Fractional order systems become increasingly popular due to their versatility in modelling and control applications across various disciplines. However, the bottleneck in deploying these tools in practice is related to their implementation on real-life systems. Numerical approximations are employed but their complexity no longer match the attractive simplicity of the original fractional order systems. This paper proposes a low-order, computationally stable and efficient method for direct approximation of general order (fractional order) systems in the form of discrete-time rational transfer functions, e.g. processes, controllers. A fair comparison to other direct discretization methods is presented, demonstrating its added value with respect to the state of art.

An efficient algorithm for low-order direct discrete-time implementation of fractional order transfer functions.

A modeling and control approach to magnetic levitation system based on state-dependent ARX model

Over the last few decades, the Brain-Computer Interfaces have been gradually making their way to the epicenter of scientific interest. Many scientists from all around the world have contributed to the state of the art in this scientific domain by developing numerous tools and methods for brain signal acquisition and processing. Such a spectacular progress would not be achievable without accompanying technological development to equip the researchers with the proper devices providing what is absolutely necessary for any kind of discovery as the core of every analysis: the data reflecting the brain activity. The common effort has resulted in pushing the whole domain to the point where the communication between a human being and the external world through BCI interfaces is no longer science fiction but nowadays reality. In this work we present the most relevant aspects of the BCIs and all the milestones that have been made over nearly 50-year history of this research domain. We mention people who were pioneers in this area as well as we highlight all the technological and methodological advances that have transformed something available and understandable by a very few into something that has a potential to be a breathtaking change for so many. Aiming to fully understand how the human brain works is a very ambitious goal and it will surely take time to succeed. However, even that fraction of what has already been determined is sufficient e.g., to allow impaired people to regain control on their lives and significantly improve its quality. The more is discovered in this domain, the more benefit for all of us this can potentially bring.

Summary of over Fifty Years with Brain-Computer Interfaces-A Review.

In this paper, the exponentially tracking control problem for a magnetic levitation system (MLS) in the presence of parameter uncertainties and external disturbances is investigated. The disturbance/uncertainties-rejecting problem for the MLS is addressed from the view of a continuous nonlinear robust control development. Another problem of the concern is the common unidirectional input constraint in the MLS (i.e., the control input, often referring to the square of the electric current, should be nonnegative during the operation of the system). By utilizing an input transformation and augmented dynamics, a virtual control input, which removes the unidirectional constraint, is affinely emerged in the dynamics. A second-order filter is introduced to provide feedback signals for the control development. A novel Lyapunov function guarantees the exponentially tracking stability of the close-loop dynamics. Finally, some numerical simulation and real-time experimental results for tracking of time-varying trajectories are presented to validate the performance of the proposed control design.

Continuous Robust Tracking Control for Magnetic Levitation System With Unidirectional Input Constraint

This paper covers a brief review of both the advantages and disadvantages of the implementation of various smoothing filters in the analysis of electroencephalography (EEG) data for the purpose of potential medical diagnostics. The EEG data are very prone to the occurrence of various internal and external artifacts and signal distortions. In this paper, three types of smoothing filters were compared: smooth filter, median filter and Savitzky–Golay filter. The authors of this paper compared those filters and proved their usefulness, as they made the analyzed data more legible for diagnostic purposes. The obtained results were promising, however, the studies on finding perfect filtering methods are still in progress.

https://www.mdpi.com/1424-8220/20/3/807/pdf

Comparison of Smoothing Filters in Analysis of EEG Data for the Medical Diagnostics Purposes.

Typical approach to non-integer order filtering consists of analogue design and implementation. Digital realization of non-integer order systems is susceptible to problems such as infinite memory requirement and sensitivity to numerical errors. The aim of this paper is to present two efficient methods for digital realization of non-integer order filters: discrete time-domain Oustaloup approximation and Laguerre impulse response approximation. Properties of both methods are investigated with use of non-integer low-pass filter. Filters realized with presented methods are then used for filtering of EEG signal. Paper concludes with discussion of merits and flaws of both methods.

/pdf/on-digital-realizations-of-non-integer-order-filters-2lvvp0vhft.pdf

On Digital Realizations of Non-integer Order Filters

Fractional band-pass filters are a promising area in the signal processing. They are especially attractive as a method for processing of biomedical signals, such as EEG, where large signal distortion is undesired. We present two structures of fractional band-pass filters: one as an analog of classical second-order filter, and one arising from parallel connection of two fractional low-pass filters. We discuss a method for filter implementation — Laguerre Impulse Response Approximation (LIRA) — along with sufficient conditions for when the filter can be realized with it. We then discuss methods of filter tuning, in particular we present some analytical results along with optimization algorithm for numerical tuning. Filters are implemented and tested with EEG signals. We discuss the results highlighting the possible limitations and potential for development.

Fractional Band-Pass Filters: Design, Implementation and Application to EEG Signal Processing

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.

Observer-based feedback for the magnetic levitation system

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.

Bi-Fractional Filters, Part 1: Left Half-Plane Case

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.

Paweł Piątek

Papers

On Digital Realizations of Non-integer Order Filters

Fractional Band-Pass Filters: Design, Implementation and Application to EEG Signal Processing

Observer-based feedback for the magnetic levitation system

Bi-Fractional Filters, Part 1: Left Half-Plane Case

Comparison of Performance Indices for Tuning of PI λ D μ Controller for Magnetic Levitation System