A Method for Optimizing Controller Parameters of DC Distribution System Considering Measurement Delay

Abstract

The present invention discloses a method for optimizing the parameters of a DC distribution system controller considering measurement delay, which relates to the fields of DC distribution systems and automatic control. First, a converter device and a controller model are established, which is the basis for constructing the control system model; secondly, according to the characteristics of the system response in different stages of the delay process, a control model of the DC / DC converter under delay is established; then, by analyzing the delay controller model, the consequences of the delay are analyzed, and the root cause of the consequences is determined. The parameter of the voltage outer loop controller G1 is unreasonable. Finally, an optimization method for the parameters of the G1 controller is proposed, and the correctness of the proposed method is verified through numerical example analysis. The present invention studies the control modes of the DC / DC converter in different stages during the delay process, determines the influence of the delay on the system through analysis, confirms the root cause of the influence based on mechanism analysis, and eliminates the influence of the delay on the operation of the DC / DC converter according to the proposed parameter optimization algorithm.

Description

Technical Field

[0001] The present invention relates to the fields of DC power distribution systems and automatic control, and particularly to a method for optimizing the parameters of a controller of a DC power distribution system considering measurement delay. Background Art

[0002] With the rapid development and large-scale application of DC power distribution systems, the delay problem has received wide attention. In a DC power distribution system, the voltage monitoring link is generally far from the control system of the converter and is vulnerable to sudden events, such as electromagnetic interference, insufficient communication bandwidth, performance limitations of signal transceivers, etc. The random delay problem is the most prominent. Relevant research points out that the safe and reliable operation of the power system increasingly depends on the information system, and communication congestion and information delay will have an adverse impact on the power system. Delays are divided into two categories: fixed delay and random delay. Fixed delay means that the delay time is fixed. For fixed delay, the suppression of delay can be achieved by using the lead-lag link of the control system to compensate for the phase angle difference. However, this method cannot be used for DC power distribution systems because the delay in DC power distribution systems is random delay. Random delay means that the delay time of the delay signal varies randomly within a certain interval. The research on random delay mainly includes two ideas.

[0003] The first idea: Use the predicted signal to update the feedback signal, and the predicted signal represents the latest state of the system. If the accuracy of the prediction algorithm is high, the predicted signal can be used to compensate for the missing signal during the delay period. Therefore, it can be approximately considered that the delay signal does not enter the control system and will not have a negative impact on the system. Commonly used prediction methods mainly include the duty cycle prediction method, the Smith prediction method, etc. The duty cycle predictor makes the predicted duty cycle closer to the true duty cycle by compensating the duty cycle signal under multiple delays, indirectly preventing the delay signal from entering the control link. Or a delay prediction controller is connected in series in the feedback loop, that is, the input signal of the prediction controller is the delay signal, but the output signal is the predicted value of the current system state, as much as possible to avoid the delay signal from entering the controller. In addition, the prediction controller and the control link are connected in anti-parallel to form a Smith predictor, which also realizes the function of supplementing the delay signal.

[0004] The second approach: Since the delay signal is random, it is inevitable to input the delay signal into the control system, which affects the operation quality of the DC power distribution system. Therefore, when the delay signal acts on the DC power grid, it is necessary to adjust the controller parameters of the system to avoid the adverse effects caused by the delay as much as possible. The most important thing is to avoid the instability of the system in the delay state. For example, a PI optimal controller is constructed based on the scattering transform, which adjusts the feedback signal of the converter with delay by using the feedback information of the converter without delay in the same area to eliminate the influence of the delay on the system instability. Or a secondary voltage regulator is designed based on PI control, which corrects the feedback voltage signal of the converter with delay according to the average voltage and current signals of the DC microgrid to improve the stability of the system under delay. Summary of the Invention

[0005] The present invention provides an optimization method for the controller parameters of a DC power distribution system considering measurement delay to solve the problem that it is difficult to eliminate the influence of the random delay in the DC power distribution system.

[0006] A typical topology of a DC power distribution system is as Figure 1 shown. The voltage on the power supply side is converted into the voltage on the load side through a DC / DC converter. Among them, during the process of transmitting the measured value of the bus voltage on the load side to the DC / DC control terminal, due to the influence of electromagnetic interference, switch performance, and communication channels, a random delay with a time scale of about 10 ms to 100 ms is generated. Due to the existence of the random delay, the output response process of the DC / DC has a "distorted" state. Therefore, it is necessary to study the control method of the converter in the delay state to ensure the safe and reliable operation of the DC power distribution system.

[0007] The present invention is realized through the following technical solutions: An optimization method for the controller parameters of a DC power distribution system considering measurement delay, which is implemented in a typical topology of a DC power distribution system. The DC / DC converter converts the voltage on the power supply side into the voltage on the load side, and includes the following steps:

[0008] Step S1: Establish a converter device and controller model, which is the basis for constructing the control system model;

[0009] Construct a DC / DC converter device model. The DC / DC converter includes two control processes. The first is the G3 link, which is the charging process of the inductor L by the bus on the power supply side. By adjusting the opening and closing times of switches S1 and S2, that is, the duty cycle D, the inductor current i input from the bus on the power supply side to the converter is controlled L ; the second is the G4 link, which is the charging / discharging process of the inductor L on the power supply side to the capacitor C on the load side, with the goal of maintaining the load side voltage v C constant, corresponding to the G4 link; the expressions of G3 and G4 are as follows:

[0010]

[0011] where Δi L (s) represents the inductor current, ΔD(s) represents the duty cycle, and Δv C (s) represents the capacitor voltage, V c represents the steady-state capacitor voltage, R represents the load, L and C represent the inductor and capacitor respectively, and D1 represents the steady-state duty cycle.

[0012] Step S2: Establish a control model of the DC / DC converter under delay according to the characteristics of the system response in different stages of the delay process;

[0013] Design a controller for the DC / DC converter, adopting double-loop control, including voltage outer loop and current inner loop control; among them, the mathematical model of the outer-loop PI controller represented by G1(s) is:

[0014]

[0015] The mathematical model of the outer-loop PI controller represented by G2(s) is:

[0016]

[0017] Using the root locus analysis method, analyze the zero-pole positions of the open-loop transfer function of the voltage-loop delay control, and then determine the change in stability. First, derive the open-loop transfer function of the voltage-loop delay control. The process is as follows. From the Pade equivalent transfer function of the delay, as shown below:

[0018]

[0019] where T + is the delay time, a n and b n are the relational expressions of the equivalent order m and the number of terms n, m = 0…3; n = 1…3; Further simplify formula (5), and the equivalent transfer function of its delay link is:

[0020]

[0021] where the real parts of P1, P2, and P3 are all less than zero, and the real parts of Z1, Z2, and Z3 are all greater than zero, and they are all related to the delay T + Related, further simplify to obtain the open-loop transfer function of the voltage-loop delay control stage:

[0022]

[0023] Multiply the links of G2(s), G3(s), G4(s), and G pade (s), and organize to obtain the following formula (8):

[0024]

[0025] As can be seen from Equation (8), the zeros of the numerator of the open-loop transfer function are [-sL / R + (1 - D1) 2 , and (s + P1)(s + P2)(s + P3). Since the real parts of P1, P2, and P3 are negative and (1 - D1) 2 is positive, it is inferred that the four zeros of the open-loop transfer function fall on the right side of the s-domain plane. Therefore, there must be four root loci crossing the imaginary axis. When the relevant parameters are set unreasonably, it will cause the roots of the converter delay control transfer function to fall on the right side of the s-domain. Therefore, the voltage loop delay control has the possibility of instability.

