\r\n\tThis book aims to offer recent informations about devices and methods to design/validate the main automotive safety features, global trends, technology vision and market strategy. \r\n\tAll readers, from specialist engineers to academic researchers, will have the opportunity to address the automotive safety problems in a detailed way that welcomes practical examples; our work aims to cover all aspects of safety contents, from new product concept/development to final validation and performance assessment. \r\n\tEvery lifesaving technology is welcome as topic in this book project: active safety (preventive role in mitigating crashes/accidents by providing advance warning or additional vehicle steering/control assistance) and passive safety (system that monitors the injuries caused to driver, passengers and pedestrians in case of accident). \r\n\t“Active and Passive Safety in Automotive Industry” aims to discuss one of the most popular and fascinating topic of modern vehicle engineering; touching several research areas: software development/testing, electrical/electronic components with body and chassis parts.
",isbn:null,printIsbn:"979-953-307-X-X",pdfIsbn:null,doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"dc9628ac29352c7bbcf119d911c7d3c6",bookSignature:"Dr. Luigi Cocco",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/9392.jpg",keywords:"Passive Safety, Crash Testing, Occupant Restraint Controller, Pedestrian Protection, Active Safety, Forward Collision Warning, Automatic Emergency Brake, Safety Technology, Radar, Camera,, Passengers Safety, Driver Drowsiness Detection, Child Detection Module, Automatic Emergency Stop, Electronic Horizon, High Density Maps, Safe Dynamic Vehicle, Stability Control, Road Wet Detection",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"August 20th 2019",dateEndSecondStepPublish:"September 10th 2019",dateEndThirdStepPublish:"November 9th 2019",dateEndFourthStepPublish:"January 28th 2020",dateEndFifthStepPublish:"March 28th 2020",remainingDaysToSecondStep:"a year",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:null,coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"112023",title:"Dr.",name:"Luigi",middleName:null,surname:"Cocco",slug:"luigi-cocco",fullName:"Luigi Cocco",profilePictureURL:"https://mts.intechopen.com/storage/users/112023/images/system/112023.jpg",biography:'Dr. Luigi Cocco has received his master\'s degree in Telecommunication Engineering and his Ph.D. in Information Engineering before to join the automotive industry. Since 2005, From the Ferrari F1 Team to Automobili Lamborghini, he has worked on Electrical/Electronics systems; he has expertise in Research & Design, Supply Quality and Product Development. Currently, he is System Responsible for Passive Safety & ADAS of Maserati vehicles. His research interests include electronic measurements and digital signal processing, he has published several papers and three books with InTech: "Modern Metrology Concerns” (2012), "New Trends and Developments in Metrology” (2016) and "Standards, methods, and solutions of Metrology” (2018).',institutionString:"Maserati S.p.A.",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"0",totalChapterViews:"0",totalEditedBooks:"3",institution:null}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"11",title:"Engineering",slug:"engineering"}],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. 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\n\t\t\t
1. Introduction
\n\t\t\t
Mixed \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\t2\n\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\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\t control has received much attention in the past two decades, see Bernstein & Haddad (1989), Doyle et al. (1989b), Haddad et al. (1991), Khargonekar & Rotea (1991), Doyle et al. (1994), Limebeer et al. (1994), Chen & Zhou (2001) and references therein. The mixed \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\t2\n\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\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\t control problem involves the following linear continuous-time systems
where,\n\t\t\t\t\t\n\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\tt\n\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\t\tR\n\t\t\t\t\t\t\t\tn\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 is the state, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tt\n\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\t\tR\n\t\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\tis the control input, \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\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tt\n\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\t\tR\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\tq\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\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\tis one disturbance input, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tw\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tt\n\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\t\tR\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\tq\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\n\t\t\t\t\t\n\t\t\t\tis another disturbance input that belongs to\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\tL\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t0,\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, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ty\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tt\n\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\t\tR\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\n\t\t\t\tis the measured output.
\n\t\t\t
\n\t\t\t\tBernstein & Haddad (1989) presented a combined LQG/\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\t control problem. This problem is defined as follows: Given the stabilizable and detectable plant (1) with \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\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and the expected cost function
which satisfies the following design criteria: (i) the closed-loop system (1) (3) is stable; (ii) the closed-loop transfer matrix \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\tT\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tz\n\t\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\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t from the disturbance input \n\t\t\t\t\t\n\t\t\t\t\t\tw\n\t\t\t\t\t\n\t\t\t\tto the controlled output \n\t\t\t\t\t\n\t\t\t\t\t\tz\n\t\t\t\t\t\n\t\t\t\tsatisfies\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\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\t\n\t\t\t\t\t\t\t\t\t\t\t\tT\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\tz\n\t\t\t\t\t\t\t\t\t\t\t\t\tw\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\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\n\t\t\t\t\t\t\t<\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t; (iii) the expected cost function \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t(\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\tc\n\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\t\tB\n\t\t\t\t\t\t\t\tc\n\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\t\tC\n\t\t\t\t\t\t\t\tc\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\tis minimized; where, the disturbance input \n\t\t\t\t\t\n\t\t\t\t\t\tw\n\t\t\t\t\t\n\t\t\t\t is assumed to be a Gaussian white noise. Bernstein & Haddad (1989) considered merely the combined LQG/\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\t control problem in the special case of \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\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\t1\n\t\t\t\t\t\t\t\tT\n\t\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\t1\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 and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t12\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t12\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 and\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\tC\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t12\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\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. Since the expected cost function \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tJ\n\t\t\t\t\t\t\t(\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\tc\n\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\t\tB\n\t\t\t\t\t\t\t\tc\n\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\t\tC\n\t\t\t\t\t\t\t\tc\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 equals the square of the \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\t2\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-norm of the closed-loop transfer matrix \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\tT\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tz\n\t\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\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t in this case, the combined LQG/\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\t problem by Bernstein & Haddad (1989) has been recognized to be a mixed \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\t2\n\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\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\t problem. In Bernstein & Haddad (1989), they considered the minimization of an “upper bound” 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\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\t\n\t\t\t\t\t\t\t\t\t\t\t\tT\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\tz\n\t\t\t\t\t\t\t\t\t\t\t\t\tw\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\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\t2\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\n\t\t\t\t subject to\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\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\t\n\t\t\t\t\t\t\t\t\t\t\t\tT\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\tz\n\t\t\t\t\t\t\t\t\t\t\t\t\tw\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\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\n\t\t\t\t\t\t\t<\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, and solved this problem by using Lagrange multiplier techniques. Doyle et al. (1989b) considered a related output feedback mixed \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\t2\n\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\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\t problem (also see Doyle et al. 1994). The two approaches have been shown in Yeh et al. (1992) to be duals of one another in some sense. Haddad et al. (1991) gave sufficient conditions for the exstence of discrete-time static output feedback mixed \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\t2\n\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\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 in terms of coupled Riccati equations. In Khargonekar & Rotea (1991), they presented a convex optimisation approach to solve output feedback mixed \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\t2\n\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\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\t problem. In Limebeer et al. (1994), they proposed a Nash game approach to the state feedback mixed \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\t2\n\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\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\t problem, and gave necessary and sufficient conditions for the existence of a solution of this problem. Chen & Zhou (2001) generalized the method of Limebeer et al. (1994) to output feedback multiobjective \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\t2\n\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\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\t problem. However, up till now, no approach has involved the combined LQG/\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\t control problem (so called stochastic mixed LQR/\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\t control problem) for linear continuous-time systems (1) with the expected cost function (2), where, \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\t≥\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tand \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t>\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tare the weighting matrices, \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\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tt\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\tis a Gaussian white noise, and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tw\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tt\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\tis a disturbance input that belongs to\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\tL\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t0,\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.
\n\t\t\t
In this chapter, we consider state feedback stochastic mixed LQR/\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\t control problem for linear discrete-time systems. The deterministic problem corresponding to this problem (so called mixed LQR/\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\t control problem) was first considered by Xu (2006). In Xu (2006), an algebraic Riccati equation approach to state feedback mixed quadratic guaranteed cost and \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\t control problem (so called state feedback mixed QGC/\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\t control problem) for linear discrete-time systems with uncertainty was presented. When the parameter uncertainty equals zero, the discrete-time state feedback mixed QGC/\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\t control problem reduces to the discrete-time state feedback mixed LQR/\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\t control problem. Xu (2011) presented respectively a state space approach and an algebraic Riccati equation approach to discrete-time state feedback mixed LQR/\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\t control problem, and gave a sufficient condition for the existence of an admissible state feedback controller solving this problem.
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On the other hand, Geromel & Peres (1985) showed a new stabilizability property of the Riccati equation solution, and proposed, based on this new property, a numerical procedure to design static output feedback suboptimal LQR controllers for linear continuous-time systems. Geromel et al. (1989) extended the results of Geromel & Peres (1985) to linear discrete-time systems. In the fact, comparing this new stabilizability property of the Riccati equation solution with the existing results (de Souza & Xie 1992, Kucera & de Souza 1995, Gadewadikar et al. 2007, Xu 2008), we can show easily that the former involves sufficient conditions for the existence of all state feedback suboptimal LQR controllers. Untill now, the technique of finding all state feedback controllers by Geromel & Peres (1985) has been extended to various control problems, such as, static output feedback stabilizability (Kucera & de Souza 1995), \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\tcontrol problem for linear discrete-time systems (de Souza & Xie 1992), \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\tcontrol problem for linear continuous-time systems (Gadewadikar et al. 2007), mixed LQR/\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\t control problem for linear continuous-time systems (Xu 2008).
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The objective of this chapter is to solve discrete-time state feedback stochastic mixed LQR/\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\t control problem by combining the techniques of Xu (2008 and 2011) with the well known LQG theory. There are three motivations for developing this problem. First, Xu (2011) parametrized a central controller solving the discrete-time state feedback mixed LQR/\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\t control problem in terms of an algebraic Riccati equation. However, no stochastic interpretation was provided. This paper thus presents a central solution to the discrete-time state feedback stochastic mixed LQR/\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\t control problem. This result may be recognied to be a stochastic interpretation of the discrete-time state feedback mixed LQR/\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\t control problem considered by Xu (2011). The second motivation for our paper is to present a characterization of all admissible state feedback controllers for solving discrete-time stochastic mixed LQR/\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\t control problem for linear continuous-time systems in terms of a single algebraic Riccati equation with a free parameter matrix, plus two constrained conditions: One is a free parameter matrix constrained condition on the form of the gain matrix, another is an assumption that the free parameter matrix is a free admissible controller error. The third motivation for our paper is to use the above results to solve the discrete-time static output feedback stochastic mixed LQR/\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\t control problem.
