Finite-Time Command Filtered Control for Oxygen-Excess Ratio of Proton Exchange Membrane Fuel Cell Systems with Prescribed Performance

Abstract

In recent years, proton exchange membrane fuel cell (PEMFC) has received growing attention as a new sustainable energy source because of its high-power density and zero-emission. In the PEMFC system, the air supply control has a significant impact on the efficiency and lifetime of the PEMFC stack. However, external disturbances and output constraints regularly have negative effects on air supply control. This paper aims to investigate a novel system analysis and advanced strategy control for the oxygen-excess ratio of a PEMFC system under the variant load current disturbance. The air-supply dynamic model is established which takes into account the supply manifolds, compressor, and the PEMFC stack. The proposed control method is designed based on finite-time command-filter control (FTCFC) to improve the tracking performance and ensure the finite-time convergence. Moreover, owing to the suggested prescribed performance function, the oxygen-excess ratio output remains in the pre-boundedness. Theoretical analysis exhibits that the closed-loop system stability is guaranteed by the Lyapunov theory. Finally, the simulation and hardware-in-loop (HIL) experiments are carried out on MATLAB environment and a 100 W power PEMFC system to validate the effectiveness of the suggested methodology.

1. Introduction

Owning the many advantages of zero emissions, low operating temperature, great stability, and low noise, the fuel cell system is applied for industry applications (including hybrid construction machinery [1], electric vehicles [2,3], tramway systems [4], etc.). Among the presented fuel cell in [5,6], the new type of energy resource, called the proton exchange membrane fuel cell, is the potential and promising energy device which has been paid a lot of attention. The PEMFC system comprises four subsystems: the hydrogen supply in the anode side, the air supply in the cathode side, the heat and stack temperature subsystem, and the humidity subsystems [7,8]. Therefore, the effective operation of PEMFC relies on the control problem of these above subsystems to against the fast load variation. It is noted that the control of the air supply system plays an important key and has great effect in the operation process of the PEMFC. Hence, the development of the air supply system becomes a challenge issue to prevent them membrane degradation, maintains high efficiency and prolongs the fuel cell life.

In the air supply system, oxygen is provided in the cathode side that can react with hydrogen via a membrane to generate continuous electricity and water. However, it is difficult to prevent oxygen starvation while hydrogen is always guaranteed [9]. Therefore, the air supply system with oxygen excess ratio (OER) received much attention. Herein, the OER denotes the ratio of the reacted amount of oxygen with the supplied oxygen. When the OER is regulated to track the optimal value with a small error, the maximum efficiency of PEMFC will be achieved. In contrast, the lack of oxygen will reduce the PEMFC’s efficiency and even jeopardize the polymeric membranes [10]. Hence, the model-based control design for the PEMFC air system has been presented to optimize the output power while providing enough oxygen. In [11], Grujicic et al. formulated a nine-order state model to retain a necessary level of oxygen for the PEMFC. In [12], a nonlinear seven-order model of a laboratory PEMFC was established to deal with the air supply control problem. However, the above high-order model is very complex, it causes some difficulties in the controller design method. By using several assumptions such as the satisfied conditions of the temperature and humidity subsystem, a simplified third-order model was presented in recent studies [13,14,15]. Nonetheless, the advanced control strategies applied to most of the above-mentioned models for the air supply system have been still limited. Therefore, it is significant and challenging to suggest a finite-time command filtered backstepping control scheme for PEMFC air-feed system within the performance constraint (PC).

As an outstanding robust control and simple design procedure, backstepping control (BC) is an effective approach to enhance the performance of the system and is widely used in various practical applications of the PEMFC system, such as [16,17,18]. In [16], the backstepping method with an extended state observer to regulate the cathode pressure subsystem of the PEMFC stack was reported. The authors in [17] presented an adaptive backstepping sliding mode controller to realize the air-feeding control. In [18], the BC was applied to an uncertain PEMFC air delivery system by using a neural network. However, the BC method often contains computational complexity due to the requirement of the derivative of the virtual control law. The command filtered control (CFC), which was first proposed by Farrell [19], not only addressed this issue but also improved the system performance. This method was applied to many applications such as a class of nonlinear system [20], dual-motor servo system [21], and so on. It is noted that the tracking error of the BC and CFC methods can only ensure asymptotic convergence, while the finite-time convergence control scheme has still been opened issue [22].

Besides, the performance constraints in real systems are regarded as an interesting research topic. To ensure prescribed tracking performance, several previous works proposed the funnel function using the backstepping method [23] and the barrier Lyapunov function controller [24]. Nevertheless, only a few works are taken into consideration for PEMFC air-feed system control. In [25], asymmetric OER constraints were used to avoid oxygen starvation, and then an adaptive control strategy was designed to tackle the system nonlinearities. However, all signals of the closed-loop system are only uniformly ultimately bounded. In addition, the validity of the presented strategy is not checked by the experimental data. How to apply these benefit methods, for instance, CFC, and finite-time control for the PEMFC to achieve the PC is still a significant problem.

Based on the literature on the relevant control algorithm, in this paper, the third-order PEMFC air delivery system is put forward and then the FTCFC control scheme is proposed. Compared with the previous works, we summarize the paper contributions as follows: (1) According to the author’s knowledge, this is the first time the FTCFC method applied for the third-order PEMFC air supply system with prescribed performance; (2) Compared with the existing conventional CFC method, the proposed strategy not only warrants the finite-time convergence of all the closed-loop system with load current variation but also enhances the tracking performance; (3) Both simulation and HIL experiment aspects is carried out to validate the effectiveness of the proposed controller.

The paper is arranged as follows. In Section 2, the nonlinear third-order PEMFC air supply system to be considered is established. In Section 3, we present the FTCFC controller design with the prescribed performance and the system stability is analyzed. In Section 4, the simulation results and experiment verification are given. Finally, the conclusion is highlighted in Section 5.

