Approximation-Based Adaptive Control of Constrained Uncertain Thermal Management Systems With Nonlinear Coolant Circuit Dynamics of PEMFCs

This paper addresses an adaptive temperature control problem for preventing the membrane dehydration and electrode flooding of nonlinear proton exchange membrane fuel cells (PEMFCs). Compared with the previous thermal control results of PEMFC temperature systems, the main contributions of this paper are two-fold: (i) nonlinear thermal management systems with nonlinear coolant circuit dynamics are firstly adopted in the temperature control field of PEMFCs and (ii) temperature constraints are considered to avoid the membrane dehydration and electrode flooding phenomena of PEMFCs. It is assumed that all system parameters and nonlinearities of thermal management systems including nonlinear coolant circuit dynamic are unknown. A recursive control design methodology is presented to guarantee the robust regulation and constraint satisfaction of the stack temperature. From the Lyapunov theorem, the stability of the resulting closed-loop system is analyzed.


I. INTRODUCTION
Proton exchange membrane fuel cells (PEMFCs) have been regarded as one of the most attractive alternative energy sources in the future because of their advantages such as low operating temperature, high energy efficiency, short charging time, and less noise [1], [2]. The PEMFCs are interconnected by multiple subsystems consisting of the hydrogen flow, the humidity, the air supply, and the thermal management systems. Among these subsystems, the control of thermal management system is important for the general operation of PEMFC in the electrochemical reaction. The temperature range for the general operation of the fuel cells is 50-100 • C while an optimal temperature is 80 • C [3]. Maintaining the optimal temperature against the abrupt change of the external load leads to improve the performance of thermal management systems and to increase the lifetime of fuel cells [4]. Thus, the control problem of thermal management systems The associate editor coordinating the review of this manuscript and approving it for publication was Lei Wang. has actively appeared. The optimal temperature control problem using the relative humidity was addressed for PEMFCs [5]. In [6], an active disturbance rejection control design was developed for achieving precise temperature regulation of thermal management systems in PEMFCs. In [7], an adaptive thermal control method was presented to control the stack temperature in a certain range. Recently, a fault-tolerant control approach using the sliding mode technique was presented for thermal management systems of PEMFCs with sensor faults [8]. However, these control strategies [5]- [8] were established without the consideration of coolant circuit models that are important for adjusting the stack temperature of PEMFCs.
Basically, PEMFCs provide electricity by an electrochemical exothermic reaction using the oxygen and hydrogen, and the heat generated at this time should be removed by a cooling system [9]. Therefore, the thermal management considering the coolant circuit model is essential for the optimization of stack performance and contributes to the PEMFC technology that is reliable for more practical applications [10]. Despite VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ this importance, some limited results have been reported for the temperature control in the presence of the coolant circuit model of PEMFCs. A proportional-integral temperature control design based on a thermal circuit was presented in [11]. In [12], a proportional-integral-derivative controller was used to validate the experimental data of water-cooled PEMFCs. A linear-quadratic-regulator-based control scheme for the minimization of the parasitic power of automotive fuel cell cooling systems was introduced in [13]. In [14], a model reference adaptive control problem was investigated to deal with system uncertainties and to control the stack and coolant inlet temperature in PEMFCs. In [15], the modular thermal modelling and model predictive control methods of water-cooled PEMFC systems were presented. However, the existing control strategies [11]- [15] are based on the linearized model of PEMFC systems and thus are only reliable in the neighborhood of the specific operating point. For more practical applications, there have been some attempts to develop thermal controllers for nonlinear PEMFC systems. The fuzzy-based PEMFC temperature and circulating coolant inlet temperature were controlled by adjusting the coolant flux and bypass valve [16]. In [17], a sliding mode control design using an extended Kalman filter was studied to regulate the temperature of PEMFCs. Despite these efforts, the aforementioned results [11]- [17] have two limitations as follows.
(L1) The existing results [11]- [17] did not consider the dynamics of the coolant pump, namely the coolant flux effects were only considered in thermal management Systems. In [16], the coolant pump model reduced to the steady state input-output representation was only considered and the stability of the closed-loop systems was not proved theoretically. Because the coolant flux for controlling the stack temperature can be adjusted by the dynamics of the coolant pump [18], the nonlinear dynamics of the coolant pump should be considered for the temperature control of PEMFCs.
(L2) The previous works [11]- [17] cannot deal with the temperature constraint problem to prevent the membrane dehydration and electrode flooding phenomena. The high stack temperature of the fuel cells may interrupt the transport effects of the reactants and cause the membrane dehydration that deteriorates the cell performance. In addition, the low stack temperature decreases the electrochemical reaction rate and may lead to the water condensation and the electrode flooding that degrade the performance of PEMFC systems [1], [9]. Thus, the total stack temperature should remain within some reasonable ranges to avoid the membrane dehydration and electrode flooding phenomena while controlling the stack temperature of PEMFCs.
Motivated by these limitations, we present an approximation-based temperature control design to deal with the membrane dehydration and electrode flooding problems of uncertain nonlinear PEMFCs. In thermal management systems, a nonlinear coolant circuit dynamics is combined with the nonlinear dynamics of the total stack temperature where temperature constraints are considered. All system parameters and nonlinearities of the thermal management systems are assumed to be unknown. An approximationbased adaptive temperature control scheme is designed by employing the barrier Lyapunov function technique [19] and the dynamic surface design technique [20]. Through the Lyapunov stability analysis, it is shown that temperature constraints are satisfied to avoid the membrane dehydration and electrode flooding and the robust regulation is achieved against unknown system parameters and nonlinearities.
The contributions of this paper are two-fold: (i) To the best of our knowledge, there are no temperature control studies for dealing with the nonlinear coolant circuit model in the nonlinear thermal model of PEMFCs although the dynamic property of the coolant pump influences the cooling stack highly. Hence, compared with the existing works [5]- [8], [11]- [17], this paper firstly considers the coolant-circuit-based uncertain nonlinear thermal management systems in the temperature control field of PEMFCs where all system parameters and nonlinearities are unknown.
(ii) Compared with the existing control designs [5]- [8], [11]- [17], this paper addresses the membrane dehydration and electrode flooding prevention problem in the temperature control of PEMFCs. Thus, the constraints of the total stack temperature are combined with the thermal control problem of PEMFCs and an adaptive control methodology is developed to ensure the stability of the closed-loop system and the constraint satisfaction of the stack temperature.
The rest of this paper is organized as follows. The stack temperature and coolant circuit models of PEMFCs are introduced in Section II. In Section III, the constrained temperature control problem is formulated for the thermal management systems with the nonlinear coolant circuit dynamics. In Section IV, an approximation-based adaptive control design is presented using the Lyapunov stability analysis. The simulation result of the resulting control system is provided in Section V. Section VI gives the conclusion of this paper.

