\r\n\tThe use of dyes and pigments is narrowly associated with everyday life. Since ancient times, people have been using various types of dyes and pigments for both aesthetic and practical applications. Typically, the coloration of various materials e.g. textiles, clay, plastics, etc. has been their main purpose. Yet, the scope of contemporary dyes and pigments has become significantly broader and there is constant interest in new products fulfilling numerous requirements parallel to their ability to act as colorants. This trend has led to the development of functional dyes.
\r\n\r\n\tIn recent years, novel dyes and pigments with hi-tech applications have been developed and there is a continuous demand for new products with better properties and/or broader application scope. Of particular interest is the development of dyes and pigments with environment-responsive aptitudes i.e. products that can undergo some structural modification as a result of external stimuli e.g. light, heat, pressure, pH-changes, etc. These stimuli-responsive functional dyes have in turn found application in sensor technologies, optical data storage, molecular switches, etc. Acknowledging these facts, this book aims to cover current state-of-the-art research and development in the remarkably important area of environment-responsive (multi)functional dyes and pigments.
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Kononowicz and Inga Hege",authors:[null]},{id:"10053",title:"Data Warehouse Technology and Application in Data Centre Design for E-government",slug:"data-warehouse-technology-and-application-in-data-centre-design-for-e-government",signatures:"Xuanzi Hu",authors:[null]},{id:"10060",title:"The Emergence of the Intelligent Government in the Second Society",slug:"the-emergence-of-the-intelligent-government-in-the-second-society",signatures:"Dennis de Kool and Johan van Wamelen",authors:[null]},{id:"10071",title:"A Lightweight SOA-based Collaboration Framework for European Public Sector",slug:"a-lightweight-soa-based-collaboration-framework-for-european-public-sector",signatures:"Adomas Svirskas, Jelena Isacenkova and Refik Molva",authors:[null]},{id:"10054",title:"Spatial Aided Decision-making System for E-Government",slug:"spatial-aided-decision-making-system-for-e-government",signatures:"Liang Wang, Rong Zhao, Bin Li, Jiping Liu and Qingpu Zhang",authors:[null]},{id:"10072",title:"Evaluating Local E-Government: a Comparative Study of Greek Prefecture Websites",slug:"evaluating-local-e-government-a-comparative-study-of-greek-prefecture-websites",signatures:"Prodromos Yannas and Georgios Lappas",authors:[null]}]}]},onlineFirst:{chapter:{type:"chapter",id:"74594",title:"Variable, Fractional-Order PID Controller Synthesis Novelty Method",doi:"10.5772/intechopen.95232",slug:"variable-fractional-order-pid-controller-synthesis-novelty-method",body:'A continuous-time proportional–integral–derivative controller (PID controller) [1] invented almost 100 years ago is one of the most widely applied controllers in the closed-loop systems [2] with many industrial applications [3, 4, 5]. Currently the continuous-time control is successively replaced by discrete-time one in which the integration is replaced by a summation and differentiation by a difference evaluation. So, in the discrete PID controller the classical integral is replaced by a sum and the derivative by a backward difference, [6]. The discrete controller’s PID algorithm is mainly realized by micro-controllers [7].
At 70s of the 20-th Century the Fractional Calculus [8] with a great success started a considerable attention in mathematics and engineering [9, 10, 11, 12]. Now, the fractional-order backward-difference (FOBD) and the fractional-order backward sum (FOBS) [6, 13] are applied in the dynamical system modeling [14] and discrete control algorithms. The continuous-time FOPID controllers are more difficult in a practical realization [15, 16, 17, 18].
There are numerous continuous and discrete-time PID and FOPID controller synthesis methods [16, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31]. One should mention that the optimisation of the closed-loop system in this case is more complicated because of the controller optimization. Apart from the three classical gains there are two additional parameters, namely, a fractional order of differentiation and summation [32]. The FOPID control characterizes by slow achieving the steady state and growing calculation “tail” [12].
