A Design Method for the Cathode Controller of a Proton Exchange Membrane Fuel Cell

Abstract

The present invention discloses a design method for a cathode controller of a proton exchange membrane fuel cell, comprising: Step 1) According to the gas dynamics equation of the cathode system of the proton exchange membrane fuel cell and the air compressor angular velocity equation, combined with the time-delay signals of the air compressor angular velocity and gas pressure caused by sensor time delay and the uncertain changes of environmental parameters, establish a cathode time-delay system model; Step 2) For the cathode time-delay system of the proton exchange membrane fuel cell, design an interference observer to estimate the interference of the motor rotor imbalance in the air compressor, and design a state feedback controller combined with robust control performance to complete the controller design of the time-delay system; Step 3) Based on the convex optimization algorithm, solve the gain matrices of the designed state feedback controller and interference observer. The present invention solves the problems of time-delay phenomenon and multi-source interference in the cathode system of the proton exchange membrane fuel cell, and effectively improves the anti-interference ability of the control system.

Description

Technical Field

[0001] The present invention relates to the technical field of hydrogen energy, and particularly relates to a design method for a cathode controller of a proton exchange membrane fuel cell. Background Art

[0002] Fuel cell technology is an advanced clean energy technology. Fuel cells can directly convert the chemical energy of fuels into electrical energy, with characteristics such as high efficiency, pollution-free, and long lifespan. Fuel cells are classified into proton exchange membrane fuel cells, alkaline fuel cells, solid oxide fuel cells, etc. according to the type of electrolyte. Compared with other types of fuel cells, PEMFCs are widely used due to their advantages such as high energy conversion efficiency, low operating temperature, and high conversion power. This type of battery mainly uses hydrogen as the main fuel for chemical reactions, and there is no pollution in the chemical reaction process. In addition to being used as the power source for transportation tools such as automobiles, PEMFCs can also be used as distributed generators in hospitals and hotels; or as the power source for portable electronic devices. Although proton exchange membrane fuel cells have many application prospects, the battery will inevitably be affected by various types of interference under working conditions. If effective control strategies cannot be implemented in a timely manner, it will often affect the working condition safety and the service life of the battery. Therefore, optimizing the control algorithm design of the battery's actuators and improving the safety and reliability of the battery during operation can greatly accelerate the application process of PEMFC fuel cells in society.

[0003] The power generation system of a proton exchange membrane fuel cell consists of a stack, anode and cathode systems, a water and heat management system, an electric energy conversion system, a control system, etc. In the anode and cathode systems (hydrogen and oxygen supply systems), reactants, hydrogen, are fed into the proton exchange membrane fuel cell stack at a certain stoichiometric ratio, where they react to generate electrical energy and water. In the composition of the cathode system, an air compressor continuously and stably transports oxygen. Through physical devices such as a gas supply manifold, a cathode flow field, a humidifier, and a cooler, it is transported to the flow field of the cathode reaction gas of the stack. If a higher oxygen pressure will increase the electrochemical reaction, and if the stoichiometric ratio of the reaction gas participating in the reaction in the cathode does not reach a certain standard, resulting in an oxygen deficiency phenomenon, it will damage the proton exchange membrane. Therefore, it is extremely important to control the oxygen stoichiometric ratio of the cathode system.

[0004] The cathode system of a proton exchange membrane fuel cell has a complex structure and has the characteristics of high coupling and strong nonlinearity. In actual application scenarios, the changes in environmental parameters and sensor measurements have error interference and time-delay interference, and the motor in the air compressor is also prone to rotor imbalance interference during operation.

[0005] The single interference suppression method only targets a certain performance of the cathode system of a proton exchange membrane fuel cell, with a large degree of conservatism and difficulty in achieving high-precision control. The control method based on an interference observer can handle multi-source interferences in the cathode system of a proton exchange membrane fuel cell, but when the control algorithm is applied to actual engineering, the situation will become complex. Summary of the Invention

[0006] The purpose of the present invention is to overcome the defects of the prior art and provide a design method for the cathode controller of a proton exchange membrane fuel cell. An interference observer is established for the unbalanced interference of the motor rotor in the air compressor in the cathode system of the proton exchange membrane fuel cell to observe the rotor unbalanced interference. For the parameter time-varying phenomenon caused by changes in environmental parameters in the model and the time-delay phenomenon generated by the angular velocity and gas pressure sensors, the gain matrix of the state feedback controller combined with robust control performance is used to suppress the influence of both on the system. Based on the convex optimization algorithm, the designed observer gain matrix and state feedback controller gain matrix are realized. The problem of multi-source interferences in the practical engineering application of the cathode control system of the proton exchange membrane fuel cell is solved, and the practicability of the control system is improved.

