Sliding mode control of biogas production by anaerobic digestion with addition of acetate

. Biogas production by anaerobic digestion with addition of acetate is considered. Sliding mode control for regulation of the biogas flow rate using the addition of acetate as a control action is proposed. The control design is carried out with direct use of nonlinear model and expert knowledge. Chattering phenomena are avoided by realizing the sliding mode with respect to the control input derivative. The state variables, external disturbance, process output and control input are varied in the known intervals. The performance of the designed sliding mode control is investigated by varying the process set point and the uncertain process parameter, which reflecting the influence of the external disturbance. The excellent performance of presented control is proved through simulation investigations in MATLAB using Simulink . .


Introduction
In the recent years, there is an increasing interest by society and business for the use of green energy [1,2]. In particular, it is necessary to note that biogas is one very useful source of green energy [3,4]. It can be seen as a promising type of biofuel.
The biogas production is typically done by anaerobic digestion (oxygen-free environment) of various organic material with the help of special microorganisms (biomass) [4,5]. This process generally carried out in continuously stirred tank bioreactors (CSTR).
The biogas consists mainly of methane (CH 4 ) and carbon dioxide (CO 2 ), and may have small amounts of hydrogen sulphide (H2S), hydrogen (H 2 ), and carbon monoxide (CO). The gases methane, hydrogen, and carbon monoxide (CO) can be combusted or oxidized with oxygen [6]. Almost all forms of organic material can be used to produce biogas. However, waste water, manure, energy crops and organic industrial waste are the most common feedstocks [7].
It is important to emphasize that the biogas production process is very complex and may sometimes become very unstable. Investigations have shown that addition of stimulating substances (acetate or glucose) in appropriate concentrations allows to stabilize the process and to increase the biogas flow rate [8,9].
The process of biogas production is usually described by a system from ordinary differential equations with uncetain parameters. For these reason it is needed the developing of sophisticated control algorithms based on nonlinear models [10][11][12][13].
It is known that the sliding mode control is effectively used in the stabilization of nonlinear and uncertain plants [14]. The guaranteed system invariance to parameter uncertainties and external disturbances is the main advantage of the sliding mode [15][16][17]. However the control signal is discontinuous in time, which leads to chattering phenomenon.
In practice, such control is hard to realize therefore various techniques for chattering attenuation are proposed (boundary layers, auxiliary input variable, fuzzy logic, input-dependent sliding surface).
The aim of this paper is to design and investigate sliding mode control of the biogas production (anaerobic digestion) with addition of acetate as a control action.

Process model
In this study, the three-stage biogas production process described by the following nonlinear system from five ordinary differential equations and one algebraic equation is considered. The proposed system has two input variables [9]: In this system (mass balance model), equation (1) describes the hydrolysis stage of the diluted organics with concentration in S 0 (g/L); equation (2) -the growth and changes of the acidogenic bacteria, with concentration 1 X (g/L), consuming the appropriate substrate, with concentration 1 S (g/L). The mass balance for this substrate is described by equation (3), where the first term reflects the consumption by the acidogenic bacteria, the second term reflects the substrate formed during the hydrolysis stage and the third one -the effluent flow rate of liquid. Equation (4) describes the growth and changes of the methane-producing (methanogenic) bacteria, with concentration 2 X (g/L), consuming acetate, with concentration 2 S (g/L). The mass balance equation for acetate (5) has four terms in his right side. The first term reflects the consumption of acetate by the methanogenic bacteria, the second one the acetate formed as a result of the activity of acidogenic bacteria, the third one the direct addition of acetate, with concentration in S 2 (g/L), (a new control input) and the last one the acetate in the effluent liquid.
The algebraic equation (6) describes the formation of biogas with flow rate Q in (day -1 ).
The specific growth rate of the acidogenic bacteria   Generally in S 0 is an unmeasurable perturbation, while in S 2 is a known constant or control input. In all cases the washout of microorganisms is undesirable, that is why changes of the total dilution rate

