Minimax Linear Regulator Problems for Positive Systems

TL;DR

This paper develops explicit solutions for a minimax linear regulator problem in positive systems, enabling robust control under disturbances with applications to large-scale water networks.

Contribution

It introduces a novel minimax LR framework for positive LTI systems with explicit solutions and a fixed-point method for infinite horizon problems.

Findings

Explicit solutions derived for finite and infinite horizons.

A fixed-point method for infinite horizon case.

Application to large-scale water management network.

Abstract

Explicit solutions to optimal control problems are rarely obtainable. Of particular interest are the explicit solutions derived for minimax problems, providing a framework to address adversarial conditions and uncertainty. This work considers a multi-disturbance minimax Linear Regulator (LR) framework for positive linear time-invariant systems in continuous time, which, analogous to the Linear-Quadratic Regulator (LQR) problem, can be utilized for the stabilization of positive systems. The problem is studied for nonnegative and state-bounded disturbances. Dynamic programming theory is leveraged to derive explicit solutions to the minimax LR problem for both finite and infinite time horizons. In addition, a fixed-point method is proposed that computes the solution for the infinite horizon case, and the minimum L1-induced gain of the system is studied. We motivate the prospective scalability properties of our framework with a large-scale water management network.