Proton exchange membrane fuel cell liquid water saturation estimation method based on fixed time adaptive observer

Abstract

The invention relates to the technical field of proton exchange membrane fuel cells, and discloses a proton exchange membrane fuel cell liquid water saturation estimation method based on a fixed time adaptive observer. According to the method, mechanism models such as a pile temperature model, a liquid water saturation model and an output voltage model of the proton exchange membrane fuel cell are established, the mechanism models are converted into a nonlinear state space model designed for an observer, a pile temperature estimation error, a reconstruction voltage error and a parameter estimation error are utilized to design feedback items respectively, and the feedback items are calculated. Acquired information is fully utilized, accurate estimation of the system state and unknown parameters is realized, the designed feedback item can ensure that the state observation error and the parameter estimation error converge within a fixed time, and the convergence time is irrelevant to the initial condition.

Description

Technical Field

[0001] The invention relates to the technical field of proton exchange membrane fuel cells, in particular to a method for estimating liquid water saturation of a proton exchange membrane fuel cell based on a fixed-time adaptive observer. Background Art

[0002] Effective water management of proton exchange membrane fuel cells can improve battery performance and is of great significance for the promotion and application of proton exchange membrane fuel cells. However, the liquid water saturation of proton exchange membrane fuel cells cannot be directly measured by sensors. At present, the estimation of liquid water saturation can be achieved by using measurement techniques such as neutron radiography or X-ray radiography. In addition, the liquid water content is estimated by analyzing the high-frequency components of the fuel cell spectral impedance. However, the above measurement techniques are expensive and the measurement process is slow. Therefore, it is necessary to study observer algorithms to realize soft measurement of liquid water saturation. For the current research on proton exchange membrane fuel cell observers, adaptive observers can only guarantee the exponential convergence of state observation errors and parameter estimation errors, and cannot provide quantitative analysis of the convergence time. However, accurate estimation of system states and unknown parameters within a fixed time can improve the performance of the control system.

[0003] The prior art lacks a means for estimating the liquid water saturation of a proton exchange membrane fuel cell at a fixed time. Summary of the invention

[0004] 1. Technical issues to be resolved

[0005] In view of the deficiencies in the prior art, the present invention provides a method for estimating liquid water saturation of a proton exchange membrane fuel cell based on a fixed-time adaptive observer, which has the advantages of achieving accurate estimation of system states and unknown parameters, and solves the above-mentioned technical problems.

[0006] (II) Technical solution

[0007] To achieve the above object, the present invention provides the following technical solution: a method for estimating liquid water saturation of a proton exchange membrane fuel cell based on a fixed-time adaptive observer, comprising a proton exchange membrane fuel cell system, comprising the following steps:

[0008] S1. Establish a proton exchange membrane fuel cell temperature model, liquid water saturation model and voltage model as a mechanism model, and transform the mechanism model into a nonlinear state space equation;

[0009] The expressions of the temperature model, liquid water saturation model and voltage model of the proton exchange membrane fuel cell are as follows:

[0010]

[0011] Vfc =n cell (E0-η act -η ohm )

[0012] Among them, T fc ,s,V fc Respectively represent the stack temperature, liquid water saturation and output voltage, Indicates the stack temperature T fc The first derivative of The first derivative of liquid water saturation s, E th represents the theoretical potential, I fc represents the load current, T amb Indicates the ambient temperature, v air represents the cathode air velocity, P ca represents cathode pressure, n cell represents the number of cells, E0 represents the ideal open circuit voltage, η act represents the activation loss, η ohm represents ohmic loss, f d and f p Represent two nonlinear functions, K1, K2, K3, K4, K s , K evap They represent system constants respectively;

[0013] The nonlinear function f d and f p The specific expression is as follows:

[0014]

[0015] f d =s 4 (1.42-4.24s+3.79s 2 )

[0016] Where exp represents an exponential function with a natural constant as the base, s represents liquid water saturation, p0 represents the pre-exponential factor, and E a represents the activation energy of evaporation; k b represents the Boltzmann parameter; p v represents the pressure of water vapor in the cathode catalyst layer, T fc Indicates the stack temperature;

[0017] The process of converting the mechanism model into a nonlinear state space equation is as follows: Select the stack temperature T fc and liquid water saturation s as state variables, and the stack temperature T in the proton exchange membrane fuel cell system is used fc and output voltage V fc Construct output signal y, load current Ifc and cathode air velocity v air As the input signal u, combined with K2, the nonlinear state space equation is obtained. The specific expression is as follows:

[0018]

[0019] y=Cx+V fc

[0020]

[0021] x=[x1 x2] T =[T fc s] T

[0022]

[0023] θ=K2

[0024] Among them, i0 represents the exchange current density, f(x,u) represents a nonlinear function, Φ(x,u) represents the regression vector, y represents the system output signal, C represents the system output matrix, A represents the system matrix, x represents the system state quantity, and x1 is used to refer to the stack temperature T fc , x2 is used to refer to the liquid water saturation s, represents the first-order derivative of x, the superscript T represents the transpose, θ represents the unknown parameter of the system, and P ca represents cathode pressure, R ohm Represents the equivalent resistance of each battery, I fc represents the load current, T fc ,s,V fc They represent the stack temperature, liquid water saturation and output voltage respectively, p0 represents the pre-exponential factor, K1, K2, K3, K4, K s , K evap They represent system constants, e represents natural constants, A geo represents the geometric area of ​​the catalyst, n cell Indicates the number of cells, E a represents the activation energy of evaporation; k b represents the Boltzmann parameter; p v represents the pressure of water vapor in the cathode catalyst layer, s represents the liquid water saturation, T amb Indicates the ambient temperature, v air represents the cathode air velocity, ln represents the logarithmic function with a natural constant as the base;

[0025] S2. Introducing feedback terms into the transformed nonlinear state space equation A fixed-time adaptive observer is obtained for estimating liquid water saturation and unknown parameters in proton exchange membrane fuel cells.

[0026] The fixed-time adaptive observer expression is as follows:

[0027]

[0028] in, express The time derivative, represents the estimated value of the system state quantity x, represents the reconstructed voltage value, represents the estimated value of the stack temperature, It means that f(x,u) is The estimated value at Indicates that Φ(x,u) is The estimated value at represents the estimated value of the unknown parameter θ of the system, h represents the high gain coefficient, h∈R, H -1 represents the inverse of the positive definite diagonal matrix with respect to h, H -2 represents the square of the inverse matrix of the positive definite diagonal matrix about h, H = diag(1,h -1 )∈R 2×2 ,K represents the observer gain matrix, K∈R 2×1 , R 2×1 Indicates that the dimension of the matrix is ​​2×1, k i and p i denote the i-th design parameter, y denotes the output signal, G denotes the extended error vector, G∈R 3×1 , R 3×1 Indicates that the dimension of the matrix is ​​3×1, represents the estimated error of the stack temperature, represents the error between the reconstructed voltage value and the true value, W represents the auxiliary vector, represents parameter estimation error, P and Q represent auxiliary variables, represents the feedback term, N and N1 represent two different auxiliary matrices, and their expressions are as follows:

[0029]

[0030]

[0031] in, represents the reconstructed voltage value, and Represents the output voltage V fc exist The partial derivatives with respect to x1 and x2;

[0032] The feedback item The expression is as follows:

[0033]

[0034] in, represents the term containing the auxiliary matrix, state estimation error and parameter estimation error, (N T G)1, (N T G)2, (N T G)3 represents N T The first, second, and third elements in G, sign(*) represents the sign function, N represents the auxiliary matrix, G represents the extended error vector, p i represents the i-th design parameter;

[0035] The expression of the adaptive law of the fixed-time adaptive observer is as follows:

[0036]

[0037] in, express The derivative with respect to time, Γ>0 represents the gain coefficient, N2 represents the auxiliary matrix, Indicated in The transposed vector of the regression vector at represents the feedback term, T fc Indicates the stack temperature, Represents the estimated value of the system state quantity x;

[0038] S3, reconstructing the output voltage value according to the estimated value of the stack temperature and the estimated value of the liquid water saturation; the reconstructing the output voltage value is the reconstruction of the voltage value The specific expression is as follows:

[0039]

[0040] in, represents the activation loss η act The estimated value of cell represents the number of cells, E0 represents the ideal open circuit voltage, η act represents activation loss, R represents the molar gas constant, α represents the charge transfer coefficient, i0 represents the exchange current density, F represents the Faraday constant, and A geo represents the geometric area of ​​the catalyst, ln represents the logarithmic function with the natural constant as the base, represents the estimated value of i0.

