Robust control design for air breathing proton exchange membrane fuel cell system via variable gain second‐order sliding mode

The nonlinear and time‐dependent characteristic and unknown modeling uncertainty of proton exchange membrane fuel cell (PEMFC) such as complex electro‐chemical, thermal, and fluid mechanic phenomena make its controller design quite challenging. In this paper, a controller based on a super twisting algorithm (STA) with variable gains is proposed to control the air breathing system of PEMFC. The strategy includes regulating the oxygen excess ratio ( λO2 ) for preventing the stack oxygen starvation and maintaining optimum net power output in spite of external disturbances and model uncertainties. The proposed algorithm has the main advantages of the fixed gain STA, such as robustness against the disturbance and parametric uncertainties with the unknown boundary, chattering reduction, and finite time convergence. The Lyapunov analysis was proposed to assess the stability of the Variable Gain Super Twisting Algorithm (VGSTA). The results verified the effectiveness of the proposed controller with attaining robust regulation against uncertainties, disturbances, and noisy circumstance compared to fixed gain SOSM controllers.


Introduction
Fuel cells are as a good alternative source to its high autonomy for the distributed generation power system, electric vehicles. This is due to high-efficiency, renewable, lower gas emissions, low environmental pollution, and consumption of fuels (methanol, hydrogen, solid oxide, etc.) [1,2]. Furthermore, they have been studied in many power applications as stand-alone power generations [3] and in transportation (cars, buses) [4] due to the high-efficiency and less greenhouse-gas emissions compared to fossil fuels [5].
A main issue in PEM fuel cell operation is the oxygen starvation phenomenon that happens during sudden increasing of load variations. The rapid load rising accelerates the oxygen consumption because of increasing of the chemical reactions in PEMFC that causes irreversible damage as a hot spot on the membrane [5][6][7]. Load change (I st ) as a measurable disturbance causes a severe decrease in O 2 and strongly affects system operation. The O 2 requires to be kept between 2 and 2.5 [8] to avoid oxygen starvation, which needs accurate control of the air breathing system. The air breathing system of the PEMFC connected to a motor and air compressor via a cathode supply manifold. Therefore, its efficiency has a high effect upon the overall dynamic efficiency and transient behavior of the fuel cell under load variation [9]. Thus, new advanced compressor control is essential to improve the performance of the PEMFC system and is a challenging control target. In the recent years, many control strategies have been proposed for adjusting the oxygen excess ratio and maximizing the output net power. A second-order sliding mode algorithm is developed for the PEM fuel cell air feed system. The oxygen excess ratio is estimated via an extended state observer from the measurements of the compressor flow rate, the load current, and the supply manifold pressure. The proposed extended state observer-based controller is

