Data-driven Model Identification and Control Using Koopman Operator: Its Application to Autonomous Vehicles

Abstract

This dissertation proposes data-driven model identification and control for autonomous driving vehicles. Typical autonomous driving consists of several essential components: perception, localization, decision-making, trajectory planning, and control. In order to conduct trajectory planning and control, prior knowledge of nonlinear vehicle dynamics is necessary to execute accurate and safe maneuvers in complex and unpredictable road environments. However, it is challenging to obtain nonlinear vehicle dynamics and control a vehicle with a nonlinear vehicle model. To tackle the problem, the author adopts the Koopman operator, which is a linear map representing nonlinear dynamics in an infinite-dimensional space. The author aims to provide the Koopman operator theory and its application to autonomous vehicles. To this end, this dissertation consists of preliminaries of the Koopman operator, a Koopman operator-based model identification and control for autonomous vehicles, uncertainty quantification of the Koopman operator, and design process of a Koopman operator-based stochastic model predictive control for lateral vehicle control.

First, a Koopman operator-based model identification and control method is proposed for a lane-keeping system. The Koopman operator is a linear mapping that can capture nonlinear dynamics but lies in an infinite-dimensional space. Thus, this work adopted the extended dynamic mode decomposition (EDMD) to approximate the Koopman operator in a finite-dimensional space. Then, the author designed a linear structure to express the nonlinear motion of full vehicle dynamics using the advantage of the Koopman approach. In the Koopman operator-based model identification, selecting the basis function for lifting the state is crucial, but how systematically to choose the basis functions is an open problem. Thus, in this study, the author made a comparative study among the typical basis functions. In addition, the author applied signal normalization to mitigate the potential problem of the EDMD approach. Furthermore, this thesis used the approximated Koopman operator to design the optimal control as a linear structure for the underlying nonlinear vehicle model. Finally, it is confirmed that the closed-loop system is uniformly ultimately bounded with the proposed controller. A full vehicle dynamic simulator, CarSim, obtains a dataset for calculating the Koopman operator. The comparative study confirmed that the position error of the proposed method was reduced by about 36% compared to other methods.

Second, this thesis proposes a method for uncertainty quantification of an autoencoder-based Koopman operator. The main challenge of using the Koopman operator is to design the basis functions for lifting the state. To this end, this dissertation builds an autoencoder to automatically search the optimal lifting basis functions with a given loss function. The author approximates the Koopman operator in a finite-dimensional space with the autoencoder, while the approximated Koopman has an approximation uncertainty. To resolve the problem, a robust positively invariant set is computed for the approximated Koopman operator to consider the approximation error. Then, the decoder of the autoencoder is analyzed by robustness certification against approximation error using the Lipschitz constant in the reconstruction phase. The forced Van der Pol model is used to show the validity of the proposed method. From the numerical simulation results, it is confirmed that the trajectory of the true state stays in the uncertainty set centered by the reconstructed state.

Third, this dissertation proposes Koopman operator-based Stochastic Model Predictive Control (K-SMPC) for enhanced lateral control of autonomous vehicles. The Koopman operator is a linear map representing the nonlinear dynamics in an infinite-dimensional space. Thus, the Koopman operator is used to represent the nonlinear dynamics of a vehicle in dynamic lane-keeping situations. The EDMD method is adopted to approximate the Koopman operator in a finite-dimensional space for practical implementation. The author considers the modeling error of the approximated Koopman operator in the EDMD method. Then, this thesis designs K-SMPC to tackle the Koopman modeling error, where the error is handled as a probabilistic signal. The recursive feasibility of the proposed method is investigated with an explicit first-step state constraint by computing the robust control invariant set. A high-fidelity vehicle simulator, i.e., CarSim, is used to validate the proposed method with a comparative study. From the results, it is confirmed that the proposed method outperforms other methods in tracking performance. Furthermore, it is observed that the proposed method satisfies the given constraints and is recursively feasible.