2. Air Supply Model Development of the PEMFC System

The PEMFC system consists of four subsystems: hydrogen supply, air supply, humidity system, and temperature system as shown in Figure 1. To facilitate and simply for procedure air supply system design, other subsystems are supposed to be in a suitable working condition. The components of air supply system are illustrated on the right side of Figure 1. The compressor generates the air which is handled by the cooler and humidified. The air then enters the supply manifold to adjust the pressure before going to the cathode of the PEMFC stack. Finally, the electricity produced by the PEMFC will be provided for the load. There are a few works that constructed the nonlinear dynamic model of the PEMFC air supply system [7,11,13]. In this paper, a reduced third-order PEMFC air supply system is developed, which is presented as follows:

2.1. Cathode Pressures Dynamics

The mass conservation and thermodynamic features are applied to model the behaviour of the air inside the cathode. The dynamics of the cathode pressures are formulated as follows [7,11]:

$$\begin{array}{l}\frac{d{p}_{\mathrm{ca}}}{dt}=\frac{R{T}_{\mathrm{ca}}}{M_{\mathrm{a},\mathrm{ca}}{V}_{\mathrm{ca}}}\left(W_{\mathrm{ca},\mathrm{in}}-W_{\mathrm{ca},\mathrm{rea} }-W_{\mathrm{ca}, \mathrm{out} }\right),\\ W_{\mathrm{ca},\mathrm{in}}=k_{\mathrm{ca},\mathrm{in}}\left(p_{\mathrm{sm}}-p_{\mathrm{ca}}\right),\\ W_{\mathrm{ca},\mathrm{rea} }=\frac{n{I}_{\mathrm{st}}{M}_{{\mathrm{O}}_2}}{4F},\\ W_{\mathrm{ca}, \mathrm{out} }=\frac{R{T}_{\mathrm{st}}}{V_{\mathrm{ca}}}\frac{C_{\mathrm{D}}{A}_{\mathrm{T}}}{\sqrt{R{T}_{\mathrm{st}}}}{\gamma }^{\frac{1}{2}}{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{2(\gamma -1)}}{p}_{\mathrm{ca}},\end{array}$$

(1)

where $W_{\mathrm{ca},\mathrm{in}} \mathrm{and} W_{\mathrm{ca},\mathrm{rea} }$, in turn, are the mass flow input and reactions of the cathode; $W_{\mathrm{ca}, \mathrm{out} }$ is the flow out of the cathode which can be obtained based on the previous works in [9,13]; p_ca is the cathode pressure, p_sm represents the air pressure in the manifold; R, T_ca, and T_st are universal gas constant, cathode, and stack operating temperatures, respectively; M_a,ca and $M_{{\mathrm{O}}_2}$ are the molar mass of air in cathode side and oxygen, respectively; I_st is the stack current; F is the Faraday’s constant; n denotes the number of cells of the PEMFC stack; V_ca is the volume of the cathode side; k_ca,in, C_D, and γ are positive constants, respectively.

2.2. Air Supply Manifold Pressure Dynamics

The manifold pressure dynamics is expressed by the following equation:

$$\frac{d{p}_{\mathrm{sm}}}{dt}=\frac{R{T}_{\mathrm{cp}}}{V_{\mathrm{sm}}}\left[W_{\mathrm{cp}}-k_{\mathrm{ca},\mathrm{in}}\left(p_{\mathrm{sm}}-p_{\mathrm{ca}}\right)\right],$$

(2)

where $T_{\mathrm{cp}}=T_{\mathrm{atm}}+\frac{T_{\mathrm{atm}}}{{\eta }_{\mathrm{cp}}}\left[{\left(\frac{p_{\mathrm{sm}}}{p_{\mathrm{atm}}}\right)}^{\frac{\gamma -1}{\gamma }}-1\right],$ W_cp and T_cp define the compressor flow map and air temperature in the compressor, respectively; η_cp represents the transfer efficiency; T_atm and p_atm are the atmospheric temperature and pressure, respectively; V_sm is the volume of the supply manifold.

2.3. Air Compressor Model

The air compressor is controlled by a direct-current (DC) motor, which supplies oxygen to the manifold before entering the fuel cell cathode side. The dynamics model of the air compressor is established by the following equation:

$$\frac{d{\omega }_{\mathrm{cp}}}{dt}=\frac{1}{J_{\mathrm{cp}}}\left({\tau }_{\mathrm{cm}}-{\tau }_{\mathrm{cp}}-{\tau }_{\mathrm{f}}\right),$$

(3)

where J_cp is the rotational inertia of the compressor rotor; τ_cm, τ_f, and τ_cp are, in turn, the electromagnetic torque generated by the permanent magnet synchronous motor, the friction torque, and the compressor load torque. They can be calculated as follows:

$$\begin{array}{l}{\tau }_{\mathrm{cm}}=k_t{\eta }_{\mathrm{cm}}{v}_{\mathrm{cm}},\\ {\tau }_{\mathrm{cp}}=\frac{C_{\mathrm{p}}}{{\omega }_{\mathrm{cp}}}\frac{T_{\mathrm{atm}}}{{\eta }_{\mathrm{cp}}}\left[{\left(\frac{p_{\mathrm{sm}}}{p_{\mathrm{atm}}}\right)}^{\frac{\gamma -1}{\gamma }}-1\right]W_{\mathrm{cp}},\\ {\tau }_{\mathrm{f}}=k_{\mathrm{f}}{\omega }_{\mathrm{cp}},\end{array}$$

(4)

where ω_cp is the rotational speed of the compressor motor; η_cm represents the transfer efficiency of compressor motor-related quantity; k_f is friction coefficient; C_p is compressor related quantity; k_t is motor constant.

Assume that the mass flow rate is proportional to the compressor speed [26], it is given as follows:

$$W_{\mathrm{cp}}=\frac{1}{2\pi }{\eta }_{v-c}{V}_{\mathrm{cpr}/\mathrm{tr}}{\rho }_a{\omega }_{\mathrm{cp}},$$

(5)

where ${\eta }_{v-c} \mathrm{and} {\rho }_a$ denote the volumetric efficiency and the air density, respectively; $V_{\mathrm{cpr}/\mathrm{tr}}$ is the compressed volume per turn.

2.4. Mathematical Model of PEMFC Air Supply System

From Equations (1) to (5), the control-oriented model of air supply system for PEMFC with the load current disturbance is presented as follows:

$$\{\begin{array}{l}{\dot{p}}_{\mathrm{ca}}=\frac{R{T}_{\mathrm{ca}}}{M_{\mathrm{a},\mathrm{ca}}{V}_{\mathrm{ca}}}\left(k_{\mathrm{ca},\mathrm{in}}\left(p_{\mathrm{sm}}-p_{\mathrm{ca}}\right)-\frac{R{T}_{\mathrm{st}}}{V_{\mathrm{ca}}}\frac{C_{\mathrm{D}}{A}_{\mathrm{T}}}{\sqrt{R{T}_{\mathrm{st}}}}{\gamma }^{\frac{1}{2}}{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{2(\gamma -1)}}{p}_{\mathrm{ca}}\right)-\frac{R{T}_{\mathrm{ca}}}{M_{\mathrm{a},\mathrm{ca}}{V}_{\mathrm{ca}}}\frac{n{M}_{{\mathrm{O}}_2}}{4F}{I}_{\mathrm{st}},\\ {\dot{p}}_{\mathrm{sm}}=\frac{R}{V_{\mathrm{sm}}}\left(T_{\mathrm{atm}}+\frac{T_{\mathrm{atm}}}{{\eta }_{\mathrm{cp}}}\left[{\left(\frac{p_{\mathrm{sm}}}{p_{\mathrm{atm}}}\right)}^{\frac{\gamma -1}{\gamma }}-1\right]\right)\left[W_{\mathrm{cp}}-k_{\mathrm{ca},\mathrm{in}}\left(p_{\mathrm{sm}}-p_{\mathrm{ca}}\right)\right],\\ {\dot{\omega }}_{\mathrm{cp}}=\frac{1}{J_{\mathrm{cp}}}\left(k_t{\eta }_{\mathrm{cm}}{v}_{\mathrm{cm}}-\frac{C_{\mathrm{p}}}{{\omega }_{\mathrm{cp}}}\frac{T_{\mathrm{atm}}}{{\eta }_{\mathrm{cp}}}\left[{\left(\frac{p_{\mathrm{sm}}}{p_{\mathrm{atm}}}\right)}^{\frac{\gamma -1}{\gamma }}-1\right]W_{\mathrm{cp}}-k_{\mathrm{f}}{\omega }_{\mathrm{cp}}\right).\end{array}$$

(6)

Define the state vector $x={\left[x_1,x_2,x_3\right]}^T={\left[p_{\mathrm{ca}},p_{\mathrm{sm}},{\omega }_{\mathrm{cp}}\right]}^T$, the model dynamics of the air supply system can be rewritten as follows:

$$\{\begin{array}{l}{\dot{x}}_1=-{\sigma }_1{x}_1+{\sigma }_2{x}_2-{\sigma }_3{I}_{\mathrm{st}},\\ {\dot{x}}_2=\left\{{\sigma }_4+{\sigma }_5\left[{\left(\frac{x_2}{{\sigma }_7}\right)}^{{\sigma }_6}-1\right]\right\}\left[{\sigma }_{12}{x}_3-{\sigma }_8\left(x_2-x_1\right)\right],\\ {\dot{x}}_3=-{\sigma }_9{x}_3-\frac{{\sigma }_{10}}{x_3}\left[{\left(\frac{x_2}{{\sigma }_7}\right)}^{{\sigma }_6}-1\right]{\sigma }_{12}{x}_3+{\sigma }_{11}u,\end{array}$$

(7)

where x₁ is the cathode pressure p_ca, x₂ is the air pressure in the manifold p_sm, x₃ is the compressor angular speed ω_cp; u = v_cm denotes the voltage control signal applied on the compressor motor; I_st is the stack current which can be directly measured; σ_i (i = 1, 2, …, 12) are predefined constants that are listed as follows:

$$\begin{array}{l}{\sigma }_1=\frac{R{T}_{\mathrm{ca}}}{M_{\mathrm{a},\mathrm{ca}}{V}_{\mathrm{ca}}}\left(k_{\mathrm{ca},\mathrm{in}}+\frac{R{T}_{\mathrm{st}}}{V_{\mathrm{ca}}}\frac{C_{\mathrm{D}}{A}_{\mathrm{T}}}{\sqrt{R{T}_{\mathrm{st}}}}{\gamma }^{\frac{1}{2}}{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{2(\gamma -1)}}\right),{\sigma }_2=\frac{k_{\mathrm{ca},\mathrm{in}}R{T}_{\mathrm{ca}}}{M_{\mathrm{a},\mathrm{ca}}{V}_{\mathrm{ca}}};{\sigma }_3=\frac{R{T}_{\mathrm{ca}}}{M_{\mathrm{a},\mathrm{ca}}{V}_{\mathrm{ca}}}\frac{n{M}_{{\mathrm{O}}_2}}{4F},\\ {\sigma }_4=\frac{R{T}_{\mathrm{atm}}}{V_{\mathrm{sm}}},{\sigma }_5=\frac{{\sigma }_4}{{\eta }_{\mathrm{cp}}},{\sigma }_6=\frac{\gamma -1}{\gamma },{\sigma }_7=p_{\mathrm{atm}},{\sigma }_8=k_{\mathrm{ca},\mathrm{in}},\\ {\sigma }_9=\frac{k_{\mathrm{f}}}{J_{\mathrm{cp}}},{\sigma }_{10}=\frac{C_{\mathrm{p}}}{J_{\mathrm{cp}}}\frac{T_{\mathrm{atm}}}{{\eta }_{\mathrm{cp}}},{\sigma }_{11}=\frac{k_t{\eta }_{\mathrm{cm}}}{J_{\mathrm{cp}}},{\sigma }_{12}=\frac{1}{2\pi }{\eta }_{v-c}{V}_{\mathrm{cpr}/\mathrm{tr}}{\rho }_a.\end{array}$$

The system (7) can be re-expressed as follows:

$$\leftrightarrow \{\begin{array}{l}{\dot{x}}_1=f_1\left(x_1\right)+b_1{x}_2+d_1,\\ {\dot{x}}_2=f_2\left(x_1,x_2\right)+b_2{x}_3,\\ {\dot{x}}_3=f_3\left(x_2,x_3\right)+b_{3}u,\end{array}$$

(8)

where d₁ is a measurable disturbance, f_i and b_i (i = 1, 2, 3), in turn, are the known nonlinear functions that are defined by

$$\begin{array}{l}{f}_1=-{\sigma }_1{x}_1,f_2=-{\sigma }_8\left(x_2-x_1\right)\left\{{\sigma }_4+{\sigma }_5\left[{\left(\frac{x_2}{{\sigma }_7}\right)}^{{\sigma }_6}-1\right]\right\},f_3=-{\sigma }_9{x}_3-\frac{{\sigma }_{10}}{x_3}\left[{\left(\frac{x_2}{{\sigma }_7}\right)}^{{\sigma }_6}-1\right]{\sigma }_{12}{x}_3,\\ b_1={\sigma }_2,b_2={\sigma }_{12}\left\{{\sigma }_4+{\sigma }_5\left[{\left(\frac{x_2}{{\sigma }_7}\right)}^{{\sigma }_6}-1\right]\right\},b_3={\sigma }_{11},d_1=-{\sigma }_3{I}_{\mathrm{st}}.\end{array}$$

The parameters of the PEMFC are described in

Table 1. H100 PEMFC parameters.

