Set membership state and parameter estimation for systems described by nonlinear differential equations

This paper investigates the use of guaranteed methods to perform state and parameter estimation for nonlinear continuous-time systems, in a bounded-error context. A state estimator based on a prediction-correction approach is given, where the prediction step consists in a validated integration of an initial value problem for an ordinary differential equation (IVP for ODE) using interval analysis and high-order Taylor models, while the correction step uses a set inversion technique. The state estimator is extended to solve the parameter estimation problem. An illustrative example is presented for each part.     

Model-order reduction for differential-algebraic equation systems with higher index

In this paper, a new model-order reduction (MOR) approach is presented for reducing large-scale differential-algebraic equation (DAE) systems with higher index. This approach is based upon the balanced truncation, single-point, and multi-point MOR methods. We decompose the DAE system into an ordinary differential equation (ODE) subsystem and a DAE subsystem. The DAE subsystem has the same index as the original DAE system. Then, the balanced truncation method is applied to the ODE subsystem. Both single-point and multi-point methods are used to reduce the DAE subsystem. In generally, the multi-point method can perform better than the single-point method across a wide-range of frequencies. Some numerical examples demonstrate the effectiveness of our approach.     

A Multigrid Method of the Second Kind for Solving Linear Systems of Odes Discretized by Continuous Runge-Kutta Methods

A Waveform Relaxation method as applied to a linear system of ODEs is the Picard iteration for a linear Volterra integral equation of the second kind ({\cal I} - {\cal K})y = b \eqno (1) called Waveform Relaxation second kind equation. A corresponding Waveform Relaxation Runge-Kutta method is the Picard iteration for a discretized version ({\cal I} - {\cal K}_l )y_l = b_l \eqno (2) of the integral equation (1), where y l is the continuous solution of the original linear system of ODE provided by the so called limit method. We consider a W-cycle multigrid method, with Picard iteration as smoothing step, for iteratively computing y l . This multigrid method belongs to the class of multigrid methods of the second kind as described in Hackbusch [3, chapter 16]. In the paper we prove that the truncation error after one iteration is of the same order of the discretization error y l @ y of the limit method and the truncation error after two iterations has order larger than the discretization error. Thus we can see the multigrid method as a new numerical method for solving the original linear system of ODE which provides, after one iteration, a continuous solution of the same order of the solution of the limit method, and after two iterations, a solution with asymptotically the same error of the solution of the limit method. On the other hand the computational cost of the multigrid method is considerably smaller than the limit method.     

Mean value theorem for boundary value problems with given upper and lower solutions

An approach based on successive application of the mean value theorem or, equivalently, a successive linear interpolation that excludes extrapolation, is described for two-point boundary value problem (BVP) associated with nonlinear ordinary differential equations (ODEs). The approach is applied to solve numerically a two-point singular BVP associated with a second-order nonlinear ODE which is a mathematical model in membrane response of a spherical cap that arises in nonlinear mechanics. The upper and lower bounds on solution for the foregoing second-order ODE are assumed known analytically. Other possible methods such as the successive bisection for the BVP associated with second-order nonlinear ODE and a multivariable Taylor series for the second or higher-order nonlinear ODEs are also discussed to solve two-point BVP. The scope/limitation of the later methods and other possible higher-order methods in the present context are stressed.     

Analytical method of predicating the instabilities of a micro arch-shaped beam under electrostatic loading

An arch-shaped beam with different configurations under electrostatic loading experiences either the direct pull-in instability or the snap-through first and then the pull-in instability. When the pull-in instability occurs, the system collides with the electrode and adheres to it, which usually causes the system failure. When the snap-through instability occurs, the system experiences a discontinuous displacement to flip over without colliding with the electrode. The snap-through instability is an ideal actuation mechanism because of the following reasons: (1) after snap-through the system regains the stability and capability of withstanding further loading; (2) the system flips back when the loading is reduced, i.e. the system can be used repetitively; and (3) when approaching snap-through instability the system effective stiffness reduces toward zero, which leads to a fast flipping-over response. To differentiate these two types of instability responses for an arch-shaped beam is vital for the actuator design. For an arch-shaped beam under electrostatic loading, the nonlinear terms of the mid-plane stretching and the electrostatic loading make the analytical solution extremely difficult if not impossible and the related numerical solution is rather complex. Using the one mode expansion approximation and the truncation of the higher-order terms of the Taylor series, we present an analytical solution here. However, the one mode approximation and the truncation error of the Taylor series can cause serious error in the solution. Therefore, an error-compensating mechanism is also proposed. The analytical results are compared with both the experimental data and the numerical multi-mode analysis. The analytical method presented here offers a simple yet efficient solution approach by retaining good accuracy to analyze the instability of an arch-shaped beam under electrostatic loading.     

