Direct bilinear Routh stability criteria for discrete systems
From the Routh conditions for Hurwitz polynomials H(s), expressed by the positivity of the coefficients of the continued fraction (CF) expansions of the tangent function of H(s), some equivalent stability conditions for discrete systems are obtained, using the bilinear transformation \(s=(z- 1)/(z+1)\); for example in Theorem 1 it is shown that a real polynomial D(z) is stable (has all its zeros inside the unit circle) iff all the coefficients of the CF in terms of \(s^{-1}=(z+1)/(z-1)\) (or in s) of the discrete tangent function \(\rho_ n(z)=(D(z)-z^ nD(z^{- 1}))/(D(z)+z^ nD(z^{-1}))\) are positive. Some mixed form (in s and \(s^{-1})\) CF expansions are emphasized and three algorithms are derived to obtain the coefficients of the CF (the first has the simplest set up and the third requires the least number of operations). It is also stated that the number of zeros of D(z) inside and outside the unit circle is equal to the number of the positive and negative coefficients of the CF.
scientific article; zbMATH DE number 3871153
scientific article