Laguerre Filters Research Articles

Adaptive GPC based on laguerre-filters modelling

A state-space model based on Laguerre filters is proposed to design an adaptive generalized predictive controller (GPC) for stable plants. A sufficient stability condition is derived to study the convergence and stability of the proposed controller in case of a plant-model match. Robustness conditions are also given for the case of plant-model mismatch. The results are applicable to GPC schemes based on moving average models. In the presence of plant-model mismatch, illustrative examples show the robustness of the controller and compare it with other GPC versions.

Non‐linear adaptive control via laguerre expansion of volterra kernels

AbstractA truncated Wiener‐Volterra functional series is used to represent the input‐output relation of a non‐linear dynamic system. By expanding Volterra kernels using orthonormal Laguerre functions, a Wiener‐type non‐linear model consisting of a Laguerre network followed by a memoryless Taylor expansion function is obtained. When functional series and Laguerre series truncations are allowed, an appropriate choice of the Laguerre filter pole permits a description of the process dynamics with a small number of parameters. Because the resulting model is linear in the parameters, those parameters can be estimated using a simple recursive least‐squares (RLS) algorithm. Furthermore, a predictive control law is derived on the basis of the model and an industrial plant adaptive control simulation is performed. This method can be used for the modelling and control of a large class of non‐linear dynamic systems.

On Determination of Laguerre Filter Pole through Step or Impulse Response Data

Any stable transfer function can be expanded by using Laguerre series. However, a finite Laguerre series is only an approximation of the true plant and the model error is a function of the filter pole. This paper summarizes some methods for the determination of Laguerre filter pole from step response or impulse response signals. The importance of a proper Laguerre filter pole for good model approximation and robust control is also discussed.

Parametric Signal Modelling using Laguerre Filters

Autoregressive (AR) modelling is generalized by replacing the delay operator by discrete Laguerre filters. The motivation is to reduce the number of parameters needed to obtain useful approximate models of stochastic processes, without increasing the computational complexity. Asymptotic statistical properties are investigated. Several AR model estimation results are extended to Laguerre models. In particular, it is shown how the choice of Laguerre time constant affects the resulting estimates. A Levinson-type algorithm for computing the Laguerre model estimates in an efficient way is also given. The Laguerre technique is illustrated by two simple examples.

Adaptive modified Laguerre filters

When designing an adaptive filter, decisions have to be made about the signal description, choice of the filter, its parameterization and the optimization criterion. These choices are interdependent in order to attain a computationally simple adaptive mechanism. We investigate appropriate signal descriptions and filters starting from an exponentially weighted squared-error criterion. Simple variations of (generalized) Laguerre filters are used. The weights in these filters can be tracked by an RLS-like algorithm. Convergence properties are discussed.

Approximation of stable systems by laguerre filters

Laguerre methods are used to develop concrete techniques and results for approximating stable infinite dimensional systems. Both Hankel norm and L ∞ approximation by Laguerre filters are studied. The infimum of the Hankel norm approximation error when the system is approximated by Laguerre filters of given maximum order is given by the norm of a certain shifted Hankel operator, and the best Laguerre filter to approximate the system is obtained essentially from the associated eigenproblem for the shifted Hankel operator. Hankel-optimal Laguerre filters are here called CF-Laguerre filters as there is a certain connection to the Caratheodory-Fejer interpolation problem, and typically provide near best solutions to related L ∞ approximation problems. A time domain compressed Laguerre shift method for solving certain Nehari problems for delay systems is used to determine analytically the achievable Hankel error norms and thus also to obtain bounds for the achievable L ∞ error norm for delay systems. A result on a Laguerre filter approximation problem in the Hilbert-Schmidt norm is given as well.

Pulse-forming networks using generalized Laguerre polynomials

A new class of all-pole transfer functions for pulse-forming networks based on the use of generalized Laguerre polynomials is introduced. It is shown that the two variable parameters, which are available with the generalized Laguerre polynomials, can always be adjusted so as to obtain an impulse response with any prescribed attenuation of transient overshoots. These filters, which may be referred to as generalized Laguerre filters, are shown to include as special cases Bessel filters, Gaussian filters and a recently derived class of transitional filters utilizing modified Bessel polynomials. A comparison with the Schiissler filters, designed for impulse response with equal ripple transient ringing is also presented. Tables are presented giving the pole locations of these filters for n = 3–7 and the transient overshoot attenuation of 20–60 dB in steps of 10 dB. Normalized element values for LC ladder realization with equal resistance terminations are also included.