New results using an integrated model and recursive algorithm for image motion estimation

A linear equation in the affine parameters used to model image motion may be derived by Taylor series expansion and truncation, and windowed spatial integration. Two methods for reducing errors in the Taylor approximation are discussed and results are presented.     

A computationally efficient technique for transient analysis of repairable Markovian systems

A technique to design efficient methods using a combination of explicit (non-stiff) and implicit (stiff) ODE methods for numerical transient analysis of repairable Markovian systems is proposed. Repairable systems give rise to stiff Markov chains due to extreme disparity between failure rates and repair rates. Our approach is based on the observation that stiff Markov chains are non-stiff for an initial phase of the solution interval. A non-stiff ODE method is used to solve the model for this phase and a stiff ODE method is used to solve the model for the rest of the duration until the end of solution interval. A formal criterion to determine the length of the non-stiff phase is described. A significant outcome of this approach is that the accuracy requirement automatically becomes a part of model stiffness. Two specific methods based on this approach have been implemented. Both the methods use the Runge-Kutta-Fehlberg method as the non-stiff method. One uses the TR-BDF2 method as the stiff method while the other uses an implicit Runge-Kutta method as the stiff method. Numerical results obtained from solving dependability models of a multiprocessor system and an interconnection network are presented. These results show that the methods obtained using this approach are much more efficient than the corresponding stiff methods which have been proposed to solve stiff Markov models.     

Different Approaches to Forecast Interval Time Series: A Comparison in Finance

An interval time series (ITS) is a time series where each period is described by an interval. In finance, ITS can describe the temporal evolution of the high and low prices of an asset throughout time. These price intervals are related to the concept of volatility and are worth considering in order to place buy or sell orders. This article reviews two approaches to forecast ITS. On the one hand, the first approach consists of using univariate or multivariate forecasting methods. The possible cointegrating relation between the high and low values is analyzed for multivariate models and the equivalence of the VAR models is shown for the minimum and the maximum time series, as well as for the center and radius time series. On the other hand, the second approach adapts classic forecasting methods to deal with ITS using interval arithmetic. These methods include exponential smoothing, the k-NN algorithm and the multilayer perceptron. The performance of these approaches is studied in two financial ITS. As a result, evidences of the predictability of the ITS are found, especially in the interval range. This fact opens a new path in volatility forecasting.     

Error analysis in the series evaluation of the exponential type integral ezE1(z)

The method of convolution algebra is used to compute values of the exponential type integral e^zE₁(z),

by expansion of the integrand in a string of Taylor series' along the real s-axis for any complex parameter z, accurate within ±1 of the last digit of seven-digit computation. Accuracy is verified by comparison with existing tables of E₁ and related integrals. This method is used to assess the accuracy of the error estimates of all subsequent computations.Three errors of a Taylor series are identified. These consist of a Taylor series truncation error, a digital truncation error, and a stability error. Methods are developed to estimate the error. By iteration a numerical radius of convergence for a given accuracy is determined.The z-plane is divided into three regions in which three different types of series are used to expand the function f(z) directly in z. Around the center the well-known Frobenius series is used. In the outer region the well-known asymptotic approximation is used. Their accuracy boundaries are determined. In the near-annular region in between, a set of Taylor series is introduced.As the result, the function f(z) can be computed fast with the appropriate series for any complex argument z, to an accuracy within less than relative error of 5 × 10⁻⁷.     

