Semi-explicit Differential-algebraic Equations Research Articles

Physics-informed neural networks for dynamic process operations with limited physical knowledge and data

In chemical engineering, process data are expensive to acquire, and complex phenomena are difficult to fully model. We explore the use of physics-informed neural networks (PINNs) for modeling dynamic processes with incomplete mechanistic semi-explicit differential–algebraic equation systems and scarce process data. In particular, we focus on estimating states for which neither direct observational data nor constitutive equations are available. We propose an easy-to-apply heuristic to assess whether estimation of such states may be possible. As numerical examples, we consider a continuously stirred tank reactor and a liquid-liquid separator. We find that PINNs can infer immeasurable states with reasonable accuracy, even if respective constitutive equations are unknown. We thus show that PINNs are capable of modeling processes when relatively few experimental data and only partially known mechanistic descriptions are available, and conclude that they constitute a promising avenue that warrants further investigation.

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.

Normal form analysis in the presence of SVC and STATCOM and the effect of their siting on damping of rotor swings

Normal form analysis is used to assess the nonlinear behavior; it can also be used for siting controllers. Normal form analysis is straight-forward for systems governed by explicit ordinary differential equations. However, the analysis becomes complicated if the governing equations are differential–algebraic. The presence of SVC/STATCOM results in the governing equations for which the normal form analysis is not as straight-forward as that for explicit differential equations. In this paper, it is shown how semi-explicit differential–algebraic equations of index 1 are amenable to normal form analysis. In particular, methods are proposed for normal form analysis in the presence of SVC and STATCOM. Normal form analysis can be used to find the location of SVC and STATCOM that results in the best damping of rotor swings; the results are analyzed by case studies and also compared with the performance obtained by locating SVC and STATCOM using linearization methods.

Fuzzy transform-based approximation method for solving fractional semi-explicit differential-algebraic equations

We present an efficient numerical method to approximate the solution of a system of fractional-order linear semi-explicit differential-algebraic equations with variable coefficients. The method is based on the use of the direct and inverse fuzzy transforms ( $${\mathcal {F}}$$ -transforms). By employing this method, we obtain an analytical approximate solution to the main problem in terms of flexible basic functions. The non-local property of fuzzy transforms helps us to have an efficient method for problems involving non-singular kernels. The error analysis and convergence evaluation of the method are demonstrated in detail. We give some examples to illustrate the significant features of the method.

Efficient Numerical Optimal Control for Highly Oscillatory Systems

We present an efficient transcription method for highly oscillatory optimal control problems. For these problems, the optimal state trajectory consists of fast oscillations that change slowly over the time horizon. Out of a large number of oscillations, we only simulate a subset to approximate the slow change by constructing a semi-explicit differential-algebraic equation that can be integrated with integration steps much larger than one period. For the solution of optimal control problems with direct methods, we provide a way to parametrize the controls and regularize the state trajectory. Finally, we utilize the method to find a fuel-optimal orbit transfer of a low-thrust satellite. Using the novel method, we reduce the size of the resulting nonlinear program by more than one order of magnitude.

Continuous Galerkin schemes for semiexplicit differential-algebraic equations

Abstract This paper studies a new class of integration schemes for the numerical solution of semiexplicit differential-algebraic equations of differentiation index 2 in Hessenberg form. Our schemes provide the flexibility to choose different discretizations in the differential and algebraic equations. At the same time they are designed to have a property called variational consistency, i.e., the choice of the discretization of the constraint determines the discretization of the Lagrange multiplier. For the case of linear constraints we prove convergence of order $r+1$, both for the state and the multiplier if piecewise polynomials of order $r$ are used. These results are also verified numerically.

Theory of index-one nonlinear complementarity systems

Generalized differentiation index-one nonlinear complementarity systems (NCSs) are mathematically regularized in this article. Strong regularity of solutions is shown to imply generalized differentiation index one holding, a concept from nonsmooth differential-algebraic equations (DAEs) theory. Well-posedness of NCSs is established, including continuation and maximal continuation of strongly-regular solutions. It is demonstrated that strongly-regular NCS solutions are lexicographically smooth (in the sense of Nesterov) with respect to parameters, implying Lipschitzian parametric dependence of solutions. These results are accomplished by relating strong regularity to complete coherent orientation from the theory of piecewise continuously differentiable functions, and an extended nonsmooth implicit function theorem useful for analyzing NCSs. The findings in this article lay the foundation for computationally-relevant sensitivity analysis theory and dynamic optimization methods for practical problems modeled using NCSs. Along the way, theory of semi-explicit DAEs with piecewise continuously differentiable right-hand side functions, which encompass a number of frameworks, is formalized.

Arnoldi Algorithms with Structured Orthogonalization

We study a stability preserved Arnoldi algorithm for matrix exponential in the time domain simulation of large-scale power delivery networks (PDN), which are formulated as semi-explicit differential algebraic equations (DAEs). The solution can be decomposed to a sum of two projections, one in the range of the system operator and the other in its null space. The range projection can be computed with one shift-and -invert Krylov subspace method. The other projection can be computed with the algebraic equations. Differing from the ordinary Arnoldi method, the orthogonality in the Krylov subspace is replaced with the semi-inner product induced by the positive semi-definite system operator. With proper adjustment, numerical ranges of the Krylov operator lie in the right half plane, and we obtain theoretical convergence analysis for the modified Arnoldi algorithm in computing phi-functions. Lastly, simulations on RLC networks are demonstrated to validate the effectiveness of the Arnoldi algorithm with structured-orthogonalization.

