On the Existence and Design of Functional Observers for LTI Systems, with Application to User Modeling

Observing the states of a linear system has been an extensively studied topic over the past few decades. Different from full-state observers and reduced-order observers which are used to reconstruct the entire state space and the unmeasured states, respectively, functional observers are developed for applications in which only a linear combination of the states is required. For linear time-invariant (LTI) systems with known and/or unknown inputs and assuming availability of the derivatives of the input and the output signals, we study the existence of a novel stable functional observer. Considering the output to be the extended vector of the output and its derivatives, we derive the existence conditions for a functional observer by making the estimation error asymptotically approach zero. The derived existence conditions are applicable to multi-input multi-output (MIMO) systems (e.g., with either known or unknown inputs, and with or without the derivatives of the signals being available). We also provide an estimate of the required observer order for the reconstruction of a desired functional and present how to design this functional observer by modifying certain current techniques. As an application, we apply the designed functional observer to model the stage of information comprehension in the user of a shared/human controlled system.