[0026] Step S3: By analyzing the delay controller model, analyze the consequences of the delay and determine the root cause of the consequences, that is, the parameters of the voltage outer-loop controller G1 are unreasonable;

[0027] When the delay occurs, the system is first in the voltage loop hold control stage. The voltage feedback signal received by the system is Δvc(s), rather than the real-time signal Δvc + (s); in the voltage loop hold control stage after the delay occurs, Δvc(s) remains unchanged; the open-loop controller G1 will adjust the output power of the converter according to Δvc(s); the running time of this stage is the same as the duration of the delay; if the duration of the delay is long, since this control is an open-loop control, it will lead to over-control, that is, the actual adjustment power of the converter has exceeded the adjustment power required for the voltage fluctuation on the load side; the delay signal Δvc + (s) × e -sT+ After entering the controller, the converter enters the voltage loop delay control operation stage; since the delayed feedback signal fails to feedback the latest voltage signal in time, but instead feedbacks the voltage signal before the delay occurs, it leads to the reverse control of the converter; reverse control means that the output power of the converter should have been reduced / increased, but in fact it has been increased / reduced; the reverse control ultimately leads to system instability; in summary, the over-control in the voltage loop hold control stage is the root cause of the instability of the voltage loop delay control; obviously, the root cause of the over-control is the relatively large parameters K vi and K vp of the G1 controller; the conclusion obtained is consistent with the conclusion of the root locus analysis.

[0028] Step S4: Propose an optimization method for the parameters of the G1 controller and verify the correctness of the proposed method through numerical example analysis:

[0029] Due to a contradiction in the optimization of the G1 parameter in a delay environment, to meet the stability of the delay process, the G1 parameter needs to be reduced, and to improve the dynamic response speed of the system, the G1 parameter needs to be increased. The G1 parameter is optimized using Lyapunov stability analysis and norm constraint methods to find the optimal value between the two contradictions;

[0030] 1) Establish a discrete-time domain state model of the delay system; to account for the variation relationships between various states as much as possible, a discrete-time domain state model is constructed according to the voltage loop delay control method; Δi R (s), Δi L (s), ΔD(s), Δv C+ (s) are the output signals of each link respectively, and they represent the state information of the control system and the converter equipment; therefore, these four parameters are used as the basic states of the converter discrete-time domain state model and are represented by x(k); to solve the converter discrete state model, the state equations of these four variables need to be determined; among them, the discrete-domain state equations of Δi L (s), Δv C+ (s), ΔD(k), Δi R (k) are as follows:

[0031] x(k + 1) = Ax(k) + A T+ x(k - T + ) + FΔU ω (k) (9)

[0032]

[0033] Among them, A, A T+ and F are relationship matrices, and the specific structures are as follows:

[0034]

[0035] Among them, the state vector x(k) = [Δi L (k) Δv C+ (k) ΔD(k) Δi R (k)] T , the quantity of the next state is: x(k + 1) = [Δi L (k + 1) Δv C+ (k + 1) ΔD(k + 1) Δi R (k + 1)] T , x(k - T + ) = [Δi L (k - T + ) Δv C+ (k - T + ) ΔD(k - T + ) Δi R (k - T+ )] T Delay system controller K T+ =[K vp K vi T ,v ω (k)=[0Δv ω (k)0 0] T is the voltage disturbance matrix;

[0036] 2) Optimization research on the parameters of controller G1: When there exists a coefficient λ greater than zero and very small (generally between 0 and 1), such that the voltage output signal Δvc + (k) and the voltage disturbance signal Δv w (k) satisfy equation (28), ||Δv c+ (k)|| 2 <λ 2 ||Δv ω (k)|| 2 holds, that is:

[0037]

[0038] It can be seen from inequality (17) that the sum of the squares of the output voltage change amounts for any state k is always less than the sum of the squares of the voltage disturbance change amounts under the constraint of λ 2 ; Therefore, when the relevant state change amounts of the converter control system satisfy the inequality, it implicitly indicates the following two aspects: 1. The converter control system is stable. 2. The overshoot of the converter control system is small and the adjustment time is short. Therefore, Theorem 1 ensures that the delay converter maximally retains the adjustment performance of the delay system under the condition of stable operation. Under the condition of stable operation of the system during the delay stage, the maximum value of the parameters of controller G1 is found; the optimal value of G1 parameters is evaluated according to inequality (17);

[0039] Subtract λ from both sides of the norm inequality ||Δv C+ (k)|| 2 <λ 2 ||Δv ω (k)|| 2 , that is: 2 ||Δv ω (k)|| 2 , that is:

[0040] J = ||Δv C+ (k)|| 2 -λ 2 ||Δv ω (k)|| 2 <0 (18)

[0041] ​From the mathematical properties of the norm, Equation (19) can be obtained as follows:

[0042]

[0043] Substituting Equation (19) into Equation (18), Equation (20) can be obtained as follows:

[0044]

[0045] To prove J < 0; the Lyapunov energy change function ΔE(k) is added to each term in the function J, where ΔE(k) = E(k + 1) - E(k). Then the equivalent form of the new function is as follows:

[0046]

[0047] Among them, ΔE(k) represents the Lyapunov energy change matrix composed of the system state x(k), and E(k) represents the Lyapunov energy matrix; it is expressed in the form of x T (k)Nx(k); N is a square matrix, and its order is the same as the number of system states;

[0048] 3) According to the theorem: as long as the energy that the system may possess has a lower bound, then it is always possible to make the possible energy greater than zero by adding a constant to the energy. At this time, the system energy has the property of positive definiteness. The physical meaning of positive definiteness is the property that the energy of the physical system is always positive. Since the converter system absorbs the energy of the power grid to supply the load, the converter always absorbs energy from the power grid. Therefore, the energy state matrix E(k) of the converter is positive definite;

[0049] According to the theorem: if the function E(k) is a positive definite function, then for any non-zero state quantity of n dimensions, there is E(k) > 0. When and only when the system is in the initial state, the energy function of the state quantity is zero, that is, E(1) = 0. Applying it to formula (21), from E(1) = 0 and E(∞ + 1) > 0, the following can be obtained:

[0050] J ≤ J1 (22)

[0051] Furthermore, the proof process of J < 0 is transformed into the proof of J1 < 0; by verifying that each term in J1 is less than zero, J1 < 0 is proved; and according to formula (22), J is less than zero is obtained; for each term in J1, that is, for a specific k, its expression L is as follows:

[0052] L = x T (k)C T Cx(k) - v ω T (k)λ 2 v ω (k) + ΔE(k) (23)

[0053] The first two terms in Equation (22) are regarded as the product among matrix transpose, constant term and matrix; for the convenience of operation, ΔE(k) in Equation (23) is changed to the same form as the first two terms; by using the Lyapunov stability analysis method, an inequality constraint model about ΔE(k) is constructed as follows:

[0054] The Lyapunov energy function is a potential energy function related to the system state x(k); in order to accurately describe the energy of the time-delay system, it is studied from the following three aspects: 1. The energy of the current system state x(k), also known as the point energy function; 2. The energy of the system under any time delay and maximum time delay states, also known as the line energy function; 3. The energy difference between the next state and the current state of the system, also known as the surface energy function; the Lyapunov energy function is constructed according to the three energy function aspects of point, line and surface;

[0055] ① Considering the energy of the current system state, construct the point energy function, where x(k) represents the system state:

[0056] E1(k) = x T (k)P T+ x(k) (24)

[0057] where P T+ is the relevant matrix of the point energy function;

[0058] ② Considering the time-delay state, construct the line energy function: T + represents the random time delay

[0059]

[0060] where the matrix Q T+ is the relevant matrix of the line energy function;

[0061] ③ Considering the maximum time-delay state, construct the line energy function: T +m represents the maximum time delay

[0062]

[0063] where the matrix Q T+m is the relevant matrix of the line energy function;

[0064] ④ Considering the energy difference between the future state and the current state, construct the surface energy function:

[0065]

[0066] where S T+ is the relevant matrix of the surface energy function;

[0067] Finally, the total energy function of the delay system is obtained as: E(k) = E1(k) + E2(k) + E3(k) + E4(k);

[0068] Next, the energy change matrices of each point, line, and surface function are solved using the difference method. The difference method is ΔE(k) = E(k + 1) - E(k); the energy change matrices of the point, line, and surface functions are shown in the following equations (28), (29), and (30):

[0069] ① The change in the point energy function ΔE1(k):

[0070]

[0071] ② The change in the line energy function ΔE2(k):

[0072]

[0073] ③ The change in the surface energy function ΔE3(k):

[0074]

[0075] Where:

[0076]

[0077] After organizing the above three energy change matrices ΔE1, ΔE2, and ΔE3, an inequality of a quadratic function about ΔE is obtained, as shown in the following equation (32):

[0078]

[0079] Where e1, e2, e3, e4, e5 are the correlation matrices of the matrix and the equivalent state matrix, as shown in equation (33);

[0080]

[0081] Further organizing formula (32), converting multiple quadratic functions into an equivalent quadratic function, as shown in the following equation (34):

[0082] ΔE(k) ≤ ξ T (k)Π T+ ξ(k) (34)

[0083] Where, Π T+ The specific structure of is as follows:

[0084]

[0085] Therefore, the following equation holds:

[0086] ΔE(k) ≤ ξ T(k)Π T+ ξ(k) (38)

[0087] where ξ(k) is the equivalent state quantity, and Π T+ is the relationship matrix of the equivalent state quantity; Equation (23) is rewritten as:

[0088] L ≤ x T (k)C T Cx(k) - v ω T (k)λ 2 v ω (k) + ξ T (k)Π T+ ξ(k) (39)

[0089] To prove L < 0, it is necessary to prove:

[0090] x T (k)C T Cx(k) - v ω T (k)λ 2 v ω (k) + ξ T (k)Π T+ ξ(k) < 0 (40)

[0091] Since Equation (41) is the sum of three quadratic functions, for convenient research, the quadratic function obtained by arranging Equation (41) is as follows:

[0092]

[0093] where Π 2T+ is the relationship matrix;

[0094] Let the quadratic function f(ξ(k)) = ξ T (k)Π 2T+ ξ(k). If f(ξ(k)) > 0 for all system states ξ(k) ≠ 0, then fξ(k) is called a positive definite quadratic form, and Π 2T+ is called a positive definite matrix, denoted as Π 2T+ > 0; if f(ξ(k)) < 0 for all system states ξ(k) ≠ 0, then fξ(k) is called a negative definite quadratic form, and Π 2T+ is called a negative definite matrix, denoted as Π 2T+ < 0; find the negative definite matrix Π 2T+ , such that the following equation holds:

[0095] f(ξ(k)) = ξ T (k)Π 2T+ ξ(k) < 0 (42)

[0096] The key to proving f(ξ(k)) < 0 is to find a negative definite matrix Π that satisfies formula (42). 2T+ ; Find Π through computer programming 2T+ , if Π that makes f(ξ(k)) less than zero cannot be found under the current system parameters 2T+ , then adjust the controller parameters until a negative definite matrix Π exists 2T+ ; The obtained Π 2T+ The derivation process is as follows;

[0097] After equivalently transforming formula (42) using matrix transformation properties, multiply the matrix Π T+ term before and after by and further organize to obtain the following formula (43):

[0098] H = ξ T (k)Π 2T+ ξ(k) (43)

[0099]

[0100] Where:

[0101]

[0102]

[0103] Among them, let A in formula (47) T+ = B×K T+ ×C = B×Z T+ , so there is:

[0104]

[0105] When a negative definite matrix is found, it can be proved that the quadratic function f(ξ(k)) < 0 holds; that is, each term in J1 forms a quadratic function value less than zero for a specific k, and combined with formula (39), the following formula holds;

[0106]

[0107] According to the negative definite matrix Π 2T+ it is obtained that J < 0, indicating that the controller G1 not only ensures the stability of the converter control system but also can maximize the retention of dynamic performance such as the overshoot and regulation speed of the system; therefore, it is necessary to solve the optimal parameters of the controller G1 according to Π 2T+ ;

[0108] Extract the information about the parameters of G1 from the negative definite matrix Π 2T+ as follows:

[0109]

[0110] Among them, is a positive definite matrix related to the time delay, is the matrix K T+ 's associated matrix, as shown in the following formula:

[0111]

[0112] Among them, K T+ =[K vp K vi ;

[0113] According to Equation (51), the optimal parameters of G1 obtained by solving are as follows:

[0114]

[0115] Compared with the prior art, the present invention has the following beneficial effects: The method for optimizing the parameters of a DC distribution system controller considering measurement delay provided by the present invention, in the context of a DC distribution network, considers the voltage quality of the DC distribution system under delay, and from both application and theoretical aspects, studies the controller design method for eliminating the influence of delay, as follows; In terms of application, for the optimization of controller parameters considering only system stability, the present invention maximizes the system response ability on the premise of ensuring system stability. In other words, a balance point is found between stability and system response ability; In terms of theory, a parameter optimization method based on norm theory is studied. This method establishes a control system model based on norm inequalities, and scales the norm inequalities through the constructed energy inequalities, determines the state energy matrix that makes the control system globally optimal, and calculates the optimal parameters of the controller. Description of the Drawings

[0116] Figure 1 is a schematic diagram of the overall structure of the DC distribution system.

[0117] Figure 2 is a schematic diagram of the operation of the converter for power supply from the power side bus to the load side.

[0118] Figure 3 is a schematic diagram of the influence of voltage disturbance signals and control signals on the output voltage.

[0119] Figure 4 is a voltage holding control block diagram.

[0120] Figure 5 is a voltage delay control block diagram.

[0121] Figure 6 is a schematic diagram of the operating state of the DC / DC converter under delay.

