An Efficient Method for Approximation of Non Rational Transfer Functions

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.

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.

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.

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.

In this paper fractional linear systems with nonlinear saturated disturbance are considered and a novel method is proposed for analyzing asymptotic stability of the given system. If the pure linear system is unstable, state feedback is used to achieve stability condition. Furthermore, when disturbance exists that is Lipschitz in state, we will obtain stability condition for the fractional-order (FO) linear system with nonlinear saturated disturbance. Since stability region plays an important role in system analysis and control, the largest stability region is found using proposed optimization problem. This analysis can be extended to FO linear systems with any nonlinear elements that are Lipschitz in stats. Finally an example of mathematical model is provided to illustrate application of analysis and the proposed optimization method.

In this study, the aeroelastic response of a wing-pylon-nacelle system in subsonic and low supersonic flow regimes is analyzed using the continuation method in conjunction with an adaptive step size control algorithm. Idealizing the pylon and nacelle as a point mass, the computed effects of a standard structural analysis of the wing together with the pylon and nacelle are compared with those of a clean wing to build a reduced-order model for analysis. The aerodynamic forces relating to different reduced frequencies are assessed using the Doublet Lattice Method (DLM) in the subsonic flow regime and supersonic lifting surface theory relying on the unsteady linearized small-disturbance potential flow model in the low supersonic flow regime. The Rational Function Approximation (RFA) method is then utilized for the state-space formulation of the system equations, appended with the continuation method for flutter prediction. Thereafter, the linearized aeroelastic equations are resolved using the continuation method with adaptive step size, the results of which are matched with those obtained from the traditional p-k method to emphasize that the continuation method exhibits a distinct advantage in achieving better accuracy in estimating the flutter speed and identifying the “mode switching” phenomenon.

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.

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

We present an interpolation method for efficient approximation of parametrized functions. The method recognizes and exploits the low‐dimensional manifold structure of the parametrized functions to provide good approximation. Basic ingredients include a specific problem‐dependent basis set defining a low‐dimensional representation of the parametrized functions, and a set of ‘best interpolation points’ capturing the spatial‐parameter variation of the parametrized functions. The best interpolation points are defined as solution of a least‐squares minimization problem which can be solved efficiently using standard optimization algorithms. The approximation is then determined from the basis set and the best interpolation points through an inexpensive and stable interpolation procedure. In addition, an a posteriori error estimator is introduced to quantify the approximation error and requires little additional cost. Numerical results are presented to demonstrate the accuracy and efficiency of the method. Copyright © 2007 John Wiley & Sons, Ltd.

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.

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

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>

Identification of linear time-varying fractional order systems using block pulse functions based on repetitive principle

In this paper, we introduce a method for multivariate function approximation using function evaluations, Chebyshev polynomials, and tensor-based compression techniques via the Tucker format. We develop novel randomized techniques to accomplish the tensor compression, provide a detailed analysis of the computational costs, provide insight into the error of the resulting approximations, and discuss the benefits of the proposed approaches. We also apply the tensor-based function approximation to develop low-rank matrix approximations to kernel matrices that describe pairwise interactions between two sets of points; the resulting low-rank approximations are efficient to compute and store (the complexity is linear in the number of points). We present an adaptive version of the function and kernel approximation that determines an approximation that satisfies a user-specified relative error over a set of random points. We extend our approach to the case where the kernel requires repeated evaluations for many values of (hyper)parameters that govern the kernel. We give detailed numerical experiments on example problems involving multivariate function approximation, low-rank matrix approximations of kernel matrices involving well-separated clusters of sources and target points, and a global low-rank approximation of kernel matrices with an application to Gaussian processes. We observe speedups up to 18X over standard matrix-based approaches.