## 1 Introduction  
The regulation of pollutant emissions in internal combustion engines has continuously become more stringent in recent years. This fact drove the engine development process, aimed at reducing the fuel consumption and the environmental impact. One of the results of this trend is the use of exhaust gas aftertreatment systems (ATS) [1]. In the case of the particulate matter, wall-flow particulate filters (PF) are nowadays a standard technology necessary to fulfil current regulations because of their high filtration efficiency, both for diesel (diesel particulate filter--DPF) [2] and gasoline (GPF) [3] engines. In this regard, PFs allow an effective reduction of the particle emission both in terms of mass and particle number, with filtration efficiency typically over 90% dependent on several parameters such as the DPF gas flow [4], the particle size, the soot mass load or the fuel [5]. The use of the particulate filter has a great impact on the engine performance and its fluid-dynamic behaviour, increasing the exhaust backpressure [6] and affecting the wave dynamics of the exhaust line [7]. The accumulation of soot inside and on the porous walls of the filter [8] aggravates these effects and leads the engine specific fuel consumption to increase [9]. For this reason, and taking into account the limited capacity of the filter for soot accumulation [10], after a certain operating time it is necessary to induce a regeneration process consisting of the oxidation of the soot particulates stored inside the filter [11]. Two different strategies are used for this purpose. Active regenerations, which are promoted periodically, make use of an external energy source [12] to trigger the soot oxidation process. Instead, passive regeneration strategies produce such oxidation without an external energy source. These passive solutions help to reduce the frequency of active events in diesel engines [13] and are even sufficient in gasoline engines [14]. In the particular case of active regenerations, different techniques can be used to increase the exhaust gas temperature until the level needed for the soot to oxidise [15]. Among them, the most common one is the use of fuel post-injections, which consists of the injection of an amount of fuel in a single or separate portions after the main injection [16] during the late expansion stroke [17]. The extra fuel injection gives as a result a significant increase of the concentration of unburned hydrocarbons (HC) in the exhaust gases. Once these hydrocarbons arrive to the diesel oxidation catalyst (DOC), their oxidation produces a significant increase in the temperature at the DPF inlet [18]. However, these strategies imply an increase of the fuel consumption as a function of the impact on the specific fuel consumption and on the regeneration duration. Thus, it is necessary to optimise them to reduce as much as possible this penalty [19]. The main parameters to define the post-injection pattern in terms of the impact on the fuel consumption are the delay of the injection command with respect to the main injection timing [20] and the fuel mass rate during this event [16]. Both parameters play a significant role in the properties of the exhaust gas arriving to the catalyst, mainly in terms of its temperature and the HC concentration [21]. In order to maximise the regeneration efficiency it is typical to combine the post-injection strategies with other engine control actuations, such as to act on the exhaust gas recirculation (EGR) valve, aimed at increasing the exhaust temperature [22] or the closure of the intake throttle valve, limiting the air flow coming into the cylinders in order to achieve richer equivalence ratio mixtures and higher HC concentration during the main combustion event [23]. [PERSON] et al. [24] studied the relationship between the exhaust gas temperature and the optimal post-injection timing for the regeneration.