Hartley Series Research Articles

Numerical solution of time-delay optimal control problems by the operational matrix based on Hartley series

In this paper, a numerical technique based on the Hartley series for solving a class of time-delayed optimal control problems (TDOCPs) is introduced. The main idea is converting such TDOCPs into a system of algebraic equations. Thus, we first expand the state and control variables in terms of the Hartley series with undetermined coefficients. The delay terms in the problem under consideration are expanded in terms of the Hartley series. Applying the operational matrices of the Hartley series including integration, differentiation, dual, product, delay, and substituting the estimated functions into the cost function, the given TDOCP is reduced to a system of algebraic equations to be solved. The convergence of the proposed method is extensively investigated. At last, the precision and applicability of the proposed method is studied through different types of numerical examples.

Identification of Multi-Axial Seismic Loadings from Several Structural Dynamic Responses

A new method is proposed to identify multi-axial seismic loadings from structural dynamic responses on limited degrees of freedom. The seismic loadings acting on structures are modeled by Hartley series approximation, and the sensitivities of structural dynamic response with respect to the unknown approximation coefficients are derived. The identification equation is set up based on best fitting structural measured and calculated responses, and is solved with the damped least-squares method. A five-story three-dimensional steel frame structure excited by El-Centro seismic accelerations is studied for validating the proposed method. Numerical simulations with measurement noise and model errors show that the proposed method can accurately identify all seismic loadings from only several responses of the structure.

Hartley series approximations for the parabolic equations

An approximate method for solving parabolic equations with a periodic boundary condition is proposed. The method is based upon using the Legendre series and the Hartley series to approximate the required solution. The parabolic equations are converted into a system of linear algebraic equations which can be solved for the unknown coefficients. A numerical example is included to demonstrate the validity and applicability of the method and a comparison is made with existing results.

The study of inrush current phenomenon using operational matrices

The paper presents a new methodology for calculating magnetizing inrush currents. Similarly to the seminal work of Rajakovic and Semlyen, all the calculations are conducted in the frequency domain, hence, circumventing the many problems encountered when time domain simulations are used to solve this problem. Unlike the work of Rajakovic and Semlyen (1989), where a pre-determined sequence of steady-state images are obtained for a sufficiently long number of cycles to capture the complete picture of the inrush phenomenon, the authors' method produces the full solution, with no loss of fidelity, of the overall inrush phenomenon. The solution is achieved by resorting to operational matrices and it is fully equivalent to that produced by time domain simulations but carried out entirely in the frequency domain. In principle, any orthogonal series expansion could be used with this method, but in this paper Hartley series have been selected. The usefulness of the methodology is illustrated with simple but practical circuits. Comparisons against time domain simulations are given.

Synchronous machine parameter estimation using the Hartley series

This paper presents a novel alternative to estimate armature circuit parameters of large utility generators using real time operating data. The proposed approach uses the Hartley series for fitting operating data (voltage and currents measurements). The essence of the method is the use of linear state estimation to identify the coefficients of the Hartley series. The approach is tested for noise corruption likely to be found in measurements. The method is found to be suitable for the processing of digital fault recorder data to identify synchronous machine parameters.

An algebraic approach for identifying operating point dependent parameters of synchronous machines using orthogonal series expansions

This paper presents a method for identifying armature and field parameters of synchronous machines from digital fault recorder (DFR) data. The method uses operational properties of orthogonal series expansions such as the Hartley, Walsh and Fourier series to transform a set of differential equations into linear algebraic equations. The algebraic formulation and use of operational calculus reduce the problem of identifying parameters to the manipulation of matrices that may be easily performed in such computational packages as Matlab. The variation of machine parameters with operating point is considered.

Parameter Estimation Using An Orthogonal Series Expansion

This paper presents an innovative alternative to estimate parameters of a system for which a dynamic model is known. The focus of this paper is the estimation of the armature circuit parameters of large utility generators using real time operating data. Other applications are possible. The alternatives considered are the use of orthogonal series expansions, in general, and the Hartley series, in particular. The main idea considers the use of orthogonal series expansions for fitting operating data (e.g., voltage and currents measurements). This allows writing a set of linear algebraic equations that can be “ solved” in the least squares sense for the unknown parameters. The method shown utilizes the pseudoinverse in the solution. The essence of the approach is linear state estimation. Several alternative types of orthogonal expansions are briefly discussed. Although solutions are the same in all domains, one wishes to employ the expansion that gives the most efficient computation. The approach may be used for static as well as dynamic problems. The approach is tested for noise corruption likely to be found in measurements. The method is found to be suitable for the processing of digital fault recorder data to identify synchronous machine parameters.

Analysis of linear time-varying systems via Hartley series

A method for finding approximated solutions of linear time-varying systems via Hartley series is proposed. The most relevant properties of the Hartley series are presented first. The operational matrix of integration together with the product and coefficient matrices are developed. They are used to transform dynamical equations of linear, time varying systems to a set of linear algebraic equations. Practical applications relating to the operation of electric networks are given.

The use of switching functions and Walsh series to calculate waveform distortion in thyristor controlled compensated power circuits

A new model based on the Walsh series for assessing the steady state operation of three-phase thyristor controlled reactors (TCRs) is presented in this paper. The model represents the TCR as an admittance matrix which interfaces easily with other power plant components in the network. Clearly, the admittance matrix representation is not confined to Walsh domain but rather it is valid for any orthogonal approximation of functions having the operational properties used in this paper. However, the switching functions which characterise the operation of TCRs are better represented by the Walsh series and they have been used in this paper. This is a new development in power systems analysis. Other orthogonal series expansions such as Fourier series and Hartley series have been used elsewhere. The Walsh transform possesses the important operational properties of the product, coefficient and integration matrices. The relevant aspects of the interaction between the TCR and the electric network are analysed. A three-phase TCR and a transmission network represented by a Thevenin equivalent are used for this purpose.

The Hartley series and its application to power quality assessment

An analog to the Fourier series (FS) that is based on the cosine-and-sine function is presented. The new series is related to the Hartley transform and is termed the Hartley series (HS). Properties and applications of the Hartley series are discussed in general. The technique is valuable for the assessment of power quality in industrial and commercial power distribution systems. In particular, the impact of switched and pulse modulated loads and AC/DC converters can be analyzed. An example is given in which an industrial distribution system impacted by a six-pulse rectifier load is analyzed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Hartley series and notch Hartley transform

A new class of Hartley transform is introduced—the Hartley series (HS). The Hartley series is appropriate in the situation that the input signal is continuous and periodic in time, and hence its Hartley transform is discrete in frequency. Through this class of Hartley transform, any continuous and periodic signal can be decomposed into the weighted sum of cas functions (where cas (·) = cos (·) + sin (·)) In order to compute these Hartley coefficients, an algorithm referred to as the notch Hartley transform (NHT) is also presented. This algorithm can be applied to the input signal composed of arbitrary frequencies, and all Hartley coefficients can be computed in advance of the end of one period of the signal