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Recent papers have established global convergence for a class of adaptive control algorithms for discrete time linear dynamic systems. However, in most cases studied to date it has been assumed that the system is stably invertible. This assumption plays a major role in the proofs of convergence. In this paper we consider the more general case in which it is not assumed that the system is either stable or stably invertible. We establish local convergence for a class of adaptive control algorithms applied to general discrete, deterministic, linear, time-invariant systems. By convergence in this context, we mean that the system inputs and outputs remain bounded for all time and the closed loop poles are effectively assigned in the limit for a given desired trajectory.