Max–plus matrix method and cycle time assignability and feedback stabilizability for min–max–plus systems
 Yuegang Tao,
 GuoPing Liu,
 Xiaowu Mu
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A variety of problems arising in nonlinear systems with timing constraints such as manufacturing plants, digital circuits, scheduling managements, etc., can be modeled as min–max–plus systems described by the expressions in which the operations minimum, maximum and addition appear. This paper applies the max–plus matrix method to analyze the cycle time assignability and feedback stabilizability of min–max–plus systems with min–max–plus inputs and max–plus outputs, which are nonlinear extensions of the systems studied in recent years. The max–plus projection matrix representation of closedloop systems is introduced to establish some structural and quantitative relationships between reachability, observability, cycle time assignability and feedback stabilizability. The necessary and sufficient conditions for the cycle time assignability with respect to a state feedback and an output feedback, respectively, and the sufficient condition for the feedback stabilizability with respect to an output feedback are derived. Furthermore, one output feedback stabilization policy is designed so that the closedloop systems take the maximal Lyapunov exponent as an eigenvalue. The max–plus matrix method based on max–plus algebra and directed graph is constructive and intuitive, and several numerical examples are given to illustrate this method.
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 Title
 Max–plus matrix method and cycle time assignability and feedback stabilizability for min–max–plus systems
 Journal

Mathematics of Control, Signals, and Systems
Volume 25, Issue 2 , pp 197229
 Cover Date
 20130601
 DOI
 10.1007/s0049801200987
 Print ISSN
 09324194
 Online ISSN
 1435568X
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Cycle time assignability
 Feedback stabilizability
 Max–plus matrix method
 Min–max–plus system
 Industry Sectors
 Authors

 Yuegang Tao ^{(1)} ^{(2)}
 GuoPing Liu ^{(3)} ^{(4)}
 Xiaowu Mu ^{(5)}
 Author Affiliations

 1. College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, 050024, People’s Republic of China
 2. Key Lab of Computational Mathematics and Applications, Hebei, Shijiazhuang, 050024, People’s Republic of China
 3. Faculty of Advanced Technology, University of Glamorgan, Pontypridd, CF37 1DL, UK
 4. CTGT Center, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China
 5. Department of Mathematics, Zhengzhou University, Henan, 450001, People’s Republic of China