[0122] Figure 7 Schematic diagram of the response ability of the DC / DC converter system under delay

[0123] Figure 8 Schematic diagram of the operating state of the DC / DC converter system under delay

[0124] Figure 9 Schematic diagram of the response of the DC / DC converter system after optimizing the G1 parameter

[0125] Figure 10 Schematic diagram of the critical time after optimizing the G1 parameter Specific implementation mode

[0126] To make the technical problems, technical solutions and advantages to be solved by the present invention clearer, the following will be described in detail with reference to the accompanying drawings and specific embodiments

[0127] A method for optimizing the parameters of a controller of a DC distribution system considering measurement delay, the optimization method is implemented in a typical DC distribution system topology, and the DC / DC converter converts the voltage on the power supply side into the voltage on the load side, including the following steps

[0128] Step S1: Establish a converter device and controller model, which is the basis for constructing the control system model

[0129] Construct a DC / DC converter device model, and the circuit diagram of the DC / DC converter is as Figure 2 shown. The DC / DC converter includes two control processes. The first is the G3 link, which is the charging process of the inductor L by the bus on the power supply side. By adjusting the opening and closing times of switches S1 and S2, that is, the duty cycle D, the inductor current i input from the bus on the power supply side to the converter is controlled L ; The second is the G4 link, which is the charging / discharging process of the inductor L on the power supply side to the capacitor C on the load side, with the goal of maintaining the voltage v C on the load side constant, corresponding to the G4 link; The expressions of G3 and G4 are as follows

[0130]

[0131] In the formula, Δi L (s) represents the inductor current, ΔD(s) represents the duty cycle, Δv C (s) represents the capacitor voltage, V c represents the steady-state capacitor voltage, R represents the load, L and C respectively represent the inductor and capacitor, and D1 represents the steady-state duty cycle

[0132] Step S2: Establish a control model of the DC / DC converter under delay according to the characteristics of the system response in different stages of the delay process

[0133] Design the controller of the DC / DC converter, adopting double-loop control, including voltage outer loop and current inner loop control; among them, the mathematical model of the outer-loop PI controller represented by G1(s) is:

[0134]

[0135] The mathematical model of the outer-loop PI controller represented by G2(s) is:

[0136]

[0137] Using the root locus analysis method, analyze the zero-pole positions of the open-loop transfer function of the voltage loop delay control, and then determine the change of stability. First, derive the open-loop transfer function of the voltage loop delay control. The process is as follows. From the Pade equivalent transfer function of the delay, as shown below:

[0138]

[0139] Among them, T + is the delay time, a n and b n are the relational expressions of the equivalent order m and the number of terms n, m = 0…3; n = 1…3; Further simplify formula (5), and the equivalent transfer function of its delay link is:

[0140]

[0141] Among them, the real parts of P1, P2, and P3 are all less than zero, and the real parts of Z1, Z2, and Z3 are all greater than zero, all related to the delay T + . Further simplify to obtain the open-loop transfer function in the voltage loop delay control stage:

[0142]

[0143] Multiply the links of G2(s), G3(s), G4(s), and G pade (s), and organize to obtain the following formula (8):

[0144]

[0145] It can be seen from formula (8) that the zeros of the numerator of the open-loop transfer function are [-sL / R+(1-D1) 2 and (s+P1)(s+P2)(s+P3). Since the real parts of P1, P2, and P3 are negative and (1-D1) 2 is positive, it is inferred that the four zeros of the open-loop transfer function fall on the right side of the s-domain plane; Therefore, there must be four root loci crossing the imaginary axis. When the relevant parameters are set unreasonably, it will make the roots of the delay control transfer function of the converter fall on the right side of the s-domain. Therefore, the voltage loop delay control has the possibility of instability.

[0146] Step S3: Analyze the consequences of the delay by analyzing the delay controller model, and determine the root cause of the consequences, that is, the parameters of the voltage outer loop controller G1 are unreasonable;

[0147] When the delay occurs, the system is first in the voltage loop hold control stage. The voltage feedback signal received by the system is Δvc(s), rather than the real-time signal Δvc + (s); in the voltage loop hold control stage after the delay occurs, Δvc(s) remains unchanged; the open-loop controller G1 will adjust the output power of the converter according to Δvc(s); the running time of this stage is the same as the duration of the delay; if the duration of the delay is long, since this control is open-loop control, it will lead to over-control, that is, the actual adjustment power of the converter has exceeded the adjustment power required by the voltage fluctuation on the load side; the delay signal Δvc + (s)×e -sT+ After entering the controller, the converter enters the voltage loop delay control operation stage; since the delayed feedback signal fails to feedback the latest voltage signal in time, but instead feedbacks the voltage signal before the delay occurs, it leads to the reverse control of the converter; reverse control means that the output power of the converter should be reduced / increased, but in fact it is increased / reduced; the reverse control ultimately leads to system instability; in summary, the over-control in the voltage loop hold control stage is the root cause of the instability of the voltage loop delay control; obviously, the root cause of the over-control is the relatively large parameters K vi and K vp of the G1 controller; the conclusion obtained is consistent with the conclusion of the root locus analysis.

[0148] Step S4: Propose an optimization method for the parameters of the G1 controller, and verify the correctness of the proposed method through numerical example analysis:

[0149] Since there is a contradiction point in the optimization of the G1 parameters in the delay environment, in order to meet the stability of the delay process, it is necessary to reduce the G1 parameters, and in order to improve the dynamic response speed of the system, it is necessary to increase the G1 parameters. Use the lyapunov stability analysis and norm constraint method to optimize the G1 parameters and find the optimal value between the two contradiction points;

[0150] 1) Establish a discrete-time domain state model of the delay system; in order to take into account the change relationship between various states as much as possible, according to the voltage loop delay control method, construct a discrete-time domain state model; Δi R (s), Δi L (s), ΔD(s), Δv C+(s) are the output signals of each link, which represent the state information of the control system and the converter equipment; therefore, these four parameters are used as the basic states of the converter discrete-time domain state model and are represented by x(k); to solve the converter discrete state model, the state equations of these four variables need to be determined; among them, Δi L (s), Δv C+ (s), ΔD(k), Δi R (k) have the following discrete-domain state equations:

[0151] x(k + 1) = Ax(k) + A T+ x(k - T + ) + FΔU ω (k) (9)

[0152]

[0153] Among them, A, A T+ , F are relationship matrices, and the specific structures are as follows:

[0154]

[0155] Among them, the state vector x(k) = [Δi L (k) Δv C+ (k) ΔD(k) Δi R (k)] T , the quantity of the next state is: x(k + 1) = [Δi L (k + 1) Δv C+ (k + 1) ΔD(k + 1) Δi R (k + 1)] T , x(k - T + ) = [Δi L (k - T + ) Δv C+ (k - T + ) ΔD(k - T + ) Δi R (k - T + )] T The delay system controller K T+ = [K vp K vi T , v ω (k) = [0 Δv ω (k) 0 0] T is the voltage disturbance matrix;

