# Temperature Control for a Proton-Exchange Membrane Fuel Cell System with Unknown Dynamic Compensations

Authors: Yashan Xing; Ramon Costa-Castelló; Jing Na
Journal: Complexity
 (2020)

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## Abstract

Numerous control strategies of temperature regulation have been carried out for proton-exchange membrane fuel cell systems including a cooling fan in order to ensure operation at the desired condition and extend the lifetime of the fuel cell stack. However, most existing control strategies are developed without considering the efficiency limitation of the cooling system such that the cooling fan may be unable to eliminate the additional heat. Moreover, there are unknown modelling errors, external disturbance and noise during modelling and experiment processes for fuel cells. Due to those unknown dynamics, the conventional control strategies may fail to achieve the expectant results. To address this issue, an alternative control strategy is proposed in this paper, which consists of a composite proportional-integral (PI) controller with an unknown system dynamics estimator. First, the control strategy is developed by reducing the temperature of input air through the humidifier and simultaneously increasing the mass flow of air in order to eliminate the excess heat that a cooling fan cannot remove. Moreover, an unknown system dynamics estimator is proposed in order to compensate the effect of the unknown dynamics. The construction of the estimator is designed through finding an invariant manifold which implies the relation between known variables and the unknown manifold. The invariant manifold is derived by applying a simple low-pass filter to the system which is beneficial to avoid the requirement of the unmeasurable state derivative. Furthermore, the proposed estimator is easily merged into the proposed PI control strategy and ensures the exponential convergence of estimated errors. Besides, the estimator is further modified such that the derivative of the desired temperature is not required in the controller. Finally, numerical simulations of the PEMFC system are provided and the results illustrate the efficacy of the proposed control strategy.


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## Body

## 1. Introduction

Proton-exchange membrane fuel cell (PEMFC) is regarded as one of the cleanest alternative energy conversions which is promising to replace traditional power generation technologies [1]. The energy conversion principle of PEMFC is to transform chemical energy into electricity, thermal energy, and water without requiring any moving part. Due to its properties of low operation temperature and fast start-up compared with other types of fuel cells, it has drawn wide attention in the fields of power station systems and vehicle systems [2, 3]. However, there are numerous stumbling blocks for widely commercial applications [4, 5].PEMFC involves a multiphysics coupling process which leads to some difficulties for control strategy design and implementation. Among those, temperature regulation of PEMFC stack is essential to ensure the high operation efficiency, avoid the thermal fatigue, and prolong the stack lifespan [6, 7]. Several control strategies regarding the temperature regulation have been carried out in recent years. In [8], an extremum seeking control with a proportional-integral (PI) controller was proposed in order to achieve the optimal temperature regulation. In [9], the temperature control strategy was proposed for a PEMFC system with a water cooling system, where the voltage of a water circulation pump was manipulated through a PI controller, and a water cooling fan was under the on/off modes. In [10], a more advanced fuzzy logic controller was applied to regulate the speed of the cooling fan and increase the convection ratio between PEMFC and atmosphere. However, the proposed control strategy is validated for a low-power PEMFC. Thus, the limitation of cooling capacity is not considered in those aforementioned control strategies [8–10]. When the generated heat from PEMFC is higher than the dissipated heat from the cooling system, the temperature of PEMFC will keep increasing. Recently, Chen et al. [11] considered the limited capacity of a water cooling system and current constraint in the PEMFC system and proposed three temperature control strategies, which developed a constrained model predictive control (MPC) strategy, and designed duty ratio split controller and current governor both based on the proportiona-integral-derivative (PID) control in order to implement the trade-off between cooling capacity and power demand.On the other hand, most control strategies mainly depend on a reliable mathematical model while more accurate models with multiple dimensions can introduce mathematical complexities. Due to those complexities, many control design methodologies cannot be applied. Moreover, the control-oriented model contains unmodelled dynamics and modelling errors due to model simplification process. Besides, external disturbances and noise from the experiment cannot be avoided. Those unknown dynamics may result in unexpected control results and even cause system unstable. Thus, it is significant to develop an estimator or an observer to eliminate the unknown disturbance and enhance the controller performance.In this line, the disturbance observer (DOB) was first proposed to address the unknown dynamics [12, 13]. Furthermore, the modified DOB for nonlinear system was designed in [14]. Moreover, the extended state observer (ESO) [15, 16] as an alternative method was developed, where the extended states of the unknown dynamics and the initial states were observed simultaneously. In [17], the ESO incorporated into a sliding mode control was used to achieve the temperature regulation in PEMFC. More recently, a modified ESO based on the Kalman-Bucy filter was incorporated into PID control to implement the temperature management for PEMFC in [18]. Besides, a new framework of unknown system dynamics estimator (USDE) [19] was proposed to deal with the unknown dynamics. It has been incorporated in many control strategies and applied in many industrial fields, such as hydraulic servo systems [20], nonlinear robotic systems [21], and spark ignition engines [22]. However, designing a proper control strategy with consideration of unknown dynamics remains as an open and interesting problem for PEMFC.This paper aims to propose an alternative control strategy of temperature regulation for a PEMFC system with consideration of the cooling fan limitation. Moreover, the unknown dynamic compensation is developed in order to enhance the performance for the proposed controller. To be specific, we first consider the limitation of the fan efficiency, where the maximum energy generated from a cooling fan is not enough to force the stack temperature into the desired value. Due to this limitation, the control strategy is proposed through increasing the mass flow of the input air and reducing the temperature of the humidifier to implement PEMFC temperature management. By considering the practical application, this control strategy is developed based on the PI controller. Furthermore, an unknown system dynamics estimator (USDE) is used in order to compensate the effect of the unknown dynamics. The construction of this estimator is through generating an invariant manifold in order to ensure the exponential convergence of the estimation errors, where the invariant manifold provides the information between known variables and the unknown dynamics. It is noticed that the derived estimator is easily merged into the proposed PI control strategy without introducing any complexity. Since the derivative of the desired temperature is not measurable, the estimator is further modified such that the derivative of the desired temperature is not required in the proposed controller. To summarize, the main contributions of the proposed control strategy are as follows:(1)
The proposed estimator for unknown dynamics is simple and easy to incorporate into the controller through finding an invariant manifold. It is significant for practical application that the construction of the invariant manifold is based on a simple low-pass filter operation without introducing the unmeasurable state derivative. Moreover, only one parameter of the bandwidth for the low-pass filter is required to be tuned.(2)
With consideration of the limited operation property of the cooling fan, the control strategy is proposed through increasing air mass flow and reducing the temperature of input air such that the voltage change is affected less and the desired temperature can be guaranteed simultaneously.The rest of the paper is organised as follows. In Section2, the model description of the PEMFC system and the relevant problem formulation are presented. Section 3 and Section 4 illustrate the control strategy for the PEMFC system with respect to the USDE and the composite PI controller. The numerical results of the proposed controller for the PEMFC system are provided in Section 5. Finally, the conclusions are given in Section 6.


## 2. Model Description of PEMFC System

### 2.1. Model Description

In this section, a PEMFC system is first introduced, and its schematic diagram is depicted in Figure1. The PEMFC system mainly consists of two humidifiers, two line heaters, a cooling fan, and a fuel cell stack. To begin with, the fuel cell stack is a low-temperature PEMFC with 8 cells with an active area of 50 cm2. The combination of anode, cathode, and electrolyte layer constitutes the core of PEMFC in the sandwich structure, as depicted in Figure 2. With respect to gas transport processes in the PEMFC system, the compressed air is first fed into the cathode channel through a humidifier and a line heater. Due to the material property of electrolyte, the PEMFC stack requires enough humidity for the electrolyte layer. Before entering into the fuel cell stack, the supplied gases need to pass through the humidifier and the line heater in order to guarantee the required humidity and maintain the gas temperature in the pipelines simultaneously. In the anode channel, the pure hydrogen is provided by a hydrogen tank and goes through another humidifier and another line heater such that the hydrogen is injected with some required water vapor. Furthermore, the hydrogen ions can pass through the electrolyte to cathode layer and react with oxygen. During this process, there is some generated water which can affect the humidity in each channel. When the relative humidity of each channel exceeds 100%, the vapor condenses into the liquid form. In this paper, we assume the liquid water cannot leave the stack until it evaporates into a gas form. Finally, the unreacted gases will be exhausted to atmosphere, and the output water vapor will be cooled down and stored in the outside vessel.Figure 1
Schematic diagram of a PEMFC stack and the PEMFC system.Figure 2
The operation principle of the PEMFC stack.For the cooling system, there is a fan (Fan 1 in Figure1) which works for regulating the PEMFC temperature. In practice, this fan is not efficient enough to regulate the PEMFC temperature when PEMFC operates at the high power condition. Moreover, the humidifier contains internal resistances to heat the injected liquid water and the input gases, while the cooling system is not included in the internal structure of the humidifier. Due to the constraint of this structure, the cooling dynamics of the humidifier is extremely slow. The outside fan (fan 2 in Figure 2) is used to help reduce the humidifier temperature at the cathode side. Therefore, the relevant modelling of the humidifier, the line heater, and PEMFC stack including cooling process of the fan will be presented in the following.


