Deliverables

Within each Working Group, the Tasks were defined and were solved by its members inside the Working Group or through efficient cooperation between the Working Groups to contribute to the defined Delibverables and also the overall Objectives of the COST Action. Although basically all Deliverables are understood to be delivered, the individual topics are not closed and still offer possibilities for further improvements and procress, just being to be reached already out of this COST Action. In total, 21 deliverables were identified and can be linked to individual Working Groups (WGs) as folows:

  • WG1 – Fractional calculus and mathematical models
    • D1 – Development and implementation of a general framework for global, non-adaptive identification of fractional and non-rational systems in general
    • D6 – Development and implementation of a general framework of adaptive identification of fractional and non-rational systems
    • D13 – Approximation of derivatives and integrals of fractional orders by new numerical and analytical methods; Fractionized models
    • D17 – Development of methods for PID controllers’ synthesis, Matlab toolboxes and discretization algorithms
  • WG2 – Fractional-order systems’ synthesis and analysis
    • D2 – Presentation of optimized low-order approximations of fractional Laplacian operator featuring lower circuit complexity
    • D7 – Presentation of tools for analysis and synthesis of the fractional-order function blocks
    • D14 – Increased approximation accuracy of fractional Laplacian operator for analogue circuit design, discrete rational approximations
    • D16 – Presentation of new fractional-order elements based on the IMPC, graphene and RC-EDP design approach
  • WG3 – Design of analogue and digital fractional-order function blocks
    • D3 – Development of integrators and differentiators, initial implementation of digital fractional order blocks
    • D9 – Development of fractional-order analogue and digital filter topologies using basic building blocks
    • D11 – Optimal dynamic control – methods for solution and numerical computation of optimal dynamic control
    • D18 – Application of FO linear and non-linear blocks in sensors, actuators and control systems
  • WG4 – Utilization of fractional-order systems in engineering and biomedical research areas
    • D4 – Development of preliminary models for automotive injection systems and of physical and mathematical knowledge on models of Havriliak
    • D8 – Detailed models and virtual prototypes; low-order models for FO controllers, development of simulation tools for systems of Havriliak-Negami type
    • D12 – Advanced FO control algorithms and realization techniques for automotive applications; model of pain pathways and corresponding software
    • D19 – Modelling and control of injection systems; Schemes for analysing propagation of electro-magnetic fields in biological tissues; Report on correlation analysis and corresponding software
  • Deliverables common for all WGs
    • D5 – Contribution of each working group to the proceedings of the Year 1 Workshop co-located with the Action meeting
    • D10 – Contribution of each working group to the proceedings of the Year 2 Workshop co-located with the Action meeting
    • D15 – Contribution of each working group to the proceedings of the Year 3 Workshop co-located with the Action meeting
    • D20 – Contribution of each working group to the proceedings of the Year 4 Workshop co-located with the Action meeting
    • D21 – Finalization of the book summarizing Action activities and scientific results achieved during its four years span

D1 – Development and implementation of a general framework for global, non-adaptive identification of fractional and non-rational systems in general (expected by July 2017)

Mainly the members and their activities done within the Working Group 1 contribute to this deliverable. This deliverable is understood to be delivered.

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 in collaboration between Serbian and Italian COST Action members. Parts of the research were also reported in doi: 10.1016/j.aeue.2017.05.036.

Next path under investigation is to develop specific forms of transfer functions 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 doi: 10.1016/j.ifacol.2017.08.2091 and doi: 10.1049/iet-cta.2018.6350.

Finally, another technique has been developed where fractional models can be derived 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.

D2 – Presentation of optimized low-order approximations of fractional Laplacian operator featuring lower circuit complexity (expected by July 2017)

Mainly the members and their activities done within the Working Group 2 contribute to this deliverable. This deliverable is understood to be delivered.

The results presented in doi: 10.1016/j.ifacol.2017.08.1422 and doi: 10.1109/ECCTD.2017.8093324 provide efficient techniques to obtain accurate and low-order approximations of fractional operators and compensators that are useful for control and other applications.