In the paper a novelty variable, the fractional-order PID (VFOPID) [6, 28, 33, 34, 35, 36, 37, 38, 39, 40, 41] controller synthesis is proposed. It consists of dividing the closed-loop system discrete-transient time division into the finite time intervals over which are defined fractional orders summation and differentiation functions. The main idea is that for the final infinite interval
Fractional-orders systems are characterized by the so called system “memory”. This, in practice, means that in every step the FOPID controller computes its output signals taking into account step-by-step linearly computed number of samples. This causes in practice the micro-controllers realization problems. It is known as “Finite memory principle” [12].
The paper is organized as follows. In Section 2 the basic information related to the fractional calculus and variable, fractional order Grünwald-Letnikov backward difference is given. The main result of the paper includes Section 3. It contains the proposed VFOPID controller synthesis method with the proposal of the order functions form. The brief description of the controller parameters evaluation algorithm is given. The investigations are supported by a numerical example presented in Chapter 4.
In the paper the following notation will be used.
In general, a fractional-order functions will be denoted by Greek letters
Definition 2.1. For
One should mention that function (1) for
The above function will be named as: the “oblivion function” or “decay function”.
Next one defines the Grünwald–Letnikov variable, fractional-order backward difference (VFOBD). For a discrete-variable bounded real-valued function
Definition 2.2. The VFOBD with an order function
Relating to (2) as the first special case of the defined above VFOBD and a constant order function
Equality (3) is valid for
where
On the base of the Grünwald-Letnikov variable, fractional-order linear time-invariant backward-difference the difference Eqs. (GL-VFOBE) for
where
According to the notation (5) Eq. (7) takes the form
The vector
and the combined Eq. (8) is of the form
or after simple transformation
To preserve the VFOBDE order one assumes that
In the transfer functions defined by the one-sided
Defining matrices
one gets
Under assumption (12)
Denoting
one gets similar to the transfer function description
or for simplicity
Remark 2.1. Though the relation (19) looks similar to the classical discrete transfer function it is different by the real discrete variables. It relates discrete SISO systems by vectors and matrices related to its dimensions
One considers a closed-loop system illustrated in Figure 1. Where a plant is described by (19) where
Closed-loop system.
The classical PID controller output is desribed by three terms
and in the convention proposed above as
which may be expressed as
where
Remark 2.2. The plant may be described by classical integer order, fractional or even variable, fractional - order difference equations. The matrix - vector description used makes it possible.
one gets a VFOPID controller transfer function-like description
To simplify the description one assumes a sensor matrix as
The closed-loop system is presented in Figure 1 from which one gets the following relations
where
A system error is evaluated by the formula
In the synthesis of the classical PID controller there are three parameters to evaluate. Namely,
In the rejection of the external disturbation one can assume that
Usually the sensor matrix
The optimal parameters (29) are evaluated due to the assumed optymality criterion. The most popular is so called ISE one (Integral of the Squared Error) or in the discrete-system case: Sum of the Squared Error (SSE).
Substitution of (31) into (32) gives
In the proposed VFOPID controller synthesis method with partially intuitive and supported by closed-loop systems synthesis experience the classical optimisation due to the performance criterion (32) is performed. The pre-defined differentiation and integration order functions orders are as follows
and
Every function
In the classical closed-loop system with PID controller there is introduced the integration part preserving the steady - state error signal tending to zero. So, in (38) there is a constant order
Now, for initially assumed order functions one applies the following algorithm based on well known Gauss method.
Chose a starting set of coefficients
Applying the classical Gauss algorithm find a minimal SSE performance index value alongside the first variable (eg. K_P),
Repeat step 2 for the next parameter,
If the SSE value is satisfactory stop else return to step 2.
Remark 3.1. Algorithm described above can be applied also to the classical discrete PID controller with three parameters.
One considers a closed-loop system depicted in Figure 1.
A plant is described by a transfer function
where
The plant is discretized with the sampling time
and controller gains
Hence, there are 5 parameters to evaluate. Due to the performance index (33) the optimal parameters are as follows
The VFOPID controller order functions are plotted in Figure 2 whereas the PID and VFOPID controllers unite step responses are given in Figure 3.
