Periodic Time-varying Coefficients Research Articles

Periodic dynamic on a class of neutral-type Rayleigh equations incorporating D operators

ABSTRACT In this study, a class of neutral-type Rayleigh equations incorporating operators and -periodic time-varying coefficients are introduced. By utilising differential inequality techniques, some new sufficient criteria are derived to guarantee the existence and global exponential stability of -periodic solutions for the addressed equations. Through numerical experiments, the validity of the main findings is verified.

A Bloch-based procedure for dispersion analysis of lattices with periodic time-varying properties

We present a procedure for the systematic estimation of the dispersion properties of linear discrete systems with periodic time-varying coefficients. The approach relies on the analysis of a single unit cell, making use of Bloch theorem along with the application of a harmonic balance methodology over an imposed solution ansatz. The solution of the resulting eigenvalue problem is followed by a procedure that selects the eigen-solutions corresponding to the ansatz, which is a plane wave defined by a frequency-wavenumber pair. Examples on spring-mass superlattices demonstrate the effectiveness of the method at predicting the dispersion behavior of linear elastic media. The matrix formulation of the problem suggests the broad applicability of the proposed technique. Furthermore, it is shown how dispersion can inform about the dynamic behavior of time-modulated finite lattices. The technique can be extended to multiple areas of physics, such as acoustic, elastic and electromagnetic systems, where periodic time-varying material properties may be used to obtain non-reciprocal wave propagation.

Symbolic Computation of Quantities Associated With Time-Periodic Dynamical Systems1

Many dynamical systems can be modeled by a set of linear/nonlinear ordinary differential equations with periodic time-varying coefficients. The state transition matrix (STM) Φ(t,α), associated with the linear part of the equation, can be expressed in terms of the periodic Lyapunov–Floquét (L-F) transformation matrix Q(t,α) and a time-invariant matrix R(α) containing a set of symbolic system parameters α. Computation of Q(t,α) and R(α) in symbolic form as a function of α is of paramount importance in stability, bifurcation analysis, and control system design. In earlier studies, since Q(t,α) and R(α) were available only in numerical forms, general results for parameter unfolding and control system design could not be obtained in the entire parameter space. In 2009, an attempt was made by Butcher et al. (2009, “Magnus' Expansion for Time-Periodic Systems: Parameter Dependent Approximations,” Commun. Nonlinear Sci. Numer. Simul., 14(12), pp. 4226–4245) to compute the Q(t,α) matrix in a symbolic form using the Magnus expansions with some success. In this work, an efficient technique for symbolic computation of Q(t,α) and R(α) matrices is presented. First, Φ(t,α) is computed symbolically using the shifted Chebyshev polynomials and Picard iteration method as suggested in the literature. Then, R(α) is computed using a Gaussian quadrature integral formula. Finally, Q(t,α) is computed using the matrix exponential summation method. Using mathematica, this approach has successfully been applied to the well-known Mathieu equation and a four-dimensional time-periodic system in order to demonstrate the applications of the proposed method to linear as well as nonlinear problems.

On general periodic time-varying bilinear processes

We study in this note the class of bilinear processes with periodic time-varying coefficients. We give necessary and sufficient conditions ensuring the existence of strict and second order stationary solutions (in periodic sense) and for the existence of higher order moments. The given conditions can be applied to periodic ARMA or periodic GARCH models. The central limit theorem and the law of iterated logarithm (LIL) for higher order sample moments are showed.

Chatter suppression in fast tool servo-assisted turning by spindle speed variation

This paper focuses on the investigation, based on theoretical analysis and through machining experiments, of using variable spindle speed machining to suppress chatter in a fast tool servo-assisted noncircular turning process. Noncircular turning is accomplished by controlling the position of the radial cutting tool in the direction normal to the surface of a workpiece with noncircular cross-sections. The process stability is thus more involved than in a general cutting process. This is mainly due to strong dynamic feedback between the machining process and the fast tool servo drive. An enhanced closed loop dynamic model of the noncircular turning process is developed, in which the fast tool servo employs an active disturbance rejection control scheme. By using the spindle’s angular position as the independent variable, the dynamics of the variable spindle speed noncircular turning process are described by a differential equation with linear periodic time-varying coefficients and a fixed delay in the angle domain. The Floquet theory is applied to determine the stability limit. Analytically predicted stability boundaries are compared with those of constant spindle speed machining generated by the Nyquist method. By investigating the effects of spindle speed variation amplitude and frequency on the stability limits, it is shown that a modest increase in noncircular turning stability is given by continuous spindle speed variations due to the limited performance of the fast tool servo controller. Experimental results are also presented, which validate the ability and explanation of increasing noncircular turning stability using spindle speed variations.

