Selection of outputs for distributed parameter systems by identifiability analysis in the time-scale domain

A method of sensor location selection is introduced for distributed parameter systems. In this method, the sensitivities of spatial outputs to model parameters are computed by a model and transformed via continuous wavelet transforms into the time-scale domain to characterise the shape attributes of output sensitivities and accentuate their differences. Regions are then sought in the time-scale plane wherein the wavelet coefficient of an output sensitivity surpasses all the others’ as indication of the output sensitivity’s distinctness. This yields a comprehensive account of identifiability each output provides to the model parameters as the basis of output selection. The proposed output selection strategy is demonstrated for a numerical case of pollutant dispersion by advection and diffusion in a two-dimensional area.     

Consistent recursive parameter estimation of partial differential equation models

This article introduces a new residual-based recursive parameter estimation algorithm for linear partial differential equations. The main idea is to replace the unmeasurable noise variables by estimates of the noise and to compute recursively both the model parameters and the noise estimates. It is proven that under some mild assumptions the estimated parameters converge to the true values with probability one. Numerical examples that demonstrate the effectiveness of the proposed approach are also provided.     

Indistinguishability and identifiability of kinetic models for the MurC reaction in peptidoglycan biosynthesis

An important question in Systems Biology is the design of experiments that enable discrimination between two (or more) competing chemical pathway models or biological mechanisms. In this paper analysis is performed between two different models describing the kinetic mechanism of a three-substrate three-product reaction, namely the MurC reaction in the cytoplasmic phase of peptidoglycan biosynthesis. One model involves ordered substrate binding and ordered release of the three products; the competing model also assumes ordered substrate binding, but with fast release of the three products. The two versions are shown to be distinguishable; however, if standard quasi-steady-state assumptions are made distinguishability cannot be determined. Once model structure uniqueness is ensured the experimenter must determine if it is possible to successfully recover rate constant values given the experiment observations, a process known as structural identifiability. Structural identifiability analysis is carried out for both models to determine which of the unknown reaction parameters can be determined uniquely, or otherwise, from the ideal system outputs. This structural analysis forms an integrated step towards the modelling of the full pathway of the cytoplasmic phase of peptidoglycan biosynthesis.     

Multi-output parameter estimation of dynamic systems by output shapes

A multi-output method of parameter estimation is introduced for dynamic systems that relies on the shape attributes of model outputs. The shapes of outputs in this method are represented by the surfaces that are generated by continuous wavelet transforms (CWTs) of the outputs in the time-scale domain. Since the CWTs also enhance the delineation of outputs and their sensitivities to model parameters in the time-scale domain, regions in the time-scale plane can be identified wherein the sensitivity of the output with respect to one model parameter dominates all the others. This allows approximation of the prediction error in terms of individual model parameters in isolated regions of the time-scale domain, thus enabling parameter estimation based on a small set of wavelet coefficients. These isolated regions of the time-scale plane also reveal numerous transparencies to be exploited for parameter estimation. It is shown that by taking advantage of these transparencies, the robustness of parameter estimation can be improved. The results also indicate the potential for improved precision and faster convergence of the parameter estimates when shape attributes are used in place of the magnitude.     

Structural identifiability and observability of compartmental models of the COVID-19 pandemic

The recent coronavirus disease (COVID-19) outbreak has dramatically increased the public awareness and appreciation of the utility of dynamic models. At the same time, the dissemination of contradictory model predictions has highlighted their limitations. If some parameters and/or state variables of a model cannot be determined from output measurements, its ability to yield correct insights – as well as the possibility of controlling the system – may be compromised. Epidemic dynamics are commonly analysed using compartmental models, and many variations of such models have been used for analysing and predicting the evolution of the COVID-19 pandemic. In this paper we survey the different models proposed in the literature, assembling a list of 36 model structures and assessing their ability to provide reliable information. We address the problem using the control theoretic concepts of structural identifiability and observability. Since some parameters can vary during the course of an epidemic, we consider both the constant and time-varying parameter assumptions. We analyse the structural identifiability and observability of all of the models, considering all plausible choices of outputs and time-varying parameters, which leads us to analyse 255 different model versions. We classify the models according to their structural identifiability and observability under the different assumptions and discuss the implications of the results. We also illustrate with an example several alternative ways of remedying the lack of observability of a model. Our analyses provide guidelines for choosing the most informative model for each purpose, taking into account the available knowledge and measurements.     

