CN113299957A - Proton exchange membrane fuel cell peroxide amount control method - Google Patents

Proton exchange membrane fuel cell peroxide amount control method Download PDF

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Publication number
CN113299957A
CN113299957A CN202110468815.2A CN202110468815A CN113299957A CN 113299957 A CN113299957 A CN 113299957A CN 202110468815 A CN202110468815 A CN 202110468815A CN 113299957 A CN113299957 A CN 113299957A
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formula
fuel cell
controller
algorithm
peroxide
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朱静
张鹏
武康
赵静欣
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Nanjing University of Aeronautics and Astronautics
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Nanjing University of Aeronautics and Astronautics
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    • H—ELECTRICITY
    • H01—ELECTRIC ELEMENTS
    • H01M—PROCESSES OR MEANS, e.g. BATTERIES, FOR THE DIRECT CONVERSION OF CHEMICAL ENERGY INTO ELECTRICAL ENERGY
    • H01M8/00—Fuel cells; Manufacture thereof
    • H01M8/04—Auxiliary arrangements, e.g. for control of pressure or for circulation of fluids
    • H01M8/04298—Processes for controlling fuel cells or fuel cell systems
    • H01M8/04992—Processes for controlling fuel cells or fuel cell systems characterised by the implementation of mathematical or computational algorithms, e.g. feedback control loops, fuzzy logic, neural networks or artificial intelligence
    • H—ELECTRICITY
    • H01—ELECTRIC ELEMENTS
    • H01M—PROCESSES OR MEANS, e.g. BATTERIES, FOR THE DIRECT CONVERSION OF CHEMICAL ENERGY INTO ELECTRICAL ENERGY
    • H01M8/00—Fuel cells; Manufacture thereof
    • H01M8/04—Auxiliary arrangements, e.g. for control of pressure or for circulation of fluids
    • H01M8/04298—Processes for controlling fuel cells or fuel cell systems
    • H01M8/04694—Processes for controlling fuel cells or fuel cell systems characterised by variables to be controlled
    • H01M8/04746—Pressure; Flow
    • H01M8/04753—Pressure; Flow of fuel cell reactants
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02E—REDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION
    • Y02E60/00—Enabling technologies; Technologies with a potential or indirect contribution to GHG emissions mitigation
    • Y02E60/30—Hydrogen technology
    • Y02E60/50—Fuel cells

Abstract

The invention discloses a proton exchange membrane fuel cell peroxide amount control method, which comprises the following steps: step 1, constructing a cascade control structure for controlling the oxygen passing amount of a proton exchange membrane fuel cell, wherein the cascade control structure comprises an inner ring part and an outer ring part, the outer ring part is used for controlling the oxygen passing amount, and the inner ring part is used for regulating the speed; and 2, introducing a high-order sliding mode operator into a controller of an outer ring part of the cascade control structure to obtain a Super-Twisting algorithm-based high-order sliding mode controller, and introducing an adaptive dynamic programming algorithm into a controller of an inner ring part of the cascade control structure to obtain an optimal tracking controller based on the adaptive dynamic programming algorithm.

Description

Proton exchange membrane fuel cell peroxide amount control method
Technical Field
The invention relates to a Super-Twisting and self-adaptive dynamic programming algorithm-based peroxide amount control method for a Proton Exchange Membrane Fuel Cell (PEMFC), and belongs to the technical fields of strategy iteration, neural network approximation and the like.
Background
Proton Exchange Membrane Fuel Cells (PEMFCs) are a new clean energy source, and have the advantages of zero pollution, high efficiency, flexible modular structure, etc., and are gaining great attention in the new energy field, and are receiving wide attention from organizations in many countries and regions. The PEMFC system can deliver oxygen and hydrogen to the inside of the reactor through an air compressor and a hydrogen pump, so that they chemically react and generate electric energy for a load. However, in practical use, the instantaneous surge in power consumption due to the increase in load may cause the supply of oxygen in the cathode manifold to temporarily fail to meet the current demand for power consumption, thereby causing oxygen starvation and damage to the proton exchange membrane inside the fuel cell reactor. This not only greatly reduces the service life of the fuel cell, but also brings about a great risk to the safe operation of vehicles, aircrafts and other moving tools which are supplied with energy by the fuel cell.
