Formula omitted]-optimal model approximation by input/output-delay structured reduced order models

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.

Optimal Approximation, Simulation and Analog Realization of the Fundamental Fractional Order Transfer FunctionThis paper provides an optimal approximation of the fundamental linear fractional order transfer function using a distribution of the relaxation time function. Simple methods, useful in systems and control theories, which can be used to approximate the irrational transfer function of a class of fractional systems for a given frequency band by a rational function are presented. The optimal parameters of the approximated model are obtained by minimizing simultaneously the gain and the phase error between the irrational transfer function and its rational approximation. A simple analog circuit which can serve as a fundamental analog fractional system is obtained. Illustrative examples are presented to show the quality and usefulness of the approximation method.

A method for rational approximation of linear fractional order systems (LFOS) is presented in the present paper. The method is computationally efficient, flexible and effective, as is illustrated by numerous examples. The proposed approach can also be used as an intermediate stage in designing indirect discrete rational approximations.

For passive networks real-part sufficiency describes the fact that their driving-point impedance or admittance is uniquely defined in terms of its real part alone. We have exploited this fact in the development of a methodology for the generation of passive rational function approximations of electromagnetic transfer functions. More specifically, by utilizing in the fitting process only data for the real part of the driving-point impedance or admittance, our methodology attempts to ensure the passivity of the rational fit by construction. This paper reports several improvements in our methodology, which provide for enhanced accuracy in the fitting of broadband data and improved robustness and computational efficiency of the fitting algorithm. The improved algorithm is demonstrated through its application to the rational fitting of the driving point impedance of interconnect circuits using both numerically-obtained and measured data over multi-GHz bandwidths.

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.

Using the Faber transformation technique, it is shown that A. A. Pekarskii's theorem on the rational approximation of functions in Hp, 1<ρ<∞ implies at once P. P. Petryshev's theorem on the rational approximation in the norm Lp[−1, l], 1<ρ<∞ and vice versa. Using the same technique, one can obtain similar results for functions that are analytic in domains with Lipschitz Jordan boundaries.

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.

For the airfoil in freestream, the pressure difference between the upper and lower surfaces and the variations in pressure gradients are significant at its leading edge area. Under reasonable deflections, the active leading edge can effectively change airfoil aerodynamic loads, which helps to improve the rotor aerodynamic performance. In this paper, a modeling method for an airfoil with an active leading edge was developed to calculate its aerodynamic loads. The pitch motion of the rotor blade and the leading edge deflections were taken into account. Firstly, simulations of steady and unsteady flow for the airfoil with an active leading edge were conducted under different boundary conditions and with different leading edge deflection movement. Secondly, the rational function approximation (RFA) was employed to establish the relationship between aerodynamic loads and airfoil/active leading edge deflections. Then, coefficient matrices of the RFA approach were identified based on a limited number of high-fidelity computational fluid dynamics (CFD) results. Finally, an aerodynamic model of the airfoil with an active leading edge was developed, and its accuracy was validated by comparing it to the high-fidelity CFD results. Comparative results reveal that the developed model can calculate the aerodynamic loads of an airfoil with an active leading edge accurately and efficiently when applied appropriately. The modeling method can be used in aerodynamic load calculations and the aeroelastic coupling analysis of a rotor with active control devices.

Chapter 4 describes the finite-state modeling techniques for deriving a linear, time-invariant state-space model of the aeroservoelastic plant. The linear systems theory is applied to the aeroelastic system in order to convert the frequency domain unsteady aerodynamics to the time domain. This requires the use of rational function approximations (RFA) in the Laplace domain for analytic continuation from harmonic curve-fits to transient response aerodynamics. The chapter presents the various RFA models, which have been developed and optimized by both gradient and non-gradient, nonlinear optimization techniques. Application of the RFA method to both typical wing section, and three-dimensional wing, with control surfaces, is presented. Illustrative finite-state model is presented for the flutter analysis of a real aircraft wing with experimental structural data, and compared with flight-flutter test results.

