System Of Nonlinear Differential-algebraic Equations Research Articles

A space of filters for the consistent initialization of differential-algebraic equations in optimization tasks

In this article, a new approach to determining consistent initial conditions for an optimization task with differential-algebraic equations (DAEs) is presented. The work is motivated by the fact that initial conditions have a significant impact on both the obtained state trajectories, as well as the progress of the optimization procedure. In the first step of the designed approach, an α-parametrization of the DAE model takes place. This gives the possibility of influencing the dynamics of the system, and in particular its total reduction. When the initial values of the differential state variables are given, then the solution of the DAE system depends only on the algebraic state variables and the input function, which should have the character of the consistent initial conditions. To obtain the consistent initial conditions and, at the same time, to optimize the considered process, in the second step of the procedure a filter space is defined and applied. The filter space is an innovative extension of Fletcher's filter method. It is built from a finite number of filters, and the dimension of the filter space depends on the input function parametrization approach. As a result of using the filter space, it is possible to obtain a reasonable initial solution to the whole optimization problem, which can be improved by available exact local algorithms. The operation of the new method is tested in the heat and mass transfer optimization task, which is described by a system of nonlinear differential-algebraic equations.

Index-Aware Model-Order Reduction for a Special Class of Nonlinear Differential-Algebraic Equations

We extend the index-aware model-order reduction method to systems of nonlinear differential-algebraic equations with a special nonlinear term mathbf{f}(mathbf{E}mathbf{x}), where mathbf{E} is a singular matrix. Such nonlinear differential-algebraic equations arise, for example, in the spatial discretization of the gas flow in pipeline networks. In practice, mathematical models of real-life processes pose challenges when used in numerical simulations, due to complexity and system size. Model-order reduction aims to eliminate this problem by generating reduced-order models that have lower computational cost to simulate, yet accurately represent the original large-scale system behavior. However, direct reduction and simulation of nonlinear differential-algebraic equations is difficult due to hidden constraints which affect the choice of numerical integration methods and model-order reduction techniques. We propose an extension of index-aware model-order reduction methods to a special class of nonlinear differential-algebraic equations without any kind of linearization. The proposed model-order reduction approach involves automatic decoupling of nonlinear differential-algebraic equations into nonlinear ordinary differential equations and algebraic equations, based on the decoupling of the linear differential equations obtained by ignoring the nonlinear term, thanks to an additional structural condition. This allows applying standard model-order reduction techniques to both parts without worrying about the index. The same procedure can also be used to simulate nonlinear differential-algebraic equations by standard integration schemes.

Extended Kalman Filter for Index-2 Nonlinear Differential Algebraic Equation Systems

Extended Kalman Filter for Index-2 Nonlinear Differential Algebraic Equation Systems

A model for discrete fracture-clay rock interaction incorporating electrostatic effects on transport

A model based on the code CrunchClay is presented for a fracture-clay matrix system that takes electrostatic effects on transport into account. The electrostatic effects on transport include those associated with the development of a diffusion potential as captured by the Nernst-Planck equation, and the formation of a diffuse layer bordering negatively charged clay particles within which partial anion exclusion occurs. The model is based on a dual continuum formulation that accounts for diffuse layer and bulk water pore space, providing a more flexible framework than is found in the classical mean electrostatic potential models. The diffuse layer model is obtained by volume averaging ion concentrations in the Poisson-Boltzmann equation, but also includes the treatment of longitudinal transport within this continuum. The calculation of transport within the bulk and diffuse layer porosity is based on a new formulation for the Nernst-Planck equation that considers averaging of diffusion coefficients and accumulation factors at grid cell interfaces. Equations for function residuals and the associated Jacobian matrix are presented such that the system of nonlinear differential-algebraic equations can be solved with Newton’s method. As an example, we consider a 2D system with a single discrete fracture within which flow and advective transport occurs that is coupled to diffusion in the clay-rich matrix. The simulation results demonstrate the lack of retardation for anions (e.g., 36Cl−) of the contaminant plume within the fracture flow system because they are largely excluded from the charged clay rock, while the migration of cations (e.g., 90Sr++) is more strongly attenuated. The diffusive loss of divalent cations in particular from the fracture is accentuated by their accumulation in the diffuse layer within the clay-rich matrix.

Lyapunov-based economic model predictive control for nonlinear descriptor systems

A large number of physical applications such as electrical circuits, chemical processes and multibody systems can be accurately described by a set of differential algebraic equations (DAEs). For instance, sustainable water desalination techniques such as membrane distillation (MD) is an example of a chemical process that is described by DAEs. Optimal control strategies such as economic model predictive control (EMPC) has many operational advantages (e.g., maximizing process economics and enhancing operational safety) for chemical applications. However, constructing EMPC schemes for a general class of nonlinear DAEs systems while ensuring closed-loop stability and recursive feasibility is an open research challenge that has not been considered. Motivated by the above considerations, this note introduces a Lyapunov-based economic model predictive control (LEMPC) design that can economically operate nonlinear descriptor systems while satisfying input and states constraints. To guarantee closed-loop stability and recursive feasibility of the proposed control design, a Lyapunov-based control law is introduced to characterize the stability region at which the LEMPC can maximize the process economics. Finally, a chemical batch process modeled by nonlinear differential algebraic equations (DAEs) is utilized to demonstrate the applicability of the proposed framework.

