Maximum Allowable Gain Calculation for Guaranteed Performance of Uncertain Systems

The goal of this work is to show a software package, based in Matlab, that simplifies the process of designing analog filters. This package is menu-oriented using the Matlab capabilities for doing so. The package is able to design lowpass, highpass, bandpass and bandreject filters. The approximation magnitude responses can be Butterworth (maximally flat magnitude), Chebyshev (equal-ripple), inverse Chebyshevi and elliptic. In the case of phase response, there is also the Thomson (maximally flat delay) also known as the Bessel response. The output for the approximation stage is a set of data that includes the transfer function poles and zeros, quadratic factors for both numerator and denominator polynomials, and the numerator and denominator polynomials. The network realizations can be either passive or active realizations. For the passive realizations, low sensitivity ladder realization is used. This realization is available for any approximation type. In the case of the active realizations, the authors have implemented the cascade of biquad stages but they are working on the active ladder realization. For the biquad stages, the Sallen-Key realization, multiple feedback, state-variable KHN, state variable Tow-Thomas and the GIC based biquad are available.

It is shown that magnitude hyperspherically symmetric transfer functions of multidimensional (MD) causal recursive digital filters must have numerator and denominator polynomials that are separately magnitude hyperspherically symmetric. Further, the exact reference-domain magnitude hyperspherically symmetric denominator polynomial is of infinite order, possessing only one free parameter, and the magnitude hyperspherically symmetric numerator polynomial itself has to be a radial even function. The corresponding MD design problem is shown to be essentially a one-dimensional design problem. Filter transfer functions having good symmetry and moderate degree can be designed by using the presented procedure. >

The paper deals with the problem of robust absolute stability of systems with parametric uncertainties. By using a 2q-convex parpolygonal value set of a family of polynomials of the form P(s, q) = a <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> (q) + a <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> (q)s +………+ a <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</inf> (q)s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> whose coefficients depend linearly on q = [q <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> , q <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> ,…, q <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</inf> ] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> and the uncertainty box is Q = {q : q <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> ∊ [q <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> , q <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> ], i = 1,2,….q}, an extremal boundary result for a transfer function, whose numerator and denominator polynomials are in the form of P(s, q), is first given. Then, using this boundary result, the robust versions of the classical absolute stability criteria of Lur'e, Popov and the Circle criterion are obtained. An example is included to illustrate the benefits of the method presented.

Provides author's reply to "Comments on 'Optimizing simultaneously over the numerator and denominator polynomials in the Youla-Kucera parameterization" (ibid., vol. 50, no. 9, pp. 1369-1374, Sep 05).

Another approach for generating reduced order discrete time system using composite method is being proposed. The proposed method is validated by considering a Linear Time Invariant Discrete System (LTIDS) having single input single output(SISO) of higher order being represented in frequency domain. Eigen Spectrum Analysis and Least Squares moment matching methods form the proposed composite method to obtain the reduced numerator and denominator polynomials. The suitability of the proposed method is judged by comparing the step, impulse responses of the original and reduced systems. The performances of the systems are measured in terms of Steady State Error (SSE). It is seen from the numerical examples that the results are quite competitive as compared to other methods thus justifying the applicability of the proposed method.

The paper deals with the problem of computing the Bode envelope of an uncertain transfer function whose numerator and denominator polynomials are multiples of independent uncertain polynomials of the form P(s, q) = l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> (q) + l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> (q) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> +.........+ l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> (q)s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> whose coefficients depend linearly on q = [q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ,..., q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> ] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> and the uncertainty-box is Q = {q : q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ϵ[q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , q <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ], i = 1,2,...., q}. Using the geometric structure of the value set of P(s, q), a powerful edge elimination procedure is proposed for computing the magnitude and phase envelopes of these uncertain systems. A numerical example is included to illustrate the benefit of the method presented.

In this brief, a new technique for designing the all-pass-based notch filter is proposed. An all-pass-based notch filter consists of an all-pass filter, which has the mirror-image symmetry relation between the numerator and denominator polynomials. With this property, a relationship between the pole angel and the zero angle of an all-pass-based notch filter is derived. Knowing this relationship, we can design an all-pass-based notch filter by assigning the pole position according to the zero position. The process of the new technique is illustrated in this brief. Moreover, experiments are presented to examine a track of the pole position at the specific frequency and to demonstrate a design example of an all-pass-based notch filter.

It is proved that the bounded rational property of multivariable functions remains invariant under simultaneous partial differentiation of numerator and denominator polynomials. The proof is simple, and all known related results follow as special cases. Other consequences are mentioned.

Path generating functions and continued fractions

An approach for locating the dominant complex natural resonances (poles) of scatterers or radiators is described. Admittance values, calculated using Richmond's thin wire computer programs, at a regular grid of complex and real frequencies are fit by a rational function. Zeros of the resultant numerator and denominator polynomials then yield, respectively, the zeros and poles of the structure. The method is illustrated for resonances of straight and bent wire structures.

ABSTRACTThis paper presents a novel arrangement of Routh table array for deriving an approximate model of a higher order z-domain uncertain system. The demand for this computation is to procure a lower order model which is easy to be exercised in comparison to their original large scale systems. Additionally, the derived model should preserve fewer dynamic characteristic of the comprehensive higher order systems. The mentioned new arrangement is achieved from the arena of different combinations of numerator and denominator polynomials. The combinations are validated by their practice over the conventional example from the literature. This precise blend is then applied to a real-time system for its rational acceptability. Both the models play a significant role in establishing the algorithm. Besides this, the limitation encountered during the foundation course of the arrangement is also taken into consideration. The paper also offers a future scope for fellow researchers.

The generation of 2-D transfer functions having non-separable denominator and numerator polynomials, starting from two 2-D all-pass transfer functions is studied. It is shown that, by a proper choice of the coefficients chosen, their magnitude response can be made to exist in two opposite quadrants only. >

In the transfer function with zeros restricted on the imaginary axis of S‐plane the phase characteristic is essentially determined by that of the denominator polynomial. As long as the denominator polynomial is a Hurwitz polynomial, the tangent of its phase characteristic, i.e., the ratio of the imaginary part (odd part) to the real part (even part), is a reactance function. This paper proposes a waveform approximation method which approximates the phase tangent of the Fourier transform of the object waveform by a reactance function using an optimization technique. The Hurwitz polynomial, which is the sum of the numerator and denominator of the reactance function, is adopted as the denominator polynomial of the approximate transfer function. The numerator polynomial of the approximate transfer function is derived by adjusting the zeros on the imaginary axis by an optimization technique to approximate the amplitude of the Fourier transform of the object waveform. An evaluation measure in the proposed approximation method is presented, describing its problems and physical implications. It is shown that the proposed method can provide a satisfactory approximation for symmetrical waveforms. For asymmetrical waveforms it is shown that the approximation can be improved by adding a delay time. An appropriate weighting function is proposed for the evaluation measure. Lastly, the proposed method is compared with other methods of approximation.

In this letter, we propose a novel all-pass (AP) fractional delay filter whose denominator polynomial is obtained by truncating the power series of a certain function. This function is derived from the frequency response of the denominator whose magnitude response is related to the desired phase response through the Hilbert transform since the denominator of a stable AP filter is of minimum phase. The target function and corresponding power series are calculated analytically and expressed in closed form. The closed-form expressions facilitate the analysis of stability. According to the properties for the coefficients of the denominator polynomial, we show that the proposed AP filter is stable for positive delay. Numerical examples indicate that the phase delays of the proposed filters are flat around DC.

Robustness of pole location in perturbed systems