\r\n\tIn sum, the book presents a reflective analysis of the pedagogical hubs for a changing world, considering the most fundamental areas of the current contingencies in education.
",isbn:"978-1-83968-793-8",printIsbn:"978-1-83968-792-1",pdfIsbn:"978-1-83968-794-5",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"b01f9136149277b7e4cbc1e52bce78ec",bookSignature:"Dr. María Jose Hernandez-Serrano",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10229.jpg",keywords:"Teacher Digital Competences, Flipped Learning, Online Resources Design, Neuroscientific Literacy (Myths), Emotions and Learning, Multisensory Stimulation, Citizen Skills, Violence Prevention, Moral Development, Universal Design for Learning, Sensitizing on Diversity, Supportive Strategies",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"September 14th 2020",dateEndSecondStepPublish:"October 12th 2020",dateEndThirdStepPublish:"December 11th 2020",dateEndFourthStepPublish:"March 1st 2021",dateEndFifthStepPublish:"April 30th 2021",remainingDaysToSecondStep:"3 months",secondStepPassed:!0,currentStepOfPublishingProcess:4,editedByType:null,kuFlag:!1,biosketch:"Dr. Phil. Maria Jose Hernandez Serrano is a tenured lecturer in the Department of Theory and History of Education at the University of Salamanca, where she currently teaches on Teacher Education. She graduated in Social Education (2000) and Psycho-Pedagogy (2003) at the University of Salamanca. Then, she obtained her European Ph.D. in Education and Training in Virtual Environments by research with the University of Manchester, UK (2009).",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"187893",title:"Dr.",name:"María Jose",middleName:null,surname:"Hernandez-Serrano",slug:"maria-jose-hernandez-serrano",fullName:"María Jose Hernandez-Serrano",profilePictureURL:"https://mts.intechopen.com/storage/users/187893/images/system/187893.jpg",biography:"DPhil Maria Jose Hernandez Serrano is a tenured Lecturer in the Department of Theory and History of Education at the University of Salamanca (Spain), where she currently teaches on Teacher Education. She graduated in Social Education (2000) and Psycho-Pedagogy (2003) at the University of Salamanca. Then, she obtained her European Ph.D. on Education and Training in Virtual Environments by research with the University of Manchester, UK (2009). She obtained a Visiting Scholar Postdoctoral Grant (of the British Academy, UK) at the Oxford Internet Institute of the University of Oxford (2011) and was granted with a postdoctoral research (in 2021) at London Birbeck University.\n \nShe is author of more than 20 research papers, and more than 35 book chapters (H Index 10). She is interested in the study of the educational process and the analysis of cognitive and affective processes in the context of neuroeducation and neurotechnologies, along with the study of social contingencies affecting the educational institutions and requiring new skills for educators.\n\nHer publications are mainly of the educational process mediated by technologies and digital competences. Currently, her new research interests are: the transdisciplinary application of the brain-based research to the educational context and virtual environments, and the neuropedagogical implications of the technologies on the development of the brain in younger students. Also, she is interested in the promotion of creative and critical uses of digital technologies, the emerging uses of social media and transmedia, and the informal learning through technologies.\n\nShe is a member of several research Networks and Scientific Committees in international journals on Educational Technologies and Educommunication, and collaborates as a reviewer in several prestigious journals (see public profile in Publons).\n\nUntil March 2010 she was in charge of the Adult University of Salamanca, by coordinating teaching activities of more than a thousand adult students. She currently is, since 2014, the Secretary of the Department of Theory and History of Education. Since 2015 she collaborates with the Council Educational Program by training teachers and families in the translation of advances from educational neuroscience.",institutionString:"University of Salamanca",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"0",totalChapterViews:"0",totalEditedBooks:"0",institution:{name:"University of Salamanca",institutionURL:null,country:{name:"Spain"}}}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"23",title:"Social Sciences",slug:"social-sciences"}],chapters:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"301331",firstName:"Mia",lastName:"Vulovic",middleName:null,title:"Mrs.",imageUrl:"https://mts.intechopen.com/storage/users/301331/images/8498_n.jpg",email:"mia.v@intechopen.com",biography:"As an Author Service Manager, my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. Whether that be identifying an exceptional author and proposing an editorship collaboration, or contacting researchers who would like the opportunity to work with IntechOpen, I establish and help manage author and editor acquisition and contact."}},relatedBooks:[{type:"book",id:"6942",title:"Global Social Work",subtitle:"Cutting Edge Issues and Critical Reflections",isOpenForSubmission:!1,hash:"222c8a66edfc7a4a6537af7565bcb3de",slug:"global-social-work-cutting-edge-issues-and-critical-reflections",bookSignature:"Bala Raju Nikku",coverURL:"https://cdn.intechopen.com/books/images_new/6942.jpg",editedByType:"Edited by",editors:[{id:"263576",title:"Dr.",name:"Bala",surname:"Nikku",slug:"bala-nikku",fullName:"Bala Nikku"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"1591",title:"Infrared Spectroscopy",subtitle:"Materials Science, Engineering and Technology",isOpenForSubmission:!1,hash:"99b4b7b71a8caeb693ed762b40b017f4",slug:"infrared-spectroscopy-materials-science-engineering-and-technology",bookSignature:"Theophile Theophanides",coverURL:"https://cdn.intechopen.com/books/images_new/1591.jpg",editedByType:"Edited by",editors:[{id:"37194",title:"Dr.",name:"Theophanides",surname:"Theophile",slug:"theophanides-theophile",fullName:"Theophanides Theophile"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3092",title:"Anopheles mosquitoes",subtitle:"New insights into malaria vectors",isOpenForSubmission:!1,hash:"c9e622485316d5e296288bf24d2b0d64",slug:"anopheles-mosquitoes-new-insights-into-malaria-vectors",bookSignature:"Sylvie Manguin",coverURL:"https://cdn.intechopen.com/books/images_new/3092.jpg",editedByType:"Edited by",editors:[{id:"50017",title:"Prof.",name:"Sylvie",surname:"Manguin",slug:"sylvie-manguin",fullName:"Sylvie Manguin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3161",title:"Frontiers in Guided Wave Optics and Optoelectronics",subtitle:null,isOpenForSubmission:!1,hash:"deb44e9c99f82bbce1083abea743146c",slug:"frontiers-in-guided-wave-optics-and-optoelectronics",bookSignature:"Bishnu Pal",coverURL:"https://cdn.intechopen.com/books/images_new/3161.jpg",editedByType:"Edited by",editors:[{id:"4782",title:"Prof.",name:"Bishnu",surname:"Pal",slug:"bishnu-pal",fullName:"Bishnu Pal"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"72",title:"Ionic Liquids",subtitle:"Theory, Properties, New Approaches",isOpenForSubmission:!1,hash:"d94ffa3cfa10505e3b1d676d46fcd3f5",slug:"ionic-liquids-theory-properties-new-approaches",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/72.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"1373",title:"Ionic Liquids",subtitle:"Applications and Perspectives",isOpenForSubmission:!1,hash:"5e9ae5ae9167cde4b344e499a792c41c",slug:"ionic-liquids-applications-and-perspectives",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/1373.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"57",title:"Physics and Applications of Graphene",subtitle:"Experiments",isOpenForSubmission:!1,hash:"0e6622a71cf4f02f45bfdd5691e1189a",slug:"physics-and-applications-of-graphene-experiments",bookSignature:"Sergey Mikhailov",coverURL:"https://cdn.intechopen.com/books/images_new/57.jpg",editedByType:"Edited by",editors:[{id:"16042",title:"Dr.",name:"Sergey",surname:"Mikhailov",slug:"sergey-mikhailov",fullName:"Sergey Mikhailov"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"371",title:"Abiotic Stress in Plants",subtitle:"Mechanisms and Adaptations",isOpenForSubmission:!1,hash:"588466f487e307619849d72389178a74",slug:"abiotic-stress-in-plants-mechanisms-and-adaptations",bookSignature:"Arun Shanker and B. 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\n\t\t\t
1. Introduction
\n\t\t\t
More than 25 years after model predictive control (MPC) or receding horizon control (RHC) appeared in industry as an effective tool to deal with multivariable constrained control problems, a theoretical basis for this technique has started to emerge, see [1-3] for reviews of results in this area.
\n\t\t\t
The focus of this chapter is on MPC of constrained dynamic systems, both linear and nonlinear, to illuminate the ability of MPC to handle constraints that makes it so attractive to industry. We first give an overview of the origin of MPC and introduce the definitions, characteristics, mathematical formulation and properties underlying the MPC. Furthermore, MPC methods for linear or nonlinear systems are developed by assuming that the plant under control is described by a discrete-time one. Although continuous-time representation would be more natural, since the plant model is usually derived by resorting to first principles equations, it results in a more difficult development of the MPC control law, since it in principle calls for the solution of a functional optimization problem. As a matter of fact, the performance index to be minimized is defined in a continuous-time setting and the overall optimization procedure is assumed to be continuously repeated after any vanishingly small sampling time, which often turns out to be a computationally intractable task. On the contrary, MPC algorithms based on discrete-time system representation are computationally simpler. The system to be controlled which usually described, or approximated by an ordinary differential equation is usually modeled by a difference equation in the MPC literature since the control is normally piecewise constant. Hence, we concentrate our attention from now onwards on results related to discrete-time systems.
\n\t\t\t
By and large, the main disadvantage of the MPC is that it cannot be able of explicitly dealing with plant model uncertainties. For confronting such problems, several robust model predictive control (RMPC) techniques have been developed in recent decades. We review different RMPC methods which are employed widely and mention the advantages and disadvantages of these methods. The basic idea of each method and some method applications are stated as well.
\n\t\t\t
Most MPC strategies consider hard constraints, and a number of RMPC techniques exist to handle uncertainty. However model and measurement uncertainties are often stochastic, and therefore RMPC can be conservative since it ignores information on the probabilistic distribution of the uncertainty. It is possible to adopt a stochastic uncertainty description (instead of a set-based description) and develop a stochastic MPC (SMPC) algorithm. Some of the recent advances in this area are reviewed.
\n\t\t\t
In recent years, there has been much interest in networked control systems (NCSs), that is, control systems close via possibly shared communication links with delay/bandwidth constraints. The main advantages of NCSs are low cost, simple installation and maintenance, and potentially high reliability. However, the use of the network will lead to intermittent losses or delays of the communicated information. These losses will tend to deteriorate the performance and may even cause the system to become unstable. MPC framework is particularly appropriate for controlling systems subject to data losses because the actuator can profit from the predicted evolution of the system. In section 7, results from our recent research are summarized. We propose a new networked control scheme, which can overcome the effects caused by the network delay.
\n\t\t\t
At the beginning of research on NCSs, more attention was paid on single plant through network. Recently, fruitful research results on multi-plant, especially, on multi-agent networked control systems have been obtained. MPC lends itself as a natural control framework to deal with the design of coordinated, distributed control systems because it can account for the action of other actuators while computing the control action of a given set of actuators in real-time. In section 8, a number of distributed control architectures for interconnected systems are reviewed. Attention is focused on the design approaches based on model predictive control.
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\n\t\t
\n\t\t\t
2. Model Predictive Control
\n\t\t\t
Model predictive control is a form of control scheme in which the current control action is obtained by solving, at each sampling instant, a finite horizon open-loop optimal control problem, using the current state of the plant as the initial state. Then the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant. This is the main difference from conventional control which uses a pre-computed control law.
\n\t\t\t
\n\t\t\t\t
2.1. Characteristics of MPC
\n\t\t\t\t
MPC is by now a mature technology. It is fair to mention that it is the standard approach for implementing constrained, multivariable control in the process industries today. Furthermore, MPC can handle control problems where off-line computation of a control law is difficult or impossible.
\n\t\t\t\t
Specifically, an important characteristic of this type of control is its ability to cope with hard constraints on controls and states. Nearly every application imposes constraints. For instance, actuators are naturally limited in the force (or equivalent) they can apply, safety limits states such as temperature, pressure and velocity, and efficiency, which often dictates steady-state operation close to the boundary of the set of permissible states. In this regard, MPC is one of few methods having applied utility, and this fact makes it an important tool for the control engineer, particularly in the process industries where plants being controlled are sufficiently slow to permit its implementation.
\n\t\t\t\t
In addition, another important characteristic of MPC is its ability to handle control problems where off-line computation of a control law is difficult or impossible. Examples where MPC may be advantageously employed include unconstrained nonlinear plants, for which on-line computation of a control law usually requires the plant dynamics to possess a special structure, and time-varying plants.
\n\t\t\t\t
A fairly complete discussion of several design techniques based on MPC and their relative merits and demerits can be found in the review article by [4].
\n\t\t\t
\n\t\t\t
\n\t\t\t\t
2.2. Essence of MPC
\n\t\t\t\t
As mentioned in the excellent review article [2], MPC is not a new method of control design. Rather, it essentially solves standard optimal control problems which is required to have a finite horizon in contrast to the infinite horizon usually employed in \n\t\t\t\t\t\t\n \n \n H\n 2\n \n \n \n\n\n\t\t\t\t\tand \n\t\t\t\t\t\t\n \n \n H\n ∞\n \n \n \n\n\t\t\t\t\tlinear optimal control. Where it differs from other controllers is that it solves the optimal control problem on-line for the current state of the plant, rather than providing the optimal control for all states, that is determining a feedback policy on-line.
\n\t\t\t\t
The on-line solution is to solve an open-loop optimal control problem where the initial state is the current state of the system being controlled which is just a mathematical programming problem. However, determining the feedback solution, requires solution of Hamilton-Jacobi-Bellman (Dynamic Programming) differential or difference equation, a vastly more difficult task (except in those cases, such as \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tlinear optimal control, where the value function can be finitely parameterized). From this point of view, MPC differs from other control methods merely in its implementation. The requirement that the open-loop optimal control problem be solvable in a reasonable time (compared with plant dynamics) necessitates, however, the use of a finite horizon and this raises interesting problems.
