*Article* **A Fractional High-Gain Nonlinear Observer Design—Application for Rivers Environmental Monitoring Model**

**Abraham Efraim Rodriguez-Mata 1,***∗* **, Yaneth Bustos-Terrones 1 , Victor Gonzalez-Huitrón 1 , Pablo Antonio Lopéz-Peréz 2, Omar Hernández-González 3 and Leonel Ernesto Amabilis-Sosa 1**


3 Conacyt-Tecnológico Nacional de México\Instituto Tecnológico de Hermosillo, Av. Tecnológico y, Periférico Poniente S/N, Sahuaro, Hermosillo 83170, Sonora, Mexico; ohernandezg@conacyt.mx

**\*** Correspondence: aerodriguez@conacyt.mx

Received: 2 June 2020; Accepted: 10 July 2020; Published: 16 July 2020

**Abstract:** The deterioration of current environmental water sources has led to the need to find ways to monitor water quality conditions. In this paper, we propose the use of Streeter–Phelps contaminant distribution models and state estimation techniques (observer) to be able to estimate variables that are very difficult to measure in rivers with online sensors, such as Biochemical Oxygen Demand (BOD). We propose the design of a novel Fractional Order High Gain Observer (FOHO) and consider the use of Lyapunov convergence functions to demonstrate stability, as it is compared to classical extended Luenberger Observer published in the literature, to study the convergence in BOD estimation in rivers. The proposed methodology was used to estimated Dissolved oxygen (DO) and BOD monitoring of River Culiacan, Sinaloa, Mexico. The use of fractional order in high-gain observers has a very effective effect on BOD estimation performance, as shown by our numerical studies. The theoretical results have shown that robust observer design can help solve problems in estimating complex variables.

**Keywords:** Streeter–Phelps model; fractional-order control; high observers; river monitoring
