Jerzy Baranowski

Laguerre Polynomial Approximation of Fractional Order Linear Systems

This paper presents a finite dimensional approximation of fractional order linear systems and its connection with transport equation. The main results show, that the linear fractional order system can be approximated by a finite number of linear differential equations. Appropriate error estimate in C 0 norm is presented. Next, the solution of fractional order linear system is presented as a linear functional of the solution of transport equation. This result establishes a connection between the semi groups theory and fractional system theory. Considerations are illustrated with simple example of fractional oscillator.

Impulse response approximation method for bi-fractional filter

In this paper method of impulse response approximation with orthogonal Laguerre functions is used for the approximation of bi-fractional filter (BFF). This filter is a new development in the field of non-integer order signal processing. Filter with three parameters allows advanced shaping of frequency response, especially the phase characteristic. Method of impulse response approximation is a very promising one because of its great conditioning and sound theoretical basis.

Robust non-integer order controller for air heater

Robust non-integer control is an emerging field in control theory. Multiple works are focused on designing controllers with flatphase and robust stability margins. In this paper design of non-integer order with robust properties is considered. This controller is designed for the model of air heating process trainer system belonging to the Department of Automatics and Biomedical Engineering of AGH University and Science and Technology and verified experimentally with that system.

Impulse Response Approximation Method for “Fractional Order Lag”

”Fractional order lag” is a system that is popular in multiple applications. In this paper, authors consider a new method for approximation of this system based on its impulse response. Certain assumptions of the approximation method are verified and algorithm is presented. Also certain problems with this system analysis are discussed, especially its realisation in the form of non-integer order differential equations.

Non-integer Order PI α D μ Control ICU-MM

The article presents dynamical system model that describes glycemia. It is based on four differential equations that simulates glucose dynamics of traumatised patient’s blood (at ICU). Authors present description of basic model and method of tuning PI α D μ controller parameters based on the integrated absolute error as the performance index.

Observer-based feedback for the magnetic levitation system

Control of active magnetic bearings is an important area of research. The laboratory magnetic levitation system can be interpreted as a model of a single axis of bearings and is a useful testbed for control algorithms. The mathematical model of this system is highly non-linear and requires careful analysis and identification. The system is observable from position measurements as long as the electromagnet is powered as shown during the research. Practically measurable signals are the position and the coil current. The velocity that is necessary for any stabilizing control usually is obtained by numerical differentiation of the position. A more sophisticated approach is to estimate the velocity with an observer. Efficient observer types for this system are high-gain and non-linear reduced observers. The velocity estimated by an observer can be effectively used instead of a derivative in PID control of the position. Such an approach substantially improves control quality and extends the range of system’s stable operation. Even greater improvement is introduced by the addition of the non-linear feedforward to the control structure. The best results, provided the model parameters are correctly identified, are obtained with a control system consisting of the PID controller, the high-gain observer and the non-linear feedforward.

Time-domain Oustaloup approximation

Paper presents a new method for implementing Oustaloup approximation of non-integer order systems in digital environment. Classical method and its limitations are presented and discussed. Method of realization is formulated and schemes for discretization are presented. Operation of method is illustrated with two examples. Work is ended with discussion of method limitations and conclusions.

Comparison of Performance Indices for Tuning of PIλDμ Controller for Magnetic Levitation System

Control of active magnetic bearings is an important area of research. The laboratory magnetic levitation system can be interpreted as a model of a single axis of bearings and is a useful testbed for control algorithms. The mathematical model of this system is highly nonlinear and requires careful analysis and identification. In this paper authors compare performance indices for tuning of PI λ D μ controller for this system. It is a part of an ongoing research on non integer controller tuning rules.

Bi-fractional filters, part 2: Right half-plane case

In this paper new non-integer order filter is proposed. Considered filter is given by a transfer function \(\cfrac{c}{s^{2\alpha}+ b s^{\alpha}+c}\), with parameters b and c chosen in a way, that locates the eigenvalues of the system in left open complex half plane. Dependence of frequency characteristic of the system on parameters α, b and c is investigated. Also method for realisation in the form of non-integer order differential equations is analysed.

Implementation of an inexpensive EEG headset for the pattern recognition purpose

There are many types of bio-signals with various control application prospects. In this work possible control application domain of electroencephalographic signal obtained from an easily available, inexpensive EEG headset - Emotiv EPOC was presented. This work also involved application of an embedded system platform. That solution caused limits in choosing an appropriate signal processing method, as embedded platforms characterise with a little efficiency and low computing power. Potential implementation of the embedded platform enables to extend the possible future application of the proposed BCI. It also gives more flexibility, as the platform is able to simulate various environments. In this work traditional, statistical methods were neither used nor described.