(711a) Observer, Filters and Moving Horizon Estimator Design for Linear Transport-Reaction Distributed Parameter Systems

Fundamental conservation laws in process control, manufacturing, transport systems are present by distributed parameter systems (DPS) models. The burden of such a general representation is that DPS models take form of partial differential equations containing higher order derivatives in space and time. The complexity of dealing with a partial differential equations (PDEs) in the case of chemical engineering relevant linear PDE models lies in necessity of modellers to account for model spatial characteristics by an approximation of underlying model through some spatial approximation arriving to a finite dimensional model representation amenable for subsequent control, observer and/or monitoring device design [1], [2].

This work provides foundation for systematic development of modelling framework for a design of observer, filters and/or moving horizon estimator for linear DPS system which which is accomplished without application of any spatial approximation and/or order reduction. Namely, we will presented a general observer and filter design methodologies which follow the seminal works of Luenberger (Luenberger observer [3]) and Kalman (Kalman Filter [4]). In particular, we will link the existing well known observer and/or filters design concepts to the infinite dimensional PDE settings and point the differences or analogies with the finite dimensional designs. Results of this novel approach are applied to the DPS emerging from the chemical transport-reaction systems varying from the convection dominated models of a plug flow reactor to diffusion dominated models of an axial dispersion reactor.
In particular, the discrete model of a distributed parameter system is obtained by using energy preserving Cayley-Tustin discretization [5]. This ultimately leads to discrete (in time and continuous in space) DPS models which are low dimensional, energy preserving and do not dissipate numerically (symplectic). In particular, infinite dimensional discrete setting is suitable to filter and moving horizon estimator design which can account for optimality and naturally present constraints in the system. Finally, the design performances are assessed by numerical simulation with application to different linear transport-reaction distributed parameter systems which are endowed with process and measurement noise.

[1] Ray, W. Harmon, Advanced Process Control, McGraw-Hill Inc.,USA, 1980

[2] Curtain, Ruth F and Zwart, Hans, An introduction to infinite-dimensional linear systems theory, Springer, 1995.

[3] David Luenberger, An Introduction to Observes, IEEE Transactions on Automatic Control, vol, 16, no 6, 1971.

[4] R. E. Kalman, A new approach to linear filtering and prediction theory, J. of Basic Engrg, Trans. ASME, Series D, vol 82, pp-34-45, 1960.

[5] V. Havu, J. Malinen, The Cayley transform as a time discretization scheme, Numerical Functional Analysis and Optimization 28 (7-8) (2007) 825-851.