Trends in identification

The paper refers to methods used for identification of linear and nonlinear systems. Deterministic and stochastic approaches are distinguished and specific features concerning parameters, structure and state estimation are briefly discussed from the point of view of possible advantages and difficulties for identification. Attention is paid to different final goals of identification with respect to the convenience of the methods in question. The most important trends in identification approaches are argued by unsolved problems of identification, by the complexity of numerical calculations and of practical applications. The significance of the uncertainty in structure, parameters or noise and the possible application of the a priori knowledge of the analysed system are taken into consideration.     

An instrumental variable method for model order identification

The paper describes a simple instrumental variable method for identifying the structure of a wide class of time-series models. The method is aimed at providing a parametrically efficient (parsimonious) model structure which will lead to a combination of low residual error variance, i.e. a good explanation of the data, and low parametric estimation error variance (as measured by some norm associated with the covariance matrix of the estimation errors). It can be applied to single input-single output and multivariable systems using either discrete or continuous-time series models. It can also function as a recursive (on-line) test for reduction in model order.     

Identification of finite dimensional models of infinite dimensional dynamical systems

The identification of finite dimensional discrete-time models of deterministic linear and nonlinear infinite dimensional systems from pointwise observations is investigated. The input and output observations are used to construct finite dimensional approximations of the solution and the forcing function which are expanded in terms of a finite element basis. An algorithm to determine a minimal basis to approximate the data is introduced. Subsequently, the resulting coordinate vectors are used to identify a finite dimensional discrete-time model. Theoretical results concerning the existence, stability and convergence of the finite dimensional representation are established. Numerical results involving identification of finite dimensional models for both linear and nonlinear infinite dimensional systems are presented.     

Parameter estimation for continuous-time models—A survey

The paper reviews the progress of research on parameter estimation for continuous-time models of dynamic systems over the period 1958–1980. Major developments are considered in historical order and within a classification system which conforms as closely as possible to that which has arisen naturally over the past two decades. While every attempt is made to consider progress in other related scientific disciplines, such as econometrics, the major accent is on research in the control and systems field.     

The extended Kalman filter as a pulmonary blood flow estiṁator

Pulmonary blood flow is represented as a time-varying parameter in a simple model of the bodily uptake of gases that are soluble in blood. The pulmonary blood flow is estimated on-line using Ljung's modification of the extended Kalman filter. The experimental technique for estimation of pulmonary blood flow is based on Zwart's soluble gas method, modified to permit time-varying ventilation and pulmonary perfusion, and also to permit binary dynamic forcing of the inhaled soluble gas. Computer simulations were performed to determine the convergence time and to choose values of algorithmic parameters. Experiments done in humans gave agreement to within the accuracy of that measured invasively by a thermal dilution technique. The method shows promise for use in many biomedical applications, including patient monitoring, physiological experimentation, and clinical heart function tests.     

Identification of partial differential equation models for a class of multiscale spatio-temporal dynamical systems

In this article, the identification of a class of multiscale spatio-temporal dynamical systems, which incorporate multiple spatial scales, from observations is studied. The proposed approach is a combination of Adams integration and an orthogonal least squares algorithm, in which the multiscale operators are expanded, using polynomials as basis functions, and the spatial derivatives are estimated by finite difference methods. The coefficients of the polynomials can vary with respect to the space domain to represent the feature of multiple scales involved in the system dynamics and are approximated using a B-spline wavelet multi-resolution analysis. The resulting identified models of the spatio-temporal evolution form a system of partial differential equations with different spatial scales. Examples are provided to demonstrate the efficiency of the proposed method.     

Local identifiability of time-invariant linear systems by periodic test signals

This paper is concerned with the parameter local identifiability of dynamic systems operating in periodic regimes. The systems dealt with are time-invariant and discrete-time linear systems, with an output white gaussian disturbance. Sets of frequencies are characterized such that the parameter vector is identifiable if and only if the spectrum of the periodic test signal does not vanish over one of these sets. The analysis is developed by taking also into consideration measurement schedule sets which are scattered along the time axis.     

