Controller reduction via stable factorization and balancing

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.

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 novel complementary filter for three-dimensional (3D) attitude estimation is proposed. In order to deals with the nonlinearity of 3D attitude, the conventional complementary filter integrated the measured angular velocity corrected by the vector measurement, but it ignored the sensor dynamics which causes the lag. Authors' previous work compensated the sensor dynamics based on the idea that it can be divided into the linear transfer function and the nonlinear coordinate transformation. The properness and stability of the total transfer function are guaranteed by the designed filter inserted between the linear and nonlinear part. However, the disturbance strongly affects the accuracy of the azimuth angle due to the coordinate transformation. This paper aims to improve the estimation accuracy by combining the idea of correcting the measured angular velocity by the vector measurement and the sensor dynamics compensation only with the proper and stable transfer function. The characteristics of the proposed method are the following two points: 1) the relaxation of the complementary condition in order to make the non-proper and unstable function proper and stable, and 2) the filtering of the estimated vector by the sensor dynamics instead of the filtering of the measured vector by the inverse dynamics in order to avoid the magnification of the high frequency noise. Through the experiment for the fast and irregular motion under 5[Hz], the validity of the proposed method is verified.

We study the class of linear differential-algebraic m-input m-output systems which have a transfer function with proper inverse. A sufficient condition for the transfer function to have proper inverse is that the system has ‘strict and non-positive relative degree’. We present two main results: first, a so-called ‘zero dynamics form’ is derived; this form is—within the class of system equivalence—a simple form of the DAE; it is a counterpart to the well-known Byrnes–Isidori form for ODE systems with strictly proper transfer function. The ‘zero dynamics form’ is exploited to characterize structural properties such as asymptotically stable zero dynamics, minimum phase, and high-gain stabilizability. The zero dynamics are characterized by (A, E, B)-invariant subspaces. Secondly, it is shown that the ‘funnel controller’ (that is a static non-linear output error feedback) achieves, for all DAE systems with asymptotically stable zero dynamics and transfer function with proper inverse, tracking of a reference signal by the output signal within a pre-specified funnel. This funnel determines the transient behaviour.

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.

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

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

A new method-based on transformations from 1-variable transfer functions is presented for the design of stable 2-variable transfer functions. As compared with already existing methods, this method results in a good response obtained through fewer numbers of iterations. The designed filters are always stable without the need of any constraints during minimization.

This paper studies the approximation of continuous functions in subsets of all causal and stable transfer functions. Such approximations play a central roll in filter design, filter bank analysis, and in sampling, since any filtering can be considered as a kind of approximation in a space defined by the filters. The present paper studies in particular the consequences resulting from the causality and stability constrain imposed on the filter process. It is shown that there exists no linear approximation method which is also causal and stable. Only if either the causality or the stability constrain is left out, a linear approximation method may exist.