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A necessary and sufficient condition for the controllability of single‐leader multi‐chain systems
Author(s) -
Hsu ShunPin
Publication year - 2016
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/rnc.3566
Subject(s) - controllability , chain (unit) , upper and lower bounds , computer science , work (physics) , control theory (sociology) , topology (electrical circuits) , mathematics , mathematical optimization , control (management) , combinatorics , engineering , mathematical analysis , physics , mechanical engineering , astronomy , artificial intelligence
Summary In this work, the controllability of single‐leader multi‐agent systems with chain structures is studied. It is shown that the necessary and sufficient condition for the multi‐chain system to be controllable is that there exist no two chain lengths in the form ℓ 1 = i + k 1 (2 i + 1) and ℓ 2 = i + k 2 (2 i + 1), where i is some natural number and k 1 and k 2 some nonnegative integers. Using this condition, the author derives an upper bound based on the length of the longest chain and proves that if the number of chains exceeds this bound, the multi‐chain system must be uncontrollable. In addition, the author investigates an augmented system constructed by connecting some follower nodes of the multi‐chain system and obtains a sufficient condition for the augmented system to be uncontrollable. Finally, the author shows how to select a minimum number of additional leaders to make an uncontrollable multi‐chain system controllable. Numerical examples are provided to illustrate the results. Copyright © 2016 John Wiley & Sons, Ltd.