Symbol	Descriptions	Value [Unit]
R	Universal gas constant	8.31447 [J/(mol.K)]
T	Operating temperature	343 [K]
F	Faraday constant	96485 [C/mol]
Tca	Temperature of cathode	298.15 [K]
Tst	Temperature of FC stack	353.15 [K]
Ma,ca	Air molar mass	28.964 × 10−3 [kg/mol]
MO2	Oxygen molar mass	32 × 10−3 [kg/mol]
CD	Nozzle discharge coefficient	0.0124
Cp	Air specific heat capacity	1004 [J/(kg.K)]
Vca	Volume of cathode	0.01 [m3]
Vsm	Volume of supply manifold	0.02 [m3]
Vcpr/tr	Compressor volume per turn	5 × 10−4 [m3/tr]
kf	Friction coefficient	0.0153 [(N.s)/rad]
kca,in	Cathode inlet constant	3.629 × 10−6 [kg/(Pa.s)]
kt	Motor constant	0.0513 [N.m/rad]
patm	Atmospheric pressure	1.01325 [atm]
AT	Nozzle opening area	0.005 [m2]
γ	Air ratio of specific heats	1.4
ηcp	Compressor efficiency	0.8
ηcm	Motor mechanical efficiency	0.98
ηv−c	Volumetric efficiency	0.98
Jcp	Compressor and motor inertia	671.9 × 10−5 [kg.m2]
ρa	Air density	1.23 [kg/m3]
n	Number of cells	20

, which is validated by experimental data and reported in our previous work [27].

Remark 1. 

The measured output variables include $V_{\mathrm{st}},x_1,x_2, and x_3$, where V_st is the PEMFC voltage; x₁and x₂are measured using pressure sensors, respectively. x₃is measured by the taking derivative of the position sensor.

The control objective is to construct the control signal that will regulate the oxygen-excess ratio around an optimal setpoint in presence of the load variation. The oxygen excess ratio is determined by the mass of the oxygen supplied into the cathode $W_{{\mathrm{O}}_2,\mathrm{in}}$ and the ratio of the consumed oxygen $W_{{\mathrm{O}}_2,\mathrm{act}}$ that is described as follows [11,26]:

$${\lambda }_{{\mathrm{O}}_2}=\frac{W_{{\mathrm{O}}_2,\mathrm{in}}}{W_{{\mathrm{O}}_2,\mathrm{act}}}=\frac{\frac{y_{{\mathrm{O}}_2}}{1+{\Omega }_{\mathrm{atm}}}{W}_{\mathrm{sm},\mathrm{out}}}{M_{{\mathrm{O}}_2}\frac{n{I}_{\mathrm{st}}}{4F}}=\frac{\frac{y_{{\mathrm{O}}_2}}{1+{\Omega }_{\mathrm{atm}}}{k}_{\mathrm{sm},\mathrm{out}}\left(p_{\mathrm{sm}}-p_{\mathrm{ca}}\right)}{M_{{\mathrm{O}}_2}\frac{n{I}_{\mathrm{st}}}{4F}}=\frac{{\sigma }_{12}\left(x_2-x_1\right)}{{\sigma }_3{I}_{\mathrm{st}}},$$

(9)

where $y_{{\mathrm{O}}_2},{\Omega }_{\mathrm{atm}}, \mathrm{and} k_{\mathrm{sm},\mathrm{out}}$ are the oxygen mole fraction, the atmospheric humidity ratio at the compressor inlet, and a gain factor, respectively.

In this paper, the desired value is set as ${\lambda }_{{\mathrm{O}}_2}^{*}=2$. To facilitate the control design, the output of Equation (8) and the desired output signal are described as:

$$\begin{array}{l}y={\lambda }_{{\mathrm{O}}_2}={\sigma }_{13}\left(x_2-x_1\right),\\ y_d={\lambda }_{{\mathrm{O}}_2}^{*}=2,\end{array}$$

(10)

where ${\sigma }_{13}={\sigma }_{12}/\left({\sigma }_3{I}_{\mathrm{st}}\right).$

Lemma 1 

([22]). For 0 < a ≤ 1 and ${\varsigma }_j\in R$, j = 1, 2, …, n, the following inequality holds:

$${\left({\displaystyle \sum _{j=1}^n\left|{\varsigma }_i\right|}\right)}^a\le {\displaystyle \sum _{j=1}^n{\left|{\varsigma }_j\right|}^a}.$$

(11)

Lemma 2 

([21]). Consider a nonlinear system $\dot{x}=f\left(x\right)$. Let V (x)≥ 0 be a continuous and smooth function. When the following inequality holds:

$$\dot{V}\left(x\right)\le -H_{1}V\left(x\right)-H_2{V}^{\beta }\left(x\right)+C,$$

(12)

where $H_1 \mathrm{and} H_2$ are the positive real numbers, respectively; 0 < β < 1 and 0 < C < ∞. Then, the equilibrium points of system $\dot{x}=f\left(x\right)$ is practical finite-time stable (PFTS), and the convergence time T meets

$$\begin{array}{c}T\le T_c=\max \{t_0+\frac{1}{{\lambda }_0{H}_1\left(1-\beta \right)}\ln \frac{{\lambda }_0{H}_1{V}^{1-\beta }\left(t_0\right)+H_2}{H_2}\\ t_0+\frac{1}{H_1\left(1-\beta \right)}\ln \frac{H_1{V}^{1-\beta }\left(t_0\right)+{\lambda }_0{H}_2}{{\lambda }_0{H}_2}\},\end{array}$$

(13)

where 0 < λ₀ < 1, t₀ is the initial time. The residual set of the solution of system $\dot{x}=f\left(x\right)$ can be determined by

$$x\in \left\{x : \underset{t\to T_c}{\lim }V(x)\le \min \left\{\frac{C}{\left(1-{\lambda }_0\right)H_1},{\left(\frac{C}{\left(1-{\lambda }_0\right)H_2}\right)}^{\frac{1}{\beta }}\right\}\right\}.$$

(14)

3. Adaptive Robust Controller and Stability Analysis

In this section, the suggested control strategies are constructed to maintain the OER at the desired value of 2 by regulating the compressor motor speed, which is controlled by the voltage control signal as shown in Figure 2. The proposed algorithm is FTCFC with the prescribed performance function that ensures not only convergence in finite-time but also remaining in the predefined boundness of the tracking error signal.

