Method for calibrating time delay feasible region of parameters of active front wheel steering time delay feedback controller

Abstract

The application discloses a kind of active front wheel steering time delay feedback controller parameter time delay feasible region's calibration method, and include steps: 1) the vehicle system model of active front wheel steering feedback controller including time delay is established;2) by constructing Lyapunov-Krasovskii functional, time delay independent stability parameter region is solved;3) by Laplace transform solving system characteristic equation, obtain stability discriminant polynomial;4) it is solved to the minimum critical time delay that can be tolerated under time delay independent stability region and time delay related stability zone under different control parameter combinations of system.This application can provide reference for controller stability design, compared with traditional active front wheel steering controller without considering time delay, can more effectively judge whether system meets stability requirement, and reduce control parameter adjustment range, so as to reduce test cost in actual engineering, guarantee the stability of vehicle control system.

Description

Technical Field

[0001] This invention belongs to the field of dynamics and control of active front wheel steering systems for automobiles, and particularly relates to a method for calibrating the feasible region of time delay of parameters of an active front wheel steering time delay feedback controller. Background Technology

[0002] Time delay is a common phenomenon in electromechanical systems, significantly affecting system performance and stability. In automotive active front-wheel steering control systems, time delay primarily originates from sensor signal sampling, ECU processing, actuator operation, and tire deformation. These time delays can degrade the design performance of the steering controller and even cause system instability, such as steering oscillations and cornering deviations, potentially threatening occupant safety in extreme cases. Therefore, reducing and optimizing the impact of time delay on the steering system, and ensuring the performance and stability of the control system under time delay conditions, is an important research topic.

[0003] Current research on time-delay effects in AFS systems is insufficient. Most existing studies do not adequately consider the distribution of controller parameter combinations in both time-delay-independent and time-delay-dependent stability regions, nor do they investigate the tolerable critical time delays for different combinations of control parameters. These issues adversely affect system stability and performance parameters, potentially leading to serious instability risks in practical engineering applications. Furthermore, current mainstream AFS control strategies are primarily based on error control methods such as PD, PID, or feedback control. The performance of these strategies is highly dependent on the selection of controller parameters or coefficients. Existing technologies have not adequately defined and analyzed their stability under time-delay conditions, meaning that selected control parameters may not maintain stability or meet vehicle performance requirements under different time-delay conditions. This necessitates extensive experimental results to determine controller parameters, increasing testing costs. Therefore, this is a problem that urgently needs to be addressed at present. Summary of the Invention

[0004] To address the shortcomings of the existing technology, this invention proposes a calibration method for the time-delay feasible region of the parameters of the Active Front Wheel Steering (AFS) time-delay feedback controller. By defining the time-delay-independent and time-delay-dependent stable regions of the controller parameter combination, the critical time delay of the AFS time-delay controller under the parameter combination can be obtained. Before selecting vehicle controller parameters, it is determined whether the control system meets the time-delay stability requirements, thereby effectively suppressing the negative impact of time delay on the AFS control system and improving the performance and stability of the controller.

[0005] The present invention adopts the following technical solution to solve the technical problem:

[0006] The method for calibrating the feasible region of time delay of parameters of an active front wheel steering time delay feedback controller according to the present invention is characterized by comprising the following steps:

[0007] Step 1: Use equation (1) to establish the motion differential equations of the 2DOF vehicle linear model:

[0008]

[0009] In equation (1): β represents the centroid sideslip angle; The first derivative of the sideslip angle β is represented by γ; γ is the yaw rate. The first derivative of γ is represented by γ; a and b are the lengths of the front and rear axles, respectively; m is the total mass of the vehicle body; V x For longitudinal vehicle speed; I z F is the moment of inertia about the Z-axis; yf F yr These are the lateral forces f for the front wheel and r for the rear wheel, respectively.

[0010] The slip angle α of the front and rear wheels is defined using equation (2). f α r :

[0011]

[0012] In equation (2): δ f Let f be the steering angle of the front wheel;

[0013] Ignoring the influence of nonlinear factors on the vehicle tire lateral force, a simplified model of the linear lateral force of the front and rear wheels of the vehicle is obtained using equation (3):

[0014] F yi =C i α i (3)

[0015] In equation (3): i = f, r represents the front and rear wheels respectively; C i Let i be the lateral stiffness of wheel i;

[0016] Step 2: Use equation (4) to establish a second-order linear vehicle time-delay system model that includes time delay:

[0017]

[0018] In equation (4): x(t) represents the state variable at time t, and x(t) = [β, γ] T ; Let represent the first derivative of x(t); τ represents the time delay of the state feedback controller, and τ≥0; Δδ f (t-τ) represents the additional front wheel steering angle considering the time delay effect; by simultaneously solving equations (1), (2), (3), and (4), we obtain the two coefficients A and B of the system's state equation, and we have:

[0019]

