A necessary and sufficient condition for the controllability of single-leader multi-chain systems
Shun‐Pin Hsu · International Journal of Robust and Nonlinear Control · 2016
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 ℓ1=i + k1(2i + 1) and ℓ2=i + k2(2i + 1), where i is some natural number and k1 and k2 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.