Distributionally robust optimal control for batch nonlinear switched system under moment uncertainty of parameter using Koopman operator

In this paper, we research optimal control problems with incomplete statistical information of parameters. Focusing on a nonlinear switched continuous-time dynamical (NSCTD) system with an uncertain system parameter to characterize the bioconversion of 1,3-propanediol (1,3-PD), we model it as a stochastic variable with known first-moment distribution. Our aim is to develop a distributionally robust optimal control (DROC) strategy to maximize 1,3-PD concentration at the terminal time, considering initial concentrations as control inputs. Solving the DROC problem is tough due to the nonlinear relation between the objective function and the uncertain system parameter. We solve this by deriving an exact Koopman representation to linearize state transitions, converting the NSCTD system into a linear parameter-varying (LPV) system. Then, using duality principles and smoothing techniques, we turn the DROC problem into a single-level optimal control (SLOC) problem. We design a gradient-based algorithm as well as validate its effectiveness and the NSCTD system's modelling ability through simulations.