Contraction Analysis of Nonlinear DAE Systems

This article studies the contraction properties of nonlinear differential-algebraic equation (DAE) systems. Such systems typically appear as a singular perturbation reduction of a multiple-time-scale differential system. In addition, a given DAE may result from the reduction of many “synthetic” differential systems. We show that an important property of a contracting DAE system is that the reduced system always contracts faster than any synthetic counterpart. At the same time, there always exists a synthetic system, whose contraction rate is arbitrarily close to that of the DAE. Synthetic systems are useful for the analysis of attraction basins of nonlinear DAE systems. As any rational DAE system can be represented in quadratic form, the Jacobian of the synthetic system can be made affine in the system variables. This allows for scalable techniques to construct attraction basin approximations, based on uniformly negative matrix measure conditions for the synthetic system Jacobian.

Modified multiple scale technique for the stability of the fractional delayed nonlinear oscillator

In the present proposal, the familiar method of the parameter expansion is combined with the multiple scales to study the stability behaviour of the Riemann–Liouville fractional derivative applied to the cubic delayed Duffing oscillator. The analysis of the modified multiple scale perturbation leads to a system of nonlinear differential-algebraic equations governing the solvability conditions. The nonlinear differential equation was reduced to the linear differential equation with the help of the algebraic one. The stability attitude of the periodic motion is determined by the steady-state analysis. Such a periodic motion is needed to better understand the dynamics of the fractional cubic delayed Duffing oscillator.

Modified Extended Kalman Filtering for Nonlinear Stochastic Differential Algebraic Systems

Abstract The extended Kalman filter (EKF) is one of the most widely used nonlinear filtering technique for a system of differential algebraic equations (DAEs). In this work we propose an alternate EKF approach for state estimation of nonlinear DAE systems that addresses shortcomings of the EKF approaches available in literature (Becerra et al., 2001; Mandela et al., 2010). The proposed approach is based on the idea that since the algebraic equations are assumed to be exact, the error covariance matrix of only the differential states needs to be directly propagated during the prediction step. The error covariance matrix for algebraic states and cross covariance matrix between the errors in differential and algebraic states, which are required to incorporate effect of prior algebraic state estimates on the update step, can be computed from the differential state error covariance matrix alone using the linearized algebraic equations. The update step of the proposed work also follows a similar philosophy and ensures that the covariance update is not approximate. The efficacy of the proposed EKF approach is evaluated using benchmark case studies of a Galvanostatic charge process and a drum boiler.

Reach-Set Estimation for DAE Systems under Uncertainty and Disturbances Using Trajectory Sensitivity and Logarithmic Norm

Abstract Trajectory sensitivity analysis is useful for analyzing the dynamic behaviour of differential-algebraic equation (DAE) systems under uncertain initial conditions and/or parameters. However, the approximate trajectories obtained using trajectory sensitivities are not accompanied by explicit error bounds. In this paper, we provide an efficient method to obtain a numerical error bound for the first-order trajectory approximation. This approach uses second-order trajectory sensitivities. A theoretical result quantifying the excursion of trajectories induced by uncertain initial conditions and external disturbances is derived based on the logarithmic norm, and is extended to DAE systems. Although this result itself provides a guaranteed over-approximation of the reach-set of nonlinear DAE systems, by combining this result with the efficient bound obtained from trajectory sensitivities, we are able to provide a much less conservative reach-set estimate for systems under uncertain initial conditions and/or parameters, and external disturbances.

Basins of Attraction and Stability of Nonlinear Systems’ Equilibrium Points

The system of differential and operator equations is considered. This system is assumed to enjoy an equilibrium point. The Cauchy problem with the initial condition with respect to one of the desired functions is formulated. The second function controls the corresponding nonlinear dynamic process. The sufficient conditions of the global classical solution’s existence and stabilisation at infinity to the equilibrium point are formulated in the main theorem. Solution can be constructed by the method of successive approximations. If the conditions of the main theorem are not satisfied, then several solutions may exist. Some solutions can blow-up in a finite time, while others stabilise to an equilibrium point. The special case of considered systems of differential-operator equations are nonlinear systems of differential-algebraic equations which model various nonlinear phenomena in power systems, chemical processes and many other processes.

