## 1 Introduction  
Reverse osmosis (RO) is the predominant technology in seawater and brackish water desalination [1]. However, this technology continues to be an intensive energy consumption process [2, 3]. Various options can be pursued with the aim of reducing the specific energy consumption (\(SEC\)) of RO [4, 5, 6], including optimizing the operation of RO desalination plants. Advances in RO membrane technology [7, 8] are a key element in the goal of improving desalination efficiency. With respect to spiral-wound membrane modules (SWMMs), studies have been made on the effect of the permeability coefficients on the performance of RO systems in terms of production and solute rejection [9, 10, 11]. Significant efforts are being made to try to inhibit the effect of fouling on permeability coefficients during operating time by improving pre-treatment processes [12] and the resistance to fouling [13]. The application of renewable energy sources (RES) to power RO systems has attracted much interest [14]. The operation of RES-driven RO systems is considerably complicated by the problem of variations in power availability and in the characteristics of the feedwater. Given the operational complexity of RO systems and the importance of taking full advantage of technological improvements, it is essential to ensure that desalination plants are working at all times under appropriate operating conditions through efficient and effective process control [15, 16]. Models that estimate the behavior of RO systems are crucial when control strategies are applied to this kind of process. [PERSON] et al. [17] proposed the first multi-loop control system for a seawaterRO (SWRO) process. The RO system had 4-inch hollow-fiber membranes (HFM) (B-10 Permasep from Dupont(r)), which are not very common nowadays. Plant modeling was carried out using transfer functions, considering feed pressure (\(p_{\text{f}}\)) and pH\({}_{\text{f}}\) as inputs and permeate flow (\(Q_{\text{p}}\)) and permeate conductivity (\(Cond_{\text{p}}\)) as outputs. [PERSON] et al. [18] presented an algorithm based on dynamic matrix control (DMC) for the control of an SWRO pilot plant. The process modeling of [PERSON] et al. [17] was used in this work. [PERSON] et al. [19] carried out a study on constrained model predictive control (CMPC) for a brackish water RO (BWRO) desalination unit. The process modeling was done considering four outputs (\(Q_{\text{p}}\), \(Cond_{\text{p}}\), trans-membrane pressure and pH\({}_{\text{f}}\)) and two inputs (rejection flow (\(Q_{\text{b}}\)) and inlet acid flow). The goal was to produce a specified \(Q_{\text{p}}\) with a desired \(Cond_{\text{p}}\), subject to the constraint that pH\({}_{\text{f}}\) and trans-membrane pressure were within specified bounds. [PERSON] [20] used a DMC algorithm with and without constraints for the control of a simulated SWRO desalination unit with HFM. The dynamic model used in their work was based on transfer functions and developed in a previous study by other authors [17]. A control system design for RO systems using advanced optimization techniques was proposed by [PERSON] et al. [21]. Trans-membrane pressure and pH\({}_{\text{f}}\) were considered as inputs, and \(Q_{\text{p}}\) and \(Cond_{\text{p}}\) as outputs in the transfer function-based model. [PERSON] et al. [22] designed and implemented a nonlinear model-based control system for a pilot-scale BWRO desalination plant. The model [23] used was based on a mass balance taken around the entire system and on an energy balance taken around the actuated concentrate valve. The model proposed by [PERSON] et al. [18] was implemented by [PERSON] et al. [24] in an optimization algorithm with an immune-genetic approach to obtain the parameters of a proportional-integral-derivative (PID) controller. The previously mentioned mass/energy-based model proposed by [PERSON] et al. [23] was used by [PERSON] et al. [25] in a simulated BWRO system with concentrate recirculation.