On Koopman Operator Framework for Semi-explicit Differential-Algebraic Equations

Abstract The so-called Koopman operator framework has attracted a lot of interest in dynamical systems theory and control theory. In this note we extend the framework to systems described by nonlinear semi-explicit Differential-Algebraic Equations (DAEs). A complete spectral characterization of DAE systems with asymptotically stable equilibrium points is presented. A difficulty of the Koopman operator framework is also pointed out for a planar DAE system with forward impasse point.

Sensitivity analysis of nonsmooth power control systems with an example of wind turbines

This paper focuses on dynamic sensitivities of nonsmooth power control systems. Recent advances in nonsmooth sensitivity theory are applicable to large-scale practical problems modeled as nonsmooth semi-explicit differential-algebraic equations (DAEs). This theory is applied here to a wind turbine control system with nonsmooth input parameter profiles. First-order sensitivity information is characterized from an auxiliary, nonsmooth sensitivity system which includes discontinuities because of nonsmooth wind speed profiles. Dynamic simulations of sensitivities are provided which give new insights on design and implementation (e.g., the system’s control limits).

Diffusive subsurface propagation of Rayleigh mode induced by embedded ultrasonic waveguide

Surface degradation in the wood surface has generated unwanted variation in the ultrasound and reduced the efficacy of ultrasonic based non-destructive evaluation (NDE). This study proposes a novel alternative by inserting a waveguide into the wooden medium to eliminate the impact of surface variation on the characteristics of the ultrasonic propagation. This technique creates a small circular insertion of 2 cm depth with 5 mm in diameter, which has been accepted by many utility inspection firms for structural health monitoring (SHM) of wooden utility grids. This work develops a closed-form analytical solution of the elastodynamic problem using the proposed embedded waveguide technique. The resulted solution based on the Navier's formulation discovered two interfering wave modes embedded in the displacement field. These two modes termed fast and slow induce diffusive displacement propagation as a function of the Poisson's ratio of the medium. The analytical result also shows a greater elastic energy penetration as a function of the insertion depth. Resulted wave characteristics are different from the traditional contact-based Rayleigh excitation. A developed numerical model using the semi-explicit differential-algebraic equation (DAE) technique with both steady-state and transient load conditions is demonstrated. The simulated displacement fields are analyzed and compared with the analytical result.

Efficient numerical methods for gas network modeling and simulation

We study the modeling and simulation of gas pipeline networks, with a focus on fast numerical methods for the simulation of transient dynamics. The obtained mathematical model of the underlying network is represented by a nonlinear differential algebraic equation (DAE). With our modeling, we reduce the number of algebraic constraints, which correspond to the $(2,2)$ block in our semi-explicit DAE model, to the order of junction nodes in the network, where a junction node couples at least three pipelines. We can furthermore ensure that the $(1, 1)$ block of all system matrices including the Jacobian is block lower triangular by using a specific ordering of the pipes of the network. We then exploit this structure to propose an efficient preconditioner for the fast simulation of the network. We test our numerical methods on benchmark problems of (well-)known gas networks and the numerical results show the efficiency of our methods.

Stability and Convergency Exploration of Matrix Exponential Integration on Power Delivery Network Transient Simulation

We propose a stability preserved Arnoldi algorithm for matrix exponential in the time domain simulation of large-scale power delivery networks (PDNs), which are formulated as semi-explicit differential-algebraic equations (DAEs). The matrix exponential and vector products (MEVPs) compose the solution of DAEs in multistep integration methods and can be efficiently approximated with the rational Krylov subspace. To produce stable simulation results for the ill-conditioned system from semi-explicit DAEs, the revised Arnoldi algorithm introduces a new structured orthogonalization process to construct the Krylov subspace. We demonstrate the performance of the new algorithm with theoretical proof and experiments. In the computation of MEVPs, we utilize the exponential related $\varphi $ functions to improve the numerical accuracy. We further explore the optimal ratio to confine the spectrum in the rational Krylov subspace. Finally, the transient framework is tested on a group of system-level PDNs, showing that matrix exponential-based algorithms could achieve high efficiency and accuracy.

Multirate implicit Euler schemes for a class of differential–algebraic equations of index-1

Systems of differential equations which consist of subsystems with widely different dynamical behaviour can be integrated by multirate time integration schemes to increase the efficiency. These schemes allow the usage of inherent step sizes according to the dynamical properties of the subsystem. In this paper, we extend the multirate implicit Euler method to semi-explicit differential–algebraic equations of index-1 where the algebraic constraints only occur in the slow changing subsystem. We discuss different coupling approaches and show that consistency and convergence order 1 can be reached. Numerical experiments validate the analytical results.