\n\t\t\t
\n\t\t
\n\t\t
\n\t\t\t
3. Linear Model Predictive Control
\n\t\t\t
MPC has become an attractive feedback strategy, especially for linear processes. By now, linear MPC theory is quite mature. The issues of feasibility of the on-line optimization, stability and performance are largely understood for systems described by linear models.
\n\t\t\t
\n\t\t\t\t
3.1. Mathematical formulation
\n\t\t\t\t
The idea of MPC is not limited to a particular system description, but the computation and implementation depend on the model representation. Depending on the context, we will readily switch between state space, transfer matrix and convolution type models [4]. In addition, nowadays in the research literature, MPC is formulated almost always in the state space. We will assume the system to be described in state space by a linear discrete-time model.
where \n\t\t\t\t\t\t\n \n x(k)∈\n ℝ\n n\n \n \n \n\n\t\t\t\t\tis the state vector at time\n\t\t\t\t\t\t\n k\n\n\t\t\t\t\t, and \n\t\t\t\t\t\t\n \n u(k)∈\n ℝ\n r\n \n \n \n\n\t\t\t\t\t is the vector of manipulated variables to be determined by the controller. The control and state sequences must satisfy\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t∈\n\t\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t∈\n\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. Usually, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\n\t\t\t\t\tis a convex, compact subset of \n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tℝ\n\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and \n X\n convex, closed subset of\n\t\t\t\t\t\t\n \n \n \n ℝ\n ^\n \n {n}\n \n MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiabl2riHoaaCaaaleqabaacbaGaa8NxaaaakmaacmaabaGaamOBaaGaay5Eaiaaw2haaaaa@3B9A@\n \n\n\t\t\t\t\t, each set containing the origin in its interior.
\n\t\t\t\t
The control objective is usually to steer the state to the origin or to an equilibrium state \n\t\t\t\t\t\t\n \n \n x\n r\n \n \n \n\n\n\t\t\t\t\t for which the output \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ty\n\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis the constant reference. A suitable change of coordinates reduces the second problem to the first which, therefore, we consider in the sequel. Assuming that a full measurement of the state \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is available at the current time\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\n\t\t\t\t\t. Then for event \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t(i.e. for state \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t at time\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\n\t\t\t\t\t), a receding horizon implementation is typically formulated by introducing the following open-loop optimization problem.
\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t≥\n\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\twhere\n\t\t\t\t\t\n\t\t\t\t\t\tp\n\t\t\t\t\t\n\t\t\t\t denotes the length of the prediction horizon or output horizon, and \n\t\t\t\t\t\n\t\t\t\t\t\tm\n\t\t\t\t\t\n\t\t\t\t denotes the length of the control horizon or input horizon. (When, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\twe refer to this as the infinite horizon problem, and similarly, when\n\t\t\t\t\t\n\t\t\t\t\t\tp\n\t\t\t\t\t\n\t\t\t\t is finite, we refer to it as a finite horizon problem). For the problem to be meaningful we assume that the origin \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t is in the interior of the feasible region.
Several choices of the objective function \n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t in the optimization eq.(2) have been reported in [4-7] and have been compared in [8]. In this Chapter, we consider the following quadratic objective
\n\t\t\t\t
\n\n\t\t\t\t\t\n\t\t\t\tE4
\n\t\t\t\t
where \n\t\t\t\t\t\n \n \n u(·):=\n \n [u\n \n (k)\n \n T\n \n ,...,u\n \n (k+m−1|k)\n \n T\n \n ]\n \n T\n \n \n \n MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaaaapeGaamyDaiaacIcacaGG3cGaaiykaiaacQdacqGH9aqpcaGGBbGaamyDaiaacIcacaWGRbGaaiyka8aadaahaaWcbeqaa8qacaWGubaaaOGaaiilaiaac6cacaGGUaGaaiOlaiaacYcacaWG1bGaaiikaiaadUgacqGHRaWkcaWGTbGaeyOeI0IaaGymaiaacYhacaWGRbGaaiyka8aadaahaaWcbeqaa8qacaWGubaaaOGaaiyxa8aadaahaaWcbeqaa8qacaWGubaaaaaa@5005@\n \n\n\n\t\n\t\t\t\t\tis the sequence of manipulated variables to be optimized; \n\t\t\t\t\t\t\n \n x(k+i|k),i=1,2,...,p\n \n\n\t\t\t\t\tdenote the state prediction generated by the nominal model (1) on the basis of the state informations at time \n\t\t\t\t\t\t\n k\n\n\t\t\t\t\tunder the action of the control sequence \n\t\t\t\t\t\t\n \n \n u(k),u(k+1|k),...,u(k+i−1|k);\n \n MathType@MTEF@5@5@+=feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr0=vqpWqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaabaaaaaaaaapeGaamyDaiaacIcacaWGRbGaaiykaiaacYcacaaMc8UaaGPaVlaadwhacaGGOaGaam4AaiabgUcaRiaaigdacaGG8bGaam4AaiaacMcacaGGSaGaaiOlaiaac6cacaGGUaGaaiilaiaadwhacaGGOaGaam4AaiabgUcaRiaadMgacqGHsislcaaIXaGaaiiFaiaadUgacaGGPaGaai4oaaaa@50C1@\n \n\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n \n \n P\n 0\n \n ,\n \n\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\tand\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\n\t\t\t\t\t are strictly positive definite symmetric weighting matrices. Let \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t...\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t be the minimizing control sequence for \n \n \n J\n \n (p,m)\n \n \n (x(k))\n \n\n subject to the system dynamics (1) and the constraint (3), and \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t be the optimizing value function.
\n\t\t\t\t
A receding horizon policy proceeds by implementing only the first control \n\t\t\t\t\t\t\n\n \n \n u\n \n (p,m)\n \n *\n \n (k|k)\n \n\n\t\t\t\t\t to obtain\n\t\t\t\t\t\t\n \n x(k+1)=Ax(k)+B\n u\n \n (p,m)\n \n *\n \n (k|k)\n \n\n\t\t\t\t\t. The rest of the control sequence \n\t\t\t\t\t\t\n \n \n u\n \n (p,m)\n \n *\n \n (i|k),i=k+1,...,k+m−1\n \n\n\t\t\t\t\t is discarded and \n\t\t\t\t\t\t\n \n x(k+1)\n \n\n\t\t\t\t\t is used to update the optimization problem (2) as a new initial condition. This process is repeated, each time using only the first control action to obtain a new initial condition, then shifting the cost ahead one time step and repeating. This is the reason why MPC is also sometimes referred to as receding horizon control (RHC) or moving horizon control (MHC). The purpose of taking new measurements at each time step is to compensate for unmeasured disturbances and model inaccuracy, both of which cause the system output to be different from the one predicted by the model. Fig.1 presents a conceptual picture of MPC.
When is the problem formulated above feasible, so that the algorithm yields a control action which can be implemented?
When does the sequence of computed control actions lead to a system which is closed-loop stable?
What closed-loop performance results from repeated solution of the specified open-loop optimal control problem?
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These questions will be explained in the following sections.
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3.2. Feasibility
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The constraints stipulated in (3) may render the optimization problem infeasible. It may happen, for example, because of a disturbance, that the optimization problem posed above becomes infeasible at a particular time step. It may also happen, that the algorithm which minimizes an open-loop objective, inadvertently drives the closed-loop system outside the feasible region.
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3.3. Closed loop stability
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In either the infinite or the finite horizon constrained case it is not clear under what conditions the closed loop system is stable. Much research on linear MPC has focused on this problem. Two approaches have been proposed to guarantee stability: one based on the original problem (1), (2), and (3) and the other where a contraction constraint is added [9, 10]. With the contraction constraint the norm of the state is forced to decrease with time and stability follows trivially independent of the various parameters in the objective function. Without the contraction constraint the stability problem is more complicated.
\n\t\t\t\t
General proofs of stability for constrained MPC based on the monotonicity property of the value function have been proposed by [11] and [12]. The most comprehensive and also most compact analysis has been presented by [13] and [14] whose arguments we will sketch here.
\n\t\t\t\t
To simplify the exposition we assume\n\t\t\t\t\t\t\n \n p=m=N\n \n\n\n\t\t\t\t\t, then \n\t\t\t\t\t\t\n \n \n J\n \n (p,m)\n \n \n =\n J\n N\n \n \n \n\n\n\t\t\t\t\tas defined in eq.(2). The key idea is to use the optimal finite horizon cost\n\t\t\t\t\t\t\n \n \n J\n N\n *\n \n \n \n\n\t\t\t\t\t, the value function, as a Lyapunov function. One wishes to show that
If it can be shown that the right hand side of (6) is positive, then stability is proven. Due to \n\t\t\t\t\t\t\n \n Q>0\n \n\n\t\t\t\t\t and\n\t\t\t\t\t\t\n \n R>0\n \n\n\t\t\t\t\t, the first term \n\t\t\t\t\t\t\n\n \n [\n x\n T\n \n (k)Qx(k)+\n u\n *\n \n \n \n T\n \n (x(k))R\n u\n *\n \n (x(k))]\n \n\n\t\t\t\t\t is positive. In general, it cannot be asserted that the second term \n\t\t\t\t\t\t\n \n [\n J\n \n N−1\n \n *\n \n (x(k+1))−\n J\n N\n *\n \n (x(k+1))]\n \n\n\n\t\t\t\t\t is nonnegative.
\n\t\t\t\t
Several approaches have been presented to assure that the right hand side of (6) is positive, please refer to [1]. The various constraints introduced to guarantee stability (end constraint for all states, end constraint for unstable modes, terminal region, etc.) may lead to feasibility problems. For instance, the terminal equality constraint may become infeasible unless a sufficiently large horizon is used.
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3.4. Open-loop performance objective versus closed loop performance
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In receding horizon control only the first of the computed control moves is implemented, and the remaining ones are discarded. Therefore the sequence of actually implemented control moves may differ significantly from the sequence of control moves calculated at a particular time step. Consequently the finite horizon objective which is minimized may have only a tentative connection with the value of the objective function as it is obtained when the control moves are implemented.
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4. Nonlinear model predictive control
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Linear MPC has been developed, in our opinion, to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control. While linear model predictive control has been popular since the 70s of the past century, the 90s have witnessed a steadily increasing attention from control theorists as well as control practitioners in the area of nonlinear model predictive control (NMPC).
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The practical interest is driven by the fact that many systems are in general inherently nonlinear and today’s processes need to be operated under tighter performance specifications. At the same time more and more constraints for example from environmental and safety considerations, need to be satisfied. In these cases, linear models are often inadequate to describe the process dynamics and nonlinear models have to be used. This motivates the use of nonlinear model predictive control.
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The system to be controlled is described, or approximated by a discrete-time model
where \n\t\t\t\t\t\n \n f(·)\n \n\n\t\t\t\t is implicitly defined by the originating differential equation that has an equilibrium point at the origin (i.e.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t). The control and state sequences must satisfy (3).
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4.1. Difficulties of NMPC
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The same receding horizon idea which we discussed in section 3 is also the principle underlying nonlinear MPC, with the exception that the model describing the process dynamics is nonlinear. Contrary to the linear case, however, feasibility and the possible mismatch between the open-loop performance objective and the actual closed loop performance are largely unresolved research issues in nonlinear MPC. An additional difficulty is that the optimization problems to be solved on line are generally nonlinear programs without any redeeming features, which implies that convergence to a global optimum cannot be assured. For the quadratic programs arising in the linear case this is guaranteed. As most proofs of stability for constrained MPC are based on the monotonicity property of the value function, global optimality is usually not required, as long as the cost attained at the minimizer decreases (which is usually the case, especially when the optimization algorithm is initialized from the previous shifted optimal sequence). However, although stability is not altered by local minimum, performance clearly deteriorates.
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The next section focuses on system theoretical aspects of NMPC. Especially the question on closed-loop stability is considered.
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4.2. Closed-loop stability
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One of the key questions in NMPC is certainly, whether a finite horizon NMPC strategy does lead to stability of the closed-loop or not. Here only the key ideas are reviewed and no detailed proofs are given. Furthermore, notice that we will not cover all existing NMPC approaches, instead we refer the reader to the overview papers [2, 15, 16]. For all the following sections it is assumed that the prediction horizon is set equal to the control horizon, that is,\n\t\t\t\t\t\t\n \n p=m\n \n\n\t\t\t\t\t.
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4.2.1. Infinite horizon NMPC
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As pointed out, the key problem with a finite prediction and control horizon stems from the fact that the predicted open and the resulting closed-loop behavior is in general different. The most intuitive way to achieve stability is the use of an infinite horizon cost [17, 18], that is, \n\t\t\t\t\t\t\t\n p\n\n\t\t\t\t\t\tin (4) is set to\n\t\t\t\t\t\t\t\n\n ∞\n\n\t\t\t\t\t\t. As mentioned in [19], in the nominal case, feasibility at one sampling instance also implies feasibility and optimality at the next sampling instance. This follows from Bellman’s Principle of Optimality [20], that is the input and state trajectories computed as the solution of the NMPC optimization problem (2), (3) and (7) at a specific instance in time, are in fact equal to the closed-loop trajectories of the nonlinear system, i.e. the remaining parts of the trajectories after one sampling instance are the optimal solution at the next sampling instance. This fact also implies closed-loop stability. When the system is both infinite-horizon and constrained, [21] considered this case for T-S fuzzy systems with PDC law and non-PDC law. New sufficient conditions were proposed in terms of LMIs. Both the corresponding PDC and non-PDC state-feedback controllers were designed, which could guarantee that the resulting closed-loop fuzzy system be asymptotically stable. In addition, the feedback controllers would meet the specifications for the fuzzy systems with input or output constraints.
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4.2.2. Finite horizon NMPC
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In this section, different possibilities to achieve closed-loop stability for NMPC using a finite horizon length have been proposed. Just as outlined for the linear case, in the proof the value function is employed as a Lyapunov function. A global optimum must be found at each time step to guarantee stability. As mentioned above, when the horizon is infinity, feasibility at a particular time step implies feasibility at all future time steps. Unfortunately, contrary to the linear case, the infinite horizon problem cannot be solved numerically.