[0007] The purpose of the present invention is realized as follows: A design method for the cathode controller of a proton exchange membrane fuel cell includes the following steps:

[0008] Step 1) Describe the mathematical model of the cathode system of the proton exchange membrane fuel cell through the change of gas pressure in the system. According to the composition of the cathode system of the proton exchange membrane fuel cell, establish the gas dynamic equations in the air compressor angular velocity, air supply manifold, cathode flow field, return manifold, and back pressure valve. Combine the time-delay signals of the air compressor angular velocity and gas pressure caused by sensor time delay and the change of environmental parameter uncertainty to establish a time-delay system model of the cathode of the proton exchange membrane fuel cell with multi-source interferences.

[0009] Step 2) Based on the time-delay system model of the cathode of the proton exchange membrane fuel cell in Step 1), design an interference observer to estimate the unbalanced interference of the motor rotor in the air compressor, and design a state feedback controller combined with robust control performance to complete the controller design of the time-delay system.

[0010] Step 3) Solve the gain matrices of the designed state feedback controller and interference observer based on the convex optimization algorithm.

[0011] As a further limitation of the present invention, in Step 1), the gas dynamic equations in the air compressor angular velocity, air supply manifold, cathode flow field, return manifold, and back pressure valve are:

[0012]

[0013] Where p smis the air pressure of the air supply manifold; p ca is the air pressure of the cathode flow field; p rm is the air pressure of the return manifold; p sat (T st ) is the saturation vapor pressure at the stack temperature; R is the gas constant; R cm is the internal resistance of the air compressor motor; T atm is the standard atmospheric temperature; T rm is the temperature of the return manifold; T st is the stack temperature; V sm is the volume of the air supply manifold; V ca is the volume of the cathode flow field; V rm is the volume of the return manifold; W cp is the output flow rate of the air compressor; v cm is the driving voltage of the air compressor; γ is the gas constant; κ is the gas fitting average constant; k sm,out is the outlet constant of the air supply manifold; k ca,out is the outlet constant of the cathode flow field; k t 、k v is the motor constant; n is the number of stacks of the fuel cell; I st is the stack current; is the molar mass of oxygen; is the molar mass of nitrogen; M a is the molar mass of the mixed gas; F is the Faraday constant; θ is the back pressure valve opening; ω cp is the angular velocity of the air compressor; ω atm is the standard humidity ratio; J cp is the moment of inertia of the air compressor; C p is the specific heat capacity at constant pressure of air; C d is the back pressure valve coefficient of the return manifold; η cp is the efficiency of the air compressor; η cm is the efficiency of the air compressor motor; A T is the area of the back pressure valve region;

[0014] Take the air pressures p sm 、p ca 、p rm and the angular velocity ω of the air compressor cp to form the state vector x(t) = [ω cp p sm p ca p rm T The stack current disturbance vector d2(t) = I st The system input vector u(t) = v cm cp Consider the angular velocity ω of the air compressor sm and the air supply manifold pressure p​For the time-delay phenomenon and the modeling error of the system, the proton exchange membrane cathode time-delay system established is as follows:

[0015]

[0016] Where: x(t) is the state variable of the system state equation, d2(t) is the stack current disturbance of the system, x(t - τ) = [ω cp (t - τ) p sm (t - τ)] T is the time-delay state variable of the system, y(t) = [ω cp p sm p ca p rm T is the output variable of the system, τ is the time-delay length; the matrices A1, A2, B, D1, C1 and f(x(t), x(t - τ)) are shown as follows:

[0017]

[0018]

[0019]

[0020] ΔA1(t), ΔA2(t) are time-varying function matrices of environmental parameters, and the matrix forms are as follows:

[0021]

[0022]

[0023] d1(t) is the unbalance disturbance of the air compressor motor rotor, which is described by the following exogenous system:

[0024]

[0025] Where, r(t) ∈ R 2×1 is the state variable of the exogenous system, W ∈ R 2×2 and V ∈ R 1×2 represent the system matrix and the output matrix of the exogenous system respectively;

[0026] The definition of the control target oxygen stoichiometry is as follows:

[0027]

[0028] The overall control target of the system is to make the desired oxygen stoichiometry To simplify the design of the system control target, the desired output is defined as follows:

[0029] ​

[0030] The C2 and D2 matrices are as follows:

[0031] D2 = 2.

