# LQG Control and Sensing Co-Design

## TL;DR

This paper addresses the joint design of sensing and control policies in LQG problems, proposing polynomial-time algorithms with suboptimality guarantees for sensor selection under cost and performance constraints.

## Contribution

It introduces the first polynomial-time algorithms with performance guarantees for LQG sensing-control co-design, leveraging a separation principle and supermodular optimization techniques.

## Key findings

- Algorithms achieve suboptimality guarantees per instance.
- Connections established between algorithm performance and control-theoretic metrics.
- Applications demonstrated in formation control and robot navigation.

## Abstract

We investigate a Linear-Quadratic-Gaussian (LQG) control and sensing co-design problem, where one jointly designs sensing and control policies. We focus on the realistic case where the sensing design is selected among a finite set of available sensors, where each sensor is associated with a different cost (e.g., power consumption). We consider two dual problem instances: sensing-constrained LQG control, where one maximizes control performance subject to a sensor cost budget, and minimum-sensing LQG control, where one minimizes sensor cost subject to performance constraints. We prove no polynomial time algorithm guarantees across all problem instances a constant approximation factor from the optimal. Nonetheless, we present the first polynomial time algorithms with per-instance suboptimality guarantees. To this end, we leverage a separation principle, that partially decouples the design of sensing and control. Then, we frame LQG co-design as the optimization of approximately supermodular set functions; we develop novel algorithms to solve the problems; and we prove original results on the performance of the algorithms, and establish connections between their suboptimality and control-theoretic quantities. We conclude the paper by discussing two applications, namely, sensing-constrained formation control and resource-constrained robot navigation.