A relative damping criterion for linear systems

Many feedback controls and other physical systems are designed on the basis of linear theory, that is, linear differential equations with constant coefficients. A. Leonhard1¿3 and A. Vazsonyi4 have given graphical methods for determining whether all roots of the characteristic equation are located within certain sectors of the complex plane. Vazsonyi´s method determines whether all dimensionless damping rates ¿ of any linear system are greater than any prescribed value. Each pair of complex conjugate roots of the characteristic equation of a linear system has its own value of ¿. Any such pair can be expressed as