## 1 Introduction  
Water contamination due to human activity is increasing significantly, hampering our future development; hence, we should take immediate action and look for sustainable technologies to treat this vital resource. It is well known that some microorganisms are useful to remediate contaminated water; for example, microalgae are capable of eliminating organic nitrogen, the normal limiting substrate in wastewater [1]. Eutrophication refers to a significant increase in the concentration of nutrients of a given aquatic ecosystem due to human industrial activities. High nutrients availability leads to considerable proliferation of certain species of algae and higher aquatic plants. Proper control of microalgae growth either in photobioreactors or in the natural ambient (basins, rivers, and lakes) can reduce the concentration of pollutants generated by human activities. In Mexico, especially in the state of Sinaloa, eutrophication is critical. Hence, local authorities are looking for technologies, such as cultivation of microalgae in open photobioreactors, that can simultaneously aid the reduction of eutrophication and can be useful in producing biofuels [2]. The development of model-based automatic controllers for the robust operation of microalgae bioreactors will aid in the mitigation of water pollution as well as in the production of sustainable biofuels [3]. Mathematical models that describe the dynamics of algae growth and biofuel production in bioreactors present several complications such as  
* strong nonlinearities; [3; 4; 5];
* high sensitivity to key variables (for example, temperature and pH variables) and to uncontrolled disturbances related to luminous intensity [6; 7; 8];
* growth rates highly sensitive to unmeasured disturbances [9; 10]; and
* biomass growth generating space distribution and time-decay of incident light [11; 12; 13]. Some traditional control and estimation techniques, which are applied in biotechnological processes, usually do not cope well with unmeasured disturbances, and in some cases, it is difficult to find conditions that assure disturbance rejection [14; 15; 16; 17]. Some previous contributions in the field guarantee a robust control [18; 19]; however, these require perfect knowledge of all the states in the system. In practice, this is extremely difficult in photobioreactors due to the lack of reliable online measurements of metabolites and biomass concentrations [20]. On the other hand, Sliding Mode Control (SMC) is a well-known technique already successfully applied to photobioreactors [19; 21]. However, this technique presents some theoretical and practical complications that hamper its application in industrial-scale bioreactors such as the monotonous switching feedback produced in the actuators, causing high-frequency flickering in the control action [21]. Given the above difficulties, it seems that fractional-order algorithms can be useful in the design of robust and scalable control systems for the culture of microalgae in photobioreactors [22]. Fractional control is an active research area, where many techniques have been developed to improve its performance and robustness as well as to combine it with other control methods [23]. Fractional control has proven to have many advantages over conventional and robust control techniques [24; 25; 26; 27], such as more flexibility, the reduction of undesirable phenomena like chattering, and the generation of small continuous and derivative control signals. When using fractional-order sliding mode control (FOSMC), the fractional contribution significantly reduces the pitching effect of the system driven by the sliding modes [28; 29; 30]. Also, internal model control (IMC) has been used successfully to control nonlinear processes in batch mode. This controller has been designed by simulation and validated experimentally in alembic batch distillation [31; 32].