This article addresses modeling systems using fractional order derivatives, highlighting three basic approaches: differential equations, operator methods, and state space representations. Each approach carries different advantages and limitations in the context of time invariant linear systems (LTI) of fractional order. The focus of the article is on Fractional Order Transfer Functions, which represent a special subject of interest because they offer practical utility with available simulation libraries (e.g., CRONE, FOMCON, NiNteger) and approximation techniques (e.g., Oustaloup, CFE, Thiele, Padé). This paper describes a transition between fractional order transfer functions (FOTF) and pseudo-rational representations of such systems. While the existing literature contains the basics of fractional differential equations and operator theory, it often omits explicit formulas for conversion between different representations. The paper fills a significant gap by proposing a novel algorithm for converting FOTF models to fractional LTI representations. Unlike previous works that implicitly assume such transitions exist, this paper provides formulas for coefficient transformations and demonstrates its effectiveness by minimizing the degree of the system.

The purpose of the article is to take a broader look at classical rational approximation methods in applications to fractional order systems. The fundamental problem is the transformation of the fractional order transfer function (FOTF) to the classical form of the linear system represented by the transfer function. The most common approach is to approximate the complex variable raised to a fractional power with one of the methods (Oustaloup, Thiele (Matsuda) and others), substitute it, and transform the FOTF to a linear system. A slightly different approach is the use of rational approximation. These methods can be applied both to the fractional power term and also straightforwardly to the entire FOTF. The two selected methods for the study are Pade approximation and rational Chebyshev approximation. According to some textbooks on numerical methods, rational Chebyshev approximation should give better results than Pade approximation. This is due to the fact that different criteria are used to calculate the approximation. This paper aims to compare these methods in the context of FOTF approximation. The work is the authors' continuation of the research using the Pade method.

This article aims to compare the time horizon of the EU hydrogen mobility policy with the potential commercialisation of this technology based on the law of market diffusion of innovation. There is growing discussion about using hydrogen as an alternative to fossil fuels in cars. This direction is further supported by EU policy and initiatives to ban the sale of new petrol and diesel cars from 2035. Alongside the development of electromobility, there is an intensification of research into technologies that will allow the widespread use of hydrogen-fuelled cars. The technology could become a future solution, primarily due to the zero-emission nature of the vehicles. However, the commercial potential of hydrogen-fuelled cars is a pertinent question. Can the interest in this technology grow sufficiently, and if so, when will it capture the early majority of the market? When can we expect hydrogen to gain a significant market share in the mobility sector? The Bass model, a reliable tool for forecasting, was used to answer these questions. The analysis of the diffusion of this innovation indicates that it is in its infancy, as its diffusion couldn’t be predicted within the time horizon adopted by the EU.

The methods of approximating fractional order systems to integer order systems include Oustaloup's method, Matsuda's method, CFE method, and M-SBL method, among others. The Padé approximation method is one of the potential approximation techniques. Its application in the literature is significantly less common than other approaches, even though it provides comparable or sometimes superior results. Therefore, the paper explores the application and properties of this method for fractional order systems. The presented analysis builds upon the authors' recent work on the Padé approximation method. This study places particular emphasis on examining the method's behavior and performance within the frequency domain. It should be noted that this method is a rational approximation expanded about the point. Therefore, the main goal of this article is to attempt to establish the relationship between the selected point of the Padé approximation and the frequency range over which the approximation achieves optimal alignment with the original system's frequency response. The research will focus on analyzing the magnitude and phase characteristics in the frequency domain (Bode plots), utilizing simulations to provide a comprehensive and precise evaluation.

This research explores various methods of approximating fractional order systems to integer order systems, such as the Padé method, the Oustaloup method, the Refined Oustaloup method and others. The main focus is on the special case of a fractional order system, a complex variable transfer function raised to a fractional power, which is the primary component of intricate fractional order systems. The novelty of this study lies in comparing these methods and presenting the approximation results, which is based on the study of the values of zeros and poles as a function of the assumed fractional power. As a result, the paper demonstrates and illustrates some key properties of the methods that concern the relationships between zeros and poles in each method. It also highlights the characteristic behavior of the values of zeros and poles across the reviewed approximation methods. Specifically, limiting scenarios have been considered, particularly instances where the fractional power of the component approaches zero or one. Both simulation and analytical techniques are used to explore these considerations.

