CRITICAL DEGENERATENESS IN LINEAR SYSTEMS,

A class of linear systems, composed of intrinsically stable elements, was studied. These linear systems were represented by the coefficient matrices of their differential equations. A sample space of these matrices was defined by specifying the nature of the distributions from which the matrix entries were selected. Matrices of given size were generated by random sampling from the defined sample space. Appropriate weighting of the distributions gave control of the degenerateness, a measure of the number of zero entries. The Hurwitz criterion was used to test whether each matrix representeded a stable system. The primary goal was to find the probabilty of stability as a function of degenerateness. It was found, even for the relatively small matrices within the range of this study, that the degenerateness is critical. For values of degenerateness less than a particular amount about 85, the system is almost certainly unstable, whereas for values of degenerateness greater than this amount, the system is almost certainly stable. Author