Synthesis of Rational Transfer Function Approximations Using a Tapped Distributed RC Line With Feedback

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.

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

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.

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.

Rational transfer function models for nitrifying trickling filters

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

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.

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.

Proportional integral (PI) and proportional integral derivative (PID) controllers have been at the heart of control engineering practice for several decades. 95 % of the controllers employed in the industry are PI/PID. In process control, one often encounters systems described by transfer functions with time delays, which become transcendental functions. The design of the controller demands the rational transfer function approximation of the time-delay term. This paper focuses on the effect of time-delay approximation techniques, viz. Taylor series expansion and Pade approximation, on the structure and performance of PI/PID controllers designed with Internal Model Control (IMC). The performance of the PI/PID controllers was tested in simulation environment on various processes with time delay. For uniform comparison, the controllers were tuned to have a same robustness measure, in terms of maximum sensitivity (M S). The results indicate, irrespective of time-delay approximation considered, the controllers provide good set point tracking and poor disturbance rejection.

Rational functions have been introduced into the performance analysis and evaluation of stochastic discrete event systems. In this paper we consider the problem of real rational function approximation. We are given a strictly proper real rational function with denominator degree equal to n, and the objective is to generate lower order rational function approximations. We suggest a system theoretic approach for generating such approximations and in addition provide error bounds. >

The characteristics of a so-called real conjugate zero pair in the biquadratic transfer function are investigated and a new active RC network is provided to implement it. Different magnitude responses are obtained where the zero pair is varied while maintaining an invariant phase response. By this feature, the network can be seen as the counterpart to a phase shifter. With the aid of a simple additional circuit, the arithmetic transformation of a real conjugate zero pair into an imaginary conjugate, or into a double zero pair at the origin, is also demonstrated. This part of the network consists of operational amplifiers and resistors, each of equal value, which can be advantageously integrated. The result is an easy-to-design second-order filter section for multi-functional use.

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 sample and hold effect, or quasi-digital (discrete) behaviour, of constant-frequency peak current-mode control (PCM) is pulse-width modulated DC-DC converters is presented in a manner consistent with the physical behaviour of the circuit. This approach develops a transfer function using control theory that captures the quasi-digital behaviour encountered when the current control loop is closed in PCM. The new transfer function accurately captures the current loop behaviour, leading the way for a compete linear time-variant system circuit for PWM DC-DC converters. This work improves upon previous derivations by remaining true to the physical behaviour of the circuit while resulting in a rational transfer function that can be used in a multiple loop complete linear system circuit to accurately predict converter behaviour due to changes in input voltage, output current, and reference voltage.