Kautz Functions Research Articles

An optimizationless stochastic volterra series approach for nonlinear model identification

Volterra series is a widely used tool for identifying physical systems with polynomial nonlinearities. In this approach, the Volterra kernels expanded using Kautz functions can be identified using several techniques to optimize the filters’ poles. This methodology is very efficient when the system observations are not subject to high noise-induced variabilities (uncertainties). However, this optimization procedure may not be effective when the uncertainty level is increased since the optimal value might be susceptible to small perturbations. Seeking to overcome this weakness, the present work proposes a new stochastic method of identification based on the Volterra series, which does not solve an optimization problem. In this new approach, the Volterra kernels are described as stochastic processes. The parameters of Kautz filters are considered independent random variables so that their probability distribution captures the variabilities. The effectiveness of the new technique is tested experimentally in a nonlinear mechanical system. The results show that the identified stochastic Volterra kernels can reproduce the nonlinear dynamics characteristics and the data variability.

Kautz basis expansion-based Hammerstein system identification through separable least squares method

Abstract This paper proposes a novel Hammerstein system identification method based on the Kautz basis expansion and the separable least squares method. In this method, to reduce the parameters to be identified, the impulse response function (IRF) of the linear subsystem is expanded by orthogonal Kautz functions, the pole parameters among which should be optimized. In addition, to improve the condition number of matrix during the identification process, the separable least squares optimization method is adopted to estimate the linear and nonlinear parameters. The separable least squares approach can simultaneously estimate the linear and nonlinear parameters in a least squares framework. Furthermore, based on the best linear approximation, an effective method for the choice of initial values of pole parameters is presented, and based on the back propagation through-time technique and the Levenberg-Marquardt algorithm, an optimization algorithm for pole and nonlinear parameters is presented in this paper. The simulation studies verify the effectiveness of the proposed Hammerstein system identification method.

Impulse response function identification of linear mechanical systems based on Kautz basis expansion with multiple poles

ABSTRACTThe impulse response function (IRF) identification of linear mechanical systems is important in many engineering applications. This paper proposes a novel IRF identification method of linear systems based on Kautz basis expansion with multiple poles. In order to reduce the parameters to be identified, the IRF is expanded in terms of orthogonal Kautz functions with multiple poles, and the poles in Kautz functions should be optimised. This allows the identification of IRF for linear mechanical systems operated under more than one mode, such as systems under the white noise excitation or the swept frequency excitation with a wide range of frequency, and can improve the identification accuracy. Furthermore, based on the backpropagation through-time technique and the expectation maximisation algorithm, a pole optimisation algorithm is presented in this paper. The simulation studies verify the effectiveness of the proposed IRF identification method.

Damage detection in an uncertain nonlinear beam

The present paper proposes the use of discrete time Volterra series expanded with Kautz filters for predicting the experimental response of a nonlinear beam, taking into account uncertainties that are inherent to the system. Nonlinear behavior is simulated considering large amplitude of vibration in a clamped-free beam, with a magnet near the free end to cause cubic sti_ness e_ect. The Volterra kernels and Kautz functions are estimated in a stochastic way, where the limits of confidence are established to the system response and used as a reference model to detect damage. The detection approach is based on a nonlinear stochastic index and hypothesis test, considering di_erent levels of simulated damage, associated with loss of mass. The results have shown that considering the uncertainties when identifying the model is very important in the process of damage detection and the nonlinear metric proved to be more appropriated to describe the system response with better results mainly in the nonlinear regime of motion.

PARAMETRIC COMPLEXITY REDUCTION OF DISCRETE-TIME LINEAR SYSTEMS HAVING A SLOW INITIAL ONSET OR DELAY

This paper is concerned with an optimal expansion of linear discrete time systems on Meixner functions. Many orthogonal functions have been widely used to reduce the model parameter number such as Laguerre functions, Kautz functions and orthogonal basis functions. However, when the system has a slow initial onset or delay, Meixner functions, which have a slow start, are more suitable in terms of providing a more accurate approximation to the system. The optimal approximation of Meixner model is ensured once the pole characterizing the Meixner functions is set to its optimal value. In this paper, a new recursive representation of Meixner model is proposed. Further we propose, from input/output measurements, an iterative pole optimization algorithm of the Meixner pole functions. The method consists in applying the Newton-Raphson’s technique in which their elements are expressed analytically by using the derivative of the Meixner functions. Simulation results show the effectiveness of the proposed optimal modeling method.

