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arXiv · 2304.04980

Structured linear quadratic control computations over 2D grids

Armaghan Zafar·Ian R. Manchester
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Abstract

In this paper, we present a structured solver based on the preconditioned conjugate gradient method (PCGM) for solving the linear quadratic (LQ) optimal control problem for $K \times N$ sub-systems connected in a two-dimensional (2D) grid structure. Our main contribution is the development of a structured preconditioner based on a fixed number of inner-outer iterations of the nested block Jacobi method. We establish that the proposed preconditioner is positive-definite. Moreover, the proposed approach retains structure in both spatial dimensions as well as in the temporal dimension of the problem. The arithmetic complexity of each PCGM step scales as $O(KNT)$, where $T$ is the length of the time horizon. The computations involved at each step of the proposed PCGM are decomposable and amenable to distributed implementation on parallel processors connected in a 2D grid structure with localized data exchange. We also provide results of numerical experiments performed on two example systems.

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Armaghan Zafar, Ian R. Manchester. 2023-04-11. Structured linear quadratic control computations over 2D grids. https://arxiv.org/abs/2304.04980

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