361 research outputs found
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
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
Online Real-time Learning of Dynamical Systems from Noisy Streaming Data
Recent advancements in sensing and communication facilitate obtaining
high-frequency real-time data from various physical systems like power
networks, climate systems, biological networks, etc. However, since the data
are recorded by physical sensors, it is natural that the obtained data is
corrupted by measurement noise. In this paper, we present a novel algorithm for
online real-time learning of dynamical systems from noisy time-series data,
which employs the Robust Koopman operator framework to mitigate the effect of
measurement noise. The proposed algorithm has three main advantages: a) it
allows for online real-time monitoring of a dynamical system; b) it obtains a
linear representation of the underlying dynamical system, thus enabling the
user to use linear systems theory for analysis and control of the system; c) it
is computationally fast and less intensive than the popular Extended Dynamic
Mode Decomposition (EDMD) algorithm. We illustrate the efficiency of the
proposed algorithm by applying it to identify the Van der Pol oscillator, the
IEEE 68 bus system, and a ring network of Van der Pol oscillators
Modeling Nonlinear Control Systems via Koopman Control Family: Universal Forms and Subspace Invariance Proximity
This paper introduces the Koopman Control Family (KCF), a mathematical
framework for modeling general discrete-time nonlinear control systems with the
aim of providing a solid theoretical foundation for the use of Koopman-based
methods in systems with inputs. We demonstrate that the concept of KCF can
completely capture the behavior of nonlinear control systems on a (potentially
infinite-dimensional) function space. By employing a generalized notion of
subspace invariance under the KCF, we establish a universal form for
finite-dimensional models, which encompasses the commonly used linear,
bilinear, and linear switched models as specific instances. In cases where the
subspace is not invariant under the KCF, we propose a method for approximating
models in general form and characterize the model's accuracy using the concept
of invariance proximity. The proposed framework naturally lends itself to the
incorporation of data-driven methods in modeling and control.Comment: 16 page
Probabilistic forecast of nonlinear dynamical systems with uncertainty quantification
Data-driven modeling is useful for reconstructing nonlinear dynamical systems
when the underlying process is unknown or too expensive to compute. Having
reliable uncertainty assessment of the forecast enables tools to be deployed to
predict new scenarios unobserved before. In this work, we first extend parallel
partial Gaussian processes for predicting the vector-valued transition function
that links the observations between the current and next time points, and
quantify the uncertainty of predictions by posterior sampling. Second, we show
the equivalence between the dynamic mode decomposition and the maximum
likelihood estimator of the linear mapping matrix in the linear state space
model. The connection provides a data generating model of dynamic mode
decomposition and thus, uncertainty of predictions can be obtained.
Furthermore, we draw close connections between different data-driven models for
approximating nonlinear dynamics, through a unified view of data generating
models. We study two numerical examples, where the inputs of the dynamics are
assumed to be known in the first example and the inputs are unknown in the
second example. The examples indicate that uncertainty of forecast can be
properly quantified, whereas model or input misspecification can degrade the
accuracy of uncertainty quantification
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