(59k) Data-Driven Linear Predictive Control of Nonlinear Processes Based on the Reduced-Order Koopman Operator

Complex industrial processes have become commonly adopted across various industries due to their potential to offer better operating safety, operational efficiency, production consistency, and product quality [1][2]. To achieve these merits, it is crucial to implement cost-effective advanced process control to appropriately regulate the process operation in real-time. However, the large scale and high nonlinearity of modern industrial processes present challenges in the development and implementation of scalable advanced control schemes [3].

The development of a successful advanced control system requires a high-fidelity model that is capable of accurately representing the dynamical behavior of the underlying industrial process. If sufficient physics information is available for establishing nonlinear first-principles models to predict the process dynamics, then nonlinear model predictive control (MPC) represents a widely recognized solution to the control of nonlinear processes. In the context of process control, various nonlinear MPC algorithms and approaches have been proposed for nonlinear processes [4][5]. However, when the process scale increases with more tightly interconnected physical units being integrated, it becomes overwhelmingly challenging to derive differential equations to accurately characterize their dynamics [6]. Therefore, the applications of the above-mentioned nonlinear MPC approaches to large and complex industrial processes are still limited. Additionally, even if a first-principles nonlinear model is accessible, the online implementation of these nonlinear MPC algorithms can be demanding. Specifically, the production capacity scales up along with the increase in the number of key states and input variables, which leads to more expensive computation in solving larger-scale (constrained) nonlinear optimization [7].

To address these two limitations, we propose to establish a data-driven linear model to account for nonlinear process dynamics and develop linear MPC schemes to conduct optimal nonlinear process operations. The Koopman theory [8] provides a promising framework for building linear models in a lifted state space to predict the dynamical behaviors of nonlinear systems/processes [9][10]. The direct application of Koopman theory to real-world nonlinear processes for simulations and monitoring/decision-making has been impractical. This has motivated the exploration of a finite-dimensional approximation of the exact Koopman operator, and powerful data-driven approximation methods were proposed [10][11]. These approximation methods have greatly facilitated the development of data-driven linear control schemes for general nonlinear systems [11][12][13]. Meanwhile, despite finite dimensions, the resulting Koopman model can still have significantly higher dimensionality as compared to the original nonlinear process, particularly when more lifting functions are selected to account for the nonlinear mapping between the original space and the lifted linear space. In such cases, it is possible that the computational complexity of Koopman-based control (e.g., predictive control) will be comparable to the nonlinear counterpart developed based on a first-principles model. When dealing with medium- to large-scale nonlinear processes, the dimension of the resulting Koopman model may become excessively high, and the significant increase in the dimensionality needs to be addressed to ensure efficient computation of this linear control paradigm.

The above observations have motivated us to reduce the dimensionality of the lifted space for the linear operator to facilitate efficient online implementation of Koopman-based predictive control solutions. In this work, we propose a data-driven reduced-order Koopman predictive control approach for general nonlinear systems. First, a Kalman-GSINDy approach is leveraged to select appropriate lifting functions from a comprehensive library. This helps to automatically find the lifted linear state-space instead of determining lifting functions manually based on trial-and-error or prior experiences. Second, proper orthogonal decomposition (POD) which has been widely used for extracting low-dimensional features of high-dimensional processes[14][15], is exploited to propose a scalable data-driven Koopman identification approach to establish reduced-order linear models. A linear robust predictive control scheme is proposed based on the reduced-order model. This approach offers benefits in that the computational efficiency can be significantly improved while maintaining comparable control performance compared to the traditional Koopman operators. The effectiveness and superiority of the proposed framework are illustrated in the simulation case.

References

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