John R. Singler

New POD Error Expressions, Error Bounds, and Asymptotic Results for Reduced Order Models of Parabolic PDEs

The derivations of existing error bounds for reduced order models of time varying partial differential equations (PDEs) constructed using proper orthogonal decomposition (POD) have relied on bounding the error between the POD data and various POD projections of that data. Furthermore, the asymptotic behavior of the model reduction error bounds depends on the asymptotic behavior of the POD data approximation error bounds. We consider time varying data taking values in two different Hilbert spaces

The Detection of Unsteady Flow Separation with Bioinspired Hair-Cell Sensors

H

Optimality of Balanced Proper Orthogonal Decomposition for Data Reconstruction

and

On the Long Time Behavior of Approximating Dynamical Systems

V

Structural Measurements for Enhanced MAV Flight

, with

Variation of the balanced POD algorithm for model reduction of linear systems

V \subset H

An HDG method for distributed control of convection diffusion PDEs

, and prove exact expressions for the POD data approximation errors considering four different POD projections and the two different Hilbert space error norms. Furthermore, the exact error expressions can be computed using only the POD eigenvalues and modes, and we prove the errors converge to zero as the number of POD modes increases. We consider the POD error estimation approaches of Kunisch and Volkwein [SIAM J. Numer. Anal., 40 (2002), pp. 492--515] and Chapelle, Gariah, an...

Boundary Control of Linear Uncertain 1-D Parabolic PDE Using Approximate Dynamic Programming

Biologists hypothesize that thousands of micro-scale hairs found on bat wings function as a network of air-flow sensors as part of a biological feedback flow control loop. In this work, we investigate hair-cell sensors as a means of detecting flow features in an unsteady separating flow over a cylinder. Individual hair-cell sensors were modeled using an EulerBernoulli beam equation forced by the fluid flow. When multiple sensor simulations are combined into an array of hair-cells, the response is shown to detect the onset and span of flow reversal, the upstream movement of the point of zero wall shear-stress, and the formation and growth of eddies near the wall of a cylinder. A linear algebraic hair-cell model, written as a function of the flow velocity, is also derived and shown to capture the same features as the hair-cell array simulation.

Output Feedback-Based Boundary Control of Uncertain Coupled Semilinear Parabolic PDE Using Neurodynamic Programming

Proper orthogonal decomposition (POD) finds an orthonormal basis yielding an optimal reconstruction of a given dataset. We consider an optimal data reconstruction problem for two general datasets related to balanced POD, which is an algorithm for balanced truncation model reduction for linear systems. We consider balanced POD outside of the linear systems framework, and prove that it solves the optimal data reconstruction problem. The theoretical result is illustrated with an example.

Adaptive Dynamic Programming Boundary Control of Uncertain Coupled Semi-Linear Parabolic PDE

In this paper we consider the impact of using “time marching” numerical schemes for computing asymptotic solutions of nonlinear differential equations. We show that stable and consistent approximating schemes can produce numerical solutions that do not correspond to the correct asymptotic solutions of the differential equation. In addition, we show that this problem cannot be avoided by placing additional side conditions on the boundary value problem, even if the numerical scheme preserves the side conditions at every step. Examples are given to illustrate the problems that can arise and the implications of using such methods in control design are discussed.