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.
Highlights
State-of-the-art advancements in the realm of industrial process operations require good understanding of complex phenomena and systems
Motivated by the superiority of condition monitoring with digital devices to account for the discrete nature of monitoring realization, we develop discrete distributed parameter systems (DPSs) models of first-order coupled hyperbolic partial different equations (PDEs) and second-order hyperbolic PDEs, which account for the infinite-dimensional nature of the pipeline hydraulic process and the wave equation
Continuous-time distributed parameter systems can be taken as good representations of complex industrial processes, while it is more practical and preferable for designers to come up with discrete-time models for observer, controller, and regulator design when it comes to actual implementation in digital devices
Summary
State-of-the-art advancements in the realm of industrial process operations require good understanding of complex phenomena and systems. Interesting work on distributed parameter observer designs for layering granulation, crystallization, and reaction-diffusion processes can be found in [36,37,38] Another important contribution arises in the backstepping approach with its wide application to control, state, and parameter estimation of infinite-dimensional systems, where Krstic and his co-workers reported constructive studies on stabilization, adaptive control, and estimation of flow, and engine and beam systems [39,40,41,42,43]. The extension of finite-dimensional Kalman filter design is provided in a closed form of a discrete infinite-dimensional distributed parameter system describing the water hammer and wave dynamics and accounts for naturally present noise in both output measurements and PDE states.
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