[0032] As a further limitation of the present invention, step 2) specifically includes:

[0033] 2-1) Construct an interference observer to estimate the unbalance interference of the motor rotor in the air compressor, and design the form of the interference observer as follows:

[0034]

[0035] where: s(t) ∈ R 2×1 is the state variable of the interference observer, is the estimated value of the unbalance interference d1(t) of the motor rotor of the air compressor in the cathode system of the proton exchange membrane fuel cell, is the estimated value of the unbalance interference r(t) of the motor rotor of the air compressor, L1 ∈ R 2×4 and L2 ∈ R 2×4 are the observer gain matrices to be designed;

[0036] 2-2) Design the controller of the cathode system of the proton exchange membrane fuel cell in the form of a state feedback controller with robust control performance as:

[0037]

[0038] where, K ∈ R 1×4 is the undetermined state feedback controller gain matrix.

[0039] As a further limitation of the present invention, step 3) specifically includes: The problem corresponding to the convex optimization algorithm containing free weight matrices is summarized as:

[0040]

[0041]

[0042]

[0043] where: Ψ1 = sym(AQ1 + BR1) + S1 + I, Ψ2 = sym[P2W + R2C1BV)] + I; λ1, λ2, α1 are positive constants; the symbol * represents the symmetric block of the corresponding part in the symmetric matrix; Q1, S1, P1, P2, P3 ∈ R 4×4 is the positive definite real matrix space; R1, R2, R3 ∈ R 4×4 is the real matrix space; M1 ∈ R 4×4 and M2 ∈ R 4×4 and M3 ∈ R4×2 is an uncertain bound parameter matrix; G1 and G2 are Lipschitz parameter matrices of the nonlinear term f(x(t), x(t - τ));

[0044] Solve the linear matrix inequality to obtain the state feedback controller gain matrix Disturbance observer gain matrix

[0045] The present invention adopts the above technical solutions. Compared with the prior art, the beneficial effects are as follows: (1) The present invention has strong robustness to disturbances. In the case of multiple sources of disturbances such as rotor imbalance disturbance, sensor time delay disturbance, and environmental parameter time-varying disturbance in the air compressor motor, the designed cathode system controller based on the disturbance observer suppresses the above disturbances; it overcomes the deficiencies of general disturbance suppression methods (such as robust control, variable structure control, etc.) with poor control accuracy and large conservativeness, and at the same time improves the reliability of the system.

[0046] (2) The requirements of the system's hybrid performance index are ensured through the design method based on convex optimization. The constructed method considers the robust control performance index; it overcomes the deficiencies of single-performance-index control methods such as anti-disturbance control and variable structure control, improves the comprehensive performance of the system; and ensures the cancellation of other forms of disturbances while meeting the control accuracy of the oxygen stoichiometric ratio. Brief Description of the Drawings

[0047] Figure 1 is the flowchart of the present invention.

[0048] Figure 2 is the structural diagram of the cathode system of the proton exchange membrane fuel cell.

[0049] Figure 3 is the composition diagram of the cathode control system of the proton exchange membrane fuel cell. Detailed Embodiment

[0050] Such as Figure 1 shown, a cathode controller of a proton exchange membrane fuel cell based on a disturbance observer, the specific implementation steps are as follows:

[0051] 1) According to the gas dynamics equation and the air compressor angular velocity equation of the cathode system of the proton exchange membrane fuel cell, establish a time-delay system model of the proton exchange membrane cathode:

[0052] Such as Figures 2 - 3 shown, is the schematic diagram of the cathode system of the proton exchange membrane fuel cell, including a supply manifold, a cathode flow channel, a cooler, a humidifier, a return manifold, and a back pressure valve. The establishment of the cathode control system of the proton exchange membrane fuel cell based on the disturbance observer is mainly based on the air flow and air pressure dynamics equations between the models. First, make the following assumptions:

[0053] (1) All gases satisfy the ideal gas law;

[0054] From the ideal gas law and the law of conservation of mass, considering the dynamic relationship between the air pressure and flow rate of each model, and considering the system time-delay signal and the time-varying phenomenon of environmental parameters, the gas equation of the cathode system of a proton exchange membrane fuel cell is obtained as follows:

[0055]

[0056] In the formula, p sm is the air pressure of the supply manifold; p ca is the air pressure of the cathode flow field; p rm is the air pressure of the return manifold; p sat (T st ) is the saturation air pressure at the stack temperature; R is the gas constant; R cm is the internal resistance of the air compressor motor; T atm is the standard atmospheric temperature; T rm is the temperature of the return manifold; T st is the stack temperature; V sm is the volume of the supply manifold; V ca is the volume of the cathode flow field; V rm is the volume of the return manifold; W cp is the output flow rate of the air compressor; v cm is the driving voltage of the air compressor; γ is the gas constant; κ is the average gas fitting constant; k sm,out is the outlet constant of the supply manifold; k ca,out is the outlet constant of the cathode flow field; k t , k v are the motor constants; n is the number of fuel cell stacks; I st is the stack current; is the molar mass of oxygen; is the molar mass of nitrogen; M a is the molar mass of the mixed gas; F is the Faraday constant; θ is the back pressure valve opening; ω cp is the angular velocity of the air compressor; ω atm is the standard humidity ratio; J cp is the moment of inertia of the air compressor; C p is the constant-pressure specific heat capacity of air; C d is the back pressure valve coefficient of the return manifold; η cp is the efficiency of the air compressor; η cm is the efficiency of the air compressor motor; A T is the area of the back pressure valve region.

[0057] Combined with the dynamic formula of the cathode system of a proton exchange membrane fuel cell, let x1(t) = ω cp , x2(t) = p sm, \(x_3(t)=p\) an , \(x_4(t)=p\) rm , \(u(t)=v\) cm , \(d_2(t)=I\) st ,

[0058]

[0059]

[0060]

[0061]

[0062] All of the above non - linear functions satisfy the Lipschitz condition, that is:

[0063] \(\left\lVert f(x(t),x(t - \tau))\right\rVert\leq\left\lVert G_1x(t)\right\rVert+\left\lVert G_2x(t - \tau)\right\rVert\)

[0064] where \(G_1\) and \(G_2\) are the upper - bound gain matrices of the non - linear function matrices.

[0065] The system output makes \(y_1=\omega\) cp , \(y_2 = p\) sm , \(y_3 = p\) ca , \(y_4 = p\) rm .

[0066] The state - space expression of the fuel - cell cathode system can be obtained:

[0067]

[0068] where: \(x(t)=[x_1\ x_2\ x_3\ x_4]\) T , \(y(t)=[y_1\ y_2\ y_3\ y_4]\) T ,

[0069]

[0070]

[0071] \(d_1(t)\) is the rotor - imbalance disturbance in the air - compressor motor, which is described by the following exogenous system:

[0072]

[0073] In the formula, \(r(t)\in R\) 2×1 is the state variable of the exogenous system, \(W\in R\) 2×2 , \(V\in R\) 1×2 represent the system matrix and the output matrix of the exogenous system respectively.

[0074] ΔA1(t) and ΔA2(t) are time-varying matrices caused by changes in environmental parameters. Considering that this time-varying matrix is bounded and unknown, it satisfies the following conditions:

[0075] [ΔA1(t) ΔA2(t)] = M1F(t)[M2 M3]

[0076] M1, M2, and M3 are known constant matrices, and F(t) is a known bounded time-varying matrix that satisfies:

[0077] F T (t)F(t) ≤ I.

[0078] Since the control objective requires the oxygen stoichiometry ratio of the cathode flow field of the fuel cell to track to its calculation expression is as follows:

[0079]

[0080] This performance index is a non-linear term, and it is not easy to separate the interference from the state variables. Therefore, the system control output is selected as:

[0081]

[0082] Matrices C2 and D2 are as follows:

[0083] D2 = 2

[0084] Combining the control output with the system state equation gives:

[0085]

[0086] 2) For the rotor imbalance interference d1(t), the environmental parameter time-varying matrices ΔA1(t) and ΔA2(t), the time-delay variable x(t - τ), and the stack current interference d2(t) in the air compressor motor of the cathode system of the proton exchange membrane fuel cell, design an interference observer as:

[0087]

[0088] where: s(t) ∈ R 2×1 is the state variable of the interference observer, is the estimated value of the rotor imbalance interference d1(t) in the air compressor motor of the cathode system of the proton exchange membrane fuel cell, is the estimated value of the rotor imbalance interference r(t) describing the air compressor motor, L1 ∈ R 2×4 and L2 ∈ R 2×4 are the observer gain matrices to be designed.

