Continuous-time Linear Model Research Articles

Design of Switching-type Dynamic Quantizers for Continuous-Time Nonlinear Systems

This paper focuses on designing discrete-valued input systems using dynamic quantizers, which convert continuous-valued signals into discrete ones. Although the optimal design of dynamic quantizer is much studied, there are some unresolved problems. This study focuses on one of the unresolved problems: the design of a switching-type discrete-time dynamic quantizer for continuous-time nonlinear systems. In this paper, we first reduced the quantizer design problem to a selection problem of continuous-time linear models for a given nonlinear system. Then, we evaluated this approach via simulations of the swing-up problem for a cart-type inverted pendulum.

Numerical approach to the drift parameter estimation in the model with two fractional Brownian motions

The article deals with numerical estimation of the drift parameter in the continuous-time linear model with two independent fractional Brownian motions. The main focus is given to the computational difficulties of the maximum likelihood approach, in particular, to the construction of the approximate solution to the Fredholm integral equation of the second kind with a singular kernel. We also introduce two alternative estimators and investigate their asymptotic properties. The performance of all estimators is studied numerically.

System Identification Based on Invariant Subspace

This article proposes a novel system identification method based on the notion of invariant subspace. It is shown that when the system input and output asymptotically converge onto an invariant subspace, a new form of regression can be obtained. New identification algorithms are then developed based on the obtained regression. The proposed method has several distinctive advantages originating from both time-domain and frequency-domain approaches. They include: 1) linear continuous-time models can be identified from slowly sampled input/output data; 2) consistency of the model parameters can be established in an error-in-variables framework; 3) the global optimum can be found by solving two linear least-square problems; and 4) the identification algorithms can be implemented online with explicit convergence rates. The theoretic results are tested by numerical examples to show the effectiveness of the proposed method.

Kernel-based identification of asymptotically stable continuous-time linear dynamical systems

ABSTRACT In many engineering applications, continuous-time models are preferred to discrete-time ones, in that they provide good physical insight and can be derived also from non-uniformly sampled data. However, for such models, model selection is a hard task if no prior physical knowledge is given. In this paper, we propose a non-parametric approach to infer a continuous-time linear model from data, by automatically selecting a proper structure of the transfer function and guaranteeing to preserve the system stability properties. By means of benchmark simulation examples, the proposed approach is shown to outperform state-of-the-art continuous-time methods, also in the critical case when short sequences of canonical input signals, like impulses or steps, are used for model learning.

Estimation of spectral functionals for Levy-driven continuous-time linear models with tapered data

The paper is concerned with the nonparametric statistical estimation of linear spectral functionals for Levy-driven continuous-time stationary linear models with tapered data. As an estimator for unknown functional we consider the averaged tapered periodogram. We analyze the bias of the estimator and obtain sufficient conditions assuring the proper rate of convergence of the bias to zero, necessary for asymptotic normality of the estimator. We prove a a central limit theorem for a suitable normalized stochastic process generated by a tapered Toeplitz type quadratic functional of the model. As a consequence of these results we obtain the asymptotic normality of our estimator.

Robust estimation for continuous-time linear models with memory

Robust estimation for continuous-time linear models with memory

CONTSID: a Matlab toolbox for standard and advanced identification of black-box continuous-time models

The CONtinuous-Time System IDentification (CONTSID) toolbox provides Matlab functions for estimating continuous-time black-box models of dynamical systems from measured data without having to fully characterize the mathematics governing the system behavior. The toolbox includes tools for standard identification of linear continuous-time models such as simple process, transfer functions and state-space models. The toolbox also provides algorithms for more advanced identification such as errors-in-variable (EIV) and closed-loop model estimation or to capture nonlinear system dynamics. This paper presents an overview of the main features of the latest release of the CONTSID toolbox and outlines some recent developments for on-line parameter and time-delay system estimation.

Adaptive estimation of continuous-time regression models using high-frequency data

We derive the asymptotic efficiency bound for regular estimates of the slope coefficient in a linear continuous-time regression model for the continuous martingale parts of two Itô semimartingales observed on a fixed time interval with asymptotically shrinking mesh of the observation grid. We further construct an estimator from high-frequency data that achieves this efficiency bound and, indeed, is adaptive to the presence of infinite-dimensional nuisance components. The estimator is formed by taking optimal weighted average of local nonparametric volatility estimates that are constructed over blocks of high-frequency observations. The asymptotic efficiency bound is derived under a Markov assumption for the bivariate process while the high-frequency estimator and its asymptotic properties are derived in a general Itô semimartingale setting. To study the asymptotic behavior of the proposed estimator, we introduce a general spatial localization procedure which extends known results on the estimation of integrated volatility functionals to more general classes of functions of volatility. Empirically relevant numerical examples illustrate that the proposed efficient estimator provides nontrivial improvement over alternatives in the extant literature.

