Root counting, phase unwrapping, stability and stabilization of discrete time systems Original Research Article

In this paper, we develop a procedure for determining the phase unwrapping of a real polynomial or rational function along the unit circle. By representing the unit circle image in terms of Tchebyshev polynomials, a formula for the unwrapped phase is determined in terms of the zeros and signs of these polynomials. The root distribution with respect to the unit circle can thus be determined in terms of the Tchebyshev representation. This result is applied to the problem of feedback stabilization of a digital control system by constant gain or by a two-parameter controller. The solution results in a determination of the entire set of stabilizing gains as a solution of sets of linear inequalities. This is in sharp contrast to the solution via classical conditions which result in nonlinear inequalities. The result also gives a new characterization of Schur stability in terms of the Tchebyshev representation which may be of independent interest.