Joint Actuator-sensor Design for Stochastic Linear Systems

TL;DR

This paper develops a gradient descent method to optimize sensor and actuator placement in stochastic linear systems, aiming to minimize long-term quadratic costs, and identifies conditions for achieving unique optimal solutions.

Contribution

It introduces a novel gradient-based optimization approach for joint actuator-sensor design in stochastic systems, addressing a non-convex problem with convergence guarantees under specific conditions.

Findings

The proposed algorithm effectively minimizes the expected quadratic cost.

Under certain conditions, the algorithm converges to a unique global minimum.

The method provides a practical solution for sensor-actuator placement in stochastic control.

Abstract

We investigate the joint actuator-sensor design problem for stochastic linear control systems. Specifically, we address the problem of identifying a pair of sensor and actuator which gives rise to the minimum expected value of a quadratic cost. It is well known that for the linear-quadratic-Gaussian (LQG) control problem, the optimal feedback control law can be obtained via the celebrated separation principle. Moreover, if the system is stabilizable and detectable, then the infinite-horizon time-averaged cost exists. But such a cost depends on the placements of the sensor and the actuator. We formulate in the paper the optimization problem about minimizing the time-averaged cost over admissible pairs of actuator and sensor under the constraint that their Euclidean norms are fixed. The problem is non-convex and is in general difficult to solve. We obtain in the paper a gradient descent algorithm (over the set of admissible pairs) which minimizes the time-averaged cost. Moreover, we show that the algorithm can lead to a unique local (and hence global) minimum point under certain special conditions.