Differential-algebraic Equations Research Articles

Reducing the Constrained Multibody Dynamics Problem to the Solution of a System of Ordinary Differential Equations Via Velocity Partitioning and Lie Group Integration

Abstract In multibody dynamics, formulating the equations of motion in absolute Cartesian coordinates results in a set of index-3 differential algebraic equations (DAEs). In this work, we present an approach that bypasses the DAE problem by partitioning the velocities in the system into dependent and independent coordinates, thereby reducing the task of producing the time evolution of the mechanical system to one of solving a set of ordinary differential equations (ODEs). In this approach, the independent coordinates are integrated directly, while the dependent coordinates are recovered through the kinematic constraint equations at the position and velocity levels. Notably, Lie group integration is employed to directly obtain the orientation matrix A at each time-step of the simulation. This eliminates the need to choose generalized coordinates to capture the orientation of a body, as the matrix A is a by-product of the solution algorithm. Herein, we outline the new approach and demonstrate it in conjunction with four mechanisms: a single pendulum, a double pendulum, a four-link mechanism, and a slider crank. We report on the convergence order behavior of the proposed method and compare its performance with an established method that combines coordinate partitioning with an Euler parameter formulation. The Python code developed to generate the reported results is open-source and available in a public repository for reproducibility studies.

A modified cell-centered nodal integral scheme for the convection-diffusion equation

The nodal integral methods (NIMs) are very efficient and accurate coarse-mesh methods for solving partial differential equations. The cell-centered NIM (CCNIM) is a simplified variant of the NIMs that has recently shown its efficiency in solving fluid flow problems but has been hampered by issues such as inapplicability to one-dimensional problems, complexities in handling Neumann boundary conditions and the formulation of a system of differential-algebraic equations (DAEs) for discrete unknowns. Here, we present a modified version of the CCNIM designed to overcome the challenges associated with its previous version. Our novel development retains the essence of CCNIM while resolving these issues. The proposed scheme, grounded in the nodal framework, achieves second-order accuracy in both spatial and temporal dimensions. Unlike its precursor, the proposed method formulates algebraic equations for discrete variables per node, eliminating the cumbersome DAE system. Neumann boundary conditions are seamlessly incorporated through a straightforward flux definition, and applicability to one-dimensional problems is now feasible. We successfully apply our approach to one and two-dimensional convection-diffusion problems with known analytical solutions to validate our approach. The simplicity and robustness of the approach lay the foundation for its seamless extension to more complex fluid flow problems.

The Analitical Solution of Linear and Non-Linear Differential-Algebraic Equations (DAEs) with Laplace-Padé Series Method

In this paper, we apply Laplace-Padé Series method to solve linear and non-linear differential-algebraic equations (DAEs). Firstly, The basic properties of the Laplace-Padé Series method are given. Secondly, we calculate the arbitrary order power series of differential-algebraic equations (DAEs), then convert it to the series form Laplace-Padé. Then, the three differential-algebraic equations (DAEs) are solved by Laplace-Padé Series method. It was seen that the method gave effective and fast results. Therefore, the method can be easily applied to linear and non-linear differential-algebraic equations (DAEs) problems in different fields.

Solvability Analysis of high-order Linear Differential-Algebraic Equations with time-varying coefficients

In this paper, we study the solvability analysis of arbitrarily high-order linear differential-algebraic equations (DAEs) with time-varying coefficients, using the algebraic-behavior approach. We propose a concept of strangeness-index and construct condensed forms for high-order linear DAEs. We also discuss other structural properties like the existence and uniqueness of a solution, consistency and smoothness requirements for an initial vector and for an inhomogeneity. Our work extends the algebraic approach for DAEs and combines this approach with the behavior approach to establish a reformulation algorithm that reveals an underlying ordinary differential equation (ODE) and all hidden constraints in the DAE. This direct treatment of the system addresses the limitations of the classical approach to transforming the system into a first-order DAE. We illustrate our theoretical results with applications in mechanical systems and electrical circuits. This comprehensive study into general high-order systems is a natural progression from the extensive study of first and second-order DAEs.