VFOPID controller order functions: νkh (in black) and μkh (in red).
VFOPID (in red) and IOPID (in black) controller unit step response.
The achieved VFOPID controller synthesis result is compared with the classical discrete-time PID controller optimized due to criterion (30). The optimal parameters are
Figure 4 contains the closed - loop systems with PID (in blue) and VFOPID (in red) controllers unit step responses. There is included a plant unit step response of the plant (in black.)
The closed-loop system response with the VFOPIS (in red) and IOPID controllers (in blue).
In Figure 5 the controlling signals are presented (PID - in black, VFOPID - in red). The controlling signals have typical shapes: first differentiation action and finally the classical integration preserving zero steady - state closed - loop system error.
The closed-loop controlling signals.
Remark 4.1. In the Numerical example proposed here the VFOPID and the classical PID controllers maximal control signal values are the same reaching assumed bounding value
Remark 4.2. In the Numerical example
One should emphasize that the proposed solution of the VFOPID controller do not guarantee the absolute optimum of the closed-loop control system synthesis. It proves that the proposal of a physically realizable VFOPID controller by micro-controller (with finite memory) leads to better results due to the assumed performance criterion.
The main idea of the proposed method is to assume a priori the order functions with unknown parameters. In the VFOPID controller synthesis essential is an assumption that the summation order equals 1 One can express the action as the assumption of skeleton order functions with unknown parameters evaluated further in an SSE optimization algorithm.
Here, it is worth mentioning that there are still open problems of the VFOPID controllers tuning.
One should define a program evaluating the order functions.
For evaluation of the VFOPID controller parameters one can apply another optimization methods. It seems that optimization methods based on the artificial intelligence will be very effective.
Another performance index may be applied. Some penalty functions may be introduced to SSE as well a term taking into account the minimal value of the error signal.
The work was supported by the National Science Center Poland by Grant no. 2016/23/B/ ST7/03686.
PID | proportional-integral-derivative controller |
FOBD | fractional-order backward difference |
FOBS | fractional-order backward sum |
FOPID | fractional-Order proportional, fractional-order integral and differential controller |
VFOPID | variable, fractional-order PID controller |
SSE | squared sum of the error |
Authors are listed below with their open access chapters linked via author name:
",metaTitle:"IntechOpen authors on the Global Highly Cited Researchers 2018 list",metaDescription:null,metaKeywords:null,canonicalURL:null,contentRaw:'[{"type":"htmlEditorComponent","content":"New for 2018 (alphabetically by surname).
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\\n\\nGang Chen 2016-18
\\n\\nJunhong Chen 2017, 2018
\\n\\nZhigang Chen 2016, 2018
\\n\\nMyung-Haing Cho 2016, 2018
\\n\\nMark Connors 2015-18
\\n\\nCyrus Cooper 2017, 2018
\\n\\nLiming Dai 2015-18
\\n\\nWeihua Deng 2017, 2018
\\n\\nVincenzo Fogliano 2017, 2018
\\n\\nRon de Graaf 2014-18
\\n\\nHarald Haas 2017, 2018
\\n\\nFrancisco Herrera 2017, 2018
\\n\\nJaakko Kangasjärvi 2015-18
\\n\\nHamid Reza Karimi 2016-18
\\n\\nJunji Kido 2014-18
\\n\\nJose Luiszamorano 2015-18
\\n\\nYiqi Luo 2016-18
\\n\\nJoachim Maier 2014-18
\\n\\nAndrea Natale 2017, 2018
\\n\\nAlberto Mantovani 2014-18
\\n\\nMarjan Mernik 2017, 2018
\\n\\nSandra Orchard 2014, 2016-18
\\n\\nMohamed Oukka 2016-18
\\n\\nBiswajeet Pradhan 2016-18
\\n\\nDirk Raes 2017, 2018
\\n\\nUlrike Ravens-Sieberer 2016-18
\\n\\nYexiang Tong 2017, 2018
\\n\\nJim Van Os 2015-18
\\n\\nLong Wang 2017, 2018
\\n\\nFei Wei 2016-18
\\n\\nIoannis Xenarios 2017, 2018
\\n\\nQi Xie 2016-18
\\n\\nXin-She Yang 2017, 2018
\\n\\nYulong Yin 2015, 2017, 2018
\\n"}]'},components:[{type:"htmlEditorComponent",content:'New for 2018 (alphabetically by surname).
