On average real sliding dynamics in linear systems
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It is well known that in implementations of sliding mode controllers using hysteresis comparators, when the hysteresis band amplitude tends to zero the real dynamics tends to the ideal sliding dynamics. However, in real systems physical limitations do not allow to effectively lower this value at will, and a steady state error is likely to appear. In this paper we relate this error with a non zero average value of the switching function in each switching period: it is shown that, in linear systems, when the controller has a constant switching frequency and the switching function is periodic, the average value of the difference between real and ideal steady state dynamics is proportional to the average value of the switching function. Hence, when this average value is non zero an average steady state error appears, while a zero average value for the switching function entails no average steady state error. The proof is carried out using a regular form approach, and the result is exemplified in a buck converter. Simulation results show that when the switching function is periodic and shows a piecewise linear behavior within the hysteresis band, thus guaranteeing zero average value, the average state error disappears. In turn, when this piecewise linear character is lost and the switching function has non zero mean value, an average steady state error arises.
CitationOlm, Josep M., Biel, D., Repecho, V., Shtessel, Y. On average real sliding dynamics in linear systems. "IFAC-PapersOnLine", 1 Juliol 2017, vol. 50, p. 1-6.