[0156] 2) Optimization research on the parameters of controller G1: When there exists a coefficient λ greater than zero and very small (generally between 0 and 1), such that the voltage output signal Δvc of the distribution system + ​(k) and voltage disturbance signal Δv w (k) When equation (28) is satisfied, ||Δv c+ (k)|| 2 <λ 2 ||Δv ω (k)|| 2 Established, that is:

[0157]

[0158] From inequality (17), we can see that the sum of the squares of the output voltage changes for any state k is always less than λ 2 The sum of the squares of the voltage disturbance changes under the constraints; therefore, when the relevant state changes of the converter control system satisfy the inequality, it implicitly indicates the following two aspects: 1. The converter control system is stable. 2. The overshoot of the converter control system is small and the adjustment time is short. Therefore, Theorem 1 ensures that the delay converter retains the adjustment performance of the delay system to the maximum extent under the condition of stable operation. Under the condition of stable operation of the system in the delay stage, the maximum value of the controller G1 parameter is found; the optimal value of the G1 parameter is evaluated according to inequality (17);

[0159] Substituting the norm inequality ||Δv C+ (k)|| 2 <λ 2 ||Δv ω (k)|| 2 Subtract λ from both sides 2 ||Δv ω (k)|| 2 , that is:

[0160] J=||Δv C+ (k)|| 2 -λ 2 ||Δv ω (k)|| 2 <0 (18)

[0161] From the mathematical properties of the norm, we can know that formula (19):

[0162]

[0163] Substituting formula (19) into formula (18), we can get formula (20):

[0164]

[0165] To prove that J<0, add the Lyapunov energy change function ΔE(k) to each term in the function J, where ΔE(k) = E(k+1)-E(k), then the equivalent form of the new function is:

[0166]

[0167] Among them, ΔE(k) represents the Lyapunov energy change matrix composed of the system state x(k), and E(k) represents the Lyapunov energy matrix; it is expressed in the form of x T (k)Nx(k); N is a square matrix, and its order is the same as the number of system states;

[0168] 3) According to the theorem: as long as the energy that the system may possess has a lower bound, then it is always possible to make the possible energy greater than zero by adding a constant to the energy. At this time, the system energy has the property of positive definiteness. The physical meaning of positive definiteness is the property that the energy of the physical system is always positive. Since the converter system absorbs the grid energy to supply the load, the converter always absorbs energy from the grid. Therefore, the energy state matrix E(k) of the converter is positive definite;

[0169] According to the theorem: if the function E(k) is a positive definite function, then for any non-zero state quantity of n dimensions, there is E(k)>0. When and only when the system is in the initial state, the energy function of the state quantity is zero, that is, E(1) = 0. Substituting it into formula (21), from E(1) = 0 and E(∞+1)>0, we can get:

[0170] J≤J1 (22)

[0171] Furthermore, the proof process of J<0 is transformed into the proof of J1<0; by verifying that each term in J1 is less than zero, it is proved that J1<0; and according to formula (22), it is concluded that J is less than zero; for each term in J1, that is, for a specific k, its expression L is as follows:

[0172] L = x T (k)C T Cx(k)-v ω T (k)λ 2 v ω (k)+ΔE(k) (23)

[0173] The first two terms in formula (22) are regarded as the product among the matrix transpose, the constant term and the matrix; for the convenience of operation, ΔE(k) in formula (23) is changed to the same form as the first two terms; using the Lyapunov stability analysis method, an inequality constraint model about ΔE(k) is constructed, and the process is as follows:

[0174] The Lyapunov energy function is a potential energy function related to the system state x(k); to accurately describe the energy of the time-delay system, it is studied from the following three aspects: 1. The energy of the current system state x(k), also known as the point energy function; 2. The energy of the system under any time delay and the maximum time delay state, also known as the line energy function; 3. The energy difference between the next state and the current state of the system, also known as the surface energy function; the Lyapunov energy function is constructed according to the three energy function aspects of point, line, and surface.

[0175] ① Considering the energy of the current system state, construct the point energy function, where x(k) represents the system state:

[0176] E1(k) = x T (k)P T+ x(k) (24)

[0177] where P T+ is the correlation matrix of the point energy function;

[0178] ② Considering the time-delay state, construct the line energy function: T + represents the random time delay

[0179]

[0180] where the matrix Q T+ is the correlation matrix of the line energy function;

[0181] ③ Considering the maximum time-delay state, construct the line energy function: T +m represents the maximum time delay

[0182]

[0183] where the matrix Q T+m is the correlation matrix of the line energy function;

[0184] ④ Considering the energy difference between the future state and the current state, construct the surface energy function:

[0185]

[0186] where S T+ is the correlation matrix of the surface energy function;

[0187] Finally, the total energy function of the time-delay system is obtained as: E(k) = E1(k) + E2(k) + E3(k) + E4(k);

[0188] Next, use the difference method to solve the energy change matrices of each point, line, and surface function. The difference method is ΔE(k) = E(k + 1) - E(k); the energy change matrices of the point, line, and surface functions are shown in the following equations (28), (29), and (30):

[0189] ① The change in the point energy function ΔE1(k):

[0190]

[0191] ② The change in the line energy function ΔE2(k):

[0192]

[0193] ③ The change in the surface energy function ΔE3(k):

[0194]

[0195] Where:

[0196]

[0197] By organizing the above three energy change matrices ΔE1, ΔE2, and ΔE3, an inequality of a quadratic form function regarding ΔE is obtained, as shown in the following formula (32):

[0198]

[0199] Where e1, e2, e3, e4, e5 are the correlation matrices of the matrix and the equivalent state matrix, as shown in formula (33):

[0200]

[0201] By further organizing formula (32), multiple quadratic form functions are converted into an equivalent quadratic form function, as shown in the following formula (34):

[0202] ΔE(k) ≤ ξ T (k)Π T+ ξ(k) (34)

[0203] Where, Π T+ The specific structure of is as follows:

[0204]

[0205]

[0206] Therefore, the following formula holds:

[0207] ΔE(k) ≤ ξ T (k)Π T+ ξ(k) (38)

[0208] In the formula, ξ(k) is the equivalent state quantity, and Π T+ is the relationship matrix of the equivalent state quantity; formula (23) is rewritten as:

[0209] L ≤ x T (k)C T Cx(k) - v ω T (k)λ 2 v ω (k) + ξ T (k)Π T+ ξ(k) (39)

[0210] To prove L < 0, it is necessary to prove that:

[0211] x T (k)C T Cx(k) - v ω T (k)λ 2 v ω (k) + ξ T (k)Π T+ ξ(k) < 0 (40)

[0212] Since equation (41) is the sum of three quadratic functions, for the convenience of research, the quadratic function obtained by arranging equation (41) is as follows:

[0213]

[0214] where Π 2T+ is the relationship matrix;

[0215] Let the quadratic function f(ξ(k)) = ξ T (k)Π 2T+ ξ(k). If f(ξ(k)) > 0 for all system states ξ(k) ≠ 0, then fξ(k) is called a positive definite quadratic form, and Π 2T+ is called a positive definite matrix, denoted as Π 2T+ > 0; if f(ξ(k)) < 0 for all system states ξ(k) ≠ 0, then fξ(k) is called a negative definite quadratic form, and Π 2T+ is called a negative definite matrix, denoted as Π 2T+ < 0; find the negative definite matrix Π 2T+ such that the following equation holds:

[0216] f(ξ(k)) = ξ T (k)Π 2T+ ξ(k) < 0 (42)

[0217] The key to proving f(ξ(k)) < 0 is to find the negative definite matrix Π 2T+ that satisfies formula (42); find Π 2T+ through computer programming. If Π 2T+ that makes f(ξ(k)) less than zero cannot be found under the current system parameters, the controller parameters are adjusted until there exists a negative definite matrix Π 2T+ ; the obtained Π 2T+ The derivation process is as follows;

[0218] After equivalently transforming formula (42) using matrix transformation properties, multiply the matrix Π T+ term before and after by and further organize to obtain the following formula (43):

[0219] H = ξ T (k)Π 2T+ ξ(k) (43)

[0220]

[0221] where:

[0222]

[0223] where, let A T+ = B×K T+ ×C = B×Z T+ , so there is:

[0224]

[0225] When a negative definite matrix is found, it can be proved that the quadratic function f(ξ(k)) < 0 holds; that is, each term in J1 forms a quadratic function value less than zero with a specific k, and combined with formula (39), the following formula holds;

[0226]

[0227] According to the negative definite matrix Π 2T+ obtain J < 0, indicating that the controller G1 not only ensures the stability of the converter control system but also can maximize the retention of dynamic performance such as overshoot and regulation speed of the system; therefore, it is necessary to solve the optimal parameters of the controller G1 according to Π 2T+ ;

[0228] Extract information about the parameters of G1 from the negative definite matrix Π 2T+ as follows:

[0229]

[0230] where, is a positive definite matrix related to the delay, is the associated matrix of the matrix K T+ , as shown in the following formula:

[0231]

[0232] Among them, K T+ = [K vp K vi ;

[0233] According to Equation (51), the optimal parameters of G1 obtained by solving are as follows:

[0234]

[0235] Example 1: Analysis of the impact of delay on the operation of the converter

[0236] In order to verify the feasibility of a parameter optimization method for a DC distribution system controller considering measurement delay, a certain smart new energy Internet in Shanxi is taken as the research object. This smart energy Internet is responsible for providing 750V DC voltage for the industrial park. The park load includes domestic load and long-distance industrial load. There is a delay of 20ms to 30ms in this park. In addition, the DC / DC converter equipment operates reliably and can be approximately considered as an ideal converter. Generally, an ideal converter model is constructed with inductors and capacitors. Table 1 shows the parameters of the DC / DC converter equipment of the energy Internet, and the default control parameter settings of the double-loop controller are shown in Table 2:

[0237] Table 1 Default parameters of the DC / DC converter

[0238]

[0239] Table 2 Default control parameters of the double-loop control

[0240]

[0241] This part is based on Figure 5 the shown control block diagram to build a simulation model of the converter system. And analyze the impact on the stability and dynamic response ability of the converter system when the delay time lengths are 21ms, 22ms, 23ms, and 24ms respectively.

[0242] The stability simulation results are as Figure 6 shown. It can be seen from the figure that when the delay time is 21ms and 22ms, the delay does not affect the system stability. When the delay time is 23ms and 24ms, the voltage on the load side becomes unstable at 2.5s and 2s respectively. Therefore, the impact of delay on the operation of the converter cannot be ignored.

[0243] Since the evaluation of the response ability is only for a stable system, the system response abilities of 21ms and 22ms are studied in this example, and the analysis results are as Figure 7As shown. It can be seen from the figure that as the delay time increases from 21 ms to 22 ms, the overshoot of the load-side voltage increases from 45 V to 65 V; the stable regulation time of the system increases from 0.25 s to 0.3 s. In summary, the increase in the delay time will reduce the dynamic response ability of the system.

[0244] Embodiment 2: Optimization of G1 parameters aiming at reducing the influence of delay

[0245] According to the proposed G1 parameter optimization method, taking the 23 ms delay with the highest occurrence frequency in the park as the research object, the research on eliminating the influence of delay is carried out. The optimized parameters obtained by solving the G1 parameters using computer programming are shown in Table 3.

[0246] After the G1 parameters are optimized, although there is still a 23 ms delay in the system, but by Figure 8 It can be seen that the delay does not affect the stability of the load-side voltage. This is because the lower G1 parameters avoid the occurrence of "over-control". The optimized parameters not only solve the stability problem of the load-side voltage, but also improve the dynamic response ability of the converter system. By Figure 9 It can be seen that the optimized 23 ms delay system is not only superior to the 22 ms delay system in terms of dynamic response, but also the differences in overshoot and regulation time from the zero-delay system are only 15 V and 0.012 s. The dynamic response ability of the optimized 23 ms delay system is almost close to that of the zero-delay system.

[0247] Table 3 G1 control parameters at λ = 0.5

[0248]

[0249] To analyze the critical stability of the system corresponding to the optimized parameters, the delay time is gradually increased, and the stability analysis results are as Figure 10 shown. It can be seen from the figure that the critical delay time corresponding to the optimized parameters is 34 ms. When the delay time reaches 35 ms, the load-side voltage becomes unstable. This means that the tolerance of the system to the delay time after parameter optimization is increased from 23 ms to 35 ms. For the application scenario of the park, the optimized parameters can not only effectively improve the operation ability of the 23 ms delay system, but also effectively avoid the influence of the delay in the range of 20 ms to 30 ms on the stable operation of the park. Therefore, for this industrial park, select K vp = 0.02, K vi = 7.31 as the optimal operating parameters of the controller G1.

[0250] The present invention studies the control modes of the DC / DC converter in different stages during the delay process, determines the impact of the delay on the system through analysis, and further confirms the root cause of the impact based on mechanism analysis. According to the proposed parameter optimization algorithm, the impact of the delay on the operation of the DC / DC converter is eliminated. The conclusions are as follows:

[0251] 1) In terms of application, aiming at the uncertainty of random delays and the variability of delay time scales in DC distribution systems, it increases the difficulty of delay control in practical applications. The greatest feature of this method is that, without adding any new devices, by adjusting the parameters, it can improve the system's tolerance to delays and reduce the cost required for delay suppression.

[0252] 2) In terms of theory, the proposed control algorithm starts from the norm inequality, and determines the optimization algorithm for the controller parameters through the scaling of the norm inequality by the Lyapunov energy function matrix. The advantage of this method is that, on the premise of ensuring the stability of the system under the action of the delay, it maximally retains the dynamic response ability of the system.

