Minimax Linear Regulator Problems for Positive Systems : with applications to multi-agent synchronization
(2025)- Abstract
- Exceptional are the instances where explicit solutions to optimal control problems are obtainable. Of particular interest are the explicit solutions derived for minimax problems, as they provide a framework for addressing challenges involving adversarial conditions and uncertainties. This thesis presents explicit solutions to a novel class of minimax optimal control problems for positive linear systems with linear costs, elementwise linear constraints in the control policy, and worst-case disturbances. We refer to this class of problems, in the absence of disturbances, as the linear regulator (LR) problem. Two types of worst-case disturbances are considered in this thesis: bounded by elementwise-linear constraints and unconstrained... (More)
- Exceptional are the instances where explicit solutions to optimal control problems are obtainable. Of particular interest are the explicit solutions derived for minimax problems, as they provide a framework for addressing challenges involving adversarial conditions and uncertainties. This thesis presents explicit solutions to a novel class of minimax optimal control problems for positive linear systems with linear costs, elementwise linear constraints in the control policy, and worst-case disturbances. We refer to this class of problems, in the absence of disturbances, as the linear regulator (LR) problem. Two types of worst-case disturbances are considered in this thesis: bounded by elementwise-linear constraints and unconstrained positive disturbances. Using dynamic programming theory, explicit solutions to the Bellman equation (in the discrete-time setting) and the Hamilton-Jacobi-Bellman equation (in the continuous-time setting) are derived for both finite and infinite horizons. For the infinite horizon case, a fixed-point method is proposed to compute the solution of the HJB equation. Furthermore, a necessary and sufficient condition for minimiz- ing the l1-induced gain of the system is derived and characterized by the disturbance penalty in the cost function of the minimax problem. This condition characterizes the solution of the l1−induced gain minimization problem and demonstrates that, if a finite solution exists for the minimax problem under the presence of worst-case, unconstrained and positive disturbances, the solution to the minimax setting reduces to that of the LR problem in the absence of disturbances.
This thesis also analyzes the stabilizability and detectability properties of the LR problem. Similar to the Linear-Quadratic Regulator (LQR) problem, the LR problem is shown to facilitate the stabilization of positive systems. A linear programming formulation is introduced to compute the associated stabilizing controller, if one exists. The scalability and practical advantages of this theoretical framework for large-scale applications are demonstrated through its implementation in an optimal voltage control problem for a DC power network and in the management of a large-scale water network.
The second important contribution of this thesis is addressing positive synchronization on undirected graphs for homogeneous discrete and continuous-time positive systems. A static feedback protocol, derived from the Linear Regulator problem, is introduced. The stabilizing policy is derived by solving the linear programming formulation of the explicit solution to the LR problem under appropriate assumptions. Necessary and sufficient conditions are provided to ensure the positivity of each agent’s trajectory for all nonnegative initial conditions. The effectiveness of this approach is illustrated through simulations on large regular graphs with varying nodal degrees.
Throughout the thesis, we demonstrate how the results can be applied to problems over networks with positive dynamics. Our results pave the way for robust networks that maintain stability and optimal performance despite adversarial conditions. By leveraging explicit solutions to minimax optimal control and multi-agent synchronization problems, this work provides a computationally efficient and scalable framework for controlling large-scale systems. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/4f1cad48-5766-47ef-a165-fb120322f1b8
- author
- Gurpegui Ramón, Alba LU
- supervisor
- organization
- publishing date
- 2025-02-28
- type
- Thesis
- publication status
- published
- subject
- pages
- 118 pages
- publisher
- Department of Automatic Control, Lund University
- project
- WASP NEST: Learning in Networks: Structure, Dynamics, and Control
- language
- English
- LU publication?
- yes
- id
- 4f1cad48-5766-47ef-a165-fb120322f1b8
- date added to LUP
- 2025-03-31 13:33:06
- date last changed
- 2025-05-15 10:16:45
@misc{4f1cad48-5766-47ef-a165-fb120322f1b8,
abstract = {{Exceptional are the instances where explicit solutions to optimal control problems are obtainable. Of particular interest are the explicit solutions derived for minimax problems, as they provide a framework for addressing challenges involving adversarial conditions and uncertainties. This thesis presents explicit solutions to a novel class of minimax optimal control problems for positive linear systems with linear costs, elementwise linear constraints in the control policy, and worst-case disturbances. We refer to this class of problems, in the absence of disturbances, as the linear regulator (LR) problem. Two types of worst-case disturbances are considered in this thesis: bounded by elementwise-linear constraints and unconstrained positive disturbances. Using dynamic programming theory, explicit solutions to the Bellman equation (in the discrete-time setting) and the Hamilton-Jacobi-Bellman equation (in the continuous-time setting) are derived for both finite and infinite horizons. For the infinite horizon case, a fixed-point method is proposed to compute the solution of the HJB equation. Furthermore, a necessary and sufficient condition for minimiz- ing the l1-induced gain of the system is derived and characterized by the disturbance penalty in the cost function of the minimax problem. This condition characterizes the solution of the l1−induced gain minimization problem and demonstrates that, if a finite solution exists for the minimax problem under the presence of worst-case, unconstrained and positive disturbances, the solution to the minimax setting reduces to that of the LR problem in the absence of disturbances. <br/> This thesis also analyzes the stabilizability and detectability properties of the LR problem. Similar to the Linear-Quadratic Regulator (LQR) problem, the LR problem is shown to facilitate the stabilization of positive systems. A linear programming formulation is introduced to compute the associated stabilizing controller, if one exists. The scalability and practical advantages of this theoretical framework for large-scale applications are demonstrated through its implementation in an optimal voltage control problem for a DC power network and in the management of a large-scale water network.<br/> The second important contribution of this thesis is addressing positive synchronization on undirected graphs for homogeneous discrete and continuous-time positive systems. A static feedback protocol, derived from the Linear Regulator problem, is introduced. The stabilizing policy is derived by solving the linear programming formulation of the explicit solution to the LR problem under appropriate assumptions. Necessary and sufficient conditions are provided to ensure the positivity of each agent’s trajectory for all nonnegative initial conditions. The effectiveness of this approach is illustrated through simulations on large regular graphs with varying nodal degrees.<br/> Throughout the thesis, we demonstrate how the results can be applied to problems over networks with positive dynamics. Our results pave the way for robust networks that maintain stability and optimal performance despite adversarial conditions. By leveraging explicit solutions to minimax optimal control and multi-agent synchronization problems, this work provides a computationally efficient and scalable framework for controlling large-scale systems.}},
author = {{Gurpegui Ramón, Alba}},
language = {{eng}},
month = {{02}},
note = {{Licentiate Thesis}},
publisher = {{Department of Automatic Control, Lund University}},
title = {{Minimax Linear Regulator Problems for Positive Systems : with applications to multi-agent synchronization}},
url = {{https://lup.lub.lu.se/search/files/218374664/Thesis_Alba_Gurpegui_Ramon.pdf}},
year = {{2025}},
}