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1. Dynamic Mode Decomposition With Gaussian Process Regression for Control of High-Dimensional Nonlinear Systems

   Tsolovikos, A; Bakolas, E; Goldstein, D (November 2024, ASME Journal of Dynamic Systems, Measurement and Control)

   In this work, we consider the problem of learning a reduced-order model of a high-dimensional stochastic nonlinear system with control inputs from noisy data. In particular, we develop a hybrid parametric/nonparametric model that learns the “average” linear dynamics in the data using dynamic mode decomposition with control (DMDc) and the nonlinearities and model uncertainties using Gaussian process (GP) regression and compare it with total least-squares dynamic mode decomposition (tlsDMD), extended here to systems with control inputs (tlsDMDc). The proposed approach is also compared with existing methods, such as DMDconly and GP-only models, in two tasks: controlling the stochastic nonlinear Stuart–Landau equation and predicting the flowfield induced by a jet-like body force field in a turbulent boundary layer using data from large-scale numerical simulations. 

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2. Control-Coherent Koopman Modeling: A Physical Modeling Approach

   Asada, Harry; Solano--Castellanos, Jose (December 2024, IEEE)

   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. 

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3. Model-Based Data-Driven System Identification and Controller Synthesis Framework for Precise Control of SISO and MISO HASEL-Powered Robotic Systems

   https://doi.org/10.1109/RoboSoft54090.2022.9762220  

   Volchko, Angella; Mitchell, Shane K.; Morrissey, Timothy G.; Humbert, J. Sean (April 2022, IEEE 5th International Conference on Soft Robotics)

   Soft robots require a complimentary control architecture to support their inherent compliance and versatility. This work presents a framework to control soft-robotic systems systematically and effectively. The data-driven model-based approach developed here makes use of Dynamic Mode Decomposition with control (DMDc) and standard controller synthesis techniques. These methods are implemented on a robotic arm driven by an antagonist pair of Hydraulically Amplified Self-Healing Electrostatic (HASEL) actuators. The results demonstrate excellent tracking performance and disturbance rejection, achieving a steady state error under 0.25% in response to step inputs and maintaining a reference orientation within 0.5 degrees during loading and unloading. The procedure presented in this work can be extended to develop effective and robust controllers for other soft-actuated systems without knowledge of their dynamics a priori. 

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4. Fractional robust data-driven control of nonlinear MEMS gyroscope

   https://doi.org/10.1007/s11071-023-08912-x  

   Rahmani, Mehran; Redkar, Sangram (November 2023, Nonlinear Dynamics)

   This research proposes a new fractional robust data-driven control method to control a nonlinear dynamic micro-electromechanical (MEMS) gyroscope model. The Koopman theory is used to linearize the nonlinear dynamic model of MEMS gyroscope, and the Koopman operator is obtained by using the dynamic mode decomposition (DMD) method. However, external disturbances constantly affect the MEMS gyroscope. To compensate for these perturbations, a fractional sliding mode controller (FOSMC) is applied. The FOSMC has several advantages, including high trajectory tracking performance and robustness. However, one of the drawbacks of FOSMC is generating high control inputs. To overcome this limitation, the researchers proposed a compound controller design that applies fractional proportional integral derivative (FOPID) to reduce the control efforts. The simulation results showed that the proposed compound Koopman-FOSMC and FOPID (Koopman-CFOPIDSMC) outperformed two other controllers, including FOSMC and Koopman-FOSMC, in terms of performance. Therefore, this research proposes an effective approach to control the nonlinear dynamic model of MEMS gyroscope. 

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5. Dynamic Mode Decomposition with Control as a Model of Multimodal Behavioral Coordination

   https://doi.org/10.1145/3462244.3479916  

   Klein, Lauren; Ardulov, Victor; Gharib, Alma; Thompson, Barbara; Levitt, Pat; Matarić, Maja J. (October 2021, ICMI '21: Proceedings of the 2021 International Conference on Multimodal Interaction)

   Observing how infants and mothers coordinate their behaviors can highlight meaningful patterns in early communication and infant development. While dyads often differ in the modalities they use to communicate, especially in the first year of life, it remains unclear how to capture coordination across multiple types of behaviors using existing computational models of interpersonal synchrony. This paper explores Dynamic Mode Decomposition with control (DMDc) as a method of integrating multiple signals from each communicating partner into a model of multimodal behavioral coordination. We used an existing video dataset to track the head pose, arm pose, and vocal fundamental frequency of infants and mothers during the Face-to-Face Still-Face (FFSF) procedure, a validated 3-stage interaction paradigm. For each recorded interaction, we fit both unimodal and multimodal DMDc models to the extracted pose data. The resulting dynamic characteristics of the models were analyzed to evaluate trends in individual behaviors and dyadic processes across infant age and stages of the interactions. Results demonstrate that observed trends in interaction dynamics across stages of the FFSF protocol were stronger and more significant when models incorporated both head and arm pose data, rather than a single behavior modality. Model output showed significant trends across age, identifying changes in infant movement and in the relationship between infant and mother behaviors. Models that included mothers’ audio data demonstrated similar results to those evaluated with pose data, confirming that DMDc can leverage different sets of behavioral signals from each interacting partner. Taken together, our results demonstrate the potential of DMDc toward integrating multiple behavioral signals into the measurement of multimodal interpersonal coordination. 

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