Adaptive Control of Proton Exchange Membrane Fuel Cell Air Supply Systems With Asymmetric Oxygen Excess Ratio Constraints

This paper investigates the oxygen starvation and parasitic loss prevention problem of the adaptive oxygen excess ratio (OER) control system of nonlinear proton exchange membrane fuel cells (PEMFCs). Asymmetric OER constraints are considered to avoid oxygen starvation and parasitic loss in the air supply system of PEMFCs. An approximation-based adaptive control strategy is established to ensure robust regulation of the OER while not violating the OER constraints, regardless of unknown system parameters, nonlinearities, and the abrupt changes of the load current. A dynamic surface design technique using an asymmetric barrier Lyapunov function is employed for a recursive control design. Compared with existing control approaches for uncertain nonlinear air supply systems of PEMFCs, this paper first considers the oxygen starvation and parasitic loss prevention problem for the regulation of optimal OER in the control field of nonlinear PEMFCs. Using the Lyapunov stability theorem, the boundedness of all closed-loop signals and the convergence of the output tracking error to the vicinity of zero are proved.


I. INTRODUCTION
Fuel cells are energy conversion devices that transform chemical energy into electrical energy via reactions between hydrogen extracted from gas and oxygen in the air. Fuel cells can provide continuous electricity as long as hydrogen and oxygen are supplied, and their by-products are heat and water. Thus, they are regarded as promising energy devices [1], [2]. Among the various type of fuel cells, proton exchange membrane fuel cells (PEMFCs) have received a considerable amount of attention for both stationary and mobile applications owing to their characteristics such as high energy efficiency, high power density, low operating temperature, and low noise [3], [4]. The structure of PEMFCs is divided into four subsystems: the air supply, the hydrogen flow, the humidity, and the stack temperature systems. Generally, the degree of the stack The associate editor coordinating the review of this manuscript and approving it for publication was Zhongyang Fei . temperature and humidity cannot change rapidly. Thus, the control problem of these two systems can be decoupled from the rest of the system [5], [6]. On the other hand, compared with simple hydrogen flow systems controlled by an electrical valve, the air supply system controlled by an electrical motor and a compressor exhibits much slower dynamics. Furthermore, the air supply system consumes more electric energy in PEMFCs, diminishing its net output power [7]. Therefore, many researchers have investigated the control problems of the air supply system of PEMFCs [8]- [10]. The control objective of the subsystem is to regulate the oxygen excess ratio (OER), which represents the ratio of consumed amount of oxygen supplied to oxygen reacted to the optimal value for obtaining the maximum efficiency of PEMFC. A linear quadratic regulator [8], a proportional-integral-derivative controller [9], and a passivity-based robust proportional-integral controller [10] were presented for controlling OER of linearized air supply systems. A main issue in controlling OER is to consider VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0/ the oxygen starvation phenomenon. Increasing the load current accelerates the amount of used oxygen in the chemical reactions, leading to oxygen starvation. This damages the membrane of the stack and reduces the power response of the fuel cell [11]- [13]. Therefore, there have been attempts to avoid oxygen starvation while regulating OER. In [14], the oxygen starvation protection problem was formulated as a constraint-enforcement problem of OER and a robust nonlinear reference governor approach was presented. In [15], a constrained model predictive control approach was presented to deal with OER constraints. Despite these successful attempts, the load governor and the model predictive controller were based on linearized air supply systems of a certain operating point. However, because the operating point can vary along operating conditions, some researchers have been motivated to develop OER control methods for nonlinear PEMFC systems. In [16], a feedback linearization technique was used to control air supply systems without uncertainties. In [13], a load governor methodology was presented to deal with the oxygen starvation problem of nonlinear PEMFC systems where system uncertainties were not considered. To consider system uncertainties, an adaptive state-observer-based controller was presented in [17] and sliding mode controllers were developed in [18]- [20]. Nevertheless, none of the control approaches [16]- [20] for nonlinear PEMFC systems considered unknown nonlinearities of nonlinear air supply systems. To deal with unknown nonlinearities, an adaptive backstepping control approach was