Root counting and phase unwrapping with respect to the unit circle with applications

In this paper we develop a new formula for counting the roots of a real polynomial inside the unit circle. This is done by representing the unit circle image of the polynomial in terms Tchebyshev polynomials, and the latter leads to a formula for the phase, unwrapped along the unit circle in terms of the zeros and signs of the Tchebyshev representation. This root counting formula can be specialized to yield a new interlacing condition for stability in terms of the Tchebyshev representation. The new formula is applied to the problem of constant gain stabilization of a digital control system and results in a determination of the entire set of stabilizing gains as a solution of sets of linear inequalities. Examples and future applications are discussed