Static Output Feedback for a Certain Class of Systems of Order Four

An outstanding, albeit important, problem in control theory is the static output feedback problem. It deals with the case when one can neither measure nor actuate all state variables and exerts control without the use of a state observer. In this paper, we consider systems of order 4, where at least 2 states are measured and also directly actuated, and present means that provide a definite answer to the question whether a stabilising static output feedback exists or not. We characterise cases, where such a control law cannot exist, and show that, for all other cases, either the answer can be provided by means of using semidefinite and linear programming or that, for surprisingly many cases, a stabilising static output feedback always exists. Finally, we show that, for many cases, the stabilising feedback can be obtained by solving a semidefinite or linear programme, outperforming off-the-shelf solutions.