[0089] The anti-interference controller of the system is:

[0090]

[0091] where K ∈ R 1×4 is the gain matrix of the state feedback controller to be determined

[0092] where K is the gain matrix of the state feedback controller to be determined; based on the design of the controller, the state space equation of the system can be expressed as:

[0093]

[0094] 3) Use the convex optimization algorithm to solve the disturbance observer and the state feedback controller:

[0095] (1) The convex optimization problem can be summarized as:

[0096]

[0097]

[0098]

[0099] where: Ψ1 = sym(AQ1 + BR1) + S1 + I, Ψ2 = sym[P2W + R2C1BV)] + I; λ1, λ2, α1 are positive constants; the symbol * represents the symmetric block of the corresponding part in the symmetric matrix; Q1, S1, P1, P2, P3 ∈ R 4×4 is the space of positive definite real matrices; R1, R2, R3 ∈ R 4×4 is the space of real matrices.

[0100] (2) Solve the cathode controller gain array and the disturbance observer gain array of the proton exchange membrane fuel cell:

[0101] Solve the linear matrix inequality to obtain the gain matrix of the state feedback controller the gain matrix of the disturbance observer

[0102] The present invention aims at the rotor imbalance interference, sensor time-delay interference and environmental parameter time-varying interference in the air compressor motor of the cathode system of a proton exchange membrane fuel cell. First, a disturbance observer is constructed to estimate the rotor imbalance interference in the air compressor motor. Secondly, a state feedback controller combined with robust control performance is designed to suppress the sensor time-delay interference and environmental parameter time-varying interference. Based on linear matrix inequalities (LMIs), the design problems of the disturbance observer and the state feedback controller gains of the cathode system of the proton exchange membrane fuel cell are transformed into convex optimization problems. Finally, the convex optimization problems are solved, and through corresponding algebraic transformations, the disturbance observer gain array and the state feedback controller gain array of the cathode system of the proton exchange membrane fuel cell are solved from the feasible solutions of the convex optimization problems.

[0103] The method of the present invention combines anti-interference control based on a disturbance observer and the time-delay phenomenon of the cathode system. While canceling multi-source interference, it ensures that the oxygen stoichiometric ratio of the system can be continuously and stably supplied to the reaction stack. This method can effectively improve the anti-interference ability and versatility of the control system.

[0104] The present invention is not limited to the above embodiments. Based on the technical solutions disclosed in the present invention, those skilled in the art can make some substitutions and deformations of some technical features without creative labor according to the disclosed technical content, and these substitutions and deformations are within the protection scope of the present invention.

