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A dynamic system is given with a transfer function of third order (one real pole, two complex poles, and two zeroes). It is desired to design a contactor control which yields a near-optimum follow-up of a given input. For zero-seeking systems and for step-inputs an "optimum" control problem can be defined easily; however, a general solution of the optimum problem is not yet available. The known particular cases yield switching surfaces in the phase space which depend on the particular pole and zero locations (parameter set). The realization of such surfaces is rather difficult and usually demands digital computer equipment. The author and her associates suggest in this paper switching functions which can be realized easily and which give good response for different parameter sets. Particular attention is given to the occurrence of discontinuities in linear switching functions and their advantages and disadvantages; to different types of chatter, which may or may not improve the system performance; to discontinuities in the switching function which can be avoided by feeding the output of the contactor back into the switching function. Analog computer tests show the performance of systems with linear switching functions. The choice of the coefficients in this linear switching function is discussed. The influence of imperfections of the control system on its performance is investigated. Comparison to linear feedback control systems are included. This paper is to be published in the Proceedings of the First IFAC Moscow Congress by Butterworth Scientific Publications in 1960.