MODELING AND ANALYSIS
Robust control design for air breathing proton exchange membrane fuel cell system via variable gain second-order sliding mode an out feedback system. This out feedback system has robustness against load variations and parametric uncertainties [10]. However, implementation of this control approach is so complex calculation and computing. In Ref. [11], theory of sliding mode control for uncertain stochastic systems and state observer designing to estimate the system state and the stochastic stability for the overall closed-loop system is analyzed. Among them, Purkrushpan and coworkers proposed a feedback control using Jacobin linearization at the nominal operating point. However, a drawback to the system is that the parametric uncertainties, the system severe nonlinearity and large disturbances; linearization control techniques cannot easily satisfy dynamic performance [8]. Niknezhadi and Fantova suggested a LQR/LQG method for adjusting of oxygen stoichiometry [12]. The LQR/LQG method is based on a linearized model at working point using its first derivative (Jacobin matrix) of the Taylor series. Even so, in that study, classic control approaches and the linear control strategy need to find precise knowledge of the model at operating point. Another drawback with this system is that finding exact model; it is impossible and so difficult when the system is a very complex and transient response is unsuitable and there is a steady-state error for tracking. Therefore, an advanced control technique requires to be developed based on the nonlinear dynamic model for the PEMFC.
Na et al. [8,13] used an advanced controller based on the exact nonlinear model. The results from this research indicated that the suggested feedback linearization controller had a better transient response. Although the problem is that the feedback linearization controller could not assure the robustness with the parametric uncertainties and the internal and external disturbances. Another issue is the requirement of an exact nonlinear model to design a controller. To solve this issue, Li et al. [14] designed a robust control strategy based on H∞ and the state feedback exact linearization. So, the steady-state error is alleviated against parameter uncertainty, and the system robustness and better transient response are guaranteed. However, the drawback in this method is that this control approach has some complex mathematical calculation, and the calculation is very time-consuming and hard to implementation.
The intelligent, low-cost, and nonlinear controller is the fuzzy controller that is profit for very complex PEMFC system [15]. Problems in this research, the rule bases are not produced in a systematic method, and the stability is not proved. Other problems are the selection of membership function type and optimization of fuzzy rules. The sliding mode control has been applied for fuel cell, such as the oxygen excess ratio control in [16] and power management in [17]. The popularity of sliding mode control method is because of its applicability to a class of nonlinear systems in the global sense, and the issues of performance and stability robustness to modeling uncertainties and disturbances. The major advantage of sliding mode is low sensitivity to plant parameter uncertainties and external/internal disturbances, which eliminates the necessity of exact modeling [18,19]. Although unfortunately, the discontinuous control in sliding mode control causes the high-frequency switching. Therefore, highfrequency vibrations or "chattering" in the output damage actuators. One of the most beneficial techniques for chattering decreasing is higher order sliding mode control, which has been developed to decline this phenomenon and maintain robustness and finite time convergence properties of the classical sliding mode under load variations and parametric uncertainties.
Because of the most attractive features such as robustness against external disturbances and high accuracy and quick convergence in sliding mode algorithm, several sliding mode observer (SMO) for fault diagnosis approaches were developed for linear systems by Edwards et al. [16,17]. However, conventional sliding mode technique induces undesirable chattering effect in control signals, which is due to the sign function. This phenomenon provides the system instability or actuator damaging. In recent years, the chattering effect was eliminated by High Order Sliding Mode (HOSM) technique because of its continuous output signal and guarantees the robustness against model uncertainties and internal or external disturbances. Therefore, Liu et al. [20], super twisting (ST) sliding mode observer or fault diagnosis and isolation (FDI) schemes to estimate the system state and the fault signal in PEMFCs system was developed. In this estimator, the fault signal is successfully reconstructed based on the ST sliding mode algorithm without chattering phenomenon.
During the last years, nonlinear approaches based on fixed gain super twisting algorithm have been proposed for three-phase power converters by Liu et al. [21]. This controller system has a cascaded configuration which consists of two loops. The outer loop adjusts capacitor voltage of the DC-link, and the control law is based on super twisting algorithm. The inner control loop adjusts the actual currents to their desired that is designed using the STA. The STA has the most attractive characteristic as the finite time asymptotic convergence to requested value and chattering reduction.
Liu et al. [22] proposed a new control approach for the three-phase two-level AC/DC converters. The outer loop which adjusts DC-link capacitor voltage used a robust H∞ controller with an extended state observer. The inner loop strategy is applied to track the reactive/active power to desired values by second-order sliding mode (SOSM) technique. The comparison results with other controllers in this paper show that the developed strategy based on SOSM controller has an exact tracking against load fluctuations.
High-order sliding mode for controlling the oxygen excess ratio (stoichiometry) in the literature [23,24,25] proposed to prevent irreversible damages in a polymer membrane due to the oxygen starvation phenomenon, in which the discontinuous control is applied on some higher time derivative of the system. The advantages of the proposed method are the robust stability, convergence to the sliding surface in finite time, and the reducing of chattering.
Kunusch et al. [7,27] used a second-order sliding mode strategy by super twisting algorithm to stabilize the system and prevent chattering phenomenon in PEMFC. The STA controller provides a smoother signal control than what offers a sliding mode. Using a second-order sliding mode control, Laghrouche et al. [23,24,28,29] suggested that may have resulted in the better transient response against fast variation of the current load. Control law based on the nonlinear optimization method to reduce chattering, stability in the tracking of the oxygen excess ratio, and optimization of the net power on PEMFC. He also proposed second-order sliding mode control in cascade configuration with a super twisting algorithm to improve output net power and maintaining of O 2 between 2 and 2.4 [6]. However, the STA and HOSM with fix gain are not able to compensate for the larger class of unknown perturbation in the severe nonlinearity of the fuel cell system.
Other options in [30] are made to control a knee exoskeleton as a wearable robot using a SOSM control. This control scheme helps the motions of the knee joint by the orthosis actuator. The SOSM strategy was developed to alleviate the chattering phenomenon consists of fast changing of the control signal that it can be dangerous for the actuators.
In [31] as a different type research, is proposed with second-order and conventional sliding mode algorithm for DC-DC buck converter. A comparison study between first-order sliding mode control and SOSM algorithm was developed. The advantages of the SOSM strategy are to suppress chattering and maintain robustness and finite time convergence comparison with conventional sliding mode control. However, in that research, fixed gain controller was considered when designing the controller. Furthermore, theoretical analysis of the system stability was not conducted.
Another application of variable gain STA strategy developed to wind energy conversion optimization is proposed by Evangelista et al. [32] and [33]. This control strategy compared with standard sliding mode strategy and concluded that mechanical stresses are reduced by this method to minimal level, and also the output chattering is reduced.
Gonzalez et al. [34] in 2012 implemented a variable gain control of a mass-spring system by providing a new technique for measuring the noise bounds. Gonzalez compared three typical such as fixed gain STA, variable gain STA, and traditional sliding mode and after implementation of these systems, they found out that the first-order sliding mode control had been chattering, and it was significantly decreased by both STA. Moreover, the result showed that compared to the classic STA, the variable gain STA could compensate for the larger class of perturbation and to reduce the chattering. Some research studies have been investigated the Lyanpunov stability theory and therefore, several approaches had been suggested to prove this theory and calculation of the finite time [35][36][37][38].
Moreno [36] proposed comprehensive of Lyapunov stability theory in super twisting algorithm that includes second-order sliding mode, and many other continuous algorithms, with constant or time-varying gains, and considering perturbations. Two distinct strict and robust Lyapunov functions are developed: a quadratic and a nonquadratic one. Each of them is useful to establish different properties of the algorithms. The quadratic Lyapunov functions are used to determine finite time convergence and to estimate the convergence time from every initial condition.

