Distributed Control in Multiple Dimensions: A Structure Preserving Computational Technique

Abstract

We consider the problem of analysis and control of discretely distributed systems in multidimensional arrays. For spatially invariant systems in one spatial dimension, we build an efficient arithmetic that preserves the rational Laurent operator structure, leading to fast iterative methods of solving Lyapunov and Riccati equations and block diagonalizations, and thus arbitrarily non-conservative stability analysis and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> distributed controller synthesis. These one-dimensional results are then used to build an efficient arithmetic and controller synthesis procedure in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensions by induction. The extension of these techniques from Laurent operators with rational symbols to sequentially semi-separable matrices yields a procedure for linear computational complexity analysis and controller synthesis for finite extent heterogeneous multidimensional systems with boundary conditions. The procedures are demonstrated on two computational examples.