Task-space control of robots using an adaptive Taylor series uncertainty estimator

An uncertainty estimation and compensation can improve the performance of control systems due to structured and unstructured uncertainty. This paper presents a robust task-space control approach using an adaptive Taylor series uncertainty estimator for electrically driven robot manipulators. It is worth noting that not only the lumped uncertainty is estimated and employed in the indirect form of robust controller, but also the upper bound of approximation error is estimated to form a robustifying term and the asymptotic convergence of tracking error and its time derivative are proven based on stability analysis. Finally, the effectiveness of the proposed controller is shown through simulation and comparison with two valuable control schemes applied on the Selective Compliance Assembly Robot Arm (SCARA) robot manipulator.     

Error analysis of two algorithms for the computation of the matrix exponential

We analyze two algorithms for the computation of the matrix exponential: the Taylor Series and the Scaling and Squaring methods. We give new upper bounds on the roundoff and truncation errors introduced, and we present some numerical experiments, comparing the actual outcomes to the theoretical error bounds. We analyze, in detail, the reasons why the Taylor Series method can compete with Scaling and Squaring method if the norm of the original matrix is less than one.     

Error Reduction of the Taylor Centered Form by Half and an Inner Estimation of the Range

We show that the overestimation error of the interval Horner method for univariate polynomials on a centered interval is reduced at least by half if the interval is split at its midpoint zero and the interval Horner method is applied to both halves separately. This observation is used to reduce the overestimation error of the Taylor centered form at least by half. Further, it can be used to compute error bounds for the Taylor centered form and an inner range estimation.     

On a posteriori pointwise error estimation using adjoint temperature and Lagrange remainder

The pointwise estimation of heat conduction solution as a function of truncation error of a finite difference scheme is addressed. The truncation error is estimated using a Taylor series with the remainder in the Lagrange form. The contribution of the local error to the total pointwise error is estimated via an adjoint temperature. It is demonstrated that the results of numerical calculation of the temperature at an observation point may thus be refined via adjoint error correction and that an asymptotic error bound may be found.     

Trigonometrically-fitted method for a periodic initial value problem with two frequencies

A second-order differential equation whose solution is periodic with two frequencies has important applications in many scientific fields. Nevertheless, it may exhibit ‘periodic stiffness’ for most of the available linear multi-step methods. The phenomena are similar to the popular Stömer-Cowell class of linear multi-step methods for one-frequency problems. According to the stability theory laid down by Lambert, ‘periodic stiffness’ appears in a two-frequency problem because the production of the step-length and the bigger angular frequency lies outside the interval of periodicity. On the other hand, for a two-frequency problem, even with a small step-length, the error in the numerical solution afforded by a P-stable trigonometrically-fitted method with one frequency would be too large for practical applications. In this paper we demonstrate that the interval of periodicity and the local truncation error of a linear multi-step method for a two-frequency problem can be greatly improved by a new trigonometric-fitting technique. A trigonometrically-fitted Numerov method with two frequencies is proposed and has been verified to be P-stable with vanishing local truncation error for a two-frequency test problem. Numerical results demonstrated that the proposed trigonometrically-fitted Numerov method with two frequencies has significant advantages over other types of Numerov methods for solving the ‘periodic stiffness’ problem.     

Error bounds in the averaging of hybrid systems

The authors analyze the error introduced by the averaging of hybrid systems. These systems involve linear systems which can take a number of different realizations based on the state of an underlying finite state process. The averaging technique (based on a formula from Lie algebras known as the Backer-Campbell-Hausdorff (BCH) formula) provides a single system matrix as an approximation to the hybrid system. The two errors discussed are: (1) the error induced by the truncation of the BCH series expansion and (2) the error between the actual hybrid system and its average. A simple sufficient stability test is proposed to check the asymptotic behavior of this error. In addition, conditions are derived that allow the use of state feedback instead of averaging to arrive at a time-invariant system matrix     

A program for perspective views of three-dimensional surfaces

A numerical method is proposed for solving linear differential equations of second order without first derivatives. The new method is superior to de Vogelaere's for this class of equations, and for non-linear equations it becomes an implicit extension of de Vogelaere's method. The global truncation error at a fixed steplength h is bounded by a term of order h⁴, and the interval of absolute stability is [?2.4, 0]. The work of Coleman and Mohamed (1978) is readily adapted to provide truncation error estimates which can be used for automatic error control. It is suggested that the new method should be used in preference to de Vogelaere's for linear equations, and in particular to solve the radial Schrödinger equation. the radial Schrödinger equation.     