Internal and External Linearization of Semi-Explicit Differential Algebraic Equations

In this paper, we study two kinds of linearization (internal and external) of nonlinear differential-algebraic equations DAEs of semi-explicit SE form. The difference of external and internal linearization is illustrated by an example of a mechanical system. Moreover, we define different levels of external equivalence for two SE DAEs. The proposed explicitation procedure allows us to treat a given SE DAE as a control system defined up to feedback transformation (a class of control systems). Then sufficient and necessary conditions, expressed via explicitation procedure, are given to describe when a given SE DAE is level-3 externally equivalent to a linear SE DAE of some specific forms. At last, we show by an example that level-2 external linearization of a DAE can be achieved if its explicitation is level-2 input-output linearizable.

High-order Legendre collocation method for fractional-order linear semi-explicit differential algebraic equations

High-order Legendre collocation method for fractional-order linear semi-explicit differential algebraic equations

Necessary Optimality Conditions for Implicit Control Systems with Applications to Control of Differential Algebraic Equations

In this paper we derive necessary optimality conditions for optimal control problems with nonlinear and nonsmooth implicit control systems. Implicit control systems have wide applications including differential algebraic equations (DAEs). The challenge in the study of implicit control system lies in that the system may be truly implicit, i.e., the Jacobian matrix of the constraint mapping may be singular. Our necessary optimality conditions hold under the so-called weak basic constraint qualification plus the calmness of a perturbed constraint mapping. Such constraint qualifications allow for singularity of the Jacobian and hence are suitable for implicit systems. Specifying these results to control of semi-explicit DAEs we obtain necessary optimality conditions for control of semi-explicit DAEs with index higher than one.

Dependence of solutions of nonsmooth differential-algebraic equations on parameters

The well-posedness of nonsmooth differential-algebraic equations (DAEs) is investigated. More specifically, semi-explicit DAEs with Carathéodory-style assumptions on the differential right-hand side functions and local Lipschitz continuity assumptions on the algebraic equations. The DAEs are classified as having differential index one in a generalized sense; solution regularity is formulated in terms of projections of generalized (Clarke) Jacobians. Existence of solutions is derived under consistency and regularity of the initial data. Uniqueness of a solution is guaranteed under analogous Carathéodory ordinary-differential equation uniqueness assumptions. The continuation of solutions is established and sufficient conditions for continuous and Lipschitzian parametric dependence of solutions are also provided. To accomplish these results, a theoretical tool for analyzing nonsmooth DAEs is provided in the form of an extended nonsmooth implicit function theorem. The findings here are a natural extension of classical results and lay the foundation for further theoretical and computational analyses of nonsmooth DAEs.

Formulation and numerical analysis of a fully-coupled dynamically deforming electromagnetic wire

An electromagnetic beam model is developed for the simulation of actuated electronic textiles. The beam is solved using a nonlinear director-based kinematic description with additional temperature and electric potential fields along its length. The three fields are fully coupled by mutual dependences on the deformation, Lorenz force, back electromotive force, temperature dependent constitutive responses, and the Seebeck effect. Instead of solving Maxwell’s equations in full detail, a quasistatic approximation is used to solve the electric potential in the presence of a moving material medium. The current-carrying beam approximation is used to further simplify the solution space for the potential. While this formulation alleviates the spatial and temporal discretization restrictions, the coupled problem is an index-1 semi-explicit Differential Algebraic Equation requiring special treatment. The time dependent problem is solved using different Runge–Kutta methods. Diagonally implicit Runge–Kutta methods and explicit Runge–Kutta methods using implicit solution of the electric potential problem are explored. The finite element model is implemented using the open source package FEniCS, which is able to automatically generate the linearizations of the multiphysics equations required for the implicit solutions. A model problem is constructed with which to test and analyze the physical formulation and numerical solution techniques. The time stepping methods are verified using the convergence orders of the higher-order Runge–Kutta methods. Runtime comparisons show that the explicit methods are generally more computationally efficient than the implicit schemes used for this problem. For the implicit schemes, a staggered solution is significantly faster than a monolithic solution at most time step sizes. However, at very large time steps, such as those that would be used for dynamic relaxation, the monolithic solution can be more efficient than the staggered solution.

High-Performance Method for Analyzing Shielding Current Density in HTS Films: Application to the Scanning Permanent-Magnet Method

A fast and stable method is proposed for calculating the shielding current density in a high-temperature superconducting film containing cracks. After discretized with the finite-element method, the initial-boundary-value problem of the shielding current density reduces to semi-explicit differential algebraic equations (DAEs). Although the DAEs can be solved with standard ordinary-differential-equation (ODE) solvers, much CPU time is required for its numerical solution. In order to shorten the CPU time, a high-speed algorithm is proposed. In the algorithm, the block LU decomposition is incorporated into function evaluations in ODE solvers. A numerical code is developed on the basis of the proposed method, and detectability of cracks by the scanning permanent-magnet method (SPM) is numerically investigated. The results of computations show that the SPM can realize high-speed crack detection. In addition, it is also found that, for multiple cracks, resolution of the SPM will be remarkably degraded.