(242b) Nonlinear Control of a Tubular Reactor with Recycle Using Modified APOD and Deim

In recent years, control of nonlinear distributed parameter system (DPS) has been of critical importance due to the simultaneous existence of diffusion, convection and reaction phenomena in a lot of chemical processes, such as catalytic reactors and heat exchangers. These phenomena lead to spatial variation of the process variables which significantly complicates the control and estimation problems. One approach is to construct reduced order models (ROMs) using Galerkin’s method by decomposing the dependent variables into temporal part and spatial part (or basis function); the latter is obtained by analytically solving the eigenfunction problem of the linear operator of the governing equation. Since this analytical approach may not be applicable in complex systems, proper orthogonal decomposition (POD) is widely used to construct basis functions. In this algorithm, an ensemble of process operation snapshots is required. However, there is no method to guarantee the system is properly excited when collecting snapshots [1]. Another disadvantage is sampling too few snapshots may lead to inaccurate ROMs, even when the system is properly excited since their relative importance will not be captured.

One solution to this issue is adaptive proper orthogonal decomposition (APOD) [2]. In this method, basis functions are updated when new snapshots become available since the optimal basis function that captures the dominant feature of the system may change over time. However, it still has limitations. The first issue is as snapshots are discarded based on their importance; new snapshots may keep being eliminated when new snapshots show significant difference from previous ones. Another disadvantage is more basis functions might be used than necessary due to the inaccuracy of eigenvalues estimated in this algorithm.

To circumvent the above problems, we propose a refinement to APOD. First, we construct a weight matrix based on the importance vector of basis functions to keep new snapshots from being eliminated. In the basis function updating part, the residual which is norm of difference between ROM and original system is used to determine whether basis functions need to be updated and whether the size of  basis needs to increase or decrease.

An issue with ROMs is that the resulting ordinary differential equations (ODEs) system may still be computationally expensive to integrate due to the way nonlinearities in the original partial differential equation(s) are handled. This can lead to delays in the computation of control action. To circumvent this issue, discrete empirical interpolation method (DEIM) [3] is adopted to reduce computational cost. In this method, the nonlinear term in governing equation in the whole domain is estimated by measurement at k points with k much less than the number of spatial grid points using nonlinear basis functions. Nonlinear basis functions are constructed off-line and the positions of those k points are determined based on DEIM algorithm.

We propose a combination of APOD and DEIM to design nonlinear controllers of reduced computational requirements to force the closed-loop DPS evolution to a desired operating point. To illustrate the performance of the proposed control scheme and investigate the stability of it, it is applied to a tubular reactor with recycle to eliminate hot spot formulation.

REFERENCES

[1] Pitchaiah, S., & Armaou, A. (2010). Output feedback control of distributed parameter systems using adaptive proper orthogonal decomposition. Industrial & Engineering Chemistry Research, 49, 10496–10509.

[2] DB Pourkargar and Antonios Armaou.(2013). Modification to adaptive model reduction for regulation of distributed parameter systems with fast transients. AIChE Journal, 59(12):4595–4611.

[3] Saifon Chaturantabut and DC Sorensen. (2010). Nonlinear model reduction via discrete empirical interpolation. SIAM Journal on Scientific Computing, 32(5):2737–2764.