88,602 research outputs found
Sample Complexity of the Robust LQG Regulator with Coprime Factors Uncertainty
This paper addresses the end-to-end sample complexity bound for learning the
H2 optimal controller (the Linear Quadratic Gaussian (LQG) problem) with
unknown dynamics, for potentially unstable Linear Time Invariant (LTI) systems.
The robust LQG synthesis procedure is performed by considering bounded additive
model uncertainty on the coprime factors of the plant. The closed-loop
identification of the nominal model of the true plant is performed by
constructing a Hankel-like matrix from a single time-series of noisy finite
length input-output data, using the ordinary least squares algorithm from
Sarkar et al. (2020). Next, an H-infinity bound on the estimated model error is
provided and the robust controller is designed via convex optimization, much in
the spirit of Boczar et al. (2018) and Zheng et al. (2020a), while allowing for
bounded additive uncertainty on the coprime factors of the model. Our
conclusions are consistent with previous results on learning the LQG and LQR
controllers.Comment: Minor Edits on closed loop identification, 30 pages, 2 figures, 3
algorithm
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
Applied Koopman Operator Theory for Power Systems Technology
Koopman operator is a composition operator defined for a dynamical system
described by nonlinear differential or difference equation. Although the
original system is nonlinear and evolves on a finite-dimensional state space,
the Koopman operator itself is linear but infinite-dimensional (evolves on a
function space). This linear operator captures the full information of the
dynamics described by the original nonlinear system. In particular, spectral
properties of the Koopman operator play a crucial role in analyzing the
original system. In the first part of this paper, we review the so-called
Koopman operator theory for nonlinear dynamical systems, with emphasis on modal
decomposition and computation that are direct to wide applications. Then, in
the second part, we present a series of applications of the Koopman operator
theory to power systems technology. The applications are established as
data-centric methods, namely, how to use massive quantities of data obtained
numerically and experimentally, through spectral analysis of the Koopman
operator: coherency identification of swings in coupled synchronous generators,
precursor diagnostic of instabilities in the coupled swing dynamics, and
stability assessment of power systems without any use of mathematical models.
Future problems of this research direction are identified in the last
concluding part of this paper.Comment: 31 pages, 11 figure
Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control
In this work, we explore finite-dimensional linear representations of
nonlinear dynamical systems by restricting the Koopman operator to an invariant
subspace. The Koopman operator is an infinite-dimensional linear operator that
evolves observable functions of the state-space of a dynamical system [Koopman
1931, PNAS]. Dominant terms in the Koopman expansion are typically computed
using dynamic mode decomposition (DMD). DMD uses linear measurements of the
state variables, and it has recently been shown that this may be too
restrictive for nonlinear systems [Williams et al. 2015, JNLS]. Choosing
nonlinear observable functions to form an invariant subspace where it is
possible to obtain linear models, especially those that are useful for control,
is an open challenge.
Here, we investigate the choice of observable functions for Koopman analysis
that enable the use of optimal linear control techniques on nonlinear problems.
First, to include a cost on the state of the system, as in linear quadratic
regulator (LQR) control, it is helpful to include these states in the
observable subspace, as in DMD. However, we find that this is only possible
when there is a single isolated fixed point, as systems with multiple fixed
points or more complicated attractors are not globally topologically conjugate
to a finite-dimensional linear system, and cannot be represented by a
finite-dimensional linear Koopman subspace that includes the state. We then
present a data-driven strategy to identify relevant observable functions for
Koopman analysis using a new algorithm to determine terms in a dynamical system
by sparse regression of the data in a nonlinear function space [Brunton et al.
2015, arxiv]; we show how this algorithm is related to DMD. Finally, we
demonstrate how to design optimal control laws for nonlinear systems using
techniques from linear optimal control on Koopman invariant subspaces.Comment: 20 pages, 5 figures, 2 code
A general stability criterion for switched linear systems having stable and unstable subsystems
We report conditions on a switching signal that guarantee that solutions of a
switched linear systems converge asymptotically to zero. These conditions are
apply to continuous, discrete-time and hybrid switched linear systems, both
those having stable subsystems and mixtures of stable and unstable subsystems
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