The PEMFC system can be mainly divided into the following four subsections: an air supply subsystem, a hydrogen supply subsystem, a temperature management subsystem, and a humidity conditioning subsystem. The air supply subsystem plays a significant role in the oxygen supply as an important part. In actual industrial control, the value of excess Oxygen (OER) is often used as an important regulatory indicator in order to characterize the instantaneous nature of oxygen supply and consumption within the reactor. When the OER is less than 1, the occurrence of oxygen starvation may cause damage to the reactor internal exchange membrane. However, the excessive oxygen supply cannot satisfy the practical economic efficiency. In production life, the electric energy required for starting and subsequent operation of the compressor is usually supplied by a fuel cell, and this ratio is often up to 30% or even higher. An excessive OER means that the air compressor discharges a large amount of compressed air into the inside of the fuel cell reactor in a short time, thereby increasing the instantaneous power consumption of the compressor, so that the actual net power generated by the fuel cell system is greatly discounted, and the use cost is increased. Therefore, it is the focus of many researchers to reasonably and quickly regulate the supply of oxygen in the cathode manifold when fluctuations occur in the load current.
In addition, among many control schemes, the sliding mode controller is widely used in the oxygen content control of the fuel cell due to its advantages of strong robustness, anti-interference and the like. When the high-order sliding mode algorithm is used for carrying out closed-loop control on the OER, a single-loop control structure based on a suboptimal sliding mode algorithm and a double-loop control structure based on a Super-Twisting algorithm can be adopted. The former can cause high-frequency and large-amplitude jitter of a control input signal while meeting the rapidity characteristic required by the peroxide amount regulation, thereby further reducing the service life of the compressor; the latter, while effectively attenuating the drastic fluctuations of the control signal, causes a certain time delay in the dynamic adjustment response of the OER. For the above reasons, how to balance the dynamic response characteristics of the OER and the relative smoothness of the input signal when using the sliding mode algorithm to adjust the excess oxygen of the fuel cell becomes a hotspot and difficulty in the actual control process.
Disclosure of Invention
The invention aims to provide a proton exchange membrane fuel cell peroxide amount control method, which aims to solve the problem that the relative smoothness of the peroxide amount regulation speed and the control signal cannot be effectively balanced when the traditional high-order sliding mode algorithm is applied to the proton exchange membrane fuel cell peroxide amount control process,
in order to achieve the purpose, the invention adopts the following technical scheme:
a proton exchange membrane fuel cell peroxide amount control method comprises the following steps:
step 1, constructing a cascade control structure for controlling the oxygen passing amount of a proton exchange membrane fuel cell, wherein the cascade control structure comprises an inner ring part and an outer ring part, the outer ring part is used for controlling the oxygen passing amount, and the inner ring part is used for regulating the speed;
step 2, introducing a high-order sliding mode operator into a controller of an outer ring part of the cascade control structure to obtain a Super-Twisting algorithm-based high-order sliding mode controller, introducing an adaptive dynamic programming algorithm into a controller of an inner ring part of the cascade control structure to obtain an optimal tracking controller based on the adaptive dynamic programming algorithm;
in the outer ring part, the actual peroxide value is made to approach a given target value as fast as possible through a high-order sliding mode controller based on a Super-twist algorithm; in the inner loop portion, the angular velocity values produced by the outer loop portion are referenced by the inner loop portion as a reference trajectory and the voltage delivered to the air compressor is regulated by an optimal tracking controller based on an adaptive dynamic programming algorithm.
