Over the last 20 years there has been increasing focus on analysis and control of processes that exhibit spatial variations. The research activity in this area has been motivated by a wealth of industrially important processes (e.g., chemical vapor deposition and etching, catalytic reaction and polymerization processes) which exhibit significant spatial variations due to the presence of strong diffusive and convective mechanisms. The key difficulty in designing control schemes for such PDE systems arises from the ``infinite-dimensional'' nature of the distributed process model. Based on this property ideas for reducing the complexity of these models were considered through the formulation of the reduced order models (ROM). One of the widely used methodologies for developing ROM is proper orthogonal decomposition (POD). Initially, one collects the experimental data or data from detailed numerical simulations (snapshots) in a ``dataset''. POD then extracts the characteristic basis (shape) functions from the collected data set. The computed basis functions are subsequently used in the framework of "method of weighted residuals" to compute the low-dimensional ROMs. The effectiveness of the POD methodology, however, is dependent on the quality of the collected dataset. Consequently, the ROMs are sufficiently accurate only in a restricted neighborhood around the state space regions where the system is sampled. On the other hand, no well defined methodology exists for constructing ROMs with a larger region of validity as this requires a ``representative'' dataset which contains all the possible spatial modes (including those that might appear during closed-loop evolution of the PDE system). To address these issues we have developed an adaptive methodology called the adaptive proper orthogonal decomposition . This methodology alleviates the requirement for the representative ensemble by recursively updating the basis functions in a computationally efficient way.
In this talk, we will present our recent results on using adaptive reduced order models (generated using APOD) in the design of model predictive controllers for stabilization of processes that are mathematically expressed as parabolic partial differential equation (PDE) systems. Initially, we construct a locally valid ROM of the PDE system employing the basis functions computed by applying the APOD methodology on a small data ensemble. This ROM is then utilized in the design of model predictive controllers (MPC) under constraints on the control action. As periodic closed-loop process data becomes available (during closed-loop operation under the constructed MPC), we recursively update the ROM by employing our computationally efficient adaptive model reduction methodology thus extending the validity of ROM over the current operating region. The effect of employing the adaptive methodology on performance of MPC will be discussed. The design of such MPC controllers is illustrated by employing the methodology on transport-reaction processes. The proposed controller successfully stabilizes the process at an open-loop unstable steady-state while minimal initial sampling is used.
1. Pitchaiah, and A. Armaou, "Feedback control of dissipative distributed parameter systems using adaptive model reduction," Ind. & Eng. Chem. Res., 55, 906-918, 2010.