This paper presents modelling of ball and plate systems based on first principles by considering balance of forces and torques. A non-linear model is derived considering the dynamics of motors, gears, ball and plate. The non-linear model is linearized near the operating region to obtain a standard state space model. This linear model is used for discrete optimal control of the ball and plate system - the trajectory of the ball is controlled by control voltages to the motor.

Some analyses state that buildings contribute to overall energy consumption by 20–40%, which, in the context of the recent geopolitical energy crisis, makes them a critical issue to study. Finding solutions for better energy management in buildings can have a significant impact on the energy sector, thus reducing EU energy dependencies and contributing to the fulfillment of the REPowerEU goals. This paper focuses on proposing a simplified model of a residential house considering the main appliances, heating and cooling, a photovoltaic system, and electric vehicle recharging. Weather and solar irradiance forecasts are taken into account. The model predicts the energy demands of a house based on online weather forecasts and the desired indoor temperature. The article also focuses on the analysis of how weather forecast uncertainty affects energy demand prediction. This model can be used to better understand and predict the energy demand of either a single house or a set of houses. A multi-objective optimization approach that takes into account the preferences of users/inhabitants is developed to provide a compromise between the price paid for the electricity and temperature comfort. The authors plan to apply the proposed model to a residential house’s real-time control system. The model will be tuned, its predictions will be tested, and it will be used for energy demand optimization.