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Iterative Thresholding and Projection Algorithms and Model-Based Deep Neural Networks for Sparse LQR Control Design

Myung Cho

TL;DR

This work addresses sparse LQR design in large-scale distributed control by formulating a regularized optimization that promotes sparse feedback while ensuring closed-loop stability. It introduces ISTA and ISPA as simple, efficient iterative methods, with model-based DNNs (DNN-ISTA, DNN-FISTA) to accelerate convergence and provide warm-start capabilities. The authors provide analytical insight into convergence, compare against ADMM and GraSP, and demonstrate substantial speedups and competitive performance across distributed multi-agent and power-system benchmarks. The practical impact lies in enabling rapid, scalable sparse controller synthesis for networks with limited communication resources, while maintaining stability and performance. By combining algorithmic innovations with data-driven acceleration, the work offers a versatile toolkit for sparse decentralized control in real-world large-scale systems.

Abstract

In this paper, we consider an LQR design problem for distributed control systems. For large-scale distributed systems, finding a solution might be computationally demanding due to communications among agents. To this aim, we deal with LQR minimization problem with a regularization for sparse feedback matrix, which can lead to achieve the reduction of the communication links in the distributed control systems. For this work, we introduce simple but efficient iterative algorithms -- Iterative Shrinkage Thresholding Algorithm (ISTA) and Iterative Sparse Projection Algorithm (ISPA). They can give us a trade-off solution between LQR cost and sparsity level on feedback matrix. Moreover, in order to improve the speed of the proposed algorithms, we design deep neural network models based on the proposed iterative algorithms. Numerical experiments demonstrate that our algorithms can outperform the previous methods using the Alternating Direction Method of Multiplier (ADMM) [2] and the Gradient Support Pursuit (GraSP) [3], and their deep neural network models can improve the performance of the proposed algorithms in convergence speed.

Iterative Thresholding and Projection Algorithms and Model-Based Deep Neural Networks for Sparse LQR Control Design

TL;DR

This work addresses sparse LQR design in large-scale distributed control by formulating a regularized optimization that promotes sparse feedback while ensuring closed-loop stability. It introduces ISTA and ISPA as simple, efficient iterative methods, with model-based DNNs (DNN-ISTA, DNN-FISTA) to accelerate convergence and provide warm-start capabilities. The authors provide analytical insight into convergence, compare against ADMM and GraSP, and demonstrate substantial speedups and competitive performance across distributed multi-agent and power-system benchmarks. The practical impact lies in enabling rapid, scalable sparse controller synthesis for networks with limited communication resources, while maintaining stability and performance. By combining algorithmic innovations with data-driven acceleration, the work offers a versatile toolkit for sparse decentralized control in real-world large-scale systems.

Abstract

In this paper, we consider an LQR design problem for distributed control systems. For large-scale distributed systems, finding a solution might be computationally demanding due to communications among agents. To this aim, we deal with LQR minimization problem with a regularization for sparse feedback matrix, which can lead to achieve the reduction of the communication links in the distributed control systems. For this work, we introduce simple but efficient iterative algorithms -- Iterative Shrinkage Thresholding Algorithm (ISTA) and Iterative Sparse Projection Algorithm (ISPA). They can give us a trade-off solution between LQR cost and sparsity level on feedback matrix. Moreover, in order to improve the speed of the proposed algorithms, we design deep neural network models based on the proposed iterative algorithms. Numerical experiments demonstrate that our algorithms can outperform the previous methods using the Alternating Direction Method of Multiplier (ADMM) [2] and the Gradient Support Pursuit (GraSP) [3], and their deep neural network models can improve the performance of the proposed algorithms in convergence speed.
Paper Structure (28 sections, 4 theorems, 52 equations, 7 figures, 4 tables, 2 algorithms)

This paper contains 28 sections, 4 theorems, 52 equations, 7 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Previous Research on Sparse LQR Control Design
  4. Iterative Shrinkage-Thresholding Algorithm
  5. Description of conventional iterative shrinkage-thresholding algorithm for sparse signal recovery
  6. Iterative shrinkage-thresholding algorithm for sparse LQR optimal control design
  7. Comparisons and merits
  8. Analysis of Iterative Shrinkage-Thresholding Algorithm for Sparse LQR Optimal Control Design
  9. Iterative Sparse Projection Algorithm for Sparse LQR Optimal Control Design
  10. Projection onto the sparsity-promoting ball
  11. Projection onto the $\ell_0$-ball
  12. Projection onto the $\ell_1$-ball
  13. Projection onto the block sparsity-ball
  14. Iterative Sparse Projection Algorithm
  15. Model-based Deep Neural Networks for Sparse LQR Optimal Control Design
  16. ...and 13 more sections

Key Result

Lemma 1

Given $\rho_{0}$ and any initial point ${\boldsymbol K}^{(0)} \in {\mathcal{F}}$, Algorithm alg:ISTA provides the non-increasing objective value, namely:

Figures (7)

  • Figure 1: Illustration of a distributed system having four agents denoted by ${\mathcal{A}}1$, ${\mathcal{A}}2$, ${\mathcal{A}}3$ and ${\mathcal{A}}4$, and its feedback signal model, where ${\boldsymbol x}_i(t)$ and ${\boldsymbol u}_i(t)$ are the state and input vectors of the $i$-th agent at time $t$ respectively and ${\boldsymbol K}$ represents the feedback matrix of the system.
  • Figure 2: Data flow of the $t$-th layer of the DNN based on the ISTA (resp. the ISPA) introduced in Algorithm \ref{['alg:ISTA']} (resp. Algorithm \ref{['alg:ISPA']}), where $w^{(t)}_1$ and $w^{(t)}_2$ are the training variables at the $t$-th layer of the DNN. For the ISPA, the operation for sparsity block is understood as the projection operation.
  • Figure 3: Illustration of the unfolded DNN based on ISTA (DNN-ISTA) for the sparse optimal control design with $l$ layers.
  • Figure 4: Illustration of the unfolded FISTA-based DNN (DNN-FISTA) for the sparse optimal control design with $l$ layers.
  • Figure 5: Comparison between the ISTA and the ADMM in the distributed multi-agent control system model. $G({\boldsymbol K}) = \| {\boldsymbol K} \|_1$ is used.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Lemma 1
  • Corollary 1
  • Theorem 1
  • Lemma 2