Motivation
In this research, we are interested in fast regulating the oxygen excess ratio and optimization of the net power output. To achieve this objective, a variable gain super twisting algorithm is used for air compressor control. Sliding mode control has already been made to fuel cellrelated applications, such as stack breathing control in [39,40] and second-order sliding mode control for stack breathing in [23,24,26]. The proposed control law developed to counteract the chattering phenomenon and to maintain robustness against the unknown bounds of uncertainties and perturbations in the PEMFC with severe nonlinearity and finite time convergence is obtained.

Contribution
The robust nonlinear adaptive gain super twisting sliding mode control is proposed for the control of an air feed system in PEMFC with a formalization of parametric uncertainties. So, the key-point of this paper is to consider the unknown bounds of uncertainties and perturbations in the PEMFC with severe nonlinearity. The accurate tracking and stability in a close-loop system of the oxygen excess ratio are proposed and a fast response time in order to avoid oxygen starvation during load variations is achieved. Furthermore, this paper provide: 1. The novel adaptive STA control law is designed for a PEMFC as a severe nonlinear system with special features such as the robustness, relative simplicity, and finite convergence time as a new strategy for PEMFC system. 2. The experimental noises and drift uncertainty terms/ disturbances with unknown boundary for PEMFC nonlinear equations are rejected by the VGSTA. 3. A controller based on Lyapunov theory is designed to assure uniform stability of a nonlinear system considering bounding functions for disturbances and uncertainties of a realistic PEMFC model. 4. This proposed controller is explored and main advantages are summarized as follows: a. Robust regulation of the oxygen excess ratio to prolong the life span with chattering effect avoiding; b. Raising transient performance features with finite convergence time; c. More robustness to drift uncertain parameters and highly variation of load current with unknown boundary compared with the first-order SM and ST with fixed gains; d. Guaranteed extended range of operation, in spite of the severe nonlinearity of plant; e. Simple controller implementation, resulting in low computational costs in real-time environment.
The structures of this paper are organized into these sections: The dynamic model of the PEMFC air breathing system is discussed in Section 2. In Section 3, methodology and design of controller law according to Lyapunov's stability is discussed, and the results are presented in Section 4. Finally, the conclusion is presented in Section 5.