In the step 2, the controller based on the Super-Twisting algorithm is as follows:
Figure BDA0003044510520000021
wherein liI belongs to {1,2,3,4} as a normal number to be selected, s (-) as a corresponding sliding mode variable, t as time, and τ as an integral of t;
Figure BDA0003044510520000022
wherein the content of the first and second substances,
Figure BDA0003044510520000023
the amount of the peroxide is the amount of the peroxide,
Figure BDA0003044510520000024
is a desired value of the excess oxygen amount;
the two sides of the formula (3.2) are subjected to derivation to obtain the following expression
Figure BDA0003044510520000031
From the above formula, it is found that the amount of the peroxide
Figure BDA0003044510520000032
With angular speed omega of the compressorcpThe relative degree of freedom between them is 1;
is provided with
Figure BDA0003044510520000033
And σ1(x, t) are bounded functions and satisfy:
Figure BDA0003044510520000034
the parameters in the Super-Twisting algorithm based controller shown in equation (3.1) are selected as follows:
Figure BDA0003044510520000035
wherein li0I e {1,2,3,4} and H are all normal numbers, according to the stability theorem, the sliding mode variable s (T) will be at a finite time T when the following equation (3.6) is satisfiedfTends towards zero and settles to this value and remains unchanged for a time thereafter;
Figure BDA0003044510520000036
in the step 2, an optimal tracking controller based on an adaptive dynamic programming algorithm is designed through the following steps:
consider an affine nonlinear system:
Figure BDA0003044510520000037
wherein x ∈ RnRepresenting the system state, f (x) and g (x) are bounded nonlinear functions;
defining an ideal tracking trajectory as:
Figure BDA0003044510520000038
then the steady state controller is calculated to:
Figure BDA0003044510520000039
subtracting the formula (4.2) from the formula (4.1) yields:
Figure BDA00030445105200000310
wherein，e＝x-xd∈Rn，fe＝f(x)-f(xd),ge＝g(x)-g(xd),ue＝u-ud
The following performance indicator functions are defined:
Figure BDA00030445105200000311
wherein the content of the first and second substances,
Figure BDA00030445105200000312
for the utility function, Q ∈ RnAnd R ∈ RnAll are positive definite matrixes to be set;
define the Hamiltonian:
Figure BDA0003044510520000041
wherein the content of the first and second substances,
Figure BDA0003044510520000042
represents the partial differential of V to the error e;
by definition, the optimal performance indicator function is expressed as:
Figure BDA0003044510520000043
combining (4.6) with (4.7) to obtain:
Figure BDA0003044510520000044
let V*(e) Can be solved and continuously can be differentiated, and (4.8) is paired with ueAnd (3) calculating a partial derivative to obtain:
Figure BDA0003044510520000045
in conjunction with (4.8) and (4.9), the optimal tracking controller is represented as:
Figure BDA0003044510520000046
in step 2, fitting a performance index function by using a neural network so as to solve a corresponding Hamiltonian; the method specifically comprises the following steps:
computing
Figure BDA0003044510520000047
The differential equation (4.6) needs to be solved to avoid solving
Figure BDA0003044510520000048
The method is characterized in that a dimension disaster is trapped in the process, and a neural network with a three-layer structure is used for approximating a performance index function (4.5):
Figure BDA0003044510520000049
wherein, Wc∈RlFor an ideal weight matrix connecting the middle and output layers, ε ∈ RlIs the approximation error generated by the neural network when approximating the function (4.5);
and (3) solving the partial derivatives of e on two sides of the formula (5.1) to obtain:
Figure BDA00030445105200000410
substituting the formula (5.2) into the formula (4.6) to obtain an approximate Hamiltonian:
Figure BDA00030445105200000411
since an ideal weight matrix is difficult to obtain, an evaluation neural network is established to approximate v (e):
Figure BDA00030445105200000412
after the formula (5.4) is substituted into the formula (4.6),
Figure BDA00030445105200000413
combining formula (5.3) and formula (5.5), yielding:
Figure BDA0003044510520000051
wherein the content of the first and second substances,
Figure BDA0003044510520000052
since the goal is to continuously iterate the weight values
Figure BDA0003044510520000053
So that the Hamiltonian (5.5) is as small as possible, thus creating an energy function
Figure BDA0003044510520000054
Continuously adjusting the weight value by adopting a gradient descent algorithm to enable the weight value to iterate towards the direction of reducing the energy function; wherein the content of the first and second substances,
Figure BDA0003044510520000055
the iterative equation of (a) is as follows:
Figure BDA0003044510520000056
wherein alpha is>0 is the learning rate of the evaluation function,
Figure BDA0003044510520000057
representing the derivation of the activation function over time;
according to (4.9) and (5.2), the ideal feedback control law can be calculated as follows:
Figure BDA0003044510520000058
and it can be further approximated as:
Figure BDA0003044510520000059
has the advantages that: the invention adopts a cascade control structure, and respectively applies controllers based on a Super-Twisting algorithm and a self-adaptive dynamic algorithm to the control of an outer ring and an inner ring. In the inner loop, by setting the corresponding performance indicator function, the speed regulation error can be made to trend towards 0 as much as possible and the input signal is controlled to iterate towards a decreasing performance indicator function. Because complex differential equations need to be calculated when the HJB equation is solved, the optimal iterative control law is obtained by adopting a method of approximating a performance index function by a neural network. The controller provided by the invention can maintain the good dynamic response characteristic of the peroxide amount and simultaneously enable the control input signal to have less chattering phenomenon.