[0253] The scope of protection required by the present invention is not limited to the above specific implementation manners. Moreover, for those skilled in the art, the present invention can have various deformations and modifications, and any modifications, improvements, and equivalent replacements made within the concept and principle of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for optimizing the parameters of a DC distribution system controller considering measurement delay, characterized in that: The optimization method is implemented in a typical DC distribution system topology. The DC / DC converter converts the voltage on the power supply side into the voltage on the load side, and includes the following steps: Step S1: Establish models of converter equipment and controllers, which are the basis for constructing the control system model; Step S2: Establish a control model of the DC / DC converter under delay according to the characteristics of the system response in different stages of the delay process; Step S3: Analyze the consequences of the delay by analyzing the delay controller model, and determine the root cause of the consequences, that is, the unreasonable parameters of the voltage outer loop controller G1; the specific process is as follows: Analysis: When the time delay occurs, the system is first in the voltage loop holding control stage. The voltage feedback signal received by the system is Δvc(s), rather than the real-time signal Δvc + (s); in the voltage loop holding control stage after the time delay occurs, Δvc(s) remains unchanged; the open-loop controller G1 will adjust the output power of the converter according to Δvc(s); the running time of this stage is the same as the duration of the time delay; if the duration of the time delay is long, since this control is open-loop control, it will cause over-control, that is, the actual adjustment power of the converter has exceeded the adjustment power required by the voltage fluctuation on the load side; the time delay signal Δvc + (s)×e -sT+ After entering the controller, the converter enters the voltage loop time delay control operation stage; because the time delay feedback signal fails to feedback the latest voltage signal in time, but feedbacks the voltage signal before the time delay occurs, resulting in the reverse control of the converter; reverse control means that the output power of the converter should be reduced / increased, but actually it is increased / reduced; the reverse control ultimately leads to system instability; in summary, the over-control in the voltage loop holding control stage is the fundamental reason for the instability of the voltage loop time delay control; obviously, the fundamental reason for the over-control is the relatively large parameters K vi and K vp ; the obtained conclusion is consistent with the conclusion of the root locus analysis; Step S4: Propose an optimization method for the parameters of the G1 controller, and verify the correctness of the proposed method through numerical example analysis.

2. The parameter optimization method for a DC power distribution system controller considering measurement delay according to claim 1, wherein: The specific process of Step S1 is as follows: Build a DC / DC converter device model. The DC / DC converter includes two control processes. The first is the G3 link, which is the charging process of the inductor L by the power-side bus. By adjusting the on and off times of switches S1 and S2, that is, the duty cycle D, the inductor current i input from the power-side bus to the converter is controlled. L The second is the G4 link, which is the charging / discharging process of the power-side inductor L to the load-side capacitor C to maintain the load-side voltage v C constant as the goal; The expressions of G3 and G4 are as follows: Where, Δi L (s) represents the inductor current, ΔD(s) represents the duty cycle, Δv C (s) represents the capacitor voltage, V c represents the steady-state capacitor voltage, R represents the load, L and C respectively represent the inductor and the capacitor, and D1 represents the steady-state duty cycle.

3. A method for optimizing the parameters of a DC power distribution system controller considering measurement delay according to claim 1, characterized in that: The specific process of Step S2 is as follows: Design the controller of the DC / DC converter, using double-loop control, including voltage outer loop and current inner loop control; among them, the mathematical model of the outer loop PI controller represented by G1(s) is: The mathematical model of the outer loop PI controller represented by G2(s) is: Use the root locus analysis method to analyze the zero-pole positions of the open-loop transfer function of the voltage loop delay control, and then determine the change of stability. First, deduce the open-loop transfer function of the voltage loop delay control. The process is as follows. From the Pade equivalent transfer function of the delay, as shown below: Among them, T + is the delay time, a n and b n are the relational expressions of the equivalent order m and the number of terms n, where m = 0...3; n = 1...3; Further simplifying formula (5), the equivalent transfer function of its delay link is: Among them, the real parts of P1, P2, and P3 are all less than zero, and the real parts of Z1, Z2, and Z3 are all greater than zero, all related to the delay T + Related, further simplified to obtain the open-loop transfer function in the voltage loop delay control stage: Multiply the links G2(s), G3(s), G4(s), and G pade (s), and after arrangement, the following equation (8) is obtained: As can be seen from Equation (8), the zeros of the numerator of the open-loop transfer function are [-sL / R+(1-D1) 2 , and (s+P1)(s+P2)(s+P3). Since the real parts of P1, P2, and P3 are negative and (1-D1) 2 is positive, it is deduced that the four zeros of the open-loop transfer function fall on the right side of the s-domain plane; therefore, there must be four root loci crossing the imaginary axis.

4. A method for optimizing the parameters of a DC power distribution system controller considering measurement delay according to claim 1, characterized in that: The specific process of Step S4 is as follows: Since there is a contradiction point in the optimization of G1 parameters under the delay environment, in order to meet the stability of the delay process, it is necessary to reduce the G1 parameters, and to improve the dynamic response speed of the system, it is necessary to increase the G1 parameters. Use the lyapunov stability analysis and norm constraint method to optimize the G1 parameters and find the optimal value between the two contradiction points; 1) Establish the discrete-time domain state model of the delay system; in order to take into account the variation relationships between various states as much as possible, a discrete-time domain state model is constructed according to the voltage loop delay control method; Δi R (s), Δi L (s), ΔD(s), Δv C+ (s) are the output signals of each link respectively, and they represent the state information of the control system and the converter equipment; therefore, these four parameters are used as the basic states of the converter discrete-time domain state model and are represented by x(k); to solve the converter discrete state model, the state equations of these four variables need to be determined; among them, Δi L (s), Δv C+ (s), ΔD(k), Δi R (k) have the following discrete-domain state equations: x(k + 1) = Ax(k) + A T+ x(k - T + ) + FΔU ω (k) (9) Among them, A, A T+ , and F are relationship matrices, and the specific structures are as follows: where the state vector x(k) = [Δi L (k) Δv C+ (k) ΔD(k) Δi R (k)] T , the quantity of the next state is: x(k + 1) = [Δi L (k + 1) Δv C+ (k + 1) ΔD(k + 1) Δi R (k + 1)] T , x(k - T + ) = [Δi L (k - T + ) Δv C+ (k - T + ) ΔD(k - T + ) Δi R (k - T + )] T The delay system controller K T+ = [K vp K vi T , v ω (k) = [0 Δv ω (k) 0 0] T is the voltage disturbance matrix;​ 2) Optimization research on the parameters of controller G1: When there exists a coefficient λ greater than zero and very small, where λ is between 0 and 1, such that the voltage output signal Δvc + (k) and the voltage disturbance signal Δv w (k) satisfy equation (28), ||Δv c+ (k)|| 2 <λ 2 ||Δv ω (k)|| 2 holds, that is: As can be seen from inequality (17), the sum of the squares of the output voltage variation for any state k is always less than the sum of the squares of the voltage disturbance variations under the constraint of λ; under the condition that the system operates stably during the delay stage, the maximum value of the controller G1 parameter is found; the optimal value of the G1 parameter is evaluated according to inequality (17); 2 Under the condition that the system operates stably during the delay stage, the maximum value of the controller G1 parameter is found; the optimal value of the G1 parameter is evaluated according to inequality (17); The norm inequality ||Δv C+ (k)|| 2 < λ 2 ||Δv ω (k)|| 2 Subtract λ from both sides, we get: 2 ||Δv ω (k)|| 2 , that is: J = ||Δv C+ (k)|| 2 -λ 2 ||Δv ω (k)|| 2 <0 (18) From the mathematical properties of the norm, it can be known that formula (19): Substitute formula (19) into formula (18), and formula (20) can be obtained: To prove J<0; add the lyapunov energy change function ΔE(k) to each term in the function J, where ΔE(k)=E(k + 1)-E(k), then the equivalent form of the new function is: Among them, ΔE(k) represents the Lyapunov energy change matrix composed of the system state x(k), and E(k) represents the Lyapunov energy matrix; it is expressed in the form of x T (k)Nx(k); N is a square matrix, and its order is the same as the number of system states; 3) The physical meaning of positive definiteness is the property that the energy of the physical system is always positive. Since the converter system absorbs the energy of the power grid to supply the load, the converter always absorbs energy from the power grid. Therefore, the energy state matrix E(k) of the converter is positive definite; If the function E(k) is a positive definite function, then for any non-zero state quantity of n dimensions, there is E(k)>0. When and only when the system is in the initial state, the energy function of the state quantity is zero, that is, E(1)=0. Substitute it into formula (21). From E(1)=0 and E(∞ + 1)>0, it can be obtained: J≤J1 (22) Furthermore, transform the proof process of J<0 into proving J1<0; prove J1<0 by verifying that each term in J1 is less than zero; and according to formula (22), it is concluded that J is less than zero; for each term in J1, that is, for a specific k, its expression L is as follows: L = x T (k)C T Cx(k) - v ω T (k)λ 2 v ω (k) + ΔE(k) (23) The first two terms in Equation (22) are regarded as the product among matrix transpose, constant term and matrix; for the convenience of operation, change ΔE(k) in Equation (23) to the same form as the first two terms; use the Lyapunov stability analysis method to construct an inequality constraint model for ΔE(k), and the process is as follows: The Lyapunov energy function is a potential energy function related to the system state x(k); to accurately describe the energy of the time-delay system, it is studied from the following three aspects:

1. The energy of the current state x(k) of the system, also known as the point energy function; 2. The energy of the system under any time delay and maximum time delay states, also known as the line energy function; 3. The energy difference between the next state and the current state of the system, also known as the surface energy function; construct the Lyapunov energy function according to the three energy function aspects of point, line and surface; ① Consider the energy of the current system state and construct the point energy function, where x(k) represents the system state: E1(k) = x T (k)P T+ x(k) (24) where, P T+ is the correlation matrix of the point energy function; ② Considering the delay state, construct the line energy function: T + Represents the random delay Among them, the matrix Q T+ is the relevant matrix of the line energy function; ③ Consider the maximum delay state and construct the line energy function: T +m Represents the maximum delay Among them, matrix Q T+m is the correlation matrix of the line energy function; ④ Consider the energy difference between the future state and the current state and construct the surface energy function: where S T+ is the correlation matrix of the surface energy function; Finally, the total energy function of the time-delay system is obtained as: E(k) = E1(k) + E2(k) + E3(k) + E4(k); Next, use the difference method to solve the energy change matrices of each point, line and surface functions. The difference method is ΔE(k) = E(k + 1) - E(k); the energy change matrices of the point, line and surface functions are shown in the following equations (28), (29) and (30): ① The change amount ΔE1(k) of the point energy function: ② The change amount ΔE2(k) of the line energy function: ③ The change amount ΔE3(k) of the surface energy function: Where: Sort out the three energy change matrices ΔE1, ΔE2, ΔE3 to obtain an inequality of the quadratic form function about ΔE, as shown in the following equation (32): Where e1, e2, e3, e4, e5 are the correlation matrices of the matrix and the equivalent state matrix, as shown in Equation (33); Further sort out Equation (32) to convert multiple quadratic form functions into an equivalent quadratic form function, as shown in the following equation (34): ΔE(k)≤ξ T (k)Π T+ ξ(k) (34) Among them, Π T+ has the following specific structure: Therefore, the following equation holds: ΔE(k) ≤ ξ T (k)Π T+ ξ(k) (38) where ξ(k) is the equivalent state quantity, and Π T+ is the relationship matrix of the equivalent state quantity; Equation (23) is rewritten as: L ≤ x T (k)C T Cx(k) - v ω T (k)λ 2 v ω (k) + ξ T (k)Π T+ ξ(k) (39) To prove that Equation L < 0, it is necessary to prove: x T (k)C T Cx(k)-v ω T (k)λ 2 v ω (k)+ξ T (k)Π T+ ξ(k) < 0 (40) Since Equation (41) is the sum of three quadratic form functions, for the convenience of research, the quadratic form function obtained by sorting out Equation (41) is as follows: Among them, Π 2T+ is the relational matrix; Let the quadratic function be \(f(\xi^{(k)})=\xi^{(k)}\Pi\xi^{(k)}\). If \(f(\xi^{(k)}) > 0\) for all \(\xi^{(k)}\neq0\) of the system state, then \(f(\xi^{(k)})\) is called a positive definite quadratic form, and \(\Pi\) is called a positive definite matrix, denoted as \(\Pi>0\). If \(f(\xi^{(k)}) < 0\) for all \(\xi^{(k)}\neq0\) of the system state, then \(f(\xi^{(k)})\) is called a negative definite quadratic form, and \(\Pi\) is called a negative definite matrix, denoted as \(\Pi<0\). Find a negative definite matrix \(\Pi\) such that the following equation holds: T (k)Π 2T+ ξ(k), if for the system state ξ(k) not equal to 0, f(ξ(k)) > 0, then fξ(k) is called a positive definite quadratic form, and Π 2T+ is called a positive definite matrix, denoted as Π 2T+ > 0; if for the system state ξ(k) not equal to 0, f(ξ(k)) < 0, then fξ(k) is called a negative definite quadratic form, and Π 2T+ is called a negative definite matrix, denoted as Π 2T+ < 0; find the negative definite matrix Π 2T+ , such that the following formula holds: f(ξ(k)) = ξ T (k)Π 2T+ ξ(k) < 0 (42) The key to proving f(ξ(k)) < 0 is to find a negative definite matrix Π that satisfies formula (42). 2T+ ; Π is found through computer programming 2T+ , if no Π that makes f(ξ(k)) less than zero is found under the current system parameters 2T+ , then the controller parameters are adjusted until a negative definite matrix Π exists 2T+ ; The derivation process of the obtained Π 2T+ is as follows; After equivalently transforming formula (42) using matrix transformation properties, multiply the matrix Π term in the formula before and after by T+ and further organize to obtain the following formula (43): T+ before and after multiplying and further organize to obtain the following formula (43): H = ξ T (k)Π 2T+ ξ(k) (43) Where: Among them, let A in Equation (47) T+ = B × K T+ × C = B × Z T+ , so we have: When a negative definite matrix is found, it can be proved that the quadratic form function f(ξ(k)) < 0 holds; that is, each term in J1 forms a quadratic form function value less than zero by a specific k, and combined with Equation (39), the following equation holds; According to the negative definite matrix Π 2T+ we get J < 0. According to Π 2T+ solve for the optimal parameters of the controller G1; Extract information about the G1 parameter from the negative definite matrix Π 2T+ as follows: Among them, is a positive definite matrix related to the time delay, is the matrix K T+ 's associated matrix, as shown in the following formula: Among them, K T+ = [K vp K vi ; According to Equation (51), the optimal parameters of G1 obtained by solution are as follows:

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