We present a machine learning-based predictive control scheme that integrates an online update of the recurrent neural network (RNN) models to capture process nonlinear dynamics in the presence of model uncertainty. Specifically, an ensemble of the RNN models are initially obtained for the nominal system, for which Lyapunov-based model predictive control (LMPC) is utilized to drive the state to the steady-state, and economic Lyapunov-based MPC (LEMPC) is applied to achieve closed-loop stability and economic optimality simultaneously. Subsequently, an event-trigger mechanism based on the decreasing rate of Lyapunov function and an error-trigger mechanism that relies on prediction errors are developed for an online model update, in which the most recent process data are utilized to derive a new ensemble of RNN models with enhanced prediction accuracy. By incorporating the event and error-triggered online RNN update within real-time machine learning-based LMPC and LEMPC, process dynamic performance is improved in terms of guaranteed closed-loop stability, optimality, and smoothness of control actions. The proposed methodology is applied to a chemical process example with time-varying disturbances under LMPC and LEMPC, respectively, to demonstrate the effectiveness of an online update of machine learning models in real-time control problems.