Fuel Cell System Dynamic Model
Dynamic models of the compressor, cathode humidifier, line heaters, and fuel cell stack channels were developed and validated. The model is suitable for nonlinear control design. Dynamic model has six states defined as: X = [ cp ,P sm ,m sm ,P O 2 ,P N 2 ,P rm ] T where motor angular speed, ω cp ; supply manifold pressure, P sm ; mass of supply manifold, m sm ; oxygen pressure, P O 2 ; nitrogen pressure, P N 2 ; and return manifold pressure, P rm . The air is mixed with these two gases such as the oxygen and nitrogen, nitrogen 79% and oxygen 21%, that is supplied to the cathode side. A schematic diagram of a PEM fuel cell stack together with some of its auxiliary is depicted in Figure 1.
The states are defined in following definitions: States Descriptions Air mass in the supply manifold Oxygen pressure in cathode side Nitrogen pressure in cathode side The state space model with six state are in equations (1-6). The complete set of equations is not included here for space reasons and can be found in [41]. They are: In this model, the compressor motor voltage (u = V cm (V)) is the input control, and the fuel cell stack current (d = I st (A)) can be measured as disturbance. The model parameters are given in Appendix. The equations (1)(2)(3)(4)(5)(6) can be written in a compact state space form which is as follows: where x ∈ ℝ 6 is the system state; f ∈ ℝ 6 → ℝ 6 is a continuous vector function representing the dynamics of the autonomous system; g ∈ ℝ 6 and φ(x) ∈ ℝ 6 are vectors representing the input and the disturbance matrices, respectively. The compressor air flow (W cp ) is: Output voltage is expressed as a function of stack current, partial pressure, and temperature of fuel cell. The output stack voltage (V cell ) will be decreased due to ohmic losses, activation losses, and concentration losses. The output voltage is the difference between open circuit voltage and voltage losses, of which voltage losses are nonlinear functions of stack current, temperature, pressure, and chemical reactions. Thermodynamic potential, E, is defined using the Nernst equation in equation (11).
Also, the net power of the fuel cell system (P net ) is the difference between stack power and motor compressor power which is calculated by the following relationships:

Oxygen excess ratio calculation
Reactants consuming in the fuel cell are faster than reactants supplied. For this reason, the starvation phenomenon occurs. Starvation happens especially during transient loads, and it is one of the main reasons of membrane damaging. Thus, fast regulation of the oxygen excess ratio by increasing the cathode mass flow is very significant. Furthermore, mass flow rising is limited by the actuator inertia. Therefore, the regulation of the oxygen excess ratio is a crucial issue [8,42].
The oxygen excess ratio cannot be measured directly and has to be concluded from another sensing. Formula (16) shows clearly that it depends upon the ratio of two mass flows.
The oxygen flow reacting can be calculated by equation (16). It is proportional to the stack current. It can be written as: The inlet air flow (W a,in ) is equal: − P a 6 x 5 6 + P a 5 x 4 6 + P a 4 x 3 6 + P a 3 x 2 6 + P a 2 x 6 + P a 1 The outlet flow in supply manifold is stated as a function of P sm and cathode pressure (P ca ).
The outlet flow in the supply manifold (W sm,out ) is obtained from a linearized nozzle equation and then the following equations can be derived.
Therefore, W sm,out is defined as follows: Thus, the inlet oxygen flow is derived by: Thus, O 2 is defined as the ratio between W O2,in and W O2,reacted in the fuel cell stack. Dependency of the stack current to the air flow W O2,reacted caused a severe drop in O 2 . In addition, the increasing of O 2 causes increasing of partial pressure of oxygen and then P Stack will be improved. Therefore, the inlet air flow is identical with the compressor air flow. Regulation of the inlet air flow is equal to regulation of O 2 against the load fluctuations.

Methodology
In this section, the controller is developed according to a new version of the super twisting algorithm. There are some steps to do for controller designing: 1. The sliding variable should be defined to achieve the desired control objective. 2. The new set of variable should be described with changing of the variable S. 3. Next step is to determine the bounding functions g 2 and the first time derivative of g 2 . 4. A Lyapunov function is selected to consider stability analysis and calculate controller adaptive gains.

