Differential Algebraic Approach Research Articles

Structural identifiability analysis of linear reaction–advection–diffusion processes in mathematical biology

Effective application of mathematical models to interpret biological data and make accurate predictions often requires that model parameters are identifiable. Approaches to assess the so-called structural identifiability of models are well established for ordinary differential equation models, yet there are no commonly adopted approaches that can be applied to assess the structural identifiability of the partial differential equation (PDE) models that are requisite to capture spatial features inherent to many phenomena. The differential algebra approach to structural identifiability has recently been demonstrated to be applicable to several specific PDE models. In this brief article, we present general methodology for performing structural identifiability analysis on partially observed reaction–advection–diffusion PDE models that are linear in the unobserved quantities. We show that the differential algebra approach can always, in theory, be applied to such models. Moreover, despite the perceived complexity introduced by the addition of advection and diffusion terms, consideration of spatial analogues of non-spatial models cannot exacerbate structural identifiability. We conclude by discussing future possibilities and the computational cost of performing structural identifiability analysis on more general PDE models.

Hybrid robust fault detection and isolation of satellite reaction wheel actuators

In this paper, a combined robust fault detection and isolation scheme is studied for satellite system subject to actuator faults, external disturbances, and parametric uncertainties. The proposed methodology incorporates a residual generation module, including a bank of filters, into an intelligent residual evaluation module. First, residual filters are designed based on an improved nonlinear differential algebraic approach so that they are not affected by external disturbances. The residual evaluation module is developed based on the suggested series and parallel forms. Further, a new ensemble classification scheme defined as blended learning integrates heterogeneous classifiers to enhance the performance. A wide range of simulations is carried out in a high-fidelity satellite simulator subject to the constant and time-varying actuator faults in the presence of disturbances, manoeuvres, uncertainties, and noises. The obtained results demonstrate the effectiveness of the proposed robust fault detection and isolation method compared to the traditional nonlinear differential algebraic approach.

A differential algebra method for aberration analysis of variable-axis lens system

A variable-axis lens (VAL) is an effective method for achieving a large field of view with high resolution because it can preserve small aberrations by shifting the optical axis according to the deflection of the electron beam, even when the beam is deflected to regions far from the central straight optical axis. However, for systems consisting of such a VAL, conventional methods for calculating aberration coefficients that refer to the central straight optical axis could be meaningless. Here, we propose a novel approach, based on the differential algebraic(DA) method, for calculating high-order aberration coefficients for VAL systems. Instead of using the straight optical axis as a reference, we trace only one electron trajectory in such a system. Then, the linear optical properties and the geometric aberration and chromatic aberration coefficients up to 5th rank are calculated (with high accuracy) and analyzed with reference to the reference trajectory. In comparison with conventional methods, the proposed DA approach shows overwhelming advantages in having analyzed the dispersion, focus error and 2nd rank aberration coefficients. A new software package was developed, and an example was shown to demonstrate the proposed approach.

Dynamical distributed control and synchronization

It is well known that most of real-world phenomena are described by partial differential equations. Nevertheless, for control design purposes it is very common to approximate them with a set of ordinary differential equations, since conventional design methods, such as calculus of variations or differential geometry, turn out to be very complex for this class of systems. However, by doing this, valuable properties are lost. In this work, we present a dynamical distributed control for nonlinear partial differential equation systems and we focus on solving the Generalized Synchronization problem, since this topic has multiple applications in the disciplines of engineering, biology, physics, etc. For the design of the control, we utilize a differential algebraic approach. The key ingredient of our design method is to find a canonical form of the given systems by means of the so-called partial differential primitive element. This representation is known as Generalized Observability Canonical Form and allows us to design a dynamical distributed control in a natural way. Additionally, to avoid a functional analysis for the stability of the resultant synchronization error, we propose to utilize tools from semi-group and spectral theory of infinite dimensional systems in a Hilbert space. As a result, we present a design approach less complex and, therefore, more accessible than most common design methods. Besides, with the proposed stability analysis, we obtain an easy criterion to select the control gains; hence, we can solve the generalized synchronization problem of partial differential equation systems in a simple way. To validate the effectiveness of the proposed control, we present two examples of generalized synchronization for reaction-diffusion systems and show their respective numerical results.

Deformations and Homotopy of Rota–Baxter Operators and $$\mathcal{O} - $$Operators on Lie Algebras

This article gives a brief introduction to some recent work on deformation and homotopy theories of Rota-Baxter operators and more generally $\mathcal{O}$-operators on Lie algebras, by means of the differential graded Lie algebra approach. It is further shown that these theories lift the existing connection between $\mathcal{O}$-operators and pre-Lie algebras to the levels of deformations and homotopy.

