System Of Differential-algebraic Equations Research Articles

A Decomposition Strategy for the Variational Inference of Complex Systems

Markov chain Monte Carlo approaches have been widely used for Bayesian inference. The drawback of these methods is that they can be computationally prohibitive especially when complex models are analyzed. In such cases, variational methods may provide an efficient and attractive alternative. However, the variational methods reported to date are applicable to relatively simple models and most are based on a factorized approximation to the posterior distribution. Here, we propose a variational approach that is capable of handling models that consist of a system of differential-algebraic equations and whose posterior approximation can be represented by a multivariate distribution. Under the proposed approach, the solution of the variational inference problem is decomposed into three steps: a maximum a posteriori optimization, which is facilitated by using an orthogonal collocation approach, a preprocessing step, which is based on the estimation of the eigenvectors of the posterior covariance matrix, and an expected propagation optimization problem. To tackle multivariate integration, we employ quadratures derived from the Smolyak rule (sparse grids). Examples are reported to elucidate the advantages and limitations of the proposed methodology. The results are compared to the solutions obtained from a Markov chain Monte Carlo approach. It is demonstrated that significant computational savings can be gained using the proposed approach. This article has supplementary material online.

Biodiesel Production by Reactive Flash: A Numerical Simulation

Reactive flash (RF) in biodiesel production has been studied in order to investigate steady-state multiplicities, singularities, and effect of biodiesel quality when the RF system approaches to bubble point. The RF was modeled by an index-2 system of differential algebraic equations, the vapor split (ϕ) was computed by modified Rachford-Rice equation and modified Raoult’s law computed bubble point, and the continuation analysis was tracked on MATCONT. Results of this study show the existence of turning points, leading to a unique bubble point manifold,(xBiodiesel,T)=(0.46,478.41 K), which is a globally stable flashing operation. Also, the results of the simulation in MATLAB® of the dynamic behavior of the RF show that conversion of triglycerides reaches 97% for a residence time of 5.8 minutes and a methanol to triglyceride molar flow ratio of 5 : 1.

Modified group preserving methods and applications in chaotic systems

The tracking control of chaotic system has been one of the research focus areas of nonlinear control in recent years, in which the vital problem is to enable chaotic system to stabilize to an equilibrium point or to track a deterministic trajectory quickly. The conventional chaos control methods make the control power unnecessarily large and generate the phenomenon of chattering easily, resulting in the instabilities of the systems.The problems above can be transformed into the solutions of differential algebraic equations effectively. Considering that the group preserving scheme not only approximates the original system, but also preserve as much as possible the geometric structure and invariants of the original system, this paper takes advantage of the group preserving method to study the control method in chaotic system from two different perspectives.A new group preserving scheme based on the fast descending control method is presented, which enables chaotic system to stabilize to an equilibrium point quickly. Firstly, we introduce a novel approach to replace the optimal control problem of nonlinear system by directly specifying a time-decaying Lagrangian function, which helps us to transform the optimal control problem into a system of differential algebraic equations. Then we derive a modified group preserving scheme for the system.Similarly, we propose a new group preserving scheme based on the sliding mode control method for chaotic system to track a deterministic trajectory quickly. Owing to numerical discretization errors, signal noises and structural uncertainties in dynamical systems, the conventional sliding mode control method cannot guarantee to maintain the trajectories on the sliding surface, unless the numerical integration method is designed to do so. On the other hand, the conventional sliding mode control method easily induces high frequency chattering of the control force. Therefore, we modify the conventional sliding mode control method and use the modified group preserving scheme to find the control force.The above two methods are the combination of traditional control method and the Lie-group method. An invariant manifold is properly designed, and the original system is transformed into the differential algebraic system, in which the modified group preserving scheme can be used to find the control force. The resulting controlled system is stable.Finally, the proposed methods are applied to the classic Lorenz system and Duffing system correspondingly. Numerical experimental results show that the new approaches are very accurate and stable. Since the two controlled methods are fast in convergence and chattering-free, each of them has a good application prospect in the tracking control of chaotic systems.