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Most of approaches modify the NMPC setup such that stability of the closed-loop can be guaranteed independently of the plant and performance specifications. This is usually achieved by adding suitable equality or inequality constraints and suitable additional penalty terms to the objective functional. These additional constraints are usually not motivated by physical restrictions or desired performance requirements but have the sole purpose to enforce stability of the closed-loop. Therefore, they are usually termed stability constraints.
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Terminal equality constraint. The simplest possibility to enforce stability with a finite prediction horizon is to add a so called zero terminal equality constraint at the end of the prediction horizon [17, 23, 28], i.e. to add the equality constraint \n\t\t\t\t\t\t\t\n \n x(k+p|k)=0\n \n\n\t\t\t\t\t\tto the optimization problem (2), (3) and (7). This leads to stability of the closed-loop, if the optimal control problem possesses a solution at\n\t\t\t\t\t\t\t\n k\n\n\n\t\t\t\t\t\t, since the feasibility at one time instance does also lead to feasibility at the following time instances and a decrease in the value function.
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The first proposal for this form of model predictive control for time-varying, constrained, nonlinear, discrete-time systems was made by [17]. This paper is particularly important, because it provides a definitive stability analysis of this version of discrete-time receding horizon control (under mild conditions of controllability and observability) and shows the value function \n\t\t\t\t\t\t\t\n \n \n J\n N\n *\n \n \n \n\n\t\t\t\t\t\t associated with the finite horizon optimal control problem approaches that of the infinite horizon problem as the horizon approaches infinity. This paper remains a key reference on the stabilizing properties of model predictive control and subsumes much of the later literature on discrete-time MPC that uses a terminal equality constraint.
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In fact, the main advantages of this version are the straightforward application and the conceptual simplicity. On the other hand, one disadvantage of a zero terminal constraint is that the system must be brought to the origin in finite time. This leads in general to feasibility problems for short prediction/control horizon lengths, i.e. a small region of attraction. From a computational point of view, the optimization problem with terminal constraint can be solved in principle, but equality constraints are computationally very expensive and can only be met asymptotically [24]. In addition, one cannot guarantee convergence to a feasible solution even when a feasible solution exists, a discomforting fact. Furthermore, specifying a terminal constraint which is not met in actual operation is always somewhat artificial and may lead to aggressive behavior. Finally, to reduce the complexity of the optimization problem it is desirable to keep the control horizon small, or, more generally, characterize the control input sequence with a small number of parameters. However, a small number of degrees of freedom may lead to quite a gap between the open-loop performance objective and the actual closed loop performance.
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Terminal constraint set and terminal cost function. Many schemes have been proposed [24, 26-28, 30, 31, 36, 37], to try to overcome the use of a zero terminal constraint. Most of them either use the so called terminal region constraint
and/or a terminal penalty term \n \n E(x(k+p|k))\n \n which is added to the cost functional. Note that the terminal penalty term is not a performance specification that can be chosen freely. Rather \n\t\t\t\t\t\t\t\n\n E\n\n\t\t\t\t\t\t and the terminal region \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\tin (8) are determined off-line such that stability is enforced.
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\n\t\t\t\t\t\t\t\tTerminal constraint set. In this version of model predictive control, \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tis a subset of \n\t\t\t\t\t\t\t\t\t\n X\n\n\t\t\t\t\t\t\t\t containing a neighborhood of the origin. The purpose of the model predictive controller is to steer the state to \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t in finite time. Inside\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t, a local stabilizing controller \n\t\t\t\t\t\t\t\t\t\n \n \n k\n f\n \n (·)\n \n\n\t\t\t\t\t\t\t\t is employed. This form of model predictive control is therefore sometimes referred to as dual mode, and was proposed. Fixed horizon versions for constrained, nonlinear, discrete-time systems are proposed in [32] and [33].
\n\t\t\t\t\t\t\t\tTerminal cost function. One of the earliest proposals for modifying (2), (3) and (7) to ensure closed-loop stability was the addition of a terminal cost. In this version of model predictive control, the terminal cost \n\t\t\t\t\t\t\t\t\t\n \n E(·)\n \n\n\t\t\t\t\t\t\t\t is nontrivial and there is no terminal constraint so that\n\t\t\t\t\t\t\t\t\t\n \n \n X\n f\n \n =\n ℝ\n n\n \n \n \n\n\n\t\t\t\t\t\t\t\t. The proposal [34] was made in the context of predictive control of unconstrained linear system. Can this technique for achieving stability (by adding only a terminal cost) be successfully employed for constrained and/or nonlinear systems? From the literature the answer may appear affirmative. However, in this literature there is an implicit requirement that \n\t\t\t\t\t\t\t\t\t\n \n x(k+p|k)∈\n X\n f\n \n \n \n\n\t\t\t\t\t\t\t\tis satisfied for every initial state \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t in a given compact set, and this is automatically satisfied if \n\t\t\t\t\t\t\t\t\t\n N\n\n\t\t\t\t\t\t\t\t is chosen sufficiently large. The constraint \n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t∈\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tthen need not be included explicitly in the optimal control problem actually solved on-line. Whether this type of model predictive control is regarded as having only a terminal cost or having both a terminal cost and a terminal constraint is a matter of definition. We prefer to consider it as belonging to the latter category as the constraint is necessary even though it is automatically satisfied if \n\t\t\t\t\t\t\t\t\t\n N\n\n\t\t\t\t\t\t\t\t is chosen sufficiently large.
\n\t\t\t\t\t
Terminal cost and constraint set. Most recent model predictive controllers belong to this category. There are a variety of good reasons for incorporating both a terminal cost and a terminal constraint set in the optimal control problem. Ideally, the terminal cost \n\t\t\t\t\t\t\t\n \n E(·)\n \n\n\t\t\t\t\t\t should be the infinite horizon value function \n\t\t\t\t\t\t\t\n \n \n J\n ∞\n *\n \n (·)\n \n\n\t\t\t\t\t\t if this were the case, then\n\t\t\t\t\t\t\t\n \n \n J\n N\n *\n \n (·)=\n J\n ∞\n *\n \n (·)\n \n\n\t\t\t\t\t\t, on-line optimization would be unnecessary, and the known advantages of an infinite horizon, such as stability and robustness, would automatically accrue. Nonlinearity and/or constraints render this impossible, but it is possible to choose \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t·\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t so that it is exactly or approximately equal to \n\t\t\t\t\t\t\t\n \n \n J\n ∞\n *\n \n (·)\n \n\n\t\t\t\t\t\t in a suitable neighborhood of the origin. Choosing \n\t\t\t\t\t\t\t\n \n \n X\n f\n \n \n \n\n\t\t\t\t\t\t to be an appropriate subset of this neighborhood yields many advantages and motivates the choice of \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t·\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t and \n\t\t\t\t\t\t\t\n \n \n X\n f\n \n \n \n\n\n\t\t\t\t\t\t in most of the examples of this form of model predictive control.
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For the case when the system is nonlinear but there are no state or control constraints, [35] use a stabilizing local control law\n\t\t\t\t\t\t\t\n \n \n k\n f\n \n (·)\n \n\n\n\t\t\t\t\t\t, a terminal cost function \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t·\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t that is a (local) Lyapunov function for the stabilized system, and a terminal constraint set \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t that is a level set of \n\t\t\t\t\t\t\t\n \n E(·)\n \n\n\t\t\t\t\t\t and is positively invariant for the system\n\t\t\t\t\t\t\t\n\n \n x(k+1)=f(x(k),\n k\n f\n \n (x(k)))\n \n\n\t\t\t\t\t\t. The terminal constraint is omitted from the optimization problem solved on-line, but it is nevertheless shown that this constraint is automatically satisfied for all initial states in a level set of\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t\t\t\tN\n\t\t\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t·\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t. The resultant closed-loop system is asymptotically (or exponentially) stabilizing with a region of attraction that is this level set of\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t\t\t\tN\n\t\t\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t·\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t.
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When the system is both nonlinear and constrained, \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t·\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\tand \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\tinclude features from the example immediately above. In [36], \n\t\t\t\t\t\t\t\n \n \n k\n f\n \n (·)\n \n\n\t\t\t\t\t\t is chosen to stabilize the linearized system\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t, where \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\t:\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t and\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t:\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t. Then the author of [36] employs a non-quadratic terminal cost \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t·\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t and a terminal constraint set \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t that is positively invariant for the nonlinear system \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t and that satisfies \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t⊂\n\t\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t and\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t⊂\n\t\t\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t.
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Variable horizon/Hybrid model predictive control. These techniques were proposed by [37] and developed by [38] to deal with both the global optimality and the feasibility problems, which plague nonlinear MPC with a terminal constraint. Variable horizon MPC also employs a terminal constraint, but the time horizon at the end of which this constraint must be satisfied is itself an optimization variable. It is assumed that inside this region another controller is employed for which it is somehow known that it asymptotically stabilizes the system. Variable horizon has also been employed in contractive model predictive control (see the next section). With these modifications a global optimum is no longer needed and feasibility at a particular time step implies feasibility at all future time steps. The terminal constraint is somewhat less artificial here because it may be met in actual operation. However, a variable horizon is inconvenient to handle on-line, an exact end constraint is difficult to satisfy, and the exact determination of the terminal region is all but impossible except maybe for low order systems. In order to show that this region is invariant and that the system is asymptotically stable in this region, usually a global optimization problem needs to be solved.
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Contractive model predictive control. The idea of contractive MPC was mentioned by [39], the complete algorithm and stability proof were developed by [40]. In this approach a constraint is added to the usual formulation which forces the actual and not only the predicted state to contract at discrete intervals in the future. From this requirement a Lyapunov function can be constructed easily and stability can be established. The stability is independent of the objective function and the convergence of the optimization algorithm as long as a solution is found which satisfies the contraction constraint. The feasibility at future time steps is not necessarily guaranteed unless further assumptions are made. Because the contraction parameter implies a specific speed of convergence, its choice comes natural to the operating personnel.
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Model predictive control with linearization. All the methods discussed so far require a nonlinear program to be solved on-line at each time step. The effort varies somewhat because some methods require only that a feasible (and not necessarily optimal) solution be found or that only an improvement be achieved from time step to time step. Nevertheless the effort is usually formidable when compared to the linear case and stopping with a feasible rather than optimal solution can have unpredictable consequences for the performance. The computational effort can be greatly reduced when the system is linearized first in some manner and then the techniques developed for linear systems are employed on-line. Some approaches have been proposed.
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Linearization theory may, in some applications, be employed to transform the original nonlinear system, using state and feedback control transformations, into a linear system. Model predictive control may be applied to the transformed system [41, 42]. [42] applies first feedback linearization and then uses MPC in a cascade arrangement for the resulting linear system. The optimal control problem is not, however, transformed into a convex problem, because the transformed control and state constraint sets and the transformed cost are no longer necessarily convex. [43, 44] employ linear transformation (\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\tis replaced by\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t, where \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t:\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t is the re-parameterized control) to improve conditioning of the optimal control problem solved on-line.
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Conclusions. MPC for linear constrained systems has been shown to provide an excellent control solution both theoretically and practically. The incorporation of nonlinear models poses a much more challenging problem mainly because of computational and control theoretical difficulties, but also holds much promise for practical applications. In this section an overview over the stability analysis of NMPC is given. As outlined some of the challenges occurring in NMPC are already solvable. Nevertheless in the nonlinear area a variety of issues remain which are technically complex but have potentially significant practical implications for stability and performance.
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5. Robust model predictive control
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MPC is a class of model-based control theories that use linear or nonlinear process models to forecast system behavior. The success of the MPC control performance depends on the accuracy of the open loop predictions, which in turn depends on the accuracy of the process models. It is possible for the predicted trajectory to differ from the actual plant behavior [45]. Needless to say, such control systems that provide optimal performance for a particular model may perform very poorly when implemented on a physical system that is not exactly described by the model (see e.g. [46]).
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When we say that a control system is robust we mean that stability is maintained and that the performance specifications are met for a specified range of model variations (uncertainty range). To be meaningful, any statement about robustness of a particular control algorithm must make reference to a specific uncertainty range as well as specific stability and performance criteria.
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Predictive controllers that explicitly consider the process and model uncertainties, when determining the optimal control policies, are called robust predictive controllers. The main concept of such controllers is similar to the idea of \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tcontrollers and consists on the minimization of worst disturbance effect to the process behavior [47]. Several applications for the formulation of robust predictive control laws began to appear in the literature in the 1990s, focusing on both model uncertainties and disturbances.
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Although a rich theory has been developed for the robust control of linear systems, very little is known about the robust control of linear systems with constraints. Most studies on robustness consider unconstrained systems. According to the Lyapunov theory, we know that if a Lyapunov function for the nominal closed-loop system maintains its descent property if the disturbance (uncertainty) is sufficiently small, then stability is maintained in the presence of uncertainty. However, when constraints on states and controls are present, it is necessary to ensure, in addition, that disturbances do not cause transgression of the constraints. This adds an extra level of complexity.
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In this section, we review two robust model predictive control (RMPC) methods and mention the advantages and disadvantages of methods below. The basic idea of each method and some method applications are stated.
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5.1. Min-Max RMPC methods
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In the main stream robust control literature, “robust performance” is measured by determining the worst performance over the specified uncertainty range. In direct extension of this definition it is natural to set up a new RMPC objective where the control action is selected to minimize the worst value the objective function can attain as a function of the uncertain model parameters. This describes the first attempt toward a RMPC algorithm which was proposed by [48]. They showed that for FIR models the optimization problem which must be solved on-line at each time step is a linear program of moderate size with uncertain coefficients and an \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\n\t\t\t\t\t-norm objective function. Unfortunately, it is well known now that robust stability is not guaranteed with this algorithm [46].
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In literature [48], the Campo algorithm fails to address the fact that only the first element of the optimal input trajectory is implemented and the whole min-max optimization is repeated at the next time step with a feedback update. In the subsequent optimization, the worst-case parameter values may change because of the feedback’s update. In the case of a system with uncertainties, the open-loop optimal solution differs from the feedback optimal solution, thereby violating the basic premise behind MPC. This is why robust stability cannot be assured with the Campo algorithm.
