Design method of engine sensor robust dynamic compensator based on hammerstein-lpv model

Abstract

The application belongs to the technical field of sensors of aero-engines, and discloses a design method of a robust dynamic compensator of an engine sensor based on a Hammerstein-LPV model. First, a Hammerstein-LPV model of the sensor is established, the model has the characteristics of dynamic and static characteristic separation, and has better dynamic characteristic fitting effect than a traditional sensor LTI model, and is more conducive to the design of a dynamic compensator. Based on the model, a sensor expected pole region is obtained by using a regional pole configuration technology, and a robust H ∞ Filter method designs a dynamic compensator of the sensor which is optimal under multiple working conditions. The design method can effectively improve the modeling accuracy of the sensor, can improve the robustness of the filter dynamic compensator design, effectively solves the problem that the compensation ability of the traditional method is greatly degraded under the condition that the working condition of the sensor changes, and has important engineering significance.

Description

Technical Field

[0001] This invention belongs to the field of sensor technology for aero-engines, specifically relating to a design method for a robust dynamic compensator for engine sensors based on the Hammerstein-LPV model. Background Technology

[0002] Sensor systems are a crucial component of aero-engine measurement systems, and their dynamic performance significantly impacts the overall control effectiveness of the aero-engine. Typically, the speed of sensor dynamic performance and its robustness are mutually exclusive. Therefore, sensors used in actual aero-engines must prioritize robustness, inevitably sacrificing rapid response. For example, in actual aero-engine operation, placing thermocouples in protective sheaths extends their lifespan, but this increases their time constant. Furthermore, due to limitations in current sensor manufacturing processes and capabilities, it is difficult to further improve sensor dynamic characteristics through manufacturing alone. Therefore, achieving dynamic compensation for sensors in the engine through software is a pressing challenge and a key technology for realizing intelligent sensors.

[0003] Currently, the design of dynamic compensators for sensors mainly focuses on the design of sensor compensators within the framework of a linear time-invariant (LTI) model under a single operating condition. The main design methods include the zero-pole placement method and the neural network method. A review of existing literature on sensor dynamic compensator design, such as Huang Junqin's "Dynamics and Applications of Test Systems," Wang Wei's "Research on Dynamic Modeling and Real-time Self-Correction of Sensor Inertia," and Zhao Yang's "Research on Nonlinear Correction and Dynamic Compensation of Sensors," reveals that the zero-pole placement method for sensor dynamic compensators is designed for compensators under a single sensor operating condition, and its performance is significantly affected by changes in the sensor model. The neural network compensation method struggles to determine the optimal network structure, and the poor convergence of the model with small training data results in low model accuracy and generalization ability. Therefore, to address this issue, this invention proposes a robust H-type engine sensor based on the Hammerstein-LPV model. ∞ The design method of dynamic compensators has important engineering significance. Summary of the Invention

[0004] To address the problems of existing zero-pole placement algorithms based on single models and data-based neural network design algorithms, this invention provides a robust H-type engine sensor based on the Hammerstein-LPV model. ∞Dynamic Compensator Design Method. In temperature measurement of aero-engines, to enhance the robustness of thermocouple operation, thicker thermocouple wires are usually selected and placed in a protective sheath, resulting in poor dynamic characteristics of the thermocouple. Therefore, this paper takes the thermocouple as an example to design a dynamic compensator. However, it should be noted that this invention is generally applicable to voltage output sensors. First, a Hammerstein-LPV model of the thermocouple is established. This model has the characteristic of separating dynamic and static characteristics, and has a better dynamic characteristic fitting effect than the traditional thermocouple LTI model, making it easier to design a dynamic compensator. Based on this model, the desired pole region of the thermocouple is obtained using the regional pole placement technique, and based on robust H... ∞ The method designs the optimal dynamic compensator for thermocouples under various operating conditions.

