Performative Control for Linear Dynamical Systems

TL;DR

This paper introduces performative control for linear dynamical systems, analyzing the existence and uniqueness of stable solutions influenced by policies, and proposes a convergent stochastic gradient descent method.

Contribution

It formalizes performative control in linear systems, establishes conditions for stable solutions, and develops a convergent optimization algorithm.

Findings

Unique performatively stable control solutions exist under certain stability conditions.

System stability influences the existence of stable solutions.

A stochastic gradient descent algorithm converges to the PSC solution.

Abstract

We introduce the framework of performative control, where the policy chosen by the controller affects the underlying dynamics of the control system. This results in a sequence of policy-dependent system state data with policy-dependent temporal correlations. Following the recent literature on performative prediction [21], we introduce the concept of a performatively stable control (PSC) solution. We first propose a sufficient condition for the performative control problem to admit a unique PSC solution with a problem-specific structure of distributional sensitivity propagation and aggregation. We further analyze the impacts of system stability on the existence of the PSC solution. Specifically, for almost surely strongly stable policy-dependent dynamics, the PSC solution exists if the sum of the distributional sensitivities is small enough. However, for almost surely unstable policy-dependent dynamics, the existence of the PSC solution will necessitate a temporally backward decaying of the distributional sensitivities. We finally provide a repeated stochastic gradient descent scheme that converges to the PSC solution and analyze its non-asymptotic convergence rate. Numerical results validate our theoretical analysis.