On sensor selection for differential algebraic systems observability

Abstract We address the problem of selecting sensors, that is, output equations, in order to endow dynamic equations of a system with some properties. Among all such desirable properties is the basic one of observability and/or identifiability. Once such a problem is solved one may ask how to choose sensors in order to improve estimation algorithms in terms of reliability, robustness, or simply, low complexity. First, what is the minimal number of sensors that make the dynamics observable? Second, when the sensors are bound to measure state components, what is their minimum number? Third, how may the observability margin be improved by selecting the sensors? In this communication we provide an overview of these questions in the differential algebraic approach of observation problems.

Orbit propagation in irregular and uncertain gravity field using differential algebra

Uncertainty propagation has been addressed extensively in space missions around the Earth, but much less for missions around small solar system bodies. Small bodies usually have irregular and weak gravity and our knowledge of their gravity, rotation speed and surrounding space environment is largely uncertain. These characteristics make the orbit propagation around these small bodies a challenging task.Focusing on the uncertainty of the small body's gravity, this paper applies the differential algebra (DA) technique to study the orbit propagation problem, and addresses its efficiency for a given the required accuracy. Different from traditional studies that focus on the uncertainty of the initial state, this study assumes an exact initial state and studies the influences that gravity model uncertainties have on the orbit. Taking the asteroid Steins as an example, the accuracy and the efficiency of the DA approach are firstly validated by comparison with the traditional Monte Carlo method. Then, the effects of gravity uncertainties on different types of orbits (prograde, retrograde and polar) are studied. The retrograde motion is found to be more robust to the gravity uncertainty than the prograde ones. For near polar orbits, the impact of gravity uncertainty on orbital motion depends significantly on the initial position, and it reaches the maximum if the initial position is near the polar regions. Moreover, short-term effects are found to play an important role in orbit deviation as a result of the gravity uncertainty. These discoveries can help mission designers assess the posed risk and design appropriate mission orbits.

Phenotypic variations in persistence and infectivity between and within environmentally transmitted pathogen populations impact population-level epidemic dynamics

BackgroundHuman pathogens transmitted through environmental pathways are subject to stress and pressures outside of the host. These pressures may cause pathogen pathovars to diverge in their environmental persistence and their infectivity on an evolutionary time-scale. On a shorter time-scale, a single-genotype pathogen population may display wide variation in persistence times and exhibit biphasic decay.MethodsWe use a transmission modeling framework to develop an infectious disease model with biphasic pathogen decay. We take a differential algebra approach to assessing model identifiability, calculate basic reproduction numbers by the next generation method, and use simulation to explore model dynamics.ResultsFor both long and short time-scales, we demonstrate that epidemic-potential-preserving trade-offs have implications for epidemic dynamics: less infectious, more persistent pathogens cause epidemics to progress more slowly than more infectious, less persistent (labile) pathogens, even when the overall risk is the same. Using identifiability analysis, we show that the usual disease surveillance data does not sufficiently inform these underlying pathogen population dynamics, even when combined with basic environmental monitoring data. However, risk could be indirectly ascertained by developing methods to separately monitor labile and persistent subpopulations. Alternatively, determining the relative infectivity of persistent pathogen subpopulations and the rates of phenotypic conversion will help ascertain how much disease risk is associated with the long tails of biphasic decay.ConclusionA better understanding of persistence–infectivity trade-offs and associated dynamics can improve our ecological understanding of environmentally transmitted pathogens, as well as our risk assessment and disease control strategies.

Structural and practical identifiability analysis of outbreak models

Estimating the reproduction number of an emerging infectious disease from an epidemiological data is becoming more essential in evaluating the current status of an outbreak. However, these studies are lacking the fundamental prerequisite in parameter estimation problem, namely the structural identifiability of the epidemic model, which determines the possibility of uniquely determining the model parameters from the epidemic data. In this paper, we perform both structural and practical identifiability analysis to classical epidemic models such as SIR (Susceptible-Infected-Recovered), SEIR (Susceptible-Exposed-Infected-Recovered) and an epidemic model with the treatment class (SITR). We performed structural identifiability analysis on these epidemic models using a differential algebra approach to investigate the well-posedness of the parameter estimation problem. Parameters of these models are estimated from different data types, namely prevalence, cumulative incidences and treated individuals. Furthermore, we carried out practical identifiability analysis on these models using Monte Carlo simulations and Fisher’s Information Matrix. Our study shows that the SIR model is both structurally and practically identifiable from the prevalence data. It is also structurally identifiable to cumulative incidence observations, but due to high correlations of the parameters, it is practically unidentifiable from the cumulative incidence data. Furthermore, we found that none of these simple epidemic models are practically identifiable from the cumulative incidence data which is the standard type of epidemiological data provided by CDC or WHO. Our analysis with simple SIR model suggest that the health agencies, if possible, should report prevalence rather than incidence data.

