An environmental control system for a building including building equipment operable to affect a variable state or condition of the building. The system includes a controller including a processing circuit. The processing circuit can obtain training data relating to operation of the building equipment and can perform a system identification process to identify parameters of a system model using the training data. The processing circuit can augment the system model with a disturbance model and estimate values of a historical heat disturbance in the training data based on the augmented system model. The processing circuit can train one or more heat disturbance models based on the training data and the estimated values. The processing circuit can predict a heat disturbance using the augmented system model along with the one or more heat disturbance models and can control the building equipment based on the predicted heat disturbance.