\n\n\n\n\n\n\n\n\n\nJocelyn Chanussot (chapter to be published soon...)
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\n\nAbdul Latif Ahmad 2016-18
\n\nKhalil Amine 2017, 2018
\n\nEwan Birney 2015-18
\n\nFrede Blaabjerg 2015-18
\n\nGang Chen 2016-18
\n\nJunhong Chen 2017, 2018
\n\nZhigang Chen 2016, 2018
\n\nMyung-Haing Cho 2016, 2018
\n\nMark Connors 2015-18
\n\nCyrus Cooper 2017, 2018
\n\nLiming Dai 2015-18
\n\nWeihua Deng 2017, 2018
\n\nVincenzo Fogliano 2017, 2018
\n\nRon de Graaf 2014-18
\n\nHarald Haas 2017, 2018
\n\nFrancisco Herrera 2017, 2018
\n\nJaakko Kangasjärvi 2015-18
\n\nHamid Reza Karimi 2016-18
\n\nJunji Kido 2014-18
\n\nJose Luiszamorano 2015-18
\n\nYiqi Luo 2016-18
\n\nJoachim Maier 2014-18
\n\nAndrea Natale 2017, 2018
\n\nAlberto Mantovani 2014-18
\n\nMarjan Mernik 2017, 2018
\n\nSandra Orchard 2014, 2016-18
\n\nMohamed Oukka 2016-18
\n\nBiswajeet Pradhan 2016-18
\n\nDirk Raes 2017, 2018
\n\nUlrike Ravens-Sieberer 2016-18
\n\nYexiang Tong 2017, 2018
\n\nJim Van Os 2015-18
\n\nLong Wang 2017, 2018
\n\nFei Wei 2016-18
\n\nIoannis Xenarios 2017, 2018
\n\nQi Xie 2016-18
\n\nXin-She Yang 2017, 2018
\n\nYulong Yin 2015, 2017, 2018
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USA, CRC Press Taylor & Francis, Asia Pacific, Trans Tech Publications Ltd., Switzerland, and Materials Science Forum, USA. He is a member of various editorial boards serving as associate editor for journals such as Environmental Chemistry Letter, Applied Water Science, Euro-Mediterranean Journal for Environmental Integration, Springer-Nature, Scientific Reports-Nature, and the editor of Eurasian Journal of Analytical Chemistry.",institutionString:"King Abdulaziz University",institution:{name:"King Abdulaziz University",country:{name:"Saudi Arabia"}}},{id:"99002",title:"Dr.",name:null,middleName:null,surname:"Koontongkaew",slug:"koontongkaew",fullName:"Koontongkaew",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"Thammasat University",country:{name:"Thailand"}}},{id:"156647",title:"Dr.",name:"A K M Mamunur",middleName:null,surname:"Rashid",slug:"a-k-m-mamunur-rashid",fullName:"A K M Mamunur Rashid",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:"MBBS, DCH, MD(Paed.), Grad. Cert. P. Rheum.(UWA, Australia), FRCP(Edin.)",institutionString:null,institution:{name:"Khulna Medical College",country:{name:"Bangladesh"}}},{id:"234696",title:"Prof.",name:"A K M Mominul",middleName:null,surname:"Islam",slug:"a-k-m-mominul-islam",fullName:"A K M Mominul Islam",position:null,profilePictureURL:"https://intech-files.s3.amazonaws.com/a043Y00000cA8dpQAC/Co2_Profile_Picture-1588761796759",biography:"Prof. Dr. A. K. M. Mominul Islam received both of his bachelor's and Master’s degree from Bangladesh Agricultural University. After that, he joined as Lecturer of Agronomy at Bangladesh Agricultural University (BAU), Mymensingh, Bangladesh, and became Professor in the same department of the university. Dr. Islam did his second Master’s in Physical Land Resources from Ghent University, Belgium. He is currently serving as a postdoctoral researcher at the Department of Horticulture & Landscape Architecture at Purdue University, USA. Dr. Islam has obtained his Ph.D. degree in Plant Allelopathy from The United Graduate School of Agricultural Sciences, Ehime University, Japan. The dissertation title of Dr. Islam was “Allelopathy of five Lamiaceae medicinal plant species”. Dr. Islam is the author of 38 articles published in nationally and internationally reputed journals, 1 book chapter, and 3 books. He is a member of the