Differential-Algebraic numerical approach to the one-dimensional Drift-Flux Model applied to a multicomponent hydrocarbon two-phase flow

Abstract This paper presents a numerical investigation of the solution of the steady-state one-dimensional Drift-Flux Model. It is proposed that these simulations, though often based on finite-volume discretizations and iterative sequential procedures, are preferably performed using established numerical methods specifically devised for Differential-Algebraic Equations (DAE) systems. Both strategies were implemented in a computer code developed for simulations of multicomponent hydrocarbon two-phase flows. The SIMPLER semi-implicit algorithm was employed in the solution of the finite-volume discretized model in order to provide comparison grounds with the adaptive BDF-implementation of DAE integration package DASSLC. Based on test simulations of a naphtha two-phase flow under varying heat-transfer conditions, the DAE approach was proved highly advantageous in terms of computational requirements and accuracy of results, both in the absence and presence of flow-pattern transitions. Numerical difficulties arising from the latter were successfully worked around by continuously switching regime-specific constitutive correlations using adjustable steep regularization functions.

Modeling Biphasic Environmental Decay of Pathogens and Implications for Risk Analysis

As the appreciation for the importance of the environment in infectious disease transmission has grown, so too has interest in pathogen fate and transport. Fate has been traditionally described by simple exponential decay, but there is increasing recognition that some pathogens demonstrate a biphasic pattern of decay—fast followed by slow. While many have attributed this behavior to population heterogeneity, we demonstrate that biphasic dynamics can arise through a number of plausible mechanisms. We examine the identifiability of a general model encompassing three such mechanisms: population heterogeneity, hardening off, and the existence of viable-but-not-culturable states. Although the models are not fully identifiable from longitudinal sampling studies of pathogen concentrations, we use a differential algebra approach to determine identifiable parameter combinations. Through case studies using Cryptosporidium and Escherichia coli, we show that failure to consider biphasic pathogen dynamics can lead to substantial under- or overestimation of disease risks and pathogen concentrations, depending on the context. More reliable models for environmental hazards and human health risks are possible with an improved understanding of the conditions in which biphasic die-off is expected. Understanding the mechanisms of pathogen decay will ultimately enhance our control efforts to mitigate exposure to environmental contamination.

Comparing numerically exact and modelled static friction

Currently there exists no mechanically consistent “numerically exact” implementation of static and dynamic Coulomb friction for general soft particle simulations with arbitrary contact situations in two or three dimension, but only along one dimension. We outline a differential-algebraic equation approach for a “numerically exact” computation of friction in two dimensions and compare its application to the Cundall-Strack model in some test cases.

Poisson brackets, Novikov-Leibniz structures and integrable Riemann hydrodynamic systems

A general differential-algebraic approach is devised for constructing multi-component Hamiltonian operators as differentiations on suitably constructed loop Lie algebras. The related Novikov-Leibniz algebraic structures are presented and a new non-associative “Riemann” algebra is constructed, which is closely related to the infinite multi-component Riemann integrable hierarchies. A close relationship to the standard symplectic analysis techniques is also discussed.

On-line estimation of VFA concentration in anaerobic digestion via methane outflow rate measurements

Abstract This paper deals with the design of a robust nonlinear observer as a software sensor to achieve the on-line estimation of the concentration of Volatile Fatty Acids (VFA) in a class of continuous anaerobic digesters (AD). Taking into account the limited availability of on-line sensors for AD process, in this contribution is assumed that only the methane outflow rate is available for on-line measurement. The estimation method is based on a modified version for a two-dimensional mathematical model of AD process. From the differential algebraic observability approach it is shown that the VFA concentration is detectable from the methane outflow rate measurements. The observer convergence is analyzed by using Lyapunov stability techniques. Numerical simulations illustrate the effectiveness of the proposed estimation method for a four-dimensional AD model with uncertainties associated with unmodeled dynamics and disturbances in the inflow composition.

