Organizing join events within the COST Action CA15225 and using its networking tools to foster interaction between the COST Action members being interested in fractional calculus, fractional systems’ and models’ design, and their utilization, the individual Tasks of each Working Group were solved to gain the expected Deliverables and achieve the Research coordination and/or Capacity building objectives as listed below:

#### Research coordination objectives:

#### O1 – Define optimization steps leading to efficient implementation of fractional-order systems

Until fractional-order elements are not readily available, it is necessary to introduce the fractional feature through suitable approximations being at least valid in specific frequency range, featuring sufficiently high accuracy of approximation and preferably not causing inadequate increase in complexity of the final circuit solution, no matter if designed as analog or digital.

Already the results presented in doi: 10.1016/j.ifacol.2017.08.1422 and doi: 10.1109/ECCTD.2017.8093324 introduce efficient techniques to obtain accurate and low-order approximations of fractional operators and compensators that are useful for control and other applications, primarily designed as digital. Approximations of the analog fractional-order differentiator and integrator operators are proposed in doi: 10.1002/cta.2598. These approximations target the realization of these operators using standard active filter transfer functions. Although the here proposed approximation techniques target analog signal processing, they may also be efficiently transformed to digital domain.

Another introduced approach to approximate the behavior of fractional-order systems is based on curve fitting. Using this approach the magnitude and phase frequency responses of the transfer function are first extracted and approximated through curve fitting-based techniques. Comparison between the conventional method and the proposed method reveals that the achieved benefit is the significant reduction of the passive and/or active components count and hence reduced circuit complexity of the final solution as it was shown e.g. in doi: 10.1109/NILES50944.2020.9257936, doi: 10.1007/s00034-020-01514-7, doi: 10.3390/fractalfract4040054, or doi: 10.1016/j.aeue.2020.153537.

For the purpose of efficient approximation of fractional-order elements, a technique based on the mixed integer-order genetic algorithm was introduced in doi: 10.1109/ACCESS.2019.2923166. The proposed technique provides phase optimization in the desired bandwidth with minimal branch number and avoids the use of negative component values, and any complex mathematical analysis. Significant improvements over the Oustaloup approximation, the Valsa recursive algorithm, and the continued fraction expansion were achieved, whereas fractional-order elements approximated in bandwidth over 4 decades and with absolute phase error below 1deg are possible.

With the end of the COST Action, the level of achievement of this Objective is undestood to be 76-100%.

#### O2 – Improve characteristics of fractional-order controllers that can be employed in different industrial loops or in electro-mechanical systems

The Matlab Toolbox FOMCON, initially developed by Aleksei Tepljakov was further improved and extended to FOMCONpy doi: **10.1109/TSP49548.2020.9163581**. Reviewing the associated mathematical concepts the FOMCONpy contains fractional-order system analysis, fractional-order system identification and mainly the fractional-order system control modules. The FOMCONpy software is available here: **https://github.com/outstandn/fomcon**.

This Objective was achieved by the progress made mainly within the Deliverables D4, D7, D8, D12 and D17 and represents the cornerstone for the following Objectives further dealing control issues, i.e. O3 and O4.

With the end of the COST Action, the level of achievement of this Objective is undestood to be 76-100%.

#### O3 – Develop tools to define dynamic simulation models, control schemes and algorithms

A general framework for global, non-adaptive identification of fractional and non-rational systems in general has been developed in several specific directions.

The first is non-parametric analysis, in which an overall shape of the frequency characteristic is analyzed first, and then a suitable explicit or implicit fractional order model is proposed and optimized by means of a PSO algorithm. Relevant study was reported in doi: 10.1016/j.ifacol.2017.08.2084 and doi: 10.1016/j.aeue.2017.05.036.

The next investigated direction to develop specific forms of transfer functions was related to various distributed-parameter systems, with or without fractional spatio-temporal dynamics. Initial results are reported in doi: 10.1109/ECCTD.2017.8093252, doi: 10.1007/s11071-016-3322-z. An important issue regarding stability of the developed fractional models, which also addresses issues of admissible values of parameters, is presented in 10.1016/j.ifacol.2017.08.2091 and doi: 10.1049/iet-cta.2018.6350.

