11 research outputs found
Shaping Pulses to Control Bistable Monotone Systems Using Koopman Operator
In this paper, we further develop a recently proposed control method to
switch a bistable system between its steady states using temporal pulses. The
motivation for using pulses comes from biomedical and biological applications
(e.g. synthetic biology), where it is generally difficult to build feedback
control systems due to technical limitations in sensing and actuation. The
original framework was derived for monotone systems and all the extensions
relied on monotone systems theory. In contrast, we introduce the concept of
switching function which is related to eigenfunctions of the so-called Koopman
operator subject to a fixed control pulse. Using the level sets of the
switching function we can (i) compute the set of all pulses that drive the
system toward the steady state in a synchronous way and (ii) estimate the time
needed by the flow to reach an epsilon neighborhood of the target steady state.
Additionally, we show that for monotone systems the switching function is also
monotone in some sense, a property that can yield efficient algorithms to
compute it. This observation recovers and further extends the results of the
original framework, which we illustrate on numerical examples inspired by
biological applications.Comment: 7 page
Shaping Pulses to Control Bistable Monotone Systems Using Koopman Operator
In this paper, we further develop a recently proposed control method to switch
a bistable system between its steady states using temporal pulses. The motivation for using
pulses comes from biomedical and biological applications (e.g. synthetic biology), where it is
generally di cult to build feedback control systems due to technical limitations in sensing and
actuation. The original framework was derived for monotone systems and all the extensions
relied on monotone systems theory. In contrast, we introduce the concept of switching function
which is related to eigenfunctions of the so-called Koopman operator subject to a xed control
pulse. Using the level sets of the switching function we can (i) compute the set of all pulses that
drive the system toward the steady state in a synchronous way and (ii) estimate the time needed
by the
ow to reach an epsilon neighborhood of the target steady state. Additionally, we show
that for monotone systems the switching function is also monotone in some sense, a property
that can yield e cient algorithms to compute it. This observation recovers and further extends
the results of the original framework, which we illustrate on numerical examples inspired by
biological applications
An Optimal Control Formulation of Pulse-Based Control Using Koopman Operator
In many applications, and in systems/synthetic biology, in particular, it is
desirable to compute control policies that force the trajectory of a bistable
system from one equilibrium (the initial point) to another equilibrium (the
target point), or in other words to solve the switching problem. 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 this problem can be solved using a static
optimization problem in the case of monotone systems. Changing the initial
point to an arbitrary state allows to build 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. One of the main advantages of
using the Koopman operator is the powerful computational tools developed for
this framework. Besides the presence of numerical solutions, the
switching/convergence problem can also serve as a building block for solving
more complicated control problems and can potentially be applied to
non-monotone systems. We illustrate this argument on the problem of
synchronizing cardiac cells by defibrillation. Potentially, our approach can be
extended to problems with different parametrizations of control signals since
the only fundamental limitation is the finite time application of the control
signal.Comment: corrected typo
Geometric Properties of Isostables and Basins of Attraction of Monotone Systems
In this paper, we study geometric properties of basins of attraction of
monotone systems. Our results are based on a combination of monotone systems
theory and spectral operator theory. We exploit the framework of the Koopman
operator, which provides a linear infinite-dimensional description of nonlinear
dynamical systems and spectral operator-theoretic notions such as eigenvalues
and eigenfunctions. The sublevel sets of the dominant eigenfunction form a
family of nested forward-invariant sets and the basin of attraction is the
largest of these sets. The boundaries of these sets, called isostables, allow
studying temporal properties of the system. Our first observation is that the
dominant eigenfunction is increasing in every variable in the case of monotone
systems. This is a strong geometric property which simplifies the computation
of isostables. We also show how variations in basins of attraction can be
bounded under parametric uncertainty in the vector field of monotone systems.
Finally, we study the properties of the parameter set for which a monotone
system is multistable. Our results are illustrated on several systems of two to
four dimensions.Comment: 12 pages, to appear in IEEE Transaction on Automatic Contro
Operator-Theoretic Characterization of Eventually Monotone Systems
Monotone systems are dynamical systems whose solutions preserve a partial
order in the initial condition for all positive times. It stands to reason that
some systems may preserve a partial order only after some initial transient.
These systems are usually called eventually monotone. While monotone systems
have a characterization in terms of their vector fields (i.e. Kamke-Muller
condition), eventually monotone systems have not been characterized in such an
explicit manner. In order to provide a characterization, we drew inspiration
from the results for linear systems, where eventually monotone (positive)
systems are studied using the spectral properties of the system (i.e.
Perron-Frobenius property). In the case of nonlinear systems, this spectral
characterization is not straightforward, a fact that explains why the class of
eventually monotone systems has received little attention to date. In this
paper, we show that a spectral characterization of nonlinear eventually
monotone systems can be obtained through the Koopman operator framework. We
consider a number of biologically inspired examples to illustrate the potential
applicability of eventual monotonicity.Comment: 13 page
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal