Including inputs and control within equation-free architectures for complex systems
The increasing ubiquity of complex systems that require control is a challenge for existing methodologies in characterization and controller design when the system is high-dimensional, nonlinear, and without physics-based governing equations. We review standard model reduction techniques such as Proper Orthogonal Decomposition (POD) with Galerkin projection and Balanced POD (BPOD). Further, we discuss the link between these equation-based methods and recently developed equation-free methods such as the Dynamic Mode Decomposition and Koopman operator theory. These data-driven methods can mitigate the challenge of not having a well-characterized set of governing equations. We illustrate that this equation-free approach that is being applied to measurement data from complex systems can be extended to include inputs and control. Three specific research examples are presented that extend current equation-free architectures toward the characterization and control of complex systems. These examples motivate a potentially revolutionary shift in the characterization of complex systems and subsequent design of objective-based controllers for data-driven models.
The top panel illustrates the data collection aspect of equation-free methods. The data can come from experiments, numerical simulations, or historical records. In this case, the historical records illustrate google flu trends in the United States with the location on the y-axis and time on the x-axis. The bottom panel presents the three recently developed perspectives on characterizing nonlinear input-output systems using equation-free techniques.