Lyapunov function and controller design
In this work, a Lyapunov function is suggested for designing of variable gain super twisting algorithm in PEM fuel cell. A time invariant and nonsmooth, Lyapunov function is selected for appropriate perturbation-bound dependent gains for the STA. This controller can be directly used to the system with a relative degree (R.D) of 1. The adaptive gain's controller provides robust performance for regulation against severe uncertainty. There is a continuous and smooth control signal. Finite convergence time is guaranteed. The super twisting algorithm is a known algorithm in second-order sliding mode control that was introduced in [43,44]. This algorithm can be written by equation (23) [43,44], where k 1 (t) and k 2 (t) are gains and ρ 1 (t, x) and ρ 2 (t, x) are perturbation terms.
where the vector T = Φ T (x) = 1 (s),u 2 , where A = −k 1 1 −k 2 0 and positive definite matrix Q = Q T > 0. The k 1 (t) and k 2 (t) can be simplified to the following equations: According to [34][35][36], variable gains are bounded to the following assumption: k 1 (t), k 2 (t), and K c > 0 are variable gains [45]. P matrix is considered as follows: Stability conditions can be obtained with the desired positive definite matrix (Q) that is determined as follows: (20) W sm,out = K sm,out ⋅ (P sm − P v,ca ).
Q would be a positive definite matrix, if: Therefore, after the change of variables, the S variable is defined as a new variable. So, control law design is based on variable sets of [Sx ′ ] and x � ∈  n−1 is replaced with x ∈  n . It can be easily separate from the part that is zeroed when S = 0 from the rest.
Proposed controller gains in equation (23) can be calculated using the Lyapunov function for equation (37) as follows: where β > 0, ε > 0, and δ > 0 are arbitrary positive constants and g 1 (t, x) and g 2 (t, x) are positive and in the process boundaries ̇2(t,x), 1 (t,x) should be calculated.

Bounds on the perturbations
The design procedure needs finding of the bounding functions is made in the equations of ρ 1 (x ′ , S, t) and the first derivative of ρ 2 (x ′ , t) considering real values for their parameters, that is, changing from nominal values, and consisting of internal and external disturbances. The sliding surface is defined in equation (38). With change of the variable, the S variable becomes a member of the new set of variables in following: As a result: Thus, the new value x 1 can be calculated as follows: As a result: A new set of conditions for x ′ = {x 2 , x 3 , x 4 , x 5 , x 6 } defined. Consider the first-order derivative S: Considering D = [B 10 + B 11 x 2 + 2B 02 h(t, x ′ , s)] and with a replacement value ̇x 1 in following: As a result: The functions ρ 1 (x ′ , S, t) and ρ 2 (x ′ , t) are extracted: 2B 02 2B 01 As a result, ρ 1 (x ′ , S, t) and ρ 2 (x ′ , t) of the extension of equation (46) are obtained and g 1 (t, x) and g 2 (t, x) can be calculated. Due to the complexity of the equations, the calculation was performed by final values of g 2 (t, x) and g 1 (t, x) that were obtained from equation (31). According to equation (37), the variable gains finally were obtained.

Results and Discussions
The aim of this section is testing and verifying the performances of the designed VGSTA and the fixed gain super twisting algorithm in the PEMFC system with unknown bounded uncertainty in the MATLAB/Simulink environment.

Fuel cell model validation TEST
The PEMFC dynamic model in this paper is based on Ref. [41]. The model was simulated by equations (1)(2)(3)(4)(5)(6)(7)(8). The net output power, the oxygen excess ratio, and compressor flow rate (eqs. 9, 15 and 16) were calculated. The model is based on simulating the relationship between the output voltage and states of an air feed system. Figures 2 and 3 show a polarization curve that is the output voltage against load current and output power versus load current at steady state for operating temperature 50°C. Using equations (1-15) of the state space and voltage and power equations of PEMFC, the polarization (V-I) and power (P-I) curves of the system indicated in Figures 2-4 were established for 80 kW. This stack is composed of N cell = 381.
The V-I and P-I curves are shown in Figure 4A and B. It is shown that the normal operation behavior of a fuel cell with temperature increasing from 30°C to 80°C causes the power to increase gradually to the maximum power point and then decrease. In order to consider the efficiency of the suggested new controller, simulation was made in the MATLAB/Simulink environment. The block diagram of the close-loop system is shown in Figure 5.
In Figure 6 is shown a block diagram of the control system. The sliding surface was calculated by the W cp and W cp,ref . Finally, k 1 (t) and k 2 (t) gains, and the control input of VGSTA were obtained. In fact, variation current loads are applied at t = 20s, 55s, 80s, and 130s. Figure 7 shows stack current that is applied as a disturbance to the system. Figure 8 shows states for the close-loop system based on variable gains super twisting controller. It is observed from the figure that the state x 1 or the angular velocity has an acceptable range, which is changed between 6261 and 25046 g. The air mass in supply manifold was changed between 0.03 and 0.05 kg with the load current variations. The pressure values of the states x 2 , x 4 , x 5 , and x 6 are in the bounded suitable pressure ranges which show that the system states have appropriate condition.
The oxygen excess ratio in the proposed controller is compared with fixed gain controller in Figure 9A. It is shown that both controllers are capable of reducing the chattering phenomenon. Furthermore, it is obviously indicated that using the VGSTA, the O 2 of PEMFC is achieved quickly with great precision and global stability of the close-loop system to the desired value of 2. However, the fixed gain controller could not track the set-point value. There is a difference in the value of O 2 for all ranges of load current, and it does not follow a desired constant value. It has a deviation from the desired values about 0.5-0.8. Figure 9B shows a comparison between the net output powers in the proposed controller and the fixed gain STA. It was concluded from this figure that the net output power increased in the applied variable gain condition,  o and the fuel cell generated an output net power of about 25-50 kW for VGSTA proposed method, but it is 25-40 kW in fixed gain STA, the highest current (it is shown at t = 55 sec). Therefore, there is a suitable saving in the output net power. There is 'start up' peak that is shown on the figures because of unexpected step load current. Because of a sudden changing of load current at t = 20, 55, 80, and 130 sec, results have a 'start up' peak at these times in oxygen excess ratio and net output power.
The variable gains k 1 (t) and k 2 (t) are shown in Figure 10, and it was clear from the figure that the gains k 1 (t) and k 2 (t) had a value about 5 and 1.5, respectively.