Finally, the third direction achieves this objective fully by developing fractional models for time-varying systems based on frequency response functions obtained at the beginning and at the end of the transformation. This technique has been used to explain the variation in dynamic response of adhesives as they undergo phase transition during the curing process. For further information see doi: 10.1016/j.apm.2019.08.021.

With the end of the COST Action, the level of achievement of this Objective is undestood to be 76-100%.

#### O4 – Design and implement fractional-order controllers for industrial processes

Using the design processes to propose fractional-order controllers with improved characteristics, the successful implementation of fractional-order PID controllers was shown. From the list of achievements, we may mention re-configurable fractional-order PI controller, suitable for brake and throttle control in an autonomous vehicle was discussed in 10.1109/ECCTD.2017.8093229 and further investigated in doi: 10.1109/MOCAST.2019.8741695. The first solution is based on designing the fractional integrator as approximation by appropriate integer-order rational function, which is implemented as OTA-C circuit. The second solution of fractional PI controller reduces significantly final circuit complexity as it already assumes the presence of fractional capacitor. An efficient design of fractional PID controllers was proposed in doi: 10.1109/CoDIT.2019.8820324. Rearranging the individual blocks in the overall set up significant reduction of the active components count was achieved. This is due to the fact that in the final circuit solution only fractional-order differentiation stage is required. Solutions leading to simplification of the final circuit realizations were also discussed in doi: 10.1109/TSP.2019.8768878 or doi: 10.1109/TSP49548.2020.9163518.

The implementation of fractional-order PD controllers as digital was also investigated. The results from doi: 10.1109/MMAR.2019.8864626 compare the features of analog and digital controllers for servo systems. The phase margin and the gain-crossover frequency have been fully fulfilled. The provided time-domain analysis confirms the appropriate closed-loop operation, while the sensitivity analysis and corner analysis results proved the robustness of the presented scheme.

Another area of fractional-order controller design was dealing with multi-loop control structures that can enhance inherent disturbance rejection performance of classical closed loop control loops. While the classical closed loop PID control loop (inner loop) deals with stability and set-point control, the additional model reference control loop of MIT rule (outer loop) can improve the disturbance rejection control performance without degrading the optimal set-point control performance. Such adaptive disturbance rejection approach, which is not influencing the set-point control performance, can be achieved by selecting reference models as transfer function of the PID control loops with ignorable time delay. This approach is researched in a fruitful collaboration between Turkey and Estonia research groups. Several variants of multi-loop Model Reference (Ml-MR) PID control designs (Ml-MR PID-MIT control) to improve disturbance rejection control were discussed doi: 10.3390/a13020038. Disturbance rejection control performance of multi-loop Model Reference FOPID control structures was reported in doi: 10.1142/S0218126618501761 and doi: 10.3390/a13080201.

A comprehensive overview or fractional-order PID controllers and their utilization in industrial applications was provided in doi: 10.1016/j.ifacol.2018.06.014.

#### O5 – Design and characterize new fractional-order elements using prospective technologies in order to obtain robust and commercial devices

From the list of possible implementations (doi: 10.1016/j.mejo.2018.12.010) fractional-order elements (FOEs) as solid-state devices, the attention was paid to hardware design of fractional-order elements based on the Resistive-Capacitive Elements with Distributed Parameters (RC-EDP). Using a software originally developed by Peter A. Ushakov, MC Observer from Russia, the initial results were presented in doi: 10.1109/ECCTD.2017.8093314 and further continued in hardware implementation of solid-state capacitive type fractional-order elements designed using the thick-film technology. The datasheets of these FOEs can be downloaded here and here. The research further continues even after the COST Action by taking the advantage of general multilayer resistive-capacitive circuits used to implement the solid-state thick-film FOEs, that is same for thin-film technology to reduce dimensions, but also for CMOS with the aim to introduce electronically controlled FOEs.

Another investigated and prospective approach to design solid-state FOEs is based on the use of carbon-based structures dispersed inside a polymeric matrix. The article doi: 10.1109/TED.2020.2965432 analyzes the material characterization of the nanocomposite employed in the fabrication of a solid-state fractional capacitor. In doi: 10.1016/j.mejo.2018.10.008 the possibility of realizing fractional capacitors by using carbon black nanostructured dielectrics is investigated. Capacitors have been realized by varying the percentage of distributed carbon black.

