  Arm locking is one of the key technologies to suppress the laser phase noise
in spaced-based gravitational waves observatories. Since arm locking was
proposed, phase margin criterion was always used as the fundamental design
strategy for the controller development. In this paper, we find that this
empirical method from engineering actually cannot guarantee the arm locking
stability. Therefore, most of the advanced arm locking controllers reported so
far may have instable problems. After comprehensive analysis of the single arm
locking's transient responses, strict analytical stability criterions are
summarized for the first time. These criterions are then generalized to dual
arm locking, modified-dual arm locking and common arm locking, and special
considerations for the design of arm locking controllers in different
architectures are also discussed. It is found that PI controllers can easily
meet our stability criterions in most of the arm locking systems. Using a
simple high gain PI controller, it is possible to suppress the laser phase
noise by 5 orders of magnitude within the science band. Our stability
criterions can also be used in other feedback systems, where several modules
with different delays are connected in parallel.
