Non-linear estimation is easy

Non-linear state estimation and some related topics, like parametric estimation, fault diagnosis, and perturbation attenuation, are tackled here via a new methodology in numerical differentiation. The corresponding basic system theoretic definitions and properties are presented within the framework of differential algebra, which permits to handle system variables and their derivatives of any order. Several academic examples and their computer simulations, with on-line estimations, are illustrating our viewpoint

A symbolic network-based nonlinear theory for dynamical systems observability

EBM and MSB acknowledge the Engineering and Physical Sciences Research Council (EPSRC), grant Ref. EP/I032608/1. ISN acknowledges partial support from the Ministerio de Economía y Competitividad of Spain under project FIS2013-41057-P and from the Group of Research Excelence URJC-Banco de Santander.Peer reviewedPublisher PD

Structural identifiability analysis of nonlinear time delayed systems with generalized frequency response functions

summary:In this paper a novel method is proposed for the structural identifiability analysis of nonlinear time delayed systems. It is assumed that all the nonlinearities are analytic functions and the time delays are constant. We consider the joint structural identifiability of models with respect to the ordinary system parameters and time delays by including delays into a unified parameter set. We employ the Volterra series representation of nonlinear dynamical systems and make use of the frequency domain representations of the Volterra kernels, i. e. the Generalized Frequency Response Functions (GFRFs), in order to test the unique computability of the parameters. The advantage of representing nonlinear systems with their GFRFs is that in the frequency domain representation the time delay parameters appear explicitly in the exponents of complex exponential functions from which they can be easily extracted. Since the GFRFs can be symmetrized to be unique, they provide us with an exhaustive summary of the underlying model structure. We use the GFRFs to derive equations for testing structural identifiability. Unique solution of the composed equations with respect to the parameters provides sufficient conditions for structural identifiability. Our method is illustrated on non-linear dynamical system models of different degrees of non-linearities and multiple time delayed terms. Since Volterra series representation can be applied for input-output models, it is also shown that after differential algebraic elimination of unobserved state variables, the proposed method can be suitable for identifiability analysis of a more general class of non-linear time delayed state space models

A Note on Delay Coordinates for Locally Observable Analytic Systems

In this short note, the problem of locally reconstructing the state of a nonlinear system is studied. To avoid computational difficulties arising in the numerical differentiation of the output, the so-called delay coordinates are considered. The assumptions of analyticity and (local) observability of the system are shown to imply that a family of mappings, induced by the delay coordinates and parameterized by a time delay parameter, gives a local diffeomorphism for generic values of such delay parameter on a certain set. A worked-out example illustrates the result

Interpolation and model reduction of nonlinear systems in the Loewner framework

This thesis studies the problem of interpolation and model order reduction for dynamical systems, with the primary objective being the development of an enhancement of the Loewner framework for general families of nonlinear differential-algebraic systems. First, an interconnection-based interpretation of the Loewner framework for linear time-invariant systems is developed. This interpretation does not rely on frequency domain notions, yielding a natural approach for enhancement of the Loewner framework to more complex systems possessing nonlinear dynamics. Next, the interconnection-based interpretation is used to develop the framework, first for systems of nonlinear ordinary differential equations, then for systems of nonlinear differential-algebraic equations, and interpolants are constructed using the so-called tangential data mappings and Loewner functions. Following this, parameterized families of systems interpolating the tangential data mappings are given. The problem of constructing interpolants from tangential data mappings and Loewner functions is considered in the most general scenario, and a dynamic extension approach to interpolant construction is developed. As a result, all systems matching the tangential data mappings, and having dimension at least as large as that of the auxiliary interpolation systems, are parameterized under mild conditions. Hence, if an interpolant exists while possessing additional desired properties, then it is contained in the dynamically extended family of interpolants. Finally, the use of behaviourally equivalent representations of a system is investigated with the goal of selecting a representation having less stringent conditions guaranteeing the existence of solution to partial differential equations. This is accomplished for a class of semi-explicit nonlinear differential-algebraic systems by making use of the explicit algebraic constraints to simplify the model of the system.Open Acces

Observability and Synchronization of Neuron Models

Observability is the property that enables to distinguish two different locations in $n$-dimensional state space from a reduced number of measured variables, usually just one. In high-dimensional systems it is therefore important to make sure that the variable recorded to perform the analysis conveys good observability of the system dynamics. In the case of networks composed of neuron models, the observability of the network depends nontrivially on the observability of the node dynamics and on the topology of the network. The aim of this paper is twofold. First, a study of observability is conducted using four well-known neuron models by computing three different observability coefficients. This not only clarifies observability properties of the models but also shows the limitations of applicability of each type of coefficients in the context of such models. Second, a multivariate singular spectrum analysis (M-SSA) is performed to detect phase synchronization in networks composed by neuron models. This tool, to the best of the authors' knowledge has not been used in the context of networks of neuron models. It is shown that it is possible to detect phase synchronization i)~without having to measure all the state variables, but only one from each node, and ii)~without having to estimate the phase

Accessibility of Nonlinear Time-Delay Systems

A full characterization of accessibility is provided for nonlinear time-delay systems. It generalizes the rank condition which is known for weak controllability of linear time-delay systems, as well as the celebrated geometric approach for delay-free nonlinear systems and the characterization of their accessibility. Besides, fundamental results are derived on integrability and basis completion which are of major importance for a number of general control problems for nonlinear time-delay systems. They are shown to impact preconceived ideas about canonical forms for nonlinear time-delay systems

Linear Control Theory with an ℋ∞ Optimality Criterion

This expository paper sets out the principal results in ℋ∞ control theory in the context of continuous-time linear systems. The focus is on the mathematical theory rather than computational methods