The first stage of designing a control system is identification, which involves creating a model of the system. For some systems, a description using a fractional-order model may be more beneficial. However, classical control theory methods are most often adapted to integer order models. Therefore, there is a need to approximate fractional order systems. Various methods used for this purpose are as follows: Oustaloup's method, CFE (Continued Fraction Expansion) method, Matsuda's method, MSBL method, and many others. The article considers the use of the Pade approximation in this case (most often used to approximate time delays in system identification and modelling). According to the principle of operation of the Pade approximation, it has wider approximation possibilities compared to the mentioned methods. It allows the designer to approximate the full form of the transfer function, which includes fractional order components. The approximation does not have to take place only for the mentioned components, as is the case with methods from the known literature. On the basis on the selected criteria, various approximation methods were compared in simulations with the Pade method.

The issue of identifying and designing control systems of fractional order models involves the necessity of their approximation to integer order. The most popular techniques include: Oustaloup's method, Matsuda's method, CFE (Continued Fraction Expansion) method, M-SBL method and different more. It is also possible to use the Pade approximation. The article examines some properties of this method. The case of the fractional power term (complex variable transfer function raised to a fractional power) is the main focus since it serves as a basis for approximating more intricate fractional order systems. The article's innovation is in using the nonlinear least squares method to obtain a mathematical description of the behavior of the zeros and poles of the chosen component after approximation. Relying on the gathered statistics, it was found that a very good fit was produced. The approximation of the behaviour of zeros and poles also allowed for the presentation of a new, alternative notation for the Pade approximation.

The article addresses the problem of identifying and controlling cascade systems using B-splines. This article proposes a classification of identification approaches at the level of experiment planning. The use of classical identification methods, based on the response of the system to a well-defined standard test signal, in some cases is not possible to use in cascade systems. A method based on B-splines, allows for system identification using any input signal. This allows for a wider range of identification scenarios. This paper tests the B-spline identification method for a heating model of a building. The article also points out a number of problems arising from the application of the method. It is not a self-sustaining procedure that guarantees obtaining a solution – in particular, with badly chosen parameters, the method may not produce a solution. The problems and some suggestions are made for the appropriate implementation of the algorithm.

This paper mainly focuses on various two-point methods of identifying the First Order Plus Time Delay (FOPTD) model, and their impact on control quality. The following criteria were used to match the identification method: ISE between the response of the real system and the model and the normalized delay time. The work used two different tuning techniques: QDR and pidtune. A three-tank interacting and non-interacting system was used as a model for the tests. All tests were simulated in Matlab environment, ISE and settling time were used to assess the quality of control.

In the face of contemporary ecological problems, including the influence of transport and mobility on the greenhouse effect, policy makers see a solution in the technology of electric vehicles (EVs). This innovation has appeared in the automotive market around the world. In this article, Rogers’ innovation diffusion concept was used to determine if and when EVs could replace combustion engine vehicles. The aim of the article is to analyze the current situation of innovative electric vehicle (EV) technology on the European Union (EU) market and the potential and possible directions of its market diffusion. As a result of the analysis of the current situation of the EU automotive markets, three groups of countries with different market conditions regarding the development of EV technology were distinguished. The classification was carried out using the K-means clustering algorithm, based on the current level of innovation and the level of development of the LPG-(Liquefied Petroleum Gas) powered car market. Then, a group of countries with the greatest advancement in EV technology was selected for further research and market diffusion scenarios were created for this innovation using the Bass model, assuming the target market size of 70–100% of all passenger cars.

The article presents a real prognostic algorithm used in the fashion industry. The method merges an econometric approach with solutions used in operations research. The Weighted Least Squares (WLS) method with constraints was used, combined with an algorithm for optimizing losses resulting from overestimating or underestimating demand.The theoretical basis of the algorithm was presented, and its arrangement in the form of a step-by-step procedure was established. An example of the procedure was then calculated, and the numerical complexity class was estimated.The order prediction algorithm represents a classical approach that can be used in any prognostic problem in which the features of the product and the sales history are known. The approach is especially dedicated when the estimated demand concerns new products with features that have already appeared in previous sales.

Personal light electric vehicles (PLEVs) are a phenomenon that can currently be observed in cities, intended to be an ecological form of transport. The authors of the paper make an attempt to determine electricity consumption by PLEVs in the context of managing a large city in accordance with the concept of sustainable development. The article is of a cognitive nature. Research questions posed against the background of the goal formulated are as follows: how strong will the demand for PLEVs be (in the example of e-motor scooters, taking into consideration the number of vehicles) and for the electricity consumed by PLEVs. The method used is a simulation model. The conducted analyses demonstrate that a dynamic growth of PLEVs will result in an increased energy demand, which must be taken into account by the cities, developing according to the sustainable development conception.

