A note on continued fraction inversion

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.

A chain of factored matrices is derived for formulating the Routh array if the first column of the array is known. The factored matrices may be used to perform the conversion of a continued fraction into a rational function. A set of tables based on the second Cauer form of continued fraction expansion is also included. These tables are the approximated rational functions for the commonly used irrational function √ and the transcendental function e s .

Uncertainty principle and sparse reconstruction in pairs of orthonormal rational function bases

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.

Rational transfer function models for nitrifying trickling filters

Basilar-membrane responses to white Gaussian noise were recorded using laser velocimetry at basal sites of the chinchilla cochlea with characteristic frequencies near 10 kHz and first-order Wiener kernels were computed by cross correlation of the stimuli and the responses. The presence or absence of minimum-phase behavior was explored by fitting the kernels with discrete linear filters with rational transfer functions. Excellent fits to the kernels were obtained with filters with transfer functions including zeroes located outside the unit circle, implying nonminimum-phase behavior. These filters accurately predicted basilar-membrane responses to other noise stimuli presented at the same level as the stimulus for the kernel computation. Fits with all-pole and other minimum-phase discrete filters were inferior to fits with nonminimum-phase filters. Minimum-phase functions predicted from the amplitude functions of the Wiener kernels by Hilbert transforms were different from the measured phase curves. These results, which suggest that basilar-membrane responses do not have the minimum-phase property, challenge the validity of models of cochlear processing, which incorporate minimum-phase behavior.

We revisit the problem of approximating a multiple-input multiple-output $p\times m$ rational transfer function $H(s)$ of high degree by another $p\times m$ rational transfer function $\widehat{H}(s)$ of much smaller degree, so that the $\mathcal{H}_2$-norm of the approximation error is minimized. We show that in the general case of higher-order poles in the reduced-order model, called the defective case, the stationary points of the $\mathcal{H}_2$-norm of the approximation error can still be characterized by tangential interpolation conditions. We also indicate that the sensitivity of the solution of this problem depends on the parameterization used.

The transfer functions G(s) for different types of heat exchangers obtained from their partial differential equations usually contain some irrational components which reflect quite well their spatio-temporal dynamic properties. However, such a relatively complex mathematical representation is often not suitable for various practical applications, and some kind of approximation of the original model would be more preferable. In this paper we discuss approximate rational transfer functions G^(s) for a typical thick-walled double-pipe heat exchanger operating in the counter-flow mode. Using the semi-analytical method of lines, we transform the original partial differential equations into a set of ordinary differential equations representing N spatial sections of the exchanger, where each nth section can be described by a simple rational transfer function matrix Gn(s), n=1,2,…,N. Their proper interconnection results in the overall approximation model expressed by a rational transfer function matrix G^(s) of high order. As compared to the previously analyzed approximation model for the double-pipe parallel-flow heat exchanger which took the form of a simple, cascade interconnection of the sections, here we obtain a different connection structure which requires the use of the so-called linear fractional transformation with the Redheffer star product. Based on the resulting rational transfer function matrix G^(s), the frequency and the steady-state responses of the approximate model are compared here with those obtained from the original irrational transfer function model G(s). The presented results show: (a) the advantage of the counter-flow regime over the parallel-flow one; (b) better approximation quality for the transfer function channels with dominating heat conduction effects, as compared to the channels characterized by the transport delay associated with the heat convection.

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.

Abstract.A distributed lag model with a rational transfer function is considered. We have demonstrated that theC‐table can be used to determine the orders of the rational transfer function and the Padé table can be used to estimate the parameters of an identifiable transfer function. In case the model is not uniquely identified, theC‐table reduces possible competing models to only a few for further examination. Padé approximant gives consistent initial values for possibly more efficient iterative procedures to estimate the coefficients of the identified model. Some asymptotic results on the estimation of theC‐table and the Padé table are given. A few examples are presented to demonstrate the effectiveness of the method.

Mapping from the s-domain to the z-domain via the magnitude-invariance method

The authors compare various methods that aim to decrease the bias in the Kumaresan-Tufts (KT) algorithm for estimating the parameters of rational transfer functions in additive white noise. A bias-reduced method for estimating rational transfer functions is shown to give improved performance over existing methods. The bias compensated method has the additional advantage of not requiring more computations than the KT method.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The three techniques of s-to-z transform, power series expansion (PSE) and signal modelling are combined to develop a new procedure for efficiently computing the fractional order derivatives and integrals of discrete-time signals. A mapping function between the s-plane and the z-plane is first chosen, and then a PSE of this mapping function raised to fractional order is performed to get the desired infinite impulse response of the ideal digital fractional operator. Finally, the desired impulse response is modelled as the impulse response of a linear invariant system whose rational transfer function is determined using deterministic signal modelling techniques. Three non-iterative techniques, namely Pade, Prony and Shanks’ methods have been considered in this paper. Using Al-Alaoui’s rule as s-to-z transform, computation examples show that both Prony and Shanks’ method can achieve more accurate fractional differentiation and integration than Pade method which is equivalent to continued fraction expansion technique.

Due to the fact that the overall geometry sizes of electronic structure in general becomes smaller and the frequencies of the signals used by these structures is increasing, many electromagnetic effects are tending to influence the behavior of the whole electronic structures. These effects do not only take place in very small features sizes in chips, but also on PCB's scale these effects are observed. Modeling of the behavior of these structures involves complicated and, more important, very large systems. Model order reduction can help reducing the size of these models, to make it computationally more flexible. In this paper, reduced order modeling of linear interval systems using multipoint continued fraction expansion is studied. Continuous time invariant systems characterized by rational transfer functions is also considered in this paper. The Kharitonov's polynomial associated with the numerators and denominators of the original interval system and interval reduced model are obtained.

In the scalar case, it is widely known that the impulseresponse sequence of a minimum-phase transfer function possesses the minimum-energy delay property, i.e., on the set of an impulse-response sequence <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H_k</tex> having the same magnitude <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|H(e^{j\omega})|</tex> , the partial energy <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon(m)</tex> defined by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon(m) = \sum_{k=0}^{m}|H_k|^2</tex> is maximum for all <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m {\geq} 0</tex> when the rational transfer function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H(z)</tex> is minimum phase [1]. In this brief, it is shown that the minimum-energy delay property is valid in the matrix case. This is proved first for the matrix-valued transfer functions of the Hardy class <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H^2</tex> , and then is verified for the matrix-valued transfer functions of the Smirnov class <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N^+</tex> . We shall see that the minimum-energy delay property is proved by using the inner-outer (or all-pass and minimum-phase) factorization of transfer functions and the Parseval identity for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L^2</tex> class functions.