The development of a high-order Taylor expansion solution to the chemical rate equation for the simulation of complex biochemical systems

A numerical method for evaluating chemical rate equations is presented. This method was developed by expressing the system of coupled, first-degree, ordinary differential chemical rate equations as a single tensor equation. The tensorial rate equation is invariant in form for all reversible and irreversible reaction schemes that can be expressed as first- and second-order reaction steps, and can accommodate any number of reactive components. The tensor rate equation was manipulated to obtain a simple formula (in terms of rate constants and initial concentrations) for the power coefficients of the Taylor expansion of the chemical rate equation. The Taylor expansion formula was used to develop a FORTRAN algorithm for analysing the time development of chemical systems. A computational experiment was performed with a Michaelis-Menten scheme in which step size and expansion order (to the 100th term) were varied; the inclusion of high-order terms of the Taylor expansion was shown to reduce truncation and round-off errors associated with Runge-Kutta methods and lead to increased computational efficiency.     

Artificial neural networks based modeling for solving Volterra integral equations system

Properly designing an artificial neural network is very important for achieving the optimal performance. This study aims to utilize an architecture of these networks together with the Taylor polynomials, to achieve the approximate solution of second kind linear Volterra integral equations system. For this purpose, first we substitute the Nth truncation of the Taylor expansion for unknown functions in the origin system. Then we apply the suggested neural net for adjusting the numerical coefficients of given expansions in resulting system. Consequently, the reported architecture using a learning algorithm that based on the gradient descent method, will adjust the coefficients in given Taylor series. The proposed method was illustrated by several examples with computer simulations. Subsequently, performance comparisons with other developed methods was made. The comparative experimental results showed that this approach is more effective and robust.     

Convergence Analysis of Iterative Laplace Transform Methods for the Coupled PDEs from Regime-Switching Option Pricing

This paper aims to analyze the convergence rates of the iterative Laplace transform methods for solving the coupled PDEs arising in the regime-switching option pricing. The so-called iterative Laplace transform methods are described as follows. The semi-discretization of the coupled PDEs with respect to the space variable using the finite difference methods (FDMs) gives the coupled ODE systems. The coupled ODE systems are solved by the Laplace transform methods among which an iteration algorithm is used in the computational process. Finally, the numerical contour integral method is used as the Laplace inversion to restore the solutions to the original coupled PDEs from the Laplace space. This Laplace approach is regarded as a better alternative to the traditional time-stepping method. The errors of the approach are caused by the FDM semi-discretization, the iteration algorithm and the Laplace inversion using the numerical contour integral. This paper provides the rigorous error analysis for the iterative Laplace transform methods by proving that the method has a second-order convergence rate in space and exponential-order convergence rate with respect to the number of the quadrature nodes for the Laplace inversion.     

On the balanced truncation and coprime factors reduction of Markovian jump linear systems

The paper deals with the balanced truncation and coprime factors reduction of Markovian jump linear (MJL) systems, which can have mode-varying state, input, and output dimensions. We develop machinery for balancing mean square stable MJL system realizations using generalized Gramians and strict Lyapunov inequalities, and provide an improved a priori upper bound on the error induced in the balanced truncation process. We also generalize the coprime factors reduction method and, in doing so, extend the applicability of the balanced truncation technique to the class of mean square stabilizable and detectable MJL systems. We provide tools to establish mean square stabilizability and detectability of the considered MJL systems. In addition, a notion of right-coprime factorization of MJL systems and methods to construct such factorizations are given. The error measure in the coprime factors reduction approach, while still norm-based, does not directly capture the mismatch between the nominal system and the reduced-order model, as is the case in the balanced truncation approach where mean square stable models are considered. Instead, the error measure is given in terms of the distance between the coprime factors realizations, and thus has an interpretation in terms of robust feedback stability. The paper concludes with an illustrative example which demonstrates how to apply the coprime factors model reduction approach.