A note on stabilization by periodic output feedback for third-order systems

Consider a linear time-invariant continuous-time control system _ x(t) = Ax(t) +Bu(t); y(t) = Cx(t) (1) with state x(t) 2 Rn , input u(t) 2 Rm and output y(t) 2 Rp . The matrices A;B and C are constant and of appropriate dimension. Under the implicit assumption that _ x(t) = Ax(t) is unstable, we want to stabilize (1) by introducing feedback into the system. It is well-known that, under mild conditions, this problem may be solved by constant state feedback. This approach however requires the full state x(t) to be available for feedback, which may be undesirable or impossible in practice. When only the output y(t) is available, different feedback strategies are possible. Usually one resorts to observer-based dynamic output feedback. Alternatively, one may consider static time-invariant output feedback u(t) = Ky(t), but the potential of this is limited in comparison with dynamic output feedback. This limitation however may be partially removed by allowing the feedback gain K to depend explicitly on time. This is the subject of the present paper.