Drawings
FIG. 1 is a PEMFC system overall framework;
FIG. 2 is a cascade control structure of the amount of peroxide;
figure 3 is an optimal tracking controller based on ADP;
FIG. 4 is an peroxide dynamic response curve;
FIG. 5 is a voltage input signal;
fig. 6 is a weight curve for evaluating a neural network.
Detailed Description
The invention is further explained below with reference to the drawings.
The invention specifically comprises the following steps:
(1) system description of air supply subsystem
The invention mainly considers the air supply subsystem in the PEMFC system and adopts a more common fourth-order model for analysis:
Figure BDA0003044510520000061
wherein x ∈ R4，
Figure BDA0003044510520000066
And is
Figure BDA0003044510520000062
Figure BDA0003044510520000063
The main physical parameters in the above system are shown in table 1.
TABLE 1 Fuel cell air supply subsystem Primary physical parameters
Figure BDA0003044510520000064
Further, Wcp＝b14ωcpAnd
Figure BDA0003044510520000065
representing the flow rate of gas flowing into the compressor and the flow rate of gas in the cathode manifold, respectively. The expression of the excess oxygen amount is as follows:
Figure BDA0003044510520000071
the invention sets the pressure in the cathode manifold and the supply manifold and the angular speed of the compressor to be measurable.
(2) Construction and description of integral control framework of PEMFC system
The present invention employs an overall control framework as shown in fig. 2 to regulate the amount of peroxide in the PEMFC system in real time. As can be seen from the figure, the control scheme employs a classical cascade control structure comprising an inner loop portion and an outer loop portion, the outer loop portion being used for oxygen control and the inner loop portion being used for speed regulation.
In the outer loop section, the actual peroxide value can be made to approach a given target value as quickly as possible by using a high-order sliding mode controller based on the Super-Twisting algorithm. Meanwhile, the angular velocity value generated by the outer ring is taken as a reference track by the inner ring, and the voltage transmitted into the air compressor is adjusted by an optimal tracking controller based on an adaptive dynamic programming algorithm to achieve the control purpose.
