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

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 indistinguishable. The latter means that 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.

Statistically Consistent Inverse Optimal Control for Linear-Quadratic Tracking with Random Time Horizon

“…2]. The former can be done via an argument similar to [12,Proof of Lem. 4.2], which relies on the bounds from Lemma 1.…”

“…The latter is known as Inverse Optimal Control (IOC) [4], and has received considerable attention. In particular, the linear-quadratic problem has been studied in many different settings, including the infinitehorizon case in both continuous time [1,5] and discrete time [6], respectively, as well as the finite-horizon case in both continuous time [7,8] and discrete time [9][10][11][12], respectively. More general underlying dynamics and objective functions have also been considered [9,[13][14][15][16][17] and applied in areas such as, e.g., machine learning [18,19], and to model and analyze human locomotion of different forms [20,21].…”