Physics-informed deep Koopman operator for Lagrangian dynamic systems

Accurate mechanical system models are crucial for safe and stable control. Unlike linear systems, Lagrangian systems are highly nonlinear and difficult to optimize because of their unknown system model. Recent research thus used deep neural networks to generate linear models of original systems by mapping nonlinear dynamic systems into a linear space with a Koopman observable function encoder. The controller then relies on the Koopman linear model. However, without physical information constraints, ensuring control consistency between the original nonlinear system and the Koopman system is tough, as the learning process of the Koopman observation function is unsupervised. This paper thus proposes a two-stage learning algorithm that uses structural subnetworks to build a physics-informed network topology to simultaneously learn the Koopman observable functions and the system energy representation. In the Koopman matrix learning session, a quadratic-constrained optimization problem is solved to ensure that the Koopman representation satisfies the energy difference matching hard constraint. The proposed energy-preserving deep Lagrangian Koopman (EPDLK) framework effectively represents the dynamics of the Lagrangian system while ensuring control consistency. The effectiveness of EPDLK is compared with those of various Koopman observable function construction methods in multistep prediction and trajectory tracking tasks. EPDLK achieves better control consistency by guaranteeing energy difference matching, which facilitates the application of the control law generated on the Koopman system directly to the original nonlinear Lagrangian system.