We present a method for constructing discrete abstractions for discrete-time, continuous-state systems. Related approaches construct a discrete bisimulation, which leaves little room for non-determinism in the outputs and quickly leads to highly complex models since all concrete behavior is covered. Our approach is to relax these requirements and build a satisficing solution: a discrete abstraction that is deterministic, robust, and as complete as possible under the given parameters. This allows us to balance granularity and computational feasibility. We leverage linearization and linear feedback control to extend the approach from globally contractive systems to systems with contractive cycles. The resulting abstraction directly induces a supervisor policy. The approach is illustrated with numerical experiments and has potential applications in various domains where system safety and reversibility are essential.