Infinite-Horizon Linear-Quadratic-Gaussian Control with Costly Measurements

TL;DR

This paper develops a method for jointly designing control and measurement strategies in infinite-horizon LQG problems with costly measurements, optimizing system performance and measurement costs.

Contribution

It analytically characterizes the optimal control and measurement strategies, revealing that the optimal measurement policy is periodic and independent of the state.

Findings

Optimal measurement strategy is periodic and independent of the state.

The optimal period depends on measurement costs and system parameters.

The approach reduces measurement overhead while maintaining system performance.

Abstract

In this paper, we consider an infinite horizon Linear-Quadratic-Gaussian control problem with controlled and costly measurements. A control strategy and a measurement strategy are co-designed to optimize the trade-off among control performance, actuating costs, and measurement costs. We address the co-design and co-optimization problem by establishing a dynamic programming equation with controlled lookahead. By leveraging the dynamic programming equation, we fully characterize the optimal control strategy and the measurement strategy analytically. The optimal control is linear in the state estimate that depends on the measurement strategy. We prove that the optimal measurement strategy is independent of the measured state and is periodic. And the optimal period length is determined by the cost of measurements and system parameters. We demonstrate the potential application of the co-design and co-optimization problem in an optimal self-triggered control paradigm. Two examples are provided to show the effectiveness of the optimal measurement strategy in reducing the overhead of measurements while keeping the system performance.