This paper studies mixed rule cellular automata: digital dynamical systems governed by two rules of simple Boolean functions. Depending on the rules, the systems exhibit a variety of binary periodic orbits. Real/potential applications include control signals of walking robots, approximation signals of time-series, and so on. In typical examples of periodic orbits, we have applied the brute force attack and have discovered the best systems that realize the strongest stability of the periodic orbits. As the stability becomes stronger, reliability/robustness of the control/approximation signals becomes stronger. The result is exact without approximation. Presenting an FPGA based hardware prototype, the strong stable periodic orbits are confirmed experimentally.