Index-aware model order reduction for differential-algebraic equations

We introduce a model order reduction (MOR) procedure for differential-algebraic equations, which is based on the intrinsic differential equation contained in the starting system and on the remaining algebraic constraints. The decoupling procedure in differential and algebraic part is based on the projector and matrix chain which leads to the definition of tractability index. The differential part can be reduced by using any MOR method, we use Krylov-based projection methods to illustrate our approach. The reduction on the differential part induces a reduction on the algebraic part. In this paper, we present the method for index-1 differential-algebraic equations. We implement numerically this procedure and show numerical evidence of its validity.     

Approximation order of bivariate spline interpolation for arbitrary smoothness

By using the algorithm of Nürnberger and Riessinger (1995), we construct Hermite interpolation sets for spaces of bivariate splines S_q^r(Δ¹) of arbitrary smoothness defined on the uniform type triangulations. It is shown that our Hermite interpolation method yields optimal approximation order for q 3.5r + 1. In order to prove this, we use the concept of weak interpolation and arguments of Birkhoff interpolation.     

On dominant poles and model reduction of second order time-delay systems

The method known as the dominant pole algorithm (DPA) has previously been successfully used in combination with model order reduction techniques to approximate standard linear time-invariant dynamical systems and second order dynamical systems. In this paper, we show how this approach can be adapted to a class of second order delay systems, which are large scale nonlinear problems whose transfer functions have an infinite number of simple poles. Deflation is a very important ingredient for this type of methods. Because of the nonlinearity, many deflation approaches for linear systems are not applicable. We therefore propose an alternative technique that essentially removes computed poles from the system?s input and output vectors. In general, this technique changes the residues, and hence, modifies the order of dominance of the poles, but we prove that, under certain conditions, the residues stay near the original residues. The new algorithm is illustrated by numerical examples.     

Numerical Treatment of Second Order Differential-Algebraic Systems

We consider the numerical treatment of systems of second order differential-algebraic equations (DAEs). The classical approach of transforming a second order system to first order by introducing new variables can lead to difficulties such as an increase in the index or the loss of structure. We show how we can compute an equivalent strangeness-free second order system using the derivative array approach and we present Runge-Kutta methods for the direct numerical solution of second order DAEs. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application of matrix polynomials to the analysis of linear differential-algebraic equations of higher order

We consider systems of linear differential-algebraic equations of order ≥ 2 and present conditions under which there exist nonsingular transformations splitting the system into systems of linear ordinary differential equations of lower order and, possibly, an algebraic part.     

Local theory of extrapolation methods

In this paper we discuss the theory of one-step extrapolation methods applied both to ordinary differential equations and to index 1 semi-explicit differential-algebraic systems. The theoretical background of this numerical technique is the asymptotic global error expansion of numerical solutions obtained from general one-step methods. It was discovered independently by Henrici, Gragg and Stetter in 1962, 1964 and 1965, respectively. This expansion is also used in most global error estimation strategies as well. However, the asymptotic expansion of the global error of one-step methods is difficult to observe in practice. Therefore we give another substantiation of extrapolation technique that is based on the usual local error expansion in a Taylor series. We show that the Richardson extrapolation can be utilized successfully to explain how extrapolation methods perform. Additionally, we prove that the Aitken-Neville algorithm works for any one-step method of an arbitrary order s, under suitable smoothness.     

On order‐reducible sinc discretizations and block‐diagonal preconditioning methods for linear third‐order ordinary differential equations

By introducing a variable substitution, we transform the two‐point boundary value problem of a third‐order ordinary differential equation into a system of two second‐order ordinary differential equations (ODEs). We discretize this order‐reduced system of ODEs by both sinc‐collocation and sinc‐Galerkin methods, and average these two discretized linear systems to obtain the target system of linear equations. We prove that the discrete solution resulting from the linear system converges exponentially to the true solution of the order‐reduced system of ODEs. The coefficient matrix of the linear system is of block two‐by‐two structure, and each of its blocks is a combination of Toeplitz and diagonal matrices. Because of its algebraic properties and matrix structures, the linear system can be effectively solved by Krylov subspace iteration methods such as GMRES preconditioned by block‐diagonal matrices. We demonstrate that the eigenvalues of certain approximation to the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the discretized linear system, and we use numerical examples to illustrate the feasibility and effectiveness of this new approach. Copyright © 2013 John Wiley & Sons, Ltd.     