Partially also delivered by presenting the comprehensive analysis of the approximation of low-pass magnitude response. These results were presented in doi: 10.1016/j.aeue.2017.04.031. The further analysis of specific transfer function types continues, see e.g. doi: 10.5755/j01.eie.24.2.20634A curve fitting based technique is introduced for approximating the behavior of fractional-order systems, and it is applied in the case of filters, controllers, and driving point impedances. The magnitude and phase frequency responses of the transfer function are first extracted and approximated through curve fitting-based techniques. A rational integer-order function is then obtained and realized using appropriately configured passive and/or active topologies.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. The concept as well as the related applications have been published e.g. in doi:  10.1109/NILES50944.2020.9257936, doi: 10.1007/s00034-020-01514-7, doi: https://doi.org/10.3390/fractalfract4040054, or doi: 10.1016/j.aeue.2020.153537.

D3 – Development of integrators and differentiators, initial implementation of digital fractional order blocks (expected by July 2017)

Mainly the members and their activities done within the Working Group 3 contribute to this deliverable. This deliverable is understood to be delivered.

Efficient circuit solutions to design fractional-order integrator and differentiators using opams, CCIIs, CFOAs and OTAs were published e.g. in doi: 10.1109/TSP.2017.8076081. Further approximations of 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. Hence, circuit implementations in integrated circuit form or in discrete component form are significantly facilitated. The concept is based on the employment of the partial fraction expansion tool and, as a result, the fractional-order transfer function is decomposed in a sum of a constant term and 1st or 2nd basic filter functions (i.e. lowpass, highpass, bandpass etc). Comparison with the literature shows that a significant reduction of the required circuit complexity is achieved. Applications of this concept have been published in a number of papers: doi: 10.1016/j.mejo.2018.11.013, doi: 10.1016/j.mejo.2019.05.002, doi: 10.1007/s00034-019-01308-6, doi: 10.3390/technologies7040085, or doi: 10.3390/electronics9010063.

Controllable fractional-order integrator, integrational-derivative two-port and practical aspects of their mutual interconnection has also been investigated. The minimal configuration blocks of fractional-order immitances with only one active element and fractional-order integrator in current mode were designed. The reached results and proposed solutions were published in  doi: 10.1109/TSP.2019.8768814, doi: 10.3390/app10010054, doi: 10.1109/TSP49548.2020.9163553 and doi: 10.1109/ICECS49266.2020.9294923.

D4 – Development of preliminary models for automotive injection systems and of physical and mathematical knowledge on models of Havriliak (expected by July 2017)

Mainly the members and their activities done within the Work Grouping 4 contribute to this deliverable. This deliverable is understood to be delivered.

Models describing individual parts that affect the injection process in advanced automotive natural gas engines were discussed in doi: 10.1016/j.ifacol.2017.08.2084. Gas engines were proposed to reduce pollution determined by combustion of Diesel or gasoline fuels, but their performance strongly depends on the metering of the air/fuel ratio, which is achieved by controlling the gas injection timing and the gas pressure in the common rail volume. The activities aimed to define an accurate model that would  represent the gas pressure dynamics in the injection system. It was identified that a fractional-order model describes the gas flow into the common rail better than ARX integer-order models of high order. Identification was made in the frequency domain by minimizing a difference criterion between the model output and real data.

Moreover, the report “Fractional-Order Modeling of Fuel Propagation in Electro-injectors Pipes” by F. Saponaro, G. Maione, P. Lino, R. Garrappa, (Workshop on current progress in fractional-order systems and their utilization – Cost Action 15225 Annual Workshop, San Sebastian, Spain, 5-6 October 2017), showed results in modeling strategic components of common rail Diesel compression-ignition engines, namely the electro-injectors used to to let the proper amount and rate of fuel enter the combustion chamber. Preliminary fractional-order models of the fuel flow inside the electro-injectors were developed to better describe certain fluid-dynamic processes associated with the high-pressure flow according to wave propagation.

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 achievements doi: 10.1007/978-3-319-45474-0_38 and doi: 10.1515/fca-2016-0060 reached by the COST Action participants. 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.

