A Minimax Optimal Controller for Positive Systems

TL;DR

This paper provides an explicit solution to the minimax optimal control problem for positive systems, revealing conditions under which solutions exist and how the problem simplifies without disturbances.

Contribution

It introduces a bound on disturbance penalty that guarantees a finite solution and links the minimax problem to a simpler disturbance-free minimization.

Findings

Derived explicit solution to Bellman equation for positive systems

Identified disturbance penalty bounds for solution existence

Showed problem reduces to minimization without disturbances when feasible

Abstract

We present an explicit solution to the discrete-time Bellman equation for minimax optimal control of positive systems under unconstrained disturbances. The primary contribution of our result relies on deducing a bound for the disturbance penalty, which characterizes the existence of a finite solution to the problem class. Moreover, this constraint on the disturbance penalty reveals that, in scenarios where a solution is feasible, the problem converges to its equivalent minimization problem in the absence of disturbances.