Multidimensional Hermite-Hadamard inequalities and the convex order

The problem of establishing inequalities of the Hermite-Hadamard type for convex functions on n-dimensional convex bodies translates into the problem of finding appropriate majorants of the involved random vector for the usual convex order. We present two results of partial generality which unify and extend the most part of the multidimensional Hermite-Hadamard inequalities existing in the literature, at the same time that lead to new specific results. The first one fairly applies to the most familiar kinds of polytopes. The second one applies to symmetric random vectors taking values in a closed ball for a given (but arbitrary) norm on Rn. Related questions, such as estimates of approximation and extensions to signed measures, also are briefly discussed.     

Non-integrability of Hamiltonian systems through high order variational equations: Summary of results and examples

This paper deals with non-integrability criteria, based on differential Galois theory and requiring the use of higher order variational equations. A general methodology is presented to deal with these problems. We display a family of Hamiltonian systems which require the use of order k variational equations, for arbitrary values of k, to prove non-integrability. Moreover, using third order variational equations we prove the non-integrability of a non-linear spring-pendulum problem for the values of the parameter that can not be decided using first order variational equations.      

A new time-discretization for delay multiple-input nonlinear systems using the Taylor method and first order hold

A new discretization method is proposed for multi-input-driven nonlinear continuous systems with time-delays, based on a combination of the Taylor series expansion and the first-order hold (FOH) assumption. The mathematical structure of the new discretization scheme is explored. On the basis of this structure, the sampled-data representation of the time-delayed multi-input nonlinear system is derived. First the new approach is applied to nonlinear systems with two inputs, and then the delayed multi-input general equation is derived. The resulting time discretization method provides a finite-dimensional representation for multi-input nonlinear systems with time-delays, thereby enabling the application of existing controller design techniques to such systems. The performance of the proposed method is evaluated using a nonlinear system with time-delays (maneuvering an automobile). Various sampling rates, time-delay values and control inputs are considered to evaluate the proposed method. The results demonstrate that the proposed discretization scheme can meet the system requirements even when using a large sampling period with precision limitations. The discretization results of the FOH method are also compared with those of the zero order hold (ZOH) method. The precision of the FOH method in the discretization procedure combined with the Taylor series expansion is much higher than that of the ZOH method except in the case of constant inputs.     

Convergence analysis of a partial differential algebraic system from coupling a semiconductor model to a circuit model

We are interested in circuit simulation including distributed semiconductor models. The circuit itself is modeled by the modified nodal analysis. The stationary drift diffusion equations are used to describe the semiconductors. The complete system is then a partial differential-algebraic system. We discretize it first in space with finite elements and the Scharfetter-Gummel discretization. The resulting semi-discrete system can be analyzed as a differential-algebraic equation with properly stated leading term. We present topological index one criteria. They coincide with previous results for the non-discretized partial differential-algebraic equation. For the time discretization we use standard BDF methods (implicit Gear formulas). Finally we derive a convergence estimate for the whole partial differential-algebraic system close to equilibrium.     

The arbitrary order implicit multistep schemes of exponential fitting and their applications

We develop the arbitrary order implicit multistep schemes of exponential fitting (EF) for systems of ordinary differential equations. We use an explicit EF scheme to predict an approximation, and then use an implicit EF scheme to correct this prediction. This combination is called a predictor–corrector EF method. We demonstrate the accuracy and efficiency of the new predictor–corrector methods via application to a variety of test cases and comparison with other analytical and numerical results. The numerical results show that the schemes are highly accurate and computationally efficient.     

Limitations of frequency domain approximation for detecting chaos in fractional order systems

In this paper, we analytically study the influences of using frequency domain approximation in numerical simulations of fractional order systems. The number and location of equilibria, and also the stability of these points, are compared between the original system and its frequency based approximated counterpart. It is shown that the original system and its approximation are not necessarily equivalent according to the number, location and stability of the fixed points. This problem can cause erroneous results in special cases. For instance, to prove the existence of chaos in fractional order systems, numerical simulations have been largely based on frequency domain approximations, but in this paper we show that this method is not always reliable for detecting chaos. This approximation can numerically demonstrate chaos in the non-chaotic fractional order systems, or eliminate chaotic behavior from a chaotic fractional order system.     

