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1. Derivative-Based Koopman Operators for Real-Time Control of Robotic Systems

   https://doi.org/10.1109/TRO.2021.3076581  

   Mamakoukas, Giorgos; L. Castano, Maria; Tan, Xiaobo; D. Murphey, Todd (May 2021, IEEE Transactions on Robotics)

   null (Ed.)

   This paper presents a generalizable methodology for data-driven identification of nonlinear dynamics that bounds the model error in terms of the prediction horizon and the magnitude of the derivatives of the system states. Using higher order derivatives of general nonlinear dynamics that need not be known, we construct a Koopman operator-based linear representation and utilize Taylor series accuracy analysis to derive an error bound. The resulting error formula is used to choose the order of derivatives in the basis functions and obtain a data-driven Koopman model using a closed-form expression that can be computed in real time. Using the inverted pendulum system, we illustrate the robustness of the error bounds given noisy measurements of unknown dynamics, where the derivatives are estimated numerically. When combined with control, the Koopman representation of the nonlinear system has marginally better performance than competing nonlinear modeling methods, such as SINDy and NARX. In addition, as a linear model, the Koopman approach lends itself readily to efficient control design tools, such as LQR, whereas the other modeling approaches require nonlinear control methods. The efficacy of the approach is further demonstrated with simulation and experimental results on the control of a tail-actuated robotic fish. Experimental results show that the proposed data-driven control approach outperforms a tuned PID (Proportional Integral Derivative) controller and that updating the data-driven model online significantly improves performance in the presence of unmodeled fluid disturbance. This paper is complemented with a video: https://youtu.be/9 wx0tdDta0. 

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2. Modeling, Reduction, and Control of a Helically Actuated Inertial Soft Robotic Arm via the Koopman Operator

   David A. Haggerty, Michael J. (January 2021, ArXivorg)

   null (Ed.)

   Soft robots promise improved safety and capabil- ity over rigid robots when deployed in complex, delicate, and dynamic environments. However the infinite degrees of freedom and highly nonlinear dynamics of these systems severely com- plicate their modeling and control. As a step toward addressing this open challenge, we apply the data-driven, Hankel Dynamic Mode Decomposition (HDMD) with time delay observables to the model identification of a highly inertial, helical soft robotic arm with a high number of underactuated degrees of freedom. The resulting model is linear and hence amenable to control via a Linear Quadratic Regulator (LQR). Using our test bed device, a dynamic, lightweight pneumatic fabric arm with an inertial mass at the tip, we show that the combination of HDMD and LQR allows us to command our robot to achieve arbitrary poses using only open loop control. We further show that Koopman spectral analysis gives us a dimensionally reduced basis of modes which decreases computational complexity without sacrificing predictive power. 

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3. Koopman-Inspired Implicit Backward Reachable Sets for Unknown Nonlinear Systems

   https://doi.org/10.1109/LCSYS.2023.3286126  

   Balim, Haldun; Aspeel, Antoine; Liu, Zexiang; Ozay, Necmiye (January 2023, IEEE Control Systems Letters)

   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. 

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4. Optimal Control for Thermal Management of a Proton Exchange Membrane Fuel Cell Stack With Koopman-Based Modeling

   https://doi.org/10.1115/1.4066011  

   Huo, Da; Hall, Carrie M (March 2025, Journal of Dynamic Systems, Measurement, and Control)

   This study presents a novel approach to optimal control utilizing a Koopman operator integrated with a linear quadratic regulator (LQR) to enhance the thermal management and power output efficiency of an open-cathode proton exchange membrane fuel cell (PEMFC) stack. First, a linear time-invariant dynamic model was derived through Koopman operator to forecast the behavior of the PEMFC stack. Second, this Koopman-based model was directly integrated with LQR for optimizing temperature, temperature variations, and output power efficiency of the PEMFC stack by regulating fan speed, with a physics-based model serving as the plant model. Finally, the performance of the Koopman-based LQRs (KLQR) was compared to a baseline proportional-integral (PI) controller across various ambient temperatures and operating conditions, focusing on temperature, temperature variations, and net power output. The results demonstrate the proposed Koopman-based approach can be seamless integration with linear optimal control algorithms, effectively minimizing temperature, temperature variations across the PEMFC stack, and the net power outputs under different ambient temperature and operating conditions. 

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5. Tensor-Based Koopman Operator and Its Application to Optimal Control Problems

   https://doi.org/10.2514/1.G008913  

   Xu, Zhi; Dai, Ran; Howell, Kathleen C (June 2025, Journal of Guidance, Control, and Dynamics)

   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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