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The literature [49] proposed the RMPC formulations which explicitly take into account uncertainties in the prediction model
where \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tw\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tq\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tw\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tw\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tq\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tw\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t
\n\t\t\t\tLet\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tw\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tbe modeled as unknown but bounded exogenous disturbances and parametric uncertainties and\n\t\t\t\t\t\n\t\t\t\t\t\tW\n\t\t\t\t\t\n\t\t\t\t, \n\t\t\t\t\t\n\t\t\t\t\t\tV\n\t\t\t\t\t\n\t\t\t\tbe polytopes respectively. A RMPC strategy often used is to solve a min-max problem that minimize the worst-case performance while enforcing input and state constraints for all possible disturbances. The following min-max control problem is referred as open-loop constrained robust optimal control problem (OL-CROC).
Other papers in the literature aim at explicitly or implicitly approximating the problem above by simplifying the objective and uncertainty description, and making the on-line effort more manageable, but still guaranteeing at least robust stability. For example, the authors of [50] use an \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\n\t\t\t\t\t-norm open-loop objective function and both assume FIR models with uncertain coefficients. A similar but more general technique has also been proposed for state-space systems with a bounded input matrix [51].
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The authors of [52] have defined a dynamic programming problem (thus accounting for feedback) to determine the control sequence minimizing the worst case cost. They show that with the horizon set to infinity this procedure guarantees robust stability. However, the approach suffers from the curse of dimensionality and the optimization problem at each stage of the dynamic program is non-convex. Thus, in its generality the method is unsuitable for on-line (or even off-line) use except for low order systems with simple uncertainty descriptions.
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These formulations may be conservative for certain problems leading to sluggish behavior because of three reasons. First of all, arbitrarily time-varying uncertain parameters are usually not a good description of the model uncertainty encountered in practice, where the parameters may be either constant or slowly varying but unknown. Second, the computationally simple open-loop formulations neglect the effect of feedback. Third, the worst-case error minimization itself may be a conservative formulation for most problems.
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The authors of [50, 53, 54] propose to optimize nominal rather than robust performance and to achieve robust stability by enforcing a robust contraction constraint, i.e.requiring the worst-case prediction of the state to contract. With this formulation robust global asymptotic stability can be guaranteed for a set of linear time-invariant stable systems. The optimization problem can be cast as a quadratic program of moderate size for a broad class of uncertainty descriptions.
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To account for the effect of feedback, the authors of [55] propose to calculate at each time step not a sequence of control moves but a state feedback gain matrix which is determined to minimize an upper bound on robust performance. For fairly general uncertainty descriptions, the optimization problem can be expressed as a set of linear matrix inequalities for which efficient solution techniques exist.
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5.2. LMI-based RMPC methods
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In the above method, a cost function is minimized considering the worst case into all the plants described by the uncertainties. Barriers of RMPC algorithms include: the computational cost, the applicability depending on the speed and size of the plant on which the control will act. In this section, we present one such MPC-based technique for the control of plants with uncertainties. This technique is motivated by developments in the theory and application (to control) of optimization involving linear matrix inequalities (LMIs) [56].
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In this regard, the authors in [55] used the formulation in LMIs to solve the optimization problem. The basic idea of LMIs is to interpret a control problem as a semi-definite programming (SDP), that is, an optimization problem with linear objective and positive-definite constraints involving symmetric matrices that are related to the decision variables.
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There are two reasons why LMI optimization is relevant to MPC. Firstly, LMI-based optimization problems can be solved in polynomial time, which means that they have low computational complexity. From a practical standpoint, there are effective and powerful algorithms for the solution of these problems, that is, algorithms that rapidly compute the global optimum, with non-heuristic stopping criteria. It is comparable to that required for the elevation of an analytical solution for a similar problem. Thus LMI optimization is well suited for on-line implementation, which is essential for MPC. Secondly, it is possible to recast much of existing robust control theory in the framework of LMIs [55].
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The implication is that we can devise an MPC scheme where, at each time instant, an LMI optimization problem (as opposed to conventional linear or quadratic programs) is solved that incorporates input/output constraints and a description of the plant uncertainty. What’s more, it can guarantee certain robustness properties.
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5.3. Our works
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In recent decades, many research results in the design of RMPC have appeared, see for examples, [55-62] and the references therein. The main drawback associated with the above-mentioned methods proposed in MPC is that a single Lyapunov matrix is used to guarantee the desired closed-loop multi-objective specifications. This must work for all matrices in the uncertain domain to ensure that the hard constraints on inputs and outputs are satisfied. This condition is generally conservative if used in time-invariant systems. Furthermore, the hard constraints on outputs of closed-loop systems cannot be transformed into a linear matrix inequality (LMI) form using the method proposed in [57, 58, 60].
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We present a multi-model paradigm for robust control. Underlying this paradigm is a linear time-varying (LTV) system.
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t∈\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tℝ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis the control input, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t∈\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tℝ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis the state of the plant and \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ty\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t∈\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tℝ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\t\ty\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis the plant output, and \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΩ\n\t\t\t\t\t\t\n\t\t\t\t\t is some pre-specified set.
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For polytopic systems, the set \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΩ\n\t\t\t\t\t\t\n\t\t\t\t\tis the polytope
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\to\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t devotes to the convex hull. In other words, if\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t∈\n\t\t\t\t\t\t\t\tΩ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, then, for some nonnegative\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t,...,\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t summing to one, we have
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t corresponds to the nominal LTI system. The system described in equation (11) subject to input and output constraints
In 2001, the authors of [63] firstly put forward the idea of using the parameter-dependent Lyapunov function to solve the problem of robust constrained MPC for linear continuous-time uncertain systems, and hereafter, this idea was applied to linear discrete-time uncertain systems in [64, 65].
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Inspired by above-mentioned work, we addressed the problem of robust constrained MPC based on parameter-dependent Lyapunov functions with polytopic-type uncertainties in [66]. The results are based on a new extended LMI characterization of the quadratic objective, with hard constraints on inputs and outputs. Sufficient conditions in LMI do not involve the product of the Lyapunov matrices and the system dynamic matrices. The state feedback control guarantees that the closed-loop system is robustly stable and the hard constraints on inputs and outputs are satisfied. The approach provides a way to reduce the conservativeness of the existing conditions by decoupling the control parameterization from the Lyapunov matrix. An example will be provided to illustrate the effectiveness of the techniques developed in [66]. As the method proposed in [55] is a special case of our results, the optimization problem should be feasible using the method proposed in our paper since it is solvable using the approach in [55]. However, the optimization may not have a solution by the result in [55], while it has a solution by our result.
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\n\t\t\t\t\tExample (Input and Output Constraints) Consider the linear discrete-time parameter uncertain system (11) with
It is shown that the optimization is infeasible with the method proposed in [55] without the constraints. However, taking output constraints with \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ty\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and input constraints with\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t0.8\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, and with uncertain parameters assumed to be\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\to\n\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t0.6\n\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\to\n\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t0.4\n\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, it is feasible using the method proposed in this paper. The simulation results are given in Figure 2 and Figure 3.
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5.4. Conclusion
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An overview of some methods on RMPC is presented. The methods are studied based on LMI and Min-Max. The basic idea and applications of methods are stated in each part. Advantages and disadvantages of methods are stated in this section too.
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6. Recent developments in stochastic MPC
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Despite the extensive literature that exists on predictive control and robustness to uncertainty, both multiplicative (e.g. parametric) and additive (e.g. exogenous), very little attention
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has been paid to the case of stochastic uncertainty. Although robust predictive control can handle constrained systems that are subject to stochastic uncertainty, it will propagate the effects of uncertainty over a prediction horizon which can be computationally expensive and conservative. Yet this situation arises naturally in many control applications. The aim of this section is to review some of the recent advances in stochastic model predictive control (SMPC).
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The basic SMPC problem is defined in Subsection 6.1. and a review of earlier work is given in Subsection 6.2.
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Figure 2.
States (\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t) (method in [66]).
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Figure 3.
Output \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ty\n\t\t\t\t\t\t\t\n\t\t\t\t\t\tand input \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t(method in [66]).
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t∈\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tℝ\n\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is the state, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t∈\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tℝ\n\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis the input and the disturbance wk are assumed to be independent and identically distributed (i.i.d.), with zero mean, known distribution, and
where\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tG\n\t\t\t\t\t\t\t\t∈\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tℝ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t\t\t\t×\n\t\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\t\t∈\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tℝ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t\t\t\t×\n\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tdenotes the \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\n\t\t\t\t\tth row of the identity matrix. This formulation covers the case of state only, input only and state/input constraints which can be probabilistic (soft) or deterministic (hard) since \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tcan be chosen for some or all\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\n\t\t\t\t\t. For each \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\n\t\t\t\t\t (17) can be invoked separately so that in this section is taken to be scalar
(where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\n\t\t\t\t\tdenotes expectation) and guarantees that the closed loop system is stable, while its state converges to a neighborhood of the origin subject to the constraint (17).
\n\t\t\t\t
As is common in the literature on probabilistic robustness (e.g. [67]), all stochastic uncertainties are assumed to have bounded support. Not only is this necessary for asserting feasibility and stability, but it matches the real world more closely than the mathematically convenient Gaussian assumption which permits \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tw\n\t\t\t\t\t\t\n\t\t\t\t\tto become arbitrarily large (albeit with small probability), since noise and disturbances derived from physical processes are finite.
\n\t\t\t
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6.2. Earlier work
\n\t\t\t\t
Stochastic predictive control (SMPC) is emerging as a research area of both practical and theoretical interest.
\n\t\t\t\t
MPC has proved successful because it attains approximate optimality in the presence of constraints. In addition, RMPC can maintain a satisfactory level of performance and guarantee constraint satisfaction when the system is subject to bounded uncertainty [2]. However, such an approach does not cater for the case in which model and measurement uncertainties are stochastic in nature, subject to some statistical regularity, and neither can it handle the case of random uncertainty whose distribution does not have finite support (e.g. normal distributions). Therefore RMPC can be conservative since it ignores information on the probabilistic distribution of the uncertainty.
\n\t\t\t\t
It is possible to adopt a stochastic uncertainty description (instead of a set-based description) and develop an MPC algorithm that minimizes the expected value of a cost function. In general, the same difficulties that plagued the set-based approach are encountered here. One notable exception is that, when the stochastic parameters are independent sequences, the true closed-loop optimal control problem can be solved analytically using dynamic programming [68]. In many cases, the expected error may be a more meaningful performance measure than the worst-case error.
\n\t\t\t\t
SMPC also derives from the fact that most real life applications are subject to stochastic uncertainty and have to obey constraints. However, not all constraints are hard (i.e. inviolable), and it may be possible to improve performance by tolerating violations of constraints providing that the frequency of violations remains within allowable limits, namely soft constraints (see e.g. [69, 70] or [71, 73]).
\n\t\t\t\t
These concerns are addressed by stochastic MPC. Early work [74] considered additive disturbances and ignored the presence of constraints. Later contributions [68, 75-78] took constraints into account, but suffered from either excessive computation or a high degree of conservativeness, or did not consider issues of closed loop stability/feasibility.
\n\t\t\t\t
An approach that arose in the context of sustainable development [70, 79] overcame some of these difficulties by using stochastic moving average models and equality stability constraints. This was extended to state space models with stochastic output maps and to inequality maps involving terminal invariant sets [81]. The restriction of model uncertainty to the output map was removed in [81], but the need to propagate the effects of uncertainty over the prediction horizon prevented the statement of results in respect of feasibility. [82] overcomes these issues through an augmented autonomous prediction formulation, and provides a method of handling probabilistic constraints and ensuring closed loop stability through the use of an extension of the concept of invariance, namely invariance with probability p.
\n\t\t\t\t
Recent work [83, 84] proposed SMPC algorithms that use probabilistic information on additive disturbances in order to minimize the expected value of a predicted cost subject to hard and soft (probabilistic) constraints. Stochastic tubes were used to provide a recursive guarantee of feasibility and thus ensure closed loop stability and constraint satisfaction. Moreover, the authors of [84] proposed conditions that, for the parameterization of predictions employed, are necessary and sufficient for recursive feasibility, thereby incurring no additional conservatism. The approach was based on state feedback, which assumed that the states are measurable. In practice this is often not the case, and it is then necessary to estimate the state via an observer. The introduction of state estimation into RMPC is well understood and uses lifting to describe the combined system and observer dynamics. In [85], these ideas are extended to include probabilistic information on measurement noise and the unknown initial plant state, and extends the approach of [84].
\n\t\t\t
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Applications
\n\t\t\t
In the next two sections, we will show that many important practical and theoretical problems can be formulated in the MPC framework. Pursuing them will assure MPC of its stature as a vibrant research area, where theory is seen to support practice more directly than in most other areas of control research.
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7. Networked control systems
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Traditionally, the different components (i.e., sensor, controller, and actuator) in a control system are connected via wired, point-to-point links, and the control laws are designed and operate based on local continuously-sampled process output measurements.
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In recent years, there has been a growing interest in the design of controllers based on the network systems in several areas such as traffic, communication, aviation and spaceflight [86]. The networked control systems (NCSs) is defined as a feedback control system where control loops are closed through a real-time network [96, 97], which is different from traditional control systems. For an overview, the readers can refer to [97], which systematically addresses several key issues (band-limited channels, sampling and delay, packet loss, system architecture) that make NCSs distinct from other control systems.
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\n\t\t\t\t
7.1. Characteristics of NCSs
\n\t\t\t\t
Advantages. Communication networks make the transmission of data much easier and provide a higher degree of freedom in the configuration of control systems. Network-based communication allows for easy modification of the control strategy by rerouting signals, having redundant systems that can be activated automatically when component failure occurs. Particularly, NCSs allow remote monitoring and adjustment of plants over the Internet. This enables the control system to benefit from the way it retrieves data and reacts to plant fluctuations from anywhere around the world at any time, see for example, [98-101] and references therein.
