Order reduction of real rational functions with error bounds

This paper describes a simple procedure for synthesizing an active distributed RC network which, by using dominant poles and zeros, realizes a very accurate approximation of an arbitrary stable rational transfer function. The network uses a single uniformly distributed RC line with taps spaced along its length. A linear combination of tap voltages is added to the input signal to form the driving voltage for the RC line; the output signal is also a linear combination of the tap voltages. The network offers a number of significant advantages. Since it realizes a nearly rational transfer function, the approximation problem can be conveniently solved using readily available results on rational function approximation. Also, the network uses only one uniform RC line, the transfer function can be changed simply by changing resistor values, and the frequency can be scaled by minor connection changes. Thus one standard network with minor modification is useful for a wide variety of applications. This paper develops the design procedure and derives the various sensitivity functions of importance. Two example designs are carried out: an approximation to a second-order low-pass transfer function and an approximation to a second-order band-pass transfer function with a Q of 100. The sensitivities for the examples are very reasonable and the measurements made on laboratory models indicate excellent agreement with theoretical predictions.

There are some different statements for necessary and sufficient conditions of strict positive realness in the control literatures. These statements are usually extracted by using state space approaches. In this paper, unlike the other works the frequency domain tools is used to prove the necessary and sufficient conditions for test strict positive realness property of real rational transfer function. The first main advantages of this approach is develop a new sufficient condition for strict positive realness and the second is prove necessary and sufficient conditions for strict positive realness property of real rational transfer function directly by using original definition in the frequency domain and without any restriction on the relative degree of transfer function by default. In this paper, the original definition of strict positive realness in the frequency domain is interpreted as a mapping from S-plane to G(s)-plane. Then the maximum modulus principle and the Taylor expansion which are the fundamental tools in the complex analysis is used to extract a new sufficiency condition for strictly positive real (SPR) transfer functions. This geometry interpretation implies a new more accurate definition for SPR transfer functions. Finally more completed and simplified version of necessary and sufficient conditions for strict positive realness is proved based on the complex analysis

We propose a rational function approximation method combining numeric and symbolic computations. Given functions or data are first interpolated by a rational function, i.e. the ratio of polynomials. Undesired poles appearing in the rational interpolant are removed by an approximate-GCD method. We call the rational approximation a Hybrid Rational Function Approximation and abbreviate it as HRFA. In this paper we give a short survey of the HRFA and then discuss its accuracy analysis by using the approximate-GCD proposed by Pan.

We prove several splitting theorems related to the following one: Every continuous real trigonometric rational function (of t, with period $2\pi $) can be written as the real part of a rational function of $z = e^{it} $ that is analytic for $|z| \leqq 1$. As an application, we obtain immediately a method for computing the conjugate function of a real trigonometric rational function.

Using methods analogous to those in diophantine approximation, the existence of good rational (function) approximations to the general solutions of each of a class of linear homogeneous differential equations is investigated. It is shown that, while many good approximations may exist, no truly extraordinary approximations exist. The proofs are via Lemmas on formal series which remain valid in a much wider context, e. g. even if the field of constants has positive characteristic.

A simplified nonlinear transient analysis method for gas bearings was recently published by the authors (Hassini, M. A., and Arghir, M., 2012, “Simplified Nonlinear Transient Analysis Method for Gas Bearings,” J. Tribol., 134(1), 011704). The method uses the fact that linearized dynamic characteristics of gas bearings, namely the impedances, can be approximated by rational transfer functions. The method gave good results if the rational transfer function approach approximated the linearized dynamic characteristics well. Indeed, each of the four complex impedances Zαβ,α,β={x,y} had one or two poles depending on the order of the rational function that were used. These poles appear as supplementary eigenvalues of the extended matrix of the homogeneous system of first order differential equations describing the model of the rotor. They govern the stability of the dynamic model in the same way as the original eigenvalues do and therefore they impose non-negligible constraints on the rational function approximation of the impedances of gas bearings. The present improvement of the method overrides this problem. The basic idea is to impose the same set of poles for Zxx, Zxy, Zyx, and Zyy. By imposing this constraint, the poles are stable and the introduction of artificial instability or erratic eigenvalues is avoided. Campbell and stability diagrams naturally taking into account the variation of the dynamic coefficients with the excitation frequency can now be easily plotted. For example, the method is used for analyzing the stability of rigid and flexible rotors supported by two identical gas bearings modeled with second order rational transfer functions. The method can be applied to any bearing or seal whose impedance is approximated by rational transfer functions.

We have derived general convergence conditions for ADF obtained by rational function approximation. These conditions are binding conditions are sufficient conditions that the transfer function of the unknown system must satisfy. These binding conditions state that the ratio of the function derived for the transfer function H*(z) of the unknown system using G*(z) and the difference function must be a positive real function. A rational and an exponential function were investigated as particular examples of transfer functions. The results confirmed that convergence is guaranteed when the transfer functions are positive real. Part of these results were also investigated for computational complexity. Since the transfer functions of hybrid circuits converting two lines to four lines in cable transmission systems, as well as any two-terminal passive impedance function, and so on are positive real, it is possible to estimate these transfer functions R(z) using ADF. For example, when using an ADF of the type H(z)=eG(z),G(z) becomes an estimation of G*(z) = log (R(z)). Further investigation continues on the analytical range of the loop gain μ0 needed for stability of the adaptive algorithm, theoretical considerations on the rate of convergence, computational comparison with the traditional methods based on polynomial approximation, and so on. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 83(12): 78–87, 2000

This paper proposes a frequency dependent model for armored power cables using FE method. The model parameters are obtained by conducting a series of studies using coupled circuit-field analysis. The frequency response of the cable was obtained by coupling the cable FE domain model and external electric circuits. This frequency response was then fitted with rational function approximation. The resulting vector fittings (VF) rational function was then represented by an equivalent electrical network. This model represents the cable high frequency dynamic behavior to account for overvoltages or current spikes.