Claims

1. A design method for the cathode controller of a proton exchange membrane fuel cell, characterized in that, It includes the following steps: Step 1) Describe the mathematical model of the cathode system of a proton exchange membrane fuel cell through the change of gas pressure in the system. According to the composition of the cathode system of the proton exchange membrane fuel cell, establish the gas dynamic equations in the air compressor angular velocity, the supply manifold, the cathode flow field, the return manifold, and the back pressure valve. Combine the changes in the air compressor angular velocity, the gas pressure time-delay signal caused by the sensor time delay, and the uncertainty of the environmental parameters to establish a time-delay system model of the proton exchange membrane fuel cell cathode with multi-source interference; Step 2) Based on the time-delay system model of the proton exchange membrane fuel cell cathode in Step 1), design an interference observer to estimate the motor rotor imbalance interference in the air compressor, and design a state feedback controller combined with robust control performance to complete the controller design of the time-delay system; Step 3) Solve the gain matrices of the designed state feedback controller and interference observer based on the convex optimization algorithm; The gas dynamic equations of the air compressor angular velocity, the supply manifold, the cathode flow field, and the return manifold described in Step 1) are: Where p sm is the air pressure of the air supply manifold; p ca is the air pressure of the cathode flow field; p rm is the air pressure of the return manifold; p sat (T st ) is the saturation air pressure at the stack temperature; R is the gas constant; R cm is the internal resistance of the air compressor motor; T atm is the standard atmospheric temperature; T rm is the temperature of the return manifold; T st is the stack temperature; V sm is the volume of the air supply manifold; V ca is the volume of the cathode flow field; V rm is the volume of the return manifold; W cp is the output flow rate of the air compressor; v cm is the driving voltage of the air compressor; γ is the gas constant; κ is the average gas fitting constant; k sm,out is the outlet constant of the air supply manifold; k ca,out is the outlet constant of the cathode flow field; k t 、k v are the motor constants; n is the number of stacks of the fuel cell; I st is the stack current; is the molar mass of oxygen; is the molar mass of nitrogen; M a is the molar mass of the mixed gas; F is the Faraday constant; θ is the back pressure valve opening; ω cp is the angular velocity of the air compressor; ω atm is the standard humidity ratio; J cp is the moment of inertia of the air compressor; C p is the constant pressure specific heat capacity of air; C d is the back pressure valve coefficient of the return manifold; η cp is the efficiency of the air compressor; η cm is the efficiency of the air compressor motor; A T is the area of the back pressure valve region; Take the pressures p of each gas sm , p ca , p rm and the angular velocity ω of the air compressor cp to form the state vector x(t) = [ω cp p sm p ca p rm T , the stack current interference vector d2(t) = I st , and the system input vector u(t) = v cm ; Considering the time-delay phenomenon of the angular velocity ω of the air compressor cp and the pressure p of the supply manifold sm and the modeling error of the system, the proton exchange membrane cathode time-delay system established is as follows:​ where: x(t) is the state variable of the system state equation, d2(t) is the fuel cell current disturbance of the system, x(t-τ) = [ω cp (t-τ)p sm (t-τ)] T is the time-delay state variable of the system, y(t) = [ω cp p sm p ca p rm T is the output variable of the system, τ is the time-delay length; the matrices A1, A2, B, D1, C1 and f(x(t), x(t-τ)) are shown as follows:​ ΔA1(t) and ΔA2(t) are time-varying function matrices of environmental parameters, and the matrix form is as follows: d1(t) is the motor rotor imbalance interference of the air compressor, which is described by the following exogenous system: where \(r(t)\in\mathbb{R}\) 2×1 is the state variable of the external system, \(W\in\mathbb{R}\) 2×2 , \(V\in\mathbb{R}\) 1×2 represent the system matrix and the output matrix of the external system, respectively; The definition of the control target oxygen stoichiometric ratio is as follows: The overall control objective of the system is to achieve the desired oxygen stoichiometry ratio To simplify the design of the system control objective, the desired outputs are defined as follows: The C2 and D2 matrices are as follows: D2＝2。 2. The design method of a cathode controller for a proton exchange membrane fuel cell according to claim 1, characterized in that, The specific content of Step 2) includes: 2-1) Construct an interference observer to estimate the motor rotor imbalance interference in the air compressor. The form of the interference observer is designed as follows: where: s(t) ∈ R 2×1 is the state variable of the disturbance observer, is the estimated value of the rotor imbalance disturbance d1(t) of the air compressor motor in the cathode system of the proton exchange membrane fuel cell, is the estimated value of the rotor imbalance disturbance r(t) of the air compressor motor, L1 ∈ R 2×4 and L2 ∈ R 2×4 are the observer gain matrices to be designed; 2-2) Use a state feedback controller with robust control performance to design the controller of the proton exchange membrane fuel cell cathode system, and the form is: where \(K\in\mathbb{R}\) 1×4 is the gain matrix of the state feedback controller to be determined.

3. A design method for a cathode controller of a proton exchange membrane fuel cell according to claim 1, characterized in that The specific content of Step 3) includes: The problem corresponding to the convex optimization algorithm with free weight matrices is summarized as: where: Ψ1 = sym(AQ1 + BR1) + S1 + I, Ψ2 = sym[P2W + R2C1BV)] + I; λ1, λ2, α1 are positive constants; the symbol * represents the symmetric block of the corresponding part in the symmetric matrix; Q1 ∈ R 4×4 , S1 ∈ R 4×4 , P1 ∈ R 4×4 , P2 ∈ R 4×4 , P3 ∈ R 4×4 is the space of positive definite real parameter matrices; R1 ∈ R 4×4 , R2 ∈ R 4×4 , R3 ∈ R 4×4 is the space of real parameter matrices; M1 ∈ R 4×4 , M2 ∈ R 4×4 , M3 ∈ R 4 ×2 is the uncertain bound parameter matrix; G1, G2 are the Lipschitz parameter matrices of the nonlinear term f(x(t), x(t - τ)); Solve the linear matrix inequality to obtain the state feedback controller gain matrix Disturbance observer gain matrix

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