IDENTIFYING RESTRICTIONS FOR FINITE PARAMETER CONTINUOUS TIME MODELS WITH DISCRETE TIME DATA

This paper revisits the question of parameter identification when a linear continuous time model is sampled only at equispaced points in time. Following the framework and assumptions of Phillips (1973), we consider models characterized by first-order, linear systems of stochastic differential equations and use a priori restrictions on the model parameters as identifying restrictions. A practical rank condition is derived to test whether any particular collection of at least $\left\lfloor {n/2} \right\rfloor$ general linear restrictions on the parameter matrix is sufficient for identification. We then consider extensions to incorporate prior restrictions on the covariance matrix of the disturbances, to identify the covariance matrix itself, and to address identification in models with cointegration.

Identifying Restrictions for Finite Parameter Continuous Time Models with Discrete Time Data

This paper revisits the question of parameter identification when a linear continuous time model is sampled only at equispaced points in time. Following the framework and assumptions of Phillips (1973), we consider models characterized by first-order, linear systems of stochastic differential equations and use a priori restrictions on the model parameters as identifying restrictions. A practical rank condition is derived to test whether any particular collection of at least floor(n/2) general linear restrictions on the parameter matrix is sufficient for identification. We then consider extensions to incorporate prior restrictions on the covariance matrix of the disturbances, to identify the covariance matrix itself, and to address identification in models with cointegration.

Consensus for linear multi-agent system with intermittent information transmissions using the time-scale theory

This paper investigates the consensus problem for linear multi-agent system with fixed communication topology in the presence of intermittent communication using the time-scale theory. Since each agent can only obtain relative local information intermittently, the proposed consensus algorithm is based on a discontinuous local interaction rule. The interaction among agents happens at a disjoint set of continuous-time intervals. The closed-loop multi-agent system can be represented using mixed linear continuous-time and linear discrete-time models due to intermittent information transmissions. The time-scale theory provides a powerful tool to combine continuous-time and discrete-time cases and study the consensus protocol under a unified framework. Using this theory, some conditions are derived to achieve exponential consensus under intermittent information transmissions. Simulations are performed to validate the theoretical results.

Version 7.0 of the CONTSID toolbox

The CONtinuous-Time System IDentification (CONTSID) toolbox to be run with Matlab includes estimation routines to determine and evaluate continuous-time models of dynamic systems directly from discrete-time input-output data. Version 7.0 of the CONTSID toolbox is now fully compatible with the latest version of the System Identification toolbox for Matlab. The new release introduces important new options for developing models. It includes additional routines for identifying continuous-time linear models from frequency-domain data, as well as linear systems with arbitrary time delay. The toolbox now supports estimation routines for the determination of simple process models from both regularly and irregularly sampled data and additional algorithms for identifying linear time-varying (LTV) models. This paper presents and illustrates these new features.

Linear theory for filtering nonlinear multiscale systems with model error.