Most antennas or antenna arrays response are not needed at a single frequency, but rather a continuous representation over a specified period is necessary. Hence, the idea of extracting the poles in the complex plane so as to cover a wider response in frequency or time domain for the antenna circular array elements are calculated using rational function approximation in the complex frequency domain of double infinite series with unknown coefficients. The solution starts using electric field integral equation, application of the method of moments and extended boundary conditions. Then the poles are calculated using Prony's (1795) method for the input impedance and admittance of each antenna element. A newly calculated window type input impedance suffering from no singularities can be obtained. Impedances' or admittances' in the complex frequency plane are calculated by repeated application of the method of moments without the need of a newly computed matrix inversion. The method used in this paper is analogous to the formal method by the same author, which was done in the real frequency and without extracting the poles but rather using the inverse of the function describing the current, impedance or admittance. Also, the current distribution assumed was not a rational function in the complex frequency domain.

A new tuning method for Fractional Order Controllers (FOC) based on Bode’s ideal transfer function as a reference model is proposed in this paper. The proposed fractional order controller parameters are obtained by a simple tuning technique in two steps. In the first step, the open loop reference fractional integrator is designed according to the desired performances, and then approximated by a rational transfer function using Charef’s or Oustaloup’s approximation method. In the second step, a standard pole placement technique is used to align each pole of the plant transfer function with the nearest one of the Rational Function Approximation (RFA). The FOC transfer function is then obtained by gathering the remained poles and zeros of the RFA. The most innovative character of the proposed method is its simplicity and its remarkable performances in terms of robustness towards the variation of the static gain. Simulation of some illustrative examples confirms and validates the proposed method.

In this paper, a method to approximate a multi-layer Green`s function is proposed based on a FDTD scheme and a rational function approximation. For a given horizontal propagation wavenumber, time domain response is calculated and then Fourier transformed to the spectral domain Green`s function. Using the rational function approximation, the pole and residue of the Green`s function can be estimated, which are crucial for a calculation of a path loss. The proposed method can provide a wideband Green`s function, while the conventional normal mode method can be applied to a single frequency problem. To validate the proposed method, We consider two problems, one of which has a analytical solution. The other is about multi-layer case, for which the proposed method is compared with the known normal mode solution, Kraken.

This paper proposes a computational high-frequency transformer model. The model parameters are obtained by using coupled-circuit-finite-element (FE) nonlinear analysis. The frequency response of the transformer was obtained by coupling the transformer FE model and external electric circuits. This technique would allow the physical representation of the nonlinear magnetization behavior of the transformer as well as the strong frequency dependence of the transformer parameters. The self capacitance of each conductor and the mutual capacitances between the turns were calculated by an electrostatic FE analysis. The capacitance order was then reduced to a lower order by shifting the capacitances connected to internal nodes (windings) to the external ones (coil nodes). The resulting reduced capacitances, along with the inductances and resistances, were then used in the circuit domain of the coupled circuit-FE analysis. The transformer frequency response was then obtained from FE analysis. This response was then fitted with rational function approximation. This rational function approximation is then used to construct a frequency-dependent branch (FDB) which is connected in parallel with the nominal frequency transformer model. The FDB branch represents the transformer high-frequency behavior over a wide frequency range. The implementation is performed on a 125-kVA transformer. The developed model terminal behavior was tested under different operating conditions. This includes different switching frequencies and connecting cable length.

Complex coefficient digital filters, with applications for processing real sequences, are examined. The method is quite general, and allows any real rational transfer function to be expressed in terms of a complex rational transfer function of reduced order. When implemented in complex hardware form, the reduction of filter order can provide an increase in computational efficiency and speed. Conventional filter structures, such as parallel, cascade, lattice, and state-space forms, are extended to the complex domain. Illustrative examples of complex coefficient filter synthesis are included, along with coefficient sensitivity comparisons between the complex coefficient filters and their real coefficient counterparts.

We develop results on interpolation of real rational matrix functions analogous to those proved in earlier work for rational matrix functions with complex coefficients. Other relevant problems are studied as well; one of them: decide (in terms of the interpolation data) if a given interpolation problem admits a real rational matrix functions interpolants. The basic methodology is rooted in the approach exposed in [BGR3].

We develop results on interpolation of real rational matrix functions analogous to those proved in earlier work for rational matrix functions with complex coefficients. Other relevant problems are studied as well; one of them: decide (in terms of the interpolation data) if a given interpolation problem admits a real rational matrix functions interpolants. The basic methodology is rooted in the approach exposed in [BGR3].MSC15A5447A5630E05