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This chapter is organized as follows: Section 2 introduces several preliminary results. In Section 3, first,we define the state feedback stochastic mixed LQR/\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\t control problem for linear discrete-time systems. Secondly, we give sufficient conditions for the existence of all admissible state feedback controllers solving the discrete-time stochastic mixed LQR/\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\t control problem. In the rest of this section, first, we parametrize a central discrete-time state feedback stochastic mixed LQR/\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\t controller, and show that this result may be recognied to be a stochastic interpretation of discrete-time state feedback mixed LQR/\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\t control problem considered by Xu (2011). Secondly, we propose a numerical algorithm for calclulating a kind of discrete-time state feedback stochastic mixed LQR/\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\t controllers. Also, we compare our main result with the related well known results. As a special case, Section 5 gives sufficient conditions for the existence of all admissible static output feedback controllers solving the discrete-time stochastic mixed LQR/\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\t control problem, and proposes a numerical algorithm for calculating a discrete-time static output feedback stochastic mixed LQR/\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\t controller. In Section 6, we give two examples to illustrate the design procedures and their effectiveness. Section 7 is conclusion.
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2. Preliminaries
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In this section, we will review several preliminary results. First, we introduce the new stabilizability property of Riccati equation solutions for linear discrete-time systems which was presented by Geromel et al. (1989). This new stabilizability property involves the following linear discrete-time systems
where,\n\t\t\t\t\t\n\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\t\n\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\tn\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 is the state, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tu\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\t\n\t\t\t\t\t\t\t\tR\n\t\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\tis the control input, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ty\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\t\n\t\t\t\t\t\t\t\tR\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\n\t\t\t\tis the measured output, \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\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≥\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tand\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t>\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. We make the following assumptions
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\n\t\t\t\tAssumption 2.1\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\tA\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\tB\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 is controllable.
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\n\t\t\t\tAssumption 2.2\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\tA\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tQ\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\t1\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\t)\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is observable.
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Define a discrete-time Riccati equation as follows:
\n\t\t\t\tGeromel & Peres (1985) showed a new stabilizability property of the Riccati equation solution, and proposed, based on this new property, a numerical procedure to design static output feedback suboptimal LQR controllers for linear continuous-time systems. Geromel et al. (1989) extended this new stabilizability property displayed in Geromel & Peres (1985) to linear discrete-time systems. This resut is given by the following theorem.
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\n\t\t\t\tTheorem 2.1 (Geromel et al. 1989) For the matrix \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t∈\n\t\t\t\t\t\t\t\n\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\tm\n\t\t\t\t\t\t\t\t\t×\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\n\t\t\t\t such that
holds, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t∈\n\t\t\t\t\t\t\t\n\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\tn\n\t\t\t\t\t\t\t\t\t×\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\n\t\t\t\tis a positive definite solution of the modified discrete-time Riccati equation
Then the matrix \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\tA\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\tB\n\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\tis stable.
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When these conditions are met, the quadratic cost function \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\tJ\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\n\t\t\t\t is given by
\n\t\t\t\tLemma 2.1 (Discrete Time Bounded Real Lemma)\n\t\t\t
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Suppose that\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t>\n\t\t\t\t\t\t\t0\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\tM\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tz\n\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\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\t\n\t\t\t\t\t\t\t\t\t\t\t\tA\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\tB\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\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\tC\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\tD\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\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\tR\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\t, then the following two statements are equivalent:
ii. There exists a stabilizing solution \n\t\t\t\t\t\n\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\t0\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\tX\n\t\t\t\t\t\t\t>\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tif \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\tC\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\tA\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\tis observable ) to the discrete-time Riccati equation
such that\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\tU\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tγ\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\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\t(\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tX\n\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t>\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t.
\n\t\t\t
Next, we will consider the following linear discrete-time systems
where,\n\t\t\t\t\t\n\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\t\n\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\tn\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 is the state, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tu\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\t\n\t\t\t\t\t\t\t\tR\n\t\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\tis the control input, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tw\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\t\n\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\tq\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\tis the disturbance input that belongs to\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\tL\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t0,\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,\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tz\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\t\n\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\tp\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 is the controlled output. Let\n\t\t\t\t\t\n\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\t0\n\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\t\tx\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\n\t\t\t\t.
\n\t\t\t
The associated with this systems is the quadratic performance index
The closed-loop transfer matrix from the disturbance input \n\t\t\t\t\t\n\t\t\t\t\t\tw\n\t\t\t\t\t\n\t\t\t\t to the controlled output \n\t\t\t\t\t\n\t\t\t\t\t\tz\n\t\t\t\t\t\n\t\t\t\t is
The following lemma is an extension of the discrete-time bounded real lemma ( see Xu 2011).
\n\t\t\t
Lemma 2.2 Given the system (10) under the influence of the state feedback (11), and suppose that\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t>\n\t\t\t\t\t\t\t0\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\tT\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tz\n\t\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\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t∈\n\t\t\t\t\t\t\tR\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\t; then there exists an admissible controller \n\t\t\t\t\t\n\t\t\t\t\t\tK\n\t\t\t\t\t\n\t\t\t\t such that \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\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\t\n\t\t\t\t\t\t\t\t\t\t\t\tT\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\tz\n\t\t\t\t\t\t\t\t\t\t\t\t\tw\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\tz\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\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\n\t\t\t\t\t\t\t<\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t if there exists a stabilizing solution \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\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\t≥\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t to the discrete time Riccati equation
such that\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\tU\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tγ\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\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\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\tT\n\t\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\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\n\t\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\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t.
\n\t\t\t
\n\t\t\t\tProof: See the proof of Lemma 2.2 of Xu (2011). Q.E.D.
\n\t\t\t
Finally, we review the result of discrete-time state feedback mixed LQR/\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\t control problem. Xu (2011) has defined this problem as follows: Given the linear discrete-time systems (10)(11) with \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tw\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\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\tL\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t[0\n\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\n\t\t\t\t\t\n\t\t\t\tand\n\t\t\t\t\t\n\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\t0\n\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\t\tx\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\n\t\t\t\t, for a given number \n\t\t\t\t\t\n\t\t\t\t\t\tγ\n\t\t\t\t\t\n\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\t0,\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t determine an admissible controller that achieves
If this controller \n\t\t\t\t\t\n\t\t\t\t\t\tK\n\t\t\t\t\t\n\t\t\t\t exists, it is said to be a discrete-time state feedback mixed LQR/\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\t controller.
\n\t\t\t
The following assumptions are imposed on the system
\n\t\t\t
\n\t\t\t\tAssumption 2.3\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\n\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\tA\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 is detectable.
\n\t\t\t
\n\t\t\t\tAssumption 2.4\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\tA\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t2\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 is stabilizable.
\n\t\t\t\tXu (2011) has provided a solution to discrete-time state feedback mixed LQR/\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\tcontrol problem, this result is given by the following theorem.
\n\t\t\t
\n\t\t\t\tTheorem 2.2 There exists a discrete-time state feedback mixed LQR/\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\t controller if the discrete-time Riccati equation (14) has a stabilizing solution \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\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\n\t\t\t\t and \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\tU\n\t\t\t\t\t\t\t\t1\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\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\tI\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tγ\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\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\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tT\n\t\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\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t>\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t.
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Moreover, this discrete-time state feedback mixed LQR/\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\tcontroller is given by
In this case, the discrete-time state feedback mixed LQR/\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\tcontroller will achieve
where,\n\t\t\t\t\t\n\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\t\n\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\tn\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 is the state, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tu\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\t\n\t\t\t\t\t\t\t\tR\n\t\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\tis the control input, \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\t0\n\t\t\t\t\t\t\t\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\t\n\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\t\n\t\t\t\t\t\t\t\t\t\tq\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\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\tis one disturbance input, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tw\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\t\n\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\t\n\t\t\t\t\t\t\t\t\t\tq\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\n\t\t\t\t\t\n\t\t\t\tis another disturbance that belongs to\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\tL\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t0,\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,\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tz\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\t\n\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\tp\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 is the controlled output, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ty\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\t\n\t\t\t\t\t\t\t\tR\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\n\t\t\t\tis the measured output.
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It is assumed that \n\t\t\t\t\t\n\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\t0\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\tis Gaussian with mean and covariance given by
The noise process \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\t0\n\t\t\t\t\t\t\t\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\n\t\t\t\t\t\n\t\t\t\t is a Gaussain white noise signal with properties
Furthermore, \n\t\t\t\t\t\n\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\t0\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\tand \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\t0\n\t\t\t\t\t\t\t\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\n\t\t\t\t\t\n\t\t\t\t are assumed to be independent, \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\t0\n\t\t\t\t\t\t\t\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\n\t\t\t\t\t\n\t\t\t\tand \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tw\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\n\t\t\t\t\t\n\t\t\t\t are also assumed to be independent, where, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tE\n\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\n\t\t\t\t\t\n\t\t\t\tdenotes expected value.
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Also, we make the following assumptions:
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\n\t\t\t\tAssumption 3.1\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\n\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\tA\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 is detectable.
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\n\t\t\t\tAssumption 3.2\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\tA\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t2\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 is stabilizable.
where,\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\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≥\n\t\t\t\t\t\t\t0\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\tR\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t>\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t , and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t>\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is a given number.
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As is well known, a given controller \n\t\t\t\t\t\n\t\t\t\t\t\tK\n\t\t\t\t\t\n\t\t\t\t is called admissible (for the plant\n\t\t\t\t\t\n\t\t\t\t\t\tG\n\t\t\t\t\t\n\t\t\t\t) if \n\t\t\t\t\t\n\t\t\t\t\t\tK\n\t\t\t\t\t\n\t\t\t\t is real-rational proper, and the minimal realization of \n\t\t\t\t\t\n\t\t\t\t\t\tK\n\t\t\t\t\t\n\t\t\t\t internally stabilizes the state space realization (15) of\n\t\t\t\t\t\n\t\t\t\t\t\tG\n\t\t\t\t\t\n\t\t\t\t.
