Optimal Approximation, Simulation and Analog Realization of the Fundamental Fractional Order Transfer Function

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.

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.

Low-order network models, commonly used to assess the thermo-acoustic stability of combustors, can be cast in a linear, time-continuous state-space representation. A standard linear eigenvalue problem for the system modes results, which can be solved in a robust and efficient manner. To represent the linear dynamics of any time-invariant flame in the state-space framework, this study presents an approximation of the distributed time-delayed flame response to acoustic velocity perturbations based on a spatially discretized propagation equation (PE). We derive the rational flame transfer function of a first-order-upwind-PE state-space model and discuss its relation to the Tustin approximation of flame transfer functions. For an exemplary discrete finite impulse response of a flame, a third-order-upwind-PE state-space model is shown to match the discrete flame frequency response with comparable accuracy as a rational approximation found by non-linear optimization. The numerical dissipation introduced by discretization of the PE ensures zero gain above the Nyquist frequency of the underlying discrete flame impulse response. Finally, we apply the PE state-space flame model to a generic Rijke tube and show that the predicted thermo-acoustic modes agree well with results obtained from a classical non-linearly optimized rational approximation of the frequency response function of the flame.

System identification is an important methodology used in control theory and constitutes the first step of control design. It is known that many real systems can be better characterized by fractional-order models. However, it is often quite complex and difficult to apply classical control theory methods analytically for fractional-order models. For this reason, integer-order models are generally considered in classical control theory. In this study, an alternative approximation method is proposed for fractional-order models. The proposed method converts a fractional-order transfer function directly into an integer-order transfer function. The proposed method is based on curve fitting that uses a quadratic system model and Equilibrium Optimizer (EO) algorithm. The curve fitting is implemented based on the unit step response signal. The EO algorithm aims to determine the optimal coefficients of integer-order transfer functions by minimizing the error between general parametric quadratic model and objective data. The objective data are unit step response of fractional-order transfer functions and obtained by using the Grünwald-Letnikov (GL) method in the Fractional-Order Modeling and Control (FOMCON) toolbox. Thus, the coefficients of an integer-order transfer function most properly can be determined. Some examples are provided based on different fractional-order transfer functions to evaluate the performance of the proposed method. The proposed method is compared with studies from the literature in terms of time and frequency responses. It is seen that the proposed method exhibits better model approximation performance and provides a lower order model.

A new way of formulating a multipoint Pade approximant of a linear system transfer function is given. It is seen to be very flexible and computationally efficient. The expansion points can be a mixture of real, complex, purely imaginary and multiple points with no complex arithmetic required. The underlying idea of the method is to convert the rational approximation problem into one of polynomial approximation. This results in simple matrix multiplications and the solution of a set of linear equations to derive the reduced model. Examples are given to demonstrate the application of the technique.

The aim of this work is to generalize to the case of weighted L 2 spaces some results about L 2 approximation by analytic and rational functions which are useful to perform the identification of unknown transfer functions of stable (linear causal time-invariant) systems from incomplete frequency data.KeywordsHardy spacesweighted rational approximationfrequency domain identification.

The paper deals with the substantiation of the topicality of the problem of the creation of the high-quality and relatively simple control systems for electric drives with Switched reluctance motor (SRM) due to the continuous increase of the field of application of SRM in industry. The mathematical simulation method with the application of Matlab System Identification toolbox was used to determine that the sufficient convergence of the results can be obtained at the identification of SRM with a fractional rational transfer function having three zeros and four poles, R-squared value for this variant makes 86%. Such a high order of the transfer function of SRM causes the complexity of its regulator design. The authors worked out a method for the identification of SRM fractional measured transfer function based on the use of FOMCON MATLAB Toolbox. This approach provides higher accuracy of SRM identification. R-squared value for the proposed method makes 94 - 96%, which allows the simplification of the controller structure and the improvement of the accuracy of SRM control in the closed-loop control system. It is shown that the structure of the fractional measured transfer function, identifying the process of SRM start, does not depend on the electric drive inertia moment.

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

The problem of estimating the transfer function of a linear system, together with the spectral density of an additive disturbance, is considered. The set of models used consists of linear rational transfer functions and the spectral densities are estimated from a finite-order autoregressive disturbance description. The true system and disturbance spectrum are, however, not necessarily of finite order. We investigate the properties of the estimates obtained as the number of observations tends to ∞ at the same time as the model order employed tends to ∞. It is shown that the estimates are strongly consistent and asymptotically normal, and an expression for the asymptotic variances is also given. The variance of the transfer function estimate at a certain frequency is related to the signal/noise ratio at that frequency and the model orders used, as well as the number of observations. The variance of the noise spectral estimate relates in a similar way to the squared value of the true spectrum.

It is shown that two properties of rational transfer functions with real coefficients are realized by means of a single complex IIR digital filter. Equivalently, the real and imaginary parts of output from a complex filter for a real input sequence correspond to real coefficient transfer function respectively. In a real signal processing, a delay operator z/sup -1/ corresponds to a real operation. In this case, a complex transfer function is formed by connecting two real transfer functions with an imaginary unit. These real functions are approximated to the desired properties and are expressed by a complex coefficient transfer function.

Closed-loop time response analysis of irrational fractional-order systems with numerical Laplace transform technique

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.

This paper introduces for the first time analog circuit implementations of two fundamental linear fractional order systems whose impulse responses called fractional order damped sine and cosine functions are the inverse Laplace transform of their irrational transfer functions. These analog circuit implementations are derived through rational function approximations of their irrational transfer functions.

In this paper, limit cycle frequency, pulse width and stability analysis are examined using different methods for relay feedback nonlinear control systems with integer or fractional order plant transfer functions. The describing function (DF), A loci, a time domain method formulated in state space notation and Matlab/Simulink simulations are used for the analysis. Comparisons of the results of using these methods are given in several examples. In addition, the work has been extended to fractional order systems with time delay. Programs have been developed in the Matlab environment for all the theoretical methods. In particular, Matlab programs have been written to obtain a graphical solution for the A loci method, which can precisely calculate the limit cycle frequency. The developed solution methods are shown in various examples. The major contribution is to look at finding limit cycles for relay feedback systems having plants with a fractional order transfer function (FOTF). However, en route to this goal new assessments of limit cycle stability have been done for a rational plant transfer function plus a time delay.