(399a) Constrained Control of Dissipative Distributed Parameter Systems Via on-Demand Data-Driven Model Reduction

Distributed parameter systems (DPSs) that frequently appear in fluid flows and diffusion-convection-reaction processes can be mathematically modeled by dissipative partial differential equations (DPDEs). The control problem of such systems is challenging due to the spatial variation and complex temporal dynamics of the state variables [1]. Model reduction is known as an auspicious method to address the control and optimization problems for dissipative DPSs, where a reduced-order model representation of the governing DPDEs is used as the basis for the optimization formulation and control design [2]. Given the required spatial basis functions, the reduced-order models can be derived by employing weighted residual methods in the form of ordinary differential equations describe the temporal dynamics [1].

Analytical methods for basis functions computation are applicable only to linear systems with simple boundary conditions. Data-driven methods such as proper orthogonal decomposition (POD) have been applied as an alternative solution to compute dominant empirical basis functions [2-5]. Specifically, POD analyzes spatially distributed profiles of the system states (aka process snapshots) obtained by high-fidelity simulations or measured by spatially distributed sensors to compute the basis functions. POD may require a large number of process snapshots to guarantee the optimality of representing the process dynamics, specifically in the presence of model uncertainty and time-varying parameters [6]. Then, the resulting empirical basis functions may need to be revised as needed due to the appearance of new trends.

In previous work, adaptive proper orthogonal decomposition (APOD) was developed to recursively update the empirical basis functions using the new process snapshots [6]. Once the empirical basis functions are updated, the current reduced-order model switches to a new one and the controller design is revised. These operations increase measurement, communication, and computation costs [7].

In this work, we develop an on-demand recursive POD to minimize the frequency of reduced-order model revisions. The proposed method hinges on an event-triggered mechanism which evaluates the constrained control Lyapunov function (CLF) of system modes estimated by a static observer. The input-constrained controller based on the reduced-order model is synthesized to force the CLF to decrease between the reduced-order model revisions to guarantee the asymptotic stability of the closed-loop switching system [8]. Then, an increase of the CLF at any time implies that the reduced-order model is inaccurate. We identify the periods when the reduced-order model must be revised by tracking the time derivative of the CLF. The recursive revision continues until the CLF value satisfies the switched system stability criteria. The proposed approach is applied to stabilize the Kuramoto-Sivashinsky equation, which exemplifies wave motion of falling liquid thin films, instability at the interface of two viscous fluids, and phase turbulence in transport-reaction processes.

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