#### 2.1.1. Auxiliary Components

The humidifier is used to ensure the desired relative humidity of input gases. Its main process is that the water in the vapor form is injected into input gases. Thus, the total mass flow of output gases from the humidifier is expressed as(1)m˙hum,jout=m˙hum,jin+m˙H2O,jinj,where j represents the anode or cathode channel in PEMFC and m˙hum,jin is the input gas flow for the humidifier at the side j. In order to achieve the desired relative humidity, the required vapor water injected into the input gases m˙H2O,jinj can be computed as(2)m˙H2O,jinj=MH2Oϕhum,jm˙hum,joutPsatThum,jM¯jPhum,jin,where ϕhum,j is the desired relative humidity of output gases; MH2O is the molar mass of water; The average molar mass of gases for the channel j is denoted as M¯j; Phum,jin represents the pressure of input gases. And PsatThum,j is the pressure of vapor saturation at the humidifier temperature Thum,j. Moreover, the saturation pressure Psatrelated to the relevant temperature T0 is approximately calculated as [23](3)log10Psat=−1.69×10−10T04+3.85×10−7T03−3.39×10−4T02+0.143T0−20.92.The output gases from the humidifier enter into PEMFC stack through a pipeline. The temperature of this pipeline has some effect on the gas pressure and relative humidity. In order to avoid vapor condensation and maintain desired relative humidify of input gases, the line heater is installed on the pipe. Based on Dalton’s law, the partial pressure of vapor water at the line heater is calculated as(4)PH2O,jlh=Tlh,jThum,jPH2O,jinj,where Tlh,j is the temperature of the line heater. The partial pressure of injected vapor is denoted as PH2O,jinj=ϕhum,jPsatThum,j. Moreover, the relative humidity of gases at the line heater is expressed as(5)ϕlh,j=PH2O,jlhϕlh,jPsatTlh,j,where Psatis the saturation pressure of vapor at the line heater temperature Tlh,j, which can be calculated by (3).


#### 2.1.2. PEMFC Stack

For control design purpose, we assume that the temperature of gases in the stack is the same as the electrolyte structure. The temperature variation along the structure dimension is not considered in this paper. Based on the energy conservation principle, the thermal energy balance of PEMFC is built as follows:(6)mfcCp,fcdTfcdt=Hcain+Hanin−Hcaout−Hanout+Hr−VfcIfc−Hfan+d,where mfc and Tfc are the mass and the temperature of PEMFC, respectively. The specific heat capacity of PEMFC is denoted as Cp,fc; Vfc is the output voltage of PEMFC; d is regarded as the unknown disturbance. The input energy values of cathode Hcain and anode layers Hanin generated by the input gases calculated by(7)Hcain=∑cawiinMi∫TrefTlh,caCp,iTdT,Hanin=∑anwiinMi∫TrefTlh,anCp,iTdT,where i denotes as each species of gases in the cathode or anode layer and Mi is the molar mass of the gas i. The mass flow rate of gas i is represented as wiin; Tref is the reference temperature. The specific heat capacity of gas i is denoted as Cp,i. It can be computed as Cp,iT=c0,i+c1,iT+c2,iT2+c3,iT3, where the heat capacity coefficients c0,i, c1,i, c2,i, and c3,i can be found in [24].Furthermore, the output energy of cathodeHcaout, anode Hanout, and the generated energy Hr from chemical reaction are expressed as [25](8)Hcaout=∑cawioutMi∫TrefTfcCp.iTdT,Hanout=∑anwioutMi∫TrefTfcCp.iTdT,Hr=−wH2rMH2ΔHro,where wiout represents the mass flow rate of gas i. The mass flow rate of reacted hydrogen is denoted as wH2r; MH2 is the molar mass of hydrogen. And the specific heat of chemical reaction is represented as ΔHro.There is a fan (Fan 1 in Figure1) close to the PEMFC stack which is used to regulate the stack temperature. The operation principle is that when the stack temperature is higher than the desired temperature, the fan turns on in order to reduce the stack temperature. Otherwise, it turns off. This procedure is expressed as the following mathematical expression:(9)Hfan=ClossAfcTfc−Tref,Tfc>Tset,0,Tfc≤Tset,where Afc is the surface area of PEMFC; Closs is the heat transfer constant; and Tset is the desired temperature of the stack.In the PEMFC, the potential difference is generated by ion flow transports from the electrochemical reaction. The theoretical voltage of PEMFC can change reversibly, which is affected by the variation of partial pressures for reactants and products. Thus, the theoretical voltage at the nonstandard condition can be determined by the Nernst equation, that is,(10)VN=VN0+Δs2FTfc−Tref+RTfc2FlnPH2+12lnPO2,where VN0 is the theoretical voltage at the standard condition; PH2 and PO2 are partial pressures of hydrogen and oxygen, respectively; Δs is the correction coefficient related to the entropy of chemical reaction; F is Faraday’s constant; and R denotes the gas constant.Moreover, there are three types of voltage losses which can affect the voltage variation irreversibly during operation processes. Considering those voltage losses, the output voltage of PEMFC is expressed by [23](11)Vfc=n⋅VN−Vact−Vohm−Vcon,where n is the number of cells for PEMFC; Vact is the activation losses; Vohm denotes the ohmic losses; and Vcon represents the concentration losses. They are calculated as follows:(12)Vact=Vact,0+Kact1−e−c1i,(13)Vohm=δmIfcηmAfc,(14)Vcon=Ifcc2IfcImaxc3,where Vact,0 is the initial activation voltage drop; The membrane thickness is denoted as Δm; Kact, c1, c2, and c3 are the empirical constants, which depend on gas partial pressure and temperature; Imax is the maximum current; and ηm denotes the membrane conductivity, which is related to the membrane water content λm and stack temperature. It can be calculated as(15)ηm=b1eb21/T0−1/Tfc,where T0=303 K is the reference temperature for membrane conductivity test. The empirical value related to the membrane water content is represented as b1=b11λm−b12 and constants b11, b12, and b2 are determined empirically. Those empirical constants have been discussed in [23].For the mass flow transports, the detailed model can be found in [23]. The reacted mass flows of hydrogen and oxygen and the generated mass flow of water are calculated based on the electrochemical relationship:(16)wH2r=NMH2Ifc2F,wO2r=NMO2Ifc4F,wH2Og=NMH2OIfc2F,where MO2 and MH2O are the molar mass of oxygen and water. It is noticed that the reacted mass flows are directly related to the current Ifc. Moreover, the input mass flows are computed as(17)wH2in=KH2wH2r,wO2in=KO2wO2r,where KH2 and KO2 are the stoichiometric ratios. Thus, the input mass flows can be determined by the desired stoichiometric ratio and the current.


### 2.2. Problem Formulation

For the PEMFC system shown in Figure1, it has been well recognized that the fan 1 can be used to regulate the temperature of the fuel cell stack. However, the fan 1 has not enough power to regulate the fuel cell temperature when it operates at relative high power. Without introducing extra cooling systems, reducing the temperature and increasing mass flows of input gases are possible ways to achieve the PEMFC temperature regulation. Besides, some parameters in the mathematical model are empirical values which can lead to some modelling errors. Furthermore, there are unmodelled dynamics derived from the model simplification during the modelling procedure and external disturbances resulted from the experiment measurement. However, ignoring those unknown dynamics and modelling errors may cause the system performance degradation or even instability when the controllers apply to the PEMFC system. Thus, this paper will develop a novel control strategy in order to address both temperature regulation problem and the unknown dynamic compensations.In the mathematical model of the PEMFC system, there are many empirically determined variables. The value of those variables may change during the fuel cell operation such that the real performance cannot be precisely presented in the mathematical model. One of the most important variables for the thermal energy balance model is the empirical specific heat of chemical reactionΔHro, which affects the generated energy Hr in the PEMFC. Moreover, there are some noises or disturbances d from the experiment. Due to those facts, we consider the reacted heat Hr and disturbances d as the unknown dynamics D1, which can be expressed as follows:(18)D1=Hr+d.From the thermal energy balance of PEMFC in (6), reducing the input energy and increasing the output energy can further achieve the temperature regulation when the energy of the cooling fan reaches the maximum. Those energies are mainly related to the input mass flow rates and the temperature of input flows, which can be adjusted by the humidifier and the mass flow controller. Without the internal cooling system, the dynamics of reducing the humidifier temperature is extremely slow. The fan 2 is used to improve the cooling process of the humidifier. However, the cooling performance of the humidifier with the fan 2 is still slow. Due to the slow dynamics, increasing the input mass flow is considered to extract the excess heat from the PEMFC stack. Furthermore, considering the economic costs, the input mass flow and the input temperature of air are chosen to regulate the temperature of PEMFC in this paper. For the control design purpose, the total extract heat by employing the additional mass flow of air is calculated as follows:(19)Qloss=−ΔwairinThum,caset−Tfc,where Δwairin is the additional mass flow of air and Thum,caset is an ideal operation point for the temperature of cathode humidifier. Therefore, the extracted heat Qloss is considered as the control input. By the simple algebraic calculus, the total mass flow of the air is the sum of the initial mass flow and the additional mass flow, which is the real control input in practice. Thus, the thermal energy balance of PEMFC with the control input u=Qloss is expressed as(20)mfcCp,fcdTfcdt=Hca,0in+Hanin−Hca,0out−Hanout+Hr−VfcIfc−Hfan+d+u,where Hca,0in and Hca,0out are the initial heat of input and output energy for the cathode layer. Furthermore, (20) is rewritten as follows for analysis purpose:(21)dTfcdt=1mfcCp,fcu+D1+F,where F=Hca,0in+Hanin−Hca,0out−Hanout−VfcIfc−Hfan is the known nonlinear function.To summarize, this paper aims to propose an alternative control strategy for a high-power PEMFC, where the cooling system cannot regulate the PEMFC temperature. The control strategy is available when the cooling fan turns on, which consists of a composite PI controller with unknown system dynamic estimator (USDE) in order to achieve the temperature regulation and eliminate the effect of the unknown dynamicsD1 through the measurable output Tfc and the known functions F. The schematic block of the proposed control strategy is shown in Figure 3.Figure 3
Schematic block of the composite PID controller.Before we illustrate the control strategy with the USDE, the following assumptions are given with respect to the thermal energy balance of PEMFC (21).Assumption 1.
The temperatureTfc and the control input u are measurable and bounded. And the function F is computable and bounded.Assumption 2.
The unknown dynamics ofD1 is continuous and its derivative is bounded such that D˙1≤η.Remark 1.
It is easy to measure the variables of stack temperatureTfc, the current Ifc, the voltage Vfc, and the input mass flows for each channel. Thus, the PEMFC can fulfill the aforementioned Assumption 1 trivially. Besides, the derivative of unknown dynamics D˙1 in Assumption 2 is not required to be known, which is used for the stability analysis of the proposed controller.