II. THERMAL CHARACTERISTICS OF PEMFC A. THERMAL MANAGEMENT MODEL OF NONLINEAR PEMFC
The thermal management model is established using molar conservation principles, the energy balance, and empirical equations [6], [16]. The dynamics of the total stack temperature T st is defined as where W in and W out are the input and output of the gas energy flow rate, respectively, W rea is the total power from the electrochemical reaction, W wc denotes the rate of the heat removal, W amb is the rate of heat loss at the stack surface, P fc is the output power of the PEMFC, m st is the mass of the PEMFC stack, and c p.s is the specific heat of the PEMFC. Each variables in (1) are defined as follows.
where λ H 2 , λ O 2 , P a , and P c are defined in Table 1, the saturation pressure function P sat (x) is given by [3] log (ii) Definition of W out : The output gas energy flow rate W out considering the water generated in the liquid state is represented by (iii) Definitions of W rea , W wc , and W amb : The total fuel energy W rea is derived from the electrochemical reaction as follows: where H is the hydrogen combustion enthalpy constant and Q rea a.H 2 is the reacted hydrogen molar flow rate defined in (3).
Since the output coolant temperature is the same as the temperature of the stack (i.e, T out wc = T st ) [6], the rate of heat removal by the coolant W wc is obtained as where k cl is a physical parameter,W cl is the coolant flux, and c l p.H 2 O and T in wc are defined in Table 1. The heat loss rate W amb at the stack surface is expressed as where R t and T amb are the thermal resistance and ambient temperatures, respectively and their values are defined in Table 1.
(iv) Definition of P fc : The output power of entire PEMFC is expressed as where n is number of the cells and I st > 0 is the load current of the fuel cell. Here, the operating voltage V c of the fuel cell is defined by combining all voltage drops associated with the activation loss and ohmic loss as follows [21], [22]: where the open circuit voltage E is defined as with the current density i = I st /A c defined by the PEMFC current I st and the active area A c , the activation overvoltages V act is given by [22], and the ohmic overvoltages V ohm is defined as V ohm = iR ohm . Here, the values of V 0 , V a , and their coefficients can be derived from the empirical process [22] and the ohmic resistance R ohm = d m /σ m is proportional to the membrane thickness d m and inversely proportional to the membrane conductivity σ m = (5.139 × 10 −3 λ m − 3.26 × 10 −3 )e (1.155−350/T st ) with the membrane water content λ m = 14 [3]. Using Definitions (i)-(iv), the thermal management system (1) can be represented by where a, φ 1 (T st , I st ), and φ 2 (T st ) are defined as with and their physical parameters are given in Table 1.