3.1. Controller Design Using FTCFC Technique

Defining the error $e_1=y-y_d$, the derivative of this error is expressed as:

$${\dot{e}}_1=\dot{y}-{\dot{y}}_d={\sigma }_{13}\left({\dot{x}}_2-{\dot{x}}_1\right)-{\dot{y}}_d={\sigma }_{13}\left(f_2+b_2{x}_3-f_1-b_1{x}_2-d_1\right)-{\dot{y}}_d.$$

(15)

An asymmetric prescribed performance can be formed as follows:

$$e_L<e_1<e_U,$$

(16)

where $e_L \mathrm{and} e_U$ denote the lower and upper bound of the error and can be represented by

$$\{\begin{array}{l}{e}_U(t)=\left({\rho }_U-{\rho }_{\infty }\right)e^{-T_{U}t}+{\rho }_{\infty },\hfill \\ e_L(t)=\left({\rho }_L+{\rho }_{\infty }\right)e^{-T_{L}t}-{\rho }_{\infty },\hfill \end{array}$$

(17)

where ${\rho }_U,{\rho }_L, \mathrm{and} {\rho }_{\infty }$ are properly positive constants, respectively; $T_U \mathrm{and} T_L$ are selected positive parameters that define the convergence rates of exponential functions, respectively.

In order to achieve the prescribed tracking performance of OER, a transformed error ε is defined as follows:

$$\begin{array}{l}\epsilon (t)=\frac{{\zeta }_1(t)}{1-{\zeta }_1^2(t)},\\ {\zeta }_1(t)=\frac{2e_1(t)-(e_U(t)+e_L(t))}{e_U(t)-e_L(t)}.\end{array}$$

(18)

The derivative of the transformed error is calculated by

$$\dot{\epsilon }(t)=s_1\left({\dot{e}}_1(t)+v_1\right),$$

(19)

where $s_1=\frac{2\left(1+{\zeta }_1^2(t)\right)}{{\left(1-{\zeta }_1^2(t)\right)}^2(e_U(t)-e_L(t))} \mathrm{and} v_1=\frac{{\zeta }_1(t)({\dot{e}}_U(t)-{\dot{e}}_L(t))-({\dot{e}}_U(t)+{\dot{e}}_L(t))}{2}.$

Remark 2: 

If the control law can guarantee that the transformed error ε(t) is bounded for ∀t ≥ 0, the prescribed performance (16) will be obtained [23].

Substituting Equation (15) to Equation (19), we obtain

$$\dot{\epsilon }(t)=s_1\left({\sigma }_{13}\left(f_2+b_2{x}_3-f_1-b_1{x}_2-d_1\right)-{\dot{y}}_d+v_1\right).$$

(20)

Defining the error $e_2=x_3-x_2{}_{d,c}$. In order to avoid the burn computation, the finite-time command filter is constructed as follows:

$$\begin{array}{l}{\dot{\phi }}_{11}=v_{11},\\ v_{11}=-r_{11}{\left|{\phi }_{11}-x_{2d}\right|}^{\frac{1}{2}}\mathrm{sign}\left({\phi }_{11}-x_{2d}\right)+{\phi }_{12},\\ {\dot{\phi }}_{12}=-r_{12}\mathrm{sign}\left({\phi }_{12}-v_{11}\right),\end{array}$$

(21)

where x_2d is the input of the filter that is designed later, $x_{2d,c}(t)={\phi }_{11}(t) \mathrm{and}\hspace{0.17em}{\dot{x}}_{2d,c}(t)=v_{11}(t)$ define the outputs of filter, respectively. $r_{11} \mathrm{and} r_{12}$ are positive constants, respectively.

The intermediate signal function $x_{2d}$ and the final control law u during the procedure of the finite-time command filtered control design is given by:

$$x_{2d}=\frac{1}{b_2}\left(-f_2+f_1+b_1{x}_2+d_1-\frac{1}{{\sigma }_{13}}\left(k_1{s}_1\epsilon -{\dot{y}}_d+v_1-{\eta }_1{s}_1{r}_1^{\lambda }\right)\right),$$

(22)

$$u=\frac{1}{b_3}\left(-f_3+{\dot{x}}_{2d,c}-k_2{e}_2-{\sigma }_{13}{b}_2{s}_1{r}_1-{\eta }_2{r}_2^{\lambda }\right),$$

(23)

where k₁, k₂, η₁, and η₂ are positive scalars, respectively. 0 < λ < 1, r₁ and r₂ are the compensated tracking error signals and denoted as:

$$r_1=\epsilon -{\xi }_1,r_2=e_2-{\xi }_2,$$

(24)

where ${\xi }_1 \mathrm{and} {\xi }_2$ are the compensation signals with ${\xi }_1\left(0\right) \mathrm{and} {\xi }_2\left(0\right)=0,$ respectively.

The derivative of the compensation signals can be constructed as

$$\begin{array}{l}{\dot{\xi }}_1=-k_1{s}_1^2{\xi }_1+b_2{\sigma }_{13}{s}_1\left(x_{2d,c}-x_{2d}\right)+b_2{\sigma }_{13}{s}_1{\xi }_2-{\mathsf{\Gamma }}_1{s}_1\mathrm{sign}\left({\xi }_1\right),\\ {\dot{\xi }}_2=-k_2{\xi }_2-{\mathsf{\Gamma }}_2\mathrm{sign}\left({\xi }_2\right),\end{array}$$

(25)

where ${\mathsf{\Gamma }}_1 \mathrm{and} {\mathsf{\Gamma }}_2$ are positive constants, respectively.

3.2. Stability Analysis

Theorem 1. 

For nonlinear PEMFC air-feed system (8) with the variation of load current, the asymmetric prescribed performance function (16), the virtual control function (22), the real final control input (23), and the compensation signals (25) can ensure that the output tracking error signals e₁(t) remain bounded within the preselected transient and steady bounds for∀t > 0, and all signals in the system are bounded. Besides, the system OER output y can be warranted to track the reference signal y_d in finite-time.

Proof of Theorem 1. 