Equations Problem statement
In this study, it is assumed that 1 D and 2 D are constants; in S 0 is the external disturbance, which is uncertain but varied in known bounds; in S 2 -the control input and Qthe output.
This allows us to simplify the dynamic model (1)-(6) for control design purpose obtaining a new model with the following dynamical part [18]: where the variable A=k 3 µ 1 X 1 is considered as a process parameter reflecting the influence of the external disturbance in S 0 . In this simplified model the new state variables are 2 X X  and 2 S S  . The specific growth rate of the methanogenic bacteria is rewritten as where  is the specific reaction rate.
The following assumptions are considerate: a1). The state variables -the biomass and substrate concentrations are positive, bounded and differentiable functions of time: where min Q and max Q are known constants; a3). The process input is the influent substrate concentration It is a differentiable function of time and varied in known boundaries: where min in S and max in S are known constants; max S is the maximum admissible value of the substrate concentration S ; a4). The specific reaction rate is known, nonlinear and differentiable function of time, depending on the state variables and the kinetics coefficients: where p is a vector of the kinetic coefficients; a5). On the basis of expert knowledge the variation interval of the process parameters A is given as: where min A and max A are known constants; a6). The state variablesubstrate concentration ) (t S is directly or indirectly measured. a7). The process output -biogas production rate is measured on-line.
In this paper, the problem is to design a sliding mode control for regulation of the biogas production rate.

Sliding model control design
Suppose it is desired to stabilize the process output . Hence, the biogas production rate error is: where g is positive constant. In order to design the sliding mode control, the system (8)-(10) is transformed into a second order differential equation of the form: where Substituting (16) into (17), the following differential equation of the biogas production rate error is obtained: According to the proposed transformation, the investigated methane fermentation process (8)-(10) is considered as plant with differentiated control input. To guarantee the invariance of the variable structure system in sliding mode to the parameter uncertainty and external disturbance, it is required that certain matching conditions should hold. Moreover, for control design aim, the model (18), is represented by the generalized observability canonical form [14]: To avoid the chattering phenomenon and to obtain a smoothed input signal, the sliding mode is realized with respect to the control input derivative. On the basis of the variable structure system theory, the control law in sliding mode is designed as follows [16]: where  i , i=1,..,4 are switching coefficients.
It is well known that the necessary and sufficient condition for existence and stability of a sliding mode on switching surface ) (t  is given by following the inequality [15,16]: In this study, the sliding surface ) (t  is defined, as follows: where 1 c and 2 c are positive constants. Taking into account (19)-(21) and (23), the stability condition (22) is guaranteed, when the switching coefficients are selected as follows: The calculation of the constant values  i  , 4 ,..., 1  i is reduced to solving the optimization problem taking into account the assumption a1)-a7) and the condition (20).
Therefore the control law (21) can be rewritten in the form: In ideal sliding dynamics using the method of equivalent control [6], the model (20) is described by following differential equation: The equation (27) includes only previous given constants 1 c and 2 c , which defines the dynamics of closed-loop system. Moreover, the system invariance in sliding mode to parameter uncertainties and external disturbance is evident.

Simulation studies
Simulation investigations in MATLAB using Simulink. are carried out to illustrate the effectiveness of the designed second order -sliding mode controller to stabilize biogas production processes (1)- (6) in presence of uncertainty and disturbances.
There are many complex functions to describe cell growth limited by a single substrate. However, Monodtype kinetics are widely used to express the specific reaction rate: In this case, the kinetic coefficients vector is written as  is the maximum specific growth rate and s k is the saturation coefficient. The following values of the parameters are adopted: The state variables, external disturbance, process output and control input are varied in the known intervals: On the basis of the conditions (25)-(26) and (28)-(30), the control law is obtained as follows: where the sliding surface is The performance of the designed sliding mode control is investigated by varying the process set point and the uncertain parameter A.
The simulation results obtained from the step changes of the set-point ( are presented in Fig. 1. (a, b, c).
The good behavior of the nonlinear process and the smooth control input are evident.
Simulation experiments where the process parameter A is changed as  Fig. 2. (a, b).
It is evident from figure that the system invariance to parameter uncertainty is guaranteed.

Conclusions
The sliding mode control for biogas flow rate regulation with addition of acetate is proposed. The control design is carried out with direct use of nonlinear model of the biogas production by anaerobic digestion with addition of acetate and expert knowledge. Smooth control input is obtained. The state variables, external disturbance, process output and control input are varied in the known intervals. The simulation results prove good performance of the developed sliding mode control with respect to set point changes and external disturbances through simulation investigations in MATLAB using Simulink.