4,943 research outputs found
Model-Based Control Using Koopman Operators
This paper explores the application of Koopman operator theory to the control
of robotic systems. The operator is introduced as a method to generate
data-driven models that have utility for model-based control methods. We then
motivate the use of the Koopman operator towards augmenting model-based
control. Specifically, we illustrate how the operator can be used to obtain a
linearizable data-driven model for an unknown dynamical process that is useful
for model-based control synthesis. Simulated results show that with increasing
complexity in the choice of the basis functions, a closed-loop controller is
able to invert and stabilize a cart- and VTOL-pendulum systems. Furthermore,
the specification of the basis function are shown to be of importance when
generating a Koopman operator for specific robotic systems. Experimental
results with the Sphero SPRK robot explore the utility of the Koopman operator
in a reduced state representation setting where increased complexity in the
basis function improve open- and closed-loop controller performance in various
terrains, including sand.Comment: 8 page
Learning Bilinear Models of Actuated Koopman Generators from Partially-Observed Trajectories
Data-driven models for nonlinear dynamical systems based on approximating the
underlying Koopman operator or generator have proven to be successful tools for
forecasting, feature learning, state estimation, and control. It has become
well known that the Koopman generators for control-affine systems also have
affine dependence on the input, leading to convenient finite-dimensional
bilinear approximations of the dynamics. Yet there are still two main obstacles
that limit the scope of current approaches for approximating the Koopman
generators of systems with actuation. First, the performance of existing
methods depends heavily on the choice of basis functions over which the Koopman
generator is to be approximated; and there is currently no universal way to
choose them for systems that are not measure preserving. Secondly, if we do not
observe the full state, then it becomes necessary to account for the dependence
of the output time series on the sequence of supplied inputs when constructing
observables to approximate Koopman operators. To address these issues, we write
the dynamics of observables governed by the Koopman generator as a bilinear
hidden Markov model, and determine the model parameters using the
expectation-maximization (EM) algorithm. The E-step involves a standard Kalman
filter and smoother, while the M-step resembles control-affine dynamic mode
decomposition for the generator. We demonstrate the performance of this method
on three examples, including recovery of a finite-dimensional Koopman-invariant
subspace for an actuated system with a slow manifold; estimation of Koopman
eigenfunctions for the unforced Duffing equation; and model-predictive control
of a fluidic pinball system based only on noisy observations of lift and drag
Data-driven system modeling and optimal control for nonlinear dynamical systems
With the increasing complexity of modern industry processes, robotics, transportation, aerospace,
power grids, an exact model of the physical systems are extremely hard to obtain whereas abundant
of time-series data can be captured from these systems. This makes it a important and
demanding research area to investigate feasibility of using data to learn behaviours of systems and
design controllers where the end goal generally evolves around stabilization. Transfer operators
i.e. Perron-Frobenius and Koopman operators play an undeniable role in advanced research of
nonlinear dynamical system stabilization. These operators have been a alternate direction of how
we generally approach dynamical systems, providing linear representations for even strongly nonlinear
dynamics. There is tremendous benet of acquiring a linear model of a system using these
models but, there remains a challenge of innite dimension for such models. To deal with it, we can
approximate a nite dimensional matrix of these operators e.g. Koopman matrix using Extended
Dynamic Mode Decomposition (EDMD) or Naturally Structured Dynamic Mode Decomposition
(NSDMD). Using duality property of Koopman and P-F operators we can derive formulation for
P-F matrix from Koopman matrix. Once we have a linear approximation of the system, Lyapunov
measure approach can be used along with a linear programming based computational framework for
stability analysis and design of almost everywhere stabilizing controller. In this work, we propose
a complete structure to stabilize a system that does not have an explicit model and only requires
black box input output time-series data. On a separate work, we show a set-oriented approach can
be used to control and stabilize systems with known dynamics model however having stochastic
parameters. Essentially, this work proposes two approaches to stabilize a nonlinear system using
both of known system model with inherent uncertainty and stabilize a black box system entirely
using input-output data
Decomposition of Nonlinear Dynamical Systems Using Koopman Gramians
In this paper we propose a new Koopman operator approach to the decomposition
of nonlinear dynamical systems using Koopman Gramians. We introduce the notion
of an input-Koopman operator, and show how input-Koopman operators can be used
to cast a nonlinear system into the classical state-space form, and identify
conditions under which input and state observable functions are well separated.
We then extend an existing method of dynamic mode decomposition for learning
Koopman operators from data known as deep dynamic mode decomposition to systems
with controls or disturbances. We illustrate the accuracy of the method in
learning an input-state separable Koopman operator for an example system, even
when the underlying system exhibits mixed state-input terms. We next introduce
a nonlinear decomposition algorithm, based on Koopman Gramians, that maximizes
internal subsystem observability and disturbance rejection from unwanted noise
from other subsystems. We derive a relaxation based on Koopman Gramians and
multi-way partitioning for the resulting NP-hard decomposition problem. We
lastly illustrate the proposed algorithm with the swing dynamics for an IEEE
39-bus system.Comment: 8 pages, submitted to IEEE 2018 AC
Data-Driven Approximation of Transfer Operators: Naturally Structured Dynamic Mode Decomposition
In this paper, we provide a new algorithm for the finite dimensional
approximation of the linear transfer Koopman and Perron-Frobenius operator from
time series data. We argue that existing approach for the finite dimensional
approximation of these transfer operators such as Dynamic Mode Decomposition
(DMD) and Extended Dynamic Mode Decomposition (EDMD) do not capture two
important properties of these operators, namely positivity and Markov property.
The algorithm we propose in this paper preserve these two properties. We call
the proposed algorithm as naturally structured DMD since it retains the
inherent properties of these operators. Naturally structured DMD algorithm
leads to a better approximation of the steady-state dynamics of the system
regarding computing Koopman and Perron- Frobenius operator eigenfunctions and
eigenvalues. However preserving positivity properties is critical for capturing
the real transient dynamics of the system. This positivity of the transfer
operators and it's finite dimensional approximation also has an important
implication on the application of the transfer operator methods for controller
and estimator design for nonlinear systems from time series data
Koopman operator-based model reduction for switched-system control of PDEs
We present a new framework for optimal and feedback control of PDEs using
Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a
linear but infinite-dimensional operator which describes the dynamics of
observables. A numerical approximation of the Koopman operator therefore yields
a linear system for the observation of an autonomous dynamical system. In our
approach, by introducing a finite number of constant controls, the dynamic
control system is transformed into a set of autonomous systems and the
corresponding optimal control problem into a switching time optimization
problem. This allows us to replace each of these systems by a K-ROM which can
be solved orders of magnitude faster. By this approach, a nonlinear
infinite-dimensional control problem is transformed into a low-dimensional
linear problem. In situations where the Koopman operator can be computed
exactly using Extended Dynamic Mode Decomposition (EDMD), the proposed approach
yields optimal control inputs. Furthermore, a recent convergence result for
EDMD suggests that the approach can be applied to more complex dynamics as
well. To illustrate the results, we consider the 1D Burgers equation and the 2D
Navier--Stokes equations. The numerical experiments show remarkable performance
concerning both solution times and accuracy.Comment: arXiv admin note: text overlap with arXiv:1801.0641
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