(3) Design of high-order sliding mode controller based on Super-Twisting algorithm
The controller based on the Super-twist algorithm is as follows:
Figure BDA0003044510520000072
wherein liI belongs to {1,2,3,4} as a normal number to be selected, s (-) as a corresponding sliding mode variable, t as time, and τ as an integral of t;
Figure BDA0003044510520000073
wherein the content of the first and second substances,
Figure BDA0003044510520000074
the amount of the peroxide is the amount of the peroxide,
Figure BDA0003044510520000075
is a desired value of the excess oxygen amount;
the two sides of the formula (3.2) are subjected to derivation to obtain the following expression
Figure BDA0003044510520000076
From the above formula, it is found that the amount of the peroxide
Figure BDA0003044510520000077
With angular speed omega of the compressorcpThe relative degree of freedom between them is 1;
is provided with
Figure BDA0003044510520000078
And σ1(x, t) are bounded functions and satisfy:
Figure BDA0003044510520000079
the parameters in the Super-Twisting algorithm based controller shown in equation (3.1) are selected as follows:
Figure BDA0003044510520000081
wherein li0I e {1,2,3,4} and H are all normal numbers, according to the stability theorem, the sliding mode variable s (T) will be at a finite time T when the following equation (3.6) is satisfiedfTends towards zero and settles to this value and remains unchanged for a time thereafter;
Figure BDA0003044510520000082
(4) designing an optimal tracking controller based on an adaptive dynamic programming algorithm:
consider an affine nonlinear system:
Figure BDA0003044510520000083
wherein x ∈ RnRepresenting the system state, f (x) and g (x) are bounded nonlinear functions;
defining an ideal tracking trajectory as:
Figure BDA0003044510520000084
then the steady state controller is calculated to:
Figure BDA0003044510520000085
subtracting the formula (4.2) from the formula (4.1) yields:
Figure BDA0003044510520000086
wherein e ═ x-xd∈Rn，fe＝f(x)-f(xd),ge＝g(x)-g(xd),ue＝u-ud
The following performance indicator functions are defined:
Figure BDA0003044510520000087
wherein the content of the first and second substances,
Figure BDA0003044510520000088
for the utility function, Q ∈ RnAnd R ∈ RnAll are positive definite matrixes to be set;
define the Hamiltonian:
Figure BDA0003044510520000089
wherein the content of the first and second substances,
Figure BDA00030445105200000810
represents the partial differential of V to the error e;
by definition, the optimal performance indicator function is expressed as:
Figure BDA0003044510520000091
combining (4.6) with (4.7) to obtain:
Figure BDA0003044510520000092
let V*(e) Can be solved and continuously can be differentiated, and (4.8) is paired with ueAnd (3) calculating a partial derivative to obtain:
Figure BDA0003044510520000093
in conjunction with (4.8) and (4.9), the optimal tracking controller is represented as:
Figure BDA0003044510520000094
(5) fitting the performance index function by using a neural network so as to solve a corresponding Hamiltonian;
computing
Figure BDA0003044510520000095
The differential equation (4.6) needs to be solved to avoid solving
Figure BDA0003044510520000096
The method is characterized in that a dimension disaster is trapped in the process, and a neural network with a three-layer structure is used for approximating a performance index function (4.5):
Figure BDA0003044510520000097
wherein, Wc∈RlFor an ideal weight matrix connecting the middle and output layers, ε ∈ RlIs the approximation error generated by the neural network when approximating the function (4.5);
and (3) solving the partial derivatives of e on two sides of the formula (5.1) to obtain:
Figure BDA0003044510520000098
substituting the formula (5.2) into the formula (4.6) to obtain an approximate Hamiltonian:
Figure BDA0003044510520000099
since an ideal weight matrix is difficult to obtain, an evaluation neural network is established to approximate v (e):
Figure BDA00030445105200000910
after the formula (5.4) is substituted into the formula (4.6),
Figure BDA00030445105200000911
combining formula (5.3) and formula (5.5), yielding:
Figure BDA00030445105200000912
wherein the content of the first and second substances,
Figure BDA00030445105200000913
since the goal is to continuously iterate the weight values
Figure BDA00030445105200000914
So that the Hamiltonian (5.5) is as small as possible, thus creating an energy function
Figure BDA00030445105200000915
Continuously adjusting the weight value by adopting a gradient descent algorithm to enable the weight value to iterate towards the direction of reducing the energy function; wherein the content of the first and second substances,
Figure BDA0003044510520000101
the iterative equation of (a) is as follows:
Figure BDA0003044510520000102
wherein alpha is>0 is the learning rate of the evaluation function,
Figure BDA0003044510520000103
representing the derivation of the activation function over time;
according to (4.9) and (5.2), the ideal feedback control law can be calculated as follows:
Figure BDA0003044510520000104
and it can be further approximated as:
Figure BDA0003044510520000105
(6) matlab simulation verification
In simulation, use
Figure BDA0003044510520000106
Indicating the tracking error for the angular speed of the compressor. Since the pressure in the cathode manifold and the supply manifold is measurable, an
Figure BDA0003044510520000107
And
Figure BDA0003044510520000108
variation of numerical value and
Figure BDA0003044510520000109
is closely related and therefore used
Figure BDA00030445105200001010
And
Figure BDA00030445105200001011
respectively, the tracking errors of these two variables.