Closed-loop Formulation for Nonlinear Dynamic Real-time Optimization

Dynamic real-time optimization (DRTO) is a higher level online strategy that exploits plant economic potential by making appropriate adjustments to the lower level controller set-point trajectories. In this work, we propose a closed-loop formulation for a nonlinear DRTO calculation in the form of a bilevel programming problem. A nonlinear differential algebraic equation (DAE) system that describes the process dynamic behavior is utilized with an embedded constrained predictive control (MPC) optimization subproblem to generate the approximate closed-loop response dynamics at the primary economic optimization layer. The bilevel DRTO problem is reformulated as a single-level mathematical program with complementarity constraints (MPCC) by replacing the MPC optimization subproblem by its Karush-Kuhn-Tucker (KKT) optimality conditions. We investigate the economics and control performance of the proposed strategy based on a polymer grade transition case study in the presence of plant-model mismatch and a disturbance, and a comparison is made with the application of a linear DRTO prediction model.

Estimation and Global Stability Analysis of PDAEs with Singular Time Derivative Matrix and Wetland Conservation Application

Some partial differential algebraic equations (PDAEs) system with singular time derivative matrices is analyzed. First, by PDE spectrum theory, this system is formulated as infinite-dimensional singular systems. Second, the state space and its properties of the system are built according to descriptor system theory. Third, the admissible property of the PDAEs is given via LMIs. Finally, the developed energy estimation method is proposed to investigate the global stability of PDAEs. The proposed approach is evaluated by an application in numerical simulations on some wetland conservation system with social behavior.

Positive Stabilization of Linear Differential Algebraic Equation System

We study in this paper the existence of a feedback for linear differential algebraic equation system such that the closed-loop system is positive and stable. A necessary and sufficient condition for such existence has been established. This result can be used to detect the existence of a state feedback law that makes the linear differential algebraic equation system in closed loop positive and stable.

Optimization of Two-Stage Pressure/Vacuum Swing Adsorption with Variable Dehydration Level for Postcombustion Carbon Capture

To investigate postcombustion capture of CO2 in the presence of water, we developed a pressure/vacuum swing adsorption (P/VSA) cycle model consisting of a system of partial differential algebraic equations incorporating mass and energy balances, the Ergun equation for pressure changes, competitive Langmuir isotherms, and the linear driving force model. Four potential adsorbents, zeolites 13X and 5A and the MOFs HKUST-1 and Ni-MOF-74, are investigated, evaluated, and compared. Using this simulation, a two-stage Skarstrom cycle, coupled with an upstream dehydration unit and a downstream compression unit, is optimized using a nondominant sorting genetic algorithm, NSGA-II, to minimize the overall cost of capturing 90% of CO2 from flue gas at a purity of 90% and compressing it for pipeline transportation at 110 bar. The results show that under dry flue gas conditions, zeolite 13X is the best performing adsorbent with an overall cost of $32.1/ton of CO2. Under humid flue gas conditions, zeolites 13X and 5A per...

Erratum to: “Numerical solution of linear differential-algebraic systems of partial differential equations”

Erratum to: “Numerical solution of linear differential-algebraic systems of partial differential equations”

From START to FINISH: computational analysis of cell cycle control in budding yeast.

In the cell division cycle of budding yeast, START refers to a set of tightly linked events that prepare a cell for budding and DNA replication, and FINISH denotes the interrelated events by which the cell exits from mitosis and divides into mother and daughter cells. On the basis of recent progress made by molecular biologists in characterizing the genes and proteins that control START and FINISH, we crafted a new mathematical model of cell cycle progression in yeast. Our model exploits a natural separation of time scales in the cell cycle control network to construct a system of differential-algebraic equations for protein synthesis and degradation, post-translational modifications, and rapid formation and dissociation of multimeric complexes. The model provides a unified account of the observed phenotypes of 257 mutant yeast strains (98% of the 263 strains in the data set used to constrain the model). We then use the model to predict the phenotypes of 30 novel combinations of mutant alleles. Our comprehensive model of the molecular events controlling cell cycle progression in budding yeast has both explanatory and predictive power. Future experimental tests of the model’s predictions will be useful to refine the underlying molecular mechanism, to constrain the adjustable parameters of the model, and to provide new insights into how the cell division cycle is regulated in budding yeast.

Erratum to: A note on the motion representation and configuration update in time stepping schemes for the constrained rigid body

The dynamics of a holonomically constrained rigid body can be modeled by Newton-Euler equations subjected to geometric constraints. This is frequently formulated as a differential-algebraic equation (DAE) system of index 1. In multibody system (MBS) dynamics it is common (1) to numerically solve this system by means of integration schemes for ordinary differential equations, and (2) to treat the rigid body motion on the direct product Lie group $$\textit{SO}\,(3) \times {\mathbb {R}}^{3}$$ , although rigid body motions form the semidirect product Lie group $$\textit{SE}\,(3) $$ . It is has been observed that the constraint satisfaction depends on which Lie group is used as configuration space (c-space). In this paper the problem is considered from a geometric perspective. It is shown that the constraints are exactly satisfied by a numerical integration scheme if they define a subgroup of the c-space. The subgroups of $$\textit{SE}\,(3) $$ have a significance for modeling mechanical systems, including lower kinematic (Reuleaux) pairs and are implicitly used in MBS modeling. It is concluded that $$\textit{SE}\,(3) $$ is the appropriate c-space for numerical DAE modeling of a constrained rigid body. This result does not immediately apply to MBS, however.