## 2.1. Model Description

In this section, a PEMFC system is first introduced, and its schematic diagram is depicted in Figure1. The PEMFC system mainly consists of two humidifiers, two line heaters, a cooling fan, and a fuel cell stack. To begin with, the fuel cell stack is a low-temperature PEMFC with 8 cells with an active area of 50 cm2. The combination of anode, cathode, and electrolyte layer constitutes the core of PEMFC in the sandwich structure, as depicted in Figure 2. With respect to gas transport processes in the PEMFC system, the compressed air is first fed into the cathode channel through a humidifier and a line heater. Due to the material property of electrolyte, the PEMFC stack requires enough humidity for the electrolyte layer. Before entering into the fuel cell stack, the supplied gases need to pass through the humidifier and the line heater in order to guarantee the required humidity and maintain the gas temperature in the pipelines simultaneously. In the anode channel, the pure hydrogen is provided by a hydrogen tank and goes through another humidifier and another line heater such that the hydrogen is injected with some required water vapor. Furthermore, the hydrogen ions can pass through the electrolyte to cathode layer and react with oxygen. During this process, there is some generated water which can affect the humidity in each channel. When the relative humidity of each channel exceeds 100%, the vapor condenses into the liquid form. In this paper, we assume the liquid water cannot leave the stack until it evaporates into a gas form. Finally, the unreacted gases will be exhausted to atmosphere, and the output water vapor will be cooled down and stored in the outside vessel.Figure 1
Schematic diagram of a PEMFC stack and the PEMFC system.Figure 2
The operation principle of the PEMFC stack.For the cooling system, there is a fan (Fan 1 in Figure1) which works for regulating the PEMFC temperature. In practice, this fan is not efficient enough to regulate the PEMFC temperature when PEMFC operates at the high power condition. Moreover, the humidifier contains internal resistances to heat the injected liquid water and the input gases, while the cooling system is not included in the internal structure of the humidifier. Due to the constraint of this structure, the cooling dynamics of the humidifier is extremely slow. The outside fan (fan 2 in Figure 2) is used to help reduce the humidifier temperature at the cathode side. Therefore, the relevant modelling of the humidifier, the line heater, and PEMFC stack including cooling process of the fan will be presented in the following.


### 2.1.1. Auxiliary Components

The humidifier is used to ensure the desired relative humidity of input gases. Its main process is that the water in the vapor form is injected into input gases. Thus, the total mass flow of output gases from the humidifier is expressed as(1)m˙hum,jout=m˙hum,jin+m˙H2O,jinj,where j represents the anode or cathode channel in PEMFC and m˙hum,jin is the input gas flow for the humidifier at the side j. In order to achieve the desired relative humidity, the required vapor water injected into the input gases m˙H2O,jinj can be computed as(2)m˙H2O,jinj=MH2Oϕhum,jm˙hum,joutPsatThum,jM¯jPhum,jin,where ϕhum,j is the desired relative humidity of output gases; MH2O is the molar mass of water; The average molar mass of gases for the channel j is denoted as M¯j; Phum,jin represents the pressure of input gases. And PsatThum,j is the pressure of vapor saturation at the humidifier temperature Thum,j. Moreover, the saturation pressure Psatrelated to the relevant temperature T0 is approximately calculated as [23](3)log10Psat=−1.69×10−10T04+3.85×10−7T03−3.39×10−4T02+0.143T0−20.92.The output gases from the humidifier enter into PEMFC stack through a pipeline. The temperature of this pipeline has some effect on the gas pressure and relative humidity. In order to avoid vapor condensation and maintain desired relative humidify of input gases, the line heater is installed on the pipe. Based on Dalton’s law, the partial pressure of vapor water at the line heater is calculated as(4)PH2O,jlh=Tlh,jThum,jPH2O,jinj,where Tlh,j is the temperature of the line heater. The partial pressure of injected vapor is denoted as PH2O,jinj=ϕhum,jPsatThum,j. Moreover, the relative humidity of gases at the line heater is expressed as(5)ϕlh,j=PH2O,jlhϕlh,jPsatTlh,j,where Psatis the saturation pressure of vapor at the line heater temperature Tlh,j, which can be calculated by (3).


### 2.1.2. PEMFC Stack

For control design purpose, we assume that the temperature of gases in the stack is the same as the electrolyte structure. The temperature variation along the structure dimension is not considered in this paper. Based on the energy conservation principle, the thermal energy balance of PEMFC is built as follows:(6)mfcCp,fcdTfcdt=Hcain+Hanin−Hcaout−Hanout+Hr−VfcIfc−Hfan+d,where mfc and Tfc are the mass and the temperature of PEMFC, respectively. The specific heat capacity of PEMFC is denoted as Cp,fc; Vfc is the output voltage of PEMFC; d is regarded as the unknown disturbance. The input energy values of cathode Hcain and anode layers Hanin generated by the input gases calculated by(7)Hcain=∑cawiinMi∫TrefTlh,caCp,iTdT,Hanin=∑anwiinMi∫TrefTlh,anCp,iTdT,where i denotes as each species of gases in the cathode or anode layer and Mi is the molar mass of the gas i. The mass flow rate of gas i is represented as wiin; Tref is the reference temperature. The specific heat capacity of gas i is denoted as Cp,i. It can be computed as Cp,iT=c0,i+c1,iT+c2,iT2+c3,iT3, where the heat capacity coefficients c0,i, c1,i, c2,i, and c3,i can be found in [24].Furthermore, the output energy of cathodeHcaout, anode Hanout, and the generated energy Hr from chemical reaction are expressed as [25](8)Hcaout=∑cawioutMi∫TrefTfcCp.iTdT,Hanout=∑anwioutMi∫TrefTfcCp.iTdT,Hr=−wH2rMH2ΔHro,where wiout represents the mass flow rate of gas i. The mass flow rate of reacted hydrogen is denoted as wH2r; MH2 is the molar mass of hydrogen. And the specific heat of chemical reaction is represented as ΔHro.There is a fan (Fan 1 in Figure1) close to the PEMFC stack which is used to regulate the stack temperature. The operation principle is that when the stack temperature is higher than the desired temperature, the fan turns on in order to reduce the stack temperature. Otherwise, it turns off. This procedure is expressed as the following mathematical expression:(9)Hfan=ClossAfcTfc−Tref,Tfc>Tset,0,Tfc≤Tset,where Afc is the surface area of PEMFC; Closs is the heat transfer constant; and Tset is the desired temperature of the stack.In the PEMFC, the potential difference is generated by ion flow transports from the electrochemical reaction. The theoretical voltage of PEMFC can change reversibly, which is affected by the variation of partial pressures for reactants and products. Thus, the theoretical voltage at the nonstandard condition can be determined by the Nernst equation, that is,(10)VN=VN0+Δs2FTfc−Tref+RTfc2FlnPH2+12lnPO2,where VN0 is the theoretical voltage at the standard condition; PH2 and PO2 are partial pressures of hydrogen and oxygen, respectively; Δs is the correction coefficient related to the entropy of chemical reaction; F is Faraday’s constant; and R denotes the gas constant.Moreover, there are three types of voltage losses which can affect the voltage variation irreversibly during operation processes. Considering those voltage losses, the output voltage of PEMFC is expressed by [23](11)Vfc=n⋅VN−Vact−Vohm−Vcon,where n is the number of cells for PEMFC; Vact is the activation losses; Vohm denotes the ohmic losses; and Vcon represents the concentration losses. They are calculated as follows:(12)Vact=Vact,0+Kact1−e−c1i,(13)Vohm=δmIfcηmAfc,(14)Vcon=Ifcc2IfcImaxc3,where Vact,0 is the initial activation voltage drop; The membrane thickness is denoted as Δm; Kact, c1, c2, and c3 are the empirical constants, which depend on gas partial pressure and temperature; Imax is the maximum current; and ηm denotes the membrane conductivity, which is related to the membrane water content λm and stack temperature. It can be calculated as(15)ηm=b1eb21/T0−1/Tfc,where T0=303 K is the reference temperature for membrane conductivity test. The empirical value related to the membrane water content is represented as b1=b11λm−b12 and constants b11, b12, and b2 are determined empirically. Those empirical constants have been discussed in [23].For the mass flow transports, the detailed model can be found in [23]. The reacted mass flows of hydrogen and oxygen and the generated mass flow of water are calculated based on the electrochemical relationship:(16)wH2r=NMH2Ifc2F,wO2r=NMO2Ifc4F,wH2Og=NMH2OIfc2F,where MO2 and MH2O are the molar mass of oxygen and water. It is noticed that the reacted mass flows are directly related to the current Ifc. Moreover, the input mass flows are computed as(17)wH2in=KH2wH2r,wO2in=KO2wO2r,where KH2 and KO2 are the stoichiometric ratios. Thus, the input mass flows can be determined by the desired stoichiometric ratio and the current.