Dynamic optimization on quantum hardware: Feasibility for a process industry use case

The quest for real-time dynamic optimization solutions in the process industry represents a formidable computational challenge, particularly within the realm of applications like model-predictive control, where rapid and reliable computations are critical. Conventional methods can struggle to surmount the complexities of such tasks. Quantum computing and quantum annealing emerge as avant-garde contenders to transcend conventional computational constraints. We convert a dynamic optimization problem, characterized by an optimization problem with a system of differential–algebraic equations embedded, into a Quadratic Unconstrained Binary Optimization problem, enabling quantum computational approaches. The empirical findings synthesized from classical methods, simulated annealing, quantum annealing via D-Wave’s quantum annealer, and hybrid solver methodologies, illuminate the intricate landscape of computational prowess essential for tackling complex and high-dimensional dynamic optimization problems. Our findings suggest that while quantum annealing is a maturing technology that currently does not outperform state-of-the-art classical solvers, continuous improvements could eventually aid in increasing efficiency within the chemical process industry.

Mitigation of numerical issues appearing in transient analyses when applying fractional derivative approximations

Due to the potential decrease of the computation time for problems with fractional order derivatives, and due to the extension of the range of applicable solvers for a given problem, the approximation of fractional derivatives (e.g., using Oustaloup’s method) is an important and frequently discussed topic. A significant problem that can occur (although one that is not discussed very often) when approximations are applied concerns the potential numerical issues that can arise when applying these approximations in a transient analysis, e.g., as shown in the paper, in a time-stepping solver. This paper examines five different methods for approximating the fractional derivative (Oustaloup’s method, a refined Oustaloup method, a modified continued fraction expansion method, Matsuda’s method and Charef’s method). An important preliminary step in the analysis was the establishment of the general form of FDAE (fractional differential-algebraic equations), the general form of the applied approximations and the general form of the DAE (differential-algebraic equations) resulting from the approximation. This allowed for a precise description of the equations according to which the transformation from the FDAE to the approximating DAE takes place. Three problems derived from studies in electrical engineering have been introduced into the analysis (a linear circuit problem, a circuit problem with nonlinear fractional elements, and a problem featuring nonlinearities and pregenerated noise). For these problems, the conditions in which numerical issues appear have been studied in detail. They are the main motivation of this study as these issues are so significant that they often result in zero-solutions or solutions tending to infinity (giving an impression as if the system is unstable). Modifications and alternatives of the transformation into the DAE, that aim at the mitigation of these numerical errors, are mentioned later. The final result is very satisfactory, where the algorithm for the transformation of the FDAE to the approximating DAE practically eliminates the most important barriers (which was the impossibility of using approximations above a certain order due to the mentioned numerical issues). The study presented in this paper is motivated by problems in electrical engineering but due to its generality it is also applicable in other fields where fractional calculus is used.

Index‐aware learning of circuits

AbstractElectrical circuits are present in a variety of technologies, making their design an important part of computer aided engineering. The growing number of parameters that affect the final design leads to a need for new approaches to quantify their impact. Machine learning may play a key role in this regard; however, current approaches often make suboptimal use of existing knowledge about the system at hand. In terms of circuits, their description via modified nodal analysis is well‐understood. This particular formulation leads to systems of differential‐algebraic equations (DAEs), which bring with them a number of peculiarities, for example, hidden constraints that the solution needs to fulfill. We use the recently introduced dissection index that can decouple a given system of DAEs into ordinary differential equations, only depending on differential variables, and purely algebraic equations, that describe the relations between differential and algebraic variables. The idea is to then only learn the differential variables and reconstruct the algebraic ones using the relations from the decoupling. This approach guarantees that the algebraic constraints are fulfilled up to the accuracy of the nonlinear system solver, and it may also reduce the learning effort as only the differential variables need to be learned.

Transforming Radical Differential Equations to Algebraic Differential Equations

In this paper, we present an algorithmic procedure that transforms, if possible, a given system of ordinary or partial differential equations with radical dependencies in the unknown function and its derivatives into a system with polynomial relations among them by means of a rational change of variables. The solutions of the given equation and its transformation correspond one-to-one. This work can be seen as a generalization of previous work on reparametrization of ODEs and PDEs with radical coefficients.

