We study state estimation of linear systems with unknown inputs. When the system is not strongly observable (strongly detectable), one cannot exactly (asymptotically) reconstruct the states without further information about the system or inputs; in this case, various formulations have been studied that require additional information about the nature of the unknown inputs (e.g., Kalman filtering and set-membership filtering). In this paper, we consider a formulation where the unknown inputs and initial condition of the system are bounded in magnitude. The objective is to construct an unknown input norm-observer which estimates an upper bound for the norm of the states. In order to characterize the existence of such an observer, we propose a notion of bounded-input-bounded-output-bounded-state (BIBOBS) stability; this concept supplements various system properties, including detectability, bounded-input-bounded-output (BIBO) stability, bounded-input-bounded-state (BIBS) stability, and input-output-to-state stability (IOSS). We provide checkable conditions on the system matrices under which a general class of linear systems is BIBOBS stable, and show that the set of modes of the system with magnitude 1 plays a key role.
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