D5 – Contribution of each working group to the proceedings of the Year 1 Workshop co-located with the Action meeting (expected by September 2017)

The members from all Working Groups have contributed the Annual Workshop to share their experience and achievements. This deliverable is understood to be delivered.

The Annual Workshop 2017 was organized in San Sebastian by the Spain representatives of the Action, Dr Karmele Lopez de Ipina and Dr. Pilar Ma Calvo, at University of the Basque Country. Following the tasks and expected deliverables described within the individual Working Groups, 19 speakers from the member countries presented the progress in the description and utilization of fractional-order systems and function blocks. Next to that also representatives of the 3 local companies – Technalia (by Hector Herrero), Stago (by Pablo Martinez Santoja) and OTRI (by Gorka Artola), gave a speech on the possible utilization of fractional-order approach in controlling their designs. For more information, please see Annual Workhop 2017.

D6 – Development and implementation of a general framework of adaptive identification of fractional and non-rational systems (expected by May 2018)

Mainly the members and their activities done within the Working Group 1 contribute to this deliverable. This deliverable is understood to be delivered.

A general framework for adaptive identification of fractional and non-rational systems in general has been developed. 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. The proposed algorithm is gradient based, and novel convergence conditions are derived generalizing well known results regarding input richness. Relevant studies and approaches have been reported in doi.org/10.1016/j.aeue.2017.04.008, 10.1109/TAC.2019.2893973.

For practical performance of controller the design methods strongly depend on the relevancy of identified models. Therefore, the mathematical models should express meaningful dynamics of real-world systems. Two fundamental numerical solution methods of fractional calculus in identification and simulation problems of One Non-Integer Order Plus Time Delay with one pole (NOPTD-I) transfer function models were discussed and utilized. The identification process is carried out by estimating parameters of a NOPTD-I type transfer function template according to the experimental step response data. The reached results were conducted within the collaboration between Estonia (University of Tallinn) and Turkey (Inonu University) groups and the results were reported in  doi: 10.1142/S1793962319410113. Part of this Deliverable is also the Matlab code, which is shared for the use of researchers in the Mathworks (https://www.mathworks.com/matlabcentral/fileexchange/88813-fractional-order-time-delay-plant-identification).

D7 – Presentation of tools for analysis and synthesis of the fractional-order function blocks (expected by July 2018)

Mainly the members and their activities done within the Working Group 2 contribute to this deliverable. This deliverable is understood to be delivered.

The FOMCON toolbox for MATLAB®, initially developed by Aleksei Teplakov, the MC member for Estonia, was further improved. The FOMCON toolbox for MATLAB® is a fractional-order calculus based toolbox for system modeling and control design. For more details about the Matlab Toolbox you may check the MathWorks® File Exchange or the FOMCON hopepage.

During the COST Action period the FOMCONpy was introduced and is a new fractional-order modelling and control toolbox for Python. It is an extension of the existing FOMCON toolbox for MATLAB, but this time aiming at Python users and the Internet of Things (IoT) community. Just like the original toolbox, it offers a set of tools for researchers in the field of fractional-order control. Similarly as FOMCON, also FOMCONpy is available for the broad research community: https://github.com/outstandn/fomcon

D8 – Detailed models and virtual prototypes; low-order models for FO controllers, development of simulation tools for systems of Havriliak-Negami type (expected by July 2018)

Mainly the members and their activities done within the Working Group 4 contribute to this deliverable. This deliverable is understood to be delivered.

