(470e) Data-Driven Control Via Dynamic-Mode Decomposition

Modeling and control of systems that exhibit spatio-temporal behavior is challenging due to their inherent high state dimensionality. Examples of such systems include flight dynamics, fluid flow control, aeroelasticity, and 3D thermal zone control in buildings [1-4]. Despite their high dimensionality, the dynamics of such systems typically resides on a low-order manifold and thus model reduction techniques can be used to create low-order and high quality model representations. Reduced-order modeling (ROM) techniques such as balanced truncation assume the explicit availability of a full-scale model; however, building such models might require an exorbitant amount of data or the models might simply not fit in memory [5]. In contrast, data-driven modeling approaches such as proper orthogonal decomposition (POD) [6] and dynamic mode decomposition (DMD) [7] can directly extract low-order models from data without the need of identifying an intermediate full-scale model. DMD has become a popular technique in recent years due to its ability to extract dominant modes directly from data and because of connections with powerful techniques such as subspace identification and Koopman operators [8]. However, characterizations of the predictive accuracy of DMD models still remains an open problem [9]; such characterizations are essential for the controller design to ensure provide convergence and closed-loop stability guarantees.

In this work we present a theoretical characterization of the prediction accuracy of DMD models. We derive an explicit error bound that reveals the effect of the model order and the number of data samples in training the DMD model and establishes conditions under which the prediction error vanishes. We propose a model-predictive control (MPC) framework based on DMD for controlling high-dimensional systems. This strategy is motivated by the need to control systems directly from real-time image and video data. We show the the proposed framework can handle high-dimensional systems in a scalable manner but also reveals interesting controllability limitations that arise in high-dimensional systems. We use a 2D heat diffusion system to illustrate the developments; this system contains 2,500 states and 36 heating/cooling actuators. It is shown that the DMD provides an accurate model (accounting for 99.5% of the information of the full-scale system) that contains only 40 states. We also show that the reduced-order MPC controller can track reference fields to high accuracy, provided that such reference fields live in a low-order controllable subspace of the original system.

References:

[1] Clarence W. Rowley. Model reduction for fluids, using balanced proper orthogonal decomposition. International Journal of Bifurcation and Chaos, 15(3): 997-1013, 2005.

[2] Mehdi Ghoreyshi, Adam Jirasek, and Russell M. Cummings. Reduced order unsteady aerodynamic modeling for stability and control analysis using computational fluid dynamics. Progress in Aerospace Science, 71: 167-217, 2014.

[3] David J. Lucia, Philip S. Beran, and Walter A. Silva. Reduced-order modeling: new approaches for computational physics. Progress in Aerospace Science, 40(1-2): 51-117, 2004.

[4] Erik A. Wolff and Siguard Skogestad. Temperature cascade control of distillation columns. Industrial & Engineering Chemistry Research, 35: 475-484, 1996.

[5] Bruce Moore. Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEE Transactions on Automatic Control, 26(1): 17-32, 1981.

[6] Athanasios C. Antoulas. Approximation of Large-scale Dynamical Systems, vol. 6, SIAM, 2015.

[7] J. Nathan Kutz, Steven L. Brunton, Bingni W. Brunton, and Joshua L. Proctor. Dynamic Mode Decomposition: Data-driven Modeling of Complex Systems, SIAM, 2016.

[8] Clarence W. Rowley. Model reduction for fluids, using balanced proper orthogonal decomposition. International Journal of Bifurcation and Chaos, 15(3): 997-1013, 2015.

[9] Hannah Lu and Daniel M. Tartakovsky. Predictive accuracy of dynamic mode decomposition. arXiv preprint: 1905.01587, 2019.