## 1 Introduction  
It is well known that the building sector uses more than 40% of world energy to maintain appropriate levels of comfort for people. Indeed, the resulting carbon emission is actually greater than that of the transportation sector [1]. In fact, this energy is mainly consumed to ensure an adequate thermal comfort level by acting on Heating, Ventilation and Air-Conditioning (HVAC) systems. Thus, energy-efficient solutions should be developed in order to decrease the total amount of carbon and emissions to help the fulfillment of the targets in the global climate change context [2]. A significant research effort has been provided by engineers and architects to influence the energy use of the new buildings through the design of the envelope and the HVAC systems selection [3,4]. However, when a building has been finished or it already exists and can be just retrofitted, the consumption of energy is determined mainly by the HVAC control [5], maintenance and by the requirements of its occupants. In any case, as renovations and isolations have high costs and are time demanding, the use of an advanced control systems can be a good trade-off between costs and benefits. Hence, it is necessary to design HVAC control systems that guarantee a high comfort level and energy saving at the same time. Regarding residential buildings, which represent the most significant case in many parts of the world, most of the control systems in rooms typically consist of thermostatic valves applied to radiators or thermostats. Often, inhabitants fail to use them optimally, by possibly overheating the room and, thus, wasting energy [6]. To improve the performance of these (low-level) control systems, proportional-integral-derivative (PID) controllers can be used with thermostatic valves. Obviously, they need to be properly tuned and, for this purpose, automatic tuning methodologies have been proposed in the literature [7,8]. Other kinds of controllers such as fuzzy logic [9] or neural networks-based controllers [10,11] have also been proposed. In any case, even if the low-level controllers guarantee a satisfactory performance in tracking the required temperature and in rejecting disturbances, the overall performance in terms of comfort and energy consumption mainly depends on the reference signals (one for each room) whose selection is indeed a critical issue if the requirements of the occupants change during the day, as it is typical in residential buildings. For example, a different temperature might be required in living rooms and in bedrooms during the day and during the night, or (during the winter) a relatively low value of the temperature can be acceptable if the room is used only for a few minutes in a time interval of a few hours. In this case, considering for simplicity a heating system (the same reasoning can be applied to a cooling system), the user selects a minimum required temperature value in a given time interval for each room and the control system must ensure that this value is always attained by taking into account the transient response of the overall system and the couplings between the rooms. The problem can be actually formulated as obtaining appropriate transient responses to track the set-point values to minimize the overall energy consumption. It is worth stressing that, in manual control, the user typically tends to waste energy for this task (for example by switching on the heating system earlier than necessary in a room to have the required temperature at a certain hour), but it is very difficult to take into account all the coupling effects simultaneously, as well as the effects of the exogenous inputs. It is clear that, to solve this problem, high-performance low-level controllers are not sufficient and an effective high-level control system is necessary to plan the set-point signals for each room to minimize the overall energy consumption by satisfying the user requirements at the same time.