Feedback control of nonstandard singularly perturbed systems

Feedback control of linear time-invariant singularly perturbed systems of the form x=A₁₁x+A ₁₂z+B₁u,ε z=A₂₁x+A₂₂ z+B₂u,y=C₁ x+C₂z Eu, where A₂₂ may be singular, is discussed. It is shown that, under stabilizability-detectability assumptions on the slow and fast models, the theory of feedback control of singularly perturbed systems can be extended to the case of singular A₂₂. Both state and output feedback results are given. In particular, a parameterization of all compensators that stabilize the system while preserving its two-time-scale structure is given. It is shown that any two-frequency-scale stabilizing compensator can be asymptotically approximated by a compensator designed via a sequential procedure. In this procedure, a fast (high-frequency) compensator is designed first to stabilize the fast model of the system. Then, a strictly proper slow (low-frequency) compensator is designed to stabilize a modified slow model. The overall compensator is taken as a parallel or cascade connection of the slow and fast compensators. The sequential procedure is applied to the design of high-gain feedback compensators for a class of multivariable plants