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Input design for the optimal control of networked moments

Philip Solimine, Anke Meyer-Baese

TL;DR

This work tackles the problem of energy-efficient input design for networks to achieve moment-based state goals, focusing on both the mean and the variance of the network state under linear dynamics.For linear moments, it derives closed-form results: optimal mean-target states and the energy to enforce mean constraints, and an eigenstructure-based rule for optimal input placement via the matrix $\Phi(\gamma)$ and its largest eigenvalue.When extending to second moments, the authors formulate ellipsoid constraints and variance-centric objectives, yielding normal equations, secular equations, and spectral bounds that relate energy to the target variance and dispersion metrics, including a generalized eigenproblem for discord regulation.To handle nonlinear outputs, they propose a Generalized Projected Gradient Method (GPGM) that iteratively optimizes input placement while enforcing the nonlinear constraint, enabling approximate local solutions and paving the way for future driver-node extensions in nonlinear settings.

Abstract

We study the optimal control of the mean and variance of the network state vector. We develop an algorithm that uses projected gradient descent to optimize the control input placement, subject to constraints on the state that must be achieved at a given time threshold; seeking to design an input that moves the moment at minimum cost. First, we solve the state-selection problem for a number of variants of the first and second moment, and find solutions related to the eigenvalues of the systems' Gramian matrices. We then nest this state selection into projected gradient descent to design optimal inputs.

Input design for the optimal control of networked moments

TL;DR

This work tackles the problem of energy-efficient input design for networks to achieve moment-based state goals, focusing on both the mean and the variance of the network state under linear dynamics.For linear moments, it derives closed-form results: optimal mean-target states and the energy to enforce mean constraints, and an eigenstructure-based rule for optimal input placement via the matrix $\Phi(\gamma)$ and its largest eigenvalue.When extending to second moments, the authors formulate ellipsoid constraints and variance-centric objectives, yielding normal equations, secular equations, and spectral bounds that relate energy to the target variance and dispersion metrics, including a generalized eigenproblem for discord regulation.To handle nonlinear outputs, they propose a Generalized Projected Gradient Method (GPGM) that iteratively optimizes input placement while enforcing the nonlinear constraint, enabling approximate local solutions and paving the way for future driver-node extensions in nonlinear settings.

Abstract

We study the optimal control of the mean and variance of the network state vector. We develop an algorithm that uses projected gradient descent to optimize the control input placement, subject to constraints on the state that must be achieved at a given time threshold; seeking to design an input that moves the moment at minimum cost. First, we solve the state-selection problem for a number of variants of the first and second moment, and find solutions related to the eigenvalues of the systems' Gramian matrices. We then nest this state selection into projected gradient descent to design optimal inputs.

Paper Structure

This paper contains 14 sections, 10 theorems, 59 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Framework
  4. Dynamics and control of linear systems
  5. Controlling the average
  6. State selection with average-state goals
  7. Input placement with average-state goals
  8. Controlling second moments
  9. State selection on an ellipsoid surface
  10. State repulsion or attraction
  11. Input placement
  12. Controlling variance; regulation and discord
  13. Input placement with nonlinear outputs
  14. Conclusions

Key Result

Lemma 1

The optimal state which can satisfy (meanconstraint) at the lowest possible energy cost for a given control input matrix $B$ and linear output $\gamma$ is given by

Figures (2)

  • Figure 1: Dynamics of a test system subject to mean control. Panel (a) gives a graphical depiction of the autonomous system. Panel (b) shows the addition of a control node representing the OMAP and its weights to existing nodes. Panels (b) and (c) show the dynamics of the system state both (c) autonomously and (d) under average control by the OMAP, with the goal $\eta=1$ on $0\leq t \leq 15$.
  • Figure 2: Dynamics of Zachary's karate club network subject to optimal variance control. Panel (a) shows the Karate club network with nodes colored by eigenvector centrality. Panel (b) shows the Laplacian dynamics of the autonomous system. Panel (c) shows the dynamics under variance control by the OMAP, with ten control nodes and the goal $\eta=1$ on $0\leq t \leq 3$. Panel (d) compares energy costs, of the OMAP with a random input set, and shows that the OMAP produces energy costs that are several orders of magnitude lower.

Theorems & Definitions (20)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 1
  • proof
  • ...and 10 more