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Disadvantages. Although the network makes it convenient to control large distributed systems, new issues arise in the design of a NCSs. Augmenting existing control networks with real-time wired or wireless sensor and actuator networks challenges many of the assumptions made in the development of traditional process control methods dealing with dynamical systems linked through ideal channels with flawless, continuous communication. In the context of networked control systems, key issues that need to be carefully handled at the control system design level include data losses due to field interference and time-delays due to network traffic as well as due to the potentially heterogeneous nature of the additional measurements (for example, continuous, asynchronous and delayed) [102]. These issues will deteriorate the performance and may even cause the system to be unstable.
\n\t\t\t\t
Hence, the main question is how to design the NCSs include the handling of data losses, time-varying delays, and the utilization of heterogeneous measurements to maintain the closed-loop stability while improving the closed-loop performance.
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7.2. Results on NCSs
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To solve these problems, various methods have been developed, e.g., augmented deterministic discrete-time model, queuing, optimal stochastic control, perturbation, sampling time scheduling, robust control, fuzzy logic modulation, event-based control, end-user control adaptation, data packet dropout analysis, and hybrid systems stability analysis. However, these methods have put some strict assumptions on NCSs, e.g., the network time delay is less than the sampling period [109, 110]. The work of [111] presents an approach for stability analysis of NCSs that decouples the scheduling protocol from properties of the network-free nominal closed-loop system. The problem of the design of robust \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tcontrollers for uncertain NCSs with the effects of both the network-induced delay and data dropout has been considered in [112] the network-induced time delay is larger than one sampling period, but there is no compensation for the time delay and data dropout.
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A common approach is to insert network behavior between the nodes of a conventional control loop, designed without taking the network behavior into account. More specifically, in [114], it was proposed to first design the controller using established techniques considering the network transparent, and then to analyze the effect of the network on closed-loop system stability and performance. This approach was further developed in [115] using a small gain analysis approach.
\n\t\t\t\t
In the last few years, however, several research papers have studied control using the \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t802.11\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and Bluetooth wireless networks, see, for example, [116-119] and the references therein. In the design and analysis of networked control systems, the most frequently studied problem considers control over a network having constant or time-varying delays. This network behavior is typical of communications over the Internet but does not necessarily represent the behavior of dedicated wireless networks in which the sensor, controller, and actuator nodes communicate directly with one another but might experience data losses. An appropriate framework to model lost data, is the use of asynchronous systems [120-122] and the process is considered to operate in an open-loop fashion when data is lost.
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The most destabilizing cause of packet loss is due to bursts of poor network performance in which case large groups of packets are lost nearly consecutively. A more detailed description of bursty network performance using a two-state Markov chain was considered in [123]. Modeling networks, using Markov chains results in describing the overall closed-loop system as a stochastic hybrid system [120]. Stability results have been presented for particular cases of stochastic hybrid systems (e.g., [124, 125]). However, these results do not directly address the problem of augmentation of dedicated, wired control systems with networked actuator and sensor devices to improve closed-loop performance.
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With respect to other results on networked control, in [126], stability and disturbance attenuation issues for a class of linear networked control systems subject to data losses modeled as a discrete-time switched linear system with arbitrary switching was studied. In [127], (see also [128-130]), optimal control of linear time-invariant systems over unreliable communication links under different communication protocols (with and without acknowledgment of successful communication) was investigated and sufficient conditions for the existence of stabilizing control laws were derived.
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Although, within control theory, the study of control over networks has attracted considerable attention in the literature, most of the available results deal with linear systems (e.g., [100, 131]).
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7.3. Our works
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MPC framework is particularly appropriate for controlling systems subject to data losses because the actuator can profit from the predicted evolution of the system. In this section, results from our works are summarized.
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Several methodologies have been reported in the open literature to handle with the problems mentioned above in networked systems. Among these papers, two basic control strategies are applied when the packet dropping happens, they are zero-input schemes, by which the actuator input is set to zero when the control packet is lost, and hold-input scheme which implies the previous control input is used again when the control packet drops. The further research is proposed in [132] by directly comparing the two control methods.
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The work of [133] presents a novel control technique combining modified MPC and modified Smith predictor to guarantee the stability of NCSs. Especially, the key point in this paper is that the future control sequence is used to compensate for the forward communication time delay and predictor is responsible for compensating the time delay in the backward channel.
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Although much research work have been done in NCSs, many of those results simply treat the NCSs as a system with time delay, which ignores NCSs features, e.g., random network delay and data transmission in packets [134]. In order to solve the problem, Markovian jump system can be used to model the random time-delay. Moreover, most work have also ignored another important feature of NCSs. This feature is that the communication networks can transmit a packet of data at the same time, which is not done in traditional control systems. We have proposed a new networked control scheme – networked predictive control which mainly consists of a control prediction generator and a network dropout/delay compensator. It is first assumed that control predictions based on received data are packed and sent to the plant side through a network. Then the network dropout/delay compensator chooses the latest control value from the control prediction sequences available on the plant side, which can compensate for the time delay and data dropouts. The structure of the networked predictive control system (NPCS) is shown as Figure 4. The random network delay in the forward channel in NCS has been studied in [135]. Some other results has been obtained in [136] and [137], where the network-induced delay is not in the form of a Markov chain.
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Figure 4.
The networked predictive control system.
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But, the random network delay in the forward and feedback channels makes the control design and stability analysis much more difficult. In [100] proposes a predictive control scheme for NCS with random network delay in both the feedback and forward channels and also provides an analytical stability criteria for closed-loop networked predictive control (NPC) systems. Furthermore, [138] can overcome the effects caused by both the unknown network delay and data dropout. Recently, [139] mainly focus on the random transmission data dropout existing in both feedback and forward channels in NCSs. So, the network-induced time delay is not discussed here.
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In fact, using the networked predictive control scheme presented in this section, the control performance of the closed-loop system with data dropout is very similar to the one without data dropout.
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8. Distributed MPC
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At the beginning of research on NCSs, more attention was paid on single plant through network. Recently, fruitful research results on multi-plant, especially, on multi-agent networked control systems have been obtained. The aim of this section is to review a classification of a number of distributed control architectures for large scale systems. Attention is focused on the design approaches based on model predictive control. The controllers apply MPC policies to their local subsystems. They exchange their predictions by communication and incorporate the information from other controllers into their local MPC problem so as to coordinate with each other. For the considered architecture, the underlying rationale, the fields of application, the merits and limitations are discussed and the main references to the literature are reported.
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8.1. Background
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Technological and economical reasons motivate the development of process plants, manufacturing systems, traffic networks, water or power networks [140] with an ever increasing complexity. In addition, there is an increasing interest in networked control systems, where dedicated local control networks can be augmented with additional networked (wired and/or wireless) actuator/sensor devices have become cheap and easy-to-install [141, 142]. These large scale systems are composed by many physically or geographically divided subsystems. Each subsystem interacts with some so called neighbouring subsystems by their states and their inputs. The technical target is to achieve some global performance of entire system (or a common goal of all subsystems). Actually, it is difficult to control with a centralized control structure due to the required inherent computational complexity, robustness and reliability problems and communication bandwidth limitations.
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For all these reasons, many distributed control structures have been developed and applied over the recent decades.
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8.2. The reasons why DMPC is adopted
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About MPC. The aim of this section is to review the distributed control approaches adopted, and to provide a wide list of references focusing the attention on the methods based on MPC. This choice is motivated by the ever increasing popularity of MPC in the process industry, see e.g. the survey papers [143, 144] on the industrial applications of linear and nonlinear MPC. Moreover, in recent years many MPC algorithms have been developed to guarantee some fundamental properties, such as the stability of the resulting closed-loop system or its robustness with respect to a wide class of external disturbances and/or model uncertainties, see e.g. the survey paper [2]. Especially, MPC is also a natural control framework to deal with the design of coordinated, distributed control systems because of its ability to handle input and state constraints, and also because it can account for the actions of other actuators in computing the control action of a given set of control actuators in real-time. Therefore, MPC is now recognized as a very powerful approach with well established theoretical foundations and proven capability to handle the problems of large scale systems.
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Other control structures
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1) Centralized Control. MPC is normally implemented in a centralized fashion. One controller is able to acquire the information of the global system, computes all the control inputs for the system, and could obtain a good global performance. In large-scale interconnected systems, such as power systems, water distribution systems, traffic systems, etc., such a centralized control scheme may not suitable or even possible apply to large scale system for some reasons: (1) there are hundreds of inputs and outputs. It requires a large computational efforts in online implementation (2) when the centralized controller fails, the entire system is out of control and the control integrity cannot be guaranteed when a control component fails (3) in some cases, e.g. in multi-intelligent vehicle system, the global information is unavailable to each controller and (4) objections to centralized control are often not computational, however, but organizational. All subsystems rely upon the central agent, making plantwide control difficult to coordinate and maintain. These obstacles deter implementation of centralized control for large-scale plants.
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In recent years, there is a trend for the development of decentralized and distributed MPC due to the disadvantages of centralized MPC mentioned above (e.g.,[145, 146]).
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2) Decentralized Control. Most large-scale and networked control systems are based on a decentralized architecture, that is, the system is divided into several subsystems, each controlled by a different agent that does not share information with the rest. Each of the agents implements an MPC based on a reduced model of the system and on partial state information, which in general results in an optimization problem with a lower computational burden. Figure 5 shows a decentralized control structure, where the system under control is assumed to be composed by two subsystems S1 and S2, with states, control and output variables (\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t,\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ty\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t) and (\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t,\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ty\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t), respectively, and the interaction between the subsystems is due the inputs and the outputs of different pairs are weak. These interactions can either be direct (input coupling) or caused by the mutual effects of the internal states of the subsystems under control, like in Figure 5.
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For example, in [147], a MPC algorithm was proposed under the main assumptions that the system is nonlinear, discrete-time and no information is exchanged between local controllers.The decentralized framework has the advantages of being flexible to system structure, error-tolerance, less computational efforts and no global information requirements [148].
\n\t\t\t\t
Figure 5.
Decentralized control of a two input (\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t)-two output (\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ty\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ty\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t) system.
\n\t\t\t\t
In plants where the subsystems interact weakly, local feedback action provided by these subsystem (decentralized) controllers may be sufficient to overcome the effect of interactions. For such cases, a decentralized control strategy is expected to work adequately. On the contrary, it is well known that strong interactions can even prevent one from achieving stability and/or performance with decentralized control, see for example [149, 150], where the role played by the so-called fixed modes in the stabilization problem is highlighted.
\n\t\t\t\t\n\t\t\t\t
Distributed MPC. While these paradigms (centralized control and decentralized Control) to process control have been successful, there is an increasing interest in developing distributed model predictive control (DMPC) schemes, where agents share information in order to improve closed-loop performance, robustness and fault-tolerance. As a middle ground between the decentralized and centralized strategies, distributed control preserves the topology and flexibility of decentralized control yet offers a nominal closed-loop stability guarantee.
\n\t\t\t\t
For each decentralized MPC, a sequence of open-loop controls are determined through the solution of a constrained optimal control problem. A local objective is used. A subsystem model, which ignores the interactions, is used to obtain a prediction of future process behavior along the control horizon. For distributed control, one natural advantage that MPC offers over other controller paradigms is its ability to generate a prediction of future subsystem behavior. If the likely influence of interconnected subsystems is known, each local controller can possibly determine suitable feedback action that accounts for these external influences. Intuitively, one expects this additional information to help improve systemwide control performance. Thus the distributed control framework is usually adopted in large-scale plants [151], in spite of that the dynamic performance of centralized frame work is better than it.
\n\t\t\t\t
Figure 6.
Distributed control of a two input (\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t)-two output (\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ty\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ty\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t) system.
\n\t\t\t\t
In distributed control structures, like the simple example shown in Figure 6, it is assumed that some information is transmitted among the local regulators (\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t in Figure 6), so that each one of them has some knowledge on the behavior of the others. When the local regulators are designed with MPC, the information transmitted typically consists of the future predicted control or state variables computed locally, so that any local regulator can predict the interaction effects over the considered prediction horizon. With reference to the simple case of Figure 6, the MPC regulators \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tare designed to control the subsystems \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, respectively. If the information was exchange among the local regulators (\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t) concerns the predicted evolution of the system states (\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t), any local regulator needs only to know the dynamics of the subsystem directly controlled (\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tand\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t).
\n\t\t\t\t\n\t\t\t\t
In any case, it is apparent that the performance of the closed-loop system depends on the decisions that all the agents take. Hence, cooperation and communication policies become very important issues.
\n\t\t\t\t
With respect to available results in this direction, several DMPC methods have been proposed in the literature that deal with the coordination of separate MPCs. These communicate in order to obtain optimal input trajectories in a distributed manner see [145, 146, 152] for reviews of results in this area. Some distributed MPC formulations are available in the literatures [153-157].
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\n\t\t\t\t
8.3. DMPC over network information exchange
\n\t\t\t\t
However, all of the above results are based on the assumption of continuous sampling of the entire plant state vector and assuming no delays and perfect communication between subsystems. In practice, individual subsystems exchange information over a communication network, especially wireless communication network, where the data is transmitted in discrete packets. These packets may be lost during communication. Moreover, the communication media is a resource that is usually accessed in a mutually exclusive manner by neighborhood agents. This means that the throughput capacity of such networks is limited.\n\t\t\t\tThus, how to improve the global performance of each subsystem with the limited network communication or limited available information is a valuable problem.
\n\t\t\t\t
Previous work on MPC design for systems subject to asynchronous or delayed measurements has primarily focused on centralized MPC design [158], [159] and little attention has been given to the design of DMPC. In [160], the issue of delays in the communication between distributed controllers was addressed. The authors of \n [161] consider the design of distributed MPC schemes for nonlinear systems in a more common setting. That is, measurements of the state are not available continuously but asynchronously and with delays.
