Optimal Safe Controller Synthesis: A Density Function Approach

TL;DR

This paper introduces a density function-based method for synthesizing optimal safe controllers, enabling straightforward incorporation of safety constraints and robust control under disturbances, demonstrated through an adaptive cruise control example.

Contribution

It presents a novel density function approach for safe control synthesis, linking density and value functions via duality, and extends it to robust control with disturbances.

Findings

Effective safety constraint handling via density functions

Comparable or improved safety performance over CBF methods

Robust control synthesis under external disturbances

Abstract

This paper considers the synthesis of optimal safe controllers based on density functions. We present an algorithm for robust constrained optimal control synthesis using the duality relationship between the density function and the value function. The density function follows the Liouville equation and is the dual of the value function, which satisfies Bellman's optimality principle. Thanks to density functions, constraints over the distribution of states, such as safety constraints, can be posed straightforwardly in an optimal control problem. The constrained optimal control problem is then solved with a primal-dual algorithm. This formulation is extended to the case with external disturbances, and we show that the robust constrained optimal control can be solved with a modified primal-dual algorithm. We apply this formulation to the problem of finding the optimal safe controller that minimizes the cumulative intervention. An adaptive cruise control (ACC) example is used to demonstrate the efficacy of the proposed, wherein we compare the result of the density function approach with the conventional control barrier function (CBF) method.