On the Existence and Computation of Minimum Attention Optimal Control Laws

TL;DR

This paper investigates the existence and computation of minimum attention control laws for nonlinear systems, providing theoretical guarantees and an iterative solution method, demonstrated on a robot arm example.

Contribution

It proves the existence of solutions under specific control structure assumptions and introduces a one-shot iterative method for computing minimum attention control laws.

Findings

Existence of solutions is guaranteed under certain control assumptions.

Derived first-order optimality conditions using Liouville equation.

Demonstrated the approach with a two degree-of-freedom robot arm example.

Abstract

Brockett's minimum attention functional \cite{Brockett} has been proposed as one means of capturing the cost of control implementation--regarded here as the rate of change of the control with respect to both state and time--for general nonlinear control systems, with applications ranging from human motor control to robotics. The main challenge in forging the minimum attention paradigm into a practical control design methodology is that the existence of solutions is not always assured, and finding numerical solutions is also difficult. In this paper we prove that, under the assumption of a control that is the sum of a time-varying feedforward term and a time-varying feedback term linear in the state, existence of a solution can be guaranteed. Under these assumptions we appeal to the Liouville equation representation of a nonlinear control system and derive the associated first-order optimality conditions. The one-shot method is then used to prove the existence of a solution and also to iteratively compute a solution. Our methodology is illustrated with an example involving a two degree-of-freedom robot arm.