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=Cxy=Cx with feedback of the form u(x)=v(x)Cxu(x)=v(x)Cx is considered, where v(x)v(x) is real-valued and has domain which is all of R2R2. It is shown that, when stabilization is possible, v(x)v(x) can be chosen to take on no more than two values throughout the entire state-space (i.e., v(x)∈{k1,k2}v(x)∈{k1,k2} for all x   and for some k1,k2k1,k2), and an algorithm for finding a specific choice of v(x)v(x) is presented. It is also shown that the classical root locus of the corresponding transfer function C(sI-A)-1BC(sI-A)-1B has a strong connection to this stabilization problem, and its utility is demonstrated through examples.