proposed for uncertain nonlinear PEMFC systems where neural networks were employed to estimate completely unknown nonlinearities [21]. Recently, an estimation-based robust control approach was presented for PEMFC systems with unknown nonlinearities [22]. However, the previous control results [16]- [22] for nonlinear air supply systems involve the following restrictions. R1) In [16]- [22], the constraint problem of OER to avoid oxygen starvation and parasitic loss was not considered for the OER control design of nonlinear air supply systems of PEMFCs. To the best of our knowledge, the oxygen starvation and parasitic loss prevention problem for OER control of uncertain nonlinear air supply systems is still unresolved. R2) In [21], a class of nonlinear air supply systems with two nonlinearities was considered, and an adaptive neuralnetwork-based controller was designed. However, only one of the nonlinearities was assumed to be unknown and was compensated by a neural network. Furthermore, it was assumed that some system parameters were exactly known, which may be impractical. R3) In [22], a lumped uncertainty including unknown nonlinear functions and disturbances of air supply systems was estimated, and an estimation-based controller was developed. However, the frequency range of the lumped uncertainty and the system parameter denoting the control coefficient of the input voltage were required for the estimation process.
Some adaptive constraint control approaches have been studied for uncertain constrained nonlinear systems [23]- [26].
In [23], an error transformation approach using multiple prescribed performance bounds was presented for switched nonlinear systems with unknown nonlinearities. In [24], an adaptive constraint control problem of uncertain active suspension systems was considered. Adaptive constraint control problems of uncertain time-delay nonlinear systems with state constraints were investigated in [25] and [26]. In [27] and [28], adaptive fuzzy output-feedback control approaches were studied for uncertain nonlinear systems with state constraints. The visual servoing control problem for a manipulator with visibility constraints was considered in [29]. However, these control designs have not been yet applied to the OER constraint problem of PEMFCs.
Based on these observations, the aim of this study is to propose an adaptive OER control strategy to deal with oxygen starvation and parasitic loss prevention problems of uncertain nonlinear air supply systems of PEMFCs. The nonlinearity denoting the air mass flow from a compressor into a supply manifold is regarded as a completely unknown function and all system parameters are assumed to be unknown. For the controller design, we transform the oxygen starvation and the parasitic loss prevention problem of OER into the OER control problem with asymmetric time-varying constraints. Constructing an asymmetric barrier Lyapunov function [30] and using the dynamic surface design technique [31], an adaptive control scheme is recursively designed where neural network approximators and adaptive tuning laws are derived to compensate for unknown system uncertainties and parameters, respectively. From rigorous stability analysis, it is shown that all the signals of the controlled closed-loop system are semi-globally ultimately uniformly bounded and the control error converges to an adjustable neighborhood of the origin while the asymmetric OER constraints are not violated. Compared with the existing OER control schemes [16]- [22] for uncertain nonlinear air supply systems of PEMFCs, the main contributions of this paper are as follows.
1) Different from the existing control results [16]- [22], this paper considers the constraint problem of OER for avoiding oxygen starvation and parasitic loss of uncertain nonlinear air supply systems. The control problem based on the OER constraints is formulated, and an adaptive control strategy is established in the presence of unknown system nonlinearities and parameters.
2) Compared with the previous adaptive neural-networkbased control approach [21] where only a nonlinear function was assumed to be unknown, this paper assumes that all system nonlinearities and parameters are unknown. In addition, the frequency information of uncertainties used in [22] is not required for the proposed control scheme.
The remainder of the paper is structured as follows. In Section II-A, the air supply system of the PEMFC is introduced and some assumptions and lemmas are presented. The background of the OER constraint is given, and the control problem based on the OER constraint is formulated in Section II-B. In Section II-C, some preliminaries for radial basis function neural networks are stated. In Sections III-A and III-B, the proposed control design and its stability analysis are presented, respectively. The effectiveness of the proposed controller is validated by the simulation in Section IV. Section V provides the conclusion.