For the controller, the parameters of the high-order sliding mode controller (3.1) are selected as follows: h1, l10＝20,l20＝20,l30＝3,l40The activation function of the neural network (5.4) was evaluated as 2000:
Figure BDA00030445105200001012
the learning rate α is 0.005. In addition, the initial values of the selected state variables and the evaluation neural network weight are x respectively0＝[60000,55000,80,20000]TAnd
Figure BDA00030445105200001013
the weight matrix in the utility function is selected to be Q-2I3∈R3And
Figure BDA00030445105200001014
given the disturbance current as:
Figure BDA00030445105200001015
the dynamic response curve of the excess oxygen amount of the system under the control of the controller proposed by the present invention is shown in fig. 4, and the voltage input signal is shown in fig. 5. The iterative update process for evaluating the weights of the neural network is shown in fig. 6.
The foregoing is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, various modifications and decorations can be made without departing from the principle of the present invention, and these modifications and decorations should also be regarded as the protection scope of the present invention.

Claims (4)

1. A proton exchange membrane fuel cell peroxide amount control method is characterized in that: the method comprises the following steps:
step 1, constructing a cascade control structure for controlling the oxygen passing amount of a proton exchange membrane fuel cell, wherein the cascade control structure comprises an inner ring part and an outer ring part;
step 2, introducing a high-order sliding mode operator into a controller of an outer ring part of the cascade control structure to obtain a Super-Twisting algorithm-based high-order sliding mode controller, introducing an adaptive dynamic programming algorithm into a controller of an inner ring part of the cascade control structure to obtain an optimal tracking controller based on the adaptive dynamic programming algorithm;
in the outer ring part, the actual peroxide value is made to approach a given target value as fast as possible through a high-order sliding mode controller based on a Super-twist algorithm; in the inner loop portion, the angular velocity values produced by the outer loop portion are referenced by the inner loop portion as a reference trajectory and the voltage delivered to the air compressor is regulated by an optimal tracking controller based on an adaptive dynamic programming algorithm.
2. The proton exchange membrane fuel cell peroxide control method as claimed in claim 1, wherein: in the step 2, the controller based on the Super-Twisting algorithm is as follows:
Figure FDA0003044510510000011
wherein liI belongs to {1,2,3,4} as a normal number to be selected, s (-) as a corresponding sliding mode variable, t as time, and τ as an integral of t;
Figure FDA0003044510510000012
wherein the content of the first and second substances,
Figure FDA0003044510510000013
the amount of the peroxide is the amount of the peroxide,
Figure FDA0003044510510000014
a desired value representing the excess oxygen amount;
the two sides of the formula (3.2) are subjected to derivation to obtain the following expression
Figure FDA0003044510510000015
From the above formula, it is found that the amount of the peroxide
Figure FDA0003044510510000016
With angular speed omega of the compressorcpThe relative degree of freedom between them is 1;
is provided with
Figure FDA0003044510510000017
And σ1(x, t) are bounded functions and satisfy:
Figure FDA0003044510510000018
the parameters in the Super-Twisting algorithm based controller shown in equation (3.1) are selected as follows:
Figure FDA0003044510510000021
wherein li0I e {1,2,3,4} and H are all normal numbers, according to the stability theorem, the sliding mode variable s (T) will be at a finite time T when the following equation (3.6) is satisfiedfTends towards zero and settles to this value and remains unchanged for a time thereafter;
Figure FDA0003044510510000022
3. the proton exchange membrane fuel cell peroxide control method as claimed in claim 1, wherein: in the step 2, an optimal tracking controller based on an adaptive dynamic programming algorithm is designed through the following steps:
consider an affine nonlinear system:
Figure FDA0003044510510000023
wherein x ∈ RnRepresenting the system state, f (x) and g (x) are bounded nonlinear functions;