System approximations based on Meixner‐like models

In this study, the authors investigate the parametric complexity reduction of the Meixner-like model for linear discrete-time system representation. The use of the Meixner-like functions is more suitable than the use of Laguerre functions and Kautz functions especially when the system have a slow initial onset or delay. The coefficients of the Meixner-like model can be estimated recursively from input–output data by the new representation. Noting that the selection of an arbitrary pole for the Meixner-like functions can raise the parameter number of the Meixner-like model. However, when the pole is set to its optimal value, an optimal expansion of transfer functions is produced. Therefore an optimisation technique is developed to generate the optimal Meixner-like pole, which is achieved by an iterative method, that consists in minimising the mean square error between the system output and the model output. Theoretical analysis and a numerical simulation show the efficiency of the approach.

Kautz filters based model predictive control for resonating systems

Model predictive control (MPC) of a special class of systems called resonating system is presented in this paper using the recursive Kautz model. Kautz functions are known to have the ability to describe resonating systems. They are formulated in a recursive fashion in order to compute the future incremental control effort of MPC. Use of Kautz functions in the optimization procedure leads to parsimonious representation. The efficacy of the proposed Kautz MPC is tested on two case studies, viz. linear highly oscillatory mechanical system and mildly nonlinear magnetic ball suspension system. Performance of Kautz MPC is compared against already published Laguerre MPC. Stability of the Kautz MPC is established through Lyapunov criteria.

Exploiting Kautz functions to improve feasibility in MPC

AbstractThis paper develops the recently published Laguerre MPC by proposing an alternative parametrization of the degrees of freedom in order to further increase the feasible region of model predictive control (MPC). Specifically, a simple but efficient algorithm that uses Kautz functions to parameterize the degrees of freedom in Optimal MPC is presented. It is shown that this modification gives mechanisms to achieve low computation burden with good feasibility and good performance. The improvements, with respect to an existing algorithm that uses a similar strategy, are demonstrated by examples.

An optimal expansion of Volterra models using independent Kautz bases for each kernel dimension

A new solution for the problem of selecting poles of the two-parameter Kautz functions in Volterra models is proposed. In general, a large number of parameters are required to represent the Volterra kernels, although this difficulty can be overcome by describing each kernel using a basis of orthonormal functions, such as the Kautz basis. This representation has a Wiener structure consisting of a linear dynamic generated by the orthonormal basis followed by a non-linear static mapping represented by the Volterra series. The resulting Wiener/Volterra model can be truncated into fewer terms if the Kautz functions are properly designed. The underlying problem is how to select the arbitrary complex poles that fully parametrize these functions. This problem has been approached in previous research by minimizing an upper bound for the error resulting from the truncation of the kernel expansion. The present paper goes even further in that each multidimensional kernel is decomposed into a set of independent Kautz bases, each of which is parametrized by an individual pair of conjugate Kautz poles intended to represent the dominant dynamic of the kernel along a particular dimension. An analytical solution for one of the Kautz parameters, valid for Volterra models of any order, is derived. A simulated example is presented to illustrate these theoretical results. The same approach is then used to model a real non-linear magnetic levitation system with oscillatory behaviour.

Identificação e controle de processos via desenvolvimentos em séries ortonormais. Parte A: identificação

O presente artigo apresenta uma visão geral do estado da arte na área de identificação de sistemas utilizando modelos dinâmicos com estrutura desenvolvida através de bases de funções ortonormais, como as funções de Laguerre, Kautz ou funções ortonormais generalizadas. Discute-se as vantagens e possíveis limitações desse tipo de estrutura bem como os fundamentos matemáticos dos modelos correspondentes nos contextos de identificação linear, linear com incertezas paramétricas (identificação robusta) e não linear, incluindo uma revisão bibliográfica abrangente sobre o tema. Diferentes realizações de modelos com funções de base ortonormal, a saber, modelos lineares, de Volterra, fuzzy e neurais, são detalhadas e discutidas comparativamente em termos de capacidade de representação, parcimônia, complexidade de projeto e interpretabilidade. Aspectos práticos da identificação desses modelos são também apresentados e ilustrados através de dois casos de estudo envolvendo um processo simulado de polimerização isotérmica.

