Convex Reformulation of LMI-Based Distributed Controller Design with Non-Block-Diagonal Lyapunov Functions

Core Concepts

This study presents novel convex LMI-based methods for designing distributed state feedback controllers for continuous-time linear time-invariant systems using a class of non-block-diagonal Lyapunov functions. The proposed approach generalizes the conventional block-diagonal relaxation and provides necessary and sufficient conditions for distributed controllers under chordal communication graphs.

Abstract

The key highlights and insights of this content are:

The authors address the distributed state feedback controller design problem for continuous-time linear time-invariant systems using linear matrix inequalities (LMIs).

They target a class of non-block-diagonal Lyapunov functions that have the same sparsity pattern as the distributed controllers, which can generalize the conventional block-diagonal relaxation.

By leveraging a block-diagonal factorization of sparse matrices and Finsler's lemma, the authors first present a nonlinear matrix inequality for stabilizing distributed controllers with such Lyapunov functions. This inequality becomes necessary and sufficient over chordal communication graphs.

The authors then derive an LMI by relaxing the nonlinear matrix inequality, which completely covers the conventional block-diagonal relaxation. They also provide analogous results for H∞ control.

Numerical examples demonstrate the efficacy of the proposed methods, showing that they outperform the conventional block-diagonal relaxation.

The computation of inverse matrices in the proposed method can be decomposed into smaller ones corresponding to subsystems formed by cliques of the communication graph, which enhances the scalability.

The authors show that the derived conditions provide a necessary and sufficient condition for distributed (H∞) controllers with the class of non-block-diagonal Lyapunov functions under chordal sparsity, allowing for easy evaluation of the conservatism.

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Convex Reformulation of LMI-Based Distributed Controller Design with a Class of Non-Block-Diagonal Lyapunov Functions

by Yuto Watanab... at arxiv.org 04-09-2024

https://arxiv.org/pdf/2404.04576.pdf

Deeper Inquiries

How can the proposed approach be extended to other fundamental control problems, such as H2 control

The proposed approach can be extended to other fundamental control problems, such as H2 control, by adapting the methodology used for H∞ control. In H2 control, the goal is to minimize the H2 norm of the transfer function matrix, which represents the energy of the system response to a disturbance input.
To extend the approach to H2 control, one would need to formulate the problem in terms of minimizing the H2 norm using a suitable Lyapunov function. This Lyapunov function would need to capture the energy behavior of the system and satisfy the necessary conditions for stability and performance. By leveraging the block-diagonal factorization of sparse matrices and Finsler's lemma, similar to the H∞ control case, one can derive a convex relaxation for H2 control that generalizes the conventional block-diagonal relaxation.

What are the potential limitations or drawbacks of the proposed non-block-diagonal Lyapunov functions compared to the conventional block-diagonal relaxation

One potential limitation of the proposed non-block-diagonal Lyapunov functions compared to the conventional block-diagonal relaxation is the computational complexity. While the non-block-diagonal approach allows for more flexibility in capturing the system dynamics and sparsity patterns, it may require more computational resources and time to solve the resulting LMIs. The block-diagonal relaxation, on the other hand, simplifies the problem by assuming a specific structure for the Lyapunov function, leading to faster computation but potentially more conservative results.
Another drawback could be the increased complexity in analyzing the conservatism of the non-block-diagonal relaxation compared to the block-diagonal approach. Since the non-block-diagonal Lyapunov functions can have a more intricate sparsity pattern, evaluating the conservatism of the resulting LMIs may require more advanced techniques and analysis.

Can the ideas and techniques developed in this work be applied to other areas of distributed optimization or control beyond the specific problem considered here

The ideas and techniques developed in this work can indeed be applied to other areas of distributed optimization or control beyond the specific problem considered here. The concept of using non-block-diagonal Lyapunov functions to design distributed controllers can be extended to various control problems in networked systems, multi-agent systems, and decentralized control architectures.
For distributed optimization, the approach of leveraging block-diagonal factorization of sparse matrices and Finsler's lemma can be applied to design distributed optimization algorithms with improved scalability and efficiency. By considering the sparsity patterns in the optimization problem and utilizing similar convex relaxations, one can develop novel distributed optimization techniques for large-scale systems.
Overall, the techniques and methodologies presented in this work have the potential to have broad applications in various fields where distributed control and optimization are essential, providing a framework for designing efficient and scalable solutions for complex systems.