Interior-Point-Based $H_{2}$ Controller Synthesis for Compartmental Systems

This article focuses on the optimal $H_{2}$ controller design for compartmental systems, with the aim of enhancing system robustness while maintaining the law of mass conservation. Through a novel problem transformation, we establish that the original problem is equivalent to a new nonconvex optimization problem with a closed polyhedral constraint. Existing works have developed various first-order methods to tackle inequality constraints. However, they often lack convergence guarantees in nonconvex scenarios, thereby reducing their reliability in practical applications. Consequently, there is a critical need to develop new and efficient algorithms with convergence guarantees. In this article, we reformulate the problem using log-barrier functions and introduce two separate approaches with convergence guarantees to address the problem: the first-order interior point method (FIPM) and the second-order interior point method (SIPM). In addition, we propose an initialization method to guarantee the interior property of initial values. Finally, we compare FIPM and SIPM through a room temperature control example and illustrate their pros and cons via simulations. They are also compared to the existing alternating direction method of multipliers across different system scales.