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Recall that the discrete-time state feedback optimal LQG problem is to find an admissible controller that minimizes the expected quadratic cost function (17) subject to the systems (15) (16) with\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tw\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\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, while the discrete-time state feedback \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\t control problem is to find an admissible controller such that \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\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\t\n\t\t\t\t\t\t\t\t\t\t\t\tT\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\tz\n\t\t\t\t\t\t\t\t\t\t\t\t\tw\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\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\n\t\t\t\t\t\t\t<\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t subject to the systems (15) (16) for a given number\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t>\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. While we combine the two problems for the systems (15) (16) with \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tw\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\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\tL\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t0,\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, the expected cost function (17) is a function of the control input \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tu\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\n\t\t\t\t\t\n\t\t\t\t and disturbance input \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tw\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\n\t\t\t\t\t\n\t\t\t\t in the case of \n\t\t\t\t\t\n\t\t\t\t\t\tγ\n\t\t\t\t\t\n\t\t\t\t being fixed and \n\t\t\t\t\t\n\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\t0\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\tbeing Gaussian with known statistics and \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\t0\n\t\t\t\t\t\t\t\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\n\t\t\t\t\t\n\t\t\t\t being a Gaussain white noise with known statistics. Thus it is not possible to pose a discrete-time state feedback stochastic mixed LQR/\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\t control problem that achieves the minimization of the expected cost function (17) subject to \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\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\t\n\t\t\t\t\t\t\t\t\t\t\t\tT\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\tz\n\t\t\t\t\t\t\t\t\t\t\t\t\tw\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\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\n\t\t\t\t\t\t\t<\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t for the systems (15) (16) with \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tw\n\t\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\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t0,\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 because the expected cost function (17) is an uncertain function depending on disturbance input\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tw\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\n\t\t\t\t\t\n\t\t\t\t. In order to eliminate this difficulty, the design criteria of discrete-time state feedback stochastic mixed LQR/\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\t control problem should be replaced by the following design criteria:
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\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\t\n\t\t\t\t\t\t\t\t\tsup\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\t∈\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\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\n\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\tinf\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\n\t\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\tE\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 subject to \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\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\t\n\t\t\t\t\t\t\t\t\t\t\t\tT\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\tz\n\t\t\t\t\t\t\t\t\t\t\t\t\tw\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\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\n\t\t\t\t\t\t\t<\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t
\n\t\t\t
because for all\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tw\n\t\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\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t0,\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, the following inequality always exists.
Based on this, we define the discrete-time state feedback stochastic mixed LQR/\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\t control problem as follows:
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Discrete-time state feedback stochastic mixed LQR/\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\t control problem: Given the linear discrete-time systems (15) (16) satisfying Assumption 3.1-3.3 with \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tw\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\t\n\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t0,\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 and the expected cost functions (17), for a given number\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t>\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, find all admissible state feedback controllers \n\t\t\t\t\t\n\t\t\t\t\t\tK\n\t\t\t\t\t\n\t\t\t\t such that
\n\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\t\n\t\t\t\t\t\t\t\t\tsup\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\t∈\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\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\n\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\t\tJ\n\t\t\t\t\t\t\t\tE\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 subject to \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\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\t\n\t\t\t\t\t\t\t\t\t\t\t\tT\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\tz\n\t\t\t\t\t\t\t\t\t\t\t\t\tw\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\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\n\t\t\t\t\t\t\t<\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t
\n\t\t\t
where, \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\tT\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tz\n\t\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\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tz\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\tis the closed loop transfer matrix from the disturbance input \n\t\t\t\t\t\n\t\t\t\t\t\tw\n\t\t\t\t\t\n\t\t\t\tto the controlled output\n\t\t\t\t\t\n\t\t\t\t\t\tz\n\t\t\t\t\t\n\t\t\t\t.
\n\t\t\t
If all these admissible controllers exist, then one of them \n\t\t\t\t\t\n\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\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 will achieve the design criteria
\n\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\t\n\t\t\t\t\t\t\t\t\tsup\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\t∈\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\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\n\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\tinf\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\n\t\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\tE\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 subject to \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\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\t\n\t\t\t\t\t\t\t\t\t\t\t\tT\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\tz\n\t\t\t\t\t\t\t\t\t\t\t\t\tw\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\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\n\t\t\t\t\t\t\t<\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t
\n\t\t\t
and it is said to be a central discrete-time state feedback stochastic mixed LQR/\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\t controller.
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\n\t\t\t\tRemark 3.1 The discrete-time state feedback stochastic mixed LQR/\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\t control problem defined in the above is also said to be a discrete-time state feedback combined LQG/\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\t control problem in general case. When the disturbance input\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tw\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\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, this problem reduces to a discrete-time state feedback combined LQG/\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\t control problem arisen from Bernstein & Haddad (1989) and Haddad et al. (1991).
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\n\t\t\t\tRemark 3.2 In the case of\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tw\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\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, it is easy to show (see Bernstein & Haddad 1989, Haddad et al. 1991) that \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\tJ\n\t\t\t\t\t\t\t\tE\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 in (17) is equivalent to the expected cost function
Define \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\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\t1\n\t\t\t\t\t\t\t\tT\n\t\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\t1\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 and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t12\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t12\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 and suppose that\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\tC\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t12\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\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, then \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\tJ\n\t\t\t\t\t\t\t\tE\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 may be rewritten as
where,\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\tT\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tw\n\t\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\t(\n\t\t\t\t\t\t\tz\n\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\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\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\tK\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\tB\n\t\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\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\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\tC\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\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\t0\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\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. If \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\t0\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 is white noise with indensity matrix \n\t\t\t\t\t\n\t\t\t\t\t\tI\n\t\t\t\t\t\n\t\t\t\t and the closed-loop systems is stable then
This implies that the discrete-time state feedback combined LQG/\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\t control problem in the special case of \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\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\t1\n\t\t\t\t\t\t\t\tT\n\t\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\t1\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 and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t12\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t12\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 and \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\tC\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t12\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\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t arisen from Bernstein & Haddad (1989) and Haddad et al. (1991) is a mixed \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\t2\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\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\t control problem.
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Based on the above definition, we give sufficient conditions for the existence of all admissible state feedback controllers solving the discrete-time stochastic mixed LQR/\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\t control problem by combining the techniques of Xu (2008 and 2011) with the well known LQG theory. This result is given by the following theorem.
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\n\t\t\t\tTheorem 3.1 There exists a discrete-time state feedback stochastic mixed LQR/ \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\t controller if the following two conditions hold:
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i. There exists a matrix \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t such that
and \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\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\n\t\t\t\t is a symmetric non-negative definite solution of the following discrete-time Riccati equation
and \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\tA\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\tc\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t−\n\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\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\tB\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\tT\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\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\tB\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\tR\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\t\t−\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\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\n\t\t\t\t\t\t\t\tT\n\t\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\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is stable and\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\tU\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tγ\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\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\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tT\n\t\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\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t>\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t;
ii. \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis an admissible controller error.
\n\t\t\t
In this case, the discrete-time state feedback stochastic mixed LQR/\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\t controller will achieve
\n\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\t\n\t\t\t\t\t\t\t\t\tsup\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\t∈\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\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\n\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\t\tJ\n\t\t\t\t\t\t\t\tE\n\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\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tlim\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\tT\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\t\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tT\n\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\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\t0\n\t\t\t\t\t\t\t\t\t\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\t\n\t\t\t\t\t\t\t\t\tt\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\n\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\tT\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\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\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\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 subject to \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\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\t\n\t\t\t\t\t\t\t\t\t\t\t\tT\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\tz\n\t\t\t\t\t\t\t\t\t\t\t\t\tw\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\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\n\t\t\t\t\t\t\t<\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t
\n\t\t\t
\n\t\t\t\tRemark 3.3 In Theorem 3.1, the controller error is defined to be the state feedback controller \n\t\t\t\t\t\n\t\t\t\t\t\tK\n\t\t\t\t\t\n\t\t\t\t minus the suboptimal controller\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\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\t=\n\t\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\t2\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\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\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\tT\n\t\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\t3\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, where, \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\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\t≥\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tsatisfies the discrete-time Riccati equation (20), that is,
where, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis the controller error, \n\t\t\t\t\t\n\t\t\t\t\t\tK\n\t\t\t\t\t\n\t\t\t\tis the state feedback controller and \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\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 is the suboptimal controller. Suppose that there exists a suboptimal controller \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\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 such that \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\tA\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\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=\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\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 is stable, then \n\t\t\t\t\t\n\t\t\t\t\t\tK\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is respectively said to be an admissible controller and an admissible controller error if it belongs to the set
\n\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\tΩ\n\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\tΔ\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\t\tA\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\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+\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t is stable \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
\n\t\t\t
\n\t\t\t\tRemark 3.4 The discrete-time state feedback stochastic mixed LQR/\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\t controller satisfying the conditions i-ii displayed in Theorem 3.1 is not unique. All admissible state feedback controllers satisfying these two conditions lead to all discrete-time state feedback stochastic mixed LQR/\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\t controllers.
\n\t\t\t
Astrom (1971) has given the mean value of a quadratic form of normal stochastic variables. This result is given by the following lemma.
\n\t\t\t
\n\t\t\t\tLemma 3.1 Let \n\t\t\t\t\t\n\t\t\t\t\t\tx\n\t\t\t\t\t\n\t\t\t\t be normal with mean \n\t\t\t\t\t\n\t\t\t\t\t\tm\n\t\t\t\t\t\n\t\t\t\t and covariance\n\t\t\t\t\t\n\t\t\t\t\t\tR\n\t\t\t\t\t\n\t\t\t\t. Then
For convenience, let\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\tA\n\t\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\tA\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t2\n\t\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\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\tB\n\t\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\t\n\t\t\t\t\t\t\t\tB\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\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\tC\n\t\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\t\n\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t1\n\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\t\tD\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t12\n\t\t\t\t\t\t\t\t\n\t\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\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\tA\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\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=\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\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 ,\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\tB\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\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=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\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\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\tC\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\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=\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\t1\n\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\t\tD\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t12\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\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, and\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\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\t=\n\t\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\t2\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\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\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\tT\n\t\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\t3\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, where, \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\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\t≥\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tsatisfies the discrete-time Riccati equation (20); then we have the following lemma.
