Set Of Differential Algebraic Equations Research Articles

Extensions of ValEncIA‐IVP for reduction of overestimation, for simulation of differential algebraic systems, and for dynamical optimization

AbstractSimulation techniques are commonly used to analyze the influence of uncertainties of initial conditions and systemparameters on the trajectories of the state variables of dynamical systems. In this context, interval arithmetic approaches are of interest. They are capable of determining guaranteed bounds of all reachable states if worst‐case bounds of the above‐mentioned uncertainties are known. Furthermore, interval algorithms ensure the correctness of numerical results in spite of rounding errors which inevitably arise if floating point operations are carried out on a computer. However, naive implementations of interval algorithms often lead to overestimation, i.e., too conservative enclosures which can make the results meaningless.In this contribution, we summarize the basic routines of ValEncIA‐IVP which computes interval enclosures of all reachable states of dynamical systems described by ordinary differential equations ODEs. ValEncIA‐IVP, VALidation of state ENClosures using Interval Arithmetic for Initial Value Problems, can be applied to the simulation of systems with both uncertain parameters and uncertain initial conditions. Advanced techniques for reduction of overestimation are demonstrated for a simplified catalytic reactor. Afirst approach to using VanEncIA‐IVP for the simulation of sets of differential algebraic equations is outlined. Finally, an outlook on the integration of ValEncIA‐IVP in an interval arithmetic framework for computation of optimal and robust control strategies for continuous‐time processes is given. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Transient stabilization of structure preserving power systems with excitation control via energy-shaping

In this paper, the transient stability of multimachine power systems based on structure preserving model (SPM) is considered. The interconnection and damping assignment passivity-based control (IDA-PBC) methodology is extended to solve the excitation regulation problem of SPM represented by a set of differential-algebraic equations. By shaping the total energy function via the introduction of a virtual coupling between the electrical and the mechanical dynamics of the power system, a decentralized excitation control law is proposed to ensure the asymptotic stability of the closed-loop system. The controller is proved to be effective in damping the oscillations and enhancing the system stability by the results of simulation research.

The discrete null space method for the energy-consistent integration of constrained mechanical systems. Part III: Flexible multibody dynamics

In the present work, the unified framework for the computational treatment of rigid bodies and nonlinear beams developed by Betsch and Steinmann (Multibody Syst. Dyn. 8, 367–391, 2002) is extended to the realm of nonlinear shells. In particular, a specific constrained formulation of shells is proposed which leads to the semi-discrete equations of motion characterized by a set of differential-algebraic equations (DAEs). The DAEs provide a uniform description for rigid bodies, semi-discrete beams and shells and, consequently, flexible multibody systems. The constraints may be divided into two classes: (i) internal constraints which are intimately connected with the assumption of rigidity of the bodies, and (ii) external constraints related to the presence of joints in a multibody framework. The present approach thus circumvents the use of rotational variables throughout the whole time discretization, facilitating the design of energy–momentum methods for flexible multibody dynamics. After the discretization has been completed a size-reduction of the discrete system is performed by eliminating the constraint forces. Numerical examples dealing with a spatial slider-crank mechanism and with intersecting shells illustrate the performance of the proposed method.

A minimal set of dynamic equations in systems modelled with bond graphs

The current paper presents a procedure for converting to a minimal set the dynamic equations of a system modelled with bond graphs starting from a large set of differential-algebraic equations including any type of topological loops and zero-order causal paths (ZCPs). This minimal set is composed only of differential equations expressed in terms of the degrees of freedom of the system. These equations are suitable for use in real-time simulations. Initially, flows corresponding to inertances and displacements associated with compliances are used to establish the dynamic equations and to find the ZCPs of the system. Two different methods are used to solve the ZCPs. With the first method, Lagrange multipliers are introduced by means of new flows and efforts as break variables of causal paths, adding constraint equations. With the second method, break variables are used directly to open the ZCPs. The procedure developed in the present paper solves the problem originated by the use of Lagrange multipliers with the introduction of break variables that imply the presence of new variables and constraint equations. This procedure, before numerical integration of the dynamic equations, converts the equations to a minimal set in order to gain computational efficiency. Several examples are also provided to illustrate the conversion steps.

Modelling and simulation of a sphere rolling on the inside of a rough vertical cylinder

A classical problem of nonholonomic system dynamics—the motion of a sphere on the inside of a rough vertical cylinder—is extended to rolling friction. The case study is modelled in independent coordinates. Due to the nonholonomic constraints imposed on the sphere, the governing equations arise as a set of differential-algebraic equations. The results of numerical simulations show the transition of the sphere from a sinusoid path on the vertical cylinder surface to a fall with slip. The physics of the ‘paradoxical’ motion is explained in detail.

