Feedback control of dissipative PDE systems using adaptive model reduction

Abstract

in Wiley InterScience (www.interscience.wiley.com).The problem of feedback control of spatially distributed processes described by highly dissipative partial differential equations (PDEs) is considered. Typically, this problem is addressed through model reduction, where finite dimensional approximations to the original infinite dimensional PDE system are derived and used for controller design. The key step in this approach is the computation of basis functions that are subsequently utilized to obtain finite dimensional ordinary differential equation (ODE) models using the method of weighted residuals. A common approach to this task is the Karhunen-Loe`ve expansion combined with the method of snapshots. To circumvent the issue of a priori availability of a sufficiently large ensemble of PDE solution data, the focus is on the recursive computation of eigenfunctions as additional data from the process becomes available. Initially, an ensemble of eigenfunctions is constructed based on a relatively small number of snapshots, and the covariance matrix is computed. The dominant eigenspace of this matrix is then utilized to compute the empirical eigenfunctions required for model reduction. This dominant eigenspace is recomputed with the addition of each snapshot with possible increase or decrease in its dimensionality; due to its small dimensionality the computational burden is relatively small. The proposed approach is applied to representative examples of dissipative PDEs, with both linear and nonlinear spatial differential operators, to demonstrate its effectiveness of the proposed methodology.

Handling state constraints and economics in feedback control of transport-reaction processes

“…Out of N possible eigenvalues from the covariance matrix of the ensemble, the most dominant m eigenvalues of the covariance matrix occupies Á energy of the whole ensemble, i.e., m j=1 j / N j=1 j ≤ Á. Then, the computational efficiency of the control system whose construction is based on the ROM with the dominant eigenfunctions will be improved due to the adaptive property of APOD [26]. Since the basis eigenfunctions are updated on-line, the initial ensemble of process snapshots may contain significantly less process solution data than POD.…”

“…Since the basis eigenfunctions are updated on-line, the initial ensemble of process snapshots may contain significantly less process solution data than POD. More details of the APOD methodology can be found in [26] and [19]. The implementation steps of the APOD methodology can be summarized as follows:…”

“…Constructing such a large ensemble of snapshots becomes a significant challenge from a practical point of view; because currently, there is no general way to realize a representative ensemble. Based on this consideration, an adaptive proper orthogonal decomposition (APOD) methodology was proposed to recursively update the ensemble of snapshots and compute on-line the new empirical eigenfunctions in the on-line closed-loop operation of PDE systems (e.g., [20,22,26,19]). While the APOD methodology of [26,19] demonstrated its ability to capture the dominant process dynamics by a relatively small number of snapshots which reduces the overall computational burden, these works did not address the issue of computational efficiency with respect to optimal control action calculation and input and state constraint handling.…”

“…Based on this consideration, an adaptive proper orthogonal decomposition (APOD) methodology was proposed to recursively update the ensemble of snapshots and compute on-line the new empirical eigenfunctions in the on-line closed-loop operation of PDE systems (e.g., [20,22,26,19]). While the APOD methodology of [26,19] demonstrated its ability to capture the dominant process dynamics by a relatively small number of snapshots which reduces the overall computational burden, these works did not address the issue of computational efficiency with respect to optimal control action calculation and input and state constraint handling. Moreover, the ROM accuracy is limited by the number of the empirical eigenfunctions adopted for the ROM; in practice, when a process faces state constraints, the accuracy of the ROM based on a limited number of eigenfunctions may not be able to allow the controller to avoid a state constraint violation.…”

Smart Manufacturing Approach for Efficient Operation of Industrial Steam-Methane Reformers

“…In the case of poor prediction, the model can be efficiently updated using adaptive POD techniques (e.g., see the work of Varshney et al 27 ), where the basis functions are recomputed efficiently in light of availability of new measurements.…”