  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.