Physics-Informed Neural Networks for Solving High-Index Differential-Algebraic Equation Systems Based on Radau Methods

As is well known, differential algebraic equations (DAEs), which are able to describe dynamic changes and underlying constraints, have been widely applied in engineering fields such as fluid dynamics, multi-body dynamics, mechanical systems, and control theory. In practical physical modeling within these domains, the systems often generate high-index DAEs. Classical implicit numerical methods typically result in varying order reduction of numerical accuracy when solving high-index systems. Recently, the physics-informed neural networks (PINNs) have gained attention for solving DAE systems. However, it faces challenges like the inability to directly solve high-index systems, lower predictive accuracy, and weaker generalization capabilities. In this paper, we propose a PINN computational framework, combined Radau IIA numerical method with an improved fully connected neural network structure, to directly solve high-index DAEs. Furthermore, we employ a domain decomposition strategy to enhance solution accuracy. We conduct numerical experiments with two classical high-index systems as illustrative examples, investigating how different orders and time-step sizes of the Radau IIA method affect the accuracy of neural network solutions. For different time-step sizes, the experimental results indicate that utilizing a 5th-order Radau IIA method in the PINN achieves a high level of system accuracy and stability. Specifically, the absolute errors for all differential variables remain as low as 10−6, and the absolute errors for algebraic variables are maintained at 10−5. Therefore, our method exhibits excellent computational accuracy and strong generalization capabilities, providing a feasible approach for the high-precision solution of larger-scale DAEs with higher indices or challenging high-dimensional partial differential algebraic equation systems.

Problem for differential-algebraic equations with a significant loads

In this article, the problem for a differential-algebraic equation with a significant loads is studied. Unlike previously studied problems for differential equations with a significant loads, in the considered equation, there is a matrix in the left part with a derivative that is not invertible. Therefore, the system of equations includes both differential and algebraic equations. To solve the problem, we propose a modification of the Dzhumabaev’s parametrization method. The considered problem is reduced to a parametric problem for the differential-algebraic equation with significant loads. We apply the Weierstrass canonical form to this problem. We obtain parametric initial value problem for a differential equations and an algebraic equations with a significant loads. The solvability conditions for the considered problem are established.

On the update of algebraic states during state estimation of differential–algebraic equation (DAE) systems

This manuscript presents a discussion on the algebraic state update step performed during recursive filtering of the differential–algebraic equation (DAE) systems. Existing DAE state estimation approaches follow a two-step state update procedure at each sampling instant. In particular, they first estimate the differential states using the Kalman update, and then update algebraic states by explicitly solving the algebraic equations. Specifically, for the case of DAE systems involving linear algebraic equations though the differential equations are nonlinear, we show that when appropriately initialized, this two-step state update procedure is not needed. It can instead be replaced with a one-step state update procedure that computes the differential and algebraic state estimates simultaneously through the Kalman update. The satisfaction of algebraic equations is guaranteed by this one-step update without it being explicitly enforced. Towards this end, we show that the error covariance matrix of augmented states, when properly initialized, satisfies a null-space property after prediction and update step at each sampling instant. This property ensures that the state estimates obtained using the proposed one-step update approach, satisfy the algebraic equations. This holds for both analytical linearization based extended Kalman filtering and statistical linearization based sigma-point filtering approaches. We also propose a heuristic-based update procedure for state estimation of DAE systems that involve nonlinear algebraic equations. This procedure draws out inferences from the case of DAE systems involving linear algebraic equations and is based on the analysis of algebraic equations residuals obtained from the updated differential and algebraic state estimates with a one-step state update. The efficacy of the proposed state update procedures is demonstrated by performing simulation studies on a benchmark drum boiler system case study. Results demonstrate that the proposed update procedures satisfactorily estimate the differential and algebraic states of a DAE system when compared to the traditional two-step update procedure.

A Posteriori Error Estimation for Parabolic Problems with Dynamic Boundary Conditions

This paper is concerned with adaptive mesh refinement strategies for the spatial discretization of parabolic problems with dynamic boundary conditions. This includes the characterization of inf–sup stable discretization schemes for a stationary model problem as a preliminary step. Based on an alternative formulation of the system as a partial differential–algebraic equation, we introduce a posteriori error estimators which allow local refinements as well as a special treatment of the boundary. We prove reliability and efficiency of the estimators and illustrate their performance in several numerical experiments.