editorial board and referee of several national and international journals. He is supervising the research of MS and Ph.D. students in areas of Agronomy. Prof. Islam is conducting research on crop management, bio-herbicides, and allelopathy.",institutionString:"Bangladesh Agricultural University",institution:{name:"Bangladesh Agricultural University",country:{name:"Bangladesh"}}},{id:"214531",title:"Mr.",name:"A T M Sakiur",middleName:null,surname:"Rahman",slug:"a-t-m-sakiur-rahman",fullName:"A T M Sakiur Rahman",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"University of Rajshahi",country:{name:"Bangladesh"}}},{id:"66545",title:"Dr.",name:"A. F.",middleName:null,surname:"Omar",slug:"a.-f.-omar",fullName:"A. F. Omar",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:"Dr. A. F. Omar obtained\nhis Bachelor degree in electrical and\nelectronics engineering from Universiti\nSains Malaysia in 2002, Master of Science in electronics\nengineering from Open University\nMalaysia in 2008 and PhD in optical physics from Universiti\nSains Malaysia in 2012. His research mainly\nfocuses on the development of optical\nand electronics systems for spectroscopy\napplication in environmental monitoring,\nagriculture and dermatology. He has\nmore than 10 years of teaching\nexperience in subjects related to\nelectronics, mathematics and applied optics for\nuniversity students and industrial engineers.",institutionString:null,institution:{name:"Universiti Sains Malaysia",country:{name:"Malaysia"}}},{id:"191072",title:"Prof.",name:"A. K. M. Aminul",middleName:null,surname:"Islam",slug:"a.-k.-m.-aminul-islam",fullName:"A. K. M. Aminul Islam",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/191072/images/system/191072.jpg",biography:"Prof. Dr. A. K. M. Aminul Islam received both of his bachelor and Master’s degree from Bangladesh Agricultural University. After that he joined as Lecturer of Genetics and Plant Breeding at Bangabandhu Sheikh Mujibur Rahman Agricultural University (BSMRAU), Gazipur, Bangladesh and became Professor in the same department of the university. He is currently serving as Director (Research) of Bangabandhu Sheikh Mujibur Rahman Agricultural University (BSMRAU), Gazipur, Bangladesh. Dr. Islam has obtained his Ph D degree in Chemical and Process Engineering from Universiti Kebangsaan Malaysia. The dissertation title of Dr. Islam was “Improvement of Biodiesel Production through Genetic Studies of Jatropha (Jatropha curcas L.)”. Dr. Islam is the author of 98 articles published in nationally and internationally reputed journals, 11 book chapters and 3 books. He is a member of editorial board and referee of several national and international journals. He is also serving as the General Secretary of Plant Breeding and Genetics Society of Bangladesh, Seminar and research Secretary of JICA Alumni Association of Bangladesh and member of several professional societies. Prof. Islam acted as Principal Breeder in the releasing system of BU Hybrid Lau 1, BU Lau 1, BU Capsicum 1, BU Lalshak 1, BU Baromashi Seem 1, BU Sheem 1, BU Sheem 2, BU Sheem 3 and BU Sheem 4. He supervised 50 MS and 3 Ph D students. Prof. Islam currently supervising research of 5 MS and 3 Ph D students in areas Plant Breeding & Seed Technologies. Conducting research on development of hybrid vegetables, hybrid Brassica napus using CMS system, renewable energy research with Jatropha curcas.",institutionString:"Bangabandhu Sheikh Mujibur Rahman Agricultural University",institution:{name:"Bangabandhu Sheikh Mujibur Rahman Agricultural University",country:{name:"Bangladesh"}}},{id:"322225",title:"Dr.",name:"A. K. M. Aminul",middleName:null,surname:"Islam",slug:"a.