defining an ideal tracking trajectory as:
Figure FDA0003044510510000024
then the steady state controller is calculated to:
Figure FDA0003044510510000025
subtracting the formula (4.2) from the formula (4.1) yields:
Figure FDA0003044510510000026
wherein e ═ x-xd∈Rn，fe＝f(x)-f(xd),ge＝g(x)-g(xd),ue＝u-ud
The following performance indicator functions are defined:
Figure FDA0003044510510000027
wherein the content of the first and second substances,
Figure FDA0003044510510000028
for the utility function, Q ∈ RnAnd R ∈ RnAll are positive definite matrixes to be set;
define the Hamiltonian:
H(e,ue,Ve)＝r(ue,e)+Ve T(fe+geud+g(x)ue)＝0 (4.6)
wherein the content of the first and second substances,
Figure FDA0003044510510000029
represents the partial differential of V to the error e;
by definition, the optimal performance indicator function is expressed as:
Figure FDA0003044510510000031
combining formula (4.6) with formula (4.7), yielding:
Figure FDA0003044510510000032
let V*(e) Can be solved and continuously can be differentiated, and (4.8) is paired with ueAnd (3) calculating a partial derivative to obtain:
Figure FDA0003044510510000033
combining equation (4.8) and equation (4.9), the optimal tracking controller is expressed as:
Figure FDA0003044510510000034
4. the proton exchange membrane fuel cell peroxide control method as claimed in claim 3, wherein: fitting the performance index function by using a neural network so as to solve a corresponding Hamiltonian; the method specifically comprises the following steps:
computing
Figure FDA0003044510510000035
The differential equation (4.6) needs to be solved to avoid solving
Figure FDA0003044510510000036
The method is characterized in that a dimension disaster is trapped in the process, and a neural network with a three-layer structure is used for approximating a performance index function (4.5):
Figure FDA0003044510510000037
wherein, Wc∈RlFor an ideal weight matrix connecting the middle and output layers, ε ∈ RlIs the approximation error generated by the neural network when approximating the function (4.5);
and (3) solving the partial derivatives of e on two sides of the formula (5.1) to obtain:
Figure FDA0003044510510000038
substituting the formula (5.2) into the formula (4.6) to obtain an approximate Hamiltonian:
Figure FDA0003044510510000039
since an ideal weight matrix is difficult to obtain, an evaluation neural network is established to approximate v (e):
Figure FDA00030445105100000310
after the formula (5.4) is substituted into the formula (4.6),
Figure FDA00030445105100000311
combining formula (5.3) and formula (5.5), yielding:
Figure FDA00030445105100000312
wherein the content of the first and second substances,
Figure FDA00030445105100000313
since the goal is to continuously iterate the weight values
Figure FDA00030445105100000314
So that the Hamiltonian (5.5) is as small as possible, thus creating an energy function
Figure FDA0003044510510000041
Continuously adjusting the weight value by adopting a gradient descent algorithm to enable the weight value to iterate towards the direction of reducing the energy function; wherein the content of the first and second substances,
Figure FDA0003044510510000042
the iterative equation of (a) is as follows:
Figure FDA0003044510510000043
wherein alpha is>0 is the learning rate of the evaluation function,
Figure FDA0003044510510000044
representing the derivation of the activation function over time;
according to (4.9) and (5.2), the ideal feedback control law can be calculated as follows:
Figure FDA0003044510510000045
and it can be further approximated as:
Figure FDA0003044510510000046
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CN114567223A (en) * 2022-04-28 2022-05-31 成都国营锦江机器厂 Position tracking method based on repeated supercoiled observer and supercoiled control

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* Cited by examiner, † Cited by third party
Title
JING ZHU等: "Online ADP based Oxygen Excess Ratio Control of the PEM Fuel Cell System Applying to UAVs", 《INTERNATIONAL CONFERENCE ON UNMANNED AIRCRAFT SYSTEMS》 *

Cited By (1)

* Cited by examiner, † Cited by third party
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CN114567223A (en) * 2022-04-28 2022-05-31 成都国营锦江机器厂 Position tracking method based on repeated supercoiled observer and supercoiled control

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