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The concept of a sampled-data nonlinearity matrix is defined. It is shown that, by means of this matrix, a direct correlation can be establihed between the nonlinearity and its equivalent gain. Thus, symmetrical single-valued and/or double-valued nonlinearities, however complicated, can be transformed directly into the respective equivalent gains. This is obtained by means of linear transformations, using numerically known matrices, i.e. matrices which are independent of the nonlinearity. Similarly, in synthesis procedures, the nonlinerities are obtained from the characteristics of the desired equivalent gain. A theorem concerning these transformations is stated and proved, and some properties of the sampled-data nonlinearity matrix are emphasised. The matrices permitting direct and reciprocal transformations are given, and two non-linearities are calculated. A stability criterion is stated, and the stability of a nonlinear system is examined. Thus, it is shown that, in the design of the system, the matrix correlations and the stability criterion allow one to obtain the edesired behaviour by a very large class of nonlinear cntrol system by acting directly on the nonlinearity.