B. NONLINEAR COOLANT CIRCUIT DYNAMICS WITH FLUX AND PUMP MODELS
In [18], the dynamic model of the coolant circuit with the 36V motor centrifugal pump was derived based on the fundamental relationships among the motor-armature current, motor speed and coolant flow rate. In this model, the coolant flux can be manipulated using a variable speed pump without a control valve. Thus, the weight and complexity of the system can be reduced [18]. The dynamics of the coolant where W cl is the coolant flux, ω r is the angular velocity of the coolant pump motor, I m is the input motor-armature current of the coolant pump, the constants J w , J com , τ f , and C are given in Table 1, and the coefficients c 1 , . . . , c 7 are given in Table 2.

III. PROBLEM FORMULATION A. THERMAL MANAGEMENT SYSTEMS WITH NONLINEAR COOLANT CIRCUIT DYNAMICS
Let us define the variables Then, the thermal management system with a 75-kW fuel cell stack and a 36V motor centrifugal coolant pump (i.e., (10) and (12)) can be rewritten by the following state-space model The system parameters a, b and the functions φ 1 , φ 2 , φ 3 , and φ 4 are unknown.
Lemma 2 [24]: For the interval −k c < z < k c with any z ∈ R and k c ∈ R, it holds that

B. CONSTRAINED TEMPERATURE CONTROL PROBLEM
The temperature management has been recognized as one of significant technical challenges of PEMFCs. The high cell temperature causes membrane dehydration because of the insufficient water supply to PEMFC and the low cell temperature may lead to electrode flooding caused by water condensation, consequently to hinder reactant mass transport with a resultant voltage loss [1], [9]. Based on [3], the optimal value of the total stack temperature is y r = 353K . Thus, it is important to consider the regulation problem of thermal management control systems with temperature constraints. To this end, the constraints of the total stack temperature y are considered as where the constant k c denotes the physical temperature constraint to prevent the membrane dehydration and the electrode flooding of PEMFCs. If the initial stack temperature y(0) does not remain within the constraints, it means that PEMFCs are under the membrane dehydration and electrode flooding phenomena at the initial time. This is not reasonable for the stable operation of PEMFCs. Thus, it is assumed that the initial stack temperature y(0) satisfies the constraints (14). Property 1: For system (13), there exists an unknown positive constant φ 2 such that 0 < φ 2 ≤ φ 2 .
Proof: The function φ 2 (x 1 ) is defined as Table 1, it holds that m st c p.s > 0, c l p.H 2 O > 0, k cl > 0, and T in wc > 0. Additionally, since the output coolant temperature is the same as the temperature of the stack and is larger than the inlet chilling coolant temperature T in wc [3], x 1 > T in wc is ensured. Therefore, Property 1 is satisfied.
Problem 1: Consider the uncertain thermal management system (13) with temperature constraints (14) of the PEMFC. The main control problem is to find approximation-based adaptive control law u so that the system output y follows the optimal value y r within the constraints (14).
Remark 1: Contrary to the previous temperature control methods for thermal management systems of PEMFCs [5]- [8], [11]- [17], the nonlinear coolant circuit dynamics (12) with flux and pump models is firstly considered with the nonlinear stack temperature dynamics (10) for the temperature control problem of PEMFCs. Furthermore, the temperature constraint problem is addressed to prevent the membrane dehydration and electrode flooding of PEMFCs. Therefore, a solution on Problem 1 cannot be suggested in the previous works [5]- [8], [11]- [17].

IV. ADAPTIVE TEMPERATURE CONTROL IN THE PRESENCE OF NONLINEAR COOLANT CIRCUIT DYNAMICS A. RADIAL BASIS FUNCTION NEURAL NETWORKS