FIR Filters with Punctured Radix-8 Symmetric Coefficients: Design and Multiplier-Free Realizations

In recent years, several approaches have been investigated to obtain multiplier-free realizations of digital filters. One approach makes use of periodically time-varying (PTV) structures. The idea is to distribute the computation in time and in space. Distribution in time provides reuse of the same hardware by means of PTV coefficients. Distribution in space increases the number of coefficients but simplifies the values of the coefficients. Computation distribution is based on radix-r number representation, and it can be carried out to the extent that each computation involves a simple coefficient that can be realized using only addition and shift (no hardware multiplier). Previous PTV realizations could not exploit the coefficient symmetry of finite impulse response (FIR) filters to reduce the number of coefficients. This paper proposes design and realizations of FIR filters with punctured radix-8 coefficients belonging to the septuple set {0, ±1, ±2, ±4), which can be implemented using only a shift operation without requiring any hardware multiplier. The realizations exploit the coefficient symmetry to reduce the hardware by about one-half. Due to a non-uniform grid of representation, we apply a modified Karmarkar's linear programming algorithm to find the optimum set of discrete coefficients that minimizes the weighted peak ripple error. Comparison with a conventional FIR filter with sum-of-powers-of-two (SOPOT) coefficients shows that the proposed filter is faster and uses less hardware than one with SOPOT coefficients. However, the punctured radix-8 system has a limit on the achievable ripple.

Multiplier-Free IIR Filter Realizations with Periodically Time-Varying Coefficients

In this paper, the periodically time-varying (PTV) structure, previously proposed for realizing FIR filters, is extended to IIR filter realization. The realization consists of ternary ({0, ±1}) or quinary ({0, ±1, ±2}) PTV coefficients with simple input and output units. Coefficient multiplications as well as the input and output units require no hardware multiplier, which helps increase the processing speed or reduce the chip area. Bit-level architectures are presented. The regularity and local interconnection of the architectures help simplify VLSI design and layout.

Multiplier-free realizations for FIR multirate converters based on mixed-radix number representation

We propose some realizations of FIR multirate converters. They are based on mixed-radix signed-digit number representation in conjunction with periodically time-varying (PTV) coefficients. These realizations have desirable properties of low complexity and regularity with simple processing elements that are suitable for easy VLSI layout. By varying some parameters, these realizations also provide a tradeoff between hardware and clock speed (or throughput). The PTV coefficients are restricted to the set {0,/spl plusmn/1} or {0,/spl plusmn/1,/spl plusmn/2} so that hardware multipliers are not needed. The coefficient precision of these proposed structures can be made as high as desired by appropriate choices of the parameters. However, the disadvantage is that a more complex timing control is required. Several examples are presented.

A class of high-precision multiplier-free FIR filter realizations with periodically time-varying coefficients

Proposes a class of high-precision, multiplier-free realizations for FIR filters. These realizations use upsampling and downsampling in conjunction with a periodically time-varying system to achieve time-invariant, multiplier-free FIR filter operation. Nonbinary encoding schemes are used for obtaining the filter coefficients, which are periodically time-varying (PTV), i.e., they vary in a periodic fashion. Each target filter coefficient is directly mapped into a set of PTV coefficients so that the realizations are easy to obtain. The values of the PTV coefficients are restricted to either the ternary set.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Signed-digit-based multiplier-free realizations for multirate converters

We propose multiplier-free realizations for FIR multirate converters. Radix-r signed-digit number representation and periodically time-varying (PTV) coefficients are used to obtain multiplier-free realizations. The PTV coefficients, together with one hold-and-scale unit at the input and one scale-add-downsample unit at the output, are utilized effectively to distribute the required computations to the extent that coefficients are highly simplified. The coefficients in the realizations are designed to take values from either the ternary set (0, /spl plusmn/1) or the quinary set (0, /spl plusmn/1, /spl plusmn/2). Their values can be easily obtained by mapping from the target filter coefficients using a radix-r signed-digit number representation. We show that fairly high precision can be achieved by these realizations. >

Approximations of IIR periodically time-varying digital filters

The paper discusses three approximations of IIR periodically time-varying (PTV) digital filters. Two conventional approximations, i.e. time-invariant difference equation and PTV difference equation approximations are first reviewed. It turns out that computation and hardware complexity of the two approximations is in proportion to the period. To reduce the computation and hardware complexity of the two approximations, an efficient PTV difference equation approximation based on extended Pade approximation is proposed. With this approximation, frequency-domain specifications are approximated by a difference equation with PTV coefficients. How to design the difference equation and how to determine the order of the difference equation to obtain the desired approximation precision are considered. A design for the difference equation is presented. As an application of the proposed approximation, a four-channel TDM-FDM transmultiplexer is simulated. It shows that the proposed PTV difference-equation approximation obtains satisfactory precision and compares favourably with above two approximations in respect of computation and hardware complexity.