The article discusses the problem of the employee suggestion system (ESS) in production companies. Based on the data from a representative international company, the system performance was assessed. A number of parameters describing the system were presented, among others: a degree of participation - the percentage of employees who participate in the system, a degree of adoption - the percentage of submitted. Over half (58%) of submitted employee ideas can be classified as suggestions referring to organizational areas, followed by ideas related to work safety and the environment 22%, quality 15%, costs of organization 5%. The article is a continuation of research on the subject of the Polish economy.

Purpose The aim of the article is to discuss the issues of the employee suggestion system (ESS) in a remanufacturing enterprise. ESS is a tool used in many modern management philosophies like continuous improvement, Kaizen, lean etc. A number of factors influencing the operation of the employee suggestion system (ESS) were examined in the literature. However, none considers the fact that within various departments, the nature of this tool may vary. Methodology/approach Based on the data from a representative international company, the system performance was assessed. Using statistical tests, it was examined whether the ESS works identically in different departments of the company. The article reviews the literature and attempts to explain the reasons for the results. Findings On the basis of statistical tests, it has been proved that despite the same implementation and occurrence of the same types of waste in various departments, the number of employee suggestions and thus the functioning of the system is different. According to the authors, the reason for this is the fact that the company performs remanufacturing processes. Managing an enterprise using lean remanufacturing has a specific character in relation to classic lean. Recommendations Since the article presents only one example of a company in which this phenomenon appeared and the lack of this issue in the literature, further analysis of this problem is necessary.

Nearly two-thirds of the emissions that cause smog come from road transport. In April 2019, the European Parliament adopted new regulations on public procurement to encourage investment in clean buses—electric, hydrogen, or gas. Directive 2009/33/EC is to apply from the second half of 2021. The aim of this article is to make an attempt to simulate the number of zero-emission buses (ZEB) in European Union (EU) member countries in two time horizons: 2025 and 2030, and to forecast the number of clean vehicles in the precise time horizons, including before and after 2050. Research questions are as follows: (1) what will be the number of ZEBs in individual EU countries over the next few years; (2) which of the EU countries will reach by 2030 the level of 95% share of ZEBs in all buses, which are a fleet of public transport buses; and (3) in which year will which EU countries reach the level of 95% share of zero-emission buses. The method used is a Bass model. The conducted analyses demonstrate that, by 2050, only four of the EU members will be able to reach 95% level of share of clean buses in the city bus transport fleets. It is likely that other countries may not achieve this even by 2050.

The paper presents the method of estimating the parameters of extreme statistics distribution by the maximum likelihood method. Knowing the distribution of extreme statistics enables the prediction of the maximum value for periods outside the analysed sample. Such forecasting has many applications. From the analysis of weather phenomena, through floods to financial data. An important part of the work are examples related to the error assessment when making decisions. The examples relate to the selection of the optimal profile when assessing the demand for clothing goods. The method of regression (or classification) trees used in this problem requires at each step the process of selecting the profiles by means of the chi square statistics. For a small number of profiles, one can view all the options and select the highest value for the chi square statistic. However, if the number of possible features increases to several dozen, finding the optimal solution is not possible in a reasonable time. The issue of choosing the optimal profile is equivalent to the problem of set partition.

. The aim of the paper is to obtain an objective opinion from the experts. Experts are evaluated on the basis of the accordance of their preferences with the total preferences. The inversion distribution was applied. This distribution is associated with inversions occurring in the tau-Kendall test. The probabilities obtained as a result of estimation are needed to construct the estimator of demand

Aim : The aim of the study is to analyze the selected areas of the company’s operations. The following aspects were examined: the benefits from the application of the concept of continuous improvement for the company and its individual employees, barriers preventing a successful implementation of the concept and self-evaluation of the company’s employees. Research method : The company was examined using a survey method. The findings are the result of the academic internship done at this company. The publication is an outcome of a science project within the framework of the Support Program of the Partnership between Higher Education and Science, and Business Activity Sector financed by the City of Wroclaw. Findings : The paper completes the set of modernization recommendations co-developed by the authors and the entrepreneur’s representative. Originality/value of the paper, scholarly contribution : In the context of current knowledge, the paper demonstrates the quality of continuous improvement based on the example of a Polish enterprise, which is a valuable source of information given the limited number of studies within this scope in the domestic strand of research. The findings may be used during the implementation of  Kaizen in other organizations. Limitations of the study( if applicable): The paper follows up on the issues concerned with the implementation of the philosophy of continuous improvement in a business entity, and hence does not encompass the full Kaizen spectrum. In the first part of the series, the findings referring to the knowledge about the concept, implementation methods, etc. were presented.