[0041] As a preferred technical solution of the present invention, the specific expressions of the system constants K1, K2, K3, and K4 are as follows:

[0042]

[0043] Among them, m fc Indicates the mass of the battery stack, C p,fc represents the average heat capacity of the stack, ρ air Represents the air density, C p,air represents the average heat capacity of air, A inlet represents the cross-sectional area of ​​the cathode casing, represents the molar mass of water, F represents the Faraday constant, A geo represents the geometric area of ​​the catalyst, σ represents the surface tension of water, cosΘ represents the cosine value of Θ, Θ represents the effective contact angle in the diffusion medium, ε represents the effective porosity, and K eff represents the effective permeability, ρ l represents the density of water, μ l represents the dynamic viscosity of water, σ CL,c Represents the width of the cathode catalyst layer.

[0044] As a preferred technical solution of the present invention, the ohmic loss η ohm The specific expression is as follows:

[0045] η ohm =R ohm I fc

[0046] Among them, R ohm Represents the equivalent resistance of each battery, I fc Indicates load current;

[0047] The η act The specific expression for activation loss is as follows:

[0048]

[0049] Where R is the molar gas constant, α is the charge transfer coefficient, i0 is the exchange current density, F is the Faraday constant, and A is geo represents the geometric area of ​​the catalyst, and ln represents the logarithmic function with a natural constant as the base.

[0050] As a preferred technical solution of the present invention, the specific expressions of the auxiliary variables P and Q are as follows:

[0051]

[0052] in, and denote the first-order derivatives of auxiliary variables P and Q, respectively, express The transpose of the filter matrix, H represents a positive definite diagonal matrix with respect to h, h represents a high gain coefficient, F l represents a matrix that satisfies the Hurwitz condition, and Satisfies the Hurwitz condition, L represents the observer gain matrix, L∈R 2×1 , R 2×1 Indicates that the dimension of the matrix is ​​2×1, and Respectively and exist The Jacobian matrix at -1 represents the inverse of the positive definite diagonal matrix about h, ι>0 represents the design parameter, the superscript T represents the transpose, T fc represents the stack temperature, and C represents the system output matrix.

[0053] As a preferred technical solution of the present invention, the expression of the auxiliary matrix N2 is as follows:

[0054] N2=[0 0P 2

[0055] Wherein, P represents an auxiliary variable.

[0056] Compared with the prior art, the present invention provides a method for estimating liquid water saturation of a proton exchange membrane fuel cell based on a fixed-time adaptive observer, which has the following beneficial effects:

[0057] The present invention transforms the mechanism models into nonlinear state space models for observer design by establishing mechanism models such as a stack temperature model, a liquid water saturation model and an output voltage model of a proton exchange membrane fuel cell, and respectively designs feedback items using the stack temperature estimation error, the reconstructed voltage error and the parameter estimation error, thereby making full use of the available information and realizing accurate estimation of the system state and unknown parameters. The designed feedback items can ensure that the state observation error and the parameter estimation error converge within a fixed time, and the convergence time is independent of the initial conditions. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 It is a schematic diagram of a fixed-time adaptive observer of the present invention;

[0059] Figure 2 Schematic diagram of input signal in the experiment of the present invention;

[0060] Figure 3 It is a schematic diagram for comparing parameter estimation results of the present invention;

[0061] Figure 4 A schematic diagram showing comparison of the results of the temperature estimation of the battery stack of the present invention;

[0062] Figure 5 It is a schematic diagram for comparing the results of the estimated value of liquid water saturation of the present invention;

[0063] Figure 6 It is a schematic diagram for comparing the results of reconstructing voltage values ​​according to the present invention. DETAILED DESCRIPTION

[0064] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0065] See also Figure 1-6 A method for estimating liquid water saturation of a proton exchange membrane fuel cell based on a fixed-time adaptive observer includes a proton exchange membrane fuel cell system, and includes the following steps:

[0066] S1. Establish a proton exchange membrane fuel cell temperature model, liquid water saturation model and voltage model as a mechanism model, and transform the mechanism model into a nonlinear state space equation;

[0067] The expressions of the temperature model, liquid water saturation model and voltage model of the proton exchange membrane fuel cell are as follows:

[0068]

[0069] V fc =n cell (E0-η act -η ohm )