II. PRELIMINARIES AND PROBLEM FORMULATION A. AIR SUPPLY SYSTEM OF NONLINEAR PEMFC
The air supply model of PEMFC systems having a 75-kW fuel cell stack fed by a 14-kW air turbo compressor was reported in [32]. Based on the fact that the molar mass of the oxygen and nitrogen are nearly the same, this model can be described using a third-order model as follows [5]: where x 1 represents the total air pressure inside the cathode comprising the sum of the pressures of oxygen, nitrogen, and vapor, x 2 is the air pressure in the supply manifold, which connected by the air compressor system and the cathode of the PEMFC stack system, x 3 is the rotational speed of the compressor motor, δ is the time-varying load current considered to be a measurable disturbance to the system, and u is the motor input voltage of the compressor. φ(x 2 , x 3 ) represents the air mass flow from the compressor to the supply manifold, which depends on compressor motor dynamics [2], [32]. Contrary to the existing literature [33] regarding air mass flow φ(x 2 , x 3 ) as a known nonlinear function, we assume that φ(x 2 , x 3 ) is a completely unknown nonlinear function. The system parameters l i , i = 1, . . . , 11 and c are given by .
Here, the detailed definitions of the used parameters are given in Table 1.
Assumption 2: The time-varying load current δ and its derivativesδ andδ are bounded and δ andδ are measurable.
Lemma 2: [30] For any k a ∈ R and k b ∈ R, it holds that where p(•) is a switching constant defined as (3)

B. CONTROL PROBLEM BASED ON OER CONSTRAINTS
The OER λ O 2 is the ratio of oxygen supplied to the stoichiometric oxygen rate defined as follows: It is well known that the maximum efficiency of the OER corresponds to λ * O 2 = 2 where λ * O 2 is an optimal value [35]. Thus, the OER control objective is regarded as the regulation problem of λ O 2 to λ * O 2 = 2. Moreover, during transients, the OER λ O 2 can change abruptly when the load current changes abruptly. When more load current is drawn from the stack, oxygen in the cathode will react instantaneously. Thus, λ O 2 drops to a very low value. Then, oxygen starvation is generated when the oxygen flow rate is less than the stoichiometric value (i.e., λ O 2 ≤ 1). This damages the fuel cell stack, shortens the cell life limit, and reduces the power response of the cell stock [11]. In the opposite situation, the reduced load current generates the surplus of oxygen in the cathode that leads to parasitic loss (i.e., λ O 2 increases quickly to a very high value) [36]. To prevent these phenomena, the constraint problem in the OER control design should be considered.
In this paper, the constrained OER is considered as where 1 < a < 2 and 2 < b < ∞ are constants denoting physical constraints of the OER. Now, let us define the system output y as y = λ 1 (x 2 − x 1 ). Then, the regulation problem of λ O 2 is transformed into the tracking problem of y to the reference signal y r = 2λ 2 δ and y is required to satisfy the following asymmetric time-varying output constraint: By defining the output y = λ 1 (x 2 − x 1 ), the constraint (6) is induced from (4) and (5). This implies that the constraint problem of OER λ O 2 can be changed to the output constraint problem. Problem 1: Consider the uncertain air supply system (1) of the PEMFC having OER constraints. Our problem is to design an adaptive control law u ensuring that the system output y tracks the reference signal y r while the time-varying constraint (5) is not violated.
Remark 1: Different from the existing OER control results for nonlinear air supply systems [16]- [22], this paper firstly considers the OER control problem in the presence of the asymmetric OER constraints, as stated in Problem 1. Thus, the existing results cannot provide a solution to Problem 1.