Modeling and simulation of milk emulsion drying in spray dryers

This work aims at modeling and simulating the drying of whole milk emulsion in spray dryers. Drops and particles make up the discrete phase and are distributed into temporal compartments following their residence time in the dryer. Air is the continuous and well-mixed phase. Mass and energy balances are developed for each phase, taking into account their interactions. Constitutive equations for describing the drop swelling and drying mechanisms as well as the heat and mass transfer between particles and hot air are proposed and analyzed. A set of algebraic-differential equations is obtained and solved by specific numerical codes. Results from experiments carried out in a pilot spray dryer are used to validate the model developed and the numerical algorithm. Comparing the simulated and experimental data, it is shown that the model predicts well the individual drop-particle history inside the dryer as well as the overall outlet air-particle temperature and humidity.

Turning dynamics and equilibrium of two-wheeled vehicles

The equations of motion of two-wheeled vehicles, e g bicycles or motorcycles, are developed by using Lagrange’s equations for quasi-coord mates The pure rolling constiatnts between the ground and the two wheels aie considered in the dynamical equations of the system For each wheel, two nonholonomic and two holonomic constraints are introduced in a set of differential-algebraic equations (DAE) The constraint Jacobian matrix is obtained by collecting all the constraint equations and converting them into the velocity form Equilibrium, an algorithm for searching for equilibrium points of two-wheeled vehicles and the associated problems are discussed Formulae foi calculating the radii of curvatures of ground-wheel contact paths and the reference point are also given

Simulations of transverse vibrations of an axially moving string: a modified difference approach

A modified finite difference approach to simulate transverse vibrations of an axially moving string is presented. The stress is treated as a new unknown in discretization of the spatial variable. A set of differential-algebraic equations is established based on the discreted governing equation and the constitutive relation. For linear vibrations, a conserved functional is employed to test the algorithm, and the 1, 2, 3, 4-term truncated modal analytical solutions are compared with the numerical solution. For the free nonlinear vibration, a new conserved functional is used to check the algorithm. Effects of the transport speed on the free and forced nonlinear vibrations are numerically investigated.

Computation of Singular and Singularity Induced Bifurcation Points of Differential-Algebraic Power System Model

In this paper, we present an efficient algorithm to compute singular points and singularity-induced bifurcation points of differential-algebraic equations for a multimachine power-system model. Power systems are often modeled as a set of differential-algebraic equations (DAE) whose algebraic part brings singularity issues into dynamic stability assessment of power systems. Roughly speaking, the singular points are points that satisfy the algebraic equations, but at which the vector field is not defined. In terms of power-system dynamics, around singular points, the generator angles (the natural states variables) are not defined as a graph of the load bus variables (the algebraic variables). Thus, the causal requirement of the DAE model breaks down and it cannot predict system behavior. Singular points constitute important organizing elements of power-system DAE models. This paper proposes an iterative method to compute singular points at any given parameter value. With a lemma presented in this paper, we are also able to locate singularity induced bifurcation points upon identifying the singular points. The proposed method is implemented into voltage stability toolbox and simulations results are presented for a 5-bus and IEEE 118-bus systems.

Modelling of a dynamic multiphase flash: the positive flashApplication to the calculation of ternary diagrams

A general and polyvalent model for the dynamic simulation of a vapor, liquid, liquid–liquid, vapor–liquid or vapor–liquid–liquid stage is proposed. This model is based on the τ -method introduced as a minimization problem by [Comput. Chem. Eng. 22 (7/8) (1998) 897] for steady-state simulation. They suggested modifying the mole fraction summation such that the same set of governing equations becomes valid for all phase regions. Thanks to judicious additional switch equations, the τ -formulation is extended to dynamic simulation and the minimization problem is transformed into a set of differential algebraic equations (DAE). Validation of the model consists in testing its capacity to overcome phase number changes and to be able to solve several problems with the same set of equations: calculation of heterogeneous residue curves, azeotropic points and distillation boundaries in ternary diagrams.

