Stabilization of A Single Chamber Single Population Microbial Fuel Cell by Using of a Novel Nonlinear Adaptive Sliding Mode Control

The importance of energy in daily life is not hidden from anyone and its increasing demand is accepted as a fundamental and vital issue, so the appropriate solutions must be provided to make it available. Although the nature of this demand varies from community to community (some communities seek to improve the quality of services and others need more energy to meet essential needs), but in principle, there is no doubt that this demand is increasing, and in any case, this demand must be answered. Renewable energy from natural processes is a reliable way to supply energy and meet growing needs. This type of energy originates from the sun, heat generated in the deep earth, wind, ocean, hydroelectric power plant, biomass, geothermal sources, biofuels and hydrogen fuels [1-8] and achieves benefits such as reducing dependence on fossil fuels, stable prices, increased competitiveness, high reliability and flexibility, reduced global warming and improved public health.

Considering the potentials of microbial fuel cells for the production of renewable energy can be examined and remarked from several perspectives:

1. The use of MFC reduces water treatment costs and greatly reduces environmental pollution.

2- MFC can be used in remote communities and also as a desalination plant.

3- Microbial fuel cell as a reliable and effective renewable energy source, produces energy from the metabolism of microorganisms and does not produce any toxic by-products. Microorganisms generate electricity using organic matter and biodegradable substrates such as wastewater, while performing biodegradation / treatment of municipal wastewater [9, 10].

MFC performance is affected by parameters such as substrate concentration, growth of microorganisms and biomass, operating and ambient temperature, and pH in the anode chamber. To investigate the effect of different parameters on MFC performance, different mathematical models have been introduced: two-compartment MFC with one type of bacterial species [11], single-compartment MFC with two bacterial species [12, 13] and single population microbial fuel cell Single compartment [14] are some of them. To achieve optimal performance, the MFC must be controlled under different conditions. To date, different control strategies have been developed based on different models to achieve optimal fuel cell performance. The problem with the above controllers is that in the presence of nonlinear terms and uncertainties, the performance of the linear controllers is not suitable and also the backstepping method is not a complete control method due to the complexity of the computations due to recursive calculations. Despite all the previous works in the field of robust control design [15-21] and the application of some of them to the fuel cell systems [22-25], the behavior of previous controllers still faces challenges, some of which are as follows:

1- The coverage of uncertainty effects and the robustness of the controller to them is not well done.

2- Under previous controllers, the system's transient state behavior suffers from shortcomings.

3- The high cost of the control signal and the lack of smoothness of the control signal are other disadvantages of the previously proposed control methods.

In this paper, a new method of adaptive sliding mode control for controlling microbial fuel cell states is presented. The microbial fuel cell is single chamber single population type and nonlinear terms and uncertainties are considered in the model under study. The sliding mode method has the ability to overcome parametric uncertainty and the adaptive method has been used to estimate nonlinear terms and uncertainties. Also, the stability of the closed-loop system and the output voltage stabilization are guaranteed using the Lyapanov criteria.

Accordingly, the second part presents the mathematical model of the microbial fuel cell and the third part describes the proposed control method. Sections 4 and 5 describe the simulation results and conclusions and suggestions for future work, respectively.

In this section, the proposed control method is described. The purpose of the adaptive sliding mode controller is to achieve optimal performance of the MFC by regulating substrate concentration to a specific set point. The dilution rate is selected as a manipulated input variable to control the MFC substrate concentration. Let us define the error as follows:

$e_{1}=x_{1}-x_{1 d}$    (5)

$e_{2}=x_{2}-x_{2 d}$    (6)

where x_1d and x_2d are the desired equilibrium points. By deriving from the error,

$\dot{e}_{1}=-\theta^{-1} Y^{-1} \frac{x_{1} x_{2}}{k_{s}+x_{1}}-x_{1} u+c_{s 0} u-\dot{x}_{1 d}$    (7)

$\dot{e}_{2}=\theta^{-1} \frac{x_{1} x_{2}}{k_{s}+x_{1}}-\left(x_{2}-1\right) u-k_{d} e_{2}-k_{d} x_{2 d}-\dot{x}_{2 d}-u$     (8)

With this definition

$u=u_{1}+u_{2}$    (9)

$f_{1}(t, x)=-\theta^{-1} Y^{-1} \frac{x_{1} x_{2}}{k_{s}+x_{1}}-x_{1} u+c_{s 0} u_{2}-\dot{x}_{1 d}$    (10)

$f_{2}(t, x)=-\theta^{-1} \frac{x_{1} x_{2}}{k_{s}+x_{1}}-\left(x_{2}-1\right) u-u_{1}-k_{d} e_{2}$$-k_{d} x_{2 d}-\dot{x}_{2 d}$   (11)

The system error equations are rewritten as follows:

$\dot{e}_{1}=f_{1}(t, x)+c_{s 0} u_{1}$    (12)

$\dot{e}_{2}=f_{2}(t, x)-u_{2}$     (13)

By selecting the sliding lines as follows:

$s_{1}(t)=e_{1}(t)$    (14)

$s_{2}(t)=e_{2}(t)$    (15)

To stabilize the error dynamics, select the u₁ and u₂ control vectors as follows:

$u_{1}(t)=\frac{1}{c_{s o}}\left(-\hat{f}_{1} \tanh \left(\frac{e_{1}}{\eta_{1}}\right)-k_{1} \operatorname{sgn}\left(s_{1}\right)\right)$     (16)

