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Minimax Linear Regulator Problems for Positive Systems

Alba Gurpegui, Mark Jeeninga, Emma Tegling, Anders Rantzer

TL;DR

The work develops a complete continuous-time minimax Linear Regulator framework for positive systems with multiple disturbances, providing explicit solutions via dynamic programming and Hamilton-Jacobi-Isaacs equations for finite and infinite horizons. It introduces a fixed-point method and a linear-programming characterization to compute the minimax solution, and links the disturbance penalties to the $L_1$-induced gain through the auxiliary vector $p$. The framework guarantees positivity and scalability, and is demonstrated on a large-scale water-flow network where the minimax controller achieves robust stabilization under worst-case disturbances and even under overestimated disturbance models. The results offer practical pathways for robust control design in large, distributed positive systems such as water networks, and lay groundwork for further improvements in fixed-point iterations and network-structure analysis.

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.

Minimax Linear Regulator Problems for Positive Systems

TL;DR

The work develops a complete continuous-time minimax Linear Regulator framework for positive systems with multiple disturbances, providing explicit solutions via dynamic programming and Hamilton-Jacobi-Isaacs equations for finite and infinite horizons. It introduces a fixed-point method and a linear-programming characterization to compute the minimax solution, and links the disturbance penalties to the -induced gain through the auxiliary vector . The framework guarantees positivity and scalability, and is demonstrated on a large-scale water-flow network where the minimax controller achieves robust stabilization under worst-case disturbances and even under overestimated disturbance models. The results offer practical pathways for robust control design in large, distributed positive systems such as water networks, and lay groundwork for further improvements in fixed-point iterations and network-structure analysis.

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.

Paper Structure

This paper contains 17 sections, 14 theorems, 79 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Problem Setup
  4. Notation and Preliminaries
  5. The Minimax Linear Regulator Problem
  6. Finite Horizon
  7. Infinite Horizon
  8. The case of positive unconstrained disturbances
  9. E-Stabilizability and Detectability Conditions
  10. Linear Programming
  11. -induced gain Minimization Problem
  12. Example: Water-flow Network
  13. Leakages: Load Disturbances
  14. Detectability
  15. Simulations
  16. ...and 2 more sections

Key Result

Theorem 1

Let $A \in \mathbb{R}^{n\times n}$, $B = \left [ B_{1} \dots B_{m} \right ] \in \mathbb{R}^{n\times m }$, $F \in \mathbb{R}^{n \times l}_+$, $H \in \mathbb{R}^{n \times c}$, $E = \left [ E_{1}^{\top} \dots E_{m}^{\top} \right ]^{\top} \in \mathbb{R}_{+}^{m \times n}$ such that $E_i^{\top} \neq \math Then the following statements are equivalent: Moreover, if the above conditions hold, the minimal

Figures (6)

  • Figure 1: Block Diagram of the closed-loop system dynamics in \ref{['dynamics']} under the presence of two types of disturbances.
  • Figure 2: State trajectories of the closed-loop system in Example \ref{['nonuniqueness_example']} with initial condition $x_0=\mathbb{1}$, under two different optimal feedback gain matrices (a) $K_1$ and (b) $K_2$.
  • Figure 3: Diagram of the water flow network in Section \ref{['sect_examples']}.
  • Figure 4: Evolution of the optimal cost $p(t)^{\top}\mathbb{1}$ over the time interval $t \in \left[0, 10\right]$ plotted on a logarithmic scale.
  • Figure 5: State trajectories $x(t)=e^{\tilde{A}t}x_0$ over the time horizon $t \in \left[0, 10\right]$, under different configurations. (a) Open-loop evolution with $\tilde{A}=A$; (b) open-loop dynamics in the presence of disturbances, $\tilde{A}=A+\left|H \right|G$; and (c) closed-loop behavior in the presence of both disturbances and control, given by $\tilde{A}=A+\left|H \right|G-BK$.
  • ...and 1 more figures

Theorems & Definitions (34)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 2
  • proof
  • proof
  • Example 1
  • Theorem 3
  • ...and 24 more