Generic pole assignability,structurally constrained controllers and unimodular completion

In this paper we assume dynamical systems are represented by linear differential-algebraic equations (DAEs) of order possibly higher than one. We consider a structured system of DAEs for both the to-be-controlled plant and the controller. We model the structure of the plant and the controller as an undirected and bipartite graph and formulate necessary and sufficient conditions on this graph for the structured controller to generically achieve arbitrary pole placement. A special case of this problem also gives new equivalent conditions for structural controllability of a plant. Use of results in matching theory, and in particular, ‘admissibility’ of edges and ‘elementary bipartite graphs’, make the problem and the solution very intuitive. Further, our approach requires standard graph algorithms to check the required conditions for generic arbitrary pole placement, thus helping in easily obtaining running time estimates for checking this. When applied to the state space case, for which the literature has running time estimates, our algorithm is faster for sparse state space systems and comparable for general state space systems.     

Strong tractability of multivariate integration of arbitrary order

We show the existence of digitally-shifted polynomial lattice rules which achieve strong tractability results for Sobolev spaces of arbitrary high smoothness. The convergence rate is shown to be best possible up to a given degree of smoothness of the integrand. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Order Reduction Approach for Singularly Perturbed Optimal Control Systems

The order reduction method for singularly perturbed optimal control systems consists of employing the system obtained while setting the small parameter to be zero. In many situations the differential-algebraic system thus obtained indeed provides an appropriate approximation to the singularly perturbed problem with a small parameter. In this paper we establish that if relaxed controls are allowed then the answer to the question whether or not this method is valid depends essentially on one simple parameter: the dimension of the fast variable, denoted n. More specifically, if n=1 then the order reduction method is indeed applicable, while if n>1 then the set of singularly perturbed optimal control systems for which it is not applicable is dense (in the L norm).     

Fault detection based on fractional order models: Application to diagnosis of thermal systems

The aim of this paper is to propose diagnosis methods based on fractional order models and to validate their efficiency to detect faults occurring in thermal systems. Indeed, it is first shown that fractional operator allows to derive in a straightforward way fractional models for thermal phenomena. In order to apply classical diagnosis methods, such models could be approximated by integer order models, but at the expense of much higher involved parameters and reduced precision. Thus, two diagnosis methods initially developed for integer order models are here extended to handle fractional order models. The first one is the generalized dynamic parity space method and the second one is the Luenberger diagnosis observer. Proposed methods are then applied to a single-input multi-output thermal testing bench and demonstrate the methods efficiency for detecting faults affecting thermal systems.     

On the order of tandem queues

We consider the optimal order of servers in a tandem queueing system withm stages, an unlimited supply of customers in front of the first stage, and a service buffer of size 1 but no intermediate storage buffers between the first and second stages. Service times depend on the servers but not the customers, and the blocking mechanism at the first two stages is manufacturing blocking. Using a new characterization of reversed hazard rate order, we show that if the service times for two servers are comparable in the reversed hazard rate sense, then the departure process is stochastically earlier if the slower server is first and the faster server is second than if the reverse is true. This strengthens earlier results that considered individual departure times marginally. We show similar results for the last two stages and for other blocking mechanisms. We also show that although individual departure times for a system with servers in a given order are stochastically identical to those when the order of servers is reversed, this reversibility property does not hold for the entire departure process.     

Optimal order for two servers in tandem

We consider two servers (serveri, i=1, 2) in tandem for which the order of servers can be changed. Server 1 has a general service time distribution and server 2 has either its shifted or truncated distribution. This permits that the service times at the two servers are overlapping. An unlimited queue is allowed in front of the first server. For the systems having zero buffer capacity between the servers, we show that the sojourn time of every customer is stochastically minimized under any arrival process if server 2 is first. For the systems with infinite buffer capacity and a Poisson arrivals, we show that this order of servers minimizes mean customer delay when traffic is light. Several numerical examples are presented to demonstrate that this optimal order is invariant under any arrival process (the interarrival times are i.i.d. r.v.'s) and mild traffic condition.Research funded by NEC Corporation C & C Laboratory.     

First order spectrum for elliptic system and nonresonance problem

In this paper we will study the first order spectrum for elliptic systems and the existence of solutions for a quasilinear elliptic system under the condition of nonresonance below the first eigensurface. This revised version was published online in June 2006 with corrections to the Cover Date.