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\n\t\t
\n\t\t\t
9. Conclusions
\n\t\t\t
Recently, there has been much interest in model predictive control which allows researchers to address problems like feasibility, stability and performance in a rigorous manner. We first give a review of discrete-time model predictive control of constrained dynamic systems, both linear and nonlinear. The min-max approach for handling uncertainties are illustrated, then the LMIs methods are showed, and the advantages and disadvantages of methods are mentioned. The basic idea of each method and some method applications are stated. Despite the extensive literature that exists on predictive control and robustness to uncertainty, very little attention has been paid to the case of stochastic uncertainty. SMPC is emerging to adopt a stochastic uncertainty description (instead of a set-based description). Some of the recent advances in this area are reviewed. We show that many important practical and theoretical problems can be formulated in the MPC framework, such as DMPC. Some considerable attention has been directed to NCSs. Although the network makes it convenient to control large distributed systems, there also exist many control issues, such as network delay and data dropout, which cannot be addressed using conventional control theory, sampling and transmission methods. Results from our recent research are summarized in Section 7. We have proposed a new networked control scheme, which can overcome the effects caused by the network delay. In the last section we review a number of distributed control architectures based on model predictive control. For the considered architectures, the underlying rationale, the fields of application, the merits and limitations are discussed.
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\n\t
Acknowledgments
\n\t\t\t
The work of Yuanqing Xia was supported by the National Basic Research Program of China (973 Program) (2012CB720000), the National Natural Science Foundation of China (60974011), Program for New Century Excellent Talents in University of China (NCET-08-0047), the Ph.D. Programs Foundation of Ministry of Education of China (20091101110023, 20111101110012), and Program for Changjiang Scholars and Innovative Research Team in University, and Beijing Municipal Natural Science Foundation (4102053,4101001). The work of Magdi S. Mahmoud is supported by the research group project no. RG1105-1 from DSR-KFUPM.
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\n',keywords:null,chapterPDFUrl:"https://cdn.intechopen.com/pdfs/37837.pdf",chapterXML:"https://mts.intechopen.com/source/xml/37837.xml",downloadPdfUrl:"/chapter/pdf-download/37837",previewPdfUrl:"/chapter/pdf-preview/37837",totalDownloads:4321,totalViews:979,totalCrossrefCites:3,totalDimensionsCites:4,hasAltmetrics:0,dateSubmitted:"March 7th 2012",dateReviewed:"June 28th 2012",datePrePublished:null,datePublished:"December 5th 2012",dateFinished:null,readingETA:"0",abstract:null,reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/37837",risUrl:"/chapter/ris/37837",book:{slug:"advances-in-discrete-time-systems"},signatures:"Li Dai, Yuanqing Xia, Mengyin Fu and Magdi S. Mahmoud",authors:[{id:"20339",title:"Prof.",name:"Meng-Yin",middleName:null,surname:"Fu",fullName:"Meng-Yin Fu",slug:"meng-yin-fu",email:"fumy@bit.edu.cn",position:null,institution:{name:"Beijing Institute of Technology",institutionURL:null,country:{name:"China"}}},{id:"145065",title:"Prof.",name:"Magdi",middleName:null,surname:"Mahmoud",fullName:"Magdi Mahmoud",slug:"magdi-mahmoud",email:"magdim@yahoo.com",position:null,institution:null},{id:"152697",title:"Dr.",name:"Yuanqing",middleName:null,surname:"Xia",fullName:"Yuanqing Xia",slug:"yuanqing-xia",email:"xia_yuanqing@163.net",position:null,institution:null},{id:"156006",title:"Dr.",name:"Li",middleName:null,surname:"Dai",fullName:"Li Dai",slug:"li-dai",email:"daili1887@gmail.com",position:null,institution:null}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Model Predictive Control",level:"1"},{id:"sec_2_2",title:"2.1. Characteristics of MPC",level:"2"},{id:"sec_3_2",title:"2.2. Essence of MPC",level:"2"},{id:"sec_5",title:"3. Linear Model Predictive Control",level:"1"},{id:"sec_5_2",title:"3.1. Mathematical formulation",level:"2"},{id:"sec_6_2",title:"3.2. Feasibility",level:"2"},{id:"sec_7_2",title:"3.3. Closed loop stability",level:"2"},{id:"sec_8_2",title:"3.4. Open-loop performance objective versus closed loop performance",level:"2"},{id:"sec_10",title:"4. Nonlinear model predictive control",level:"1"},{id:"sec_10_2",title:"4.1. Difficulties of NMPC",level:"2"},{id:"sec_11_2",title:"4.2. Closed-loop stability",level:"2"},{id:"sec_11_3",title:"4.2.1. Infinite horizon NMPC",level:"3"},{id:"sec_12_3",title:"4.2.2. Finite horizon NMPC",level:"3"},{id:"sec_15",title:"5. Robust model predictive control",level:"1"},{id:"sec_15_2",title:"5.1. Min-Max RMPC methods",level:"2"},{id:"sec_16_2",title:"5.2. LMI-based RMPC methods",level:"2"},{id:"sec_17_2",title:"5.3. Our works",level:"2"},{id:"sec_18_2",title:"5.4. Conclusion",level:"2"},{id:"sec_20",title:"6. Recent developments in stochastic MPC",level:"1"},{id:"sec_20_2",title:"6.1. Basic SMPC problem ",level:"2"},{id:"sec_21_2",title:"6.2. Earlier work ",level:"2"},{id:"sec_23",title:"Applications",level:"1"},{id:"sec_24",title:"7. Networked control systems",level:"1"},{id:"sec_24_2",title:"7.1. Characteristics of NCSs",level:"2"},{id:"sec_25_2",title:"7.2. Results on NCSs",level:"2"},{id:"sec_26_2",title:"7.3. Our works",level:"2"},{id:"sec_28",title:"8. Distributed MPC",level:"1"},{id:"sec_28_2",title:"8.1. Background",level:"2"},{id:"sec_29_2",title:"8.2. The reasons why DMPC is adopted",level:"2"},{id:"sec_30_2",title:"8.3. DMPC over network information exchange",level:"2"},{id:"sec_32",title:"9. Conclusions",level:"1"},{id:"sec_33",title:"Acknowledgments",level:"1"}],chapterReferences:[{id:"B1",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMorari\n\t\t\t\t\t\t\tM.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tLee\n\t\t\t\t\t\t\tJ. 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P.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2007\n\t\t\t\t\tRobust distributed model predictive control.\n\t\t\t\t\tInternational Journal of Control\n\t\t\t\t\t80\n\t\t\t\t\t9\n\t\t\t\t\t1517\n\t\t\t\t\t1531\n\t\t\t\t\n\t\t\t'},{id:"B157",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tVenkat\n\t\t\t\t\t\t\tA. N.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tRawlings\n\t\t\t\t\t\t\tJ. B.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tWright\n\t\t\t\t\t\t\tS. J.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2007\n\t\t\t\t\tDistributed model predictive control of large-scale systems. In: Proceedings of the assessment and future directions of nonlinear model predictive control.\n\t\t\t\t\t Berlin Heidelberg\n\t\t\t\t\tSpringer\n\t\t\t\t\t591\n\t\t\t\t\t605\n\t\t\t\t\n\t\t\t'},{id:"B158",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tLiu\n\t\t\t\t\t\t\tJ.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMuñoz la\n\t\t\t\t\t\t\tPeña. D.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tChristofides\n\t\t\t\t\t\t\tP. D.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tDavis\n\t\t\t\t\t\t\tJ. F.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2009\n\t\t\t\t\tLyapunovbased model predictive control of nonlinear systems subject to time-varying measurement delays.\n\t\t\t\t\t International Journal of Adaptive Control and Signal Processing\n\t\t\t\t\t23\n\t\t\t\t\t788\n\t\t\t\t\n\t\t\t'},{id:"B159",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMuñoz la\n\t\t\t\t\t\t\tPeña. D.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tChristofides\n\t\t\t\t\t\t\tP. D.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2008\n\t\t\t\t\tLyapunov-based model predictive control of nonlinear systems subject to data losses.\n\t\t\t\t\tIEEE Transactions on Automatic Control\n\t\t\t\t\t53\n\t\t\t\t\t2076\n\t\t\t\t\t2089\n\t\t\t\t\n\t\t\t'},{id:"B160",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tFranco\n\t\t\t\t\t\t\tE.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMagni\n\t\t\t\t\t\t\tL.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tParisini\n\t\t\t\t\t\t\tT.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tPolycarpou\n\t\t\t\t\t\t\tM. M.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tRaimondo\n\t\t\t\t\t\t\tD. M.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2008\n\t\t\t\t\tCooperative constrained control of distributed agents with nonlinear dynamics and delayed information exchange: a stabilizing receding-horizon approach.\n\t\t\t\t\t IEEE Transactions on Automatic Control\n\t\t\t\t\t53\n324\n\t\t\t\t\t338\n\t\t\t\t\n\t\t\t'},{id:"B161",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tLiu\n\t\t\t\t\t\t\tJ.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tde la Peña\n\t\t\t\t\t\t\tD. M.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tChristofides\n\t\t\t\t\t\t\tP. D.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2010\n\t\t\t\t\tDistributed model predictive control of nonlinear systems subject to asynchronous and delayed measurements.\n\t\t\t\t\tAutomatica\n\t\t\t\t\t4652\n\t\t\t\t\t61\n\t\t\t\t\n\t\t\t'}],footnotes:[],contributors:[{corresp:null,contributorFullName:"Li Dai",address:null,affiliation:'
School of Automation, Beijing Institute of Technology, China
School of Automation, Beijing Institute of Technology, China
'},{corresp:null,contributorFullName:"Magdi S. Mahmoud",address:null,affiliation:'
Systems Engineering Department, King Fahd University of Petroleum and Minerals, Saudi Arabia
'}],corrections:null},book:{id:"3133",title:"Advances in Discrete Time Systems",subtitle:null,fullTitle:"Advances in Discrete Time Systems",slug:"advances-in-discrete-time-systems",publishedDate:"December 5th 2012",bookSignature:"Magdi S. Mahmoud",coverURL:"https://cdn.intechopen.com/books/images_new/3133.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"145065",title:"Prof.",name:"Magdi",middleName:null,surname:"Mahmoud",slug:"magdi-mahmoud",fullName:"Magdi Mahmoud"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},chapters:[{id:"38179",title:"Stochastic Mixed LQR/H∞ Control for Linear Discrete-Time Systems",slug:"stochastic-mixed-lqr-h-control-for-linear-discrete-time-systems",totalDownloads:988,totalCrossrefCites:0,signatures:"Xiaojie Xu",authors:[{id:"20349",title:"Dr.",name:"Xiaojie",middleName:null,surname:"Xu",fullName:"Xiaojie Xu",slug:"xiaojie-xu"}]},{id:"40676",title:"Robust Control Design of Uncertain Discrete-Time Descriptor Systems with Delays",slug:"robust-control-design-of-uncertain-discrete-time-descriptor-systems-with-delays",totalDownloads:1680,totalCrossrefCites:0,signatures:"Jun Yoneyama, Yuzu Uchida and Ryutaro Takada",authors:[{id:"6944",title:"Dr.",name:"Jun",middleName:null,surname:"Yoneyama",fullName:"Jun Yoneyama",slug:"jun-yoneyama"},{id:"15518",title:"Dr.",name:"Yuzu",middleName:null,surname:"Uchida",fullName:"Yuzu Uchida",slug:"yuzu-uchida"},{id:"153401",title:"Mr.",name:"Ryutaro",middleName:null,surname:"Takada",fullName:"Ryutaro Takada",slug:"ryutaro-takada"}]},{id:"39065",title:"Delay-Dependent Generalized H2 Control for Discrete-Time Fuzzy Systems with Infinite-Distributed Delays",slug:"delay-dependent-generalized-h2-control-for-discrete-time-fuzzy-systems-with-infinite-distributed-del",totalDownloads:1105,totalCrossrefCites:0,signatures:"Jun-min Li, Jiang-rong Li and Zhi-le Xia",authors:[{id:"153140",title:"Prof.",name:"Junmin",middleName:null,surname:"Li",fullName:"Junmin Li",slug:"junmin-li"},{id:"153987",title:"Dr.",name:"Jiangrong",middleName:null,surname:"Li",fullName:"Jiangrong Li",slug:"jiangrong-li"},{id:"153988",title:"Dr.",name:"Zhile",middleName:null,surname:"Xia",fullName:"Zhile Xia",slug:"zhile-xia"}]},{id:"37837",title:"Discrete-Time Model Predictive Control",slug:"discrete-time-model-predictive-control",totalDownloads:4321,totalCrossrefCites:3,signatures:"Li Dai, Yuanqing Xia, Mengyin Fu and Magdi S. Mahmoud",authors:[{id:"145065",title:"Prof.",name:"Magdi",middleName:null,surname:"Mahmoud",fullName:"Magdi Mahmoud",slug:"magdi-mahmoud"},{id:"20339",title:"Prof.",name:"Meng-Yin",middleName:null,surname:"Fu",fullName:"Meng-Yin Fu",slug:"meng-yin-fu"},{id:"152697",title:"Dr.",name:"Yuanqing",middleName:null,surname:"Xia",fullName:"Yuanqing Xia",slug:"yuanqing-xia"},{id:"156006",title:"Dr.",name:"Li",middleName:null,surname:"Dai",fullName:"Li Dai",slug:"li-dai"}]},{id:"41280",title:"Stability Analysis of Nonlinear Discrete-Time Adaptive Control Systems with Large Dead-Times - Theory and a Case Study",slug:"stability-analysis-of-nonlinear-discrete-time-adaptive-control-systems-with-large-dead-times-theory-",totalDownloads:1085,totalCrossrefCites:0,signatures:"Mario A. Jordan, Jorge L. Bustamante and Carlos E. Berger",authors:[{id:"152460",title:"Dr.",name:"Mario",middleName:"Alberto",surname:"Jordán",fullName:"Mario Jordán",slug:"mario-jordan"},{id:"164662",title:"Dr.",name:"Jorge",middleName:null,surname:"Bustamante",fullName:"Jorge Bustamante",slug:"jorge-bustamante"},{id:"164663",title:"Mr.",name:"Carlos",middleName:null,surname:"Berger",fullName:"Carlos Berger",slug:"carlos-berger"}]},{id:"41300",title:"Adaptive Step-Size Orthogonal Gradient-Based Per-Tone Equalisation in Discrete Multitone Systems",slug:"adaptive-step-size-orthogonal-gradient-based-per-tone-equalisation-in-discrete-multitone-systems",totalDownloads:1179,totalCrossrefCites:2,signatures:"Suchada Sitjongsataporn",authors:[{id:"16087",title:"Dr.",name:"Suchada",middleName:null,surname:"Sitjongsataporn",fullName:"Suchada Sitjongsataporn",slug:"suchada-sitjongsataporn"}]},{id:"39667",title:"An Approach to Hybrid Smoothing for Linear Discrete-Time Systems with Non-Gaussian Noises",slug:"an-approach-to-hybrid-smoothing-for-linear-discrete-time-systems-with-non-gaussian-noises",totalDownloads:955,totalCrossrefCites:0,signatures:"Gou Nakura",authors:[{id:"18676",title:"Dr.",name:"Gou",middleName:null,surname:"Nakura",fullName:"Gou Nakura",slug:"gou-nakura"}]},{id:"41301",title:"Discrete-Time Fractional-Order Systems: Modeling and Stability Issues",slug:"discrete-time-fractional-order-systems-modeling-and-stability-issues",totalDownloads:2686,totalCrossrefCites:7,signatures:"Saïd Guermah, Saïd Djennoune and Maâmar Bettayeb",authors:[{id:"153595",title:"Dr.",name:"Saïd",middleName:null,surname:"Guermah",fullName:"Saïd Guermah",slug:"said-guermah"}]},{id:"37761",title:"Investigation of a Methodology for the Quantitative Estimation of Nursing Tasks on the Basis of Time Study