Padé methods of Hurwitz polynomial approximation with application to linear system reduction

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.

We present an approach to generate multiscroll attractors via destabilization of piecewise linear systems based on Hurwitz matrix in this paper. First we present some results about the abscissa of stability of characteristic polynomials from linear differential equations systems; that is, we consider Hurwitz polynomials. The starting point is the Gauss–Lucas theorem, we provide lower bounds for Hurwitz polynomials, and by successively decreasing the order of the derivative of the Hurwitz polynomial one obtains a sequence of lower bounds. The results are extended in a straightforward way to interval polynomials; then we apply the abscissa as a measure to destabilize Hurwitz polynomial for the generation of a family of multiscroll attractors based on a class of unstable dissipative systems (UDS) of affine linear type.

A necessary and sufficient condition for designing formation of discrete-time multi-agent systems with delay

Investigation and improvement of the Mini-Max Polynomial Approximation method for solving burnup equations

In this paper, stabilizing problems in control design are addressed for linear discrete-time systems, reflecting equality constraints tying together some state variables. Based on an enhanced representation of the bounded real lemma for discretetime systems, the existence of a state feedback control for such conditioned stabilization is proven, and an LMI-based design procedure is provided. The control law gain computation method used circumvents generally an ill-conditioned singular design task. The principle, when compared with previously published results, indicates that the proposed method outperforms the existing approaches, guarantees feasibility, and improves the steady-state accuracy of the control. Furthermore, better performance is achieved with essentially reduced design effort. The approach is illustrated on simulation examples, where the validity of the proposed method is demonstrated using one state equality constraint.

This paper presents a novel reduction technique for higher-order systems. By using the suggested hybrid approach, the lower-order system is achieved. In this proposed work, the denominator polynomial of the reduced order system is identified using the Routh approximation technique, and the numerator polynomial of the reduced order system is determined using the Pade method. The proposed approach is theoretically straightforward and always creates a stable reduced system for stable the original systems. The performance of the anticipated technique is examined through various numerical illustrations.

The state feedback design for singularly perturbed systems described in Delta operator is considered. The composite state feedback controller for slow and fast subsystems is designed by using the direct method. The obtained results can bring previous conclusions of continuous and discrete time systems into the unified Delta framework. A simulation example is presented to demonstrate the validity and efficiency of the design.

State estimation for two-dimensional (2-D) discrete-time systems is considered. Results for both the Givone-Roesser local state model and the Porter-Aravena wave advance process model are established. The analysis avoids the assumption of stationarity of the system and separability of the characteristic polynomial.

International audience

In this thesis a certain estimation problem for nonlinear stochastic dynamical systems in discrete time, known as filtering in the literature, is considered. The objective is to reconstruct the current state of the system by means of observations. The observations are noise corrupted measurements of a function of the state.In the first chapter we will provide concepts from probability theory necessary to define and investigate the nonlinear filtering problem. The definition of the nonlinear filter is presented in the second section of this chapter.The second chapter explains in detail why filtering is an extremely difficult problem in a nonlinear context. It turns out that a finite dimensional representation of the filter is possible in very special circumstances, only. This chapter summarizes known results and is intended mainly as a motivation for the necessity of investigating approximation schemes for the nonlinear filter, considered in the following two chapters.Numerical approximation methods are then investigated from a general point of view in the third chapter. A general framework to obtain error bounds for approximation schemes is presented. Necessary for this investigation are metrics for probability distributions we will make extensive use of. Furthermore, an essential property of the filter required to give a bounded approximation error turns out to be a negative Lyapunov exponent of the filter dynamics.The fourth chapter provides some classes of approximation methods. Common to all these methods are projection techniques on parametrized families of probability distributions. This approach was carried out for continuous time systems already by several authors. However, the error analysis carried out in this thesis is, to the best of our knowledge, new.The last two chapters present (aside from two small sections devoted to a Monte Carlo approach), two interesting and important applications of nonlinear filtering. The first is estimation of an unknown parameter in the dynamics. This problem is very important in all branches of science, and we will present two numerical examples. The second application is reconstruction of a sent message in telecommunications. To this problem a chapter is devoted, including results on the bit error probability obtained using methods from nonlinear filtering theory for a simple transmitter model.To summarize, the aim of this thesis is threefold. First, to show that filtering of nonlinear dynamical system is a nontrivial and interesting problem from a mathematical point of view, second to show methods to overcome the difficulties arising in applications, and third to show that filtering is not a purely artificial mathematical problem but has a great significance in science and engineering.