$\left.u_{2}(t)=\hat{f}_{2} \tanh \left(\frac{e_{2}}{\eta_{2}}\right)+k_{2} \operatorname{sgn}\left(s_{2}\right)\right)$     (17)

where $\hat{f}_{1}$ and $\hat{f}_{2}$ are adaptive estimators for estimating the uncertainties and nonlinear terms of the system. To prove stability, we select the Lyapunov function as follows:

$V(\mathrm{t})=\frac{1}{2} s_{1}^{2}(t)+\frac{1}{2} s_{2}^{2}+\frac{1}{2} \tilde{f}_{1}^{2}+\frac{1}{2} \tilde{f}_{2}^{2}$      (18)

where

$\tilde{f}_{1}=f_{1}-\hat{f}_{1}$

$\tilde{f}_{2}=f_{2}-\hat{f}_{2}$

By deriving from the Lyapunov function and replacing u₁ and u₂, it is obtained

$\begin{aligned} \dot{V}(t)=s_{1}(t) \dot{s}_{1}(t)+& s_{2}(t) \dot{s}_{2}(t)-\tilde{f}_{1} \dot{\hat{f}}_{1}-\tilde{f}_{2} \hat{\hat{f}}_{2} \\ &=s_{1}(t)\left(f_{1}\right.\\ &-\hat{f}_{1} \tanh \left(\left(\frac{e_{1}}{\eta_{1}}\right)-k_{1} \operatorname{sgn}\left(s_{1}\right)\right) \\ &+s_{2}(t)\left(f_{2}-\hat{f}_{2} \tanh \left(\frac{e_{2}}{\eta_{2}}\right)\right.\\ &\left.+k_{2} \operatorname{sgn}\left(s_{2}\right)\right)-\tilde{f}_{1} \dot{\hat{f}}_{1}-\tilde{f}_{2} \dot{\hat{f}}_{2} \end{aligned}$    (19)

By simplifying the above equation is obtained

$\dot{V}(t) \leq\left|s_{1}(t)\right| f_{1}-s_{1}(t) f_{1} \tanh \left(\frac{e_{1}}{\eta_{1}}\right)+s_{1}(t) \tilde{f}_{1} \tanh \left(\frac{e_{1}}{\eta_{1}}\right)$

$-k_{1} s_{1}(t) \operatorname{sgn}\left(s_{1}(t)\right)-\tilde{f}_{1} \dot{\hat{f}}_{1}+\left|s_{2}(t)\right| f_{2}$

$-s_{2}(t) f_{2} \tanh \left(\frac{e_{2}}{\eta_{2}}\right)+s_{2}(t) \tilde{f}_{2} \tanh \left(\frac{e_{2}}{\eta_{2}}\right)$

$-k_{2} s_{2}(t) \operatorname{sgn}\left(s_{2}(t)\right)-\tilde{f}_{2} \dot{\hat{f}}_{2}$     (20)

Considering inequation $0 \leq|x|-x \tanh \left(\frac{x}{\mu}\right) \leq 0.2785 \mu$ [27], the above equation becomes:

$\begin{array}{rl}\dot{V}(t) \leq 0.2785 \eta_{1} & f_{1}+\tilde{f}_{1}\left(s_{1}(t) \cdot \tanh \left(\frac{s_{1}}{\eta_{1}}\right)-\dot{\hat{f}}_{1}\right) \\ & -k_{1} s_{1}(t) \operatorname{sgn}\left(s_{1}(t)\right) \\ & +0.2785 \eta_{2} f_{2} \\ & +\tilde{f}_{2}\left(s_{2}(t) \cdot \tanh \left(\frac{s_{2}}{\eta_{2}}\right)-\dot{\hat{f}}_{2}\right) \\ & -k_{2} s_{2}(t) \operatorname{sgn}\left(s_{2}(t)\right)\end{array}$      (21)

Now by defining the adaptive laws as follows

$\dot{\hat{f}}_{1}=s_{1}(t) \tanh \left(\frac{s_{1}(t)}{\eta_{1}}\right)$    (22)

$\dot{\hat{f}}_{2}=s_{2}(t) \tanh \left(\frac{S_{2}(t)}{\eta_{2}}\right)$    (23)

We achieve

$\begin{aligned} \dot{V} \leq 0.2785 \eta_{1} f_{1}+& 0.2785 \eta_{2} f_{2} \\ &-k_{1} s_{1}(t) \operatorname{sgn}\left(s_{1}(t)\right) \\ &-k_{2} s_{2}(t) \operatorname{sgn}\left(s_{2}(t)\right) \end{aligned}$     (24)

By defining $\sigma=0.2785 \eta_{1} f_{1}+0.2785 \eta_{2} f_{2}$ and k=min (k₁, k₂), it is obtained

$\dot{V} \leq-k\left(s_{1}^{2}(t)+s_{2}^{2}(t)\right)+\sigma$     (25)

By integrating the above equation on the region $\xi \in[0, T]$

$V(T)-V(0) \leq-k \int_{0}^{T}\left(s_{1}^{2}(t)+s_{2}^{2}(t)\right) d \xi+\int_{0}^{T} \sigma d \xi$    (26)

Considering $(T) \geq 0$, it can be obtained that:

$\int_{0}^{T}\left(s_{1}^{2}(t)+s_{2}^{2}(t)\right) d \xi \leq \frac{1}{k} V(0)+\frac{1}{k} \int_{0}^{T} \sigma d \xi$    (27)

From the above equation, it is identified that the closed-loop system is stable and ultimately bounded, which is in-line with tracking error. Therefore, we have developed a combined adaptive sliding mode control scheme for the MFC which behaves in a preferred manner.