The results shown in the previous publications doi: 10.1016/j.ifacol.2016.08.071, doi: 10.1109/CDC.2015.7403160, doi: 10.3182/20150218-3-AU-30250//978-3-902823-71-70075, doi: 10.3182/20140824-6-ZA-1003.00889 are based on simulation models developed in the Matlab/Simulink environment or on virtual prototypes built by the AMESim software package. The mentioned tools also allow an easy and fast prototyping of the control systems on the basis of a model-based approach that guarantees accuracy and effectiveness of the results. Namely, the implemented simulation models allow an accurate representation of the complex, nonlinear, time-varying dynamical processes that occur in the considered common rail injection systems and characterize their operation in different working points. In this sense, the models can be considered very close to the hardware and real systems they represent. More specifically, the adoption of fractional order models for CNG injection systems provides a compact mathematical representation of the only most significant characteristics of the injection process. In fact, unlike the classical high integer-order ARX models, a simple model structure can be obtained in the form of a fractional-order transfer function by neglecting the secondary effects and  capturing only the relevant features for control design, providing a reliable prediction of the injection pressure at the same time. The identification procedure is based on a frequency-domain method and on classical and efficient convex techniques applied to experimental data.

In doi: 10.3390/fractalfract4030037, a serial structure of cascaded, shifted, fractional-order, lead compensators was proposed as a new type of fractional-order controller. Two stages are connected in series and introduce their respective phase leads in shifted adjacent frequency ranges. The obtained compensator shows a nearly flat phase diagram in a large frequency range and can be easily realized by low-order rational transfer functions, each stage being a second-order transfer function with limited sensitivity of coefficients to parametric variations. However, the number of stages and their free parameters can be changed such that these new controllers can be flexible and suitable for solving difficult control problems. On this basis, a method is introduced to design a robust controller for a class of benchmark plants that are difficult to compensate due to monotonically increasing lags. The main design strategy consists in compensating the rapidly and monotonically increasing phase lag by the lead introduced by the compensator in the same frequency range. Desired set-point response, stability robustness to gain and parameter variations, and compensation of dead-time to satisfy strict specifications can be achieved much better than by integer-order controllers.

To simulate Havriliak-Negami models in the time domain, efficient numerical schemes were developed. Firstly, a convolution quadrature rule was derived on the basis of the Laplace transform representation of the response function. The method allows to discretize fractional Havriliak-Negami models in the time domain and then obtain a numerical approximation useful to simulate the time-domain response of these models. Secondly, a Prabhakar function was employed to describe anomalous relaxation properties of dielectric materials with Havriliak-Negami type of behaviour, doi: 10.1007/s11071-020-05897-9, doi: 10.1515/fca-2020-0002.

D9 – Development of fractional-order analogue and digital filter topologies using basic building blocks (expected by August 2018)

Mainly the members and their activities done within the Working Group 3 contribute to this deliverable. This deliverable is understood to be delivered.

Analogue or digitally controlled analogue fractional-order filters providing the low-, band- and high-pass frequency response. Some of the basic results reached by Action members can be found e.g. in doi: 10.1016/j.aeue.2017.04.031, doi: 10.5755/j01.eie.24.2.20634, doi: 10.1109/TSP.2018.8441421, or doi: 10.1515/jee-2018-0001. During the design, the attention was paid to proposing circuit solutions featuring also optimized circuit complexity. This was partially reached by designing and experimental verification and optimization of RC structures with distributed parameters and by designing suitable values of fractional-order series in order to cover ranges required by circuit applications. Comprehensive research on the design of functional fractional-order blocks, their verification and optimization based on the target features was made. Moreover, analysis of fractional-order transfer functions of various filter types was delivered. Several reconfigurable and reconnection-less reconfigurable filtering structures have been also designed. Finally, we have studied building blocks suitable for applications in fractional-order domain. Results relevant to this deliverable and reached by COST Action members can be found in: doi: 10.5755/j01.eie.25.3.23673, doi: 10.1016/j.jare.2020.06.022, doi: 10.1109/TSP.2019.8769089, doi: 10.1109/TSP49548.2020.9163400.

D10 – Contribution of each working group to the proceedings of the Year 2 Workshop co-located with the Action meeting (expected by September 2018)

The members from all Working Groups have contributed the Annual Workshop to share their experience and achievements. This deliverable is understood to be delivered.

The Annual Workshop 2018 was organized in Bialystok by the Polish representative of the Action, Dr Dorota Mozyrska, at University of Bialystok. Following the tasks and expected deliverables described within the individual Working Groups, 16 speakers from the member countries presented the progress in analysis and design of optimized fractional-order function blocks and system control. For more information, please see Annual Workshop 2018.