Load Margin Assessment of Power Systems Using Physics-Informed Neural Network with Optimized Parameters

Challenges in the operation of power systems arise from several factors such as the interconnection of large power systems, integration of new energy sources and the increase in electrical energy demand. These challenges have required the development of fast and reliable tools for evaluating the operation of power systems. The load margin (LM) is an important index in evaluating the stability of power systems, but traditional methods for determining the LM consist of solving a set of differential-algebraic equations whose information may not always be available. Data-Driven techniques such as Artificial Neural Networks were developed to calculate and monitor LM, but may present unsatisfactory performance due to difficulty in generalization. Therefore, this article proposes a design method for Physics-Informed Neural Networks whose parameters will be tuned by bio-inspired algorithms in an optimization model. Physical knowledge regarding the operation of power systems is incorporated into the PINN training process. Case studies were carried out and discussed in the IEEE 68-bus system considering the N-1 criterion for disconnection of transmission lines. The PINN load margin results obtained by the proposed method showed lower error values for the Root Mean Square Error (RMSE), Mean Square Error (MSE) and Mean Absolute Percentage Error (MAPE) indices than the traditional training Levenberg-Marquard method.

A Lie group variational integrator in a closed-loop vector space without a multiplier

Abstract. As a non-tree multi-body system, the dynamics model of four-bar mechanism is a differential algebraic equation. The constraints breach problem leads to many problems for computation accuracy and efficiency. With the traditional method, constructing an ODE-type dynamics equation for it is difficult or impossible. In this exploration, the dynamics model is built with geometry mechanic theory. The kinematic constraint variation relation of a closed-loop system is built in matrix and vector space with Lie group and Lie algebra theory respectively. The results indicate that the attitude variation between the driven body and the follower body has a linear recursion relation, which is the basis for dynamics modelling. With the Lie group variational integrator method, the closed-loop system Lagrangian dynamics model is built in vector space, with Legendre transformation. The dynamics model is reduced to be the Hamilton type. The kinematic model and dynamics model are solved using Newton iteration and the Runge–Kutta method respectively. As a special case of a crank and rocker mechanism, the dynamics character of a parallelogram mechanism is presented to verify the good structure conservation character of the closed-loop geometry dynamics model.

Simplified Reactor Model for Microreactor Coupled with Helium Closed Brayton Cycle

Microreactors comprise a new actively developing class of very small advanced reactors that have the potential to be an alternative to carbon-intensive energy technologies. A microreactor based on high-temperature gas reactor (HTGR) technology is a very promising advanced reactor with inherent safety, and it can couple with a closed Brayton cycle for higher efficiency. Since dynamics characteristics are fundamental to analyzing a power generation system and a reactor is the main source of the dynamics characteristics of a system, it is necessary to study a microreactor model suitable for system analysis. The main goal is to simulate the performance of the previously mentioned integrated system, focusing on the details of the power conversion unit while still ensuring acceptable calculation times. Hence, a simplified reactor model is needed that could supply sufficiently accurate values of pressure drop and heat transfer across the core. In this paper, by simplifying the physical processes in a microreactor, a dynamic model described by differential algebraic equations is obtained based on the lumped parameter modeling methodology and the basic conservation of fluid mass, momentum, and energy. Coupling thermal hydraulics with neutron kinetics, the temperature coefficient of reactivity and xenon poisoning are considered. Finally, the model is programmed and calculated using Modelica language. The transient responses of the main parameters under typical perturbations are analyzed, and the results show that the responses are correct. Because of the effect of reactivity feedback, fluctuations of the main parameters caused by microperturbations eventually tend to stabilize. In addition, the effects of negative reactivity introduced by xenon poisoning under two typical dynamic processes are analyzed. In power regulation, excess reactivity is required to compensate for the negative reactivity introduced by 135Xe. The model and results can properly predict the systematic parameters and serve as a basis for system analysis of microreactor coupling with the helium closed Brayton cycle.

Inertial Parameter Identification for Closed-Loop Mechanisms: Adaptation of Linear Regression for Coordinate Partitioning

Abstract This study investigates the use of linear-regression-based identification in rigid multibody system applications. A multibody system model, originally described with differential-algebraic equations (DAE), is transformed into a set of ordinary differential equations using coordinate partitioning. This allows the identification framework (where the system is described with ordinary differential equations) to be applied to rigid multibody systems described with nonminimal coordinates. The methodology is demonstrated via numerical and experimental validation on a slider–crank mechanism. The results show that the presented methodology is capable of accurately identifying the system's inertial parameters even with a short motion trajectory used for training. The presented linear-regression-based identification approach opens new opportunities to develop more accurate multibody models. The resulting updated multibody models can be considered especially useful for state-estimation and the control of multibody systems.