Acceleration of implicit schemes for large systems of nonlinear differential-algebraic equations

When solving large systems of nonlinear differential-algebraic equations by implicit schemes, each integration step requires the solution of a system of large nonlinear algebraic equations. The latter is solved by an inexact Newton method which, in its turn, leads to a set of large linear systems commonly solved by a Krylov subspace iterative method. The efficiency of the whole process depends on the initial guesses for the inexact Newton and the Krylov subspace methods. An inexpensive approach is proposed and justified that computes good initial guesses for these methods. It requires a subspace of small dimension and the use of line search and trust region for the inexact Newton method and Petrov-Galerkin for the Krylov subspace method. Numerical examples are included to illustrate the effectiveness of the proposed approach.

Bicycle dynamics and its circular solution on a revolution surface

In this paper, we study the dynamics of an idealized benchmark bicycle moving on a surface of revolution. We employ symbolic manipulations to derive the contact constraint equations from an ordered process, and apply the Lagrangian equations of the first type to establish the nonlinear differential algebraic equations (DAEs), leaving nine coupled differential equations, six contact equations, two holonomic constraint equations and four nonholonomic constraint equations. We then present a complete description of hands-free circular motions, in which the time-dependent variables are eliminated through a rotation transformation. We find that the circular motions, similar to those of the bicycle moving on a horizontal surface, nominally fall into four solution families, characterized by four curves varying with the angular speed of the front wheel. Then, we numerically investigate how the topological profiles of these curves change with the parameter of the revolution surface. Furthermore, we directly linearize the nonlinear DAEs, from which a reduced linearized system is obtained by removing the dependent coordinates and counting the symmetries arising from cyclic coordinates. The stability of the circular motion is then analyzed according to the eigenvalues of the Jacobian matrix of the reduced linearized system around the equilibrium position. We find that a stable circular motion exists only if the curvature of the revolution surface is very small and it is limited in small sections of solution families. Finally, based on the numerical simulation of the original nonlinear DAEs system, we show that the stable circular motion is not asymptotically stable.

Theoretical and Experimental Investigation of Thermal Dynamics of Steinhart–Hart Negative Temperature Coefficient Thermistors

The temperature-dependent dynamics of a negative temperature coefficient (NTC) thermistor conducting variable electric current is modeled using the differential approach. The thermistor is assumed to follow the Steinhart–Hart resistance-temperature equation. The developed mathematical model consists of a nonlinear differential-algebraic equations system, and it was analyzed by the Adomian decomposition method (ADM) and its time-marching version known as the multistage Adomian decomposition method (MADM) as well as the Dormand–Prince (DP) numerical method. Five sets of experiments were conducted on five different NTC thermistors and the laboratory measurements were compared with the model predictions. It is demonstrated that the proposed model, when combined with the MADM, can accurately simulate the thermal behavior of the NTC thermistors. The MADM reproduces the experimental temperature dynamics of the five NTC thermistors with an average absolute relative error of about 2.601% while the corresponding errors for the DP method and the classic ADM are 8.122% and 51.255%, respectively. Also, it is shown that the MADM is highly efficient in terms of computational efficiency and it is approximately 6.5 times faster than the classic DP method, when tuned appropriately.

The effect of material density on load rate sensitivity in nonlinear viscoelastic material models

This work presents the results of a numerical sensitivity analysis of material density on crack propagation in nonlinear viscoelastic softening material models. Two basic material models are analyzed: Maxwell and Kelvin material models, under the loading with changing rate (slow or fast impact loading). The material is discretized as a lattice model, and the analysis is performed on simple examples consisting of several lattice bars. The mathematical description is based on the theory of dynamical systems, i.e., it is a system of nonlinear differential equations for Kelvin, and a system of nonlinear differential-algebraic equations for Maxwell material model. Sensitivity of displacements on mass and load rate accompanies the analysis.

Time Stepping Methods and Solvers for Battery Models

There are a wide range of battery models at different scales, from empirical models to molecular dynamics models, that can describe the battery behavior. The Pseudo 2-Dimensional (P2D) model considers the porous electrode theory, concentrated solution theory, Ohm’s law, kinetic relationships, as well as charge and material balances.1 These physic-based behaviors are described by a set of stiff nonlinear partial differential algebraic equations (DAEs) which can be only solved numerically. In this talk, we discuss and review different methods to simulate the battery models, specifically about integration in the time domain. As the P2D model discretized in spatial coordinates by using any suitable method such as finite difference2-3, finite volume4 and spectral methods5-8, it results in a system of nonlinear DAEs. Typically, nonlinear DAEs can be solved based on Runge-Kutta (RK) methods (explicit or implicit) or multistep methods. The different orders of accuracy will affect the accuracy of the numerical solutions for each time step and it will further affect the computational efficiency. However, most of the time the stability region becomes smaller when the order of accuracy of the method increases. For the multistep method, Backward Differentiation Formula (BDF) method is commonly used, because we can get more than second order of accuracy without increasing the number of variables.9 Both RK and BDF methods will be reviewed for simulating battery models. Subtle differences among different methods and efficiency improvement and robustness of all these methods will be analyzed and discussed. In particular, the compromise between, stability, accuracy, and ease of programming will be discussed. Implementation in different programming languages and platforms will also be compared. Acknowledgements The authors are thankful for the financial support of this work by the Clean Energy Institute (CEI) at the University of Washington.

