Differential-algebraic Systems Research Articles

Decentralized and performance-constrained state-estimated fuzzy control for nonlinear differential-algebraic interconnected systems

This paper introduces a decentralized and mixed performance-constrained state-estimated fuzzy control strategy designed for nonlinear differential-algebraic interconnected systems. Initially, the approach employs a Takagi-Sugeno (T-S) fuzzy model to approximate these complex systems. To address potential limitations within the interconnected systems, a Proportional-plus-Derivative (PD) state control method is integrated into the proposed control scheme. Additionally, an observer is constructed to estimate system states, enhancing the feasibility of the control method. System stability analysis is conducted using the free-weighing Lyapunov function approach, allowing designers to customize the weight matrix for desired control performance and facilitate controller design. Subsequently, a fuzzy controller is developed to ensure stability and meet both Passive-H∞ Performance (PHP) and H2 Control Performance (HCP) criteria. Finally, the effectiveness of the proposed design approach is validated through simulation results.

Nonlinear state estimation of a power plant superheater by using the extended Kalman filter for differential algebraic equation systems

A nonlinear estimator was developed for process monitoring of a power plant superheater system to estimate the transient temperature profiles of the critical measured and unmeasured variables under load-following operations. The model used in the estimator was a detailed, distributed, first-principles, differential-algebraic equation model. The superheater geometry is characterized based on an operating natural gas combined cycle power plant. The model included mass and energy balances for the operating fluids, i.e., steam and flue gas, with due consideration of tube wall temperature dynamics. The dynamic model was used for state and parameter estimation by using a modified extended Kalman filter. When results are compared with the data from a natural gas power plant, the estimator yielded root mean squared error of 0.2 °C for the main steam temperature compared to 1 °C obtained from the first-principles model. While the first-principles model yielded as high as 2.5 °C absolute instantaneous error for the tube temperatures, the estimator reduced the absolute instantaneous error below 0.3 °C for most variables. Performance of the estimator for estimating critical variables like the main steam temperature and flue gas temperature was evaluated in the presence of high model mismatch and noisy measurement data by simulating load-following operation of the plant. The estimator yielded maximum absolute instantaneous error of 3.4 °C for the flue gas and 1 °C for the main steam temperature.

Modular differential equations and algebraic systems

Modular differential equations and algebraic systems

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.

Estimation of the Region of Attraction of Polynomial Swing Equation Using Sum of Squares Theory

Estimation of the region of attraction of the power system using sum of squares (SOS) programming theory is investigated in this paper. Firstly, the Taylor expansion formula is applied to explore the inherent polynomial structure of the swing equation. The single machine projection equation for a multi-machine system is proposed based on the amplitude characteristic of the rotor angle response in decoupled space. Secondly, the theory of SOS programming originating from the polynomial system is derived, and the theory about estimation of the region of attraction for the differential algebraic system is summarized. Thirdly, the method for estimating the region of attraction for the polynomial swing equation based on a V-S two-stage iterative procedure is proposed. Finally, a single machine infinite bus power system, an IEEE 3-machine power system, and an IEEE 4-machine power system are employed to validate the effectiveness and accuracy of the proposed method and analysis.

Modeling carbon nutrition and photosynthesis taking into account plant stomatal resistance

A differential-algebraic system of equations of carbon dioxide transport and photosynthesis in vegetation is formulated for mathematical analysis of the influence on productivity and water regime of plant populations. A simple semi-empirical algebraic approximation of the true photosynthesis function is constructed for the quasi-stationary solution of the system, including the influence of stomatal regulation, leaf index, absorbed photosynthetically active radiation, and temperature.