For the oxygen excess ratio subsystem, the Lyapunov function is selected as

$$V_1=\frac{1}{2}{r}_1^2.$$

(26)

The time derivative of V₁ is computed as

$$\begin{array}{l}\dot{V}=r_1{\dot{r}}_1=r_1\left(\dot{\epsilon }-{\dot{\xi }}_1\right)=r_1\left(s_1\left({\sigma }_{13}\left(f_2+b_2{x}_3-f_1-b_1{x}_2-d_1\right)-{\dot{y}}_d+v_1\right)-{\dot{\xi }}_1\right)\\ =r_1\left(s_1\left({\sigma }_{13}\left(f_2+b_2\left(x_{2d,c}+e_2\right)-f_1-b_1{x}_2-d_1\right)-{\dot{y}}_d+v_1\right)-{\dot{\xi }}_1\right)\\ =r_1\left(s_1\left({\sigma }_{13}\left(f_2+b_2{e}_2+b_2{x}_{2d}+b_2\left(x_{2d,c}-x_{2d}\right)-f_1-b_1{x}_2-d_1\right)-{\dot{y}}_d+v_1\right)-{\dot{\xi }}_1\right).\end{array}$$

(27)

By substituting the virtual control signal $x_3{}_d$ in (22) and the derivative of compensation signal ξ₁ in Equation (25) into Equation (27), one yields

$${\dot{V}}_1=r_1\left(-{\eta }_1{s}_1^2{r}_1^{\lambda }-k_1{s}_1^2{r}_1+s_1{\sigma }_{13}{b}_2{r}_2+{\mathsf{\Gamma }}_1{s}_1\mathrm{sign}\left({\xi }_1\right)\right).$$

(28)

For the second subsystem, the Lyapunov function V₂ is designed as

$$V_2=V_1+\frac{1}{2}{r}_2^2.$$

(29)

Based on Equation (8) and the compensated tracking error signal Equation (24), taking the derivative of V₂ obtains

$$\begin{array}{l}{\dot{V}}_2={\dot{V}}_1+r_2{\dot{r}}_2={\dot{V}}_1+r_2\left({\dot{e}}_2-{\dot{\xi }}_2\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}={\dot{V}}_1+r_2\left(f_3+b_{3}u-{\dot{x}}_{2d,c}-{\dot{\xi }}_2\right).\end{array}$$

(30)

By substituting Equations (23) and (25) into Equation (30), it follows that:

$$\begin{array}{l}{\dot{V}}_2={\dot{V}}_1+r_2\left(-k_2{e}_2-{\sigma }_{13}{b}_2{s}_1{r}_1-{\eta }_2{r}_2^{\lambda }+k_2{\xi }_2+{\mathsf{\Gamma }}_2\mathrm{sign}\left({\xi }_2\right)\right)\\ =r_1\left(-{\eta }_1{s}_1^2{r}_1^{\lambda }-k_1{s}_1^2{r}_1+{\mathsf{\Gamma }}_1{s}_1\mathrm{sign}\left({\xi }_1\right)\right)+r_2\left(-k_2{r}_2-{\eta }_2{r}_2^{\lambda }+{\mathsf{\Gamma }}_2\mathrm{sign}\left({\xi }_2\right)\right)\\ =-k_1{s}_1^2{r}_1^2-k_2{r}_2^2-{\eta }_1{s}_1^2{r}_1^{\lambda +1}-{\eta }_2{r}_2^{\lambda +1}+{\mathsf{\Gamma }}_1{s}_1{r}_1\mathrm{sign}\left({\xi }_1\right)+{\mathsf{\Gamma }}_2{r}_2\mathrm{sign}\left({\xi }_2\right).\end{array}$$

(31)

Applying the Young’s inequality, one yields

$$\begin{array}{l}{\mathsf{\Gamma }}_1{s}_1{r}_1\mathrm{sign}\left({\xi }_1\right)\le \frac{{\mathsf{\Gamma }}_1}{2}{\left(s_1{r}_1\right)}^2+\frac{{\mathsf{\Gamma }}_1}{2}\left|\mathrm{sign}\left({\xi }_1\right)\right|\le \frac{{\mathsf{\Gamma }}_1}{2}{\left(s_1{r}_1\right)}^2+\frac{{\mathsf{\Gamma }}_1}{2},\\ {\mathsf{\Gamma }}_2{r}_2\mathrm{sign}\left({\xi }_2\right)\le \frac{{\mathsf{\Gamma }}_2}{2}{r}_2^2+\frac{{\mathsf{\Gamma }}_2}{2}\left|\mathrm{sign}\left({\xi }_2\right)\right|\le \frac{{\mathsf{\Gamma }}_2}{2}{r}_2^2+\frac{{\mathsf{\Gamma }}_2}{2}.\end{array}$$

(32)

Substituting Equation (32) into Equation (31), and combining with Lemma 1 we obtain

$$\begin{array}{l}{\dot{V}}_2\le -\left(k_1-\frac{{\mathsf{\Gamma }}_1}{2}\right){\left(s_1{r}_1\right)}^2-\left(k_2-\frac{{\mathsf{\Gamma }}_2}{2}\right)r_2^2-{\eta }_1{s}_1^2{r}_1^{\lambda +1}-{\eta }_2{r}_2^{\lambda +1}+{\displaystyle \sum _{i=1}^2\frac{{\mathsf{\Gamma }}_i}{2}}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\le -{\alpha }_1{V}_2-{\alpha }_2{V}_2^{\frac{\lambda +1}{2}}+{\alpha }_3,\end{array}$$

(33)

where ${\alpha }_1=\min \left(2k_i-{\mathsf{\Gamma }}_i\right),{\alpha }_2=\min \left(2^{\frac{\lambda +1}{2}}{\eta }_i\right), \mathrm{and} {\alpha }_3={\displaystyle \sum _{i=1}^2\frac{{\mathsf{\Gamma }}_i}{2}},i=1,2$.

According to Lemma 2, if the following conditions $2k_i-{\mathsf{\Gamma }}_i>0, i=1,2$ are satisfied, then, r₁, r₂ will reach to the small region $\left|r_i\right|\le \min \left\{\sqrt{\frac{2{\alpha }_3}{\left(1-T_0\right){\alpha }_1}},\sqrt{2{\left(\frac{{\alpha }_3}{\left(1-T_0\right){\alpha }_2}\right)}^{\frac{2}{\lambda +1}}}\right\}$ in finite-time, where $0<T_0<1$. The convergence time T₁ is determined by

$$T_1\le \max \{\frac{1}{T_0{\alpha }_1\left(1-\frac{1+\lambda }{2}\right)}\ln \frac{T_0{\alpha }_1{V}^{1-\frac{1+\lambda }{2}}(0)+{\alpha }_2}{{\alpha }_2},\frac{1}{{\alpha }_1\left(1-\frac{1+\lambda }{2}\right)}\ln \frac{{\alpha }_1{V}^{1-\frac{1+\lambda }{2}}(0)+T_0{\alpha }_2}{T_0{\alpha }_2}\}.$$

(34)

This completes the proof. □

4. Simulation and Experiment Results

4.1. Numerical Simulation

4.1.1. Simulation Setup

In this section, we built the 100 W PEMFC model-based simulation platform to demonstrate the effectiveness and generalization of the proposed finite-time command filtered control with the prescribed performance (FTCFCPP) method. The simulation studies are performed in MATLAB/Simulink environment. Furthermore, the PID controller, and the conventional command-filtered controller with prescribed constrain (CFCPC) in [28] are given to compare the advantages of the suggested control strategy under various conditions.

