Optimal control formulation of pulse-based control using Koopman operator. (May 2018)

• Record Type: Journal Article
• Authors: Sootla, Aivar; Mauroy, Alexandre; Ernst, Damien
• Is Part Of: Automatica. Volume 91(2018)
• Page Start: 217
• Page End: 224
• Publication Date: 2018-05
• Subjects: Monotone systems -- Koopman operator -- Isostables -- Generalized repressilator -- Genetic toggle switch; Automatic control -- Periodicals; Automation -- Periodicals; 629.805
• DOI: 10.1016/j.automatica.2018.01.036
• Languages: English
• ISSNs: 0005-1098

Abstract

Abstract: In many applications, and in systems/synthetic biology, in particular, it is desirable to solve the switching problem, i.e., to compute control policies that force the trajectory of a bistable system from one equilibrium (the initial point) to another equilibrium (the target point). It was recently shown that for monotone bistable systems, this problem admits easy-to-implement open-loop solutions in terms of temporal pulses (i.e., step functions of fixed length and fixed magnitude). In this paper, we develop this idea further and formulate a problem of convergence to an equilibrium from an arbitrary initial point. We show that the convergence problem can be solved using a static optimization problem in the case of monotone systems. Changing the initial point to an arbitrary state allows building closed-loop, event-based or open-loop policies for the switching/convergence problems. In our derivations, we exploit the Koopman operator, which offers a linear infinite-dimensional representation of an autonomous nonlinear system and powerful computational tools for their analysis. Our solutions to the switching/convergence problems can serve as building blocks for other control problems and can potentially be applied to non-monotone systems. We illustrate this argument on the problem of synchronizing cardiac cells by defibrillation.