Abstract
While state estimation techniques are routinely applied to systems represented by ordinary differential equation (ODE) models, it remains a challenging task to design an observer for a distributed parameter system described by partial differential equations (PDEs). Indeed, PDE systems present a number of unique challenges related to the space-time dependence of the states, and well-established methods for ODE systems do not translate directly. However, the steady progresses in computational power allows executing increasingly sophisticated algorithms, and the field of state estimation for PDE systems has received revived interest in the last decades, also from a theoretical point of view. This paper provides a concise overview of some of the available methods for the design of state observers, or software sensors, for linear and semilinear PDE systems based on both early and late lumping approaches.
Highlights
For many physical systems, the states, inputs, and outputs may depend on spatial coordinates, which define a position in a multidimensional space
These systems are modeled by partial differential equations (PDEs) and, along with systems modeled by integral equations and delay differential equations, are called distributed parameter systems (DPSs)
In the late-lumping approach, the distributed nature of the system model is preserved for state estimation resulting in an infinite-dimensional state observer, which is later lumped for implementation purposes
Summary
The states, inputs, and outputs may depend on spatial coordinates, which define a position in a multidimensional space These systems are modeled by partial differential equations (PDEs) and, along with systems modeled by integral equations and delay differential equations, are called distributed parameter systems (DPSs). The on-line state estimation of DPSs modeled by PDEs is a delicate problem in view of the system dimensionality and the fact that providing a comprehensive set of sensors is either physically impossible or too costly [18] In such a case, the internal states have to be estimated on the basis of the mathematical model and (available) online measurements provided by sensors located at strategic positions in the spatial domain. A brief introduction to PDE systems is presented in Section 2, while Sections 3 and 4 survey the early and late lumping approaches, respectively, Section 5 describes the applied perspectives of the subject, and Section 6 concludes this paper
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