Series solution of nonlinear differential equations by a novel extension of the Laplace transform method

A technique for extending the Laplace transform method to solve nonlinear differential equations is presented. By developing several theorems, which incorporate the Adomian polynomials, the Laplace transformation of nonlinear expressions is made possible. A number of well-known nonlinear equations including the Riccati equation, Clairaut's equation, the Blasius equation and several other ones involving nonlinearities of various types such as exponential and sinusoidal are solved for illustration. The proposed approach is analytical, accurate, and free of integration.     

Frequency-domain identification of continuous-time ARMA models from sampled data

The subject of this paper is the direct identification of continuous-time autoregressive moving average (CARMA) models. The topic is viewed from the frequency domain perspective which then turns the reconstruction of the continuous-time power spectral density (CT-PSD) into a key issue. The first part of the paper therefore concerns the approximate estimation of the CT-PSD from uniformly sampled data under the assumption that the model has a certain relative degree. The approach has its point of origin in the frequency domain Whittle likelihood estimator. The discrete- or continuous-time spectral densities are estimated from equidistant samples of the output. For low sampling rates the discrete-time spectral density is modeled directly by its continuous-time spectral density using the Poisson summation formula. In the case of rapid sampling the continuous-time spectral density is estimated directly by modifying its discrete-time counterpart.     

有色噪声干扰下多变量系统的辅助模型辨识方法

针对有色噪声干扰的多输入多输出系统模型辨识问题,提出了多变量系统的FIR辅助模型辨识方法;其辨识思想是把传递函数矩阵中的子模型等价成辅助模型—有限脉冲相应(FIR)模型,然后利用辨识得到的辅助模型估计输出向量的子子模型,最后利用递推最小二乘算法或帕德近似化方法得到子子模型的参数估计,并通过仿真来验证算法的性能。     

Adaptive state observation of linear time-varying systems with delayed measurements and unknown parameters

In this article, we address the problem of adaptive state observation of linear time-varying systems with delayed measurements and unknown parameters. Our new developments extend the results reported in our recently works. The case with known parameters has been studied by many researchers. However in this article we show that the generalized parameter estimation-based observer design provides a very simple solution for the unknown parameter case. Moreover, when this observer design technique is combined with the dynamic regressor extension and mixing estimation procedure the estimated state and parameters converge in fixed-time imposing extremely weak excitation assumptions.     

Maximum likelihood and prediction error methods

The basic ideas behind the parameter estimation methods are discussed in a general setting. The application to estimation or parameters in dynamical systems is treated in detail using the prototype problem of estimating parameters in a continuous time system using discrete time measurements. Computational aspects are discussed. Theoretical results in consistency, asymptotic normality and efficiency are covered. Model validation and selection of model structures are discussed. An example is given which illustrates some properties of the methods and shows the usefulness of interactive computing. Additional examples illustrate what happens when the data has different artefacts.     

Mathematical model of heart rate regulation during exercise

A model of the heart rate regulation during physical load and recovery has been developed. The model contains four parameters with the following physiological meanings: Heart rate increment related to the load of 1 watt, a load which can be coped with by vagal inhibition, a time constant of the vagal component and a time constant of the slower complex neurohumoral sympathetic component. All these parameters were estimated uniquely with relatively small reliability intervals for a group of 15 healthy men. The results are useful practically for the evaluation of the physical efficiency of individuals and whole groups of healthy subjects.     

A method for parameter estimation of a class of non-linear distributed systems under noisy observations

A method is presented for estimating an unknown parameter of a distributed parameter system which depends on the system state. The system considered is modelled by a class of non-linear partial differential equations of a parabolic type. Noisy observations are assumed to be taken through an arbitrary number of sensors allocated on the spatial region. First, the explicit form of the stationary solution of the state equation is discussed. Second, use is made of the maximum likelihood approach to obtain the optimal estimate of the unknown parameter. Consistency properties w.p.1 of the optimal estimate obtained are also shown. Finally, results of digital simulation experiments are included to support the theoretical aspects.