…”

“…The  ∞ -optimal combination of the sensor measurements is presented in the following method. 12 , and ∈ ℝ >0 that minimize subject to (13), (17), and (18).…”

“…8,9 Different approaches focus on selecting sensors or actuators to optimize open-loop system properties. For example, maximizing the controllability or observability of the open-loop system, [10][11][12][13][14][15][16][17] minimizing parameter estimation error, 18 minimizing the minimum real part of the open-loop system's transmission zeros, 11 or maximizing the pitch, roll, and yaw moments of an aircraft. 19 An alternative to sensor selection is sensor combination, where multiple sensor measurements are combined together to yield a blended measurement that can then be input into a feedback controller.…”

Coupled Inertial Navigation and Flush Air Data Sensing Algorithm for Atmosphere Estimation

“…Sensor placement optimization for observability was first suggested in Ref. 36, and has subsequently been studied for a wide range of applications such as structural vibration, 37 chemical reactors, 38 airfoil wake estimation, 39 and Mars entry navigation. 40 Another approach, suggested in Ref.…”

Insect and insect-inspired aerodynamics: unsteadiness, structural mechanics and flight control

“…With the majority of the sensilla placed proximally, they are also more protected from degraded observability owing to wing damage (e.g. [59]) although the more distal sensilla at the trailing edge are informative for fine-scale control of aerodynamic loads at that location [10,60]. …”