A method for the computation of self-sustained oscillations in systems with piecewise linear elements

In this paper a study of the self-sustained oscillations in systems which contain a nonlinear element of the piecewise linear type with or without memory is presented. A method is developed by which one may systematically detect possible symmetric limit cycles, and subsequently identify the oscillations completely. Vector-matrix techniques are employed to transform certain points between regions of the phase space. The boundaries of the regions of phase space correspond to the break points of the piecewise linear characteristic. A set of equations which determine points on a possible closed trajectory are derived using the transformations. Certain switching conditions which are necessary for the existence of an oscillation are applied to these equations. From the solution of the resulting equations, points on the closed trajectory are determined. Using the values of the coordinates of these points, one computes the waveform of the oscillation. Applications of the method to control systems with various configurations are discussed in theory and illustrative examples are presented.