\n\t\t\t
\n\t\t\t\tLemma 3.2 Suppose that the conditions i-ii of Theorem 3.1 hold, then the both \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\tA\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\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\n\t\t\t\t and \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\tA\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 are stable.
\n\t\t\t
Proof: Suppose that the conditions i-ii of Theorem 3.1 hold, then it can be easily shown by using the similar standard matrix manipulations as in the proof of Theorem 3.1 in de Souza & Xie (1992) that
where,\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\tB\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\t1\n\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\t\tγ\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\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\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tT\n\t\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\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\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\n\t\t\t\t.Thus we have
Since the discrete-time Riccati equation (20) has a symmetric non-negative definite solution \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\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\n\t\t\t\t and \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\tA\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\tc\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t−\n\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\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\tB\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\tT\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\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\tB\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\tR\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\t\t−\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\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\n\t\t\t\t\t\t\t\tT\n\t\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\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is stable, and we can show that \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\tA\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\tc\n\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\t\tA\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\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+\n\t\t\t\t\t\t\n\t\t\t\t\t\n\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\tγ\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\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\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\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\n\t\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\t1\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\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\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\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\n\t\t\t\t\t\t\t\tT\n\t\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\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\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\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, the discrete-time Riccati equation (21) also has a symmetric non-negative definite solution \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\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\n\t\t\t\t and \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\tA\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\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+\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tγ\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\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\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\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\n\t\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\t1\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\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\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\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\n\t\t\t\t\t\t\t\tT\n\t\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\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\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\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 also is stable. Hence, \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\n\t\t\t\t\t\t\t\tU\n\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\t−\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\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\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\n\t\t\t\t\t\t\t\tT\n\t\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\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\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\n\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\t\tA\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\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)\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis detectable. Based on this, it follows from standard results on Lyapunov equations (see Lemma 2.7 a), Iglesias & Glover 1991) that \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\tA\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\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\n\t\t\t\t is stable. Also, note that \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is an admissible controller error, so \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\tA\n\t\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\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\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\n\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\t\tB\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is stable. Q. E. D.
\n\t\t\t
Proof of Theorem 3.1: Suppose that the conditions i-ii hold, then it follows from Lemma 3.2 that the both \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\tA\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\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\n\t\t\t\t and \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\tA\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 are stable. This implies that\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\tT\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tz\n\t\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\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t∈\n\t\t\t\t\t\t\tR\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\t.
\n\t\t\t
Define\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tV\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\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\tT\n\t\t\t\t\t\t\t\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\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\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\n\t\t\t\t\t\n\t\t\t\t, where, \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\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\n\t\t\t\tis the solution to the discrete-time Riccati equation (20), then taking the difference\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\tV\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, we get
On the other hand, we can rewrite the discrete-time Riccati equation (20) by using the same standard matrix manipulations as in the proof of Lemma 3.2 as follows:
It follows from Lemma 2.2 that\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\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\t\n\t\t\t\t\t\t\t\t\t\t\t\tT\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\tz\n\t\t\t\t\t\t\t\t\t\t\t\t\tw\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\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\n\t\t\t\t\t\t\t<\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t.Completing the squares for (22) and substituting (23) in (22), we get
Note that \n\t\t\t\t\t\n\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\t∞\n\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\t\t\n\t\t\t\t\t\t\t\t\tlim\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\tT\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\tx\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tand
\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\t\n\t\t\t\t\t\t\t\t\tsup\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\t∈\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\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\n\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\t\tJ\n\t\t\t\t\t\t\t\tE\n\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\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tlim\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\tT\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\t\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tT\n\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\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\t0\n\t\t\t\t\t\t\t\t\t\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\t\n\t\t\t\t\t\t\t\t\tt\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\n\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\tT\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\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\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\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 subject to \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\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\t\n\t\t\t\t\t\t\t\t\t\t\t\tT\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\tz\n\t\t\t\t\t\t\t\t\t\t\t\t\tw\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\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\n\t\t\t\t\t\t\t<\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t Q.E.D.
\n\t\t\t
In the rest of this section, we give several discussions.
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\n\t\t\t\tA. A Central Discrete-Time State Feedback Stochastic Mixed LQR/\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\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\t\n\t\t\t\tController\n\t\t\t
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We are to find a central solution to the discrete-time state feedback stochastic mixed LQR/\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\t control problem.This central solution involves the discrete-time Riccati equation
where, \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\tB\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\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\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\tγ\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\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\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\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\t1\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\tB\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\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\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\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\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\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\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\tI\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\t0\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\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\t0\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\tR\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\tI\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\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. Using the similar argument as in the proof of Theorem 3.1 in Xu (2011), the expected cost function \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\tJ\n\t\t\t\t\t\t\t\tE\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 can be rewritten as:
If\n\t\t\t\t\t\n\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\t\n\t\t\t\t\t\t\t\tU\n\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\t−\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\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\tT\n\t\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\t3\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, then we get that
by using Lemma 3.1 and the similar argument as in the proof of Theorem 3.1. Thus, we have the following theorem:
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\n\t\t\t\tTheorem 3.2 There exists a central discrete-time state feedback stochastic mixed LQR/\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\t controller if the discrete-time Riccati equation (24) has a stabilizing solution \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\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\t≥\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and\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\tU\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tγ\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\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\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tT\n\t\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\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t>\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t.
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Moreover, if this condition is met, the central discrete-time state feedback stochastic mixed LQR/ \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\t controller is given by
In this case, the central discrete-time state feedback stochastic mixed LQR/\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\t controller will achieve
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\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\t\n\t\t\t\t\t\t\t\t\tsup\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\t∈\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\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\n\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\tinf\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\n\t\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\tE\n\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\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tlim\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\tT\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\t\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tT\n\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\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\t0\n\t\t\t\t\t\t\t\t\t\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\t\n\t\t\t\t\t\t\t\t\tt\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\n\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\tT\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\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\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\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 subject to \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\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\t\n\t\t\t\t\t\t\t\t\t\t\t\tT\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\tz\n\t\t\t\t\t\t\t\t\t\t\t\t\tw\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\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\n\t\t\t\t\t\t\t<\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t
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\n\t\t\t\tRemark 3.5 When\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, Theorem 3.1 reduces to Theorem 3.2.
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\n\t\t\t\tRemark 3.6 Notice that the condition displayed in Theorem 3.2 is the same as one displayed in Theroem 2.2. This implies that the result given by Theorem 3.2 may be recognied to be a stochastic interpretation of the discrete-time state feedback mixed LQR/\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\t control problem considered by Xu (2011).
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\n\t\t\t\tB. Numerical Algorithm\n\t\t\t
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In order to calculate a kind of discrete-time state feedback stochastic mixed LQR/ \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\t controllers, we propose the following numerical algorithm.
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\n\t\t\t\tAlgorithm 3.1\n\t\t\t
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Step 1: Fix the two weighting matrices \n\t\t\t\t\t\n\t\t\t\t\t\tQ\n\t\t\t\t\t\n\t\t\t\t and\n\t\t\t\t\t\n\t\t\t\t\t\tR\n\t\t\t\t\t\n\t\t\t\t, set\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0\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\tΔ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0\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\tU\n\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\t(\n\t\t\t\t\t\t\t\t\ti\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\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tand a small scalar\n\t\t\t\t\t\n\t\t\t\t\t\tδ\n\t\t\t\t\t\n\t\t\t\t, and a matrix \n\t\t\t\t\t\n\t\t\t\t\t\tM\n\t\t\t\t\t\n\t\t\t\twhich is not zero matrix of appropriate dimensions.
for \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\tX\n\t\t\t\t\t\t\t\ti\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 symmetric non-negative definite such that
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\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\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\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tc\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\t=\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t−\n\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\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\tB\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\tT\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\ti\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\tB\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\tR\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\t\t−\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\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\n\t\t\t\t\t\t\t\tT\n\t\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\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is stable and\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\tU\n\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\ti\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\tI\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tγ\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\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\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tT\n\t\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\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t>\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t.
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Step 3: Calculate\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\tU\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\ti\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, \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\tU\n\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\t(\n\t\t\t\t\t\t\t\t\ti\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\tand \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\tK\n\t\t\t\t\t\t\t\ti\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 by using the following formulas
Step 4: Let \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\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\ti\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\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t(or\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\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\ti\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\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t) and\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\tU\n\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\t(\n\t\t\t\t\t\t\t\t\ti\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\n\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\t\tU\n\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\t(\n\t\t\t\t\t\t\t\t\ti\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.
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Step 5: If \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\tA\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\ti\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 is stable, that is, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\ti\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\tis an admissible controller error, then increase \n\t\t\t\t\t\n\t\t\t\t\t\ti\n\t\t\t\t\t\n\t\t\t\t by\n\t\t\t\t\t\n\t\t\t\t\t\t1\n\t\t\t\t\t\n\t\t\t\t, goto Step 2; otherwise stop.
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Using the above algorithm, we obtain a kind of discrete-time state feedback stochastic mixed LQR/\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\t controllers as follows:
\n\t\t\t\tC. Comparison with Related Well Known Results\n\t\t\t
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Comparing the result displayed in Theorem 3.1 with the earlier results, such as, Geromel & Peres (1985), Geromel et al. (1989), de Souza & Xie (1992), Kucera & de Souza (1995) and Gadewadikar et al. (2007); we know easily that all these earlier results are given in terms of a single algebraic Riccati equation with a free parameter matrix, plus a free parameter constrained condition on the form of the gain matrix. Although the result displayed in Theorem 3.1 is also given in terms of a single algebraic Riccati equation with a free parameter matrix, plus a free parameter constrained condition on the form of the gain matrix; but the free parameter matrix is also constrained to be an admissible controller error. In order to give some interpretation for this fact, we provided the following result of discrete-time state feedback stochastic mixed LQR/\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\t control problem by combining directly the proof of Theorem 3.1, and the technique of finding all admissible state feedback controllers by Geromel & Peres (1985) ( also see Geromel et al. 1989, de Souza & Xie 1992, Kucera & de Souza 1995).