For the online approximation of unknown nonlinear functions W i , i = 1, 2, 3, to be defined in the controller design, radial basis function neural networks (RBFNNs) are used. Using the universal approximation property of the RBFNN [25], [26], for continuous real-valued function W i ( i ) : D i → R with a compact set D i ⊂ R q i , there exists the ideal weight vector θ * i with a sufficiently large q i such that where i = 1, 2, 3, i = [ i,1 , . . . , i,q i ] ∈ D i and ψ i are the input vector and the network reconstruction error, VOLUME 8, 2020 respectively, θ * i ∈ R r i is the optimal weighting vector defined as θ . . , r i denotes the Gaussian function with the center of the receptive field c i,j = [c i,j,1 , . . . , c i,j,q i ] ∈ R q i and the width ι i,j ∈ R.

B. DESIGN OF ADAPTIVE CONTROLLER
In this section, an approximation-based adaptive controller design strategy is presented for system (13) with the output constraint (14). For the dynamic surface design [20], the error surfaces and the boundary layer errors are defined as where z 1 , z 2 and z 3 are control error surfaces, 1 and 2 are boundary layer errors, α 1 and α 2 are the virtual control laws, and α 1f and α 2f are the signals derived from the first-order filters with the time constants ν 1 , ν 2 > 0 as follows: The recursive control design consists of three steps. In the first step, the adaptive virtual control law α 1 is designed to stabilize the dynamics of the first error surface z 1 for the regulation of the stack temperature x 1 while the stack temperature constraints (14) are satisfied. To this end, the Lyapunov stability analysis strategy using a barrier function is established. In the second step, the adaptive virtual control law α 2 is designed to stabilize the dynamics of the second error surface z 2 based on the flux dynamics of the nonlinear coolant circuit (12). In the third step, the adaptive actual control law u denoting the input motor-armature current of the coolant pump is designed to stabilize the dynamics of the third error surface z 3 . For the stable control design, the Lyapunov stability theorem [27] is used in these design steps. In addition, the first-order low-pass filters (17) based on the dynamic surface design technique are employed to avoid the calculation of the time derivative of the virtual control laws in the recursive design.
Step 1: The time derivative of z 1 along the first equation of (13) is given bẏ The output constraint problem (14) can be redefined as the constraint problem of z 1 as follows: From (19), we consider the following barrier Lyapunov function where log denotes the natural logarithm.
Substituting (49)-(52) into (48), we geṫ Remark 4: In the third step, the actual control law (49) with adaptive laws (50)-(52) is designed using the Lyapunov stability theorem. In the Lyapunov-based control design, the adaptive laws (41) and (42) are derived to tune the weighting vector of the neural-network-based function approximator θ 3 ξ 3 and to compensate for the unknown reconstruction error ψ 3 , respectively. Furthermore, the adaptive law (52) is designed to compensate for the unknown parameterb.
Remark 5: The previous temperature control results [11]- [17] for PEMFC thermal management systems did not consider the dynamic property of the coolant pump and the temperature constraint problem to prevent the membrane dehydration and electrode flooding. However, we consider the nonlinear dynamics of the coolant circuit (12) with the non-affine and affine nonlinearities φ 3 and φ 4 in the constrained thermal management systems (13). By deriving two physical properties (i.e., Properties 1 and 2) for the recursive control design, we construct the approximationbased adaptive temperature control scheme (i.e., (28)-(31), (40)-(42) and (49)-(52)), as shown in Fig. 1. Moreover, all system parameters and nonlinearities can be compensated by the proposed adaptive approximation control scheme, compared to [11]- [17].

