RICCATI-BASED BOUNDARY FEEDBACK STABILIZATION OF INCOMPRESSIBLE NAVIER-STOKES FLOWS

被引:35
作者
Baensch, Eberhard [1 ]
Benner, Peter [2 ,3 ]
Saak, Jens [2 ,3 ]
Weichelt, Heiko K. [3 ]
机构
[1] Univ Erlangen Nurnberg, Res Grp Appl Math AMIII 3, D-91058 Erlangen, Germany
[2] Tech Univ Chemnitz, Res Grp Math Ind & Technol MiIT, D-09126 Chemnitz, Germany
[3] Max Planck Inst Dynam Complex Tech Syst Magdeburg, Res Grp Computat Methods Syst & Control Theory CS, D-39106 Magdeburg, Germany
关键词
flow control; Navier-Stokes equations; Riccati-based feedback; BALANCED TRUNCATION; DESCRIPTOR SYSTEMS; LYAPUNOV EQUATIONS; MODEL-REDUCTION; ALGORITHMS;
D O I
10.1137/140980016
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
In this article a boundary feedback stabilization approach for incompressible Navier-Stokes flows is studied. One of the main difficulties encountered is the fact that after space discretization by a mixed finite element method (because of the solenoidal condition) one ends up with a differential algebraic system of index 2. The remedy here is to use a discrete realization of the Leray projection used by Raymond [J.-P. Raymond, SIAM J. Control Optim., 45 (2006), pp. 790-828] to analyze and stabilize the continuous problem. Using the discrete projection, a linear quadratic regulator (LQR) approach can be applied to stabilize the (discrete) linearized flow field with respect to small perturbations from a stationary trajectory. We provide a novel argument that the discrete Leray projector is nothing else but the numerical projection method proposed by Heinkenschloss and colleagues in [M. Heinkenschloss, D. C. Sorensen, and K. Sun, SIAM J. Sci. Comput., 30 (2008), pp. 1038-1063]. The nested iteration resulting from applying this approach within the Newton-ADI method to solve the LQR algebraic Riccati equation is the key to compute a feedback matrix that in turn can be applied within a closed-loop simulation. Numerical examples for various parameters influencing the different levels of the nested iteration are given. Finally, the stabilizing property of the computed feedback matrix is demonstrated using the von Karman vortex street within a finite element based flow solver.
引用
收藏
页码:A832 / A858
页数:27
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