To design environmental friendly solid-state FOEs, the possibility of using Bacterial Cellulose (BC), a bio-derived polymer, for the realization of fractional order electronic devices was proposed in doi: 10.1016/j.aeue.2019.152927. Bacterial Cellulose, unlike plant-derived cellulose, is produced by some genera of bacteria, if a suitable culture is maintained. Compared to plant-derived cellulose, BC can be obtained with a green and low-energy production process, which does not produce pollutants nor carbon composites. Bacterial Cellulose is used as the bulk in a capacitor-like structure. The device impedance has been investigated and experimental evidence of its fractional nature is given.

#### O6 – Utilize fractional-order adjustment rule to model reference adaptive control in engineering applications

A general framework for adaptive identification of fractional and non-rational systems in general has been developed first. The research has targeted algorithms that are capable of identifying unknown parameters in transfer functions of arbitrary form, including rational, fractional, transfer functions, transfer functions with “fractional delay”, and other forms of transfer functions, which are derived from partial differential equations describing distributed parameter systems. Relevant studies and approaches have been reported in doi: 10.1016/j.aeue.2017.04.008, doi: 10.1109/TAC.2019.2893973.

In recent papers doi: 10.1016/j.ejcon.2020.06.005 and doi: 10.23919/ECC.2019.8796163, classes of fractional-order PID controllers and distributed-order PID controllers were proposed and successfully applied to permanent magnet synchronous motors used in industry. Optimization was based on a cost function considering both performance and robustness indexes, i.e. the maximum sensitivity, maximum noise sensitivity, maximum resonant peak, while guaranteeing closed-loop stability. Results show that the proposed controllers can successfully replace usual PI/PID controllers both in reference tracking and disturbance rejection.

Moreover, other results regarded fractional-order control of robotic manipulators. In particular, in doi: 10.1109/SMC.2019.8914031, a general procedure was introduced to design fractional-order controllers for independent-joint control of DC motors actuating robot joints, for a 5DOF robotic manipulator. Both position and speed are controlled by employing feedback and feedforward actions.

Another successful utilization of a fractional-order lag network or a fractional-order PI controller for the motion control of three revolute joints of a manipulator was presented in doi: 10.1016/j.ifacol.2020.12.2050. The introduced fractional compensators are designed by using the symmetrical optimum principle and by parameters optimization or by frequency-domain loop shaping, respectively. The reached results show effectiveness and robustness of the approach.

#### O7 – Implement fractional-order digital/analogue function blocks especially in medical signal processing

Significant contribution to achieve this objective was done by the Greek/UAE group by presenting the Design of CMOS Analog Integrated Fractional-Order Circuits: Applications in Medicine and Biology (doi: 10.1007/978-3-319-55633-8). This book represents a very good summary, whereas first the attention is paid to the design of analogue fractional-order filters operating in different modes (voltage/current) and the possibilities to implement fractional-order elements. From this viewpoint, this book also contributes to Objective O1. As next, the successful and compared to integer-order systems advantageous implementations of the fractional-order function blocks in medical signal processing is shown in this book by presenting the results of filtering very noisy signals, being very often characteristic for biomedical signals.

To analogue fractional-order filter design is also devoted our chapter “Analog Filters With Arbitrarily Adjustable Frequency Response” that is part of the book “Fractional Order Systems Optimization, Control, Circuit Realizations and Applications” (doi: 10.1016/C2017-0-04459-2).

The fractional-order signal processing in digital domain was also investigates and applied for the purpose of evaluating the level of Parkinsonic dysgraphia that affects the majority of Parkinson’s disease (PD) patients as a result of handwriting abnormalities mainly caused by motor dysfunctions. Several effective approaches of quantitative PD dysgraphia analysis, such as online handwriting processing, have been successfully utilized; doi: 10.1109/TSP.2018.8441293, doi: 10.1109/ICUMT.2018.8631265, doi: 10.3390/app8122566, or doi: 10.23919/EUSIPCO.2019.8903088. The theoretical concepts were also successfully applied to evaluate graphomotor disabilities (GD) of school-aged children. Although the basic kinematic features such as velocity, acceleration, and jerk were proved to effectively quantify these symptoms, a recent body of research identified that the theory of fractional calculus can be used to even improve the objective GD assessment (doi: 10.1109/ACCESS.2020.3042591).

