Data-Driven Approximations of Dynamical Systems Operators for Control

Kaiser, Eurika; Kutz, J. Nathan; Brunton, Steven L.

doi:10.1007/978-3-030-35713-9_8

Cited by 35 publications

(19 citation statements)

References 193 publications

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“…Delay coordinates also define intrinsic coordinates for the Koopman operator [ 53 ], which provides a simple linear embedding of nonlinear systems [ 76 , 77 ]. Koopman models have recently been used for MPC [ 24 , 25 ] and have been identified using SINDy regression [ 78 ] and subsequently used for optimal control [ 78 ]. Recently, SINDY has been extended to modify an existing model based on new incoming measurements to enable rapid model recovery from abrupt changes to the system [ 30 ].…”

Section: Discussionmentioning

confidence: 99%

Sparse identification of nonlinear dynamics for model predictive control in the low-data limit

2018

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Data-driven discovery of dynamics via machine learning is pushing the frontiers of modelling and control efforts, providing a tremendous opportunity to extend the reach of model predictive control (MPC). However, many leading methods in machine learning, such as neural networks (NN), require large volumes of training data, may not be interpretable, do not easily include known constraints and symmetries, and may not generalize beyond the attractor where models are trained. These factors limit their use for the online identification of a model in the low-data limit, for example following an abrupt change to the system dynamics. In this work, we extend the recent sparse identification of nonlinear dynamics (SINDY) modelling procedure to include the effects of actuation and demonstrate the ability of these models to enhance the performance of MPC, based on limited, noisy data. SINDY models are parsimonious, identifying the fewest terms in the model needed to explain the data, making them interpretable and generalizable. We show that the resulting SINDY-MPC framework has higher performance, requires significantly less data, and is more computationally efficient and robust to noise than NN models, making it viable for online training and execution in response to rapid system changes. SINDY-MPC also shows improved performance over linear data-driven models, although linear models may provide a stopgap until enough data is available for SINDY. SINDY-MPC is demonstrated on a variety of dynamical systems with different challenges, including the chaotic Lorenz system, a simple model for flight control of an F8 aircraft, and an HIV model incorporating drug treatment.

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Section: Discussionmentioning

confidence: 99%

Sparse identification of nonlinear dynamics for model predictive control in the low-data limit

2018

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“…where ξ n is the n-th eigenvector, w n is the n-th left eigenvector of K scaled so w T n ξ n = 1, and B is the matrix of appropriate weighting vectors so that x = (ΨB) T [14]. Now we can describe the evolution of the original nonlinear system using the estimated Koopman operator by plugging expressions (4) back to (1). Control inputs can be readily incorporated to the definition of Ψ as an augmented state [25].…”

Section: A Koopman Operator Theory and Extended Dynamic Mode Decompos...mentioning

confidence: 99%

“…K OOPMAN operator theory and associated numerical methods have been widely applied for system identification, state estimation, and control (e.g., [1]- [4]). In an effort to handle approximate models (or lack thereof) that serve as a target for motion control of (nonlinear) robotic systems, methods based on Koopman operator theory are increasingly used in the context of robotics.…”

Section: Introductionmentioning

confidence: 99%

ACD-EDMD: Analytical Construction for Dictionaries of Lifting Functions in Koopman Operator-based Nonlinear Robotic Systems

Shi¹,

Karydis²

2021

Preprint

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Koopman operator theory has been gaining momentum for model extraction, planning, and control of datadriven robotic systems. The Koopman operator's ability to extract dynamics from data depends heavily on the selection of an appropriate dictionary of lifting functions. In this paper we propose ACD-EDMD, a new method for Analytical Construction of Dictionaries of appropriate lifting functions for a range of data-driven Koopman operator based nonlinear robotic systems.The key insight of this work is that information about fundamental topological spaces of the nonlinear system (such as its configuration space and workspace) can be exploited to steer the construction of Hermite polynomial-based lifting functions. We show that the proposed method leads to dictionaries that are simple to implement while enjoying provable completeness and convergence guarantees when observables are weighted bounded. We evaluate ACD-EDMD using a range of diverse nonlinear robotic systems in both simulated and physical hardware experimentation (a wheeled mobile robot, a two-revolute-joint robotic arm, and a soft robotic leg). Results reveal that our method leads to dictionaries that enable high-accuracy prediction and that can generalize to diverse validation sets. The associated GitHub repository of our algorithm can be accessed at https: //github.com/UCR-Robotics/ACD-EDMD.

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“…It turns out to be a challenging problem, since the lifting argument in the presence of control is no longer valid. Regardless of the progress that has been made in this direction during the last few years [11][12][13][14], a principled data-driven framework for nonlinear control synthesis is not yet available.…”

Section: Introductionmentioning

confidence: 99%

A Convex Data-Driven Approach for Nonlinear Control Synthesis

2021

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We consider a class of nonlinear control synthesis problems where the underlying mathematical models are not explicitly known. We propose a data-driven approach to stabilize the systems when only sample trajectories of the dynamics are accessible. Our method is built on the density-function-based stability certificate that is the dual to the Lyapunov function for dynamic systems. Unlike Lyapunov-based methods, density functions lead to a convex formulation for a joint search of the control strategy and the stability certificate. This type of convex problem can be solved efficiently using the machinery of the sum of squares (SOS). For the data-driven part, we exploit the fact that the duality results in the stability theory can be understood through the lens of Perron–Frobenius and Koopman operators. This allows us to use data-driven methods to approximate these operators and combine them with the SOS techniques to establish a convex formulation of control synthesis. The efficacy of the proposed approach is demonstrated through several examples.

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Data-Driven Approximations of Dynamical Systems Operators for Control

Cited by 35 publications

References 193 publications

Sparse identification of nonlinear dynamics for model predictive control in the low-data limit

Sparse identification of nonlinear dynamics for model predictive control in the low-data limit

ACD-EDMD: Analytical Construction for Dictionaries of Lifting Functions in Koopman Operator-based Nonlinear Robotic Systems

A Convex Data-Driven Approach for Nonlinear Control Synthesis

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