Analytical and numerical stability analysis of a toroidal wheel with nonholonomic constraints

In this paper, a detailed and comprehensive linear stability analysis of a rolling toroidal wheel is performed. The wheel is modeled as a rigid toroid-shaped body rolling without slipping on a horizontal surface. The nonlinear equations of motion constitute a Differential-Algebraic Equations system, given by the dynamic equilibrium equations augmented with the nonholonomic constraints, which arise from the no-slip condition. The circular steady motion and the linearized equations of motion along this relative equilibrium are obtained, for both the solid and hollow tori. The expressions of the linearized equations and the corresponding eigenvalues are derived analytically as a function of the torus aspect ratio. The variation of the stability boundary with the torus aspect ratio is shown. A comparison of the results obtained in the solid and hollow scenarios is included, and all the results are validated with the rolling hoop, which corresponds to a degenerate torus with zero aspect ratio. In the particular case of the steady straight-line rolling and spinning about a vertical diameter, which constitute limit motions of the circular steady motion, the critical rotational and angular speeds required for stabilization are obtained.

Physical modeling and simulation of electro-mechanical actuator-based TVC systems for reusable launch vehicles

This paper presents the high-fidelity physical modeling of a Thrust Vector Control (TVC) system operated by Electro-Mechanical Actuators (EMAs) for Reusable Launch Vehicles (RLVs). In contrast to simplified, often linear, models, high-fidelity physical models enable a better assessment of the actual system performance and a deeper verification of the on-board software robustness against disturbances, unmodeled dynamics or degradation. This aspect is essential to enable the reusability concept as driven by long-term reliability requirements. Moreover, critical systems like Fault Detection, Isolation and Recovery (FDIR) logic, whose robustness and efficacy are often difficult to prove, can be thoroughly tested without requiring hardware experiments with eventual invasive modifications for fault injections. This work captures the whole dynamics of the EMA and TVC components, including the power drive electronics, the electrical motor, and the mechanical transmission for the EMA, as well as the mechanical properties of the engine nozzle acting as a load. The resulting differential algebraic equation system is modeled in the Modelica acausal object-oriented modeling language. An ad-hoc framework is implemented to guarantee flexibility and modularity, which allows models with different fidelity levels to be easily exchanged to achieve the simulation objectives and needed accuracy level. The impact of the different models on the closed-loop TVC dynamics is analyzed in both the frequency and time domain, and then benchmarked with a realistic RLV mission. The results show that the high-fidelity physical models provide a better understanding of the more complex effects governing the TVC dynamics and can, in turn, effectively improve its requirement definition process.

On the construction and parameterization of interpolants in the Loewner framework

We develop a general method for the construction of interpolants in the Loewner framework for nonlinear differential–algebraic systems. The approach involves building a family of systems preserving the properties of Loewner equivalence and matching of tangential data functions. It is shown that under mild conditions this family of systems parameterizes all interpolants of sufficiently large dimension matching the tangential data while possessing tangential generalized controllability and observability functions with full column and row rank Jacobians. As a result, this family of systems provides all possible degrees of freedom that an interpolant can have. The results are also discussed in the linear setting and, when taken in combination with existing results, provide a broader framework for the construction of linear interpolating systems.

Operator splitting based dynamic iteration for linear differential-algebraic port-Hamiltonian systems

A dynamic iteration scheme for linear differential-algebraic port-Hamiltonian systems based on Lions–Mercier-type operator splitting methods is developed. The dynamic iteration is monotone in the sense that the error is decreasing and no stability conditions are required. The developed iteration scheme is even new for linear port-Hamiltonian systems governed by ODEs. The obtained algorithm is applied to a multibody system and an electrical network.

