Stabilization of discrete-time systems by first-order controllers

In this note, we consider the problem of stabilizing a given but arbitrary linear time invariant discrete-time system with transfer function P(z), by a first-order discrete-time feedback controller C(z)=(zx₁+x₂)/(z+x₃). The complete set of stabilizing controllers is determined in the controller parameter space [x₁, x₂, x₃]. The solution involves the Chebyshev representation of the characteristic equation on the unit circle. The set is shown to be computable explicitly, for fixed x₃ by solving linear equations involving the Chebyshev polynomials in closed form, and the three-dimensional set is recovered by sweeping over the scalar parameter x₃. This result gives a constructive solution of: 1) the problem of "first-order stabilizability" of a given plant; 2) simultaneous stabilization of a set of plants P_i(z); and 3) stable or minimum phase first-order stabilization of a plant. The solution is facilitated by the fact that it is based on linear equations and the intersection of sets can be found by adding more equations. In each case, the complete set of solutions is found and this feature is essential and important for imposing further design requirements. Illustrative examples are included.