(266e) Dynamic Estimation and Control of Dissipative Pde Systems Using Extended Kalman Filter
Many processes relevant to the chemical process industry necessitate the consideration of transport phenomena (momentum, heat and mass transfer), often coupled with chemical reactions. Examples range from reactive distillation in petroleum processing to plasma enhanced chemical vapor deposition, etching and metallorganic vapor phase epitaxy (MOVPE) in semiconductor manufacturing. Mathematical descriptions of these transport-reaction processes can be derived from dynamic conservation equations and usually involve highly dissipative (typically parabolic) partial differential equation systems (PDEs).
The feedback control problem of these processes is nontrivial owing to the spatially distributed description of their dynamics. The state in the corresponding control problem is infinite dimensional when expressed in appropriate spaces. However the discrete nature of actuators, finite precision and limited memory capacity of the computers implementing the control action requires the approximation of the dynamics of the infinite dimensional PDE system by a finite dimensional dynamical system. The traditional approach to achieve this is based on discretization of the underlying PDEs using finite differences/elements to yield a set of ordinary differential equations (ODEs) which are subsequently used in formulating the feedback control problem for the processes. However, this approach in general leads to very high dimensionality of the controllers and especially cannot be applied to problems which have complicated geometries and/or have sharp variations of solutions, since use of finer discretization to accurately capture the dynamics is then required.
Some model reduction techniques capitalize on the fact that the eigenspectrum of the spatial differential operators in the parabolic PDE systems can be partitioned into a finite-dimensional slow subspace and an infinite-dimensional fast subspace. In other words, the
long-term dynamics of the dissipative PDEs are finite-dimensional and therefore a finite dimensional model would accurately capture the dynamics of the original infinite dimensional problem. One such approach is Galerkin's method, where the solution of the system is expanded using the eigenfunctions of the spatial differential operator. This yields a system of ordinary differential equations (ODEs) that accurately describe the dominant (slow) modes of the PDEs, and is subsequently used to design feedback controllers.
In this work, the problem of dynamic estimation and control of these PDEs when noisy measurements are available (from limited number of sensors) is investigated using Galerkin's method combined with an extended Kalman filter (EKF). We present an application of the above approach to control the temperature in a catalytic rod where an exothermic reaction is taking place. We assume that the noisy measurement data is available from three measurement sensors. The effect of location of these sensors on the performance of EKF is studied through Monte-Carlo simulations of the estimator. The role of process noise covariance and the measurement noise covariance towards the effectiveness of EKF is elucidated through numerical simulations. It is observed from numerical simulations that the 2-norm of the estimation error asymptotically converges to zero as more measurements from the process are made available to the estimator. A discrete time nonlinear feedback controller, based on the estimated states, is designed which successfully stabilizes the process to an open-loop unstable steady-state.