Parameters variations as unknown uncertainty
In order to validate the robustness of the proposed controller, some PEMFC parameters were changed under permissible bound of variation as the uncertainty in Table 1 [41].
In real condition, the actual and nominal value of the parameters into the PEMFC system cannot be accurately achieved. They are sensitive to the ambient circumstances (e.g., temperature, pressure, etc.) and structural characteristics (e.g., manifold volumes, rotor inertia, motor resistance, motor inductance, etc.). The main issue of the PEMFC model is the existence of unknown parametric uncertainties and variations that motivate using of new control strategy. Although, the PEMFC system is a severe non-linear model and the proposed novel VGSTA is suitable for this system. The Figure 11A shows that the net power against the load current variation in the presence or absence of uncertainty is almost identical variation in both of them. Figure 11B shows the behavior of the oxygen excess ratio under load variations and parametric uncertainties, compared to behavior of the nominal system under the same controller gains. It can be seen that the controller has robustness against the uncertainty in the system and the O 2 tracks the desired value 2. Variable gains in the perturbed condition are shown in Figure 12.

Realistic measurement noise test
A noise test has been applied in addition to the parameter uncertainty. Considering that the common values of signal/  noise (SNR) are about 28 [dB], all measurements were with power transmission equal to PSD = 0.22 [W], and then a distribution noise is Gaussian with mean value = 0.0023237, variance = 0.13362, and covariance = 0.0179 [41]. Figure 13 shows states profile in the case of the variable gain controller under realistic noise condition. Figure 14 indicates the noise and noise-free profiles of the measured quantities. It shows the appearance of small variation of the O 2 . It confirms the proposed controller regulates accurately also in the simultaneous presence of noise and parameter uncertainties.

Conclusion
In this paper, a variable gain super twisting controller as a second-order sliding mode for the air breathing system of the PEM fuel cell is proposed. The PEMFC model is on sixth order and severe nonlinear model. The proposed controller is designed via Lyapunov theory and by using simulation, taking into account uncertainty, and experimental noises are validated and provided acceptable results. The suitability, accuracy, and high performance of the suggested controller through comparison with fixed gain was shown. This design method needs the calculation of bound of disturbance for controller design, and this method reduces online calculations against another nonlinear and robust controller such as H∞ and adaptive control law that are very complex. Structure of the algorithm is simple,   thus low online computation is required. The variable gain controller is very precise in set-point tracking of the oxygen excess ratio and improves the net output power compared with fixed gain controller. In result section, extra uncertainty has been incorporated in several parameters of the system (Table 1) and the robustness against the realistic measurement noise, and uncertainty was considered and verified. Reduction of chattering, simplicities of implementation, and finite time convergence are the advantages of the proposed controller. A formal proof of stability based on the Lyapunov function and also calculation of the adaptive gains is presented. The simulation results show that regulation of the O 2 is more robust than fixed gain super twisting algorithm with the better transient response under parametric uncertainties, load variations, and noisy circumstances to avoid oxygen starvation during load variations. 3.49×10 −7 kg/sec 2 /bar B 11 3.55×10 −13 kg/sec B 02 −4.11×10 −10 kg/sec/bar