D11 – Optimal dynamic control – methods for solution and numerical computation of optimal dynamic control (expected by July 2019)

Mainly the members and their activities done within the Working Group 3 contribute to this deliverable. This deliverable is understood to be delivered.

The activities within this deliverable mainly base on further development of the FOMCON toolbox that was originally developed by Aleksei Teplakov (https://de.mathworks.com/matlabcentral/fileexchange/66323-fomcon-toolbox-for-matlab). During the COST Action period the FOMCONpy was introduced and is a new fractional-order modelling and control toolbox for Python and is also available for the community for further usage and design of fractional controllers: https://github.com/outstandn/fomcon

D12 – Advanced FO control algorithms and realization techniques for automotive applications; model of pain pathways and corresponding software (expected by July 2019)

Mainly the members and their activities done within the Working Group 4 contribute to this deliverable. This deliverable is understood to be delivered.

It was shown that some advanced fractional-order control techniques can be applied to automotive engines using the common rail injection system technology and compressed natural gas, which gives a solution to reduce emissions of polluting gases and particulate matter. In this case, the injection process is strongly non-linear, time-variant and highly coupled, so suitable control systems must be designed to guarantee the desired performance. The main controlled variable affecting emissions and consumption of the engines is the common rail pressure in the injection system.

An approach was made available to synthesize and realize fractional order controllers. Synthesis of the controller is based on a loop-shaping technique, which is applied on the open-loop transfer function to achieve frequency-domain performance and robustness specifications. The technique pursues an optimal feedback system in a specified bandwidth and takes advantage of the fractional integrator to achieve enhanced robustness. Moreover, the design approach is reinforced by the D-decomposition methodology that guarantees robust stability of the closed-loop system. Finally, the design formulas are specified by closed-form expressions. As regards the realization of the synthesized controllers, accuracy and simplicity are both considered, to allow an efficient and easy implementation as required by industry. Last but not least, the realization formulas guarantee stability and minimum-phase properties of the controllers.

The performance indexes, the robustness (sensitivity to parametric changes) and disturbance rejection capability are tested by simulation of virtual prototypes that are based on very accurate non-linear models of the considered injection systems. Results indicate that fractional-order controllers allow a higher accuracy in metering the injected fuel and better promptness in setting the rail pressure to the desired reference values.

The variation between different working points of the injection system (in terms of the reference rail pressure) is compensated by a model-based fractional-order gain scheduling control strategy, which allows switching from one controller to another each time the working point associated to the rail pressure changes. In case of small variations, only one switch is necessary; if large variations occur, then more controllers are considered such that variations of the injection timings and the rail pressure are limited. In this way, nonlinearity effects, oscillations and instability problems in the rail pressure are prevented.

New results were obtained for the fractional-order control of the common rail pressure affecting the injection process, to increase the performance of an advanced common rail compressed natural gas engine.The reached results contributing to this deliverable were discussed e.g. in doi: 10.1016/j.ifacol.2017.08.2084 and as chapter are part of the Handbook of Fractional Calculus with Applications, Vol. 6, Applications in Control, 2019, Ed. I. Petráš (chapter on Fractional-Order Controllers for Mechatronics and Automotive Applications by P. Lino, G. Maione, pp. 267-292). Additionally, it is expected to publish a manuscript including the latest results in the performance increasement of an advanced common rail compressed natural gas engine in an international peer-reviewed journal relevant to the control engineering community.

D13 – Approximation of derivatives and integrals of fractional orders by new numerical and analytical methods; Fractionized models (expected by September 2019)

Mainly the members and their activities done within the Working Group 1 contribute to this deliverable. This deliverable is understood to be delivered.