C. RADIAL BASIS FUNCTION NEURAL NETWORKS
Radial basis function neural networks (RBFNNs) are utilized to approximate unknown nonlinear functions during the control design steps. Based on the universal approximation property of the RBFNN [37], [38], given a continuous real-valued function Z ( ) : → R with a compact set ⊂ R q , if q is sufficiently large, then there exists an ideal weight vector θ * such that where = [ 1 , . . . , q ] ∈ denotes the input vector, ψ represents the network reconstruction error, θ * ∈ R r with node number r > 1 is the optimal weighting vector defined as ;θ is an estimate of θ * , and ξ ( ) = [ξ 1 ( ), ξ 2 ( ), . . . , ξ r ( )] ∈ R r is a radial basis function. Here, ξ i ( ) are chosen as the following Gaussian functions where c i = [c i,1 , . . . , c i,q ] ∈ R q is the center of the receptive field and ι ∈ R is the width of the Gaussian functions. Assumption 3: [37] θ * and ψ are bounded as θ * ≤θ and |ψ| ≤ ψ * , respectively, whereθ > 0 and ψ * > 0 are unknown constants.

III. MAIN RESULT A. CONTROLLER DESIGN
The PEMFC air supply systems (1) can be rewritten aṡ Note that 1 and 2 are unknown nonlinear functions owing to unknown nonlinearity φ and system parameters l i , i = 1, . . . , 11. This section focuses on an adaptive controller design for the system (9) with the asymmetric time-varying output constraint (6). The adaptive controller is recursively designed via the dynamic surface design technique [31]. For the controller design procedure, we define error surfaces as follows: where z 1 and z 2 are control error surfaces, is a boundary layer error, α is the virtual control law, and x 3r is the filtered virtual control law computed from the following first-order low-pass filter with the time constant ν > 0.
Step 1: Consider the first error surface z 1 . The time derivative of z 1 along the first and second equations of (9) is given byż From the mean value theorem [39], we represent the func-tion¯ 1 as Property 1: The function g χ is unknown, its sign is positive, and there exists an unknown positive constant g χ 0 such that 0 < g χ 0 ≤ g χ .
Considering (17), an asymmetric barrier Lyapunov function (ABLF) is defined as where ϕ a = z 1 /k a , ϕ b = z 1 /k b , log denotes the natural logarithm, and p(z 1 ) is a switching constant defined in Lemma 2. Note that V z 1 is zero when z 1 is zero and goes to infinity if z 1 (t) goes to the upper time-varying constraint k b (t) or to the lower time-varying constraint −k a (t). Thus, (18) is employed to design the adaptive controller satisfying the constraint (17). For simplicity of presentation, we write p(z 1 ) as p.
The time derivative of V z 1 is given bẏ Thus,V z 1 along (16) becomeṡ

Remark 4:
For Assumptions 1-4, it is pointed out that (i) Assumption 1 is reasonable because the system parameters l i , i = 1, . . . , 11 and c cannot be exactly known in practical engineering systems and the air mass flow φ is generally adopted by fitting experimental results [40]. Thus, l i , c, and φ are assumed to be unknown.
(ii) As reported in [21] and [32], the load current δ is measurable for the regulation problem of OER, and δ and its derivatives are bounded signals because of the physical meaning of δ. Thus, Assumption 2 is a reasonable assumption.
(iii) Owing to the function approximation property of RBFNN [37], [38], the optimal weights and reconstruction error of RBFNN are bounded. In this paper, their bounds are assumed to be unknown. Thus, Assumption 3 is a general assumption in the neural network control field.
(iv) Assumption 4 means that the time derivative of g χ is bounded in the compact set D x ⊂ R 4 . Here, g χ is well-defined function, as discussed in Remark 2, and is differentiable because of x 2 > l 6 and x 3 > 0. Therefore, Assumption 4 is reasonable.
Remark 5: The design parameters of the proposed adaptive controller can be chosen via the proof of Theorem 1. The guidelines for selecting design parameters are provided as follows: 1) Increasing ζ 1 and ζ 2 and decreasing ν help to increase L, which subsequently reduces the bound 1 − e −2ḡ χ (C/L) of ψ j where j = a, b. Thus, the tracking error z 1 can be reduced. 2) Decreasing w i , κ 1 , and κ 2 helps to reduce C which leads to reduce the bound 1 − e −2ḡ χ (C/L) of ψ j . This helps to reduce the tracking error z 1 .
3) Fixing σ i as small constants and increasing γ i help to increase the tuning speed of the adaptive parameters θ 1 , θ 2 , ψ 1 , ψ 2 , l 3 , and l 11 .