-k.-m.-aminul-islam",fullName:"A. K. M. Aminul Islam",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/no_image.jpg",biography:"Prof. Dr. A. K. M. Aminul Islam received both of his bachelor's and Master’s degree from Bangladesh Agricultural University. After that he joined as Lecturer of Genetics and Plant Breeding at Bangabandhu Sheikh Mujibur Rahman Agricultural University (BSMRAU), Gazipur, Bangladesh, and became Professor in the same department of the university. He is currently serving as Director (Research) of Bangabandhu Sheikh Mujibur Rahman Agricultural University (BSMRAU), Gazipur, Bangladesh. Dr. Islam has obtained his Ph.D. degree in Chemical and Process Engineering from Universiti Kebangsaan Malaysia. The dissertation title of Dr. Islam was 'Improvement of Biodiesel Production through Genetic Studies of Jatropha (Jatropha curcas L.)”. Dr. Islam is the author of 99 articles published in nationally and internationally reputed journals, 11 book chapters, 3 books, and 20 proceedings and conference paper. He is a member of the editorial board and referee of several national and international journals. He is also serving as the General Secretary of Plant Breeding and Genetics Society of Bangladesh, Seminar, and research Secretary of JICA Alumni Association of Bangladesh and a member of several professional societies. Prof. Islam acted as Principal Breeder in the releasing system of BU Hybrid Lau 1, BU Lau 1, BU Capsicum 1, BU Lalshak 1, BU Baromashi Seem 1, BU Sheem 1, BU Sheem 2, BU Sheem 3 and BU Sheem 4. He supervised 50 MS and 3 PhD students. Prof. Islam currently supervising the research of 5 MS and 3 PhD students in areas Plant Breeding & Seed Technologies. Conducting research on the development of hybrid vegetables, hybrid Brassica napus using CMS system, renewable energy research with Jatropha curcas.",institutionString:"Bangabandhu Sheikh Mujibur Rahman Agricultural University",institution:{name:"Bangabandhu Sheikh Mujibur Rahman Agricultural University",country:{name:"Bangladesh"}}},{id:"91977",title:"Dr.",name:"A.B.M. Sharif",middleName:null,surname:"Hossain",slug:"a.b.m.-sharif-hossain",fullName:"A.B.M. Sharif Hossain",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"University of Malaya",country:{name:"Malaysia"}}},{id:"97123",title:"Prof.",name:"A.M.M.",middleName:null,surname:"Sharif Ullah",slug:"a.m.m.-sharif-ullah",fullName:"A.M.M. Sharif Ullah",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/97123/images/4209_n.jpg",biography:"AMM Sharif Ullah is currently an Associate Professor of Design and Manufacturing in Department of Mechanical Engineering at Kitami Institute of Technology, Japan. He received the Bachelor of Science Degree in Mechanical Engineering in 1992 from the Bangladesh University of Engineering and Technology, Dhaka, Bangladesh. In 1993, he moved to Japan for graduate studies. He received the Master of Engineering degree in 1996 from the Kansai University Graduate School of Engineering in Mechanical Engineering (Major: Manufacturing Engineering). He also received the Doctor of Engineering degree from the same institute in the same field in 1999. He began his academic career in 2000 as an Assistant Professor in the Industrial Systems Engineering Program at the Asian Institute of Technology, Thailand, as an Assistant Professor in the Industrial Systems Engineering Program. 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