Set-Based State Estimation Approaches for Descriptor Systems

Abstract

This chapter proposes a general set-based framework for robust state estimation of discrete-time descriptor systems, which builds a bridge to fault diagnosis and control design problems. Specifically, a set-membership state estimator and a zonotopic Kalman observer are investigated. The considered LTI descriptor systems are affected by three types of system uncertainties: unknown inputs and unknown-but-bounded system disturbances and measurement noise. One limitation for the use of zonotopic approaches in real applications is that some system disturbances are unknown and it may not be possible to bound them in a predefined zonotope as a priori. To overcome this problem, two classes of unknown system disturbances are considered: (i) bounded disturbances in a zonotope; (ii) unbounded disturbances, which are considered to be unknown inputs and can be decoupled in the observer design. As shown in Fig. 2.1, two set-based approaches with different criteria are studied and therefore the relationship between both approaches is also established. In particular, it is proved that the zonotopic observer in the current estimation type is equivalent to the set-membership approach. Besides, the set-membership approach is extended for discrete-time LPV descriptor systems, where a new zonotope minimization criterion based on the \( \mathcal {L}_{\infty } \) norm is defined.