TY - JOUR
T1 - Optimal control formulation of pulse-based control using Koopman operator
AU - Sootla, Aivar
AU - Mauroy, Alexandre
AU - Ernst, Damien
N1 - Funding Information:
This work was performed while A. Sootla held the F.R.S.-FNRS fellowship (Chargé de recherches) at University of Liège. Currently, Aivar Sootla is supported by the EPSRC Grant EP/M002454/1. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Kok Lay Teo under the direction of Editor Ian R. Petersen.
Publisher Copyright:
© 2018 Elsevier Ltd
Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.
PY - 2018/5/1
Y1 - 2018/5/1
N2 - 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.
AB - 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.
KW - monotone systems
KW - Koopman operator
KW - isostables
KW - generalized repressilator
KW - genetic toggle switch
KW - Genetic toggle switch
KW - Monotone systems
KW - Isostables
KW - Generalized repressilator
UR - http://www.scopus.com/inward/record.url?scp=85042140784&partnerID=8YFLogxK
U2 - 10.1016/j.automatica.2018.01.036
DO - 10.1016/j.automatica.2018.01.036
M3 - Article
SN - 0005-1098
VL - 91
SP - 217
EP - 224
JO - Automatica
JF - Automatica
ER -