Effect of Finite-Dimensional Approximations on Observability Analysis of Distributed Parameter Models

This paper investigates the effect of different discretization schemes for observability analysis of distributed parameter models. While it is common knowledge that approximating an infinite-dimensional model can introduce non-physical numerical diffusion or numerical oscillation in the dynamics of finite dimensional approximation of the distributed parameter system, less attention has been paid to the effect that these approximations have on conclusions drawn about observability and controllability of the system. This paper addresses this point and presents a detailed analysis of results obtained for three approximation schemes for first-order hyperbolic partial differential equations (PDEs). The different, and sometimes misleading, conclusions that can be drawn for the three approximation schemes are discussed in detail and the analysis is illustrated with a numerical example. The case study illustrates the point that a model which may approximate the dynamic behavior of the distributed system accurately may not necessarily correctly reflect observability of the original distributed system.