Inverse linear-quadratic discrete-time finite-horizon optimal control for indistinguishable homogeneous agents: a convex optimization approach

TL;DR

This paper addresses the inverse optimal control problem for homogeneous agents in discrete-time, finite-horizon settings, proposing convex optimization methods to recover cost functions and system matrices from observational data, even with noise.

Contribution

It introduces a convex optimization framework for globally identifying the quadratic cost and system matrices in inverse LQ control with homogeneous agents, including noisy data scenarios.

Findings

Unique global solutions are achievable via convex optimization.

The method is statistically consistent under noisy observations.

Numerical examples validate the approach's effectiveness.

Abstract

The inverse linear-quadratic optimal control problem is a system identification problem whose aim is to recover the quadratic cost function and hence the closed-loop system matrices based on observations of optimal trajectories. In this paper, the discrete-time, finite-horizon case is considered, where the agents are also assumed to be homogeneous and indistinguishable. The latter means that the agents all have the same dynamics and objective functions and the observations are in terms of "snap shots" of all agents at different time instants, but what is not known is "which agent moved where" for consecutive observations. This absence of linked optimal trajectories makes the problem challenging. We first show that this problem is globally identifiable. Then, for the case of noiseless observations, we show that the true cost matrix, and hence the closed-loop system matrices, can be recovered as the unique global optimal solution to a convex optimization problem. Next, for the case of noisy observations, we formulate an estimator as the unique global optimal solution to a modified convex optimization problem. Moreover, the statistical consistency of this estimator is shown. Finally, the performance of the proposed method is demonstrated by a number of numerical examples.