[0070] Among them, T fc ,s,V fc Respectively represent the stack temperature, liquid water saturation and output voltage, Indicates the stack temperature T fc The first derivative of The first derivative of liquid water saturation s, E th represents the theoretical potential, I fc represents the load current, T amb Indicates the ambient temperature, v air represents the cathode air velocity, P ca represents cathode pressure, n cell represents the number of cells, E0 represents the ideal open circuit voltage, ηact represents the activation loss, η ohm represents ohmic loss, f d and f p Represent two nonlinear functions, K1, K2, K3, K4, K s , K evap They represent system constants respectively;

[0071] The specific expressions of system constants K1, K2, K3, and K4 are as follows:

[0072]

[0073] Among them, m fc Indicates the mass of the battery stack, C p,fc represents the average heat capacity of the stack, ρ air Represents the air density, C p,air represents the average heat capacity of air, A inlet represents the cross-sectional area of ​​the cathode casing, represents the molar mass of water, F represents the Faraday constant, A geo represents the geometric area of ​​the catalyst, σ represents the surface tension of water, cosΘ represents the cosine value of Θ, Θ represents the effective contact angle in the diffusion medium, ε represents the effective porosity, and K eff represents the effective permeability, ρ l represents the density of water, μ l represents the dynamic viscosity of water, σ CL,c represents the width of the cathode catalyst layer;

[0074] f d and f p The specific expression is as follows:

[0075]

[0076] f d =s 4 (1.42-4.24s+3.79s 2 )

[0077] Where exp represents an exponential function with a natural constant as the base, s represents liquid water saturation, p0 represents the pre-exponential factor, and E a represents the activation energy of evaporation; k b represents the Boltzmann parameter; p v represents the pressure of water vapor in the cathode catalyst layer, T fc Indicates the stack temperature;

[0078] Ohmic loss η ohm The specific expression is as follows:

[0079] η ohm=R ohm I fc

[0080] Among them, R ohm Represents the equivalent resistance of each battery, I fc Indicates load current;

[0081] η act The specific expression for activation loss is as follows:

[0082]

[0083] Where R is the molar gas constant, α is the charge transfer coefficient, i0 is the exchange current density, F is the Faraday constant, and A is geo represents the geometric area of ​​the catalyst, ln represents the logarithmic function with the natural constant as the base;

[0084] The process of converting the mechanism model into a nonlinear state space equation is as follows: Select the stack temperature T fc and liquid water saturation s as state variables, and the stack temperature T in the proton exchange membrane fuel cell system is used fc and output voltage V fc Construct output signal y, load current I fc and cathode air velocity v air As the input signal u, combined with K2, the nonlinear state space equation is obtained. The specific expression is as follows:

[0085]

[0086] y=Cx+V fc

[0087]

[0088] x=[x1 x2] T =[T fc s] T

[0089]

[0090] θ=K2

[0091] Among them, i0 represents the exchange current density, f(x,u) represents a nonlinear function, Φ(x,u) represents the regression vector, y represents the system output signal, C represents the system output matrix, A represents the system matrix, x represents the system state quantity, and x1 is used to refer to the stack temperature T fc , x2 is used to refer to the liquid water saturation s, represents the first-order derivative of x, the superscript T represents the transpose, θ represents the unknown parameter of the system, and P carepresents cathode pressure, R ohm Represents the equivalent resistance of each battery, I fc represents the load current, T fc ,s,V fc They represent the stack temperature, liquid water saturation and output voltage respectively, p0 represents the pre-exponential factor, K1, K2, K3, K4, K s , K evap They represent system constants, e represents natural constants, A geo represents the geometric area of ​​the catalyst, n cell Indicates the number of cells, E a represents the activation energy of evaporation; k b represents the Boltzmann parameter; p v represents the pressure of water vapor in the cathode catalyst layer, s represents the liquid water saturation, T amb Indicates the ambient temperature, v air represents the cathode air velocity, ln represents the logarithmic function with a natural constant as the base;

[0092] S2. Introducing feedback terms into the transformed nonlinear state space equation A fixed-time adaptive observer is obtained for estimating liquid water saturation and unknown parameters in proton exchange membrane fuel cells.

