Discrete-Time Kalman Filter Design for Linear Infinite-Dimensional Systems

Abstract

As the optimal linear filter and estimator, the Kalman filter has been extensively utilized for state estimation and prediction in the realm of lumped parameter systems. However, the dynamics of complex industrial systems often vary in both spatial and temporal domains, which take the forms of partial differential equations (PDEs) and/or delay equations. State estimation for these systems is quite challenging due to the mathematical complexity. This work addresses discrete-time Kalman filter design and realization for linear distributed parameter systems. In particular, the structural- and energy-preserving Crank–Nicolson framework is applied for model time discretization without spatial approximation or model order reduction. In order to ensure the time instance consistency in Kalman filter design, a new discrete model configuration is derived. To verify the feasibility of the proposed design, two widely-used PDEs models are considered, i.e., a pipeline hydraulic model and a 1D boundary damped wave equation.

Regional enlarged controllability of a fractional derivative of an output linear system

“…Then, using ( 16), we get χ RL ω D α x L T (u ⋆ ) = y ⋆ . Hence, with (10,) we deduce (11). Applying inequality (15), we get…”

“…), g(. )]controllable on ω, then (y ⋆ , µ ⋆ ) is a unique solution of system (11), where r > 0 is suitably chosen.…”

“…In this section, we solve the equations ( 10) and (11) numerically and propose an Uzawa-type algorithm to evaluate the effectiveness of the Lagrangian method (see [18], page 3).…”

“…As the optimal linear filter and estimator, the Kalman Filter design for linear infinitedimensional systems has been widely employed for state estimation and prediction in the realm of lumped parameter systems (see [11]). Besides that, fractional derivatives have been applied to the modelling of combustion processes, offering unique insights into the dynamics and features of these systems.…”