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\n\t\t\t\tTheorem 3.3 There exists a state feedback stochastic mixed LQR/\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\tcontroller if there exists a matrix \n\t\t\t\t\t\n\t\t\t\t\t\tL\n\t\t\t\t\t\n\t\t\t\t such that
and \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\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\n\t\t\t\t is a symmetric non-negative definite solution of the following discrete-time Riccati equation
and \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\tA\n\t\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\t\n\t\t\t\t\t\t\t\tγ\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\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\t\n\t\t\t\t\t\t\t\tB\n\t\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\t\tU\n\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\t−\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\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\tT\n\t\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\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\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 is stable and\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\tU\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tγ\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\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\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tT\n\t\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\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t>\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t.
Note that\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\tA\n\t\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\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\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\n\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\t\tB\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\tK\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\tK\n\t\t\t\t\t\t\t=\n\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\t+\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and
This implies that \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\tA\n\t\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\t\n\t\t\t\t\t\t\t\tγ\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\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\t\n\t\t\t\t\t\t\t\tB\n\t\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\t\tU\n\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\t−\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\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\tT\n\t\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\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\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 is stable if \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\tA\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\tc\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 is stable and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis an admissible controller error. Thus we show easily that in the case of\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, there exists a matrix \n\t\t\t\t\t\n\t\t\t\t\t\tL\n\t\t\t\t\t\n\t\t\t\t such that (29) holds, where, \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\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\n\t\t\t\tis a symmetric non-negative definite solution of discrete-time Riccati equation (30) and \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\tA\n\t\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\t\n\t\t\t\t\t\t\t\tγ\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\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\t\n\t\t\t\t\t\t\t\tB\n\t\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\t\tU\n\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\t−\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\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\tT\n\t\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\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\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 is stable if the conditions i-ii of Theorem 3.1 hold.
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At the same time, we can show also that if \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is an admissble controller error, then the calculation of the algotithm 3.1 will become easilier. For an example, for a given admissible controller error\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\ti\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, the step 2 of algorithm 3.1 is to solve the discrete-time Riccati equation
for \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\tX\n\t\t\t\t\t\t\t\ti\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 being a stabilizing solution, where,\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\tQ\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\t\n\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tT\n\t\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\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\tT\n\t\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\t2\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\ti\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\tΔ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\ti\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. Since \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\tA\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\tc\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\t=\n\t\t\t\t\t\t\tA\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\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\tB\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\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\tB\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\tT\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\ti\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\tB\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\tR\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\t\t−\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\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\n\t\t\t\t\t\t\t\tT\n\t\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\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is stable and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\ti\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 is an admissible controller error, so \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\tA\n\t\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\t\n\t\t\t\t\t\t\t\tγ\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\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\t\n\t\t\t\t\t\t\t\tB\n\t\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\t\tU\n\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\ti\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\t−\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\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\tT\n\t\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\ti\n\t\t\t\t\t\t\t\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\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 is stable. This implies the condition ii displayed in Theorem 3.1 makes the calculation of the algorithm 3.1 become easier.
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4. Static Output Feedback
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This section consider discrete-time static output feedback stochastic mixed LQR/\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\t control problem. This problem is defined as follows:
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Discrete-time static ouput feedback stochastic mixed LQR/\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\t control problem: Consider the system (15) under the influence of static output feedback of the form
with\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tw\n\t\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\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t0,\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, for a given number\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t>\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, determine an admissible static output feedback controller \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\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t such that
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\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\t\n\t\t\t\t\t\t\t\t\tsup\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\t∈\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\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\n\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\t\tJ\n\t\t\t\t\t\t\t\tE\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 subject to \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\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\t\n\t\t\t\t\t\t\t\t\t\t\t\tT\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\tz\n\t\t\t\t\t\t\t\t\t\t\t\t\tw\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\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\n\t\t\t\t\t\t\t<\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t
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If this admissible controller exists, it is said to be a discrete-time static output feedback stochastic mixed LQR/\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\t controller. As is well known, the discrete-time static output feedback stochastic mixed LQR/\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\t control problem is equivalent to the discrete-time state feedback stochastic mixed LQR/\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\t control problem for the systems (15) (16), where, \n\t\t\t\t\t\n\t\t\t\t\t\tK\n\t\t\t\t\t\n\t\t\t\tis constrained to have the form of\n\t\t\t\t\t\n\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\t\tF\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\t\tC\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\n\t\t\t\t. This problem is also said to be a structural constrained state feedback stochastic mixed LQR/\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\t control problem.Based the above, we can obtain all solution to discrete-time static output feedback stochastic mixed LQR/\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\t control problem by using the result of Theorem 3.1 as follows:
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\n\t\t\t\tTheorem 4.1 There exists a discrete-time static output feedback stochastic mixed LQR/ \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\tcontroller if the following two conditions hold:
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i.There exists a matrix \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t such that
and \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\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\n\t\t\t\t is a symmetric non-negative definite solution of the following discrete-time Riccati equation
and \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\tA\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\tc\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t−\n\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\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\tB\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\tT\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\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\tB\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\tR\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\t\t−\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\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\n\t\t\t\t\t\t\t\tT\n\t\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\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is stable and\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\tU\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tγ\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\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\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tT\n\t\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\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t>\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t.
ii. \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis an admissible controller error.
\n\t\t\t
In this case, the discrete-time static output feedback stochastic mixed LQR/\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\t controller will achieve
\n\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\t\n\t\t\t\t\t\t\t\t\tsup\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\t∈\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t2\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\n\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\t\tJ\n\t\t\t\t\t\t\t\tE\n\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\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tlim\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\tT\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\t\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tT\n\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\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\t0\n\t\t\t\t\t\t\t\t\t\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\t\n\t\t\t\t\t\t\t\t\tt\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\n\t\t\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\tT\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\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\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\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 subject to \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\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\t\n\t\t\t\t\t\t\t\t\t\t\t\tT\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\tz\n\t\t\t\t\t\t\t\t\t\t\t\t\tw\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\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\n\t\t\t\t\t\t\t<\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t
\n\t\t\t
\n\t\t\t\tRemark 4.1 In Theorem 4.1, define a suboptimal controller as\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\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\t=\n\t\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\t2\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\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\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\tT\n\t\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\t3\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, then\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\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\t\tF\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\t\tC\n\t\t\t\t\t\t\t\t2\n\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\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. As is discussed in Remark 3.1, suppose that there exists a suboptimal controller \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\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 such that \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\tA\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\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=\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\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 is stable, then \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is an admissible controller error if it belongs to the set:
\n\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\tΩ\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{\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t:\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t2\n\t\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\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\t2\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 is stable}
\n\t\t\t
It should be noted that Theorem 4.1 does not tell us how to calculate a discrete-time static output feedback stochastic mixed LQR/\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\t controller\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\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. In order to do this, we present, based on the algorithms proposed by Geromel & Peres (1985) and Kucera & de Souza (1995), a numerical algorithm for computing a discrete-time static output feedback stochastic mixed LQR/\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\t controller \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\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and a solution \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\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\n\t\t\t\t to discrete-time Riccati equation (32). This numerical algorithm is given as follows:
\n\t\t\t
\n\t\t\t\tAlgorithm 4.1\n\t\t\t
\n\t\t\t
Step 1: Fix the two weighting matrices \n\t\t\t\t\t\n\t\t\t\t\t\tQ\n\t\t\t\t\t\n\t\t\t\t and\n\t\t\t\t\t\n\t\t\t\t\t\tR\n\t\t\t\t\t\n\t\t\t\t, set\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0\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\tΔ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tK\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, and\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\tU\n\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\t(\n\t\t\t\t\t\t\t\t\ti\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\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t.
for \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\tX\n\t\t\t\t\t\t\t\ti\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 symmetric non-negative definite such that
\n\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\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\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tc\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\t=\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t−\n\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\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\tB\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\tT\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\ti\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\tB\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\tR\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\t\t−\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\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\n\t\t\t\t\t\t\t\tT\n\t\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\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is stable and\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\tU\n\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\ti\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\tI\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tγ\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\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\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tT\n\t\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\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t>\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t.
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Step 3: Calculate\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\tU\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\ti\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\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\tU\n\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\t(\n\t\t\t\t\t\t\t\t\ti\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\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\tand \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\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\ti\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\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 by using the following formulas
Step 4: If \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tΔ\n\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\ti\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\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 is an admissible controller error, then increase \n\t\t\t\t\t\n\t\t\t\t\t\ti\n\t\t\t\t\t\n\t\t\t\t by\n\t\t\t\t\t\n\t\t\t\t\t\t1\n\t\t\t\t\t\n\t\t\t\t, and goto Step 2; otherwise stop.
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If the four sequences\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\tX\n\t\t\t\t\t\t\t\t0\n\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\t\tX\n\t\t\t\t\t\t\t\t1\n\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\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\ti\n\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\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\tU\n\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\t1\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\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\t1\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\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\t⋯\n\t\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\t1\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\ti\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\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\tU\n\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\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\n\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\t\tU\n\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\t(\n\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\t,\n\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\t\tU\n\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\t(\n\t\t\t\t\t\t\t\t\ti\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\t⋯\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t , and \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\tU\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t3\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\n\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\t\tU\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t3\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\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\t⋯\n\t\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\t3\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\ti\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\t⋯\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t converges, say to\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\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\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\tU\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\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\tU\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\n\t\t\t\t and\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\tU\n\t\t\t\t\t\t\t\t3\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, respectively; then the both two conditions displayed in Theorem 4.1 are met. In this case, a discrete-time static output feedback stochastic mixed LQR/\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\t controllers is parameterized as follows:
In this chapter, we will not prove the convergence of the above algorithm. This will is another subject.
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5. Numerical Examples
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In this section, we present two examples to illustrate the design methods displayed in Section 3 and 4 respectively.
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\n\t\t\t\tExample 5.1 Consider the following linear discrete-time system (15) under the influence of state feedback of the form\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tu\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\tK\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\n\t\t\t\t\t\n\t\t\t\t, its parameter matrices are
The above system satisfies Assumption 3.1-3.3, and the open-loop poles of this system are \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\tp\n\t\t\t\t\t\t\t\t1\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\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\t2.7302\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\tp\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t2.9302\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t; thus it is open-loop unstable.
