A distributionally robust optimization model for batch nonlinear switched time-delay system considering uncertain output measurements

In this paper, we consider a nonlinear switched time-delay (NSTD) system with unknown switching times and unknown system parameters, where the output measurement is uncertain. This system is the underling dynamical system for the batch process of glycerol bioconversion to 1,3-propanediol induced by Klebsiella pneumoniae. The uncertain output measurement is regarded as a stochastic vector (whose components are stochastic variables) and the only information about its distribution is the first-order moment. The objective of this paper is to identify the unknown quantities of the NSTD system. For this, a distributionally robust optimization problem (a bi-level optimization problem) governed by the NSTD system is proposed, where the relative error under the environment of uncertain output measurements is involved in the cost functional. The bi-level optimization problem is transformed into a single-level optimization problem with non-smooth term through the application of duality theory in probability space. By applying the smoothing technique, the non-smooth term is approximated by a smooth term and the convergence of the approximation is established. Then, the gradients of the cost functional with respect to switching times and system parameters are derived. A hybrid optimization algorithm is developed to solve the transformed problem. Finally, we verify the obtained switching times and system parameters, as well as the effectiveness of the proposed algorithm, by solving this distributionally robust optimization problem.