[0005] The technical solution of this invention:

[0006] A robust dynamic compensator design method for engine sensors based on the Hammerstein-LPV model, comprising the following steps:

[0007] Step 1: Establish the Hammerstein-LPV model of the thermocouple

[0008] Step 1.1: Using T s For the sampling period, N operating point temperatures are selected within the entire operating range of the thermocouple to be modeled. A thermocouple dynamic calibration instrument is used to generate the standard temperature input u of the required thermocouple. The thermocouple is fed into the corresponding position of the thermocouple dynamic calibration instrument, which is regarded as the thermocouple receiving a step excitation signal with an amplitude of the set temperature of the thermocouple dynamic calibration instrument. At the same time, the output voltage value y of the thermocouple is collected. out ;

[0009] Step 1.2: Analyze the output voltage value y of the thermocouple acquired in Step 1.1. out The data is processed as follows: First, outliers are removed using the 3σ principle, and then mean filtering is performed to obtain the dynamic output voltage data of the thermocouple.

[0010] Step 1.3: Obtain the corresponding temperature-voltage meter based on the thermocouple's calibration number, and perform polynomial fitting on the thermocouple's temperature and voltage values ​​to obtain y. out =f(u)=c2u 2 +c1u+c0, where c2, c1, and c0 are fitting coefficients, and this fitting polynomial is used as the static model in the Hammerstein-LPV model of the thermocouple.

[0011] Step 1.4: Based on the geometry of the thermocouple, the heat transfer effect inside the thermocouple is equivalent to a one-dimensional heat conduction problem. According to the heat conduction differential equation in the heat transfer theory, the thermocouple is divided into several unit cells to obtain the discrete state space model of the thermocouple. The order of the discrete state space model is the number of unit cells p.

[0012] Based on the normalized dynamic output voltage data of the thermocouple obtained in step 1.2, parameter identification is performed to obtain the normalized discrete state-space model of the thermocouple:

[0013]

[0014] in,

[0015]

[0016] C di =cx(θ)*(0 0 0 ... ...0 0 1)∈R 1×p ;

[0017] Where u(k) is the input of the thermocouple's state-space model, representing the temperature value of the thermocouple's operating condition; y(k) is the output of the thermocouple's state-space model, representing the thermocouple's output voltage value; x(k) is the state vector of the thermocouple's state-space model, representing the temperature value of each unit cell after the thermocouple is discretized; and θ is the normalization exponent of the thermocouple's operating temperature. v max u is the maximum measurement temperature of the thermocouple. min θ represents the minimum measurement temperature of the thermocouple; ax(θ), bx(θ), and cx(θ) represent the heat transfer coefficient of adjacent thermocouple units, the heat transfer coefficient between the hot junction of the thermocouple and the outside world, and the thermocouple output coefficient, respectively, which are affected by the scheduling parameter θ.

[0018] Step 1.5: Rewrite ax(θ), bx(θ), and cx(θ) in polynomial form as follows:

[0019] ax(θ) = a0 + a1θ + a2θ 2 +a3θ 3

[0020] bx(θ) = b0 + b1θ + b2θ 2 +b3θ 3

[0021] cx(θ) = c0 + c1θ + c2θ 2 +c3θ 3

[0022] Where a0, a1, a2, a3, b0, b1, b2, b3, c0, c1, c2, and c3 are the coefficients to be determined;

[0023] This yields the dynamic model in the Hammerstein-LPV model of the thermocouple. In the Hammerstein model, the static nonlinear model is placed first, and the dynamic linear model is placed later. Connecting the static model and the dynamic model obtained in step 1.3 in series yields the Hammerstein-LPV model of the thermocouple.

[0024] Step 2: Using the regional pole placement technique, by solving a set of linear matrix inequalities (LMIs), the state feedback vector K is obtained, and a reference model for the thermocouple normalized continuous state-space model is obtained.

