Optimal control formulation of pulse-based control using Koopman operator

Abstract

a b s t r a c tIn 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.

Data-Driven Approximations of Dynamical Systems Operators for Control

“…4.3). Optimal control formulations have also been considered for switching problems [166,165,136]. Based on a global bilinearization, the underlying dynamical system can be stabilized using feedback linearization [65].…”

Pulse-Based Control Using Koopman Operator Under Parametric Uncertainty

“…At the first glance, the problem of convergence to B ε (x • ) (i.e., Problem 1) does not seem to be easier than the problem of convergence to the ε-ball around x • (e.g., {x ∈ R n | x − x • 2 ≤ ε}). As shown in [8], however, Problem 1 can be solved using the following static optimization program under Assumptions A1 -A5:…”

“…The computation of the function r can be performed using the methods to compute s 1 [8], which may not be numerically cheap for large-scale and/or stiff systems. However, solving an optimal control problem using dynamic programming or maximum principle generally requires (at least) a comparable computational effort.…”

Data-Driven Nonlinear Stabilization Using Koopman Operator

“…We use Extended Dynamic Mode Decomposition (EDMD) algorithm for the approximation of U 1 ∆t and U 0 ∆t thereby approximating A and B in Eqs. (29) and (28) respectively [35]. For this purpose let the time-series data generated by the dynamical system (24) be given by…”