C. STABILITY ANALYSIS
This section focuses on the stability analysis of the proposed control system. The dynamics of the boundary layer errors are follow aṡ and Consider the following total Lyapunov function V Remark 6: The total Lyapunov function V in (55) consists of the Lyapunov functions V 1 , V 2 , and V 3 used in the design steps and the boundary layer errors 1 and 2 for the firstorder filtering of the virtual control laws. The function V is Theorem 1: Consider the uncertain nonlinear thermal management system (13) with the temperature constraint (14). For any initial conditions satisfying V (0) ≤ l with a constant l > 0 and y r − k c < y(0) < y r + k c , the proposed adaptive control scheme consisting of (28)-(31), (40)-(42) and (49)-(52) ensures that all the signals in the closed-loop system are uniformly ultimately bounded and the control error converges to a neighborhood of the origin while the stack temperature remains within constraints.
Remark 7: Based on the proof of Theorem 1, some guidelines for the choice of the design parameters are given as follows.
2) As κ i decreases, C is reduced. Subsequently, the bound 1 − e −2φ 2 (C/K ) of ϕ c can be reduced, namely, the control error z 1 is reduced.
The adaptive regulation results of the stack temperature to y r = 353 are compared in Fig. 3. While the temperature response using the controller [6] violates the temperature constraint according to the change of the load current, the output  response of the proposed adaptive thermal control system is ensured within stack temperature constraints. The coolant flux and motor speed of coolant circuit of the proposed adaptive control system are depicted in Fig. 4. The pump motor current for the control input of the proposed control system is displayed in Fig. 5. Fig. 6 shows the outputs of the RBFNNs and the parameter estimates used in the proposed control system. In these figures, we can see that the proposed adaptive control has good regulation performance while the stack temperature remains within constraints for preventing the membrane dehydration and electrode flooding of nonlinear PEMFCs. Furthermore, the control result reveals that the dynamic load current and the parametric and nonparametric uncertainties can be overcome by establishing the approximation-based adaptive control strategy although the uncertain nonlinear coolant circuit dynamics is considered in the thermal management systems. . RBFNN outputs and adaptive parameters of the proposed control system (a)θ 1 ξ 1 ,θ 2 ξ 2 , andθ 3 ξ 3 (b)ψ 1 ,ψ 2 , andψ 3 (c)â andb.

VI. CONCLUSION
The paper has established the adaptive temperature control strategy for avoiding the membrane dehydration and electrode flooding of uncertain thermal management systems with nonlinear coolant circuit dynamics of nonlinear PEMFCs. The coolant circuit model including the nonlinear coolant flux and pump dynamics has been firstly combined with the nonlinear stack temperature dynamics in temperature control field. Then, an adaptive temperature control scheme has been constructed to guarantee the robust temperature regulation within the stack temperature constraints even though all system parameters and nonlinearities are unknown and the external load changes suddenly. A Lyapunov-based analysis method has been derived to prove the convergence of the control error while all closed-loop signals remain bounded.