Stabilized state-based peridynamics for elasticity emanating from constrained Lagrangian

State-based peridynamics for elasticity paves the way to bridge the local theory with the inherent capabilities of general elasticity with non-local effects. However, the original non-ordinary state-based peridynamics with correspondence materiel models suffers from high-frequency oscillations due to zero-energy mode instability. To address this issue, we start from the fundamental Lagrangian with the introduction of a constraint potential energy that minimizes the numerical error and suppresses the zero-energy mode oscillations. A stabilized non-ordinary state-based peridynamics is then derived from this constrained Lagrangian that forms an integral–differential–algebraic system. In specifics, the equation of motion for particles is described by integral-differential equations while the constraint suppressing the zero-energy mode instability is expressed as algebraic equations (zero-error constraint). An energy–momentum conserving implicit time-stepping scheme for this integral–differential–algebraic system is then derived. End-to-end 2D and 3D simulations, including scalar wave propagation, crack path, and large deformation problems, are presented to show that the proposed method is promising for these applications in terms of accurately capturing the crack path and large deformation, efficiently eliminating the numeral oscillations, conserving the total energy for long duration simulations, and preserving the zero-error constraint.

Geometrically transformed spectral methods to solve partial differential equations in circular geometries, application for multi-phase flow

Circular geometries are ubiquitously encountered in science and technology, and the polar coordinate provides the natural way to analyze them; However, its application is limited to symmetric cases, and it cannot be applied to segments that are formed in multiphase flow problems in pipes. To address that, spectral discretization of circular geometries via orthogonal collocation technique is developed using geometrical mapping. Two analytical mappings between the circle and square geometries, namely, elliptical and horizontally squelched mappings, are employed. Accordingly, numerical algorithms are developed for solving PDEs in circular geometries with different boundary conditions for both steady state and transient problems. Various implementation issues are thoroughly discussed, including vectorization and strategies to avoid solving the differential–algebraic system of equations. Moreover, several case studies for symmetric and asymmetric Poisson equations with different boundary conditions are performed to evaluate several aspects of these techniques, such as error properties, condition number, and computational time. For both steady state and transient solvers, it was revealed that the computation time scales quadratically with respect to the grid size for both mapping and polar discretization techniques. However, due to the presence of the second mixed derivative, mapping techniques are more computationally costly. Finally, the squelched mapping was successfully employed to discretize the two, and three-phase gravity flows in sloped pipes.

On Stabilizability of Nonbilinear Perturbed Descriptor Systems

One way in which nonlinear descriptor systems of (index-k) naturally arise is through semiexplicit differential-algebraic equations. The study considers the nonbilinear dynamical systems which are described by the class of higher-index differential-algebraic equations (DAEs). Their nature is analysed both quantitatively and qualitatively, and stability characteristics are presented for their solution. Higher-index differential-algebraic systems seem to show inherent shaky around their solution manifolds. The often use of logarithmic norms is for the estimation of stability and perturbation bounds in linear ordinary differential equations (ODEs). The question of how to apply the notation of logarithmic norms to nonlinear DAEs has long been an open question. Other problem extensions including nonlinear dynamics and nonbilinear DAEs need subtle modification of the logarithmic norms. The logarithmic norm is combined by conceptual focus with the finite-time stability criterion in order to treat nonbilinear DAEs with the aim of covering some unbounded operators. This means we obtain the perturbation bounds from differential inequalities for a norm by the use of the relationship between Dini derivatives and semi-inner products. A numerical result obtained when tested on the nonbilinear mechanical system with a larger scale showed that the method was highly efficient and accurate and particularly suitable for nonbilinear DAEs.

Fixed-Time Control for a Class of Nonlinear PH-DAE Systems

The globally fixed-time stabilization and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty}$</tex-math> </inline-formula> control for the port-Hamiltonian differential algebraic equation (PH-DAE) is addressed by applying the structural properties of the port-Hamiltonian (PH) systems, and the interconnection and damping assignment passivity-based control (IDA-PBC) technique. For fixed-time control design of the differential algebraic equation systems, the main obstacle lies that the algebraic variables are hidden in the system and their dynamics are ambiguous. We propose an available and effective approach to overcome this obstacle. The system variables are decomposed into differential component and the algebraic component, and the explicit representation of the dynamics of algebraic component is obtained. Fist, we investigate the locally fixed-time stabilization control as well as globally attractive for PH-DAE. Second, we focus on solving the globally fixed-time stabilization and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty}$</tex-math> </inline-formula> control problems by jointly taking advantage of the locally fixed-time stabilization control for the origin and the globally fixed-time attractivity control for a predeterminate region. Finally, in order to address fixed-time stabilization control and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H_{\infty}$</tex-math> </inline-formula> control for PH-DAE, new controllers are designed. Moreover, the convergence time of systems can be easily estimated after external disturbance vanishes. The efficiency of the theoretical results acquired in present article is verified via two examples with numerical simulation.