## 2.1.1. Auxiliary Components

The humidifier is used to ensure the desired relative humidity of input gases. Its main process is that the water in the vapor form is injected into input gases. Thus, the total mass flow of output gases from the humidifier is expressed as(1)m˙hum,jout=m˙hum,jin+m˙H2O,jinj,where j represents the anode or cathode channel in PEMFC and m˙hum,jin is the input gas flow for the humidifier at the side j. In order to achieve the desired relative humidity, the required vapor water injected into the input gases m˙H2O,jinj can be computed as(2)m˙H2O,jinj=MH2Oϕhum,jm˙hum,joutPsatThum,jM¯jPhum,jin,where ϕhum,j is the desired relative humidity of output gases; MH2O is the molar mass of water; The average molar mass of gases for the channel j is denoted as M¯j; Phum,jin represents the pressure of input gases. And PsatThum,j is the pressure of vapor saturation at the humidifier temperature Thum,j. Moreover, the saturation pressure Psatrelated to the relevant temperature T0 is approximately calculated as [23](3)log10Psat=−1.69×10−10T04+3.85×10−7T03−3.39×10−4T02+0.143T0−20.92.The output gases from the humidifier enter into PEMFC stack through a pipeline. The temperature of this pipeline has some effect on the gas pressure and relative humidity. In order to avoid vapor condensation and maintain desired relative humidify of input gases, the line heater is installed on the pipe. Based on Dalton’s law, the partial pressure of vapor water at the line heater is calculated as(4)PH2O,jlh=Tlh,jThum,jPH2O,jinj,where Tlh,j is the temperature of the line heater. The partial pressure of injected vapor is denoted as PH2O,jinj=ϕhum,jPsatThum,j. Moreover, the relative humidity of gases at the line heater is expressed as(5)ϕlh,j=PH2O,jlhϕlh,jPsatTlh,j,where Psatis the saturation pressure of vapor at the line heater temperature Tlh,j, which can be calculated by (3).


## 2.1.2. PEMFC Stack

For control design purpose, we assume that the temperature of gases in the stack is the same as the electrolyte structure. The temperature variation along the structure dimension is not considered in this paper. Based on the energy conservation principle, the thermal energy balance of PEMFC is built as follows:(6)mfcCp,fcdTfcdt=Hcain+Hanin−Hcaout−Hanout+Hr−VfcIfc−Hfan+d,where mfc and Tfc are the mass and the temperature of PEMFC, respectively. The specific heat capacity of PEMFC is denoted as Cp,fc; Vfc is the output voltage of PEMFC; d is regarded as the unknown disturbance. The input energy values of cathode Hcain and anode layers Hanin generated by the input gases calculated by(7)Hcain=∑cawiinMi∫TrefTlh,caCp,iTdT,Hanin=∑anwiinMi∫TrefTlh,anCp,iTdT,where i denotes as each species of gases in the cathode or anode layer and Mi is the molar mass of the gas i. The mass flow rate of gas i is represented as wiin; Tref is the reference temperature. The specific heat capacity of gas i is denoted as Cp,i. It can be computed as Cp,iT=c0,i+c1,iT+c2,iT2+c3,iT3, where the heat capacity coefficients c0,i, c1,i, c2,i, and c3,i can be found in [24].Furthermore, the output energy of cathodeHcaout, anode Hanout, and the generated energy Hr from chemical reaction are expressed as [25](8)Hcaout=∑cawioutMi∫TrefTfcCp.iTdT,Hanout=∑anwioutMi∫TrefTfcCp.iTdT,Hr=−wH2rMH2ΔHro,where wiout represents the mass flow rate of gas i. The mass flow rate of reacted hydrogen is denoted as wH2r; MH2 is the molar mass of hydrogen. And the specific heat of chemical reaction is represented as ΔHro.There is a fan (Fan 1 in Figure1) close to the PEMFC stack which is used to regulate the stack temperature. The operation principle is that when the stack temperature is higher than the desired temperature, the fan turns on in order to reduce the stack temperature. Otherwise, it turns off. This procedure is expressed as the following mathematical expression:(9)Hfan=ClossAfcTfc−Tref,Tfc>Tset,0,Tfc≤Tset,where Afc is the surface area of PEMFC; Closs is the heat transfer constant; and Tset is the desired temperature of the stack.In the PEMFC, the potential difference is generated by ion flow transports from the electrochemical reaction. The theoretical voltage of PEMFC can change reversibly, which is affected by the variation of partial pressures for reactants and products. Thus, the theoretical voltage at the nonstandard condition can be determined by the Nernst equation, that is,(10)VN=VN0+Δs2FTfc−Tref+RTfc2FlnPH2+12lnPO2,where VN0 is the theoretical voltage at the standard condition; PH2 and PO2 are partial pressures of hydrogen and oxygen, respectively; Δs is the correction coefficient related to the entropy of chemical reaction; F is Faraday’s constant; and R denotes the gas constant.Moreover, there are three types of voltage losses which can affect the voltage variation irreversibly during operation processes. Considering those voltage losses, the output voltage of PEMFC is expressed by [23](11)Vfc=n⋅VN−Vact−Vohm−Vcon,where n is the number of cells for PEMFC; Vact is the activation losses; Vohm denotes the ohmic losses; and Vcon represents the concentration losses. They are calculated as follows:(12)Vact=Vact,0+Kact1−e−c1i,(13)Vohm=δmIfcηmAfc,(14)Vcon=Ifcc2IfcImaxc3,where Vact,0 is the initial activation voltage drop; The membrane thickness is denoted as Δm; Kact, c1, c2, and c3 are the empirical constants, which depend on gas partial pressure and temperature; Imax is the maximum current; and ηm denotes the membrane conductivity, which is related to the membrane water content λm and stack temperature. It can be calculated as(15)ηm=b1eb21/T0−1/Tfc,where T0=303 K is the reference temperature for membrane conductivity test. The empirical value related to the membrane water content is represented as b1=b11λm−b12 and constants b11, b12, and b2 are determined empirically. Those empirical constants have been discussed in [23].For the mass flow transports, the detailed model can be found in [23]. The reacted mass flows of hydrogen and oxygen and the generated mass flow of water are calculated based on the electrochemical relationship:(16)wH2r=NMH2Ifc2F,wO2r=NMO2Ifc4F,wH2Og=NMH2OIfc2F,where MO2 and MH2O are the molar mass of oxygen and water. It is noticed that the reacted mass flows are directly related to the current Ifc. Moreover, the input mass flows are computed as(17)wH2in=KH2wH2r,wO2in=KO2wO2r,where KH2 and KO2 are the stoichiometric ratios. Thus, the input mass flows can be determined by the desired stoichiometric ratio and the current.


## 2.2. Problem Formulation

For the PEMFC system shown in Figure1, it has been well recognized that the fan 1 can be used to regulate the temperature of the fuel cell stack. However, the fan 1 has not enough power to regulate the fuel cell temperature when it operates at relative high power. Without introducing extra cooling systems, reducing the temperature and increasing mass flows of input gases are possible ways to achieve the PEMFC temperature regulation. Besides, some parameters in the mathematical model are empirical values which can lead to some modelling errors. Furthermore, there are unmodelled dynamics derived from the model simplification during the modelling procedure and external disturbances resulted from the experiment measurement. However, ignoring those unknown dynamics and modelling errors may cause the system performance degradation or even instability when the controllers apply to the PEMFC system. Thus, this paper will develop a novel control strategy in order to address both temperature regulation problem and the unknown dynamic compensations.In the mathematical model of the PEMFC system, there are many empirically determined variables. The value of those variables may change during the fuel cell operation such that the real performance cannot be precisely presented in the mathematical model. One of the most important variables for the thermal energy balance model is the empirical specific heat of chemical reactionΔHro, which affects the generated energy Hr in the PEMFC. Moreover, there are some noises or disturbances d from the experiment. Due to those facts, we consider the reacted heat Hr and disturbances d as the unknown dynamics D1, which can be expressed as follows:(18)D1=Hr+d.From the thermal energy balance of PEMFC in (6), reducing the input energy and increasing the output energy can further achieve the temperature regulation when the energy of the cooling fan reaches the maximum. Those energies are mainly related to the input mass flow rates and the temperature of input flows, which can be adjusted by the humidifier and the mass flow controller. Without the internal cooling system, the dynamics of reducing the humidifier temperature is extremely slow. The fan 2 is used to improve the cooling process of the humidifier. However, the cooling performance of the humidifier with the fan 2 is still slow. Due to the slow dynamics, increasing the input mass flow is considered to extract the excess heat from the PEMFC stack. Furthermore, considering the economic costs, the input mass flow and the input temperature of air are chosen to regulate the temperature of PEMFC in this paper. For the control design purpose, the total extract heat by employing the additional mass flow of air is calculated as follows:(19)Qloss=−ΔwairinThum,caset−Tfc,where Δwairin is the additional mass flow of air and Thum,caset is an ideal operation point for the temperature of cathode humidifier. Therefore, the extracted heat Qloss is considered as the control input. By the simple algebraic calculus, the total mass flow of the air is the sum of the initial mass flow and the additional mass flow, which is the real control input in practice. Thus, the thermal energy balance of PEMFC with the control input u=Qloss is expressed as(20)mfcCp,fcdTfcdt=Hca,0in+Hanin−Hca,0out−Hanout+Hr−VfcIfc−Hfan+d+u,where Hca,0in and Hca,0out are the initial heat of input and output energy for the cathode layer. Furthermore, (20) is rewritten as follows for analysis purpose:(21)dTfcdt=1mfcCp,fcu+D1+F,where F=Hca,0in+Hanin−Hca,0out−Hanout−VfcIfc−Hfan is the known nonlinear function.To summarize, this paper aims to propose an alternative control strategy for a high-power PEMFC, where the cooling system cannot regulate the PEMFC temperature. The control strategy is available when the cooling fan turns on, which consists of a composite PI controller with unknown system dynamic estimator (USDE) in order to achieve the temperature regulation and eliminate the effect of the unknown dynamicsD1 through the measurable output Tfc and the known functions F. The schematic block of the proposed control strategy is shown in Figure 3.Figure 3
Schematic block of the composite PID controller.Before we illustrate the control strategy with the USDE, the following assumptions are given with respect to the thermal energy balance of PEMFC (21).Assumption 1.
The temperatureTfc and the control input u are measurable and bounded. And the function F is computable and bounded.Assumption 2.
The unknown dynamics ofD1 is continuous and its derivative is bounded such that D˙1≤η.Remark 1.
It is easy to measure the variables of stack temperatureTfc, the current Ifc, the voltage Vfc, and the input mass flows for each channel. Thus, the PEMFC can fulfill the aforementioned Assumption 1 trivially. Besides, the derivative of unknown dynamics D˙1 in Assumption 2 is not required to be known, which is used for the stability analysis of the proposed controller.


