1,931 research outputs found
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
Network Reconstruction from Intrinsic Noise
This paper considers the problem of inferring an unknown network of dynamical
systems driven by unknown, intrinsic, noise inputs. Equivalently we seek to
identify direct causal dependencies among manifest variables only from
observations of these variables. For linear, time-invariant systems of minimal
order, we characterise under what conditions this problem is well posed. We
first show that if the transfer matrix from the inputs to manifest states is
minimum phase, this problem has a unique solution irrespective of the network
topology. This is equivalent to there being only one valid spectral factor (up
to a choice of signs of the inputs) of the output spectral density.
If the assumption of phase-minimality is relaxed, we show that the problem is
characterised by a single Algebraic Riccati Equation (ARE), of dimension
determined by the number of latent states. The number of solutions to this ARE
is an upper bound on the number of solutions for the network. We give necessary
and sufficient conditions for any two dynamical networks to have equal output
spectral density, which can be used to construct all equivalent networks.
Extensive simulations quantify the number of solutions for a range of problem
sizes. For a slightly simpler case, we also provide an algorithm to construct
all equivalent networks from the output spectral density.Comment: 11 pages, submitted to IEEE Transactions 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
Inverse Problems for Matrix Exponential in System Identification: System Aliasing
This note addresses identification of the -matrix in continuous time
linear dynamical systems on state-space form. If this matrix is partially known
or known to have a sparse structure, such knowledge can be used to simplify the
identification. We begin by introducing some general conditions for solvability
of the inverse problems for matrix exponential. Next, we introduce "system
aliasing" as an issue in the identification of slow sampled systems. Such
aliasing give rise to non-unique matrix logarithms. As we show, by imposing
additional conditions on and prior knowledge about the -matrix, the issue of
system aliasing can, at least partially, be overcome. Under conditions on the
sparsity and the norm of the -matrix, it is identifiable up to a finite
equivalence class.Comment: 7 page
Linear Shape Deformation Models with Local Support Using Graph-based Structured Matrix Factorisation
Representing 3D shape deformations by linear models in high-dimensional space
has many applications in computer vision and medical imaging, such as
shape-based interpolation or segmentation. Commonly, using Principal Components
Analysis a low-dimensional (affine) subspace of the high-dimensional shape
space is determined. However, the resulting factors (the most dominant
eigenvectors of the covariance matrix) have global support, i.e. changing the
coefficient of a single factor deforms the entire shape. In this paper, a
method to obtain deformation factors with local support is presented. The
benefits of such models include better flexibility and interpretability as well
as the possibility of interactively deforming shapes locally. For that, based
on a well-grounded theoretical motivation, we formulate a matrix factorisation
problem employing sparsity and graph-based regularisation terms. We demonstrate
that for brain shapes our method outperforms the state of the art in local
support models with respect to generalisation ability and sparse shape
reconstruction, whereas for human body shapes our method gives more realistic
deformations.Comment: Please cite CVPR 2016 versio
Bayesian variable selection in linear dynamical systems
We develop a method for reconstructing regulatory interconnection networks
between variables evolving according to a linear dynamical system. The work is
motivated by the problem of gene regulatory network inference, that is, finding
causal effects between genes from gene expression time series data. In
biological applications, the typical problem is that the sampling frequency is
low, and consequentially the system identification problem is ill-posed. The
low sampling frequency also makes it impossible to estimate derivatives
directly from the data. We take a Bayesian approach to the problem, as it
offers a natural way to incorporate prior information to deal with the
ill-posedness, through the introduction of sparsity promoting prior for the
underlying dynamics matrix. It also provides a framework for modelling both the
process and measurement noises. We develop Markov Chain Monte Carlo samplers
for the discrete-valued zero-structure of the dynamics matrix, and for the
continuous-time trajectory of the system.Comment: 19 page
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