Choosing the reference OER signal $y_d={\lambda }_{{\mathrm{O}}_2}^{*}=2$ and the PID controller as the comparison is designed as follows:

$$u_{pid}=k_{p}e\left(t\right)+k_i{\displaystyle \underset{0}{\overset{t}{\int }}e\left(t\right)dt}+k_d\frac{de\left(t\right)}{dt},$$

(35)

where the control gains of the PID controller are chosen: k_p = 30, k_i = 16, and k_d = 0.5.

The parameters of the PF are selected as ${\rho }_U=1, {\rho }_L=-1, {\rho }_{\infty }=0.15, T_U=20, T_L=20$. The gain parameters of the proposed controller are set as: $k_1=18, k_2=20, {\eta }_1=1.9,$ ${\eta }_2=3.5; r_{11}=200; r_{12}=50; {\mathsf{\Gamma }}_1=0.5, {\mathsf{\Gamma }}_2=0.1$. Meanwhile, the control parameters of the CFCPC are chosen with the same parameters of the proposed controller.

Remark 3: 

For a fair comparison, the control parameters of the CFCPC and the proposed method are adopted as the same values as each other. In the comparison with the CFCPC controller, the FTCFC approach not only has the advantages of the conventional CFC but also ensures finite-time convergence. Meanwhile, the PID gain parameters (i.e., k_p, k_i, k_d) are carefully designed to obtain the best OER tracking qualification with less overshoot.

The varying step of load current are considered to inspect the control performance as follows:

$$I_{\mathrm{st}}=\{\begin{array}{ll}0.5\hspace{0.17em}\hspace{0.17em}\mathrm{A}& 0\hspace{0.17em}\mathrm{s}\le t<40\hspace{0.17em}\mathrm{s},\\ 2.25\hspace{0.17em}\hspace{0.17em}\mathrm{A}& 40\hspace{0.17em}\mathrm{s}\le t<80\hspace{0.17em}\mathrm{s},\\ 1.0\hspace{0.17em}\hspace{0.17em}\mathrm{A}& 80\hspace{0.17em}\mathrm{s}\le t<120\hspace{0.17em}\mathrm{s},\\ 4.5\hspace{0.17em}\hspace{0.17em}\mathrm{A}& 120\hspace{0.17em}\mathrm{s}\le t\le 170\hspace{0.17em}\mathrm{s},\\ 0.75\hspace{0.17em}\mathrm{A}& 170\hspace{0.17em}\mathrm{s}\le t\le 210\hspace{0.17em}\mathrm{s},\\ 1.75\hspace{0.17em}\mathrm{A}& 210\hspace{0.17em}\mathrm{s}\le t\le 240\hspace{0.17em}\mathrm{s},\\ 0.5\hspace{0.17em}\mathrm{A}& 240\hspace{0.17em}\mathrm{s}\le t\le 270\hspace{0.17em}\mathrm{s},\\ 3.5\hspace{0.17em}\mathrm{A}& 270\hspace{0.17em}\mathrm{s}\le t\le 300\hspace{0.17em}\mathrm{s},\\ 1.5\hspace{0.17em}\mathrm{A}& 300\hspace{0.17em}\mathrm{s}\le t\le 330\hspace{0.17em}\mathrm{s},\\ 5\hspace{0.17em}\mathrm{A}& 330\hspace{0.17em}\mathrm{s}\le t\le 350\hspace{0.17em}\mathrm{s}.\end{array}$$

(36)

4.1.2. Simulation Results

The simulation results of the suggested control strategy in this paper are depicted in Figure 3, Figure 4, Figure 5 and Figure 6, where the load current profile is shown in Figure 3. From Figure 3, the load current disturbance is given as a variant step between 1 and 12 A during the simulation time 350 s. The trajectories of OER y and its desired value y_d of three controllers are displayed in Figure 4. It is obvious that the suggested controller outperforms the other two controllers with the best performance. The PID controller brings the worst performance with a large overshoot and low convergence rate. Meanwhile, in comparison with the conventional CFCPC, our suggested approach not only has a faster convergence rate but also obtains better tracking performance. Especially, the zoom in 120−140 s, corresponding with the suddenly adjusted load current, reconfirms the effectiveness and the finite-time convergence of the presented strategy. Figure 5 illustrates the comparative tracking errors of three controllers. It reveals that the transformed error of the proposed controller, CFCPC, and PID controller are all bounded. However, only the proposed controller, CFCPC can ensure that the tracking error satisfies the prescribed performance. The predetermined error state constraints are violated in the case of the PID controller. Furthermore, it can be seen from the partially enlarged view in Figure 5 that the tracking error of the proposed controller is the smallest and shortens the response time among the three controllers. The control input signals are plotted in Figure 6. From Figure 6, it can be seen that the control input of these controllers varies between 0 and 9 V. The control efforts are regulated to cope with the changing of the load current.

To derive a fair quantitative comparison, the detailed performance indices of the simulation trials are displayed in Figure 7, where μ_rms, M_e, and SD describe the root mean square error, the maximal absolute value, and the standard deviation of tracking error [24]: ${\mu }_{rms}=\sqrt{\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{\left(e_1\left(i\right)\right)}^2}},M_e=\underset{i=1,\dots ,n}{\max }\left\{\left|e_1\left(i\right)\right|\right\},SD=\sqrt{\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{\left(e_1\left(i\right)-{\overline{e}}_1\right)}^2}}$.

From Figure 7, the root means square error μ_rms of the proposed controller is about 23.5% and 49.5% of those from CFCPC and PID controller. The SD result is the same with μ_rms in which the SD of the proposed controller is the smallest. The maximal absolute value M_e of the suggested controller (0.3827) is smaller than that of the other controllers (0.4107 and 0.830).

4.2. Experiment Verification

4.2.1. HIL Experiment Description

The proposed FTCFC with prescribed performance has been performed on a HIL test bench in presence of the load current variation. The HIL experimental test bench of the PEMFC air supply system is displayed in Figure 8. The fan plays a role as the compressor which is steered by a DC servo motor through a bridge motor driver. The fan speed is proportional to the air-flow rate and is regulated by the signal control voltage. Meanwhile, the H100 PEMFC stack has the maximum power of 100 W which is fabricated by Horizon Fuel Cells Technologies. The PEMFC stack has consisted of 20 cells, each membrane with a dimension of 10.4 cm × 9.4 cm. Herein, a pressurized hydrogen tank at 2 atm supplies to the anode side, whilst oxygen is provided by the air-feed system in the cathode side. Two pressure sensors are deployed to measure the cathode pressure and the air pressure in the manifold. The load current disturbance is generated by using Chroma’s 63600 Modular DC Electronic Load. The proposed control program is implemented on the MATLAB/Simulink software and then incorporated with the Simulink Desktop Real-Time. The listed devices of the practical PEMFC system are described in

Table 2. Devices setting of the practical PEMFC system.