## 3. Controller Design of PEMFC System

In this section, we first present the USDE forD1. Then, it will be incorporated into the PI controller. The stability analysis for USDE and the composite PI controller will be provided separately.


### 3.1. Estimator of Unknown Dynamics

In order to derive an invariant manifold, a low-pass filter operation is applied to the variablesTfc, F and u. The filtered variables Tfc,f, Ff and uf are expressed as follows:(22)κT˙fc,f+Tfc,f=Tfc,Tfc,f0=0,κF˙f+Ff=F,Ff0=0,κu˙f+uf=u,uf0=0,where κ>0 is a filter constant, which is the only parameter that needs to be tuned for USDE.Then, the invariant manifold is provided in the following lemma, which illustrates the relation between the unknown dynamicsD1 with the measurable parameter Tfc and the filtered variables Tfc,f, Ff, and uf.Lemma 1.
From the thermal energy balance of PEMFC in (21) and filtered vectors in (22), we can derive the following manifold:(23)limt⟶∞limκ⟶0Tfc−Tfc,fκ−1mfcCp,fcuf+Ff+D1=0.For any small constantκ>0, it is invariant and exponentially attractive for the derived manifold (23).Proof.
The auxiliary coordinate for analyzing the manifold invariant is defined as(24)z1t=Tfc−Tfc,fκ−1mfcCp,fcuf+Ff+D1.
By differentiating (24), we can derive that(25)z˙1=T˙fc−T˙fc,fκ−1mfcCp,fcu˙f+F˙f+D˙1=−1κz1+κmfcCp,fcD˙1.
Then, we selectVz1=z12/2 as Lyapunov function to show the bound of z1. Thus, the time derivative of Vz1 is(26)V˙z1=z1z˙1=−1κz12−1mfcCp,fcz1D˙1.
From Assumption2 and Young’s inequality, (26) can be further be derived as(27)V˙z1≤−12κz12+κ2mfc2Cp,fc2D˙12≤−1κVz1+κ2mfc2Cp,fc2η2.
Furthermore, we apply the integration into (27) and we can obtain Vz1t≤e−t/κVz10+κ2η2/2mfc2Cp,fc2. Thus, it is concluded that zt converges exponentially to a small compact around zero, which is expressed as(28)z1t≤z120e−t/κ+κ2η22mfc2Cp,fc2.
Based on Assumption 2, the upper bound ofD˙1 is η. Due to this fact, (28) implies that limt˗∞z1t=κη/mfcCp,fc. Then, we can further obtain that κD˙1⟶0 and κ2η2⟶0 as κ⟶0. Thus, z1t will converge to zero for κ⟶0 and/or η⟶0. Finally, the manifold z1=0 in (23) is invariant and exponentially attractive.
Based on the rendered manifold in (23), the estimator for unknown dynamics D1 can be designed as(29)D^1=mfcCp,fcκTfc−Tfc,f−uf−Ff.
The following theorem is summarized the convergence of the estimator errors for the proposed estimator in (29).Theorem 1.
The estimator (29) is used to obtain the unknown dynamics D1 in the thermal energy balance of PEMFC (21). The estimation error D˜1=D1−D^1 will exponentially converge to a small compact set around zero.Proof.
To avoid requiring the information of temperature derivativeT˙fc, we first apply a low-pass filter to the equation (21). Then, we get(30)κκs+1Tfc=1mfcCp,fc1κs+1u+1κs+1F+1κs+1D1+ε1,where ε1 is the exponentially vanishing term from the filtered initial value of state Tfc0, which can be neglected.
Moreover, the filtered variableD1,f is defined as(31)κD˙1,f+D1,f=D1,D1,f0=0.
Since the effect of the exponentially vanishing termε is not considered, the equation (30) can be rewritten as the following form based on (22) and (31):(32)T˙fc,f=Tfc−Tfc,fκ=1mfcCp,fcuf+Ff+D1,f.
From the comparison between (24) and (32), it is implied that D^1=D1,f. Hence, we compute the derivative of the estimation error D˜1, that is,(33)D˜˙1=D˙1−D^˙1=D˙1−D˙1,f=D˙1−D−D1,fκ=−1κD˜1+D˙1.
To show the boundedness of the estimation errorD˜1, we choose Lyapunov function as VD1=D˜12/2. Its derivative is computed as(34)V˙D1=D˜1D˜˙1=−1κD˜12+D˜1D˙1.
Considering Young’s inequality and Assumption2, we get(35)V˙D1≤−12κD˜1+κ2D˙12≤−1κVD1+κ2η2.
Then, the integration of (35) is VD1≤e−t/κVD10+κ2η2/2. Based on this, it can be derived that the estimation error D˜1 will exponentially converge to a small compact:(36)D˜1≤D˜120e−t/κ+κ2η2.
Finally,D˜1 will converge to zero when κ⟶0 and η⟶∞Remark 2.
For the unknown system dynamics estimator (29), there are two main advantages in comparison with other estimation (i.e., [14, 26]). First, only one parameter κ is required to be tuned. From (22) and (31), κ determines the bandwidth of the low-pass filter such that it has an impact on robustness. Moreover, the boundedness of the unknown dynamics derivative η can affect the estimation performance while the constant κ can be used to reduce the effect of this boundedness η. Besides, the convergence speed of the estimation error is related to this constant κ, which can be found in the proof of Theorem 1. Hence, the selection of this parameter κ need to consider the trade-off between the convergence speed of the estimation error and the robustness against disturbance. Moreover, the information of the state derivative T˙fc,f is not required due to the low-pass filter operations. The discontinuities and chattering phenomena can be avoided compared with the sliding mode-based estimator.


### 3.2. Composite PI Controller with Unknown Dynamic Compensation

In this section, a composite PI controller based on the USDE is presented. From Theorem1, the estimator (29) will exponentially converge to a small compact, even to zero when κ⟶0. Hence, it can be used to compensate the unknown dynamics and enhance the control performance. Based on the framework of controller in Figure 3, a composite PI control is designed as(37)u=Qloss=kper+ki∫0terτdτ−D^1+F−mfcCp,fcT˙fc,r,where Tfc,r denotes the desired temperature of PEMFC; er=Tfc,r−Tfc is the control error; and the proportional and integral gains are represented as kp>0 and ki>0, respectively. Since the dynamics of PEMFC temperature is slow, the desired temperature is usually considered as a constant or slow-varying value. Thus, the derivative of the desired temperature T˙fc,r can be neglected.Remark 3.
The proposed control (37) is combined with the estimator of unknown dynamics (29), which has the function to compensate the unknown dynamics. The PI controller is widely used for the PEMFC system in practice. Hence, the proposed control strategy has an advantage to enhance the predesigned PI controller for practical applications.
Moreover, the convergence of the tracking error and the estimation error for the proposed controller are summarized in the following theorem.Theorem 2.
To regulate the temperature of the PEMFC system, the control strategy of the composite PI controller (37) combined with an unknown dynamic compensation is proposed as depicted in Figure 3. The control error er and the estimation D˜1 will exponentially converge to a small compact set around zero.Proof.
Based on the thermal balance equation (21) and the controller (37), the derivative of control error is calculated as(38)e˙r=T˙fc,r−T˙fc=T˙fc,r−1mfcCp,fcu−1mfcCp,fcD1−1mfcCp,fcF=−1mfcCp,fckper+kieri+D˜1,where eri=∫0terτdτ denotes the integral error. To analyze the stability of the controller, the error including the control error and the integral error is defined as e=erierT. Thus, the equation (38) is reformulated as(39)e˙=e˙rie˙r=01−kimfcCp,fc−kpmfcCp,fcerier+0−1mfcCp,fcD˜1=A1e+B1D˜1.
In the matrixA1, parameters mfc and Cp,fc and control gains ki and kp are positive constants. In order to guarantee the stability of the control system, the control gains ki and kp are chosen such that the matrix A1 is a Hurwitz matrix. According to the Lyapunov theory, there are symmetric and positive definite matrices P1>0 and Q1>0 such that the matrix A1 satisfies the equation P1A1+A1TP1=−Q1. Moreover, we choose the Lyapunov function as Ve=eTP1e/2+D˜12/2. Then, its derivative is expressed as(40)V˙e=12eTP1e˙+12e˙TP1e+D˜1D˜˙1=12eTP1A1e+12eTA1TP1e+eTP1BD˜1+D˜1D˜˙1=−12eTQ1e+eTP1BD˜1−1κD˜12+D˜1D˙1.
Based on Young’s inequality with a positive constantm>0, we can further derive that(41)V˙e≤−12λminQ1−m2e2−1κ−1+P1B22mD˜12+mη22≤−ψ1V+ζ1,where there are positive constants of ψ1=minλminQ1−m/2/λmaxP1,21/κ−1+P1B2/2m and ζ1=mη2/2 when constants m and κ are satisfied, m≤2λminQ1 and κ≤2m/1+P1B2; the maximum eigenvalue of the matrix P1 is denoted as λmaxP1; And λminQ1 is the minimum eigenvalue of the matrix Q1. Then, we can further derive that Vt≤e−ψ1tV0+ζ1/ψ1, which implies that the control error e and the estimation error D˜1 will exponentially converge to the following small compact set around zero.(42)Ω≔e,D˜1|e≤ζ1ψ1,D˜1≤ζ1ψ1.


