Properties of recoverable sets for input and state constrained systems

This paper investigates the recoverability properties of linear time-invariant systems with state and input constraints. A state is said to be recoverable if it can be driven to zero while respecting the constraints. In the paper, general properties of the set of recoverable states are established. The equivalence between recoverability and the existence of a time-optimal control is exploited to develop a procedure to calculate recoverable sets. The main theorem of the paper establishes properties of the boundary of the recoverable set for two-dimensional systems. It is shown that the edges of the recoverable set are either edges of the admissible state set or trajectories resulting from the use of maximal control inputs. Further, it is shown that at least one of the end points of the maximum control trajectories must satisfy certain-tangency conditions