[0025] Step 2.1: Set the performance evaluation index of the thermocouple, and obtain the expected region D = D of the thermocouple poles based on the correspondence between poles and dynamic performance. (H，Q) ;

[0026] Step 2.2: Based on D obtained in Step 2.1 (H，Q) The corresponding characteristic functions are as follows:

[0027]

[0028] Where s is an interior pole. H and Q are the conjugate poles of s; H and Q are the conjugate poles of D. (H，Q) The feature matrix;

[0029] Step 2.3: Based on the bilinear transformation method, transform the discrete state-space model of the thermocouple obtained in Step 1.4 into a normalized continuous state-space model of the thermocouple:

[0030]

[0031] Among them, A i B i C i D i These are the system matrix, control matrix, output matrix, and direct transfer matrix of the thermocouple normalized continuous state-space model, respectively.

[0032] Step 2.4: Design a state feedback control law for the normalized continuous state-space model of the thermocouple obtained in Step 2.3:

[0033] u = Kx + v

[0034] Where x is the state vector of the thermocouple, K is the state feedback vector, and v is the input of the thermocouple after state feedback.

[0035] Based on the generalized Lyapunov theorem, we obtain

[0036]

[0037] In this approach, we introduce variable substitution W = KP, where P is a symmetric positive definite matrix and K is the state feedback vector to be determined. Represents the Kronecker product;

[0038] The state feedback vector K is obtained by solving the above LMIs and variable substitution.

[0039] Step 2.5: Substitute the state feedback control law obtained in Step 2.4 into the thermocouple normalized continuous state space model obtained in Step 2.3 to obtain the reference model of the thermocouple normalized continuous state space model;

[0040]

[0041] Among them, A refi B refi C refi D refi These are the system matrix, control matrix, output matrix, and direct transfer matrix of the reference model for the normalized continuous state-space model of the thermocouple;

[0042] Step 3: Based on robust H ∞ Technical design of dynamic compensators for thermocouples

[0043] Step 3.1: Combine the reference model of the thermocouple normalized continuous state-space model obtained in Step 2.5 and the thermocouple normalized continuous state-space model obtained in Step 2.3 into an augmented model:

[0044]

[0045] in, C augi =[C i 0], D augi =D i L = [0 C refi ]

[0046] Step 3.2: Obtain the robustness H of the thermocouple ∞ The design conditions for the dynamic compensator are the following set of linear matrix inequalities (LMIs):

[0047] X > 0

[0048] RX > 0

[0049]

[0050] Among them, R, X, M, N, Z, Df Let γ be the decision variables, and I be the identity matrix; if and only if the decision variables R, X, M, N, Z, D f When γ exists, the LMIs given above have solutions;

[0051] Step 3.3: Solve the system of linear matrix inequalities (LMIs) given in Step 3.2 to obtain the state-space equations of the dynamic compensator:

[0052] A f =X -1 M, B f =X -1 Z, C f =N

[0053] Among them, (A) f B f C f D f ) for robust H ∞ The state-space equations of a dynamic compensator;

[0054] Step 3.4: Apply the robust H obtained in Step 3.3 ∞ The state-space equation of the dynamic compensator is based on the sampling time T. s Discretization.

[0055] The beneficial effects of this invention: The robust H-model based on the thermocouple Hammerstein-LPV designed in this invention... ∞ The design method of dynamic compensator for filters can effectively improve the modeling accuracy of thermocouples, enhance the robustness of dynamic compensator design, and effectively solve the problem of significant degradation of compensation capability under changing thermocouple operating conditions in traditional methods, which has important engineering significance. Attached Figure Description

[0056] Figure 1 This is a schematic diagram of the process of the present invention.

[0057] Figure 2 This is the Hammerstein-LPV model for the sensor.

[0058] Figure 3 This is a schematic diagram of the sensor dynamic compensation principle.

[0059] Figure 4(a) shows the fitted curve of ax(θ).

[0060] Figure 4(b) shows the fitted curve of bx(θ).

[0061] Figure 4(c) shows the fitted curve of cx(θ).