Modelling of a dynamic multiphase flash: the positive flash: Application to the calculation of ternary diagrams

A general and polyvalent model for the dynamic simulation of a vapor, liquid, liquid–liquid, vapor–liquid or vapor–liquid–liquid stage is proposed. This model is based on the τ-method introduced as a minimization problem by [Comput. Chem. Eng. 22 (7/8) (1998) 897] for steady-state simulation. They suggested modifying the mole fraction summation such that the same set of governing equations becomes valid for all phase regions. Thanks to judicious additional switch equations, the τ-formulation is extended to dynamic simulation and the minimization problem is transformed into a set of differential algebraic equations (DAE). Validation of the model consists in testing its capacity to overcome phase number changes and to be able to solve several problems with the same set of equations: calculation of heterogeneous residue curves, azeotropic points and distillation boundaries in ternary diagrams.

A Geometric Approach to Solving Problems of Control Constraints: Theory and a DAE Framework

The paper deals with controlled mechanical systems in which the number of control inputs is equal to the number of desired system outputs, and is smaller than the number of degrees of freedom of the system. The determination of control input strategy that force the underactuated system to complete the partly specified motion is a challenging problem. In the present formulation, the outputs (performance goals), expressed in terms of system states, are treated as constraints on the system—called control or program constraints as distinct from contact constraints in the classical sense, and a mathematical resemblance of the inverse control problem to the constrained system dynamics is exploited. However, while the reactions of contact constraints act in the directions orthogonal to the respective constraint manifold, the available control reactions may have arbitrary directions with respect to the program constraint manifold, and in the extreme may be tangent. A specific methodology must then be developed to find the solution of such “singular” problems, related to a class of control tracking problems such as position control of elastic joint robots, control of cranes, and aircraft control in prescribed trajectory flight. The governing equations of the problem arise as a set of differential-algebraic equations (DAEs), and an effective method for solving the DAEs, based on backward Euler method, is proposed. The open-loop control formulation obtained this way is then extended by a closed-loop control law to provide stable tracking of the required reference trajectories in the presence of perturbations. Some examples of applications of the theory and results of numerical simulations are reported.

Symbolic Numeric Index Analysis Algorithm for Differential Algebraic Equations

Mathematical modeling of dynamic chemical engineering systems such as flow-sheet simulation, transient analysis of reaction and separation networks, computational fluid dynamic, or control problems often leads to systems of differential−algebraic equations (DAEs). Knowing the index of a DAE is an important prerequisite for its consistent initialization and its numerical solution. For large sets of DAEs occurring in chemical engineering practice, index analysis requires computational procedures. This paper presents a new approach for automatic index analysis. Earlier index analysis algorithms were based on purely structural calculus. Unfortunately, structural methods may actually arbitrarily over- or underestimate the correct DAE index. To overcome the shortcomings of existing approaches, this paper proposes a novel algorithm expanding DAE analysis to include both symbolic and numerical information. The novel symbolic numerical analysis accounts for variable cancellation and processes linear expressions ac...

Power System Modeling for Inverse Problems

Large disturbances in power systems often initiate complex interactions between continuous dynamics and discrete events. The paper develops a hybrid automaton that describes such behavior. Hybrid systems can be modeled in a systematic way by a set of differential-algebraic equations, modified to incorporate impulse (state reset) action and constraint switching. This differential-algebraic impulsive-switched (DAIS) model is a realization of the hybrid automaton. The paper presents a practical object-oriented approach to implementing the DAIS model. Each component of a system is modeled autonomously. Connections between components are established by simple algebraic equations. The systematic nature of the DAIS model enables efficient computation of trajectory sensitivities, which in turn facilitate algorithms for solving inverse problems. The paper outlines a number of inverse problems, including parameter uncertainty, parameter estimation, grazing bifurcations, boundary value problems, and dynamic embedded optimization.

A Complete Dynamic Model for Twin Screw Extruders

A one-dimensional physically motivated dynamic model of a twin-screw extruder for reactive extrusion has been developed. This model can predict extruder behaviour such as pressure, filling ratio, temperature and molar conversion as well profiles as residence time distribution under various operating conditions such as feed rate, screw speed, monomer/initiator ratio and heat flux supplied to the barrel. The model consists of a cascade of perfectly stirred reactors which can be either fully or partially filled with backflow. We consider the mass balance coupled with the momentum balance for the calculation of the pressure profile and the flows between the different reactors. At each reactor is associated the concentration in monomer, the temperature of the matter, the temperatures of the associated piece of screw and barrel. The final model consists of a set of differential algebraic equations. The experimental validation is only made on flow aspects of the model by using experimental RTD's.

