7 research outputs found
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
Linear identification of nonlinear systems: A lifting technique based on the Koopman operator
We exploit the key idea that nonlinear system identification is equivalent to
linear identification of the socalled Koopman operator. Instead of considering
nonlinear system identification in the state space, we obtain a novel linear
identification technique by recasting the problem in the infinite-dimensional
space of observables. This technique can be described in two main steps. In the
first step, similar to the socalled Extended Dynamic Mode Decomposition
algorithm, the data are lifted to the infinite-dimensional space and used for
linear identification of the Koopman operator. In the second step, the obtained
Koopman operator is "projected back" to the finite-dimensional state space, and
identified to the nonlinear vector field through a linear least squares
problem. The proposed technique is efficient to recover (polynomial) vector
fields of different classes of systems, including unstable, chaotic, and open
systems. In addition, it is robust to noise, well-suited to model low sampling
rate datasets, and able to infer network topology and dynamics.Comment: 6 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 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
Robustness of networked systems to unintended interactions with application to engineered genetic circuits
A networked dynamical system is composed of subsystems interconnected through
prescribed interactions. In many engineering applications, however, one
subsystem can also affect others through "unintended" interactions that can
significantly hamper the intended network's behavior. Although unintended
interactions can be modeled as disturbance inputs to the subsystems, these
disturbances depend on the network's states. As a consequence, a disturbance
attenuation property of each isolated subsystem is, alone, insufficient to
ensure that the network behavior is robust to unintended interactions. In this
paper, we provide sufficient conditions on subsystem dynamics and interaction
maps, such that the network's behavior is robust to unintended interactions.
These conditions require that each subsystem attenuates constant external
disturbances, is monotone or "near-monotone", the unintended interaction map is
monotone, and the prescribed interaction map does not contain feedback loops.
We employ this result to guide the design of resource-limited genetic circuits.
More generally, our result provide conditions under which robustness of
constituent subsystems is sufficient to guarantee robustness of the network to
unintended interactions