Koopman operator-based model reduction for switched-system control of PDEs

Abstract

We present a new framework for optimal and feedback control of PDEs using Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a linear but infinite-dimensional operator which describes the dynamics of observables. A numerical approximation of the Koopman operator therefore yields a linear system for the observation of an autonomous dynamical system. In our approach, by introducing a finite number of constant controls, the dynamic control system is transformed into a set of autonomous systems and the corresponding optimal control problem into a switching time optimization problem. This allows us to replace each of these systems by a K-ROM which can be solved orders of magnitude faster. By this approach, a nonlinear infinite-dimensional control problem is transformed into a low-dimensional linear problem. In situations where the Koopman operator can be computed exactly using Extended Dynamic Mode Decomposition (EDMD), the proposed approach yields optimal control inputs. Furthermore, a recent convergence result for EDMD suggests that the approach can be applied to more complex dynamics as well. To illustrate the results, we consider the 1D Burgers equation and the 2D Navier-Stokes equations. The numerical experiments show remarkable performance concerning both solution times and accuracy.

Digital Twin: Values, Challenges and Enablers From a Modeling Perspective

“…This ROM concept (also called surrogate modeling or metamodeling) allows one to emulate complex full-order models or processes, and lies at the interface of data and domain specific sciences. It has been systematically explored for decades in many different areas including computational mechanics [191]- [194], combustion [195]- [197], cosmology [198]- [200], electrodynamics [201]- [204], meteorology [205]- [208], fluid mechanics [209]- [215], heat transfer [216]- [218] and various other multiphysics processes [219]- [225], as well as systems and control theories [226]- [231] in order to provide computational feasible surrogate models. Although the concept can be traced back to the works done by Fourier (1768-1830), there exist many recent monographs [232]- [239] and review articles [240]- [253].…”

Data-driven approximation of the Koopman generator: Model reduction, system identification, and control

“…The same approach was also used in combination with MPC in [52]. This state augmentation significantly increases the data requirements (all combinations of states and control inputs should be covered), such that an alternative transformation was proposed in [53,54] by restricting u(t) to a finite set of inputs {u 1 , . .…”

Data-Driven Approximations of Dynamical Systems Operators for Control

“…Parameterized linear models: Another class of linear parameter-varying models have been considered to disambiguate the approximation of the Koopman operator of the unforced system from the effect of exogenous forcing [190] and for model predictive control of switched systems [136]. In particular, a set of linear models parametrized by the control input is considered as an approximation to the non-autonomous Koopman operator:…”

“…(35) is a combinatorial problem, which can be efficiently solved for low-dimensional problems using iterative algorithms from dynamic programming [18]. In [136], the MPC problem for the sequence of control inputs is transformed into a time switching optimization problem assuming a fixed sequence of consecutive discrete control inputs. The model can further be modified as a bilinear continuous, piecewise-smooth variant with constant system matrices, which does not suffer from the curse of dimensionality and allows for continuous control inputs via linear interpolation [137].…”

“…4.1) have been increasingly used in combination with optimal control for trajectory steering, e.g. with LQR [29], SDRE [76], and MPC [91,136,8,79]. Besides theoretical developments, Koopman-based MPC has been increasingly applied in realistic problems, such as power grids [92], highdimensional fluid flows [69], and experimental robotic systems [2,1].…”