Non-Asymptotic State and Input Estimation for Smooth Linear Parameter Varying Systems

This note explains the advantage of employing integral system representations of linear systems in application to state and input estimation for a broad class of LPV systems. Integral (kernel) representations of linear systems are seen as vehicles for retaining local information about the system's input-output behaviour in which the notion of initial or boundary conditions play no role. An important by-product of kernel system representations is their capacity for reconstruction of time derivatives of the measured system output by using the kernel derivatives. These attributes immediately lead to the construction of kernel-based dead-beat state and parameter estimators for linear systems. The approach is extended here to LPV systems with measured scheduling parameters, or else LPV systems in which the scheduling parameters cannot be measured directly, but whose values may be inferred from the output observations of other, possibly nonlinear, dynamical systems. It is shown that the lack of knowledge of the system input does not prejudice successful state estimation provided that the system has strong observability properties that effectively permit input reconstruction in a suitable B-spline basis. The method also applies to nonlinear smooth systems that can be transformed to LPV systems with dynamically varying parameters. An example of a strongly nonlinear system is presented for which the extended Kalman filter is known to fail.