In this paper, we study filtering of multiscale dynamical systems with model error arising from limitations in resolving the smaller scale processes. In particular, the analysis assumes the availability of continuous-time noisy observations of all components of the slow variables. Mathematically, this paper presents new results on higher order asymptotic expansion of the first two moments of a conditional measure. In particular, we are interested in the application of filtering multiscale problems in which the conditional distribution is defined over the slow variables, given noisy observation of the slow variables alone. From the mathematical analysis, we learn that for a continuous time linear model with Gaussian noise, there exists a unique choice of parameters in a linear reduced model for the slow variables which gives the optimal filtering when only the slow variables are observed. Moreover, these parameters simultaneously give the optimal equilibrium statistical estimates of the underlying system, and as a consequence they can be estimated offline from the equilibrium statistics of the true signal. By examining a nonlinear test model, we show that the linear theory extends in this non-Gaussian, nonlinear configuration as long as we know the optimal stochastic parametrization and the correct observation model. However, when the stochastic parametrization model is inappropriate, parameters chosen for good filter performance may give poor equilibrium statistical estimates and vice versa; this finding is based on analytical and numerical results on our nonlinear test model and the two-layer Lorenz-96 model. Finally, even when the correct stochastic ansatz is given, it is imperative to estimate the parameters simultaneously and to account for the nonlinear feedback of the stochastic parameters into the reduced filter estimates. In numerical experiments on the two-layer Lorenz-96 model, we find that the parameters estimated online, as part of a filtering procedure, simultaneously produce accurate filtering and equilibrium statistical prediction. In contrast, an offline estimation technique based on a linear regression, which fits the parameters to a training dataset without using the filter, yields filter estimates which are worse than the observations or even divergent when the slow variables are not fully observed. This finding does not imply that all offline methods are inherently inferior to the online method for nonlinear estimation problems, it only suggests that an ideal estimation technique should estimate all parameters simultaneously whether it is online or offline.

Improved algebraic method for linear continuous-time model identification

Recently a novel algebraic method was proposed for linear continuous-time model identification, which has attracted extensive attention in the literature. This work reveals its connection to classic identification methods, discusses a limitation and presents a useful modification of the method. The discussions are supported by analysis and numerical experiments.

Developments for the CONTSID toolbox

This paper describes the latest developments for the CONtinuous-Time System IDentification (CONTSID) toolbox to be run with Matlab. The toolbox supports time-domain identification methods for estimating continuous-time linear and nonlinear models directly from regularly or irregularly sampled data. It now includes additional routines for identifying continuous-time linear models in closed loop, as well as efficient routines for identifying both LPV and Hammerstein-Wiener continuous-time models.

Using Surrogate Data for Nonlinear Identification: A Case Study

The use of surrogate data for nonlinear identification is illustrated using a case study. It is shown how the surrogate data approach can be used to detect nonlinear features in the experimental data, and how this approach can be used to determine whether a model is insufficient for describing the detected nonlinear predictability. In order to develop a nonlinear model, the structure of an identified continuous time linear model is used. The developed continuous time nonlinear models are then evaluated, and some deemed adequate for capturing the nonlinear predictability of the data.

Continuous-Time Linear Models

I translate familiar concepts of discrete-time time-series to continuous-time equivalent. I cover lag operators, ARMA models, the relation between levels and differences, integration and cointegration, and the Hansen-Sargent prediction formulas.

A Continuous-Time Linear System Identification Method for Slowly Sampled Data

Both direct and indirect methods exist for identifying continuous-time linear systems. A direct method estimates continuous-time input and output signals from their samples and then use them to obtain a continuous-time model, whereas an indirect method estimates a discrete-time model first. Both methods rely on fast sampling to ensure good accuracy. In this paper, we propose a more direct method where a continuous-time linear model is directly fitted to the available samples. This method produces an exact model asymptotically, modulo some possible aliasing ambiguity, even when the sampling rate is relatively slow. We also state conditions under which the aliasing ambiguity can be resolved, and we provide experiments showing that the proposed method is a valid option when a slow sampling frequency must be used but a large number of samples is available.

Identification de modèles à temps continu Méthodes et développements récents

This article aims at presenting an overview of direct parametric continuous-time linear model identification from sampled data. The main approaches for transfer function model estimation are described such as the traditional state-variable filter approach, the optimal instrumental variable method and the output error technique. The CONTSID toolbox that supports most of the techniques developed over the last thirty years is briefly presented and the main advantages of direct data-based modelling are exposed. A few more advanced aspects are then discussed including state-space model identification by subspace techniques, closed-loop and errors-in-variables model identification.

Robust FIR Filters for Linear Continuous-Time State-Space Models With Uncertainties

This letter proposes robust finite impulse response (FIR) filters for linear continuous-time statespace models with bounded uncertainties. A set of all reachable current states under bounded uncertainties is determined from inputs and outputs on a recent finite time interval. If some condition is met, this set is shown to be represented in an ellipsoidal form. The derivation procedure is much simplified by utilizing the result on the optimal tracking control with an indefinite cost function. In order to minimize the maximum estimation error due to uncertainties, the center of the reachable ellipsoidal set is chosen as an estimated state. It is shown through simulation that the proposed robust FIR filter achieves a more robust performance than existing robust infinite impulse response (IIR) filters.