Optimal $H_2$ Decentralized Control of Cone Causal Spatially Invariant Systems

TL;DR

This paper derives an explicit, optimal decentralized H2 controller for cone causal spatially invariant systems, transforming the problem into solvable model matching problems and providing a constructive, closed-form solution.

Contribution

It introduces the first explicit formula for optimal decentralized H2 control of cone causal systems, simplifying controller design and implementation.

Findings

Explicit formula for optimal decentralized controller derived

Controller expressed as a positive feedback scheme

Numerical example demonstrates effectiveness of the method

Abstract

This paper presents an explicit solution to decentralized control of a class of spatially invariant systems. The problem of optimal $H_2$ decentralized control for cone causal systems is formulated. Using Parseval's identity, the optimal $H_2$ decentralized control problem is transformed into an infinite number of model matching problems with a specific structure that can be solved efficiently. In addition, the closed-form expression (explicit formula) of the decentralized controller is derived for the first time. In particular, it is shown that the optimal decentralized controller is given by a specific positive feedback scheme. A constructive procedure to obtain the state-space representation of the decentralized controller is provided. A numerical example is given and compared with previous works which demonstrate the effectiveness of the proposed method.