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Let\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t1\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\tQ\n\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\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\t1\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\t0\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\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\t0\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\t1\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\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\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t9.5\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\tδ\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0.01\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\tM\n\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\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\t0.04\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\t1.2\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\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; by using algorithm 3.1, we solve the discrete-time Riccati equation (20) to get\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\tX\n\t\t\t\t\t\t\t\ti\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\tU\n\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\ti\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 , \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\tK\n\t\t\t\t\t\t\t\ti\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\ti\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0,1,2,\n\t\t\t\t\t\t\t⋯\n\t\t\t\t\t\t\t,10\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\tand the corresponding closed-loop poles. The calculating results of algorithm 3.1 are listed in Table 1.
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It is shown in Table 1 that when the iteration index\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t10\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\tX\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t10\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\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and \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\tU\n\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\t10\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\t−\n\t\t\t\t\t\t\t0.2927\n\t\t\t\t\t\t\n\t\t\t\t\t\n\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\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, thus the discrete-time state feedback stochastic mixed LQR/\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\t controller does not exist in this case. Of course, Table 1 does not list all discrete-time state feedback stochastic mixed LQR/\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\t controllers because we do not calculating all these controllers by using Algorithm in this example. In order to illustrate further the results, we give the trajectories of state of the system (15) with the state feedback of the form \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tu\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\tK\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\n\t\t\t\t\t\n\t\t\t\t for the resulting discrete-time state feedback stochastic mixed LQR/\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\t controller \n\t\t\t\t\t\n\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\n\t\t\t\t\t\n\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\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\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\t0.3071\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\t2.0901\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\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. The resulting closed-loop system is
\n\t\t\t\t\t\t\tSolution of DARE \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\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\n\t\t\t\t\t\t
State FeedbackController\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\ti\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
Let\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tw\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\t\n\t\t\t\t\t\t\t\tγ\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\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\t\n\t\t\t\t\t\t\t\tU\n\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\t−\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\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tT\n\t\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\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t2\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\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\t(\n\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, then the trajectories of mean values of states of resulting closed-loop system with \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\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\t0\n\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\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\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\t3\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\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\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\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\tT\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 are given in Fig. 1.
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Figure 1.
The trajectories of mean values of states of resulting system in Example 5.1.
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\n\t\t\t\tExample 5.2 Consider the following linear discrete-time system (15) with static output feedback of the form\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tu\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\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\n\t\t\t\t\t\t\ty\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\n\t\t\t\t\t\n\t\t\t\t, its parameter matrices are as same as Example 5.1.
\n\t\t\t
When \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\tC\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\n\t\t\t\t is quare and invertible, that is, all state variable are measurable, we may assume without loss of generality that \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\tC\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t ; let\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t6.5\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\tR\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tand\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\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\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\t1\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\t0\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\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\t0\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\t1\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\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, by solving the discrete-time Riccati equation (24), we get that the central discrete-time state feedback stochastic mixed LQR/\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\t controller displayed in Theorem 3.2 is
and the poles of resulting closed-loop system are\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\tp\n\t\t\t\t\t\t\t\t1\n\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\t0.1923\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\tp\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0.0205\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t.
\n\t\t\t
When\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\tC\n\t\t\t\t\t\t\t\t2\n\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\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\t\n\t\t\t\t\t\t\t\t\t\t\t\t1\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\t5.4125\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\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, let\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tγ\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t6.5\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\tR\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t1\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\tQ\n\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\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\t1\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\t0\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\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\t0\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\t1\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\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, by using Algorithm 4.1, we solve the discrete-time Riccati equation (32) to get
Thus the discrete-time static output feedback stochastic mixed LQR/\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\t controller displayed in Theorem 4.1 is\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\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t0.3727\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. The resulting closed-loop system is
Let\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tw\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\t\n\t\t\t\t\t\t\t\tγ\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\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\t\n\t\t\t\t\t\t\t\tU\n\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\t−\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\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tT\n\t\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\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\t\t2\n\t\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\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\t2\n\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\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\t(\n\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, then the trajectories of mean values of states of resulting closed-loop system with \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\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\t0\n\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\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\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\t1\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\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t2\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\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\tT\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 are given in Fig. 2.
\n\t\t\t
Figure 2.
The trajectories of mean values of states of resulting system in Example 5.2.
\n\t\t
\n\t\t
\n\t\t\t
6. Conclusion
\n\t\t\t
In this chapter, we provide a characterization of all state feedback controllers for solving the discrete-time stochastic mixed LQR/\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\t control problem for linear discrete-time systems by the technique of Xu (2008 and 2011) with the well known LQG theory. Sufficient conditions for the existence of all state feedback controllers solving the discrete-time stochastic mixed LQR/\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\t control problem are given in terms of a single algebraic Riccati equation with a free parameter matrix, plus two constrained conditions: One is a free parameter matrix constrained condition on the form of the gain matrix, another is an assumption that the free parameter matrix is a free admissible controller error. Also, a numerical algorithm for calculating a kind of discrete-time state feedback stochastic mixed LQR/\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\t controllers are proposed. As one special case, the central discrete-time state feedback stochastic mixed LQR./\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\t controller is given in terms of an algebraic Riccati equation. This provides an interpretation of discrete-time state feedback mixed LQR/\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\t control problem. As another special case, sufficient conditions for the existence of all static output feedback controllers solving the discrete-time stochastic mixed LQR/\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\t control problem are given. A numerical algorithm for calculating a static output feedback stochastic mixed LQR/\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\t controller is also presented.
\n\t\t
\n\t\n',keywords:null,chapterPDFUrl:"https://cdn.intechopen.com/pdfs/38179.pdf",chapterXML:"https://mts.intechopen.com/source/xml/38179.xml",downloadPdfUrl:"/chapter/pdf-download/38179",previewPdfUrl:"/chapter/pdf-preview/38179",totalDownloads:1012,totalViews:73,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,dateSubmitted:"March 4th 2012",dateReviewed:"June 25th 2012",datePrePublished:null,datePublished:"December 5th 2012",dateFinished:null,readingETA:"0",abstract:null,reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/38179",risUrl:"/chapter/ris/38179",book:{slug:"advances-in-discrete-time-systems"},signatures:"Xiaojie Xu",authors:[{id:"20349",title:"Dr.",name:"Xiaojie",middleName:null,surname:"Xu",fullName:"Xiaojie Xu",slug:"xiaojie-xu",email:"xiaojiexu@whu.edu.cn",position:null,institution:{name:"Wuhan University",institutionURL:null,country:{name:"China"}}}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Preliminaries",level:"1"},{id:"sec_3",title:"3. State Feedback",level:"1"},{id:"sec_4",title:"4. Static Output Feedback",level:"1"},{id:"sec_5",title:"5. Numerical Examples",level:"1"},{id:"sec_6",title:"6. Conclusion",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\tAstrom\n\t\t\t\t\t\t\tK. J.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1970\n\t\t\t\t\tIntroduction to stochastic control theory\n\t\t\t\t\tAcademic Press\n\t\t\t\t\tINC\n\t\t\t\t\n\t\t\t'},{id:"B2",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\tAthans\n\t\t\t\t\t\t\tM.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1971\n\t\t\t\t\tThe role and use of thr stochastic linear-quadratic-Gaussian problem in control system design\n\t\t\t\t\tIEEE Trans. Aut. 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S.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tChang\n\t\t\t\t\t\t\tB. C.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1992\n\t\t\t\t\tNecessary and sufficient conditions for mixed H2 and H∞ optimal control\n\t\t\t\t\tIEEE Trans. Aut. Control\n\t\t\t\t\t37\n\t\t\t\t\t3\n\t\t\t\t\t355\n\t\t\t\t\t358\n\t\t\t\t\n\t\t\t'},{id:"B26",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\tZhou\n\t\t\t\t\t\t\tK.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tDoyle\n\t\t\t\t\t\t\tJ. C.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tGlover\n\t\t\t\t\t\t\tK.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1996\n\t\t\t\t\tRobust and optimal control\n\t\t\t\t\tPrentice-Hall, INC.\n\t\t\t\t\n\t\t\t'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Xiaojie Xu",address:"xiaojiex@public.wh.hb.cn",affiliation:'
School of Electrical Engineering, Wuhan University, P. R. China
'}],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:1012,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:1697,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:1119,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:4393,totalCrossrefCites:3,signatures:"Li Dai, Yuanqing Xia, Mengyin Fu and Magdi S. 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1. Introduction
Decision making is a problem solving process that produces a goal of factors such as subjectivity and linguistics which tend to be presented in real life to a lower or greater level [1]. Difficulties are often encountered when a problem involves several alternatives and the factors that influence it (criteria), to overcome this problem, it is able to use the Multi-Attribute Decison Making (MADM) method. The results of these methods still contain uncertainty so that in this case fuzzy logic plays an important role in overcoming problems that contain uncertainty. Fuzzy logic is the basis of a system that can implement a problem and solve sharp problems [2]. However, Fuzzy MADM is only able to solve the problem of uncertainty in the data presented and numbers of diverse attributes is usually conflicting, thus to make a decision there needs to be a classic MADM method, so that decisions are more precise and more accurate [3], besides this method can also be used to provide input to the doctor so that there is no mistake in diagnosing dengue disease. One of the classic MADM methods that can be used is Simple Additive Weighting.
Simple Additive Weighting is often referred as a method with weighted sum. The basic concept of SAW method is to find a weighted sum of performance branches on each alternative of all attributes [4]. One of the problems that can be solved by this method is the misdiagnosis of DHF. DHF is a type of infectious disease caused by the degue virus which is transmitted through the bite of the Aedes aegypti and Aedes albopictus mosquitoes. DHF is often misdiagnosed with Typoid Fever, Morbili, ARI, Ensafalitis and Acute Pharyngitis. These errors occur because the initial symptoms that arise from the five diseases are almost the same as DHF [5]. However, in this case the application of SAW method is less effective if a Decision Support System is made so that a development method is needed. The development method that can be used is the FMADM method with its development or often called Fuzzy Decision Making (FDM). This method is development method of the classic MADM method. The results of SAW method will be used as a level of importance or input on the FDM method. The combination of these two methods will produce more optimal output.