[0093] The fixed-time adaptive observer expression is as follows:

[0094]

[0095] in, express The time derivative, represents the estimated value of the system state quantity x, represents the reconstructed voltage value, represents the estimated value of the stack temperature, It means that f(x,u) is The estimated value at Indicates that Φ(x,u) is The estimated value at represents the estimated value of the unknown parameter θ of the system, h represents the high gain coefficient, h∈R, H -1 represents the inverse of the positive definite diagonal matrix with respect to h, H -2 represents the square of the inverse matrix of the positive definite diagonal matrix about h, H = diag(1,h -1 )∈R 2×2 ,K represents the observer gain matrix, K∈R 2×1 , R 2×1 Indicates that the dimension of the matrix is ​​2×1, k i and p irepresent the i-th design parameter, k i ∈R, (i=1,2) and 0<p1<1,p2>1 are design parameters, y represents the output signal, G represents the extended error vector, G∈R 3×1 , R 3×1 Indicates that the dimension of the matrix is ​​3×1, represents the estimated error of the stack temperature, represents the error between the reconstructed voltage value and the true value, W represents the auxiliary vector, represents parameter estimation error, P and Q represent auxiliary variables, represents the feedback term, N and N1 represent two different auxiliary matrices, and their expressions are as follows:

[0096]

[0097] in, represents the reconstructed voltage value, and Represents the output voltage V fc exist The partial derivatives with respect to x1 and x2;

[0098] The specific expressions of auxiliary variables P and Q are as follows:

[0099]

[0100] in, and denote the first-order derivatives of auxiliary variables P and Q, respectively, express The transpose of the filter matrix, H represents a positive definite diagonal matrix with respect to h, h represents a high gain coefficient, F l represents a matrix that satisfies the Hurwitz condition, and Satisfies the Hurwitz condition, K represents the observer gain matrix, K∈R 2×1 , R 2×1 Indicates that the dimension of the matrix is ​​2×1, and Respectively and exist The Jacobian matrix at -1 represents the inverse of the positive definite diagonal matrix about h, ι>0 represents the design parameter, the superscript T represents the transpose, T fc represents the stack temperature, C represents the system output matrix;

[0101] Feedback The expression is as follows:

[0102]

[0103] in, represents the term containing the auxiliary matrix, state estimation error and parameter estimation error, (N T G)1, (N T G)2, (N T G)3 represents N T The first, second, and third elements in G, sign(*) represents the sign function, N represents the auxiliary matrix, G represents the extended error vector, p i represents the i-th design parameter;

[0104] The expression of the adaptive law of the fixed-time adaptive observer is as follows:

[0105]

[0106] in, express The derivative with respect to time, Γ>0 represents the gain coefficient, N2 represents the auxiliary matrix, express, represents the feedback term, T fc Indicates the stack temperature, Represents the estimated value of the system state quantity x;

[0107] The expression of the auxiliary matrix N2 is as follows:

[0108] N2=[0 0P 2

[0109] Among them, P represents the auxiliary variable;

[0110] S3, reconstructing the output voltage value based on the estimated value of the stack temperature and the estimated value of the liquid water saturation;

[0111] Reconstruct the output voltage value, that is, reconstruct the voltage value The specific expression is as follows:

[0112]

[0113] in, represents the activation loss η act The estimated value of cell represents the number of cells, E0 represents the ideal open circuit voltage, η act represents activation loss, R represents the molar gas constant, α represents the charge transfer coefficient, i0 represents the exchange current density, F represents the Faraday constant, and A geo represents the geometric area of ​​the catalyst, ln represents the logarithmic function with the natural constant as the base, represents the estimated value of i0.

[0114] In order to verify the effectiveness of the proposed fixed-time adaptive observer method, the present invention collects experimental data through the H-100 fuel cell system and conducts experimental verification of the fixed-time adaptive observer proposed by the present invention. Figure 2 As shown, the design parameters of the fixed-time adaptive observer are h = 3, k1 = 2, k2 = 2, p1 = 0.8, p2 = 1.2, Γ = 8 × 10 4 , the initial conditions are

[0115] In addition, a high-gain adaptive observer method is used as a comparative experiment. Figure 2 The required design parameters are consistent with the corresponding parameters in the fixed-time adaptive observer, and the initial conditions are consistent with the fixed-time adaptive observer. The comparison of the integrated square error, maximum absolute error and standard deviation of the reconstructed voltage value between the method of the present invention and the high-gain adaptive observer is shown in Table 1. The statistical time interval of the error is 10s-5000s; the comparison of parameter estimation results is shown in the figure below. Figure 3 As shown; the comparison of the stack temperature estimation results is shown in Figure 4 As shown; the comparison diagram of liquid water saturation estimation is as follows Figure 5 As shown; the comparison of the reconstructed voltage results is shown in Figure 6 As shown. It can be seen from this that the fixed-time adaptive observer algorithm proposed in the present invention has a faster convergence speed and higher estimation accuracy.

