On the stabilizability of two-dimensional linear systems via switched output feedback

The problem of stabilizing a second order SISO LTI system of the form x^˙ = Ax + Bu, y = Cx with feedback of the form u(x) = v(x)Cx is considered, where v(x) is real-valued and has domain which is all of R². It is shown that, when stabilization is possible, v(x) can be chosen to take on no more than two values throughout the entire state space (i.e., v(x) ∈ {v₁, v₂} for all x and for some v₁, v₂) and an algorithm for finding a specific choice of v(x) is presented. It is also shown that the classical root locus of the corresponding transfer function C(sl - A)^-1 B has a strong connection to this stabilization problem, and its utility is demonstrated through several design examples.