Numerous novel approaches for the numerical handling of fractional operators have been proposed recently by research groups within and outside of the COST action. In a detailed discussion of some of these methods doi: 10.3390/math8030324, a significant number of problems have been identified and improvements or alternatives that avoid these issues have been suggested. Furthermore, a new class of numerical methods has been proposed. These novel methods differ in their structure and in the interpretation of their respective parameters greatly from the traditional approaches. Therefore a comparison of the performance is difficult and appropriate concepts for such a sensible comparison need to be designed. This work is currently ongoing. We expect to have publishable results within the two years after the end of the action. The outcome will then be published in suitable international peer reviewed journals. Moreover, a first guide to the evaluation of fractional integrals and derivatives (according to different definitions) of the main elementary functions, has been provided, doi: 10.3390/math7050407, thus filling a gap in the scientific literature on the subject.

Regarding the fractionized models, their efficient applications in tumor-growth, tumor-immune surveillance and epidemiology were studied, see e.g.  doi: 10.1063/1.5096159, doi: 10.11121/ijocta.01.2020.00862, or doi: 10.1016/j.chaos.2021.110654.

D14 – Increased approximation accuracy of fractional Laplacian operator for analogue circuit design, discrete rational approximations (expected by September 2019)

Mainly the members and their activities done within the Working Group 2 contribute to this deliverable. This deliverable is understood to be delivered.

One of the contributions to this deliverable is in the comprehensive analysis of the Oustaloup approximation and defining the equations to determine the initial parameters of this approximation technique to obtain a response that satisfies the designers’ requirements of approximation error in magnitude and/or phase in a specific frequency range for the minimal possible order N of the approximation as presented in doi 10.1109/ICUMT.2018.8631227.

Other results relevant to this deliverable were published in doi: 10.1109/CoDIT.2019.8820521. A link was established between the Lagrange’s continued fraction expansion (CFE) and two other CFEs introduced for approximating the fractional Laplacian operator s^ν, with 0 < ν < 1, and their discrete realizations. On their turn, the two novel CFEs are linked with each other. Zeros and poles of these new approximations alternate on the negative real half-axis of the s-plane (for analog realizations) and on the real segment inside the unit circle of the z-plane (for discrete realizations). Discrete approximations the fractional operator have poles and zeros enjoying a nice symmetrical distribution on the z-plane, namely with respect to the origin of the z-plane. These properties are obtained for any order of realization (i.e. degree of numerator and denominator of the approximation, or number of zeros and poles).

An indirect approach in two steps was proposed in doi: 10.1109/SMC.2019.8914260 to obtain discrete rational transfer functions (TFs) for implementing the fractional-order Tustin operator (FTO). The polynomial coefficients of the rational discrete TF approximation of the FTO are given by closed-form expressions. In this way, an easy computation is possible, which is a remarkable new feature. The proposed coefficients expressions are the basis for proving the zero-pole interlacing of the discrete FTO. The interlaced zero-pole pattern shows a symmetrical configuration on the z-plane.

D15 – Contribution of each working group to the proceedings of the Year 3 Workshop co-located with the Action meeting (expected by September 2019)

The members from all Working Groups have contributed the Annual Workshop to share their experience and achievements. This deliverable is understood to be delivered.

The Annual Workshop 2019 was organized in Ghent by the Belgium representative of the Action, Dr Dana Copot, at Ghent University. Following the tasks and expected deliverables described within the individual Working Groups, 14 speakers from the member countries presented the progress in analysis and design of optimized fractional-order function blocks and system control and discussed their results and future plans. The topics covered the areas from mathematical description to practical utilization in analysis, modelling, classification and/or control of different applications. Newly, the possible usage of fractional approach in cryptography was presented by Fatih Ozkaynak from University of Firat, Turkey. The fractional chaotic systems provide higher entropy and are suitable for more robust cryptography protocols. For more information, please visit Annual Workshop 2019.

D16 – Presentation of new fractional-order elements based on the IMPC, graphene and RC-EDP design approach (expected by June 2020)

Mainly the members and their activities done within the Working Group 2 contribute to this deliverable. This deliverable is understood to be delivered.

The hardware design of fractional-order elements based on the Resistive-Capacitive Elements with Distributed Parameters (RC-EDP) was performed by implementing thick-film technology based fractional-order elements. A software design tool was developed by prof. Ushakov and presented at doi: 10.1109/ECCTD.2017.8093314 and later further described in detail doi: 10.1016/j.jare.2020.06.021.

