## 1 Introduction  
Model predictive control (MPC) is a successful technique with increasing industrial applications [1]. Nevertheless, the computational requirements that involve a centralized control MPC are, in some cases, unfeasible from a practical point of view due to the large-scale nature of the actual engineering infrastructure. Therefore, distributed model predictive control (DMPC) appears as a possible solution to these problems since a complex problem can always be seen as a set of coupled simpler problems, with a clear structure that represents the global one [2]. In addition, DMPC allows the exploitation of this structure to formulate control laws based on local information and some communication between the agents to achieve global performance and stability. In this case, negotiation among agents is a typical approach. The critical task of solving the negotiation problem in multi-agent DMPC systems comes from designing distributed control protocols based on local information from each agent and its neighbors. Consequently, the main concern of this negotiation problem is achieving global decisions based on local information. Therefore, strategies that allow agents to change their local decisions online according to changes in the environment or the behavior of other agents to achieve a joint agreement on a global interest is a priority concern. Negotiation to achieve consensus is focused on addressing this problem. An interesting approach is using deep neural networks as a negotiation manager trained in the machine learning method, which offers many advantages and motivates the development of many algorithms for decision making and control, studying all the properties of stability, convergence, and feasibility. Within artificial intelligence, machine learning techniques have proven to be a powerful tool when using knowledge extracted from data for supervision and the control of complex processes [3; 4; 5; 6; 7]. Technological advances and data availability have boosted this methodology. Deep learning is another field that uses the large amount of data provided by intelligent sensors. This learning is based on neural networks that capture complex behavior, such as [8], where deep learning-based models as soft-sensors to forecast WWTP key features have been developed. Among them, there is one approach, reinforcement learning (RL) [9] in which an RL agent learns by interacting with the environment in order to determine, based on a policy, what action to take given an environment state, aiming at the maximization of the expected cumulative reward. Ultimately, the RL agent learns the policy to follow when the environment is in a certain state, adapting to variations. In other words, reinforcement learning has as its main objective the training of an RL agent to complete a task in an uncertain environment which it gets to know through the states that the environment assumes due to the actions [10; 11]. Valuable surveys about the RL algorithms and their applications can be found in [9; 12]. Several online model-free value-function-based reinforcement learning algorithms that use the expected cumulative reward criterion (Q-learning, Sarsa, and actor-critic methods) are described. Further information can also be found in [13; 14; 15]. However, several policy search and policy gradient algorithms have been proposed in [16; 17]. Many traditional reinforcement learning algorithms have been designed to solve problems in the discrete domain. However, real-world problems are often continuous, making the learning process to select a good or even an optimal policy quite a complex problem. Two RL algorithms that address continuous problems are the deep deterministic policy gradient (DDPG), which uses a deterministic policy, and policy gradient (PG), which assumes a stochastic policy. Despite the significant advances over the last few years, many issues still need to improve the ability of reinforcement learning methods in complex and continuous domains that can be tackled.