Choice of free parameters in expansions of discrete-time Volterra models using Kautz functions

This work tackles the problem of modeling nonlinear systems using Volterra models based on Kautz functions. The drawback of requiring a large number of parameters in the representation of these models can be circumvented by describing every kernel using an orthonormal basis of functions, such as the Kautz basis. The resulting model, so-called Wiener/Volterra model, can be truncated into a few terms if the Kautz functions are properly designed. The underlying problem is how to select the free-design complex poles that fully parameterize these functions. A solution to this problem has been provided in the literature for linear systems, which can be represented by first-order Volterra models. A generalization of such strategy focusing on Volterra models of any order is presented in this paper. This problem is solved by minimizing an upper bound for the error resulting from the truncation of the kernel expansion into a finite number of functions. The aim is to minimize the number of two-parameter Kautz functions associated with a given series truncation error, thus reducing the complexity of the resulting finite-dimensional representation. The main result is the derivation of an analytical solution for a sub-optimal expansion of the Volterra kernels using a set of Kautz functions.

A parameter-reduced volterra model for dynamic RF power amplifier modeling based on orthonormal basis functions

A nonlinear dynamic behavioral model for radio frequency power amplifiers is presented. It uses orthonormal basis functions, Kautz functions, with complex poles that are different for each nonlinea ...

SYSTEM IDENTIFICATION OF ACOUSTIC CHARACTERISTICS OF ENCLOSURES WITH RESONANT SECOND ORDER DYNAMICS

This research concerns offline identification of acoustic characteristics of enclosures with second-order resonant dynamics and their modeling as linear dynamic systems. The applied models can be described by basis function expansions. The practical problem of acoustic echo in enclosures is used as the target problem to be addressed. It has been found out that the classical filters are ineffective filter structures for approximating an echo generating system, due to their many required parameters. In order to reduce the number of estimated parameters, alternative methods for modeling the room impulse response need to be investigated. Out of various available techniques impulse response identification is utilized. With the help of given experimental data, the enclosures' impulse response is modeled using special orthonormal basis functions called Kautz functions. As another improved approximation, hybrid multistage system identifiers have been used in which the simplicity of classical filter structures and fast convergence of orthonormal structures is utilized as an advantage.

Application of orthonormal basis functions for identification of flexible-link manipulators

The advantage of generalized orthonormal bases, as compared to other orthonormal basis functions such as the Laguerre and Kautz functions, is the ability to include a priori knowledge of the real and complex pole locations of the model. In this paper, the implementation of generalized orthonormal basis functions (GOBF) for system identification is discussed and applied to identification of the dynamics of a single flexible-link manipulator. The plant is non-minimum phase with real and complex poles. A global optimization strategy for selecting the location of the poles of the basis functions is proposed. This method is evaluated through simulation and experimental studies and shows superior performance as compared to ARMAX and FIR identification.

CHOICE OF FREE PARAMETERS IN EXPANSIONS OF DISCRETE-TIME VOLTERRA MODELS USING KAUTZ FUNCTIONS

The present paper involves the approximation of nonlinear systems using Wiener/Volterra models with Kautz orthonormal functions. It focuses on the problem of selecting the free complex pole which characterizes these functions. The problem is solved by minimizing an upper bound of the error arising from the truncated approximation of Volterra kernels using Kautz functions. An analytical solution for the optimal choice of one of the parameters related to the Kautz pole is thus obtained, with the results valid for any-order Wiener/Volterra models. An example illustrates the application of the mathematical results derived.