Control systems are pervasive in our lives. Our homes have environmental controls. The appliances we use, such as the washing machine, microwave, etc. carry embedded controllers in them. We fly in airplanes and drive automobiles that extensively use control systems. The industrial plants that produce consumer goods run on process control systems. The recent drive toward automation has increased our reliance on control systems technology. This book discusses control systems design from a model-based perspective for dynamic system models of single-input single-output type. The emphasis in this book is on understanding and applying the techniques that enable the design of effective control systems in multiple engineering disciplines. The book covers both time-domain and the frequency-domain design methods, as well as controller design for both continuous-time and discrete-time systems. MATLAB© and its Control Systems Toolbox are extensively used for design. Technical topics discussed in the book include: • Mathematical models of physical systems • Analysis of transfer function and state variable models • Control systems design objectives • Control system design with root locus • Control system design in the state-space • Control system design of sampled-data systems • Compensator design with frequency response methods

Implicit planar curve and surface fitting to a set of scattered points plays an important role in solving a wide variety of problems occurring in computer graphics modelling, computer graphics animation, and computer assisted surgery. The fitted implicit surfaces can be either algebraic or non‐algebraic. The main problem with most algebraic surface fitting algorithms is that the surface fitted to a given data set is often unbounded, multiple sheeted, and disconnected when a high degree polynomial is used, whereas a low degree polynomial is too simple to represent general shapes. Recently, there has been increasing interest in non‐algebraic implicit surface fitting. In these techniques, one popular way of representing an implicit surface has been the use of radial basis functions. This type of implicit surface can represent various shapes to a high level of accuracy. In this paper, we present an implicit surface fitting algorithm using radial basis functions with an ellipsoid constraint. This method does not need to build interior and exterior layers for the given data set or to use information on surface normal but still can fit the data accurately. Furthermore, the fitted shape can still capture the main features of the object when the data sets are extremely sparse. The algorithm involves solving a simple general eigen‐system and a computation of the inverse or psedo‐inverse of a matrix, which is straightforward to implement.

NP-hard cases of the single-item capacitated lot-sizing problem have been the topic of extensive research and continue to receive considerable attention. However, surprisingly few theoretical results have been published on approximation methods for these problems. To the best of our knowledge, until now no polynomial approximation method is known that produces solutions with a relative deviation from optimality that is bounded by a constant. In this paper we show that such methods do exist by presenting an even stronger result: the existence of fully polynomial approximation schemes. The approximation scheme is first developed for a quite general model, which has concave backlogging and production cost functions and arbitrary (monotone) holding cost functions. Subsequently we discuss important special cases of the model and extensions of the approximation scheme to even more general models.

Graph Neural Networks (GNNs) exploit signals from node features and the input graph topology to improve node classification task performance. Recently proposed GNNs work across a variety of homophilic and heterophilic graphs. Among these, models relying on polynomial graph filters have shown promise. We observe that polynomial filter models need to learn a reasonably high degree polynomials without facing any over-smoothing effects. We find that existing methods, due to their designs, either have limited efficacy or can be enhanced further. We present a spectral method to learn a bank of filters using a piece-wise polynomial approach, where each filter acts on a different subsets of the eigen spectrum. The approach requires eigendecomposition only for a few eigenvalues at extremes (i.e., low and high ends of the spectrum) and offers flexibility to learn sharper and complex shaped frequency responses with low-degree polynomials. We theoretically and empirically show that our proposed model learns a better filter, thereby improving classification accuracy. Our model achieves performance gains of up to $$\sim $$ 6% over the state-of-the-art (SOTA) models while being only $$\sim $$ 2x slower than the recent spectral approaches on graphs of sizes up to $$\sim $$ 169K nodes.

Critical stability constraints are a small set of conditions that are enough to maintain the stability of a system when some parameters are perturbed from a nominal stable setting. The paper uses a recently introduced efficient integer-preserving (IP) form of the Bistritz test to derive critical constraints for stability of discrete-time linear systems. The new procedure produces polynomial (rather than rational) constraints of particularly low degree whose variates are the free parameters (or the literal coefficients) of the system's characteristic polynomial. Comparison with the modified Jury test, also an efficient IP stability test, shows that the constraints are obtained with less computation and, more contributing to simplicity, the constraints appear as polynomials with degrees lower by a factor of two.