[0116] Table 1 Error comparison results

[0117] Integrated square error Maximum absolute error Standard Deviation High Gain Method <![CDATA[2.1483×10 -4 ]]> <![CDATA[1.4538×10 -4 ]]> <![CDATA[3.1894×10 -8 ]]> Method of the present invention <![CDATA[6.3171×10 -6 ]]> <![CDATA[1.3909×10 -5 ]]> <![CDATA[5.6465×10 -10 ]]>

[0118] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for estimating liquid water saturation of a proton exchange membrane fuel cell based on a fixed-time adaptive observer, comprising a proton exchange membrane fuel cell system, characterized in that: The following steps are involved: S1. Establish a proton exchange membrane fuel cell temperature model, liquid water saturation model and voltage model as a mechanism model, and transform the mechanism model into a nonlinear state space equation; The expressions of the temperature model, liquid water saturation model and voltage model of the proton exchange membrane fuel cell are as follows: V fc =n cell (E0-h act -or ohm ) Among them, T fc ,s,V fc Respectively represent the stack temperature, liquid water saturation and output voltage, Indicates the stack temperature T fc The first derivative of The first derivative of liquid water saturation s, E th represents the theoretical potential, I fc represents the load current, T amb Indicates the ambient temperature, v air represents the cathode air velocity, P ca represents cathode pressure, n cell represents the number of cells, E0 represents the ideal open circuit voltage, η act represents the activation loss, η ohm represents ohmic loss, f d and f p Represent two nonlinear functions, K1, K2, K3, K4, K s , K evap They represent system constants respectively; The nonlinear function f d and f p The specific expression is as follows: f d =s 4 (1.42-4.24s+3.79s 2 ) Where exp represents an exponential function with a natural constant as the base, s represents liquid water saturation, p0 represents the pre-exponential factor, and E a represents the activation energy of evaporation; k b represents the Boltzmann parameter; p v represents the pressure of water vapor in the cathode catalyst layer, T fc Indicates the stack temperature; The process of converting the mechanism model into a nonlinear state space equation is as follows: Select the stack temperature T fc and liquid water saturation s as state variables, and the stack temperature T in the proton exchange membrane fuel cell system is used fc and output voltage V fc Construct output signal y, load current I fc and cathode air velocity v air As the input signal u, combined with K2, the nonlinear state space equation is obtained. The specific expression is as follows: y=Cx+V fc x=[x1 x2] T =[T fc s] T θ=K2 Among them, i0 represents the exchange current density, f(x,u) represents a nonlinear function, Φ(x,u) represents the regression vector, y represents the system output signal, C represents the system output matrix, A represents the system matrix, x represents the system state quantity, and x1 is used to refer to the stack temperature T fc , x2 is used to refer to the liquid water saturation s, represents the first-order derivative of x, the superscript T represents the transpose, θ represents the unknown parameter of the system, and P ca represents cathode pressure, R ohm Represents the equivalent resistance of each battery, I fc represents the load current, T fc ,s,V fc They represent the stack temperature, liquid water saturation and output voltage respectively, p0 represents the pre-exponential factor, K1, K2, K3, K4, K s , K evap They represent system constants, e represents natural constants, A geo represents the geometric area of ​​the catalyst, n cell Indicates the number of cells, E a represents the activation energy of evaporation; k b represents the Boltzmann parameter; p v represents the pressure of water vapor in the cathode catalyst layer, s represents the liquid water saturation, T amb Indicates the ambient temperature, v air represents the cathode air velocity, ln represents the logarithmic function with the natural constant as the base; S2. Introducing feedback terms into the transformed nonlinear state space equation A fixed-time adaptive observer is obtained for estimating liquid water saturation and unknown parameters in proton exchange membrane fuel cells. The fixed-time adaptive observer expression is as follows: in, express The time derivative, represents the estimated value of the system state quantity x, represents the reconstructed voltage value, represents the estimated value of the stack temperature, It means that f(x,u) is The estimated value at Indicates that Φ(x,u) is The estimated value at represents the estimated value of the unknown parameter θ of the system, h represents the high gain coefficient, h∈R, H -1 represents the inverse of the positive definite diagonal matrix with respect to h, H -2 represents the square of the inverse matrix of the positive definite diagonal matrix about h, H = diag(1,h -1 )∈R 2 ×2 ,K represents the observer gain matrix, K∈R 2×1 , R 2×1 Indicates that the dimension of the matrix is ​​2×1, k i and p i denote the i-th design parameter, G denotes the extended error vector, G∈R 3×1 , R 3×1 Indicates that the dimension of the matrix is ​​3×1, represents the estimated error of the stack temperature, represents the error between the reconstructed voltage value and the true value, W represents the auxiliary vector, represents parameter estimation error, P and Q represent auxiliary variables, represents the feedback term, N and N1 represent two different auxiliary matrices, and their expressions are as follows: in, represents the reconstructed voltage value, and Represents the output voltage V fc exist The partial derivatives with respect to x1 and x2; The feedback item The expression is as follows: in, represents the term containing the auxiliary matrix, state estimation error and parameter estimation error, (N T G)1, (N T G)2, (N T G)3 respectively represent N T The first, second, and third elements in G, sign(*) represents the sign function, N represents the auxiliary matrix, G represents the extended error vector, p i represents the i-th design parameter; The expression of the adaptive law of the fixed-time adaptive observer is as follows: in, express The derivative with respect to time, Γ>0 represents the gain coefficient, N2 represents the auxiliary matrix, Indicated in The transposed vector of the regression vector at represents the feedback term, T fc Indicates the stack temperature, Represents the estimated value of the system state quantity x; S3, reconstructing the output voltage value based on the estimated value of the stack temperature and the estimated value of the liquid water saturation; The reconstructed output voltage value, i.e., the reconstructed voltage value The specific expression is as follows: in, represents the activation loss η act The estimated value of cell represents the number of cells, E0 represents the ideal open circuit voltage, η act represents activation loss, R represents the molar gas constant, α represents the charge transfer coefficient, i0 represents the exchange current density, F represents the Faraday constant, and A geo represents the geometric area of ​​the catalyst, ln represents the logarithmic function with the natural constant as the base, represents the estimated value of i0.

