Unknown-input state observers for hybrid dynamical structures

This chapter investigates the unknown-input state observation problem for hybrid dynamical systems with state jumps. The problem considered is that of deriving an asymptotic estimate of a linear function of the state of a given system in the presence of unknown inputs. The systems addressed are hybrid dynamical systems which exhibit a continuous-time linear behaviour except at isolated points of the time axis, where their state shows abrupt changes ruled by an algebraic linear equation. The systems belonging to this class are also known as linear impulsive systems. It will be assumed that the time interval between two consecutive state discontinuities is lower bounded by a positive real constant. The presence of unknown inputs precludes, in general, the possibility of asymptotically estimating the whole system state, but may permit the asymptotic estimation of a linear function of the state. Thus, the objective of this chapter is to state and prove necessary and sufficient conditions for the existence of solutions to this problem, in the context of linear impulsive systems. The methodology adopted is structural, based on the use of properly defined geometric objects and properties. A general necessary and sufficient condition is proven first. Then, under more restrictive, yet acceptable in contexts of practical interest, assumptions, a constructive necessary and sufficient condition is shown. The latter condition can be checked by an algorithmic procedure, since it is based on subspaces which can be easily computed and on properties which can be systematically ascertained. Special attention is paid to the synthesis of asymptotic observers whose state has the minimal possible dimension, briefly referred to as minimal-order observers.