[0062] Figure 5 For the desired region D of the thermocouple poles(n，Q) .

[0063] Figure 6(a) shows a comparison curve of the thermocouple signal after compensation using the proposed method and the unprocessed thermocouple signal when a certain type of aero-engine is operating under full afterburner conditions and the thermocouple at a certain cross section is at a temperature of about 550°C.

[0064] Figure 6(b) shows a comparison curve of the thermocouple signal after compensation using the proposed method and the thermocouple signal after compensation using the zero-pole placement method when a certain type of aero-engine is operating under full afterburner conditions and the thermocouple at a certain cross section is at a temperature of about 550°C.

[0065] Figure 7(a) shows a comparison curve of the thermocouple signal after compensation using the proposed method and the unprocessed thermocouple signal when a certain type of aero-engine is operating in cruise mode and the thermocouple at a certain section is at a temperature of about 400°C.

[0066] Figure 7(b) shows a comparison curve of the thermocouple signal after compensation using the proposed method and the thermocouple signal after compensation using the zero-pole placement method when a certain type of aero-engine is operating in cruise mode and the thermocouple at a certain cross section is at an operating temperature of about 400°C. Detailed Implementation

[0067] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings and technical solutions.

[0068] The flowchart of this invention is shown below. Figure 1 As shown, the specific steps are as follows:

[0069] Step 1.1: Using T s With a sampling period of 0.01s, five operating temperatures of 200℃, 300℃, 400℃, 500℃, and 600℃ are selected within the full operating range of the thermocouple to be modeled. A thermocouple dynamic calibration instrument is used to generate the required standard temperature input u for the thermocouple. The thermocouple is then fed into the corresponding position on the thermocouple dynamic calibration instrument, which is considered as the thermocouple receiving a step excitation signal with an amplitude equal to the set temperature of the thermocouple dynamic calibration instrument. Simultaneously, the output voltage value y of the thermocouple is acquired. out ;

[0070] Step 1.2: Analyze the output voltage value y of the thermocouple acquired in Step 1.1. out The data is processed as follows: First, outliers are removed using the 3σ principle, and then mean filtering is performed to obtain the dynamic output voltage data of the thermocouple.

[0071] Step 1.3: Based on the temperature-voltage meter corresponding to the K-type thermocouple used, perform polynomial fitting on the temperature-voltage values ​​of the thermocouple to obtain y. out =f(u)=c2u 2+c1u+c0, where c2=-1.89×10 -6 c1 = 0.0434 and c0 = -0.395 are the fitting coefficients. This fitting polynomial is used as the static model in the Hammerstein-LPV model of the thermocouple.

[0072] Step 1.4: Based on the dynamic output voltage data of the thermocouple obtained in Step 1.2, normalize the data and then perform parameter identification. Set the order of the discrete state-space model to p = 4 to obtain the normalized discrete state-space model of the thermocouple:

[0073]

[0074] in,

[0075]

[0076] C di =cx(θ)*(0 0 0 … … 0 0 1)∈R 1×p ;

[0077] Where u(k) is the input of the thermocouple's state-space model, representing the temperature value of the thermocouple's operating condition; y(k) is the output of the thermocouple's state-space model, representing the thermocouple's output voltage value; x(k) is the state vector of the thermocouple's state-space model, representing the temperature value of each unit cell after the thermocouple is discretized; and θ is the normalization exponent of the thermocouple's operating temperature. u max u is the maximum measurement temperature of the thermocouple. min θ represents the minimum measurement temperature of the thermocouple; ax(θ), bx(θ), and cx(θ) represent the heat transfer coefficient of adjacent thermocouple units, the heat transfer coefficient between the hot junction of the thermocouple and the outside world, and the thermocouple output coefficient, respectively, which are affected by the scheduling parameter θ.