A differential algebraic approach for the modeling of polycrystalline ferromagnetic hysteresis with minor loops and frequency dependence

In the current paper, a nonlinear differential algebraic approach is proposed for the modeling of hysteretic dynamics of polycrystalline ferromagnetic materials. The model is constructed by employing a phenomenological theory to the magnetization orientation switching. For the modeling of hysteresis in polycrystalline ferromagnetic materials, the single crystal model is applied to each magnetic domain along its own principal axis. The overall dynamics of the polycrystalline materials is obtained by taking a weighted combination of the dynamics of all magnetic domains. The weight function for the combination is taken as the distribution function of the principal axes. Numerical simulations are performed and comparisons with its experimental counterparts are presented. The hysteretic dynamics caused by orientation switching processes is accurately captured by the proposed model. Minor hysteresis loops associated with partial-amplitude loadings are also captured. Rate dependence of the hysteresis loops are inherently incorporated into the model due to its differential nature.

LPV/H∞ suspension robust control adaption of the dynamical lateral load transfers based on a differential algebraic estimation approach

This paper is concerned with a new global chassis strategy combining the LPV/H∞ control framework and the differential algebraic estimation approach. The main objective is to enhance the vehicle performances by adapting its control to the dynamical lateral load transfers using a very efficient algebraic dynamical behaviour estimation strategy. Indeed, the lateral load transfers influence considerably the vehicle dynamical behaviour, stability and safety especially in dangerous driving situations. It is important to emphasize that the dynamical load transfers are different from the static ones generated mainly by the bank of the road. The computation of these dynamics must be based on the effective lateral acceleration and roll behaviour of the car. Such effective data cannot be given directly by the hardware sensors (which give correlated measures).The information on the real dynamical lateral load transfers is very important to ensure a good adaptation of the vehicle control and performances to the considered driving situation. A very interesting differential algebraic estimation approach allows to provide the effective needed measures for the control strategy using only sensors available on most of commercial cars. It is based on the differential flatness property of nonlinear systems in an algebraic context.Then, thanks to this estimation approach, the dynamical lateral load transfers can be calculated and used to adapt the vertical performances of the vehicle using the LPV/H∞ for suspension systems control. Simulations performed on non linear vehicle models with data collected on a real car are used to validate the proposed estimation and control approaches. Results show the efficiency of this vehicle control strategy.

Closure to Discussion on “A Differential Algebraic Estimator for Sensorless Permanent-Magnet Synchronous Machine Drive”

This paper is a response to the comments on the paper “A Differential Algebraic Estimator for Sensorless Permanent-Magnet Synchronous Machine Drive”. The authors clarifies some points regarding the derivation of their position estimators and the extensibility of the differential algebraic approach to salient permanent magnet synchronous motor (PMSM).

Integrability Analysis of a Two-Component Burgers-Type Hierarchy

The Lax integrability of a two-component polynomial Burgers-type dynamical system is analyzed by using a differential-algebraic approach. Its linear adjoint matrix Lax representation is constructed. A related recursive operator and an infinite hierarchy of nonlinear Lax integrable dynamical systems of the Burgers–Korteweg–de-Vries type are obtained by the gradient-holonomic technique. The corresponding Lax representations are presented.

A Differential Algebraic Estimator for Sensorless Permanent-Magnet Synchronous Machine Drive

This paper describes a sensorless control for a high-performance permanent-magnet synchronous machine (PMSM) drive. A differential algebraic approach is used to estimate the position (and speed). The relationship thus obtained for analytical estimation of the position is based on the control voltages, the measured currents, and their derivatives. Unlike the other estimators like the extended Kalman filter (EKF) that is complex and computationally expensive, the differential algebraic estimator is simple with a 1-D model. Global convergence of the estimator is demonstrated, and experimental results show the effectiveness of the method and its robustness to current sensor fault.

Integrability of and differential–algebraic structures for spatially 1D hydrodynamical systems of Riemann type

A differential–algebraic approach to studying the Lax integrability of a generalized Riemann type hydrodynamic hierarchy is revisited and a new Lax representation is constructed. The related bi-Hamiltonian integrability and compatible Poissonian structures of this hierarchy are also investigated using gradient-holonomic and geometric methods.The complete integrability of a new generalized Riemann hydrodynamic system is studied via a novel combination of symplectic and differential–algebraic tools. A compatible pair of polynomial Poissonian structures, a Lax representation and a related infinite hierarchy of conservation laws are obtained.In addition, the differential–algebraic approach is used to prove the complete Lax integrability of the generalized Ostrovsky–Vakhnenko and a new Burgers type system, and special cases are studied using symplectic and gradient-holonomic tools. Compatible pairs of polynomial Poissonian structures, matrix Lax representations and infinite hierarchies of conservation laws are derived.