A matrix for evaluating Schwarz's form

Schwarz's form is fundamental and effective in constructing Liapuaov functions, in proving the Hurwitz criterion, and in evaluating performance measures in system analysis. However, the procedures developed thus far for obtaining the Schwarz form are complicated. This paper establishes a basic transformation matrix by which a phase-variable form is easily converted into a Schwarz form. When the new transformation matrix is used, Kalman-Bertram's Liapunov function is simplified and Ralston's symmetric matrix formulation of the Hurwitz criterion is derived in a completely different but much more sophisticated way. Finally, to the authors' knowledge, this is the first time that practical use has been made of the second, third, etc., columns of Routh's array.