Solving Differential-Algebraic Equation Systems: Alternative Index-2 and Index-1 Approaches for Constrained Mechanical Systems

Regarding constrained mechanical systems, we are faced with index-3 differential-algebraic equation (DAE) systems. Direct discretization of the index-3 DAE systems only enforces the position constraints to be fulfilled at the integration-time points, but not the hidden constraints. In addition, order reduction effects are observed in the velocity variables and the Lagrange multipliers. In literature, different numerical techniques have been suggested to reduce the index of the system and to handle the numerical integration of constrained mechanical systems. This paper deals with an alternative concept, called collocated constraints approach. We present index-2 and index-1 formulations in combination with implicit Runge–Kutta methods. Compared with the direct discretization of the index-3 DAE system, the proposed method enforces also the constraints on velocity and—in case of the index-1 formulation—the constraints on acceleration level. The proposed method may very easily be implemented in standard Runge–Kutta solvers. Here, we only discuss mechanical systems. The presented approach can, however, also be applied for solving nonmechanical higher-index DAE systems.

On a nonlinear partial differential algebraic system arising in the technical textile industry: analysis and numerics

In this paper we explore a numerical scheme for a nonlinear fourth order system of partial differential algebraic equations that describes the dynamics of slender inextensible elastica as they arise in the technical textile industry. Applying a semi-discretization in time, the resulting sequence of nonlinear elliptic systems with the algebraic constraint for the local length preservation is reformulated as constrained optimization problems in a Hilbert space setting that admit a solution at each time level. Stability and convergence of the scheme are proved. The numerical realization is based on a finite element discretization in space. The simulation results confirm the analytically predicted properties of the scheme.

Methane/nitrogen separation through pressure swing adsorption process from nitrogen-rich streams

This work presents a novel approach for the treatment of nitrogen contaminated natural gas streams. Literature states that for small and medium scale production wells (<400MNm3/day), such stream can be treated by PSA. For larger scale natural gas production, the only mature option available is cryogenic distillation. Computer simulations were performed to evaluate the usage of silicalite in a PSA process for the separation of a nitrogen rich N2/CH4 mixture. This unit is designed for coupling to a cryogenic distillation process. The differential algebraic equations system, composed of mass and energy balances as well as other constitutive equations, was solved in the spatial dimension using the finite volumes method. Numerical methods employed were analyzed for numerical dispersion and oscillation. In order to test the numerical approach, parameters and model, data from the literature was correlated. A Rotated Central Composite Design (RCCD) was performed in order to determine optimum operational pressures. In the chosen set of operational values, the process presented nitrogen purity greater than 96% and a recovery beyond 50%. The nitrogen rich outlet stream purity is specified for venting, while the methane rich stream is proper for recycling to the cryogenic process.

Continuity of the path delay operator for dynamic network loading with spillback

This paper establishes the continuity of the path delay operators for dynamic network loading (DNL) problems based on the Lighthill–Whitham–Richards model, which explicitly capture vehicle spillback. The DNL describes and predicts the spatial-temporal evolution of traffic flow and congestion on a network that is consistent with established route and departure time choices of travelers. The LWR-based DNL model is first formulated as a system of partial differential algebraic equations. We then investigate the continuous dependence of merge and diverge junction models with respect to their initial/boundary conditions, which leads to the continuity of the path delay operator through the wave-front tracking methodology and the generalized tangent vector technique. As part of our analysis leading up to the main continuity result, we also provide an estimation of the minimum network supply without resort to any numerical computation. In particular, it is shown that gridlock can never occur in a finite time horizon in the DNL model.