Simple algorithm for judging equivalence of differential-algebraic equation systems

Mathematical formulas play a prominent role in science, technology, engineering, and mathematics (STEM) documents; understanding STEM documents usually requires knowing the difference between equation groups containing multiple equations. When two equation groups can be transformed into the same form, we call the equation groups equivalent. Existing tools cannot judge the equivalence of two equation groups; thus, we develop an algorithm to judge such an equivalence using a computer algebra system. The proposed algorithm first eliminates variables appearing only in either equation group. It then checks the equivalence of the equations one by one: the equations with identical algebraic solutions for the same variable are judged equivalent. If each equation in one equation group is equivalent to an equation in the other, the equation groups are judged equivalent; otherwise, non-equivalent. We generated 50 pairs of equation groups for evaluation. The proposed method accurately judged the equivalence of all pairs. This method is expected to facilitate comprehension of a large amount of mathematical information in STEM documents. Furthermore, this is a necessary step for machines to understand equations, including process models.

Load- and Renewable-Following Control of Linearization-Free Differential Algebraic Equation Power System Models

Electromechanical transients in power networks are mostly caused by a mismatch between power consumption and production, causing generators to deviate from the nominal frequency. To that end, feedback control algorithms have been designed to perform frequency and load/renewable-following control. In particular, the literature addressed a plethora of grid-and frequency-control challenges with a focus on linearized, differential equation models whereby algebraic constraints i.e., power flows (PFs) are eliminated. This is in contrast to the more realistic nonlinear differential algebraic equation (NDAE) models. Yet, as grids are increasingly pushed to their limits via intermittent renewables and varying loads, their physical states risk escaping operating regions due to either a poor prediction or sudden changes in renewables or demands—deeming a feedback controller based on a linearization point virtually unusable. In lieu of linearized differential equation models, the objective of this article is to design a simple, purely decentralized, linearization-free, feedback control law for the NDAE models of power networks. The aim of such a controller is to primarily stabilize frequency oscillations after a significant, unknown disturbance in renewables or loads. Although the controller design involves advanced NDAE system theory, the controller itself is as simple as a decentralized proportional or linear quadratic regulator (LQR) in its implementation. Case studies demonstrate that the proposed controller is able to stabilize dynamic and algebraic states under significant disturbances.

Control-oriented modeling of a LiBr/H[formula omitted]O absorption heat pumping device and experimental validation

Absorption heat pumping devices (AHPDs, comprising absorption heat pumps and chillers) are devices that use thermal energy instead of electricity to generate heating and cooling, thereby facilitating the use of waste heat and renewable energy sources such as solar or geothermal energy. Despite this benefit, widespread use of AHPDs is still limited. One reason for this is partly unsatisfactory control performance under varying operating conditions, which can result in poor modulation and part load capability. A promising approach to tackle this issue is using dynamic, model-based control strategies, whose effectiveness, however, strongly depend on the model being used. This paper therefore focuses on the derivation of a viable dynamic model to be used for such model-based control strategies for AHPDs such as state feedback or model-predictive control. The derived model is experimentally validated, showing good modeling accuracy. Its modeling accuracy is also compared to alternative model versions, that contain other heat transfer correlations, as a benchmark. Although the derived model is mathematically simple, it does have the structure of a nonlinear differential–algebraic system of equations. To obtain an even simpler model structure, linearization at an operating point is discussed to derive a model in linear state space representation. The experimental validation shows that the linear model does have slightly worse steady-state accuracy, but that the dynamic accuracy seems to be almost unaffected by the linearization. The presented new modeling approach is considered suitable to be used as a basis for the design of advanced, model-based control strategies, ultimately aiming to improve the modulation and part load capability of AHPDs.