 The limitations of assumed thick-film technology process were investigated and layout optimization recommendations described. Following these optimization steps the solid-state capacitive FOEs were produced and are available as utility samples; capFOE_045 and capFOE_05.

 Next to the investigation of solid-state FOE design based on the RC-EDP theory, with respect to the realization of analogue hardware devices, with intrinsic, fractional-order structure, another two different design technologies have been analyzed. 

The first approach 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. The studies on the nanocomposite characterization include the Fourier-transform infrared (FTIR) spectroscopy spectra, the Raman spectra, the X-ray powder diffraction (XRD) spectrum, the transmission electron microscopy (TEM), and the scanning electron microscopy (SEM) images, while in, doi: 10.1016/j.mejo.2018.10.008, the possibility of realizing fractional capacitors by using carbon black nanostructured dielectrics was investigated. Capacitors have been realized by varying the percentage of distributed carbon black. The frequency analysis of the capacitors has been, therefore, performed. The Bode diagrams outline that this class of devices shows a non integer order behavior. Moreover, a dependance between the curing temperature and the fractional order has been shown.

In the next approach,  doi: 10.1016/j.aeue.2019.152927,  the team proposes and demonstrates the possibility of using Bacterial Cellulose (BC), a bio-derived polymer, for the realization of fractional-order electronic devices. BC, 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. BC 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. A model is proposed and a possible explanation of the involved phenomena is provided.

D17 – Development of methods for PID controllers’ synthesis, Matlab toolboxes and discretization algorithms (expected by July 2020)

Mainly the members and their activities done within the Working Group 1 contribute to this deliverable. This deliverable is understood to be delivered.

Numerical methods for the solution of fractional differential equations, in particular for multi-order systems, i.e. systems in which each differential equation has a different order,  and multi-term equations (when in the same equation there are several fractional derivatives) has been discussed, doi: 10.3390/math6020016 and  doi: 10.3390/math8030324. As a result of this investigation a set of robust Matlab codes have been released. These are general purposes codes, with a similar usage to those of other built-in Matlab codes for classical ordinary differential equations, and their use is therefore very simple and possible also by users with no particular experience in numerical analysis.

Other results contributing to this deliverable deal with the investigation of multi-loop control structures. The multi-loop control structures 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 structure may deal with the design tradeoff between set-point and disturbance rejection performances for low time-delay systems. This approach is researched in a fruitful collaboration with Dr. Aleksei Tepljakov and Prof. Eduard Petlenkov from Tallinn University. 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. The Matlab code delivery was shared in the Mathworks (https://www.mathworks.com/matlabcentral/fileexchange/88823-multi-loop-model-reference-pid-control).

In recent papers, see 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. The design of the controllers’ parameters was made by a generalized particle swarm optimization, which was applied to the controllers in two nested loops of the electrical drive. 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. Design formulas provide the controllers’ parameters as a function of frequency-domain specifications. A detailed simulation model allows to verify that better performance is achieved with respect to integer-order controllers, even in presence of disturbances and plant nonlinearities.

A paper doi: 10.1016/j.ifacol.2020.12.2050, investigated the use of a fractional-order lag network or a fractional-order PI controller for the motion control of three revolute joints of a manipulator. The introduced fractional compensators are designed by using the symmetrical optimum principle and by parameters optimization or by frequency-domain loop shaping, respectively. In both cases, the same performance and robustness specifications were considered. Simulation results and frequency response show effectiveness and robustness of the approach.

Finally, another activity (see doi: 10.2298/TAM201203016L) supporting this deliverable addressed the problem of finite-time stability for uncertain neutral nonhomogeneous fractional-order systems with time-varying delays. A robust finite-time stability test procedure based on the extended form of the generalized Grönwall inequality was suggested. The sufficient condition for robust finite-time stability of such systems was established. Numerical examples show the effectiveness of the procedure.

