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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.
Zhang et al. (Tue,) studied this question.