Recursive identification of acoustic echo systems using orthonormal basis functions

In hands-free telephone or video conference application, there exists an acoustic feedback coupling between the loudspeaker and microphone in an enclosed environment, which creates the acoustic echo. FIR filters are commonly used in acoustic echo cancellers because of their simple structure. However, in this paper, the Kautz and Laguerre filter structures are shown to be more efficient echo cancellers than the FIR filters, because they can describe accurately the acoustic echo system with fewer parameters. These filters are built from their respective orthonormal Kautz and Laguerre basis functions. The proposal is motivated by some theoretical and numerical results that the time-varying acoustic echo path is basically due to its time-varying zeros and not its time-invariant acoustical poles. Therefore, the poles of the Kautz and the Laguerre filters are estimated, and can be kept fixed or updated occasionally if required. The poles are estimated by a batch Gauss-Newton algorithm. Then, the coefficients of the Kautz and Laguerre filters can be estimated by most recursive algorithms that are suitable for linear regression models, e.g., the normalized LMS algorithm. Generally, it is shown that the proposed Kautz and Laguerre filters, as the filter structures in an acoustic echo canceller, have better convergence and tracking properties than the FIR and IIR filters.

Orthonormal Basis Functions in Time and Frequency Domain: Hambo Transform Theory

The class of finite impulse response (FIR), Laguerre, and Kautz functions can be generalized to a family of rational orthonormal basis functions for the Hardy space H2 of stable linear dynamical systems. These basis functions are useful for constructing efficient parameterizations and coding of linear systems and signals, as required in, e.g., system identification, system approximation, and adaptive filtering. In this paper, the basis functions are derived from a transfer function perspective as well as in a state space setting. It is shown how this approach leads to alternative series expansions of systems and signals in time and frequency domain. The generalized basis functions induce signal and system transforms (Hambo transforms), which have proved to be useful analysis tools in various modelling problems. These transforms are analyzed in detail in this paper, and a large number of their properties are derived. Principally, it is shown how minimal state space realizations of the system transform can be obtained from minimal state space realizations of the original system and vice versa.

Orthogonal Rational Functions: A Transformation Analysis

Finite impulse response (FIR) models are among the most basic tools in control theory and signal processing and are routinely used in almost all fields of application. The connections to orthogonal polynomials are well known. However, infinite impulse response (IIR) models often provide much more compact descriptions and in many cases give improved performance. The objective of this paper is to present a simple framework for the derivation and analysis of orthogonal IIR transfer functions, which are directly related to orthogonal rational functions. Orthogonality simplifies approximation analysis and leads to improved numerical properties. The basic idea is to use a fractional transformation to map the problem to a new domain, where an FIR description is most appropriate. This FIR representation is then mapped back to the original domain to give an orthogonal IIR representation. It is then straightforward to extend many results for FIR models to IIR model structures with arbitrary stable poles; i.e., properties of orthogonal polynomials are easily generalized to orthogonal rational functions. Much of the theory to be presented is classical, e.g., Laguerre and Kautz functions, and we will make use of well-known results in orthogonal filter theory. However, our main contribution is to present a uniform and transparent theory which also covers more novel results that have mainly been presented in the signals, systems, and control literature in the last decade.

Pertinent choice of parameters for discrete Kautz approximation

Kautz functions have received much attention in the recent mathematical modeling and identification literature. These functions which involve free parameters can approximate efficiently signals with strong oscillatory behavior. We consider here the choice of the free parameters in discrete (two-parameter) Kautz approximation. Using a key relationship between Kautz and Laguerre expansions we derive an upper bound for the quadratic truncation error. Minimization of this upper bound yields pertinent parameters, whose computation then requires reduced knowledge of the function to be modeled.

On approximation of stable linear dynamical systems using Laguerre and Kautz functions

Approximation of stable linear dynamical systems by means of so-called Laguerre and Kautz functions, which are the Laplace transforms of a class of orthonormal exponentials, is studied. Since the impulse response of a stable finite dimensional linear dynamical system can be represented by a sum of exponentials (times polynomials), it seems reasonable to use basis functions of the same type. Assuming that the transfer function of a system is bounded and analytic outside a given disc, it is shown that Laguerre basis functions are optimal in a mini-max sense. This result is extended to the “two-parameter” Kautz functions which can have complex poles, while the poles of Laguerre functions are restricted to the real axis. By conformai mapping techniques the “two-parameter” Kautz approximation problem is recast as two Laguerre approximation problems. Thus, the well-developed theory of Laguerre functions can be applied to analyze Kautz approximations. Unilateral shifts are used to further develop the connection between Laguerre functions and Kautz functions. Results on þ 2 and þ ∞ approximation using Kautz models are given. Furthermore, the weighted L 2 Kautz approximation problem is shown to be equivalent to solving a block Toeplitz matrix equation.