Components	Descriptions	Specification
PEMFC	Type	FCS-C100
	Rated power	100 [W]
	Humidification	Self-humidified
	Max. stack temperature	65 [°C]
	Reactants	Hydrogen and Air
DC motor	Type	Maxon DCX26L-GB-KL
	Rated rotation speed	5060 [rpm]
	Gear ratio	35:1
	Absolute encoder	4096 [pulses/rev]
Bridge motor drivers	Type	H-Bridges MD04
	Analog input	0-10 [V]
	H-Bridge/Motor voltage	60/24 [V]
Gas pressure sensor	Model	SDE-D10-G2-W18-L-PU-M8
	Rated output	1.5 [bar/V]
	Pressure range	0 to 10 [bar]
Electronic load	Model	Chroma 63600
	Voltage range	Up to 600 [V]
	Max. power	100 [W]
Power Supply	Model	PWS-3010D
	Output voltage	0 to 30 [V]
	Output current	0 to 10 [A]
Data acquisition Card	Type	NI PCI-6014
	Resolution	16AI/2AO: 16-bits

.

The goal of the HIL experiments is to exhibit the real applicability of the suggested control strategy for avoiding oxygen starvation of the PEMFC air supply system. We assume that the stack temperature, and gas pressure inlet the cathode/anode side are well regulated. Besides, it helps to verify the effectiveness of the proposed controller as compared with other control schemes. The multistep load demand is also to fully test the control effects in the experiment which is conducted by Chroma programming [29]. It then is applied for Electronic Load. The measurement data of load current is collected via a current sensor and oscilloscope which is depicted in Figure 9. The experiment time is 0–400 s which is presented by the following I(t) = 3 A if t ∈ [0, 40), 0.5 A if t ∈ [40, 90), 2.25 A if t ∈ [90, 130), 1 A if t ∈ [130, 170), 4.5 A if t ∈ [170, 220), 0.75 A if t ∈ [220, 260), 1.75 A if t ∈ [260, 290), and 0.5 A if t ∈ [290, 320), 3.5 A if t ∈ [320, 350), 1.5 A if t ∈ [350, 380), 5 A if t ∈ [380, 400).

In the experiment working conditions, the trial-and-error strategy is adopted to choose the suitable design parameters of the relevant controllers. The parameters of the PF are reselected as ${\rho }_U=2.5, {\rho }_L=-2.5, {\rho }_{\infty }=0.75,T_U=20, T_L=20$. The gain parameters of the proposed method are set as $k_1=27, k_2=25, {\eta }_1=3.9, {\eta }_2=2.5, {\mathsf{\Gamma }}_1=0.35, {\mathsf{\Gamma }}_2=0.05$. The control gains of the PID controller are chosen: k_p = 36, k_i = 8, and k_d = 0.06.

Remark 4: 

It is noted that the PEMFC air supply system parameters are determined through the identification process as shown in Table 1. Therefore, in the experiment, all the parameters of the PEMFC air system, as well as the control gains (i.e., the parameters of the prescribed function ${\rho }_U, {\rho }_L, {\rho }_{\infty }, T_U, \mathrm{and} T_L$; the option of CFC parameters as $k_1, k_2, {\eta }_1, {\eta }_2$, and the filter error compensation parameters ${\mathsf{\Gamma }}_1, {\mathsf{\Gamma }}_2$), are firstly set similar to the corresponding values in the simulation. However, there is some difference between the simulation and experimental results. Because the effects of noise, the deviation of the identification process, and the appearance of other unknown factors in practice have not been sufficiently considered which is the reason why the difference between simulation and experimental results. Hence, other sets of values are properly adjusted and tested in the experiment.

4.2.2. Experimental Results

The experiment results are shown in Figure 10, Figure 11 and Figure 12. Figure 10 describes the OER tracking performance of three controllers. As shown in Figure 10, the proposed controller provides the best performance under the variations of the load current. The OER tracking errors of the three controllers are displayed in Figure 11. From Figure 11, the control performance can be appraised that the maximum error different from the desired signal of the PID controller is close to 0.8922. This value appears since the load current change from 1 A to 4.5 A at the time t = 170 s. In addition, the error constraint is violated several times, for instance, t = 170 s, 320 s, and 380 s. On the contrary, owning to the prescribed performance function, the tracking error of the CFCPC and the proposed control method are in required constraints. The maximum error is held less than 0.5306 and 0.3526, which can be seen from the enlarged picture in Figure 11. Nonetheless, it is worth mentioning that the proposed controller brings the smallest tracking error and fast convergence in comparison with CFCPC thanks to the advantages of the FTCFC. The control input signals are shown in Figure 12, which are adjusted to adopt the varying load current.

Table 3. Performance indexes of three controllers in experiment.

Controller	PID	CFCPC	Proposed
${\mu }_{rms}$	0.1642	0.0821	0.0585
$M_e$	0.8922	0.5306	0.3526
$SD$	0.1640	0.0819	0.0584

presents the performance indexes of the three controllers in the experiment. It is observable in Table 3 that the RMS value of the OER tracking error for the suggested method is the smallest. This demonstrates the outstanding of the proposed controller with other control schemes. Meanwhile, the PID and CFCPC show larger RMS, maximum, and SD values during the working process.

It can be concluded from the above observations that the proposed control method in this paper effectively extinguishes the overshoot of the system, achieves high accuracy, and obtains a fast convergence time. Moreover, it is beneficial and feasible for the long-term effectiveness of the PEMFC air supply system.

5. Conclusions

This paper suggests the use of FTCFC to achieve finite-time tracking control for PEMFC air supply system with prescribed performance, where the prescribed constraint problem is solved by the error transformation approach. The FTCFC method can not only handle the computational explosion issue, but also improve the system performance with finite-time convergence. The numerical simulations and experimental results prove that the OER tracking error converges to the origin in finite-time, and the predefined state constraints are not violated. Future work of this paper will consider the impact of faults on the PEMFC air supply system. Moreover, we will develop a fault identification and an advanced fault-tolerant control strategy to cope with faults.