An exact solution to the stabilization of discrete systems using a first-order controller

An exact solution is derived for stabilizing a given but arbitrary, linear time-invariant discrete system by a first-order discrete-time feedback controller, which has received considerable attention in the past few years. An approach has been recently proposed to compute the first-order controllers, given in the form of C(z)=(zx₁+x₂)/(z+x₃). This approach derives the stabilizing set in the x₁-x₂ plane by fixing x₃, and then repeat the procedure by sweeping over all possible values of x₃. In this note, from the geometrical point of view, we present an exact solution to the problem.