## 3.1. Estimator of Unknown Dynamics

In order to derive an invariant manifold, a low-pass filter operation is applied to the variablesTfc, F and u. The filtered variables Tfc,f, Ff and uf are expressed as follows:(22)κT˙fc,f+Tfc,f=Tfc,Tfc,f0=0,κF˙f+Ff=F,Ff0=0,κu˙f+uf=u,uf0=0,where κ>0 is a filter constant, which is the only parameter that needs to be tuned for USDE.Then, the invariant manifold is provided in the following lemma, which illustrates the relation between the unknown dynamicsD1 with the measurable parameter Tfc and the filtered variables Tfc,f, Ff, and uf.Lemma 1.
From the thermal energy balance of PEMFC in (21) and filtered vectors in (22), we can derive the following manifold:(23)limt⟶∞limκ⟶0Tfc−Tfc,fκ−1mfcCp,fcuf+Ff+D1=0.For any small constantκ>0, it is invariant and exponentially attractive for the derived manifold (23).Proof.
The auxiliary coordinate for analyzing the manifold invariant is defined as(24)z1t=Tfc−Tfc,fκ−1mfcCp,fcuf+Ff+D1.
By differentiating (24), we can derive that(25)z˙1=T˙fc−T˙fc,fκ−1mfcCp,fcu˙f+F˙f+D˙1=−1κz1+κmfcCp,fcD˙1.
Then, we selectVz1=z12/2 as Lyapunov function to show the bound of z1. Thus, the time derivative of Vz1 is(26)V˙z1=z1z˙1=−1κz12−1mfcCp,fcz1D˙1.
From Assumption2 and Young’s inequality, (26) can be further be derived as(27)V˙z1≤−12κz12+κ2mfc2Cp,fc2D˙12≤−1κVz1+κ2mfc2Cp,fc2η2.
Furthermore, we apply the integration into (27) and we can obtain Vz1t≤e−t/κVz10+κ2η2/2mfc2Cp,fc2. Thus, it is concluded that zt converges exponentially to a small compact around zero, which is expressed as(28)z1t≤z120e−t/κ+κ2η22mfc2Cp,fc2.
Based on Assumption 2, the upper bound ofD˙1 is η. Due to this fact, (28) implies that limt˗∞z1t=κη/mfcCp,fc. Then, we can further obtain that κD˙1⟶0 and κ2η2⟶0 as κ⟶0. Thus, z1t will converge to zero for κ⟶0 and/or η⟶0. Finally, the manifold z1=0 in (23) is invariant and exponentially attractive.
Based on the rendered manifold in (23), the estimator for unknown dynamics D1 can be designed as(29)D^1=mfcCp,fcκTfc−Tfc,f−uf−Ff.
The following theorem is summarized the convergence of the estimator errors for the proposed estimator in (29).Theorem 1.
The estimator (29) is used to obtain the unknown dynamics D1 in the thermal energy balance of PEMFC (21). The estimation error D˜1=D1−D^1 will exponentially converge to a small compact set around zero.Proof.
To avoid requiring the information of temperature derivativeT˙fc, we first apply a low-pass filter to the equation (21). Then, we get(30)κκs+1Tfc=1mfcCp,fc1κs+1u+1κs+1F+1κs+1D1+ε1,where ε1 is the exponentially vanishing term from the filtered initial value of state Tfc0, which can be neglected.
Moreover, the filtered variableD1,f is defined as(31)κD˙1,f+D1,f=D1,D1,f0=0.
Since the effect of the exponentially vanishing termε is not considered, the equation (30) can be rewritten as the following form based on (22) and (31):(32)T˙fc,f=Tfc−Tfc,fκ=1mfcCp,fcuf+Ff+D1,f.
From the comparison between (24) and (32), it is implied that D^1=D1,f. Hence, we compute the derivative of the estimation error D˜1, that is,(33)D˜˙1=D˙1−D^˙1=D˙1−D˙1,f=D˙1−D−D1,fκ=−1κD˜1+D˙1.
To show the boundedness of the estimation errorD˜1, we choose Lyapunov function as VD1=D˜12/2. Its derivative is computed as(34)V˙D1=D˜1D˜˙1=−1κD˜12+D˜1D˙1.
Considering Young’s inequality and Assumption2, we get(35)V˙D1≤−12κD˜1+κ2D˙12≤−1κVD1+κ2η2.
Then, the integration of (35) is VD1≤e−t/κVD10+κ2η2/2. Based on this, it can be derived that the estimation error D˜1 will exponentially converge to a small compact:(36)D˜1≤D˜120e−t/κ+κ2η2.
Finally,D˜1 will converge to zero when κ⟶0 and η⟶∞Remark 2.
For the unknown system dynamics estimator (29), there are two main advantages in comparison with other estimation (i.e., [14, 26]). First, only one parameter κ is required to be tuned. From (22) and (31), κ determines the bandwidth of the low-pass filter such that it has an impact on robustness. Moreover, the boundedness of the unknown dynamics derivative η can affect the estimation performance while the constant κ can be used to reduce the effect of this boundedness η. Besides, the convergence speed of the estimation error is related to this constant κ, which can be found in the proof of Theorem 1. Hence, the selection of this parameter κ need to consider the trade-off between the convergence speed of the estimation error and the robustness against disturbance. Moreover, the information of the state derivative T˙fc,f is not required due to the low-pass filter operations. The discontinuities and chattering phenomena can be avoided compared with the sliding mode-based estimator.


## 3.2. Composite PI Controller with Unknown Dynamic Compensation

In this section, a composite PI controller based on the USDE is presented. From Theorem1, the estimator (29) will exponentially converge to a small compact, even to zero when κ⟶0. Hence, it can be used to compensate the unknown dynamics and enhance the control performance. Based on the framework of controller in Figure 3, a composite PI control is designed as(37)u=Qloss=kper+ki∫0terτdτ−D^1+F−mfcCp,fcT˙fc,r,where Tfc,r denotes the desired temperature of PEMFC; er=Tfc,r−Tfc is the control error; and the proportional and integral gains are represented as kp>0 and ki>0, respectively. Since the dynamics of PEMFC temperature is slow, the desired temperature is usually considered as a constant or slow-varying value. Thus, the derivative of the desired temperature T˙fc,r can be neglected.Remark 3.
The proposed control (37) is combined with the estimator of unknown dynamics (29), which has the function to compensate the unknown dynamics. The PI controller is widely used for the PEMFC system in practice. Hence, the proposed control strategy has an advantage to enhance the predesigned PI controller for practical applications.
Moreover, the convergence of the tracking error and the estimation error for the proposed controller are summarized in the following theorem.Theorem 2.
To regulate the temperature of the PEMFC system, the control strategy of the composite PI controller (37) combined with an unknown dynamic compensation is proposed as depicted in Figure 3. The control error er and the estimation D˜1 will exponentially converge to a small compact set around zero.Proof.
Based on the thermal balance equation (21) and the controller (37), the derivative of control error is calculated as(38)e˙r=T˙fc,r−T˙fc=T˙fc,r−1mfcCp,fcu−1mfcCp,fcD1−1mfcCp,fcF=−1mfcCp,fckper+kieri+D˜1,where eri=∫0terτdτ denotes the integral error. To analyze the stability of the controller, the error including the control error and the integral error is defined as e=erierT. Thus, the equation (38) is reformulated as(39)e˙=e˙rie˙r=01−kimfcCp,fc−kpmfcCp,fcerier+0−1mfcCp,fcD˜1=A1e+B1D˜1.
In the matrixA1, parameters mfc and Cp,fc and control gains ki and kp are positive constants. In order to guarantee the stability of the control system, the control gains ki and kp are chosen such that the matrix A1 is a Hurwitz matrix. According to the Lyapunov theory, there are symmetric and positive definite matrices P1>0 and Q1>0 such that the matrix A1 satisfies the equation P1A1+A1TP1=−Q1. Moreover, we choose the Lyapunov function as Ve=eTP1e/2+D˜12/2. Then, its derivative is expressed as(40)V˙e=12eTP1e˙+12e˙TP1e+D˜1D˜˙1=12eTP1A1e+12eTA1TP1e+eTP1BD˜1+D˜1D˜˙1=−12eTQ1e+eTP1BD˜1−1κD˜12+D˜1D˙1.
Based on Young’s inequality with a positive constantm>0, we can further derive that(41)V˙e≤−12λminQ1−m2e2−1κ−1+P1B22mD˜12+mη22≤−ψ1V+ζ1,where there are positive constants of ψ1=minλminQ1−m/2/λmaxP1,21/κ−1+P1B2/2m and ζ1=mη2/2 when constants m and κ are satisfied, m≤2λminQ1 and κ≤2m/1+P1B2; the maximum eigenvalue of the matrix P1 is denoted as λmaxP1; And λminQ1 is the minimum eigenvalue of the matrix Q1. Then, we can further derive that Vt≤e−ψ1tV0+ζ1/ψ1, which implies that the control error e and the estimation error D˜1 will exponentially converge to the following small compact set around zero.(42)Ω≔e,D˜1|e≤ζ1ψ1,D˜1≤ζ1ψ1.


## 4. Alternative Controller Design of PEMFC System

In Section3.2, the controller requires the information of the desired temperature derivative T˙fc,r. Since the desired temperature of the PEMFC system is usually a constant or slow time-varying value, the influence of the desired temperature derivative T˙fc,r has a small impact on the proposed controller. In order to further eliminate this small impact, an alternative composite PI controller is proposed to further avoid requiring the derivation of the desired temperature T˙fc,r in this section.


