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An Accelerated Proximal Alternating Direction Method of Multipliers for Optimal Decentralized Control of Uncertain Systems

Bo Yang, Xinyuan Zhao, Xudong Li, Defeng Sun

TL;DR

Numerical experiments confirm that the accelerated algorithm outperforms the well-known COSMO, MOSEK, and SCS solvers in efficiently solving large-scale CP problems, particularly those arising from H2\documentclass[12pt]{minimal}-guaranteed cost ODC problems.

Abstract

To ensure the system stability of the $\bf{\mathcal{H}_{2}}$-guaranteed cost optimal decentralized control problem (ODC), an approximate semidefinite programming (SDP) problem is formulated based on the sparsity of the gain matrix of the decentralized controller. To reduce data storage and improve computational efficiency, the SDP problem is vectorized into a conic programming (CP) problem using the Kronecker product. Then, a proximal alternating direction method of multipliers (PADMM) is proposed to solve the dual of the resulted CP. By linking the (generalized) PADMM with the (relaxed) proximal point algorithm, we are able to accelerate the proposed PADMM via the Halpern fixed-point iterative scheme. This results in a fast convergence rate for the Karush-Kuhn-Tucker (KKT) residual along the sequence generated by the accelerated algorithm. Numerical experiments further demonstrate that the accelerated PADMM outperforms both the well-known CVXOPT and SCS algorithms for solving the large-scale CP problems arising from $\bf{\mathcal{H}_{2}}$-guaranteed cost ODC problems.

An Accelerated Proximal Alternating Direction Method of Multipliers for Optimal Decentralized Control of Uncertain Systems

TL;DR

Numerical experiments confirm that the accelerated algorithm outperforms the well-known COSMO, MOSEK, and SCS solvers in efficiently solving large-scale CP problems, particularly those arising from H2\documentclass[12pt]{minimal}-guaranteed cost ODC problems.

Abstract

To ensure the system stability of the -guaranteed cost optimal decentralized control problem (ODC), an approximate semidefinite programming (SDP) problem is formulated based on the sparsity of the gain matrix of the decentralized controller. To reduce data storage and improve computational efficiency, the SDP problem is vectorized into a conic programming (CP) problem using the Kronecker product. Then, a proximal alternating direction method of multipliers (PADMM) is proposed to solve the dual of the resulted CP. By linking the (generalized) PADMM with the (relaxed) proximal point algorithm, we are able to accelerate the proposed PADMM via the Halpern fixed-point iterative scheme. This results in a fast convergence rate for the Karush-Kuhn-Tucker (KKT) residual along the sequence generated by the accelerated algorithm. Numerical experiments further demonstrate that the accelerated PADMM outperforms both the well-known CVXOPT and SCS algorithms for solving the large-scale CP problems arising from -guaranteed cost ODC problems.
Paper Structure (17 sections, 6 theorems, 86 equations, 3 figures, 5 tables, 3 algorithms)

This paper contains 17 sections, 6 theorems, 86 equations, 3 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Main Contribution
  3. Optimal decentralized control problem
  4. Problem statement
  5. An approximated CP model of the ODC problem
  6. An accelerated PADMM for solving the CP model
  7. PADMM for solving the CP model
  8. Acceleration of the PADMM scheme
  9. A fast implementation for solving the subproblems of APADMM
  10. Numerical experiments
  11. Implementation details
  12. Verification of decentralized structure
  13. Numerical results for the lifting technique and acceleration
  14. Numerical results for APADMM in comparison with COSMO, MOSEK, and SCS
  15. Conclusions and future work
  16. ...and 2 more sections

Key Result

Lemma 2.1

JMa For $i=1,\ldots,M,$ the function $\mathcal{F}_{i}:\mathbb{S}^{p}\rightarrow \mathbb{R}^{n\times n}$ is defined as follows: where Based on the partition of the decentralized structure of the feedback gain matrix $K$ defined in a3, we can define the set and Then,

Figures (3)

  • Figure 1: System response under the feedback matrix $K$.
  • Figure 2: The performance of APADMM with various restart intervals.
  • Figure 3: The performance comparison of APADMM, sGS_PADMM, and GPADMM.

Theorems & Definitions (8)

  • Lemma 2.1
  • Remark 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Remark 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Proposition B.1