Rafal Stanislawski

A Comparative Analysis of Laguerre-Based Approximators to the Grünwald-Letnikov Fractional-Order Difference

This paper provides a series of new results in both steady-state accuracy and frequency-domain analyses for two Laguerre-based approximators to the Grunwald-Letnikov difference. In a comparative study, the Laguerre-based approximators are found superior to the classical Tustin- and Al-Alaoui-based approximators, which is illustrated in simulation examples.

Normalized finite fractional differences

This paper presents a series of new results in finite and infinite-memory modeling of discrete-time fractional differences. The introduced normalized finite fractional difference is shown to properly approximate its fractional difference original, in particular in terms of the steady-state properties. A stability analysis is also presented and a recursive computation algorithm is offered for finite fractional differences. A thorough analysis of computational and accuracy aspects is culminated with the introduction of a perfect finite fractional difference and, in particular, a powerful adaptive finite fractional difference, whose excellent performance is illustrated in simulation examples.

Modeling of open-loop stable linear systems using a combination of a finite fractional derivative and orthonormal basis functions

This paper presents new results in modeling of linear open-loop stable systems by means of discrete-time finite fractional orthonormal basis functions, in particular the Laguerre functions. New stability conditions are offered and a useful modification to finite fractional derivative is introduced, called normalized finite fractional derivative. Simulation examples illustrate the usefulness of the new modeling methodology.

Identification of open-loop stable linear systems using fractional orthonormal basis functions

Abstract This paper presents new results in identification of linear open-loop stable systems modeled by means of discrete-time fractional orthonormal basis functions (OBF), in particular the Laguerre functions. Fractional OBF are used to model 1) various simulated plants and 2) a fuel preparation system for the electric power boiler. Advantages and drawbacks of the fractional modeling approach are analysed.

Modeling and identification of a fractional-order discrete-time SISO Laguerre-Wiener system

This paper presents a new implementable strategy for modeling and identification of a fractional-order discrete-time nonlinear block-oriented SISO Wiener system. The concept of modeling of a linear dynamics by means of orthonormal basis functions (OBF) is employed to separate linear and nonlinear submodels, which enables a linear regression formulation of the parameter estimation problem. Finally, discrete-time Laguerre filters are uniquely embedded in modeling of the fractional-order dynamics, eliminating the disastrous bilinearity issue. Simulation experiments show a very good identification performance for a fractional-order Laguerre-based Wiener model, both in terms of low prediction errors and accurate reconstruction of the actual system characteristics.

A new form of a σ-inverse for nonsquare polynomial matrices

This paper presents a new simple form of a polynomial matrix σ-inverse introduced as a result of research works on minimum variance control (MVC) for LTI MIMO nonsquare systems. A new approach to construction of a σ-inverse of a nonsquare polynomial matrix can result in e.g. pole-free design of MVC, which is provided by specially selected degrees of freedom of the σ-inverse. A simulation example in the Matlab® environment illustrates theoretical achievements of the paper.

Finite approximations of a discrete-time fractional derivative

This paper presents new results in finite-memory modeling of a discrete-time fractional derivative. The introduced normalized finite fractional derivative is shown to properly approximate its fractional derivative original, in particular in terms of the steady-state properties. A stability analysis is also presented as well as a recursive computation algorithm is offered for finite fractional derivatives.

Modeling of discrete-time fractional-order state space systems using the balanced truncation method

This paper presents a new approach to approximation of linear time-invariant (LTI) discrete-time fractional-order state space SISO systems by means of the SVD-originated balanced truncation (BT) method applied to an FIR-based representation of the fractionalorder system. This specific representation of the system enables to introduce simple, analytical formulas for determination of the Cholesky factorizations of the controllability and observability Gramians, which contributes to significant improvement of the computational efficiency of the BT method. As a model reduction result for the fractional-order systems we obtain a low-order rational (integer-order) state space system. Simulation experiments show a high efficiency of the introduced methodology both in terms of the approximation accuracy of the model and low time complexity of the approximation algorithm.

Modeling and Identification of Fractional-Order Discrete-Time Laguerre-Based Feedback-Nonlinear Systems

This paper presents a new implementable strategy for modeling and identification of a fractional-order discrete-time block-oriented feedback-nonlinear system. Two different concepts of orthonormal basis functions (OBF) are used to model a linear dynamic part, namely ”regular” OBF and inverse IOBF. It is shown that the IOBF concept enables to separate linear and nonlinear submodels, which leads to a linear regression formulation of the parameter estimation problem, with the detrimental bilinearity effect totally eliminated. Finally, Laguerre filters are uniquely embedded in modeling of the fractional-order dynamics. Unlike for regular OBF, simulation experiments show a very good identification performance for an IOBF-structured, fractional-order Laguerre-based feedback-nonlinear model, both in terms of low prediction errors and accurate reconstruction of the actual system characteristics.

Fractional-Order Discrete-Time Laguerre Filters: A New Tool for Modeling and Stability Analysis of Fractional-Order LTI SISO Systems

This paper presents new results on modeling and analysis of dynamics of fractional-order discrete-time linear time-invariant single-input single-output (LTI SISO) systems by means of new, two-layer, “fractional-order discrete-time Laguerre filters.” It is interesting that the fractionality of the filters at the upper system dynamics layer is directly projected from the lower Laguerre-based approximation layer for the Grunwald-Letnikov difference. A new stability criterion for discrete-time fractional-order Laguerre-based LTI SISO systems is introduced and supplemented with a stability preservation analysis. Both the stability criterion and the stability preservation analysis bring up rather surprising results, which is illustrated with simulation examples.