**1. Introduction**

Among the main problems of humanity are those related to water availability and pollution. Unfortunately, industrial development and population growth lead to an increase in pollutant discharges which have a negative impact mostly on aquatic ecosystems [1]. With the accelerated urbanization in the world, and especially in Mexico, environmental water problems have become even more considerable. Long-term ineffective governance has driven to aggravate water pollution in some areas more than ever before [2]. In recent years, with importance on environmental water quality, water pollution has been gradually brought under control. In addition, research has been centred on the field of water quality exploration and pollutant diffusion simulation in rivers and watersheds via mathematical modeling and novel tech [3].

Related to aquatic environments researches, the water quality model is a fundamental strategy for the water study and its forecasts. Streeter–Phelps model is distinguished as the first dynamic and spatial water quality model. Therefore, almost all modern timeline researchers have done an enormous amount of research in improving and developing water quality models [4]. Due to their relevance, experts in environmental river control are still employing it to obtain mathematical models of water quality from rivers and lakes [5–9].

The Streeter–Phelps model relates the two main mechanisms that define dissolved oxygen in a lotic water Biochemical Oxygen Demand (BOD) receiving wastewater tributary. These mechanisms are a decomposition of organic matter and oxygen aeration [5]. It is also used to estimate BOD and Dissolved oxygen (DO) transport, which is achieved by feeding data on hydromorphological and water quality parameters [10].

Usually, automated water quality control in rivers and lakes is still done traditionally, i.e., employing complicated laboratory techniques, which implies time, consumable resources, and delays in decision-making [11]. Some robotic prototypes have been proposed to obtain a better environmental control [12–14]; these are an excellent tool for active monitoring of water quality since this type of robotic device has embedded sensors of physicochemical parameters of remote-controlled GPS position [15–17]. Currently, existing commercial sensors can measure variables, such as turbidity, pH, temperature, and DO, which help measure water quality in water bodies [14]. Currently, there are remarkable scientific advances in relation to mobile and fixed stations for environmental monitoring of rivers [18–20]. Still, there are water quality variables that are not commonly measured by commercial sensors, such as chemical oxygen demand and BOD [11]. Using the information about the *Streeter-Phelps* model and algorithms called Observers, it is possible to estimate BOD via the information of other sensors, such as the available DO and longitude (GPS signal). Observers have received several versions and improvements in their performance, over time, in the difficult task of state estimation and parameters in complex systems, but these are still being improved in all their different applications. For example, some outstanding works of each technique, from the first generations linear [21], non-linear [22], adaptive [23], sliding mode [24], non-uniformly observable [25], and discreet observers who can provide good results [26], based on the frequency [27] and high-gain systems [28], have been proposed. There are also applications of its functions and fuzzy techniques for diagnosing sensor faults and multiple applications [29,30]. Currently, the use of fractional techniques and their implementation in state estimation via observers is being arduously studied, since the use of fractional techniques allows to add degrees of freedom in the tuning of the observers [31].

From the knowledge of the authors, there are few published works about the use of observers for *Streeter-Phelps* states estimation; there is a very outstanding work where it was used as a linear technique and a Luenberger observer for the BOD estimation [32]. This remarkable work was possible due to change in the basis to obtain an observable model. However, given the time in which it was published, obsolete techniques are used that are not robust to parametric changes and in the presence of disturbances.

In this paper, we present a robust Fractional High-Gain Nonlinear Observer design for state estimation in the Streeter–Phelps model published in Reference [32]. Besides, we analyze parametric sensibility via the use of Lyapunov convergence functions. The proposed novel fractional observer design is based on a Lyapunov analysis. The proposal is directed to future work where this kind of algorithms can be embedded and programmed in robotic watercraft to estimate environmental rivers quality water variables. MATLAB Simulink simulations are made to show the advantages of our algorithm over those proposed for similar models. Finally, in conclusion, there are some perspectives and thoughts open to investigation for possible tasks in real time.

#### **2. Mathematical Model and Problem Statement**

The most common nutrient and oxygen distribution pollutant mathematical model is the *Streeter-Phelps* system. Since 1925, it has described the oxygen balance in rivers, lakes, and water sources. Although the appearance of more complex models, the *Streeter-Phelps* model (and its extensions) continues to be one of the most widely used, since it is a relatively simple mathematical model. However, it injects much essential information; despite almost a century since its presentation, it continues to be used [4–10]: Under some assumptions, *Streeter-Phelps* model helps to estimate and

measure the main variables for environmental studies, such as BOD and DO, via aquatic robots and algorithms observers programmed, thus improving the environmental monitoring of rivers. Together, BOD and DO are the parameters that represent the largest number of pollutants in water bodies. Indeed, the higher the BOD, the lower the amount of OD, which in turn poses a greater risk to aquatic life. Furthermore, the study of these parameters allows us to know the resilience of a river and/or the degree of pollution received by polluting discharges [33]. We propose to use the model in this work under the following assumptions.
