An official website of the United States government
The Koopman operator theory provides a global linearization framework for general nonlinear dynamics, offering significant advantages for system analysis and control. However, practical applications typically involve approximating the infinite-dimensional Koopman operator in a lifted space spanned by a finite set of observable functions. The accuracy of this approximation is the key to effective Koopman operator-based analysis and control methods, generally improving as the dimension of the observables increases. Nonetheless, this increase in dimensionality significantly escalates both storage requirements and computational complexity, particularly for high-dimensional systems, thereby limiting the applicability of these methods in real-world problems. In this paper, we address this problem by reformulating the Koopman operator in tensor format to break the curse of dimensionality associated with its approximation through tensor decomposition techniques. This effective reduction in complexity enables the selection of high-dimensional observable functions and the handling of large-scale datasets, which leads to a precise linear prediction model utilizing the tensor-based Koopman operator. Furthermore, we propose an optimal control framework with the tensor-based Koopman operator, which adeptly addresses the nonlinear dynamics and constraints by linear reformulation in the lifted space and significantly reduces the computational complexity through separated representation of the tensor structure.
more »
« less
Koopman-Inspired Implicit Backward Reachable Sets for Unknown Nonlinear Systems
Koopman liftings have been successfully used to learn high dimensional linear approximations for autonomous systems for prediction purposes, or for control systems for leveraging linear control techniques to control nonlinear dynamics. In this paper, we show how learned Koopman approximations can be used for state-feedback correct-by-construction control. To this end, we introduce the Koopman over-approximation, a (possibly hybrid) lifted representation that has a simulation-like relation with the underlying dynamics. Then, we prove how successive application of controlled predecessor operation in the lifted space leads to an implicit backward reachable set for the actual dynamics. Finally, we demonstrate the approach on two nonlinear control examples with unknown dynamics.
more »
« less
- Award ID(s):
- 1931982
- PAR ID:
- 10481566
- Publisher / Repository:
- IEEE Control Systems Letters
- Date Published:
- Journal Name:
- IEEE Control Systems Letters
- Volume:
- 7
- ISSN:
- 2475-1456
- Page Range / eLocation ID:
- 2245 to 2250
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
The modeling of nonlinear dynamics based on Koopman operator theory, originally applicable only to autonomous systems with no control, is extended to nonautonomous control system without approximation of the input matrix. Prevailing methods using a least square estimate of the input matrix may result in an erroneous input matrix, misinforming the controller. Here, a new method for constructing a Koopman model that yields the exact input matrix is presented. A set of state variables are introduced so that the control inputs are linearly involved in the dynamics of actuators. With these variables, a lifted linear model with the exact input matrix, called a Control-Coherent Koopman Model, is constructed by superposing control input terms, which are linear in local actuator dynamics, to the Koopman operator of the associated autonomous nonlinear system. As an example, the proposed method is applied to multi degree-of-freedom robotic arms, which are controlled with Model Predictive Control (MPC). It is demonstrated that the prevailing Dynamic Mode Decomposition with Control (DMDc) using an approximate input matrix does not provide a satisfactory result, while the Control-Coherent Koopman Model performs well with the correct input matrix, even performing better than the bilinear formulation of the Koopman operator.more » « less
-
Abstract In this work, we propose the integration of Koopman operator methodology with Lyapunov‐based model predictive control (LMPC) for stabilization of nonlinear systems. The Koopman operator enables global linear representations of nonlinear dynamical systems. The basic idea is to transform the nonlinear dynamics into a higher dimensional space using a set of observable functions whose evolution is governed by the linear but infinite dimensional Koopman operator. In practice, it is numerically approximated and therefore the tightness of these linear representations cannot be guaranteed which may lead to unstable closed‐loop designs. To address this issue, we integrate the Koopman linear predictors in an LMPC framework which guarantees controller feasibility and closed‐loop stability. Moreover, the proposed design results in a standard convex optimization problem which is computationally attractive compared to a nonconvex problem encountered when the original nonlinear model is used. We illustrate the application of this methodology on a chemical process example.more » « less
-
null (Ed.)Koopman decomposition is a nonlinear generalization of eigen-decomposition, and is being increasingly utilized in the analysis of spatio-temporal dynamics. Well-known techniques such as the dynamic mode decomposition (DMD) and its linear variants provide approximations to the Koopman operator, and have been applied extensively in many fluid dynamic problems. Despite being endowed with a richer dictionary of nonlinear observables, nonlinear variants of the DMD, such as extended/kernel dynamic mode decomposition (EDMD/KDMD) are seldom applied to large-scale problems primarily due to the difficulty of discerning the Koopman-invariant subspace from thousands of resulting Koopman eigenmodes. To address this issue, we propose a framework based on a multi-task feature learning to extract the most informative Koopman-invariant subspace by removing redundant and spurious Koopman triplets. In particular, we develop a pruning procedure that penalizes departure from linear evolution. These algorithms can be viewed as sparsity-promoting extensions of EDMD/KDMD. Furthermore, we extend KDMD to a continuous-time setting and show a relationship between the present algorithm, sparsity-promoting DMD and an empirical criterion from the viewpoint of non-convex optimization. The effectiveness of our algorithm is demonstrated on examples ranging from simple dynamical systems to two-dimensional cylinder wake flows at different Reynolds numbers and a three-dimensional turbulent ship-airwake flow. The latter two problems are designed such that very strong nonlinear transients are present, thus requiring an accurate approximation of the Koopman operator. Underlying physical mechanisms are analysed, with an emphasis on characterizing transient dynamics. The results are compared with existing theoretical expositions and numerical approximations.more » « less
-
This paper presents a data-driven methodology for linear embedding of nonlinear systems. Utilizing structural knowledge of general nonlinear dynamics, the authors exploit the Koopman operator to develop a systematic, data-driven approach for constructing a linear representation in terms of higher order derivatives of the underlying nonlinear dynamics. With the linear representation, the nonlinear system is then controlled with an LQR feedback policy, the gains of which need to be calculated only once. As a result, the approach enables fast control synthesis. We demonstrate the efficacy of the approach with simulations and experimental results on the modeling and control of a tail-actuated robotic fish and show that the proposed policy is comparable to backstepping control. To the best of our knowledge, this is the first experimental validation of Koopman-based LQR control.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1109/LCSYS.2023.3286126
