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A Unified Alternating Optimization Framework for Joint Sensor and Actuator Configuration in LQG Systems

Nachuan Yang, Yuzhe Li, Ling Shi, Tongwen Chen

TL;DR

This work addresses the problem of jointly configuring sensors and actuators in $LQG$ systems, where components are designed from scratch under general costs and constraints. The authors derive analytical gradients of the $LQG$ performance with respect to the actuator and sensor matrices via algebraic Riccati equations and establish first-order optimality conditions using KKT-like normal cone formulations. They propose a unified ADMM-based alternating optimization framework that decouples the coupled design variables and tailor it to three representative scenarios: sparsity-promoting, low-rank promoting, and structure-constrained configurations, each with efficient subproblem solutions. Through simulations on a chemical reactor benchmark, the approach demonstrates favorable tradeoffs between estimation/control performance and configuration costs, while ensuring stabilizability and detectability of the resulting designs. Overall, the framework enables scalable, adaptable co-design of sensors and actuators for networked $LQG$ systems, with potential extensions to nonlinear and real-time settings.

Abstract

This paper fills a gap in the literature by considering a joint sensor and actuator configuration problem under the linear quadratic Gaussian (LQG) performance without assuming a predefined set of candidate components. Different from the existing research, which primarily focuses on selecting or placing sensors and actuators from a fixed group, we consider a more flexible formulation where these components must be designed from scratch, subject to general-form configuration costs and constraints. To address this challenge, we first analytically characterize the gradients of the LQG performance with respect to the sensor and actuator matrices using algebraic Riccati equations. Subsequently, we derive first-order optimality conditions based on the Karush-Kuhn-Tucker (KKT) analysis and develop a unified alternating direction method of multipliers (ADMM)-based alternating optimization framework to address the general-form sensor and actuator configuration problem. Furthermore, we investigate three representative scenarios: sparsity promoting configuration, low-rank promoting configuration, and structure-constrained configuration. For each scenario, we provide in-depth analysis and develop tailored computational schemes. The proposed framework ensures numerical efficiency and adaptability to various design constraints and configuration costs, making it well-suited for integration into numerical solvers.

A Unified Alternating Optimization Framework for Joint Sensor and Actuator Configuration in LQG Systems

TL;DR

This work addresses the problem of jointly configuring sensors and actuators in systems, where components are designed from scratch under general costs and constraints. The authors derive analytical gradients of the performance with respect to the actuator and sensor matrices via algebraic Riccati equations and establish first-order optimality conditions using KKT-like normal cone formulations. They propose a unified ADMM-based alternating optimization framework that decouples the coupled design variables and tailor it to three representative scenarios: sparsity-promoting, low-rank promoting, and structure-constrained configurations, each with efficient subproblem solutions. Through simulations on a chemical reactor benchmark, the approach demonstrates favorable tradeoffs between estimation/control performance and configuration costs, while ensuring stabilizability and detectability of the resulting designs. Overall, the framework enables scalable, adaptable co-design of sensors and actuators for networked systems, with potential extensions to nonlinear and real-time settings.

Abstract

This paper fills a gap in the literature by considering a joint sensor and actuator configuration problem under the linear quadratic Gaussian (LQG) performance without assuming a predefined set of candidate components. Different from the existing research, which primarily focuses on selecting or placing sensors and actuators from a fixed group, we consider a more flexible formulation where these components must be designed from scratch, subject to general-form configuration costs and constraints. To address this challenge, we first analytically characterize the gradients of the LQG performance with respect to the sensor and actuator matrices using algebraic Riccati equations. Subsequently, we derive first-order optimality conditions based on the Karush-Kuhn-Tucker (KKT) analysis and develop a unified alternating direction method of multipliers (ADMM)-based alternating optimization framework to address the general-form sensor and actuator configuration problem. Furthermore, we investigate three representative scenarios: sparsity promoting configuration, low-rank promoting configuration, and structure-constrained configuration. For each scenario, we provide in-depth analysis and develop tailored computational schemes. The proposed framework ensures numerical efficiency and adaptability to various design constraints and configuration costs, making it well-suited for integration into numerical solvers.

Paper Structure

This paper contains 15 sections, 13 theorems, 143 equations, 8 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear Quadratic Gaussian Control
  4. Algebraic Riccati Equation Method
  5. Sensor and Actuator Configuration Problem
  6. Main Results
  7. Optimization-based Sensor and Actuator Configuration
  8. Scenario I: Sparsity Promoting Configuration
  9. Scenario II: Low-Rank Promoting Configuration
  10. Scenario III: Structure-Constrained Configuration
  11. Simulations
  12. Simulation I: Sparsity Promoting Configuration
  13. Simulation II: Low-Rank Promoting Configuration
  14. Simulation III: Structure-Constrained Configuration
  15. Conclusions

Key Result

Theorem 1

Given a constant sensor matrix $C \in \mathcal{D}$, for any actuator matrix $B \in \mathcal{S}$, the gradient of the LQG performance $J_{\text{LQG}}$ with respect to matrix $B$ is given by Here, $G_1$ and $G_2$ are the unique symmetric solutions to the following Lyapunov matrix equations with $K^* = -R^{-1}B^{\mathop{\mathrm{\mathrm{T}}}\limits}P$ denoting the optimal controller gain.

Figures (8)

  • Figure 1: Convergence process of Algorithm 2 in Problem S1.
  • Figure 2: The LQG performance $J_{\rm LQG}$ versus the configuration cost $\tilde{J}_{\rm SAC}$ under different relative weights $\gamma$.
  • Figure 3: The configuration costs defined using the $\ell_0$ norm and $\ell_1$ norm under different relative weights $\gamma$.
  • Figure 4: Convergence process of Algorithm 2 in Problem S2.
  • Figure 5: The LQG performance $J_{\rm LQG}$ versus the configuration cost $\tilde{J}_{\rm SAC}$ under different relative weights $\gamma$.
  • ...and 3 more figures

Theorems & Definitions (17)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 2
  • Proposition 1
  • Remark 3
  • Remark 4
  • Theorem 4
  • Corollary 1
  • ...and 7 more