Reduction of Physical and Constraint Degrees-of-Freedom of Redundant Formulated Multibody Systems

Commercial multibody system simulation (MBS) tools commonly use a redundant coordinate formulation as part of their modeling strategy. Such multibody systems subject to holonomic constraints result in second-order d-index three differential algebraic equation (DAE) systems. Due to the redundant formulation and a priori estimation of possible flexible body coordinates, the model size increases rapidly with the number of bodies. Typically, a considerable number of constraint equations (and physical degrees-of-freedom (DOF)) are not necessary for the structure's motion but are necessary for its stability like out-of-plane constraints (and DOFs) in case of pure in-plane motion. We suggest a combination of both, physical DOF and constraint DOF reduction, based on proper orthogonal decomposition (POD) using DOF-type sensitive velocity snapshot matrices. After a brief introduction to the redundant multibody system, a modified flat Galerkin projection and its application to index-reduced systems in combination with POD are presented. The POD basis is then used as an identification tool pointing out reducible constraint equations. The methods are applied to one academic and one high-dimensional practical example. Finally, it can be reported that for the numerical examples provided in this work, more than 90% of the physical DOFs and up to 60% of the constraint equations can be omitted. Detailed results of the numerical examples and a critical discussion conclude the paper.

A note on the motion representation and configuration update in time stepping schemes for the constrained rigid body

The dynamics of a holonomically constrained rigid body can be modeled by Newton-Euler equations subjected to geometric constraints. This is frequently formulated as a differential-algebraic equation (DAE) system of index 1. In multibody system (MBS) dynamics it is common (1) to numerically solve this system by means of integration schemes for ordinary differential equations, and (2) to treat the rigid body motion on the direct product Lie group $$\textit{SO}\,(3) \times {\mathbb {R}}^{3}$$ , although rigid body motions form the semidirect product Lie group $$\textit{SE}\,(3) $$ . It is has been observed that the constraint satisfaction depends on which Lie group is used as configuration space (c-space). In this paper the problem is considered from a geometric perspective. It is shown that the constraints are exactly satisfied by a numerical integration scheme if they define a subgroup of the c-space. The subgroups of $$\textit{SE}\,(3) $$ have a significance for modeling mechanical systems, including lower kinematic (Reuleaux) pairs and are implicitly used in MBS modeling. It is concluded that $$\textit{SE}\,(3) $$ is the appropriate c-space for numerical DAE modeling of a constrained rigid body. This result does not immediately apply to MBS, however.

Solution of the incompressible Navier–Stokes equations by the method of lines

SummaryThe numerical method of lines (NUMOL) is a numerical technique used to solve efficiently partial differential equations. In this paper, the NUMOL is applied to the solution of the two‐dimensional unsteady Navier–Stokes equations for incompressible laminar flows in Cartesian coordinates. The Navier–Stokes equations are first discretized (in space) on a staggered grid as in the Marker and Cell scheme. The discretized Navier–Stokes equations form an index 2 system of differential algebraic equations, which are afterwards reduced to a system of ordinary differential equations (ODEs), using the discretized form of the continuity equation. The pressure field is computed solving a discrete pressure Poisson equation. Finally, the resulting ODEs are solved using the backward differentiation formulas.The proposed method is illustrated with Dirichlet boundary conditions through applications to the driven cavity flow and to the backward facing step flow. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Numerical solution of linear differential-algebraic systems of partial differential equations

A linear differential-algebraic system of partial differential equations of an arbitrary index is examined. The index of a differential-algebraic system is determined by the maximum degree (over the corresponding domain) of the elementary divisors corresponding to the zero and infinite roots of the characteristic polynomial of the matrix pencil constructed from the coefficients of this system. An iterative algorithm for the numerical solution of a differential-algebraic system is proposed. It is based on a special splitting of the associated matrix pencil and an application of the method of successive approximations to the split system. The stability of this method is shown, and its efficiency is demonstrated through test examples.

An Improved KM Algorithm for Computing the Structural Index of DAE System

Modeling and simulation technology is widely used to design complex products in industry. The problem of solving DAEs(Differential Algebraic Equations) is a key part of modeling and simulation technology, and computing the structural index of DAEs correctly and efficiently is very important to solve DAEs. The traditional algebraic method to compute the structural index is very costly. In this paper, we firstly convert the problem of computing the structural index of DAEs into the maximum weighted matching problem of bipartite graph, reducing a mass of symbolic manipulations; and then, we present an improved KM algorithm(called as Greedy_KM in this paper) based on the properties of DAEs to solve this matching problem. In order to solve the matching problem efficiently, it firstly computes matches as much as possible using greedy strategy, and then call KM algorithm to search the matches for the unmatched vertices after the step of greedy strategy. This paper also gives a set of numerical experiments to evaluate the time performance of our method. The results show that the time performance of Greedy_KM algorithm is significantly improved compared with the traditional Gaussian elimination algorithm and classical KM algorithm.

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.