Data",slug:"investigation-of-a-methodology-for-the-quantitative-estimation-of-nursing-tasks-on-the-basis-of-time",totalDownloads:1054,totalCrossrefCites:0,signatures:"Atsue Ishii, Takashi Nakamura, Yuko Ohno and Satoko Kasahara",authors:[{id:"37861",title:"Prof.",name:"Yuko",middleName:null,surname:"Ohno",fullName:"Yuko Ohno",slug:"yuko-ohno"},{id:"52651",title:"Associate Prof.",name:"Atsue",middleName:null,surname:"Ishii",fullName:"Atsue Ishii",slug:"atsue-ishii"},{id:"90323",title:"MSc.",name:"Satoko",middleName:null,surname:"Kasahara",fullName:"Satoko Kasahara",slug:"satoko-kasahara"},{id:"153889",title:"Dr.",name:"Takashi",middleName:null,surname:"Nakamura",fullName:"Takashi Nakamura",slug:"takashi-nakamura"}]}]},relatedBooks:[{type:"book",id:"4457",title:"Finite Element Analysis",subtitle:null,isOpenForSubmission:!1,hash:"285298bf100533ffcdc1a1a969b29177",slug:"finite-element-analysis",bookSignature:"David Moratal",coverURL:"https://cdn.intechopen.com/books/images_new/4457.jpg",editedByType:"Edited by",editors:[{id:"9850",title:"Dr.",name:"David",surname:"Moratal",slug:"david-moratal",fullName:"David Moratal"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"},chapters:[{id:"11974",title:"Finite Element Analysis on Strains of Viscoelastic Human Skull and Duramater",slug:"finite-element-analysis-on-strains-of-viscoelastic-human-skull-and-duramater",signatures:"Xianfang Yue",authors:[null]},{id:"11975",title:"Application of Finite Element Analysis in Dentistry",slug:"application-of-finite-element-analysis-in-dentistry",signatures:"Ming-Lun Hsu and Chih-Ling Chang",authors:[null]},{id:"11977",title:"Finite Element Analysis for Dental Prosthetic Design",slug:"finite-element-analysis-for-dental-prosthetic-design",signatures:"Akikazu Shinya and Daiichiro Yokoyama",authors:[null]},{id:"11985",title:"Application of Finite Element Analysis in Root Canal Therapy",slug:"application-of-finite-element-analysis-in-root-canal-therapy",signatures:"Tao Hu, Ran Cheng, Meiying Shao, Hui Yang, Ru Zhang, Qianhua Gao and Liyang Guo",authors:[null]},{id:"11986",title:"Finite element simulation. Applications in Orthopaedics and Traumatology",slug:"finite-element-simulation-applications-in-orthopaedics-and-traumatology-",signatures:"Antonio Herrera",authors:[null]},{id:"11987",title:"Finite Element Analysis in Orthopaedic Biomechanics",slug:"finite-element-analysis-in-orthopaedic-biomechanics",signatures:"Daniel Kluess",authors:[null]},{id:"11992",title:"Orthopaedic Biomechanics: A Practical Approach to Combining Mechanical Testing and Finite Element Analysis",slug:"orthopaedic-biomechanics-a-practical-approach-to-combining-mechanical-testing-and-finite-element-ana",signatures:"Rad Zdero and Habiba Bougherara",authors:[null]},{id:"11993",title:"Finite Element Modeling for a Morphometric and Mechanical Characterization of Trabecular Bone from High Resolution Magnetic Resonance Imaging",slug:"finite-element-modeling-for-a-morphometric-and-mechanical-characterization-of-trabecular-bone-from-h",signatures:"Angel Alberich-Bayarri, Luis Marti-Bonmati, M. 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1. Introduction
Two and a half billion years ago a natural fission reactor operated on the Earth (Oklo). The discovery of this natural energy source created a series of theories and had implications yet to be evaluated both on the man-made artifacts of similar type and on some fundamentals considered so far as improbable to be challenged in quantum physics, biology, ecology, nuclear reactor theory. It also has an impact on knowledge management, on the epistemology and ethics. Aspects of the implications for mankind and the lessons learnt so far on the actions to build a sustainable civilization are presented in this chapter.
In 1972 the international community involved in the research, design and operation of MMES of fission type reactors was surprised and challenged by a discovery of the remains of an ancient natural fission reactor, in Oklo (Gabon). It was a NES type reactor (NES_Oklo).
However the discovery was predicted long time before by PK Kuroda [1]. The reactor in Gabon operated, intermittently, two and a half billion years ago for about two hundreds millions year and had an approximate power of 100 kW. It operated with uranium ore (using the isotope U235) and water [2, 3, 4, 5].
As the reactor physics classic results show, this would not be possible, provided the concentration of U235 (considered as a constant for the whole universe) being presently 0.71% was not higher (around 3.3%) by the time the reactor started operating. And this is not all. The reactor had to have a concentrated amount of U235 in a place forming a geometry and a configuration of cooling (with cooling water) of a very specific precise type. Apparently cyanobacteria concentrated the uranium and the water from the underground, pushed by the geological moves by that time (Africa and South America were splitting apart) created actually the reactor core, as called in the nuclear engineering. Even more than that, the type of soil assured the retention of the radioactive elements resulted from fission, which actually did not migrate further than the site.
All those aspects were very troubling for the nuclear community. In addition the calculations for the MMES reactors were seriously challenged when they were used to describe NES_Oklo.
Findings did not stop here, as series of other theories were developed, as for instance:
Theories related to how the oxygen formation (taking place exactly by that time) were related to the activity of the geyser nuclear reactor splitting water vapors, as water got overheated, to the atmosphere.
As for the biology the time of NES_Oklo operation is also coincident with the appearance of eukaryotes, living beings having cells with nucleus in a membrane, to which we also belong.
As a top of troubling discoveries, the site evaluations challenged some fundamentals of quantum mechanics and relativity, related to the alpha constant and the speed of light.
Not to mention the fact that new theories and observations started to assume that, may be even the Earth core is a nuclear fission reactor and may be Oklo was not the only surface reactor.
More than that evidence on existence of fission reactors is found also in our neighboring planets (Mars), all taking place at a certain time of evolution of energy chains of the universe, of the solar system and of our Earth. Operation of such NES reactors appears to give serious inputs on how an ecological type of such source of energy might be designed by mankind. All those aspects are really of high interest and researches are going on.
A troubling set of correlations and coincidences illustrate for this particular case how various phenomena with their lifecycles, their appearance, and development are connected to each other and how Mother Nature gives us lessons on how to manage complicated lifecycles of high energies without damaging it.
There is a vast literature on Oklo reactor, of which the references are representative in our view. The references could be started with the works of PK Kuroda, who predicted the first the possibility of the existence of a natural fission reactor on Earth.
2. Evaluation method
This chapter will focus only on the lessons learnt so far. However, there are more than only natural sciences implications, but also on the manner we acquire knowledge, on how we build models and interact with their reality and how we related to their lifecycles.
Therefore the chapter will not address the details of the researches on Oklo, but rather the lessons learnt to the humanity for such a discovery. The approach adopted in the presentation of Oklo lessons in this chapter is also based on some author’s researches on the philosophy of science and models proposed to consider, model and interact with the energy sources, by describing their creation/emergence, their lifecycle and their interaction with mankind and its knowledge.
For this endeavor, a systematic approach was adopted and presented previously [6, 7, 8, 9]. Based on this approach the NES and MMES are evaluated in their interaction and development/transformation from one to another in a systematic manner, which is based on some assumptions, as follows:
Energy sources create systems, which might be considered Complex Systems (CS) [6] These systems are composed of elements and connectors between them defined as categories, in the mathematical sense [6].
For the ES considered as CS, defined by NES and MMES, because they have a behavior of topological nature and for their models, a topological description is possible, as they
are described by invariants, that preserve their nature after transformations,
create complex networks fractal like structures and
their emergence/transformation from one phase/state/form/source to another takes place step by step [10].
The KP of a given ES for a given NES cannot be predicted in detail, but in its general features. The proposed approach considers that the KP generates a topological structure (K(i)) based on a set of relationships between the objects modeled and it is developed in accordance with a certain Theory (Th(K(i))). The topological structure resulting from the KP is in isomorphism with the topological structure describing the emergence rules of the NES from one state to another. The method is based on three principles [10]:
Principle 1: The topological structure K(i) is described by the notion of category considered as:
reflecting a hierarchical “matrioshka” type of structure
being a general description of cybernetic description of objects and models as “black-boxes” for each level of construction and for each object.
being described by objects, morphisms, and identity morphisms
Principle 2: KP is performed in iterations on the categories for each object and each level up to the moment of reaching a critical status due to number and type of paradoxes that result at each step.
The set of invariants (syzygies) is continuously optimized from diverse points of view (using tools from different sciences) and based on the existing results on them a final set of minimal syzygies for a given model—using a given scientific tool—is reached (Hilbert’s syzygy theorem).
The process of reaching a status for a set of syzygies is therefore predictable and has an end. However the end state described by the resultant set of syzygies in that KP phase may not correspond to the real object. Therefore, a new iteration using another type of methods—analogy from another science that the previous iteration—is used for a new iteration.
The KP with these new tools will lead to another set of syzygies and have a status of paradoxes in comparison with the real object that will require a new iteration etc.
An example of NES group is presented in this paragraph. NES are assumed to consist of the following levels of energy sources (NES):
Subquantic (SQ)
Quantic (Q)
Electromagnetic (EM)
Molecular (MO)
Molecular and life (MOL)
Conscious planetary life (CPL)
Stellar and universe not alive (SUNA)
Stellar and universe life (SUA)
Conscious stellar and universe (CSU)
Principle 3: KP is asymptotically stable and complete. However the resultant final structure of this process, which is a CAS, may not be known by its detailed phenomenological characteristics, nor predicted, but rather known for its dominant syzygies.
The invariants are called syzygies and they are in the format described by formulas (1) and (2).
GenNES=EnThEnISyEmNlnCxFrE1
SyzygyNES=fGenNESE2
There are some specific generators (in the sense of syzygy theory) for a K(i) structure built for NES:
Exergy (Ex) of a NES (defined as the maximum useful work possible during a process that brings the system into equilibrium with a heat reservoir), as a measure of the efficiency of an energy conversion process. This generator has some specific characteristics:
It is conserved only when all processes of the system and the environment are reversible
It is destroyed whenever an irreversible process occurs.
Entropy in a thermodynamic (EnTh) interpretation as a measure of disorder
Information entropy (EnI) (as a measure of knowledge limits themselves)
Synergy (Sy) as a measure of a resultant set of features for a NES appearing from the existence and interaction of all systems and subsystems, leading to a set of characteristics for the whole NES than exist in the sum of its parts
Emergence (Em) from one level to another (in the example for NES presented from SQ to CSU) a process in which larger entities, patterns, and regularities arise through interactions among smaller or simpler entities that themselves do not exhibit such properties and evolve to new levels.
Nonlinearity (even for simple systems) and/or complexity (NlnCx) of NES as sources of chaotic structure and behavior
Features of CAS—fractals type of structure (Fr) of NES and K(i) knowledge topological structures built for them.
The physical meaning of the dominating syzygies, defining the phase change of ES (NES and MMES) is that they are a triadic set of characteristics of the state of the ES/syzygies and are [10, 11]:
Energy (E)
Mass (m)
Entropy (ψ)
These are optimal descriptors of each ES state and are described by the formulas (3)–(5)
Ek=E0k+∑j=18Ejk∗ijkE3
mk=m0k+∑j=18mjk∗ijkE4
ψk=ψ0k+∑j=18ψjk∗ijkE5
where
E0, m0, ψ0 –and E1(k)*i1(k); m1(k)*i1(k) ψ 1(k)*i1(k) (Noted for the states 0 and 1) define the term called real energy/mass/entropy; examples of energy in such states are the energies perceived at Earth level by a human observer (including such as NES_Oklo), defining the Real Reality.
indexes 2 and 3 the simple complex part (for the states 2 and 3); examples of states of this type are the paranormal phenomena, energies, information channels perceived by a human observer becoming part of the observed object, defining the Intuition Reality of the second level Realm (cosmic) and
the rest of components are the hyper-complex part (for the states 4–8); examples are states of paradoxical situations coming from other realities and totally unexplainable for a human observer, but managing them by enantiotropy feedback chain (entropy of states of the triadic ES) and they are our connection to the Universe Realm and diverse realities (Universes) (formula (6))
The entropy has the following dominant syzygies for each state, as follows [11]:
Thermodynamic entropy, for the states 0 and 1 for the real states
Shannon entropy for the states 2 and 3, for the simple complex states
Enantiotropy for the states 4–8
The triadic set of syzygies defined the set of Realities (as in formula (6))
Rk=R0k+∑j=18Rjk∗ijkE6
ES and their models define topological algebraic spaces, which might be represented as polyhedral type, describing their states and illustrating the optimal cases.
