(555a) Optimal Constrained State Estimation of Parabolic PDE Systems

This work focuses on an optimal constrained state estimation of linear parabolic partial differential equations (PDEs) with output constraints. It is evident from the latest developments of distributed parameter systems estimation theory, that the consideration of state constraints and/or output constraints has been often neglected due to difficulties to include constraints into the structure of an optimal filter [1]. Hence, neglecting the PDE constraints in the estimation algorithm may cause an error in the state estimation and therefore affect computed control law in the feedback structure. For this reason, it is necessary to explore the optimal state estimation with output and state constraints present within the linear PDE setting. In this work, we focus on the design of an optimal constrained state estimator for the parabolic PDE with  the output constraints. We use Galerkin’s method to reformulate the PDE system as the infinite-dimensional system in an appropriate Hilbert space [2], and modal decomposition technique is used to derive the finite-dimensional system that captures the dominant dynamics of the infinite-dimensional system [3]. Then, the resulting slow mode finite-dimensional system and the upper bound of the stable fast mode infinite-dimensional system evolution are used as basis for the synthesis of  the proposed optimal state estimator for the original infinite-dimensional system. Finally, we consider an example: the temperature distribution in a long thin rod being heated in a multizone furnace [4]. An easily implementable proposed optimal state estimator subject to output constraints is derived and demonstrated, via simulation, to achieve the state estimation objective. The results are demonstrated as comparison among unconstrained and constrained optimal state estimation for linear parabolic PDE systems.

Reference

[1] D. Simon (2010). Kalman filtering with state constraints, a survey of linear and nonlinear algorithms. Control Theory and Applications, IET, vol.4, no.8, pp.1303-1318.

[2] R. F. Curtain. and A. J. Pritchard (1978). Infinite Dimensional Linear Systems Theory. Berlin: Springer-Verlag.

[3] S. Dubljevic, P. Mhaskar, N. H. El-Farra, and P. D. Christofides (2005). Predictive control of transport reaction processes. Computers and Chemical Engineering, vol.29, pp.2335-2345.

[4] W. H. Ray (1981). Advanced Process Control. New York: McGraw-Hill.