Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control

By Milan Korda, Igor Mezi'c. at - Automatisierungstechnik, 2016

1,254 citations recorded by the source corpus.

This paper presents a class of linear predictors for nonlinear controlled dynamical systems. The basic idea is to lift (or embed) the nonlinear dynamics into a higher dimensional space where its evolution is approximately linear. In an uncontrolled setting, this procedure amounts to numerical approximations of the Koopman operator associated to the nonlinear dynamics. In this work, we extend the Koopman operator to controlled dynamical systems and apply the Extended Dynamic Mode Decomposition (EDMD) to compute a finite-dimensional approximation of the operator in such a way that this approximation has the form of a linear controlled dynamical system. In numerical examples, the linear predictors obtained in this way exhibit a performance superior to existing linear predictors such as those based on local linearization or the so called Carleman linearization. Importantly, the procedure to construct these linear predictors is completely data-driven and extremely simple – it boils down to a nonlinear transformation of the data (the lifting) and a linear least squares problem in the lifted space that can be readily solved for large data sets. These linear predictors can be readily used to design controllers for the nonlinear dynamical system using linear controller design methodologies. We focus in particular on model predictive control (MPC) and show that MPC controllers designed in this way enjoy computational complexity of the underlying optimization problem comparable to that of MPC for a linear dynamical system with the same number of control inputs and the same dimension of the state-space. Importantly, linear inequality constraints on the state and control inputs as well as nonlinear constraints on the state can be imposed in a linear fashion in the proposed MPC scheme. Similarly, cost functions nonlinear in the state variable can be handled in a linear fashion. We treat both the full-state measurement case and the input–output case, as well as systems with disturbances/noise. Numerical examples demonstrate the approach.

Subfields: Computational Mechanics, Control and Systems Engineering, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/dbb73c3f14600e3a746c643c2e8bece117de6be4

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Citations in the corpus

On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator (2017, methodology): “, [13]), where model is often known, or for numerical studies of Koopman operator approximations on”
On Numerical Approximations of the Koopman Operator (2020, background): “It has a number of extensions (many of which are summarized in [23]), for example, Exact DMD [24]; Bayesian/subspace DMD [25]; Optimized DMD [26,27]; Recursive DMD [28]; Variational DMD [29]; DMD with control [30,31]; sparsity promoting DMD [32]; DMD for noisy systems [33–35].”
Generalizing Koopman Theory to Allow for Inputs and Control (2016, neutral): “Further, the formulation of this problem could have utilized uk+1 = 0 or the shift operator from [26].”
Sparse identification of nonlinear dynamics for model predictive control in the low-data limit (2017, methodology): “Koopman models have recently been used for MPC [23] and have been identified using SINDy regression [22] and subsequently used for optimal control [22].”
Network resilience (2020, methodology): “For high-dimensional networks, in addition to the traditional linearization method [36, 33], dimension-reduction method based on meanfield theory has been verified as a powerful tool to predict the tipping points [12, 54]; or we could use the”
Data-driven approximation of the Koopman generator: Model reduction, system identification, and control (2019, methodology): “The same approach was also used in combination with MPC in [54].”
Deep Dynamical Modeling and Control of Unsteady Fluid Flows (2018, positive): “Recent work has sought to incorporate reduced-order models with robust- and optimal-control techniques to devise controllers for nonlinear systems [6], [13], [14].”
Extended dynamic mode decomposition with dictionary learning: A data-driven adaptive spectral decomposition of the Koopman operator. (2017, methodology): “This is an important advantage of the Koopman framework for current challenges in model reduction, prediction, data fusion, and system control 7–12 .”
Koopman Lyapunov‐based model predictive control of nonlinear chemical process systems (2019, background): “However, making a shrewd choice of the basis functions for unknown systems is still an active research area and a few studies suggest that canonical choices like radial basis functions and Hermite polynomials are a good starting point.(14,19) Several other approaches propose using deep learning meth”
Data-Driven Sparse Sensor Placement for Reconstruction: Demonstrating the Benefits of Exploiting Known Patterns (2017, methodology): “Recently, Koopman analysis has been used to develop nonlinear esti-5 mators [79], [80] and controllers [81], although establishing rigorous connections to control theory is an ongoing effort [82], [83], [84].”
Error bounds for kernel-based approximations of the Koopman operator (2023, methodology): “Koopman methods have found widespread application in system identification [5], control [29, 49, 30, 22, 56], sensor placement [38], molecular dynamics [57, 51, 42, 43, 23, 64], and many other fields.”
Data-driven spectral analysis of the Koopman operator (2017, neutral): “For example, global stability is analyzed in [27], whereas [28] deals with the so-called isostables and isochrones; ergodic partition and mixing properties are analyzed in [7], and [18, 6] utilize the Koopman operator approximations for control whereas [29] for model reduction.”