Asymptotic solutions of singularly perturbed linear differential-algebraic equations with periodic coefficients

The paper deals with the problem of constructing asymptotic solutions for singular perturbed linear differential-algebraic equations with periodic coefficients. The case of multiple roots of a characteristic equation is studied. It is assumed that the limit pencil of matrices of the system has one eigenvalue of multiplicity n, which corresponds to two finite elementary divisors and two infinite elementary divisors whose multiplicity is greater than 1.A technique for finding the asymptotic solutions is developed and n formal linearly independent solutions are constructed for the corresponding differential-algebraic system. The developed algorithm for constructing formal solutions of the system is a nontrivial generalization of the corresponding algorithm for constructing asymptotic solutions of a singularly perturbed system of differential equations in normal form, which was used in the case of simple roots of the characteristic equation.The modification of the algorithm is based on the equalization method in a special way the coefficients at powers of a small parameter in algebraic systems of equations, from which the coefficients of the formal expansions of the searched solution are found. Asymptotic estimates for the terms of these expansions with respect to a small parameter are also given.For an inhomogeneous differential-algebraic system of equations with periodic coefficients, existence and uniqueness theorems for a periodic solution satisfying some asymptotic estimate are proved, and an algorithm for constructing the corresponding formal solutions of the system is developed. Both critical and non-critical cases are considered.

Gradient-enhanced physics-informed neural networks for power systems operational support

The application of deep learning methods to speed up the challenging power system problems has recently shown very encouraging results. However, power system dynamics are not snapshot, steady-state operations. These dynamics must be considered to ensure that the optimal solutions provided by these models adhere to practical constraints to avoid frequency fluctuations and grid instabilities. Unfortunately, dynamic system models based on ordinary or partial differential equations are frequently unsuitable for direct application in control or state estimates due to their high computational costs. To address these challenges, this paper introduces a machine learning method to approximate the behavior of power systems dynamics in near real-time. The proposed framework is based on gradient-enhanced physics-informed neural networks (gPINNs) and encodes the underlying physical laws governing power systems. A key characteristic of the proposed gPINN is its ability to train without the need of generating expensive training data. The paper illustrates the potential of the proposed approach in both forward and inverse problems in a single-machine infinite bus system and a three-bus power network for predicting rotor angles and frequency, and uncertain parameters such as inertia and damping to showcase its potential for a range of power systems applications. The model exhibited high accuracy in predicting the variables, achieving a range of 0.533–4.092 and an average L2 relative error improvement of up to 13.30× compared to the PINN model. The computational performance of the proposed gPINN model was compared to a conventional solver, revealing a remarkable speed-up of 31 to 171 times faster in solving differential–algebraic systems of equations in power systems.

MORpH: Model reduction of linear port-Hamiltonian systems in MATLAB

Abstract We present a novel software toolbox MORpH for the efficient storage, analysis, interconnection and structure-preserving model order reduction (MOR) of linear port-Hamiltonian differential-algebraic equation systems (pH-DAEs). The model class of pH-DAEs enables energy-based modeling and a flexible coupling of models across different physical domains. This makes them particularly suited for the simulation and control of complex technical systems. To promote the use of recent theoretical findings in engineering practice, efficient software solutions are required. In this work, we illustrate how possibly large-scale pH-DAEs can be efficiently stored and interconnected in MATLAB in an object-oriented way. We discuss three structure-preserving MOR strategies that are supported by MORpH and demonstrate the application and performance of selected MOR algorithms by means of two benchmark examples.