(560d) Adaptive Model Reduction for Dissipative PDE Systems with Strong Convective Phenomena

Many industrial chemical processes exhibit spatial variation due to diffusion and convection. Examples include plasma enhanced, chemical vapor/physical vapor deposition and tubular reactors. Controlling these processes is of significant safety and economic importance. Since these processes are infinite dimensional in nature because of the spatial variation, the observation and control questions become nontrivial. These systems are called distributed parameter system (DPS) and most of them can be mathematically described by dissipative partial differential equations (PDEs). Since the dynamic behavior of systems described by dissipative PDEs can be approximated by finite dimensional systems, low order models can be obtained via method of weighted residuals (MWR). However, as convection becomes increasingly more dominant compared to diffusion, the accuracy of the reduced order model of a certain order decreases since the part of the system energy that is captured by the modes of the dominant model and the ones of the neglected model become closer. In the extreme case in which convection is dominant and diffusion is negligible, this approach is no longer applicable, since all the modes in WMR are needed to capture the system energy.

To deal with the accuracy issue of such model order reduction methods for processes with significant convection behavior, we propose to use discrete adaptive proper orthogonal decomposition (DAPOD) to recursively update the reduced order model (ROM) when new observations become available. We first use MWR and proper orthogonal decomposition (POD) to construct a reduced order model offline. When new snapshots become available online, the ROM is updated by DAPOD. DAPOD can mitigate the impact of historic observations compared with POD and has lower computational cost than applying POD iteratively.

Improving the accuracy of ROM for systems with strong convective phenomena can also improve the process controllability. Actuation in PDEs can be divided into 3 categories: source terms/interior control, boundary control and coefficients control. Most of the studies focus on source terms and boundary control, while control via coefficients has been rarely discussed. Solving the accuracy issue makes control by manipulating the convection coefficient feasible.

In this work, the performance of DAPOD in systems with strong convective phenomena is illustrated through a tubular reactor example. First, a ROM of a system with time-varying velocity is constructed via DAPOD to demonstrate the improvement in the accuracy of ROM. Then velocity is actuated to control the system. It's also demonstrated that APOD can improve the accuracy of the ROM when unexpected disturbances occur.

REFERENCES

[1] Amit Varshney, Sivakumar Pitchaiah, and Antonios Armaou. Feedback control of dissipative PDE systems using adaptive model reduction. AIChE journal, 55(4), 2009.

[2] Manda Yang and Antonios Armaou. Revisiting APOD accuracy for nonlinear control of transport reaction processes: A spatially discrete approach. Chemical Engineering Science, 181:146–158, 2018.