The description of emergence/transformation of one source in another or of passage from one phase to another is based on the method presented in [6, 10].
ES and their models exist in two types of interconnections, with:
Other natural phenomena
At a given level of civilization
For instance NES_Oklo appeared 2.5 billion years ago, while the “Reactor designer” had at its disposal:
A certain geological configuration
A certain status of living beings
A certain status of interface with cosmos
No existing civilization
Environment as we know being under construction
However, the interpretation we make of this source is done at a certain level of our civilization (in its very early beginnings, judging by the criteria of what kind of energy we could harness) [6]. We are far away by several centuries before being able to harness the energy of our sun, which is quite a primitive phase. On the other side, our KP is based on an extremely advanced tool (the interdisciplinary and trans disciplinary one) which may push us to advance much faster than we may envisage now. However, the stronger the forces we harness, the higher the risk to get to the finish of civilization by self-destruction.
We are at a crossroad of the evolution and lessons from NES like Oklo are extremely useful, as they show us how to harness better high energy with high risk sources [6].
In our present knowledge the KP assumes for the ES cases a set of assumptions generated by the paradigms, creating paradoxes, as for instance [6]:
Paradigm 1-ES as a CS: A modeling system has to be built in order to represent Risk Analyses for ES (RES) as a complex system, too. RES is converging to a stable unique real state. However the KP results, including those RES are limited by our present knowledge, as described by the real Earth level mentioned above.
Paradigm 2: ES model involves knowledge of the risks associated to a certain source of energy. However, usually we actually are not aware of the real risks and we know very little about the interconnections of lifecycle dangers for interfering processes (energy level, emergence correlated with civilization one or with geological one etc.)
Paradigm 3: Details of ES and their lessons learnt. We design ES (MMES) for which Nature already indicated the optimal solutions. However, due to our reduced technical and scientific level at a certain moment we cannot understand the lessons from the beginning, but step by step.
Paradigm 4: Understanding the ES risks (RES) and defining them is a difficult task as we design first of a kind MMES and as we are not aware of all the aspects of the lifecycle. The MMES are challenged inevitably by serious events, which apparently test the design continuously.
Paradigm 5: ES risk analyses results are seen as inputs to decision making risk calculation results are used for decisions. However we are facing decisions under high uncertainties and the use of lateral thinking is decisive.
Paradigm 6: In the ES risk analyses results there are limits and biases specific to the level of knowledge of that issue, but also there are “hidden” biases due to the level of KP in the whole civilization at that moment. Inter and trans disciplinarity is not just a desired option, but a mandatory one to minimize such biases.
Paradigm 7: RES results evaluation for further iterations in the.KP is an iterative process and the Principle 3 mentioned above applies. The result could be a better understanding by the use of diverse tools, as for instance the information one can get by “backward engineering” from natural examples.
3. Lessons learnt from NES_Oklo
NES_Oklo sends to us messages. By diverse evaluations one could mention so far messages as the following:
The issue of the meaning of risk analyses for ES is very important, as the lessons learnt from NES_Oklo show. NES_Oklo was a combined non-live living organisms operation to produce energy. This is a high important topic for the future MMES to be designed by assuming the use of Artificial Intelligence, may be also natural and living organisms, etc. The evolution of our civilization and/or possible future interactions at cosmic level require a clear strategy on how to proceed if combined (natural, artificial, living non-living, etc.) energy sources production is to be evaluated and designed.
NES_Oklo teaches us on the absolute importance of intrinsic safety (the reactor operated, got decommissioned without being of any harm to its environment, but on the contrary, being part of the evolution “plan”).
NES_Oklo has the following features of importance for future evolutive MMES to be designed, built and operated by the mankind:
The limits of NES_Oklo were very well defined for all its lifecycle phases
During operation
Geometry stability of the core assured by the rocks configuration (the concrete part of any MMES)
Climate was stable in the parameters of the period
Interface with living organism was designed to be not only harmless, but also useful for both sides (cyanobacteria were prosperous for several millions of years).
During decommissioning
There was no migration beyond the site of the heavy radioactive solid waste.
The aerosols were actually part of the plan to rebuild the Earth atmosphere and generate new living beings—eukaryotes.
Apparently the design assumed how to better decommission it at the end of the lifecycle. Thinking of decommissioning from the research phase is a mandatory requirement for a well-designed MMES.
There is a fractal like design of the whole NES_Oklo reactor, as for instance the manner the following reactor functions were assured, as reflection at lower levels of the same principles:
Fuel load (uranium 235) to the reactor core, assured by cyanobacteria, as an intrinsic self-regulated process, in mirror with the operation of the whole reactor.
Diffusion of small distances in the specific rock of the site (several meters for more than 2 billion years [12]).
Radioactive radio-sols were part of the creation of new living organisms; therefore the containment was the whole atmosphere, without damaging it, but helping it.
There was an intrinsic safety assured by delayed neutrons, preventing transformation of the reactor into a bomb
The validity of reactor physics codes used for MMES was highly challenged. Although it seems so far that they could reproduce the reactor core design, there are yet issues to be clarified.
NES_Oklo has a direct impact on the lifecycle preparation of existing and future MMES, as follows:
Review the type of best plant control—centralized versus decentralized
Review of the safety analyses models for all the lifecycles and especially for decommissioning
Review existing researches on the future man machine interface for new reactors, role of artificial intelligence and the role of KP and generations to operate the plants
Set the goal of maximum simplification of MMES, counting to the highest extent possible on passive features and intrinsic safety protection.
Review the manner various phenomena are modeled for the reactor in coupled computer codes and either use higher computing capacities or simplify them
Design MMES as part of regional/global energy sources systems, integrated in the environment, based on ecological principles.
Several aspects from fundamental quantum mechanics and theory of relativity are yet to be reviewed, as the NES_Oklo measurements are challenging some of them
How constant is the alpha constant and the role of the amazing number 137 in the architecture of the universe
It appears that some constants are not so constant (for instance speed of light). If so the impact is very high on many aspects already considered confirmed and taboo to be challenged. An epistemic revolution is to be generated in Physics on the way to change the existing paradigms.
There is an amazing set of coincidences to have a reactor core designed (geological, biological, cosmic, etc.). If the rare coincidence might be more or less accepted, the troubling finding that the NES_Oklo is not the only one of this type leads to the debate about anaphatic and kataphatic approaches to the understanding of the Designer of the world.
The NES_Oklo operated from the design to decommissioning phase as a cybernetic machine understandable with high level cybernetics considering all the three levels from formulas (3)–(6)—real, simple complex and hyper-complex. The hyper-cybernetics, governed by the feedback control via the enantiotropy (entropy of the optimal ES states) is a very possible answer to previous questions. High level cybernetics—the cybernetics of CS states is indicated as describing such systems.
NES_Oklo raises a series of philosophical debates, too:
The evolution of life on Earth, the meaning of life and the role of randomness (if any) in its emergence and evolution.
The future of our civilization and how to use better the lessons so that to avoid destroying ourselves by the time we harness more and more powerful energy sources.
Why and how was it possible at a certain moment in time to have NES_Oklo? How to explain strange coincidences of NES_Oklo with eukaryotes, Earth terraforming and conditions for us to appear in the evolution (or what?) chain.
How to understand/manage messages for which we do not have yet the capability to understand, as they are from the category of hyper complex reality?
4. Conclusions
NES_Oklo had so far a significant impact on nuclear physics and nuclear engineering. However, its impact is yet to be completed, as new investigations and interdisciplinary works discover unexpected facts of the lessons transmitted by Oklo to us.
NES_Oklo is an example of how to build and operate an optimal, environmental friendly, for all lifecycle phases, nuclear fission reactor.
Summarizing, its lessons are related to:
Improvement of the design strategies for new MMES
Lessons on how to solve the waste management problem
The high advantages of using combined live-non alive elements in the fuel cycle
Foster the fundamental research in quantum mechanics, as the lessons are that, we are not yet understanding even basic aspects (as for instance the role of various universal constants)
Review the models we build for the Physics and ES and improve the KP for those aspects by using systematic approaches
\n',keywords:"natural energy system (NES), man made energy systems (MMES), risk, nuclear reactor, quantum mechanics, philosophy of science, speed of light, ecology, energy systems life cycle (ES_LC), knowledge process (KP), topological structure (K(i)), theory of a given topological structure (Th(K(i)), Backward engineering)",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/70864.pdf",chapterXML:"https://mts.intechopen.com/source/xml/70864.xml",downloadPdfUrl:"/chapter/pdf-download/70864",previewPdfUrl:"/chapter/pdf-preview/70864",totalDownloads:100,totalViews:0,totalCrossrefCites:0,dateSubmitted:"October 24th 2019",dateReviewed:"December 14th 2019",datePrePublished:"January 27th 2020",datePublished:null,dateFinished:null,readingETA:"0",abstract:"The chapter presents in a systematic manner the lessons learnt from the natural energy systems (NES) and their specific features. The conclusions are based on the evaluation of the risk impact on environment and for the improvement of the risk evaluation methodologies of such systems. A specific feature of the NES is the interdependence between them and society/mankind and the environment. Risk analyses for such systems have specific features underlined while compared with the features of the artificial (man-made) energy systems (MMES). Previous works illustrated in detail the NES versus MMES differences. This chapter presents the main aspects of such a review, when applied to a specific NES, the natural nuclear fission reactor in Oklo, Gabon (NES_Oklo). NES_Oklo operated about two billion years ago for about two hundreds millions of years. The lessons drawn from studying how this reactor was built, operated and self-decommissioned are of high importance for nuclear energy and not only. There are conclusions drawn from the study of Oklo reactor, which seem to shake some taboo issues in Physics, like for instance the light speed limit and other fundamental aspects of Quantum Mechanics, which have also important philosophical implications.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/70864",risUrl:"/chapter/ris/70864",signatures:"Dan Serbanescu",book:{id:"9385",title:"Renewable Energy - Technologies and Applications",subtitle:null,fullTitle:"Renewable Energy - Technologies and Applications",slug:null,publishedDate:null,bookSignature:"Associate Prof. Tolga Taner",coverURL:"https://cdn.intechopen.com/books/images_new/9385.jpg",licenceType:"CC BY 3.0",editedByType:null,editors:[{id:"197240",title:"Associate Prof.",name:"Tolga",middleName:null,surname:"Taner",slug:"tolga-taner",fullName:"Tolga Taner"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:[{id:"200401",title:"Dr.",name:"Dan",middleName:null,surname:"Serbanescu",fullName:"Dan Serbanescu",slug:"dan-serbanescu",email:"dan.serbanescu1953@yahoo.com",position:null,institution:null}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Evaluation method",level:"1"},{id:"sec_3",title:"3. Lessons learnt from NES_Oklo",level:"1"},{id:"sec_4",title:"4. Conclusions",level:"1"}],chapterReferences:[{id:"B1",body:'Kuroda PK. The Origin of the Chemical Elements and the Oklo Phenomenon. Springer; 1982. ISBN: 978-3-642-68667-2'},{id:"B2",body:'Gil L. IAEA Office of Public Information and Communication, Meet Oklo, the Earth’s Two-Billion-Year-Old Only Known Natural Nuclear Reactor. 2018. Available from: https://www.iaea.org/newscenter/news/meet-oklo-the-earths-two-billion-year-old-only-known-natural-nuclear-reactor'},{id:"B3",body:'https://en.wikipedia.org/wiki/Oklo'},{id:"B4",body:'Meshik AP. The Workings of an Ancient Nuclear Reactor. Scientific American on January 26, 2009, originally appeared in the October 2005. Available from: https://www.scientificamerican.com/article/ancient-nuclear-reactor/'},{id:"B5",body:'Lederman LM. Symmetry and the Beautiful Universe. Prometheus Books; 2004. ISBN: 9781591022428'},{id:"B6",body:'Serbanescu D. Selected Topics in Risk Analyses for Some Energy Systems. Lambert; 2015'},{id:"B7",body:'Spiridon LV, Serbanescu D, Sticlaru G. O Privire Asupra Unor Lectii de Cunoastere Date de Cavalcada Modelelor in f Zica. IYL; 2015'},{id:"B8",body:'Serbanescu D. Energetica si Fizica Nucleara Descoperiri, Accidente, Lectii ale Naturii, Comitetul Roman de Istoria si Filozofia Ştiinţei şi Tehnicii (CRIFST) Curs de Initiere in Istoria si Filozofia Stiintei Seria a IXoa. 2015'},{id:"B9",body:'Serbanescu D. On Some Natural Energy Systems and Lessons Learnt from Their Analysis [Despre Unele Sisteme Energetice Naturale si Invatamintele Studierii]. ISBN: 9783668669192; ISBN (Book): 9783668669208'},{id:"B10",body:'Șerbănescu D. An integrated perspective on knowledge and existence. In: Noema XVI; 2017. pp. 185-216'},{id:"B11",body:'Serbanescu D. On Some Unifying Approaches in Physics and Mind Sciences [Despre Unele Abordări Integratoare ale Fizicii și Științelor Minții—(A Main Text in Romanian with Extended Presentations in English of the Models and Results)]. DOI: 10.13240/RG.2.2.29959'},{id:"B12",body:'Yucca Mountain Project (YMP). Oklo: Natural Nuclear Reactors. Office of Civilian Radioactive Waste Management (OCRWM); 2004. Fact sheets. DOE. YMP-0010. Archived from the original on 2004'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Dan Serbanescu",address:"dan.serbanescu1953@yahoo.com;, dserbanescu@nuclearelectrica.ro",affiliation:'
Division of Logic and Models in Science and Technology of the Romanian Committee for Science and Technology, Romanian Academy, Romania
Romanian National Nuclear Electricity Company Nuclearelectrica SA, Romania
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