2. Methodology and realization
2.1 Designing FMADM with SAW and FDM
The data used are primary and secondary data, primary data obtained from the results of doctor interviews and secondary data is data on patients with DHF, secondary data will be used to validate the system. Completion of cases of dengue diagnosis will be through SAW method then the results of SAW method are used in the FDM method.
The first method will use one crisp value with 1 degree membership and use preference weight multiplication while the second method uses 3 crisp values namely right boundary, left boundary and crisp value with 1 membership degree which will later go through the aggregation process and total integral value.
2.2 The FMADM method with SAW to diagnose a type of disease
The weight of the criteria is obtained from triangular fuzzy numbers which are then converted into the form of crisp.
2.2.2.1 Fever
The author defines the universal value for the criteria for fever is [0,1] and divides it into 5 categories of fuzzy triangle sets, which are normal (N), low fever (DR), moderate fever (DS), high fever (DT), very high fever (DST).
By using the concept of the Likert scale and the defuzzy method, Large of Maximum, Table 1 is obtained as the weight of the criteria for fever.
Fever
Fuzzy Set
Crisp Value (weight)
36°C-37,5°C
Normal (N)
0
37,5–38°C
Low Fever (DR)
0.25
38°C-39,5 °C
Moderate Fever (DS)
0.5
39,5–40°C
High Fever (DT)
0.75
>40°C
Very High Fever (DST)
1
Table 1.
Weight of fever.
2.2.2.2 Spots (Petheciae)
The author defines the universal value for the criteria of spots is [0,1] and divides them into 5 categories of fuzzy triangle sets which are none (TA), few (SDK), somewhat a lot (ABYK) many (BYK), very much (SBYK). By using the concept of the Likert scale and the defuzzy method, Large of Maximum, the Table 2 is obtained as the weight of the criteria for spots:
Spots
Fuzzy Set
Crisp Value (weight)
0–10 spots
None (TA)
0
10–20 spots
few (SDK)
0.25
20–30 spots
Somewhat a lot (ABYK)
0.5
30–50 spots
Many (BYK)
0.75
>50 spots
Very Much (SBYK)
1
Table 2.
Weigth of spots.
2.2.2.3 Bleeding gum
We are defines the universal value for bleeding gum criteria is [0,1] and divides it into 2 fuzzy triangle set categories namely never (TP), ever (P). By using the concept of the Likert scale and the defuzzy method, Large Of Maximum, the Table 3 is obtained as the weight of the bleeding gum criteria.
Bleeding Gums
Fuzzy Sets
Crisp Value (weight)
0 No
Never (TP)
0
Once or More
Ever (P)
1
Table 3.
Weight of bleeding gum.
2.2.2.4 Nausea
The author defines the universe value for the nausea criteria is [0.1] and divides it into 4 fuzzy triangle set categories namely never (TP), ever (P), rare (J) and often (S). By using the concept of the Likert scale and the defuzzy method, Large of Maximum, Table 4 is obtained as the weight of the criteria for nausea.
Nausea
Fuzzy Sets
Crisp Value (weight)
0
Never (TP)
0
1 time a day
Ever (P)
0.25
2–3 times a day
Rare (J)
0.5
>3 times a day
Often (S)
0.75
Table 4.
Weigth of nausea.
2.2.2.5 Headache
The author defines the universal value for the headache criteria is [0,1] and divides it into 4 fuzzy triangle set categories namely never (TP), ever (P), rarely (J) and often (S). By using the concept of the Likert scale and the defuzzy method, namely Large Of Maximum, Table 5 is obtained as the weight of the headache criteria.
Headache
Fuzzy Set
Crisp Value (weight)
0
Never (TA)
0
1 time a day
Ever (P)
0.25
3–4 times a day
Rare (J)
0.5
4–5 times a day
Often (S)
0.75
Table 5.
Weight of headache.
2.2.2.6 Defecation disorder
The author defines the universal value for the criteria for defecation disorder is [0,1] and divides it into 3 categories of fuzzy triangles, namely normal (N), difficult to do defecation (SB) and diarrhea (D). By using the concept of the Likert scale and the defuzzy method, Large of Maximum, Table 6 is obtained as the weight of the criteria for BAB defects.
Defecation Disorder
Fuzzy Set
Crisp Value (weight)
1–2 times a day
Normal (N)
0.5
1–2 days unable to do defecation
Hard to do Defecation (SB)
0.75
>3 times a day
Diarrhea (D)
1
Table 6.
Defecation disorder weight.
2.2.3 Determine the suitability rating of each alternative on each criterion
Interview results from an expert (doctor) on Table 7.
c1
c2
c3
c4
c5
c6
a1
DT
SDK
TP
J
P
SB
a2
DT
SBYK
P
J
J
D
a3
DT
TA
TP
T
J
N
a4
DST
SDK
TP
P
S
D
a5
DS
TA
TP
S
P
N
a6
DST
TA
TP
J
S
SB
Table 7.
Linguistics data.
From the Table 8, the match rating value is obtained as follows:
c1
c2
c3
c4
c5
c6
a1
0.75
0.25
0
0.5
0.25
1
a2
0.75
1
1
0.5
0.5
0.75
a3
0.75
0.25
0
0
0.5
0.5
a4
1
0
0
0.75
0.75
0.75
a5
0.5
0
0
0.5
0.25
0.5
a6
1
0
0
0.5
0.75
1
Table 8.
Match rating value.
The compatibility rating in this method is also called the decision matrix which will be normalized.
2.2.4 The determination of the preference weight
The determination of the preference weight is stated in Table 9 as follows:
2.2.6 Finding preference values obtained from multiplication of weights W with normalized matrix R
Vj=∑j=1nwjrijE3
The results of the calculation are shown in Table 10 as follows.
V1 (Morbili)
V2 (DBD)
V3 (ARI)
V4 (Typhoid Fever)
V5 (Acute pharyngitis)
V6 (Encephalitis)
0.5
0.83
0.42
0.58
0.30
0.57
Rank 4
Rank 1
Rank 6
Rank 2
Rank 5
Rank 3
Table 10.
Preference value.
The highest value achieved by the second alternative (V2) is DBD so someone will be stated to suffer from DHF if they experience symptoms of high fever, spots (petheciae) very much, have experienced bleeding gums if they have entered a severe stage, rarely nausea, rarely headaches and have diarrhea, but to be sure to be able to use laboratory tests again.
In this case, SAW method is not appropriate if it is used to make a decision support system thus the author tries to use a method developed by Joo (2004) [6], namely the FMADM method with development or FDM.
2.3 The FMADM method with SAW to diagnose a type of disease
2.3.1 Representation of the problem
Consists of 3 stages, namely:
Objective Identification
The purpose of this decision is to determine or diagnose an illness that is suffered based on the initial symptoms experienced.
Identification of Criteria and Alternatives.
The criteria used are still 6 types of diseases and 6 criteria (symptoms).
The hierarchical structure that determines the disease is shown in the Figure 1.
Figure 1.
Hierarchy Structure.
2.3.2 Evaluation of Fuzzy Sets
Consists of 4 stages, namely:
Selecting the set of ratings for the criteria weights. There are two things that must be done, namely determining the degree of importance and determining the degree of compatibility. T (importance) W = {c1 = {N, DR, DS, DT, DST}, c2 = {TA, DK, ABYK, BYK, SBYK}, c3 = {TP, P}, c4 =, c5 = {TP, P, J, S}, c6 = {NR, D, SB}}. T (match) S = {Very Low (SR), Low (R), Enough (C), High (T), Very High (ST)}.
The parameters of each level of interest are as follows:
The degree of compatibility of each decision criteria as follows:
VeryLowSR=000.25,
LowR=00.250.5,
EnoughC=0.250.50.75,
HeightT=0.50.751
Very HighST=0.7511
Based on this, the degree of compatibility of each alternative is obtained to the decision criteria in Table 11 and the branch of interest for the decision criteria in Table 12.
Aggregate the weight of criteria and the degree of compatibility of each alternative with its criteria, using the following equation:
c1
c2
c3
c4
c5
c6
a1
T
R
SR
C
R
ST
a2
T
ST
R
C
C
T
a3
T
ST
SR
SR
C
C
a4
ST
R
SR
T
T
T
a5
C
SR
SR
C
C
C
a6
ST
SR
SR
C
T
ST
Table 11.
The degree of compatibility of each alternative to the decision criteria.
Fever
Spot
Bleeding gum
Nausea
Headache
Defecation Disorder
High
Very Much
Ever
Rare
Rare
Diarrhea
(0.5,0.75, 1)
(0.75,1, 1)
(0, 1, 1)
(0.25, 0.5, 0.75)
(0.25, 0.5, 0.75)
(0.5, 0.75, 1)
Table 12.
Branch of interest for decision criteria.
Yi=1k∑t=1koitaiE4
Qi=1k∑t=1kpitbiE5
Zi=1k∑t=1kqitciE6
The result is compatibility index obtained from the aggregation of the weight of the criteria and the degree of compatibility of each alternative with its criteria that’s shown in Table 13.
Alternative
Compatibility Rate
Fuzzy Compatibility Index
c1
c2
c3
c4
c5
c6
Yi
Qi
Zi
a1
T
R
SR
C
R
ST
0.1146
0.3229
0.6146
a2
T
ST
R
C
C
T
0.1979
0.4792
0.7708
a3
T
ST
SR
SR
C
C
0.1667
0.3646
0.6250
a4
ST
R
SR
T
T
T
0.1458
0.3854
0.7083
a5
C
SR
SR
C
C
C
0.0625
0.2083
0.5208
a6
ST
SR
SR
C
T
ST
0.1563
0.3542
0.6354
Table 13.
Compatibility index.
2.3.3 Selecting optimal alternatives
Prioritizing decision alternatives based on aggregation results by substituting the fuzzy match index value into the following equation:
ITαF=12αc+b+1−αaE7
By taking optimism degree (α), namely: α = 0 (not optimistic), α = 0.5 (optimistic) and α = 1 (very optimistic). The following results are obtained on Table 14.