#### O8 – Utilize fractional-order models and systems in bioengineering and biomedical applications

In the area of modeling systems stemming from bioengineering the achievements fostering this objective are supported by design of numerous emulators describing and preserving in time the fractional features of systems being observed. As examples, the following may be mentioned: Fractional order circuit for emulating the mechanical impedance model of the human respiratory system (doi: 10.1109/PACET.2017.8259963), model of the biceps tissue during fatigue exercise (dois: 10.1109/TSP49548.2020.9163499, 10.1109/ICUMT51630.2020.9222426), Electrical Models of Young and Old Dentines (doi: 10.1109/EMBC44109.2020.9175916).

Preliminary mathematical developments and properties regarding Havriliak-Negami type of time-domain or frequency-domain models were obtained by putting together the best results from the following previous publications doi: 10.1007/978-3-319-45474-0_38 and doi: 10.1515/fca-2016-0060. Some advancements were proposed in the current output doi: 10.1515/fca-2020-0002. In details, numerical methods are now available to give an approximate but accurate solution to differential equations in which the Prabhakar derivative is used to better describe anomalous relaxation in Havriliak-Negami models of dielectric materials or biological tissues. Moreover, the time-domain relaxation and response functions of the most common materials that show anomalous relaxation are now well known and described.

A mathematical model of nociception pathway and a prototype device for pain measurement have been developed. Such mathematical models capturing essential dynamics of pain mechanism are necessary to develop suitable management policies and drug delivery assist devices for analgesia. The main achievements in this area were summarized in doi: 10.1515/9783110571905-004.

A fractional-order mathematical model for a tumor-immune surveillance mechanism was presented in doi: 10.1063/1.5096159. Here the interactions between various tumor cell populations and immune system via a system of fractional differential equations (FDEs) were analyzed. An efficient numerical procedure is suggested to solve these FDEs by considering singular and nonsingular derivative operators. An optimal control strategy for investigating the effect of chemotherapy treatment on the proposed fractional model is also provided. The reached results show that the new presented model based on the fractional operator with Mittag–Leffler kernel represents various asymptomatic behaviors that tracks the real data more accurately than the other fractional- and integer-order models. Numerical simulations also verify the efficiency of the proposed optimal control strategy and show that the growth of the naive tumor cell population is successfully declined.

#### O9 – Characterize preservation properties of non-integer order control and dynamical systems under discretization

To foster this objective, the main achievements were reached by join cooperation between the Romanian (West University of Timisoara) and Polish (Bialystok University of Technology) groups. Based on the initial results described “Stability and numerical solutions to the variable-order Caputo fractional difference equations with order values from (1; 2]” and presented on NODYCON 2019 conference in Rome, the stability issues of two-dimensional systems of fractional-order difference equations or variable-Order Biquadratic Difference Equations were further investigated and applied on FitzHugh–Nagumo neuronal model, Rulkov neuronal model, which describe the biological neuron’s spiking behavior (dois: 10.3390/math8101751, 10.1007/978-3-030-34724-6_31, 10.1007/978-3-030-34724-6_30).

The join study on Caputo–Fabrizio fractional delta derivatives on an arbitrary time scale made between the Polish (Bialystok University of Technology) and Portugal (University of Aveiro) teams and presented in doi: 10.1016/j.nahs.2018.12.001 also contributes to achieving this objective. Here, the behavior of solutions to initial value problems with the Caputo–Fabrizio fractional delta derivative on an arbitrary time scale were investigated. In particular, the exponential stability of linear systems were studied and necessary and sufficient condition for the exponential stability of linear systems with the Caputo–Fabrizio fractional delta derivative on time scales were presented. Using the Banach fixed point theorem, the existence and uniqueness of solution to a nonlinear initial value problem with the Caputo–Fabrizio fractional delta derivative on time scales was proved.

The further results on discrete-time fractional, variable-order PID controller for a plant with delay (dois: 10.3390/e22070771, 10.24425/124261); stability of fractional variable order difference systems (dois: 10.1515/fca-2019-0044, 10.5755/j01.eie.24.5.21846, 10.1007/s40435-016-0239-9, 10.2139/ssrn.3270846) also significantly contribute to successful achievement of this objective.

