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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.
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Modularized Bilinear Koopman Operator for Modeling and Predicting Transients of Microgrids
Modularized Koopman bilinear form (M-KBF) is presented to model and predict the transient dynamics of microgrids in the presence of disturbances. As a scalable data-driven approach, M-KBF divides the identification and prediction of the high-dimensional nonlinear system into the individual study of subsystems, and thus, alleviates the difficulty of intensively handling high volume data and overcomes the curse of dimensionality. For each subsystem, Koopman bilinear form is established to efficiently identify its model by identifying isotypic eigenfunctions via the Extended Dynamic Mode Decomposition (EDMD) method with an eigenvalue-based order truncation. Extensive tests show that M-KBF can provide accurate transient dynamics prediction for the nonlinear microgrids and verify the plug-and-play modeling and prediction function, which offers a potent tool for identifying high-dimensional systems with reconfiguration feature. The modularity feature of M-KBF enables the provision of fast and precise prediction for the power grid operation and control, paving the way towards online applications.
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- Award ID(s):
- 2229435
- PAR ID:
- 10525092
- Publisher / Repository:
- IEEE
- Date Published:
- Journal Name:
- IEEE Transactions on Smart Grid
- ISSN:
- 1949-3053
- Page Range / eLocation ID:
- 1 to 1
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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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
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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
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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
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1109/TSG.2024.3399076