### 4.1. Alternative Estimator of Unknown Dynamics

Before we present the USDE, we first calculate the derivative of control error in order to find the relation containing the derivative of the desired temperature. Thus, the derivative of control error is computed as(43)e˙r=T˙fc,r−T˙fc=T˙fc,r−1mfcCp,fcu−1mfcCp,fcD1−1mfcCp,fcF=D2T˙fc,r,D1−1mfcCp,fcu−1mfcCp,fcF,where D2, including T˙fc,r and D1, is considered as a new unknown dynamics in this section. For analysis, we assume this unknown dynamics is continuous and its derivative is bounded such that D˙2≤η2. Therefore, T˙fc,r and D1 as unknown dynamics will be estimated in this subsection though the feedback control error er, control input u, and known function F.Following the previous procedure of designing the estimator in Section3, we define the filtered variable er,f with respect to er in order to avoid requiring the information of derivative of control error e˙r in the USDE. We obtain(44)κe˙r,f+er,f=er,er,f0=0.Then, we can find the following invariant manifold.Lemma 2.
Based on the equation of control error (43), the filtered variables er,f, Ff, and uf defined in (22) and (44), the manifold is derived as follows:(45)limt⟶∞limκ⟶0er−er,fκ+1mfcCp,fcuf+Ff−D2=0.This manifold is invariant and exponentially attractive for any small positive constantκ>0.Proof.
we define the auxiliary coordinate for analyzing the manifold invariant:(46)z2t=er−er,fκ+1mfcCp,fcuf+Ff−D2.
Then, we differentiate (46) with respect to time, that is,(47)z˙2=e˙r−e˙r,fκ+1mfcCp,fcu˙f+F˙f−D˙2=−1κz2+κD˙2.
Moreover, we selectVz2=z22/2 as Lyapunov function to show the bound of z2. Thus, the time derivative of Vz2 is(48)V˙z2=z2z˙2=−1κz22−κz2D˙2≤−12κz22+κD˙12≤−1κVz2+κη22.
Furthermore, (48) is integrated such that we can get Vz2t≤e−t/κVz20+κ2η22/2. Thus, it is concluded that z2t converges exponentially to a small compact around zero, which is expressed as(49)z2t≤z220e−t/κ+κ2η222.
Following the same analysis in Lemma1, we can draw the conclusion that z2 will converge to zero for κ⟶0 and/or η2⟶0 along the time. Finally, the manifold z2=0 in (45) is invariant and exponentially attractive.
Therefore, the estimator is provided based on the invariant manifold in (45)(50)D^2=1κer−er,f+1mfcCp,fcuf+1mfcCp,fcFf.Theorem 3.
Based on the proposed estimator (50) and the derivative equation of control error (43), the estimation error D˜2=D2−D^2 will exponentially converge to a small compact set around zero.Proof.
for the analysis purpose, the filtered variableD2,f is defined as(51)κD˙2,f+D2,f=D2,D2,f0=0.
Then, a low-pass filter is applied to the equation (43). We get(52)κκs+1er=−1mfcCp,fc1κs+1u+1κs+1F+1κs+1D2+ε2,where the exponentially vanishing term ε2 from the filtered initial value of state er0 is neglected.
By substituting (22) and (51), the equation (52) is reformulated as(53)e˙r,f=er−er,fκ=−1mfcCp,fcuf+Ff+D2,f.
By the comparison of (46) and (53), we can get the fact D^2=D2,f. Before we provide the bound of the estimation error D˜2, the derivative of the estimation error is first computed as follows:(54)D˜˙2=D˙2−D^˙2=D˙2−D˙2,f=D˙2−D−D2,fκ=−1κD˜2+D˙2.
Furthermore, we choose Lyapunov function asVD2=D˜22/2. The derivative of this function is(55)V˙D2=D˜2D˜˙2=−1κD˜22+D˜2D˙2≤−12κD˜2+κ2D˙22≤−1κVD2+κ2η22.
Then, the integration of (55) is VD2≤e−t/κVD20+κ2η22/2. Based on this, it can be derived that the estimation error D˜2 will exponentially converge to a small compact:(56)D˜2≤D˜220e−t/κ+κ2η22.
Finally,D˜2 will converge to zero when κ⟶0 and η⟶∞.


### 4.2. Alternative Composite PI Controller with Unknown Dynamic Compensation

In this section, an alternative composite PI controller is proposed without requiring the derivative of the desired temperature. Since the alternative estimator is designed based on the equation (43), the derivative of the desired temperature T˙fc,r and the unknown dynamics D1 are estimated by estimator (56). Thus, the composite PI controller can be designed as(57)u=kper+ki∫0terτdτ+mfcCp,fcD^2−F.Then, the following theorem illustrates the convergence of the tracking error and the estimation error for the proposed controller (57).Theorem 4.
To regulate the temperature of the PEMFC system, the control strategy of the composite PI controller (57) is combined with an unknown dynamic compensation (50). The control error er and the estimation D˜2 will exponentially converge to a small compact set around zero.Proof.
by substituting (57) into (43), we can get(58)e˙r=−1mfcCp,fckper+kieri+D˜2,where eri=∫0terτdτ denotes the integral error. To analyze the stability of the controller, the errors including the control error and the integral error are defined as e=erierT. Thus, the equation (58) is reformulated as (59)e˙=e˙rie˙r=01−kimfcCp,fc−kpmfcCp,fcerier+01D˜2=A2e+B2D˜2.
Moreover, there are symmetric and positive definite matricesP2 and Q2 such that the matrix A2 fulfills the equation P2A2+A2TP2=−Q2. Then, we choose the Lyapunov function as Ve2=eTP2e/2+D˜22/2. Its derivative is expressed as(60)V˙e2=12eTP2e˙+12e˙TP2e+D˜2D˜˙2=12eTP2A2e+12eTA2TP2e+eTP2B2D˜2+D˜2D˜˙2=−12eTQ2e+eTP2B2D˜2−1κD˜22+D˜2D˙2.
Based on Young’s inequality with a positive constantm>0, we can further derive that(61)V˙e2≤−12λminQ2−m2e2−1κ−1+P2B222mD˜22+mη222≤−ψ2V+ζ2,where there are positive constants of ψ2=minλminQ2−m/2/λmaxP2,21/κ−1+P2B22/2m and ζ1=mη22/2 when constants m and κ are satisfied, m≤2λminQ2 and κ≤2m/1+P2B22; the maximum eigenvalue of the matrix P2 is denoted as λmaxP2 and λminQ2 is the minimum eigenvalue of the matrix Q2. Then, we can further derive Vt≤e−ψ2tV0+ζ2/ψ2, which implies that the control error e and the estimation error D˜2 will exponentially converge to the following small compact set around zero:(62)Ω≔e,D˜2|e≤ζ2ψ2,D˜2≤ζ2ψ2.


## 4.1. Alternative Estimator of Unknown Dynamics

Before we present the USDE, we first calculate the derivative of control error in order to find the relation containing the derivative of the desired temperature. Thus, the derivative of control error is computed as(43)e˙r=T˙fc,r−T˙fc=T˙fc,r−1mfcCp,fcu−1mfcCp,fcD1−1mfcCp,fcF=D2T˙fc,r,D1−1mfcCp,fcu−1mfcCp,fcF,where D2, including T˙fc,r and D1, is considered as a new unknown dynamics in this section. For analysis, we assume this unknown dynamics is continuous and its derivative is bounded such that D˙2≤η2. Therefore, T˙fc,r and D1 as unknown dynamics will be estimated in this subsection though the feedback control error er, control input u, and known function F.Following the previous procedure of designing the estimator in Section3, we define the filtered variable er,f with respect to er in order to avoid requiring the information of derivative of control error e˙r in the USDE. We obtain(44)κe˙r,f+er,f=er,er,f0=0.Then, we can find the following invariant manifold.Lemma 2.
Based on the equation of control error (43), the filtered variables er,f, Ff, and uf defined in (22) and (44), the manifold is derived as follows:(45)limt⟶∞limκ⟶0er−er,fκ+1mfcCp,fcuf+Ff−D2=0.This manifold is invariant and exponentially attractive for any small positive constantκ>0.Proof.
we define the auxiliary coordinate for analyzing the manifold invariant:(46)z2t=er−er,fκ+1mfcCp,fcuf+Ff−D2.
Then, we differentiate (46) with respect to time, that is,(47)z˙2=e˙r−e˙r,fκ+1mfcCp,fcu˙f+F˙f−D˙2=−1κz2+κD˙2.
Moreover, we selectVz2=z22/2 as Lyapunov function to show the bound of z2. Thus, the time derivative of Vz2 is(48)V˙z2=z2z˙2=−1κz22−κz2D˙2≤−12κz22+κD˙12≤−1κVz2+κη22.
Furthermore, (48) is integrated such that we can get Vz2t≤e−t/κVz20+κ2η22/2. Thus, it is concluded that z2t converges exponentially to a small compact around zero, which is expressed as(49)z2t≤z220e−t/κ+κ2η222.
Following the same analysis in Lemma1, we can draw the conclusion that z2 will converge to zero for κ⟶0 and/or η2⟶0 along the time. Finally, the manifold z2=0 in (45) is invariant and exponentially attractive.
Therefore, the estimator is provided based on the invariant manifold in (45)(50)D^2=1κer−er,f+1mfcCp,fcuf+1mfcCp,fcFf.Theorem 3.
Based on the proposed estimator (50) and the derivative equation of control error (43), the estimation error D˜2=D2−D^2 will exponentially converge to a small compact set around zero.Proof.
for the analysis purpose, the filtered variableD2,f is defined as(51)κD˙2,f+D2,f=D2,D2,f0=0.
Then, a low-pass filter is applied to the equation (43). We get(52)κκs+1er=−1mfcCp,fc1κs+1u+1κs+1F+1κs+1D2+ε2,where the exponentially vanishing term ε2 from the filtered initial value of state er0 is neglected.
By substituting (22) and (51), the equation (52) is reformulated as(53)e˙r,f=er−er,fκ=−1mfcCp,fcuf+Ff+D2,f.
By the comparison of (46) and (53), we can get the fact D^2=D2,f. Before we provide the bound of the estimation error D˜2, the derivative of the estimation error is first computed as follows:(54)D˜˙2=D˙2−D^˙2=D˙2−D˙2,f=D˙2−D−D2,fκ=−1κD˜2+D˙2.
Furthermore, we choose Lyapunov function asVD2=D˜22/2. The derivative of this function is(55)V˙D2=D˜2D˜˙2=−1κD˜22+D˜2D˙2≤−12κD˜2+κ2D˙22≤−1κVD2+κ2η22.
Then, the integration of (55) is VD2≤e−t/κVD20+κ2η22/2. Based on this, it can be derived that the estimation error D˜2 will exponentially converge to a small compact:(56)D˜2≤D˜220e−t/κ+κ2η22.
Finally,D˜2 will converge to zero when κ⟶0 and η⟶∞.