**Capacity building objectives:**

#### O10 – Establishment of European-wide scientific and technology knowledge platform in order to instigate interdisciplinary interaction for the development of innovative fractional-order systems

At the beginning of the COST Action CA15225 already 23 countries and their representatives started to be active participants willing to contribute to the tasks, deliverables and objectives of the COST Action. Dealing with the activities of the Action and using the COST Networking tools, the platform further increased. With the end of the COST Action 25 European countries represented by their MC Members/Substitutes together with 8 MC Observers from Canada, Egypt, India, Russian Federation, United States (USA), and United Arab Emirates (UAE) were part of the COST Action.

Hence, during its lifetime the COST Action CA15225 managed to address a significant number of researchers spanning from mathematicians to applied research engineers being active in different R&D areas. The regularly organized events, such as MC Meetings, Annual Workshops, Training Schools and Short-Term Scientific Missions, enabled to initiate and/or strengthen the interaction between the participants dealing not just the in common topics being discussed within the individual Work Groups, but also triggered the join discussions and interaction between the Work Groups to further support the Objective O11 of the COST Action.

#### O11 – Bridging separate research fields and disciplines to present interdisciplinary approach to scientific research and foster multidisciplinary breakthroughs

The organization of and active participation at the events enabled to better interact in between and provided an insight into the activities being dealt within individual Work Groups. The regularly organized Training Schools and Study Group event are identified as the most efficient networking tools to bring together participants initially “isolated” in their Work Groups and being primarily focused on their research area. Thanks to the mentioned events, but also thanks to regular MC Meetings and Annual Workshops, discussions and brainstorming between people, who would not normally meet together at one place, was enabled. Such interactions gradually bridged the individual research fields of fractional calculus, system modelling and emulation, hardware design, and signal processing development. The cooperation and multidisciplinary research activities is supported not just by numerous join publications co-authored by participants/members from different Work Groups of the COST Action but also by the number of research projects aiming to continue in the research activities on a specific topic.

The join publications, research projects being proposed (basically from the view point of capacity building no matter if supported or not), and the face-to-face meetings during the events organized within the COST Action represent a significant and valuable precursor for notinues cooperation of already smaller but efficient groups that may provide novel and multidisciplinary breakthroughs expected in the near future. Even if the COST Action is already at its end, it provided the base required to achieve this objective.

#### O12 – Ensure Early Career Investigators to participate in the Action within dedicated dissemination and formation activities such as workshops or STSMs and give them the best possible return in terms of scientific knowledge, research direction and coordination skills

At the events regularly organized by the COST Action the Early Career Investigators (ECIs) and/or already current PhD students were supported to participate and share their knowledge with other young colleagues but also gain experience from the senior researchers. The Short-Term Scientific Missions (STSMs), as one of the COST networking tools, proved to be very beneficial for individuals, who visited a partner institution, got familiar with the local research team, and discussed not just the current STSM topic but also possible future research activities. The ECIs and PhD students also significantly participated at the Training Schools to take advantage of becoming aware of new or different views on fractional calculus and its utilization in engineering and also to create contacts with other Trainees (and also the Trainers) and to discuss their current and planned research topics.

The young researchers also accepted the ICT Conference Grants, the newly introduced COST networking tool. Similarly to STSMs, the ICT Conference Grants proved to be useful first for active researchers and their presentation of results at international conferences, second also to disseminate the ideas of the network calling itself FRACTIONAL, the COST Action CA15225.

#### O13 – Increase the gender balance in terms of researchers involved in Action activities, both in terms of technical and scientific contribution as well as of research direction and Action governance

At the beginning of the COST Action, the Male:Female (M:F) ratio was identified to be 74:26, which was a bit away from the optimal ratio spanning from 60:40 to 40:60 as it is understood by COST Association. At the level of Action governance, the M:F ratio was 88:22. Starting to regularly organize events within individual Grant Periods, using other COST networking tools, and accepting other MC Members/Substitutes and MC Observers from countries newly joining the COST Action, the M:F ratio improved to 65:35. Also at the level of Action governance the M:F ratio improved to 67:33. Even if it is not within the optimal range according to COST Association, we understand the improvement in gender balance as a significant achievement, as the trend in M:F ratio was significantly increasing during the second half of the COST Action lifetime.

Last update: May 6, 2021