Dynamic analysis of beam-soil interaction systems with material and geometrical nonlinearities

In this paper a Hybrid Domain Boundary Element Method is developed for the geometrically nonlinear dynamic analysis of inelastic Euler-Bernoulli beams of arbitrary doubly symmetric simply or multiply connected constant cross-section, resting on viscous inelastic Winkler foundation. The beam is subjected to the combined action of arbitrarily distributed or concentrated transverse dynamic loading and bending moments in both directions as well as to axial loading, while its edges are subjected to the most general boundary conditions. A displacement based formulation is developed and inelastic redistribution is modelled through a distributed plasticity (fibre) approach. A uniaxial hysteretic law is considered for the evolution of the plastic part of the normal stress following the phenomenological hysteresis model, while hysteretic force-displacement model is also employed in order to describe the inelastic behaviour of the Winkler springs. Numerical integration over the beam cross sections is performed in order to resolve the hysteric parts of the stress resultants. Application of the boundary element technique yields a system of nonlinear Differential-Algebraic Equations, which are written in state-space form and solved by an incremental–iterative solution strategy. Numerical examples are worked out confirming the accuracy and the computational efficiency of the proposed beam formulation, as well as the significant influence of material and geometrical nonlinearities in the response of beam-soil interaction systems.

A Multi-Decomposition Approach for Accelerated Time-Domain Simulation of Transient Stability Problems

Time-domain simulation (TDS) is the most accurate and reliable method for solving transient stability problem. This approach relies on the solution of nonlinear differential-algebraic equations (DAEs) using numerical integration, wherein a system of nonlinear equations is solved at each integration time step. Very DisHonest Newton (VDHN) method is one of the fastest available approaches for solving the system of nonlinear equations in the TDS. However, VDHN does not guarantee convergence and its sequential nature makes it difficult to take advantage of available multicore processors. This paper presents a new multi-decomposition approach (MDA) to achieve fast TDS, wherein the nonlinear system of DAEs is decomposed into three linear subsystems. An adaptive scheme is developed for updating the linear subsystems to ensure that approximation remains sufficiently accurate. The linear subsystems can be solved in parallel using multi-processor architecture. The MDA is demonstrated using IEEE 145-bus test system, and the results are verified against a commercial transient stability problem, Powertech Labs' DSATools.

Analytical Solution of a Nonlinear Index-Three DAEs System Modelling a Slider-Crank Mechanism

The slider-crank mechanism (SCM) is one of the most important mechanisms in modern technology. It appears in most combustion engines including those of automobiles, trucks, and other small engines. The SCM model considered here is an index-three nonlinear system of differential-algebraic equations (DAEs), and therefore difficult to integrate numerically. In this work, we present the application of the differential transform method (DTM) to obtain an approximate analytical solution of the SCM model in convergent series form. In addition, we propose a posttreatment of the power series solution with the Padé resummation method to extend the domain of convergence of the approximate series solution. The main advantage of the proposed technique is that it does not require an index reduction and does not generate secular terms or depend on a perturbation parameter.

Analysis of hollow fibre membrane systems for multicomponent gas separation

This paper analysed the performance of a membrane system over key design/operation parameters. A computation methodology is developed to solve the model of hollow fibre membrane systems for multicomponent gas feeds. The model represented by a nonlinear differential algebraic equation system is solved via a combination of backward differentiation and Gauss–Seidel methods. Natural gas sweetening problem is investigated as a case study. Model parametric analyses of variables, namely feed gas quality, pressure, area, selectivity and permeance, resulted in better understanding of operating and design optima. Particularly, high selectivities and/or permeabilities are shown not to be necessary targets for optimal operation. Rather, a medium selectivity (<60 in the given example) combined with medium permeance (∼300–500×10−10mol/sm2Pa in the given case study) is more advantageous. This model-based membrane systems engineering approach is proposed for the synthesis of efficient and cost-effective multi-stage membrane networks.

Solving system of DAEs by Modified Homotopy Perturbation Method

This paper presents using a new modified homotopy perturbation method (NHPM) for solving linear and nonlinear systems of differential-algebraic equations (DAEs) of index one or higher index without index reduction. By using this scheme, explicit exact solution is calculated in the form of a convergent power series with easily computable components. Some examples are given to illustrate the simplicity and reliability of the new method. The obtained results are found to be in good agreement with the exact solutions known.