Feedback Control For Interval Max-Plus Systems Subject To State Restriction

A number of problems arising in computer networks, digital circuits, and automatedmanufacturing plants, etc., can be modeled as max-plus systems. Max-plus systems arebased on max-plus algebra. Max-plus algebra is an algebraic structure, which is obtainedby replacing the operations addition and multiplication in general algebraic structurewith the operations maximum and addition. Max-plus algebra is a special dioid, and theset of all the intervals on max-plus algebra, endowed with two operations, is an intervaldioid.In this paper, the feedback control for interval max-plus systems subject to staterestriction is researched. For a manufacturing system with input structure, we need toconsider the input time of raw materials, the transportation time and processing time.On the basis of satisfying the system requirements, we want to control the system, andmake the state of this system meet given restrictions. By computing the feedback intervalmatrices, we can adjust the input, and then control the system, when the processing timeof the machines is interval. In order to compute the feedback interval matrices, we studythe solutions and algorithm of equation A ? x = B ? y.In the part of introduction, we introduce the related background and status of intervalmax-plus systems.In the first chapter, basic concepts are introduced, such as dioids, max-plus algebra,intervals, interval matrices, and interval max-plus systems, etc. We emphatically provesome properties of intervals and interval matrices. The operations between matriceson interval dioid are introduced by examples. These concepts and properties providetheoretical support for the later chapters.In the second chapter, the solutions and algorithm of the equation A ? x = B ? yare further studied. We mainly investigate the solutions and stable solutions of theequation, give an algorithm for it, and analyse the relations between the run numbers ofthe algorithm and the solutions of the equation above, when the lower and upper boundsof all elements of interval matrices A and B are integers. The algorithm provides amethod for computing the feedback interval matrices in the third chapter.The third chapter is the core of this thesis. By applying the properties of intervalsand interval matrices and using algebraic method, feedback control for interval max-plus systems is researched. A sufficient and necessary condition for the existence of thesolutions of the feedback control for interval max-plus systems is presented, when theinitial conditions satisfy given restrictions. Then the problem of solving the solutions offeedback control is transformed into computing an equation. By adding some additionalconditions, a sufficient condition for the existence of the solutions of the feedback controlfor interval max-plus systems is obtained. In the end, a numerical example illustrates ourresults.In the part of conclusion, we summarize the main conclusions of this thesis, and raisethe issue which needs further study.