Statistical properties of a class of pseudorandom sequences

The class of sequences discussed in the paper consists of those sequences which can be produced by inversion of alternate values in binary maximal-length sequences. The statistical properties of such sequences are analysed in terms of their multidimensional higher-order autocorrelation functions, and the results are compared with the statistical properties of a truly random binary sequence. The odd-order autocorrelation functions are identically zero. Some general properties of the even-order autocorrelation functions are derived. A sampled-data Laguerre filter chain is used to assess the statistical properties of some typical sequences. The paper ends with a discussion of some techniques which can be used to form multilevel sequences.

A sampled-data nonlinear filter

The paper describes a method of constructing a general form of nonlinear filter in a digital computer. The filter is easily realised and optimised, and the technique provides for the gathering of the necessary statistical data. The memory part of the filter is based on a sampled-data Laguerre filter chain, which is particularly convenient in the expansion of nonlinear operators. The filter memory can be readily adjusted and auxiliary processing can be included for this purpose. An important relationship between product averages of linear filter outputs is described which is used to give economies in computation. The paper ends with the description of some results obtained in the solution of a nonlinear identification problem.

Seismological applications of Laguerre expansions

abstract If we establish a set of orthogonal functions with resonant and decay characteristics quite similar to those of seismic signals, then a few, or at most a few dozen, terms of the series may match significant phases of seismic events. A few hundred terms of a Fourier series might be required for an equivalent match of these same events. One such set of orthogonal functions are those based on Laguerre polynomials. This paper describes the Laguerre theory which leads to better Fourier transforms (spectra). Laguerre functions are defined over all time, and their frequency responses are known over all frequencies. Therefore, if the Laguerre series converges quickly, then only a few points in time determine the spectra over the entire frequency range. Conversely, a few points in frequency can determine a transient response over all time. Seismic applications include the transient calibration of seismometers yielding magnifications and phase responses over all frequencies, the automatic Laguerre analysis of Rayleigh waves by an analog computer, and the use of the Laguerre filter as a correlator to enhance pP phases on seismograms.

Bayes' optimum filters derived using Wiener canonical forms

Canonical forms and circuit structures are derived for Bayes' optimum decision rules utilizing Wiener-like functionals. The formalism is valid for loss functions whose derivatives with respect to the estimates exist, ( L_2 over the space of unknowns ), The structures derived consist of orthonormal filter sets (usually Laguerre filters ) followed by combinations of \nu^{th} -law-devices (Hermite polynomial generators of the Laguerre coefficients), amplifiers of specified gains, and then summing and division circuits. The amplifier gains can be partially preadjusted via a sample of the observable and unknown or by a pre-computation; the remainder of the gain adjustment is, in the general case, obtained via feedback through a function generator. If the gains are adjusted via the first method, ergodicity is required and self-adaptive features are implied. As a first step in the exposition, analogous canonical forms and structures, where the past of the observable is characterized by its time-samp]es rather than by its Iaguerre coefficients, are obtained. The derivation does not stop here because of the limitations of the description, analysis, and equipment-realization of a stochastic process in terms of a finite number of its time samples. The advantage of the structural forms derived over previous ones is that they are formally valid for all values of signal-to-noise ratio and are always physically realizable and time-invariant whereas this has not usually been true in the past. Several examples of structures are given. Existence and convergence problems are discussed in the appendices. Perhaps more important than the explicit results obtained here are the implications involved in the procedure. The fact that expansions of the probability measures for a significant class of stochastic processes has been obtained in terms of canonical expansions of the Wiener process is felt to be a major accomplishment.

Fourier Coefficients of Speech Power Spectra as Measured by Autocorrelation Analysis

Time-domain speech processors, that analyze the short-time autocorrelation function in terms of a set of orthogonal functions, are shown to yield parameters that are the coefficients of a generalized Fourier cosine series representation of the speech power spectrum. Two examples are described, one using a Laguerre filter and one using a tapped delay line as in the autocorrelation Vocoder. In the case of the autocorrelation Vocoder, the derived power spectrum representation reduces to a simple Fourier cosine series in the frequency domain.