Dual-Loop Robust Control of Biased Koopman Operator Model by Noisy Data of Nonlinear Systems

The Koopman operator approach for data-driven control design of a nonlinear system is on the rise because of its capability to capture the behaviours of global dynamics. However, the measurement noises of inputs and outputs will bias the Koopman model identification and cause model mismatch from the actual nonlinear dynamics. The current work evaluates the bounds of the noise-induced model bias of the Koopman operator model and proposes a data-driven robust dual-loop control framework (Koopman based robust control-KROC) for the biased model. First, the model mismatch is found bounded under radial basis functions (RBF) and the bounded noises, and the bound of model mismatch is assessed. Second, the pitfalls of linear quadratic Gaussian (LQG) control based on the biased Koopman model of Van Der Pol oscillator are shown. Motivated from the pitfalls, the dual-loop control is proposed, which consist of an observer-based state-feedback control based on the nominal Koopman model and an additional robust loop to compensate model mismatch. A linear matrix inequality (LMI) is derived, which can guarantee robust stability and performance under bounded noises for the finite-dimensional Koopman operator model. Finally, the proposed framework is implemented to a nonlinear Van Der Pol oscillator to demonstrate enhanced control performance by the dual-loop robust control.