## 4.2. Alternative Composite PI Controller with Unknown Dynamic Compensation

In this section, an alternative composite PI controller is proposed without requiring the derivative of the desired temperature. Since the alternative estimator is designed based on the equation (43), the derivative of the desired temperature T˙fc,r and the unknown dynamics D1 are estimated by estimator (56). Thus, the composite PI controller can be designed as(57)u=kper+ki∫0terτdτ+mfcCp,fcD^2−F.Then, the following theorem illustrates the convergence of the tracking error and the estimation error for the proposed controller (57).Theorem 4.
To regulate the temperature of the PEMFC system, the control strategy of the composite PI controller (57) is combined with an unknown dynamic compensation (50). The control error er and the estimation D˜2 will exponentially converge to a small compact set around zero.Proof.
by substituting (57) into (43), we can get(58)e˙r=−1mfcCp,fckper+kieri+D˜2,where eri=∫0terτdτ denotes the integral error. To analyze the stability of the controller, the errors including the control error and the integral error are defined as e=erierT. Thus, the equation (58) is reformulated as (59)e˙=e˙rie˙r=01−kimfcCp,fc−kpmfcCp,fcerier+01D˜2=A2e+B2D˜2.
Moreover, there are symmetric and positive definite matricesP2 and Q2 such that the matrix A2 fulfills the equation P2A2+A2TP2=−Q2. Then, we choose the Lyapunov function as Ve2=eTP2e/2+D˜22/2. Its derivative is expressed as(60)V˙e2=12eTP2e˙+12e˙TP2e+D˜2D˜˙2=12eTP2A2e+12eTA2TP2e+eTP2B2D˜2+D˜2D˜˙2=−12eTQ2e+eTP2B2D˜2−1κD˜22+D˜2D˙2.
Based on Young’s inequality with a positive constantm>0, we can further derive that(61)V˙e2≤−12λminQ2−m2e2−1κ−1+P2B222mD˜22+mη222≤−ψ2V+ζ2,where there are positive constants of ψ2=minλminQ2−m/2/λmaxP2,21/κ−1+P2B22/2m and ζ1=mη22/2 when constants m and κ are satisfied, m≤2λminQ2 and κ≤2m/1+P2B22; the maximum eigenvalue of the matrix P2 is denoted as λmaxP2 and λminQ2 is the minimum eigenvalue of the matrix Q2. Then, we can further derive Vt≤e−ψ2tV0+ζ2/ψ2, which implies that the control error e and the estimation error D˜2 will exponentially converge to the following small compact set around zero:(62)Ω≔e,D˜2|e≤ζ2ψ2,D˜2≤ζ2ψ2.


## 5. Simulation Results

In this section, the proposed control strategy for the PEMFC system is conducted in the MATLAB/Simulink environment. For the simulation of PEMFC, the stoichiometric ratios of oxygen and hydrogen are set asKO2=4 and KH2=2.5, respectively. And the current of fuel cell is set as Ifc=15A, where the cooling fan cannot remove enough heat to maintain the temperature of PEMFC. Moreover, the disturbance is set as a bounded signal dt=0.005sin0.1te−0.001t+0.005. The temperature of the humidifier at the anode side is set as Thum,anset=40°C. The initial temperature of the cathode humidifier is 34.5°C. The ideal operation temperature of the humidifier at the cathode side is set as Thum,caset=29.5°C. The temperature profile of the cathode humidifier in the simulation is depicted in Figure 4. It is illustrated that the humidifier performs a slow dynamics of reducing its temperature to reach the idea operation temperature. Due to lack of the effective cooling system, the humidifier takes almost 2000 s to reduce its temperature, which implies that reducing the temperature of input gases is not effective enough to regulate PEMFC temperature. The temperature of line heaters is chosen as the average value between the humidifier temperature with the desired PEMFC temperature, which fulfills the purpose of line heaters to maintain the gas temperature in the pipelines.Figure 4
The profile of the humidifier temperature at the cathode side.Based on the USDE in (29) and (50), only one tuning constant is set as κ=10−3. For the PI controller, the gain coefficients are kp=4 and ki=1, which are determined based on the tuning rule in the frequency domain [27] in order to maintain the stable performance. The comparison of temperature profiles regulated by the proposed composite PI controllers ((37) and (57)) and the classical PI is shown in Figure 5. It is illustrated that the controllers ((37) and (57)) and the classical PI can achieve the temperature regulation. In the steady-state, the PEMFC temperature can reach to the desired value by the controllers. However, it is noted that the composite PI control performs much better than the classical PI control in the transient performance. The control errors in the composite PI controllers are small and bounded. Moreover, the convergence speed of the composite PI controllers is faster than that of the classical PI controller.Figure 5
The temperature profiles regulated by the proposed composite PI controller ((37) and (57)) and the classical PI.Before the control input enters into the system, the anti-windup scheme is used to avoid the input mass flows in the negative value or over the system bounds. The control inputu of the total energy is given in Figure 6. In practice, the additional mass flow of the air is considered as a practical control input. Based on the relation in (19), the additional mass flow of the air can be obtained and the profile of the additional air mass flow is depicted in Figure 7. It is shown that the anti-windup has a good performance such that the additional mass flow of the air cannot exceed the system constraint.Figure 6
The profiles of the control input by classical PI and composite PI ((37) and (57)).Figure 7
The profiles of the additional air mass flows by classical PI and composite PI ((37) and (57)).Since the mass flow of the air is changed during the control process, the partial pressure of oxygen will be changed which has an impact on the PEMFC voltage of (10). Moreover, there are many variables in the PEMFC voltage related to the temperature, which has been discussed in (12)–(15). The voltage profile affected by the proposed composite PI controllers ((37) and (57)) and the classical PI is depicted in Figure 8, which implies that the proposed control strategy has a small impact to the PEMFC voltage. Besides, the profile of the cooling energy generated by fan 1 is shown in Figure 9. It is noticed that the fan turns on when the PEMFC temperature reduces. Thus, there are no coupling effects between the controller and the fan operation mode. Figure 10 shows the result profile of estimated unknown dynamics D1 and D2. It is illustrated that the USDE can achieve accurate estimation. There are some sharp transient peaks in D2, which are affected by the change of the desired temperature and the fan operation condition.Figure 8
The voltage profiles regulated by the proposed composite PI controllers ((37) and (57)) and the classical PI.Figure 9
Simulation profile of the cooling energy by fan 1.Figure 10
Estimation result of the unknown dynamicsD1 and D2 by the estimators ((29) and (50)).
(a)(b)For the presented simulation results above, the anti-windup scheme plays an important role in the proposed control strategy. In order to further show the efficacy of composite PI controllers, the amplitude of the desired temperature is set as a small variation. The control inputu for this case is given in Figure 11. It is shown that the unknown system dynamics estimators ((29) and (50)) can compensate the unknown dynamics in the composite PI controllers ((37) and (57)). The regulated temperature profiles by the proposed composite PI controllers ((37) and (57)) and the classic PI controller are depicted in Figure 12. It is shown that the temperature errors by the composite PI controllers ((37) and (57)) are always small while the result by the classic PI controller is affected by the unknown dynamics D1 and D2 such that the temperature cannot be regulated to the desired temperature.Figure 11
The profiles of the control input by classical PI and composite PI ((37) and (57)) for the small variation of temperature regulation.Figure 12
The small variation of temperature regulation profiles by the proposed composite PI controller ((37) and (57)) and the classical PI.


## 6. Conclusions

In this paper, a composite PI control with the USDE is proposed for a PEMFC system. Considering the limitation of the cooling fan system, an alternative control strategy is developed by reducing the temperature of input air through the humidifier and simultaneously increasing the mass flow of air in order to eliminate the excess heat that a cooling fan cannot remove. Then, a low-pass filter is applied to the system in order to obtain an invariant manifold. This invariant manifold contains the information between the unknown dynamics with the known variables, which can be directly used to construct the estimator. Moreover, the requirement of the state derivative can be avoided which is suitable for practical application. The derived estimator can ensure the exponential convergence and is easily incorporated into the proposed control strategy such that the performance of the PI control is improved and the effect of unknown dynamics can be eliminated. Finally, numerical simulations of the PEMFC system are provided. The simulation results demonstrate the efficacy of the proposed control strategy and the improvement of control performance.



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*Source: 8822835-2020-10-23.xml*