Differential Fault and Algebraic Equation Combined Analysis on PICO

In modern information technology, research on block cipher security is imperative. Concerning the ultra lightweight block cipher PICO, there has been only one study focused on recovering its complete master key, with a large search space of 264, and no fault analysis yet. This paper proposes a new fault analysis approach, combining differential fault and algebraic equation techniques. It achieved the recovery of PICO’s entire master key with 40 faults in an average time of 0.57 h. S-box decomposition was utilized to optimize our approach, reducing the time by a remarkable 75.83% under the identical 40-fault condition. Furthermore, PICO’s complete master key could be recovered with 28 faults in an average time of 0.78 h, indicating a significant 237 reduction in its search space compared to the previous study. This marks the first fault analysis on PICO. Compared to conventional fault analysis methods DFA (differential fault analysis) and AFA (algebraic fault analysis), our approach outperforms in recovering PICO’s entire master key, highlighting the cruciality of key expansion complexity in block cipher security. Therefore, our approach could serve to recover master keys of block ciphers with comparably complicated key expansions, and production of more secure block ciphers could result.

Effective numerical simulations of synchronous generator system

Synchronous generator system is a complicated dynamic system for energy transmission, which plays an important role in modern industrial production. In this article, we propose some predictor-corrector methods and structure-preserving methods for a generator system based on the first benchmark model of subsynchronous resonance, among which the structure-preserving methods preserve a Dirac structure associated with the so-called port-Hamiltonian descriptor systems. To illustrate this, the simplified generator system in the form of index-1 differential-algebraic equations has been derived. Our analyses provide the global error estimates for a special class of structure-preserving methods called Gauss methods, which guarantee their superior performance over the PSCAD/EMTDC and the predictor-corrector methods in terms of computational stability. Numerical simulations are implemented to verify the effectiveness and advantages of our methods.

A robust Gated-PINN to resolve local minima issues in solving differential algebraic equations

Physics-Informed Neural Network (PINN) has emerged as a promising tool for solving various physical problems with differential equations. However in practice, PINN often suffers from the local minima issue while solving problems with minimal initial conditions. In this paper, we propose a robust Gated-PINN to address this issue, which blends numerical methods and PINN. The Gated-PINN overcomes the local minima issue by controlling the flow of information of the neural network similar to a numerical method. We first investigate the use of conventional PINN to the Cartesian-coordinate simple pendulum problem with minimal initial conditions, as one of basic differential algebraic equation (DAE) problem. Our results show the advantage of PINN over existing numerical methods in that it does not require sophisticated mathematical techniques such as index reduction. But we also observe that conventional PINN can lead to inaccurate solutions; such solutions partially satisfy the differential equation requirement, but does not meet the given initial condition and fail to further improve. We demonstrate the effectiveness of the proposed Gated-PINN by showing that it yields accurate solutions, such that for a pendulum of length 1, the mean Euclidean error between the Gated-PINN model and the traditional numerical method model is less than 0.01 and pointwise maximum Euclidean error is less than 0.04. Moreover, Gated-PINN can operate without any complicated index reduction, and unlike conventional PINN, accurate solution can be obtained consistently without falling into a local minima. Overall, our study presents the potential of the Gated-PINN for solving DAE problems and provides a valuable insight into the challenges and limitations of using PINN for solving physical problems.

Solutions to the output regulation problem of time-varying descriptor systems

This paper studies the output regulation problem of time-varying descriptor systems and the problem of designing state feedback and dynamic measurement output feedback control laws which asymptotically achieves output regulation and disturbance rejection is considered. New regulator equations are proposed for time-varying descriptor systems in the form of differential-algebraic matrix equations. The unique solution of the proposed regulator equations is given as well. We prove that the output regulation problem of time-varying descriptor systems is solvable if and only if the given regulator equations are solvable. Based on the solution of the regulator equations, the state feedback and dynamic measurement output feedback control laws are designed to solve the output regulation problem. The work extends the existing results of output regulation problem for time-varying linear systems to the time-varying descriptor systems. Numerical examples are given to show the effectiveness of our methodology.