An Algorithm for Optimizing the Unsteady Chemical Processes by Simulated Annealing

In this paper, we propose an algorithm based on basic simulated annealing (SA) to optimize unsteady chemical systems. By parametrizing the control inputs, the system described by a set of differential-algebraic equations is first transferred into a nonlinear programming (NLP) problem. By using a linear function, we furthermore convert discretized control inputs and time grids into the control profile with a variable time length to improve numerical quality. Thus, the discretized control inputs and the corresponding execution time lengths will be considered as a set of decision variables. Then, these decision variables are globally determined by our algorithm with the help of a special integrator to optimize the performance index. In order to exhibit the facility of the proposed algorithm, several typical examples are provided.

Reactive transport in surface sediments. II. Media: an object-oriented problem-solving environment for early diagenesis

The MEDIA (Modelling Early DIAgenesis) software package comprises a flexible and extensible software system that provides problem-solving assistance for simulating 1D reactive transport in surface sediments. MEDIA allows multiple diagenetic models to be built by extending a model template with new model components from a toolbox of available objects (elements, species, parameters, reactions). A detailed review is given of the transport and reaction components available for model construction. Upon assemblage, the model is channelled to the numerical subunit of the MEDIA package. Via a canonical transformation, the user-defined mixed kinetic-equilibrium model is rearranged into a proper set of differential algebraic equations (DAE), for which both steady state and transient solutions can be calculated. Steady-state profiles are obtained either directly using a Newton–Raphson method, or alternatively, as the asymptotic result of a dynamic simulation. Dynamic simulations involve a global implicit procedure based on the stiff-ODE solver package VODE, employing a direct substitution approach to reduce the number of equations in the DAE system. Verification of the MEDIA code was accomplished (1) by comparison with analytic models and (2) by emulating the model formulation and output of the existing diagenetic model code STEADYSED. As an example application, a diagenetic model was constructed to analyse an extensive dataset collected from a marine sediment in the Santa Barbara Basin (California). The different pathways of organic matter mineralization were modelled, and the coupling among the biogeochemical cycles of C, O, N, S, Mn and Fe was investigated. Depth profiles of both porewater and solid-phase constituents could be reproduced with great accuracy.

Dynamic Modelling and Identification of a Waterjet Cutting System

This paper describes a new dynamic model for a waterjet cutting system that includes a double-acting reciprocating intensifier pump. Since the system operates at high pressures the fluid flow is assumed to be compressible. The dynamic model includes the characteristics of the intensifier pump, the check valves, the accumulator, the system piping and compressible jet flow through the nozzle. The system model is presented as a set of differential-algebraic equations. Experimental results for an actual system are used to identify the discharge coefficient of the nozzle, certain unknown parameters associated with the check valve, and to determine the velocity profile of the piston in the intensifier pump. This is accomplished by formulating and solving a parameter optimization problem. The paper also includes numerical simulation results that validate the dynamic model.

Foundations of a query and simulation system for the modeling of biochemical and biological processes.

The analysis of large amounts of data, produced as (numerical) traces of in vivo, in vitro and in silico experiments, has become a central activity for many biologists and biochemists. Recent advances in the mathematical modeling and computation of biochemical systems have moreover increased the prominence of in silico experiments; such experiments typically involve the simulation of sets of Differential Algebraic Equations (DAE), e.g., Generalized Mass Action systems (GMA) and S-systems. In this paper we reason about the necessary theoretical and pragmatic foundations for a query and simulation system capable of analyzing large amounts of such trace data. To this end, we propose to combine in a novel way several well-known tools from numerical analysis (approximation theory), temporal logic and verification, and visualization. The result is a preliminary prototype system: simpathica/xssys. When dealing with simulation data simpathica/xssys exploits the special structure of the underlying DAE, and reduces the search space in an efficient way so as to facilitate any queries about the traces. The proposed system is designed to give the user possibility to systematically analyze and simultaneously query different possible timed evolutions of the modeled system.

H/sub ∞/ control for nonlinear descriptor systems

In this paper, we study the H/sub /spl infin// control problem for nonlinear descriptor systems governed by a set of differential-algebraic equations (DAEs) of the form Ex/spl dot/ = F(x, w, u), z = Z(x, w, u), y = Y(x, w, u), where E is, in general, a singular matrix. Necessary and sufficient conditions are derived for the existence of a controller solving the problem. We first give various sufficient conditions for the solvability of H/sub /spl infin// control problem for DAEs. Both state-feedback and output-feedback cases are considered. Then, necessary conditions for the output feedback control problem to be solvable are obtained in terms of two Hamilton-Jacobi inequalities plus a weak coupling condition. Moreover, a parameterization of a family of output feedback controllers solving the problem is also provided.