This technical note finds that a conventional 2-rate controller (which receives feedback at T -but provides control at 2T -intervals) has a 2-periodic core, and augments this core to have a generalized 2-rate control that has complete loop-zero (and pole) placement capability. An algorithm for controller parameter evaluation for pole-zero placement using only linear algebraic steps has been developed. The controller also ensures zero steady-state and improved transient ripples. Its superiority in respect of gain-margin compensation is illustrated for a third-order plant with an unstable pole and a non-minimum-phase zero.