Optimal Hankel-norm approximation of stable systems with first-order stable weighting 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.

Autotuners represent a combination of a relay feedback identification test and some control design method. In this contribution, models with up to three parameters are estimated by means of a single asymmetrical relay experiment. Then a stable low order transfer function with a time delay term is identified by a relay experiment. Autotuning principles then combine asymmetrical relay feedback tests with a control synthesis. Two algebraic control syntheses then are presented in this paper. The first one is based on the ring of proper and stable rational functions R PS. The second one utilizes a special ring R MS, a set of RQ-meromorphic functions. In both cases, controller parameters are derived through a general solution of a linear Diophantine equation in the appropriate ring. A final controller can be tuned by a scalar real parameter m>0. The presented philosophy covers a generalization of PID controllers and the Smith-like control structure. This contribution deals with the implementation of proposed autotuners and presents some illustrative examples. A Matlab toolbox for automatic design and simulation was developed and various simulations performed and analyzed.

A method for implementing Gaussian low-pass filter by using switched-current technique has been presented in this paper. The stable rational transfer function for Gaussian-shaped low-pass filter is approximated by the multiplication of the follower-integrator functions based on central limit theorem. An SI filter structure constructed from the inverse follow-the-leader feedback configuration has been proposed to synthesize the obtained transfer function, in which the switched-current bilinear integrator is employed as the building cell. Explicit design formulas are also derived. The 8th-order Gaussian low-pass filter is selected as the example to elaborate the design details. Simulation result shows that the proposed design method has high approximation precision and low magnitude sensitivity.

This paper proposes a new coefficient identification procedure to stabilize rational function predistorters. Rational functions are inherently capable of superior nonlinear modelling and extrapolative prediction, which make them ideal candidates for power amplifier (PA) predistortion. However their susceptibility to model divergence can lead to instability and linearization failure. In this paper, the root cause of model divergence is first determined. A constrained coefficient identification procedure is then proposed to identify unconditionally stable rational functions. Finally, the potential of the stable rational function is demonstrated by linearizing a GaN Doherty PA transmitting wideband modulated signals.

This paper studies the generation of stable transfer functions for which the real or imaginary part takes prescribed values at discrete uniformly spaced points on the unit circle. Formulas bounding the error between a particular interpolating function and any function consistent with the data are presented, which have the desirable property that the error goes to zero exponentially fast with the number of interpolating points. The paper also examines generation of stable minimum phase transfer functions for which the magnitude takes prescribed values at uniformly spaced points on the unit circle, and presents error bounds for this problem. Connection with the discrete Hilbert transform is made. The effect of uncertainty in the original data is also examined.

The contribution is focused on brief description of proposed control design process and primarily on practical application of obtained simple robust control algorithms. The synthesis is based on general solutions of Diophantine equations in the ring of proper and stable rational functions, Youla-Kučera parameterization and divisibility conditions and one of its main advantages can be seen in existence of single scalar tuning parameter. The real-time control behaviour has been tested on laboratory model of hot-air tunnel which has been mathematically described by various transfer functions containing parametric type of uncertainty.

On the Problem of Decoupling

This study describes the use of a biofeedback method for the noninvasive study of baroreflex mechanisms. Five previously untrained healthy male participants learned to control oscillations in heart rate using biofeedback training to modify their heart rate variability at specific frequencies. They were instructed to match computer-generated sinusoidal oscillations with oscillations in heart rate at seven frequencies within the range of 0.01-0.14 Hz. All participants successfully produced high-amplitude target-frequency oscillations in both heart rate and blood pressure. Stable and predictable transfer functions between heart rate and blood pressure were obtained in all participants. The highest oscillation amplitudes were produced in the range of 0.055-0.11 Hz for heart rate and 0.02-0.055 Hz for blood pressure. Transfer functions were calculated among sinusoidal oscillations in the target stimuli, heart rate, blood pressure, and respiration for frequencies at which subjects received training. High and low target-frequency oscillation amplitudes at specific frequencies could be explained by resonance among various oscillatory processes in the cardiovascular system. The exact resonant frequencies differed among individuals. Changes in heart rate oscillations could not be completely explained by changes in breathing. The biofeedback method also allowed us to quantity characteristics of inertia, delay, and speed sensitivity in baroreflex system. We discuss the implications of these findings for using heart rate variability biofeedback as an aid in diagnosing various autonomic and cardiovascular system disorders and as a method for treating these disorders.

It is shown that a p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</sup> m transfer function G(s) with p⩾m can be decomposed as G = (I-ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> ...(I-ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r+2</sub> Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r+2</sub> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> (I-ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r=1</sub> Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r=1</sub> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</sub> where Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> are stable all-pass transfer functions, 1=ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ..=ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> ≫ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r=1</sub> ..≫ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> ≫0 are the Hankel-singular-values of GW* <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> where G*G=WW* with W stable and minimum phase. Results on the McMillan degree of G <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Λ</sup> :=(I-ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> )..(I-ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> )G then show that G <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Λ</sup> gives a good low order approximation to G in the sense of relative error.

Simultaneous stabilization using stable system inversion

UDC 517.5 We study the relationship between certain Fibonacci polynomials and the Schwarz lemma within the framework of the complex analysis and control engineering. The Fibonacci polynomials are traditionally associated with combinatorial mathematics and recursive sequences. At the same time, they also play a significant role in the power series expansions of analytic functions. We present a version of the Schwarz lemma based on the first Fibonacci polynomial. By considering the boundary version of the Schwarz lemma, we deduce new inequalities that provide deeper insights into the relationship between these two mathematical concepts. Beyond their theoretical significance, we also present practical implications of our results. Specifically, we show that the deduced results can be applied to get marginally stable discrete-time transfer functions in digital control systems.

An earlier paper showed that the generalised Nyquist approach, although intrinsically a frequency-response approach, can be deployed to produce an effective multivariable self-tuning methodology. The results, however, were confined to the case of plants with open-loop stable poles and characteristic transfer functions whose branch points are all stable. The purpose of the present paper is to develop the framework that applies to the general case. In particular, consideration is given to fractional representations for the branches of the characteristic transfer functions, and algorithms are developed for both the computation of these representations and their implementation within a self-tuning configuration. Consideration is also given to issues of stability. The results of the paper are illustrated by means of numerical examples and the paper is concluded by a design study based on the open-loop unstable chemical reactor.

Conditions for a Class of Rational Approximants of Fractional Differentiators/Integrators to Enjoy the Interlacing Property

Stray free switched capacitor loop biquad that realizes different generic transfer functions

l1 Robust Identification of Fractional Transfer Functions