Homeostatic and adaptive control mechanisms are essential for keeping organisms structurally and functionally stable. Integral feedback is a control theoretic concept, which has long been known to keep a controlled variable A robustly at a given set-point Aset by feeding the integrated error back into the process which generates A. However, how integral feedback and robust homeostasis is realized in biochemical systems is still not fully understood. We have recently identified two reaction kinetic conditions that permits to build integral control into different chemical negative feedback structures and which allow to generate biochemical models of robust homeostatic controllers. In their simplest form such "homeostats" consist of two compounds: the homeostatic controlled variable A and a controller molecule E, where the concentration of E is related to the integrated error between the concentration of A and its set-point Aset. In the here presented work we extend the concept of homeostasis to include sustained oscillatory conditions. We show how oscillatory homeostats can maintain stability in A by keeping the average concentration of A at the controller's set-point Aset due to a change in the oscillator's frequency and amplitude. In addition, dependent on the negative feedback structure of the oscillator an increase or decrease in the average concentration of E is observed. Because in oscillatory homeostats the average level of the manipulated variable E is associated with the frequency of the compensatory flux, we show how robust frequency control may be achieved by controlling the average level of E. The occurrence of such behaviors in oscillatory signaling and in biological clock rhythms is discussed.