Analytic determination of stabilizing gains for digital control systems

We develop an analytical criterion for determining the stabilizing gains of a digital control system. The criterion is based on analysis of the behaviour of a real polynomial X(u) constructed from the plant transfer function G(z). It is shown that the range of stabilizing gain K can be completely determined by the real zeros u_i of X(u) lie in [-1, +1] and the signs of appropriate derivatives of X(u) evaluated at u=u_i. The analytical solution given here is computationally simpler than the Nyquist criterion and is a first step towards investigating stabilizability by higher order controllers