Optimal control formulation of pulse-based control using Koopman operator

Aivar Sootla, Alexandre Mauroy, Damien Ernst

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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.

Original languageEnglish
Pages (from-to)217-224
Number of pages8
JournalAutomatica
Volume91
DOIs
Publication statusPublished - 1 May 2018

Keywords

  • monotone systems
  • Koopman operator
  • isostables
  • generalized repressilator
  • genetic toggle switch
  • Genetic toggle switch
  • Monotone systems
  • Isostables
  • Generalized repressilator

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