Alternative
Integral Total Value
α = 0
α = 0.5
α = 1
a1
0.22
0.34
0.47
a2
0.34
0.48
0.63
a3
0.27
0.38
0.49
a4
0.27
0.47
0.55
a5
0.14
0.25
0.36
a6
0.26
0.37
0.49
Table 14.
Integral total value.
Based on the results above, it can be seen that regardless of the degree of optimism, the alternative a2 is that DHF has the greatest value compared to other alternatives.
2.4 System Implementation
2.4.1 Algorithm of decision support system
The following figure (Figure 2) is a flowchart that shows how decision support system works.
Figure 2.
Decision Support System Algorithm.
2.4.2 Implementation in MATLAB
Based on the matlab program algorithm, we must first do the FMADM process with SAW by making a coding in the editor according to the FMADM algorithm with SAW, then the results of the method will be used as input for the next method using the Graphical User Interface (GUI) that will be shown in the Figure 3.
Figure 3.
View of GUI (Opening).
Figure 4 is the appearance of the two matlab programs with a GUI that contains: self-identity, symptoms experienced, save, clean, close, diagnosis, output, for self-identity and symptoms must be filled. The second display looks like the following picture:
Figure 4.
GUI Display (Form Filling).
Following are the steps to diagnose a type of disease: Fill in the biodata form and symptoms, then click the diagnosis button then click the save button. The results of the diagnosis are obtained as follows like what’s shown in Figure 5.
Figure 5.
GUI Display (Diagnose Result).
The storage results are displayed in a form of what’s shown in Figure 6.
Figure 6.
GUI Display (Data Base).
2.4.3 System accuracy testing
The accuracy of the FMADM decision support system with MAPE obtained the following equation
The Accuracy=∑Dataujibenar∑Totalujix100%E8
Obtained from 30 data is as follows:
The accuracy=30−240x100%=93%E9
3. Conclusion
Based on the method in the first stage, the FMADM method with SAW rank 1 was obtained in the second alternative V2 so that someone can be confirmed to suffer from dengue if they experience the initial symptoms of high fever at 39.5° C - 40° C many spots appear during the lumple leed test (> 50 petheciae), bleeding gums, rarely experiencing nausea and headaches, then experiencing diarrhea. In the second method, the results of the first method will be the input for the second method, then the total integral value will be obtained with the degree of optimism α = 1, from the second method or FMADM with Development (FDM). Then the results of the accuracy of the decision support system with MAPE obtained 93% of 100% consisting of 40 patients suffering from DHF.
\n',keywords:"diagnosing a type of disease, FDM, FMADM, SAW",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/74830.pdf",chapterXML:"https://mts.intechopen.com/source/xml/74830.xml",downloadPdfUrl:"/chapter/pdf-download/74830",previewPdfUrl:"/chapter/pdf-preview/74830",totalDownloads:34,totalViews:0,totalCrossrefCites:0,dateSubmitted:"June 5th 2020",dateReviewed:"October 23rd 2020",datePrePublished:"January 29th 2021",datePublished:null,dateFinished:"January 18th 2021",readingETA:"0",abstract:"Fuzzy logic is widely applied to daily life with various methods. One method is fuzzy multi-attribute decision making (FMADM). FMADM is able to select the best alternative from a number of alternatives. In FMADM there is a supporting method so that the results obtained are accurate and optimal, namely the classic MADM method. One method in classic MADM is the Simple Additive Weighting (SAW) method. The SAW method is precisely used to minimize diagnostic errors, but if a decision support system is made, the SAW method still requires a further development method, one of which is the FMADM method with its development. The purposes of this study are to describe the steps of SAW method and the development of FDM in theory, implement SAW method and the development of FDM to diagnose a type of disease and implement it in a decision support system using GUI matlab. The completion step of those two methods is through two stages, the first one will go through FMADM stage with SAW, which is weighted sum, then the output will be used as input to the FDM method based on total integral values. The result of this study is proven by patient experienced initial symptoms of high fever at a temperature of 39.5° C - 40° C, very much spots appear in rumple leed test (> 50 petheciae), bleeding gums, rarely got nausea and headache, as well as diarrhea. Accuracy for the decision support system using MAPE was obtained 93% so that the decision support system with FMADM method to diagnose the disease was feasible to use.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/74830",risUrl:"/chapter/ris/74830",signatures:"Sugiyarto Surono and Mustika Sari",book:{id:"9976",title:"Fuzzy Systems",subtitle:null,fullTitle:"Fuzzy Systems",slug:null,publishedDate:null,bookSignature:"Prof. Constantin Volosencu",coverURL:"https://cdn.intechopen.com/books/images_new/9976.jpg",licenceType:"CC BY 3.0",editedByType:null,editors:[{id:"1063",title:"Prof.",name:"Constantin",middleName:null,surname:"Volosencu",slug:"constantin-volosencu",fullName:"Constantin Volosencu"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:null,sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Methodology and realization",level:"1"},{id:"sec_2_2",title:"2.1 Designing FMADM with SAW and FDM",level:"2"},{id:"sec_3_2",title:"2.2 The FMADM method with SAW to diagnose a type of disease",level:"2"},{id:"sec_3_3",title:"2.2.1 Determine alternative sets and criteria",level:"3"},{id:"sec_4_3",title:"Table 1.",level:"3"},{id:"sec_4_4",title:"Table 1.",level:"4"},{id:"sec_5_4",title:"Table 2.",level:"4"},{id:"sec_6_4",title:"Table 3.",level:"4"},{id:"sec_7_4",title:"Table 4.",level:"4"},{id:"sec_8_4",title:"Table 5.",level:"4"},{id:"sec_9_4",title:"Table 6.",level:"4"},{id:"sec_11_3",title:"Table 7.",level:"3"},{id:"sec_12_3",title:"Table 9.",level:"3"},{id:"sec_13_3",title:"2.2.5 Normalization of the matrix",level:"3"},{id:"sec_14_3",title:"Table 10.",level:"3"},{id:"sec_16_2",title:"2.3 The FMADM method with SAW to diagnose a type of disease",level:"2"},{id:"sec_16_3",title:"2.3.1 Representation of the problem",level:"3"},{id:"sec_17_3",title:"Table 11.",level:"3"},{id:"sec_18_3",title:"Table 14.",level:"3"},{id:"sec_20_2",title:"2.4 System Implementation",level:"2"},{id:"sec_20_3",title:"2.4.1 Algorithm of decision support system",level:"3"},{id:"sec_20_4",title:"2.4.2 Implementation in MATLAB",level:"4"},{id:"sec_22_3",title:"2.4.3 System accuracy testing",level:"3"},{id:"sec_25",title:"3. Conclusion",level:"1"}],chapterReferences:[{id:"B1",body:'Kusumadewi Sri. 2003, Artificial Intelligence (Teknik dan Aplikasinya). Yogyakarta : Graha Ilmu.'},{id:"B2",body:'J. Ross,Timothy. 1997. Fuzzy Logic With Engneering Applications. Singapore ; Mc. Grow. Hill, Inc.'},{id:"B3",body:'Chen, Shu- Jen & Chin- Lai Hwang. 1992.Fuzzy Multi Attribute Decision Making : Methods and Applications, Berlin : Spinger- Varlag'},{id:"B4",body:'Kusumadewi, Sri, Sri Hartati, Agus Harjoko & Retantyo Wardoyo. 2006. Fuzzy Multi-Attribute Decision Making (Fuzzy MADM). Yogyakarta : Graha Ilmu'},{id:"B5",body:'Masjoer, Arif. 2000. Kapita Selekta Kedokteran. Jakarta:Media Aesculapus.'},{id:"B6",body:'Joo. Hyun Monn & Kang, Chang Sung. 2009. Application of Fuzzy Decision Making Method to the Evaluation of spent fuel stronge Options: Korea'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Sugiyarto Surono",address:"sugiyarto@math.uad.ac.id",affiliation:'
'}],corrections:null},book:{id:"9976",title:"Fuzzy Systems",subtitle:null,fullTitle:"Fuzzy Systems",slug:null,publishedDate:null,bookSignature:"Prof. Constantin Volosencu",coverURL:"https://cdn.intechopen.com/books/images_new/9976.jpg",licenceType:"CC BY 3.0",editedByType:null,editors:[{id:"1063",title:"Prof.",name:"Constantin",middleName:null,surname:"Volosencu",slug:"constantin-volosencu",fullName:"Constantin Volosencu"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}}},profile:{item:{id:"135896",title:"Dr.",name:"Slimane",middleName:null,surname:"Laref",email:"slimane.laref@ens-lyon.fr",fullName:"Slimane Laref",slug:"slimane-laref",position:null,biography:null,institutionString:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",totalCites:0,totalChapterViews:"0",outsideEditionCount:0,totalAuthoredChapters:"1",totalEditedBooks:"0",personalWebsiteURL:null,twitterURL:null,linkedinURL:null,institution:{name:"University of Lyon System",institutionURL:null,country:{name:"France"}}},booksEdited:[],chaptersAuthored:[{title:"Opto-Electronic Study of SiC Polytypes: Simulation with Semi-Empirical Tight-Binding Approach",slug:"opto-electronic-study-of-sic-polytypes-simulation-with-semi-empirical-tight-binding-approach",abstract:null,signatures:"Amel Laref and Slimane Laref",authors:[{id:"55638",title:"Dr.",name:"Amel",surname:"Laref",fullName:"Amel Laref",slug:"amel-laref",email:"larefamel@yahoo.com"},{id:"135896",title:"Dr.",name:"Slimane",surname:"Laref",fullName:"Slimane Laref",slug:"slimane-laref",email:"slimane.laref@ens-lyon.fr"}],book:{title:"Silicon Carbide",slug:"silicon-carbide-materials-processing-and-applications-in-electronic-devices",productType:{id:"1",title:"Edited Volume"}}}],collaborators:[{id:"44315",title:"Dr.",name:"Kun",surname:"Xue",slug:"kun-xue",fullName:"Kun Xue",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/44315/images/system/44315.jpeg",biography:"Dr. Kun Xue received her PhD degree from Tsinghua University in 2009 and joined Beijing Institute of Technology the same year as associate professor. 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