D18 – Application of FO linear and non-linear blocks in sensors, actuators and control systems (expected by July 2020)

Mainly the members and their activities done within the Working Group 3 contribute to this deliverable. This deliverable is understood to be delivered.

The activities on this deliverable were initiated by utilizing fractional-order function blocks in the PID control: see e.g. doi: 10.1142/S0218126618501761, doi: 10.1016/j.ifacol.2018.06.151, or do: 10.1109/TSP.2018.8441247.

Fractional-order PID controllers and fractional controllers with various distributed orders were further designed, realized, and applied to control electrical drives and robots, in several applications – see doi: 10.1016/j.ejcon.2020.06.005, doi: 10.2298/TAM201203016L, doi: 10.1109/JAS.2017.7510325, doi: 10.1007/978-3-030-17344-9_11, doi: 10.1109/SMC.2019.8914031, doi: 10.1016/j.ifacol.2018.06.154, whereas detail description of the PID controllers and their purpose and application is part of Deliverable D17.

D19 – Modelling and control of injection systems; Schemes for analysing propagation of electro-magnetic fields in biological tissues; Report on correlation analysis and corresponding software (expected by July 2020)

Mainly the members and their activities done within the Working Group 4 contribute to this deliverable. This deliverable is understood to be delivered.

The contribution to accurate modelling of the electro-injectors that are used in common rail injection systems of Diesel engines was made and presented e.g. in doi: 10.1016/j.ifacol.2016.08.071. The model takes into account the fuel properties, the nonlinear dynamics of the fuel flow, the electro-hydraulic elements and the mechanical components subject to displacement and deformation. But it also considers fractional-order representation of the high-pressure fuel propagation inside a peculiar annular pipe of the electro-injectors, which is given by fractional-order differential equations. To this aim, partial differential equations with fractional-order time derivatives are obtained by starting from the conventional integer-order continuity and momentum equations. It is demonstrated that conservation laws are not violated, thanks to a physical interpretation since the injector is not a closed system hence it loses energy (fractional mass conservation) and on the usage of fractional viscosity (for a fractional momentum balance). The absence of analytical solutions in closed form pushes us to adopt a numerical procedure to solve and simulate the system of time–fractional PDEs by time and space discretization. The specific modelling and control results related to this new approach were presented in a paper submitted to an international peer-reviewed journal. The new model shows a better prediction capability than a rigid body model, which is based on assuming that some relevant coupled mechanical elements behave as rigid bodies, and a nominal model, which uses nominal values of the parameters that are here fixed by conventional expressions but that can be subject to optimization. Model-based simulation shows the improvement in prediction by the help of real data.

D20 – Contribution of each working group to the proceedings of the Year 4 Workshop co-located with the Action meeting (expected by September 2020)

The members from all Working Groups have contributed the Annual Workshop to share their experience and achievements. This deliverable is understood to be delivered.

The final Annual Workshop 2020 was postponed to March 2021 as based on our request the COST Action period was extended by 6 months due to the COVID-19 pandemic situation. Even if originally planned to be organized as face-2-face in Brno Czech Republic, it was necessary to organize the final Annual Workshop online using the MS Teams platform due to lasting COVID-19 situation. With the end of the COST Action, there were over 50 participants, whereas only 8 active speakers gave their presentations. Even if the numner of active speakers was lower compated to previsous Annual Workshop, within each presentation there was a fruitful discussion about the recent results and mainly their new research objectives and goals. For more informatation, please visit final Annual Workshop 2021.

D21 – Finalization of the book summarizing Action activities and scientific results achieved during its four years span (expected by September 2020)

The members from all Working Groups have contributed the Action Book. Currently, it is not delivered but delivery is foreseen within 2 years of the end of the Action (to be specific by August 2021).

The Action Book started to be prepared in the year 2019 by collecting the inputs from the COST Action participants and currently is being finalized within the cooperation with MC members/substitutes/observers and other participants being supported from the COST Action CA15225 and contributing to the Action’s Objectives.

The final version of the Action Book will be made available online and free to download from the COST Action webpage.

Last update: May 5, 2021