2. The method for estimating liquid water saturation of a proton exchange membrane fuel cell based on a fixed-time adaptive observer according to claim 1, characterized in that: The specific expressions of the system constants K1, K2, K3, and K4 are as follows: Among them, m fc Indicates the mass of the battery stack, C p,fc represents the average heat capacity of the stack, ρ air Represents the air density, C p,air represents the average heat capacity of air, A inlet represents the cross-sectional area of ​​the cathode casing, represents the molar mass of water, F represents the Faraday constant, A geo represents the geometric area of ​​the catalyst, σ represents the surface tension of water, cosΘ represents the cosine value of Θ, Θ represents the effective contact angle in the diffusion medium, ε represents the effective porosity, and K eff represents the effective permeability, ρ l represents the density of water, μ l represents the dynamic viscosity of water, σ CL,c Represents the width of the cathode catalyst layer.

3. The method for estimating liquid water saturation of a proton exchange membrane fuel cell based on a fixed-time adaptive observer according to claim 2, characterized in that: The ohmic loss η ohm The specific expression is as follows: or ohm =R ohm I fc Among them, R ohm Represents the equivalent resistance of each battery, I fc Indicates load current; The η act The specific expression for activation loss is as follows: Where R is the molar gas constant, α is the charge transfer coefficient, i0 is the exchange current density, F is the Faraday constant, and A is geo represents the geometric area of ​​the catalyst, and ln represents the logarithmic function with a natural constant as the base.

4. The method for estimating liquid water saturation of a proton exchange membrane fuel cell based on a fixed-time adaptive observer according to claim 1, characterized in that: The specific expressions of the auxiliary variables P and Q are as follows: in, and denote the first-order derivatives of auxiliary variables P and Q, respectively, express The transpose of the filter matrix, H represents a positive definite diagonal matrix with respect to h, h represents a high gain coefficient, F l represents a matrix that satisfies the Hurwitz condition, and Satisfies the Hurwitz condition, K represents the observer gain matrix, K∈R 2×1 , R 2×1 Indicates that the dimension of the matrix is ​​2×1, and Respectively and exist The Jacobian matrix at -1 represents the inverse of the positive definite diagonal matrix about h, l>0 represents the design parameter, the superscript T represents the transpose, T fc represents the stack temperature, and C represents the system output matrix.

5. The method for estimating liquid water saturation of a proton exchange membrane fuel cell based on a fixed-time adaptive observer according to claim 4, characterized in that: The expression of the auxiliary matrix N2 is as follows: N2=[0 0 P 2 ] Wherein, P represents an auxiliary variable.

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