Including inputs and control within equation-free architectures for complex systems

Eur. Phys. J. Special Topics 225, 2413–2434 (2016) © The Author(s) 2016 DOI: 10.1140/epjst/e2016-60057-9 THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS Review Including inputs and control within equation-free architectures for complex systems Joshua L. Proctor1,a , Steven L. Brunton2 , and J. Nathan Kutz3 1 2 3 Institute for Disease Modeling, Bellevue, WA 98004, USA Mechanical Engineering, University of Washington, Seattle, WA 98195, USA Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA Received 19 February 2016 / Received in final form 28 June 2016 Published online 22 November 2016 Abstract. 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. 1 Introduction The characterization and control of complex systems permeate classic physical, biological, and engineering sciences and enable modern applications such as the eradication of Poliomyelitis, control of internet traffic, optimizing energy infrastructures, and social media advertising. The rapidly expanding capabilities in computing power, data storage, and data transfer rates have generated enormous data sets, novel experiments, and offered inspiration to control complex systems in real-time. Adapting traditional mathematical and engineering methods to these high-dimensional, nonlinear systems has presented substantial challenges. Moreover, the design of controllers and the characterization of these complex systems remain an open-challenge requiring a e-mail: 2414 The European Physical Journal Special Topics the development of new quantitative methods. We review the traditional approaches to data-driven analysis while highlighting a number of recent methodological advances toward the control of complex systems. Despite the dimensionality and complexity, it is often possible to find a lowdimensional model to represent the input-output characteristics of the system. This observation is well-known in the field of dynamical systems, specifically within the model reduction community. Identifying the correct, qualitative types of solutions from complex systems is the primary objective of normal forms [1]. The analysis of solutions from higher dimensional systems such as those found in fluid dynamics led to model reduction techniques such as the Proper Orthogonal Decomposition (POD) [2–8] which have been extended to numerous other fields such as animal locomotion [9, 10], vibrational analysis [11–13], and damage analysis [14, 15]. Combining POD with known governing equations via Galerkin projections produces ReducedOrder Models (ROMs) that have been used to analyze fluid flows [3, 4, 6, 16], optical systems [17–19], and fluid flows with control [20–22]. Recent advances have illustrated how POD/Galerkin projections can be extended to more efficiently handle the model reduction component involving the nonlinearities of partial differential equations [23–25]. For engineered systems, discovering advantageous dynamical regimes is important for objective-based control. The rich literature of system identification within the control theoretic community has taken a parallel perspective to POD. In fact, the Singular Value Decomposition (SVD), which is the dimensionality-reduction technique utilized in POD, is also prominently used to construct low-dimensional subspaces where controller design is computationally tractable [6,26–33]. Balanced truncation is a foundational technique that produces ROMs that are constructed to balance input-output characteristics such as controllability and observability [26]. Generalizations of this method, such as Balanced Proper Orthogonal Decomposition (BPOD), utilize the SVD on high-dimensional measurement data to produce balanced ROMs of complex systems, but the method still requires a difficult linear adjoint calculation [29, 34–36]. System identification methods do not rely on this adjoint calculation and were developed to aid in the discovery of input-output models for systems with control [27, 37–41]. Further, these system identification methods can be considered equation-free since they do not rely on a set of governing equations such as with POD/Galerkin projection and BPOD. A fundamentally important observation by Ma et al. demonstrated that the Eigensystem Realization Algorithm (ERA), a system identification method, reproduces balanced input-output models similar to BPOD, thus linking equation-based and equation-free methods [42]. We briefly review POD and BPOD in the Background section because of the historical context and the direct relationship to current equation-free methods. Other system identification methods called subspace identification methods bypass the identification of Markov parameters to produce input-output models from measurement data [43–46]. Dynamic Mode Decomposition (DMD) is a data-driven methods that operates on snapshot measurement only and is considered an equation-free architecture [47–52]. Early success for DMD has been achieved on fluid dynamics problems [52–57], and has been subsequently extended to epidemiology [58], neuroscience [59], and foreground/background separation in video streams [60]. DMD has been shown to be connected to system identification methods such as ERA [51] and equation-based methods such as BPOD [42]. ERA has been predominantly used on systems where the number of measurements is assumed to be low and the system linear whereas DMD excels on high-dimensional measurement data from complex, nonlinear systems [61]. The architecture also lends itself to enabling extensions that take advantage of innovative sampling strategies in space and time [57, 62–64], multi-resolution/ Temporal and Spatio-Temporal Dynamic Instabilities 2415 multi-scale phenomenon [65], de-noising [66, 67], and data fusion [68], extended and kernel DMD [69, 70]. The method was also extended to handle data from complex systems with inputs [61]. DMD has been shown (...truncated)