[0078] Step 1.5: Rewrite ax(θ), bx(θ), and cx(θ) in polynomial form as follows:

[0079] ax(θ) = a0 + a1θ + a2θ 2 +a3θ 3

[0080] bx(θ) = b0 + b1θ + b2θ 2 +b3θ 3

[0081] cx(θ) = c0 + c1θ + c2θ 2 +c3θ 3

[0082] Among them, a0, a1, a2, a3, b0, b1, b2, b3, c0, c1, c2, and c3 are the coefficients to be determined; the fitting curves of ax(θ), bx(θ), and cx(θ) are shown in Figure 4.

[0083] This yields the dynamic model in the Hammerstein-LPV model of the thermocouple. In the Hammerstein model, the static nonlinear model is placed first, and the dynamic linear model is placed later. Connecting the static model and the dynamic model obtained in step 1.3 in series yields the Hammerstein-LPV model of the thermocouple.

[0084] Step 2.1: Convert the corresponding index into the desired region D=D of the thermoelectric dipole. (H，Q) Let α = 5 and r = 250. Among them, D1={x+jy|x<-α}, D2={x+jy|x 2 +y 2 <r 2}, D3={x+jy||y|<-xtanθ}, D=D1∩D2∩D3, D=D (H，Q) like Figure 5 As shown.

[0085] Step 2.2: Based on D obtained in Step 2.1 (H，Q) The corresponding characteristic functions are as follows:

[0086]

[0087] Where s is an interior pole. H and Q are the conjugate poles of s; H and Q are the conjugate poles of D. (H，Q) The characteristic matrices are H = diag(H1, H2, H3) and Q = diag(Q1, Q2, Q3), respectively, where H1 = 2α = 10 and Q1 = 1. H3 = 0,

[0088] Step 2.3: Based on the bilinear transformation method, transform the discrete state-space model of the thermocouple obtained in Step 1.4 into a normalized continuous state-space model of the thermocouple:

[0089]

[0090] Among them, A i B i C i D i These are the system matrix, control matrix, output matrix, and direct transfer matrix of the thermocouple normalized continuous state-space model, respectively.

[0091] Step 2.4: Design a state feedback control law for the normalized continuous state-space model of the thermocouple obtained in Step 2.3:

[0092] u = Kx + v

[0093] Where x is the state vector of the thermocouple, K is the state feedback vector, and v is the input of the thermocouple after state feedback.

[0094] Based on the generalized Lyapunov theorem, we obtain

[0095]

[0096] In this approach, we introduce variable substitution W = KP, where P is a symmetric positive definite matrix and K is the state feedback vector to be determined. Represents the Kronecker product;

[0097] Solving the above optimization problem using the YALMIP optimization toolbox yields W and P, where the state feedback control law K = WP. -1 ;

[0098] Step 2.5: Substitute the state feedback control law obtained in Step 2.4 into the thermocouple normalized continuous state space model obtained in Step 2.3 to obtain the reference model of the thermocouple normalized continuous state space model;

[0099]

[0100] Where A refi B refi C refi D refi These are the system matrix, control matrix, output matrix, and direct transfer matrix of the above model, respectively.

[0101] Step 3.1: Combine the reference model of the thermocouple normalized continuous state-space model obtained in Step 2.5 and the thermocouple normalized continuous state-space model obtained in Step 2.3 into an augmented model:

[0102]

[0103] in C augi =[C i 0], D augi =D i L = [0 C refi ]

[0104] Step 3.2: Obtain the robustness H of the thermocouple ∞ The design conditions for the dynamic compensator are the following set of linear matrix inequalities (LMIs):

[0105] X > 0

[0106] RX > 0

[0107]

[0108] Among them, R, X, M, N, Z, D f Let γ be the decision variables, and I be the identity matrix; if and only if the decision variables R, X, M, N, Z, D f When γ exists, the LMIs given above have solutions;

[0109] Step 3.3: Use the YALMIP optimization toolbox to solve the linear matrix inequalities (LMIs) given in Step 3.2 to obtain the state-space equations of the dynamic compensator:

[0110] A f =X -1 M, B f =X -1 Z, C f =N

[0111] Among them, (A) f B f C f D f ) for robust H ∞ The state-space equations of a dynamic compensator;

[0112] Step 3.4: Apply the robust H obtained in Step 3.3 ∞ The state-space equation of the dynamic compensator is based on the sampling time T. s =0.01s discretization. Thermocouple compensation principle is as follows: Figure 3 As shown;

[0113] Figure 6(a) shows the simulation results of a certain type of engine operating under full afterburner conditions, where the thermocouple at a certain cross-section of the engine operates at approximately 550°C. The results are obtained after compensation using the proposed method and after calculation of the unprocessed thermocouple signal. Figure 6(b) shows the simulation results of the compensator designed using the zero-pole placement method and the compensator obtained using the proposed method, both operating under the above conditions. The simulation results demonstrate that the robust H-type thermocouple based on the Hammerstein-LPV model designed using the proposed method achieves good performance. ∞ The proposed method, using a dynamic compensator, reduces the thermocouple time constant by 74.99% and the rise time by 68.56% compared to the original signal. Compared to the zero-pole placement method, the proposed method reduces the thermocouple time constant by 20.34% and the rise time by 47.1%.

[0114] Figure 7(a) shows the simulation results of the thermocouple signal after compensation using the proposed method and the unprocessed thermocouple signal when a certain type of engine is operating in cruise mode, with the thermocouple at a certain cross-section of the engine operating at approximately 400°C. Figure 7(b) shows the simulation results of the thermocouple signals after compensation using the zero-pole placement method and the proposed method when the thermocouple is operating under the above conditions. Simulation results show that the robust H-type thermocouple based on the Hammerstein-LPV model designed using the method proposed in this invention is effective. ∞ The proposed dynamic compensator design method reduces the thermocouple time constant by 56.79% and rise time by 28.71% compared to the original signal. Compared to the zero-pole placement method, the proposed method reduces the thermocouple time constant by 20.73% and rise time by 9.52%.

[0115] Simulation results show that the robust H designed based on this method... ∞ The response speed of the thermocouple after dynamic compensator treatment is significantly better than that of the thermocouple after compensator treatment designed by zero-pole placement method and the response speed of the thermocouple without compensator direct output, and it is also more robust to changes in thermocouple model.

Claims

1. A design method for a robust dynamic compensator for engine sensors based on the Hammerstein-LPV model, characterized in that, The steps are as follows: Step 1: Establish the Hammerstein-LPV model of the thermocouple Step 1.1: with The sampling period is selected within the entire operating range of the thermocouple to be modeled. The required operating point temperature is determined using a thermocouple dynamic calibration instrument to generate the standard temperature input for the desired thermocouple. When the thermocouple is inserted into the corresponding position of the thermocouple dynamic calibration instrument, it is considered that the thermocouple receives a step excitation signal with an amplitude equal to the set temperature of the thermocouple dynamic calibration instrument, and the output voltage value of the thermocouple is collected simultaneously. ; Step 1.2: Analyze the output voltage value of the thermocouple acquired in Step 1.