IV. SIMULATION RESULTS
To illustrate the effectiveness of the proposed control scheme, we consider the PEMFC air supply system (1) whose system parameters are given in Table 1. In addition, compared with an adaptive controller designed without using ABLF, the constraint satisfaction of the proposed ABLF-based controller is shown where the adaptive controller designed without using ABLF is designed by redefining V z 1 in (18) as V z 1 = z 2 1 . We simulate two cases according to the type of the load current.
The output tracking results of the proposed ABLF-based controller and those of the controller designed without using ABLF are compared in Fig. 2. Fig. 3 shows the OER regulation results of the proposed controller and the controller designed without using ABLF. While the output response of the proposed ABLF-based control system remains within the asymmetric constraints (see Fig. 2(a)), the output response of the control system designed without using ABLF violates the asymmetric constraints in the transient response Comparison of the tracking results for case 1 (a) y , y r , and constraints of the proposed control system (b) y , y r , and constraints of the control system designed without using ABLF.
(see Fig. 2(b)). These results are also observed in Fig. 3 where the OER response of the control system designed without using ABLF gets out of the lower bound b in the transient response (see Fig. 3(b)). This means that the control system designed without using ABLF may experience oxygen starvation. Thus, both the OER regulation and the prevention of oxygen starvation and parasitic loss can be achieved with the proposed control approach. Fig. 4 displays the control input voltage u of the proposed control scheme. It is shown that the control input (i.e, motor compressor voltage) consistently changes because of the time-varying load current δ. Fig. 5 shows the RBFNN outputs and the adaptive parameters of the proposed control scheme where the RBFNN outputŝ θ 1 ξ 1 andθ 2 ξ 2 compensate for the nonlinearitiesZ 1 and Z 2 , respectively.
Case 2: We set to a = 1.6 and b = 2.5 for the constraints on y and the load current δ is defined as constant values between 200A to 310A, as illustrated in Fig. 6. The initial values of , and constraints of the control system designed without using ABLF. the state variables are chosen as x 1 (0) = 1.6 × 10 5 , x 2 (0) = 1.81 × 10 5 , and x 3 (0) = 6.7 × 10 3 . The design parameters are chosen as ζ 1 = 2, ζ 2 = 2, ν = 0.01, w 0 = w 1 = 1, κ 1 = κ 2 = 1, γ 1 = diag [15], γ 2 = 0.5, γ 3 = 10 4 , γ 4 = diag[1],   , and constraints of the control system designed without using ABLF. In Figs. 7 and 8, the output tracking and OER regulation results of the proposed ABLF-based controller and the controller designed without using ABLF are compared. Figs. 7(b) and 8(b) show that the controller designed without using ABLF cannot ensure OER constraint satisfaction while regulating λ O 2 to λ * O 2 . However, the OER response λ O 2 of the proposed control system is confined within the asymmetric constraints although the load current δ changes abruptly and the air supply system (1) includes unknown system parameters and nonlinearities. Fig. 9 shows the control input u of the proposed control system where u varies as δ varies. Fig. 10 depicts the RBFNN outputs and the adaptive parameters.

V. CONCLUSION
This paper has presented an adaptive OER control strategy to prevent oxygen starvation and parasitic loss of uncertain nonlinear air supply systems of PEMFCs. An approximation-based adaptive controller using ABLF has been designed to ensure OER regulation within OER constraints where neural network approximators and adaptation parameters were employed to compensate for system uncertainties. The concerned design problem has been the first trial in the OER control field of nonlinear air supply systems of PEMFCs. The stability and OER constraint satisfaction of the resulting closed-loop system have been analyzed via the Lyapunov stability theorem. The simulation results have demonstrated the effectiveness of the proposed control scheme against changes in the load current and system uncertainties.