Optimal Model Reduction and Robust Feedback Control for Aerodynamics AFoSR

In this project, methods of model reduction that integrate feedback active flow control with applications to nonlinear convection and turbulent flows governed by Navier-Stokes equations are developed. A new methodology which extracts boundary conditions in reduced order proper orthogonal decomposition (POD) and finite difference models is developed. A new model reduction method based on empirical data and balanced truncation was developed and applied to nonlinear Galerkin models. Based on this method a new empirical Hankel norm model reduction algorithm is proposed. These methods are applied to a prototype nonlinear convective problem governed by the two-dimensional (2D) Burgers’ equation. The reduced models are used in the design of robust boundary controllers that achieve tracking, and implemented on the full order Computational Fluid Dynamics (CFD) models. POD and balanced truncation are shown to be optimal in the sense of distance minimizations in spaces of Hilbert-Schmidt (HS) operators. POD has been shown to be optimal in a broader sense than is reported in the literature. The optimality is in the sense of distance minimization in a space of integral operators under the Hilbert-Schmidt norm. A connection with balanced truncation is found. In particular, balanced truncation is shown to be optimal in the sense of distance minimization albeit in a different space of integral operators under the Hilbert-Schmidt norm when the so-called 'Curtain-Glover' balanced realization is used. This is a novel discovery as balanced truncation is usually known to be not optimal in any sense. An algorithm that combines the extended Kalman filter and expectation maximization to estimate the model coefficients from particle image velocimetry (PIV) measurements for turbulent 2D flows is developed. The algorithm is recursive and convenient for on-line implementation. The method is applied to a 2D flow control problem over the NACA 4412 airfoil using experimental data obtained from Prof. Glauser’s flow control group at the University of Syracuse. The motivation for this problem is minimization of aero-optic distortion in a bluff-body flow. POD and balanced truncation are shown to minimize different n-widths of partial differential equation solutions including the Kolmogorov, Gelfand, linear and Bernstein n-widths. The nwidths are notions from metric complexity theory. They quantify inherent and representation errors due to lack of data and loss of information. Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 29 MAR 2010 2. REPORT TYPE Final 3. DATES COVERED 01-08-2008 to 30-06-2010 4. TITLE AND SUBTITLE OPTIMAL MODEL REDUCTION AND ROBUST FEEDBACK CONTROL FOR AERODYNAMICS 5a. CONTRACT NUMBER FA9550-08-1-0450