This paper addresses the switching control synthesis problem of discrete-time switched linear systems. A particular class of matrix inequalities, the so-called Lyapunov-Metzler inequalities is modified to provide conditions for stability analysis and output feedback control synthesis under a relaxed min-switching logic. The switching rule combined with switching output feedback controllers are designed to stabilize the switched closed-loop system and satisfy a pre-specified ℓ₂ gain performance. The proposed switching control approach is to reduce the high frequency switches commonly observed in min-switching strategy based designs. The effectiveness of the proposed approach is illustrated through a numerical example.