1. Processing is carried out; firstly, 3... The principle is to remove outliers from the data, and then perform mean filtering to obtain the dynamic output voltage data of the thermocouple. Step 1.3: Obtain the corresponding temperature-voltage meter based on the thermocouple's calibration number, and perform polynomial fitting on the thermocouple's temperature and voltage values ​​to obtain... ,in , , The fitting coefficients are used as the static model in the Hammerstein-LPV model of the thermocouple. Step 1.4: Based on the thermocouple's geometry, the heat transfer effect inside the thermocouple is equivalent to a one-dimensional heat conduction problem. According to the heat conduction differential equation in heat transfer theory, the thermocouple is divided into several unit cells, resulting in a discrete state-space model of the thermocouple. The order of the discrete state-space model is the number of unit cells. ; Based on the normalized dynamic output voltage data of the thermocouple obtained in step 1.2, parameter identification is performed to obtain the normalized discrete state-space model of the thermocouple: ， in, ; ； ； in, Input to the state-space model of the thermocouple, representing the temperature value of the thermocouple's operating condition; This is the output of the thermocouple's state-space model, representing the thermocouple's output voltage value. is the state vector of the thermocouple's state-space model, representing the temperature value of each unit cell after the thermocouple is discretized; It is the normalized exponent of the operating temperature of the thermocouple. ; This is the maximum measurement temperature of the thermocouple. This is the minimum measurement temperature of the thermocouple; , , These represent the heat transfer coefficients of adjacent thermocouple units, the heat transfer coefficient between the thermocouple hot junction and the external environment, and the thermocouple output coefficient, respectively, subject to scheduling parameters. The impact; Step 1.5: [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require the full context.] , , In polynomial form, it is as follows: in, , , , , , , , , , , , These are the coefficients to be determined; This yields the dynamic model in the Hammerstein-LPV model of the thermocouple. In the Hammerstein model, the static nonlinear model is placed first, and the dynamic linear model is placed later. Connecting the static model and the dynamic model obtained in step 1.3 in series yields the Hammerstein-LPV model of the thermocouple. Step 2: Using the regional pole placement technique, the state feedback vector is obtained by solving a set of linear matrix inequalities (LMIs). Thus, a reference model for the normalized continuous state-space model of the thermocouple is obtained; Step 2.1: Set the performance evaluation index of the thermocouple, and obtain the desired region of the thermocouple poles based on the correspondence between poles and dynamic performance. ; Step 2.2: Based on the results obtained in Step 2.1 The corresponding characteristic functions are as follows: in, For internal poles, for The conjugate poles; , for The feature matrix; Step 2.3: Based on the bilinear transformation method, transform the discrete state-space model of the thermocouple obtained in Step 1.4 into a normalized continuous state-space model of the thermocouple: ， in, , , , These are the system matrix, control matrix, output matrix, and direct transfer matrix of the thermocouple normalized continuous state-space model, respectively. Step 2.4: Design a state feedback control law for the normalized continuous state-space model of the thermocouple obtained in Step 2.3: in, Let be the state vector of the thermocouple. For the state feedback vector, The thermocouple receives state feedback input; Based on the generalized Lyapunov theorem, we obtain ， Among them, variable substitution is introduced. , It is a symmetric positive definite matrix. Let be the state feedback vector to be determined. Represents the Kronecker product; The state feedback vector is obtained by solving the above LMIs and by variable substitution. ; Step 2.5: Substitute the state feedback control law obtained in Step 2.4 into the thermocouple normalized continuous state space model obtained in Step 2.3 to obtain the reference model of the thermocouple normalized continuous state space model; ， in, , , , These are the system matrix, control matrix, output matrix, and direct transfer matrix of the reference model for the normalized continuous state-space model of the thermocouple; Step 3: Based on robustness Technical design of dynamic compensators for thermocouples Step 3.1: Combine the reference model of the thermocouple normalized continuous state-space model obtained in Step 2.5 and the thermocouple normalized continuous state-space model obtained in Step 2.3 into an augmented model: ， in, , , , , Step 3.2: Obtain the robustness of the thermocouple The design conditions for the dynamic compensator are the following set of linear matrix inequalities (LMIs): in, , , , , , , As decision variables, It is an identity matrix; if and only if the decision variables , , , , , , If they exist, the LMIs given above have solutions; Step 3.3: Solve the system of linear matrix inequalities (LMIs) given in Step 3.2 to obtain the state-space equations of the dynamic compensator: in, For robust The state-space equations of a dynamic compensator; Step 3.4: Apply the robustness obtained in Step 3.3 The state-space equation of the dynamic compensator according to the sampling time Discretization.

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