Structural and thermodynamic properties of multicomponent hard-sphere fluids at odd dimensions have recently been derived in the framework of the rational function approximation (RFA) [Rohrmann and Santos, Phys. Rev. E 83, 011201 (2011)]. It is demonstrated here that the RFA technique yields the exact solution of the Percus-Yevick (PY) closure to the Ornstein-Zernike (OZ) equation for binary mixtures at arbitrary odd dimensions. The proof relies mainly on the Fourier transforms c(ij)(k) of the direct correlation functions defined by the OZ relation. From the analysis of the poles of c(ij)(k) we show that the direct correlation functions evaluated by the RFA method vanish outside the hard core, as required by the PY theory.

This paper presents an aeroelastic analysis procedure to highlight the influence of wind velocity on the structural damping and frequency of a long span cable-stayed bridge. Frequency dependent self-excited forces in terms of flutter derivatives are expressed as continuous functions using rational function approximation technique. The aeroelastically modified structural equation of motion is expressed in terms of frequency independent modal state-space parameters. The modal logarithmic dampings and frequencies corresponding to a particular wind speed are then determined from the eigen solution of the state matrix.

Accurate modeling of cables is important to study the behavior of high frequency disturbances in power converter systems. This paper reviews and compares two popular methodologies to model energy cables – an improved per unit length parameters based model and a Laplace SPICE element based model. The two models presented take into account the frequency dependence of the parameters of the cable. A ladder network is used for this purpose in the per unit length based model. The Laplace SPICE element model is generated from from a rational function approximation for the admittance parameters that are frequency dependant. The rational function approximation is obtained using a well known vector fitting algorithm. The time and frequency domain solutions of a two wire energy cable, obtained from the two models, agree well.

The purpose of this paper is to construct an integral rational Fourier operator based on the system of Chebyshev – Markov rational functions and to study its approximation properties on classes of Markov functions. In the introduction the main results of well-known works on approximations of Markov functions are present. Rational approximation of such functions is a well-known classical problem. It was studied by A. A. Gonchar, T. Ganelius, J.-E. Andersson, A. A. Pekarskii, G. Stahl and other authors. In the main part an integral operator of the Fourier – Chebyshev type with respect to the rational Chebyshev – Markov functions, which is a rational function of order no higher than n is introduced, and approximation of Markov functions is studied. If the measure satisfies the following conditions: suppμ = [1, a], a &gt; 1, dμ(t) = ϕ(t)dt and ϕ(t) ἆ (t − 1)α on [1, a] the estimates of pointwise and uniform approximation and the asymptotic expression of the majorant of uniform approximation are established. In the case of a fixed number of geometrically distinct poles in the extended complex plane, values of optimal parameters that provide the highest rate of decreasing of this majorant are found, as well as asymptotically accurate estimates of the best uniform approximation by this method in the case of an even number of geometrically distinct poles of the approximating function. In the final part we present asymptotic estimates of approximation of some elementary functions, which can be presented by Markov functions.

The approximation of a fractional order PIλDμ-controller transfer function using a chain fraction theory is considered. Analytical expressions for the approximation of s±α components of the transfer functions of PIλDμ-controllers were obtained through the application of the chain fraction theory. Graphs of transition functions and frequency characteristics of Dμ (α = μ = 0.5) and Iλ (α = λ = −0.5) parts for five different decomposition orders were obtained and analyzed. The results showed the possibility of applying the approximation of the PIλDμ-controller transfer function by the method of chain fractions with different valuesof λ and μ. For comparison, the transfer functions with the same order polynomials, obtained by the methods of Oustaloup transformation and chain fractions, were approximated for α = ±0.5. The analysis proved the advantages of using the chain fraction method to approximate the transfer function of the PIλDμ-controller. The performed approximation opens up the possibility of developing engineering methods for the technical implementation of PIλDμ-controllers. The accuracy of the same order transfer function approximation is higher when the method of chain fractions is used. It has been established that the adequacy of the frequency characteristics of the transfer functions obtained by the chain fraction method also depends on the approximation order.