[0020] In equation (5): C r C represents the lateral stiffness of the rear wheel r; f Let f be the lateral stiffness of the front wheel;

[0021] Step 3: Design a vehicle AFS controller with time-delay feedback control using equation (6):

[0022]

[0023] In equation (6), K is the gain of the state feedback controller; x(t-τ) represents the state variable considering time delay, and x(t-τ)=[β(t-τ),γ(t-τ)] T ;k p k d This represents the proportional control coefficient to be determined in the state feedback controller;

[0024] Using equation (7), the 2DOF closed-loop control model of the vehicle is obtained:

[0025]

[0026] Step 4: Construct the Lyapunov-Krasovskii functional V(t) at time t using equation (8):

[0027]

[0028] In equation (8), P and S are two n×n dimensional constant matrices; and P = P T >0, S=S T >0;P T S represents the transpose of P. T S represents the transpose of S; s represents the time variable; x(s) represents the state of the AFS time-delay control system at a past time point s; x T (s) denotes the transpose of x(s); x T (t) denotes the transpose of x(t);

[0029] The first derivative of V(t) is obtained using equation (9). Therefore, the stability condition can be constructed using equation (10):

[0030]

[0031]

[0032] In equation (9), x T (t-τ) denotes the transpose of x(t-τ);

[0033] Equation (10) is solved using the LMI toolbox in MATLAB to obtain the results of the time-delay-independent stability parameter region of the control system.

[0034] Step 5: Using the Laplace transform shown in equation (11), obtain the characteristic equation D(λ) of the vehicle active front wheel steering time delay controller with feedback control:

[0035] D(λ)=P(λ)+Q(λ)e -λτ =0 (11)

[0036] In equation (11), λ represents the eigenvalue of the characteristic equation; P(λ) and Q(λ) are second-order and first-order real coefficient polynomials, respectively;

[0037] Step 6: When τ = 0, the characteristic equation of the vehicle AFS time delay controller is obtained using equation (12):

[0038] D(λ)=P(λ)+Q(λ)=0 (12)

[0039] Satisfying the result in equation (12) yields one of the conditions for the time-delay-independent stability parameter region of the system.

[0040] Step 7: When τ>0, let the pure imaginary root λ=iω0, and use equation (13) to separate the real and imaginary parts of equation (11):

[0041] P R (ω0)+iP I (ω0)+[Q R (ω0)+iQ I (ω0)][cos(τω0)-isin(τω0)]=0 (13)

[0042] In equation (13), λ represents the root of the complex number; i represents the imaginary unit; ω0 represents the angular frequency; P R (ω0) represents the real part coefficient of P(λ); P I (ω0) represents the imaginary part coefficient of P(λ); Q R (ω0) represents the real part coefficients of Q(λ); Q I (ω0) represents the imaginary part coefficient of Q(λ);

[0043] Setting both the real and imaginary parts of equation (13) to 0, we obtain equation (14), which simplifies to equation (15):

[0044]

[0045]

[0046] In equation (14), R e(ω0) represents the real coefficients of the characteristic equation; I m (ω0) represents the imaginary part coefficient of the characteristic equation;

[0047] Eliminating the trigonometric functions in equation (14), we can obtain the polynomial equation F(ω0) using equation (16):

[0048]

[0049] Step 8: Solve the equation F(ω0)=0 using the built-in functions of MATLAB. If there are no positive real roots other than 0 in the solution, it means that the vehicle AFS time-delay controller is stable for any time delay and the time-delay-independent stability region is obtained; if there is a positive real root ω0, it means that the AFS time-delay control system is in the time-delay-dependent stability region. Substitute the positive real root ω0 into equation (15) to obtain the current longitudinal vehicle speed V. x The critical time delay τ0 is given below the critical time delay τ0; and τ0>0, and then the minimum critical time delay τ at different periodic angular frequencies is obtained using equation (17). min :

[0050]

[0051] In equation (17), j represents the angular frequency of different periods;

[0052] Step 9: Change different control parameters k p k d Then, return to step 3 and execute sequentially to obtain the minimum critical time delay feasible region that the controller can encompass in the time delay-independent stability region and the time delay-dependent stability region of the AFS time delay feedback controller under different proportional control coefficients;

[0053] Step 10: Change the longitudinal speed V x Then, return to steps 3 to 9 in sequence to obtain the minimum critical time delay feasible region that the AFS feedback controller can cover in the time delay-independent stability region and time delay-dependent stability region under different proportional control coefficients under different vehicle speeds.

[0054] The present invention provides an electronic device, including a memory and a processor, wherein the memory is used to store a program that supports the processor in executing the calibration method, and the processor is configured to execute the program stored in the memory.

[0055] The present invention discloses a computer-readable storage medium on which a computer program is stored, wherein the computer program is executed by a processor to perform the steps of the calibration method.

[0056] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0057] 1. This invention defines the time-delay-independent stability region and the time-delay-dependent stability region of the AFS controller parameters, effectively analyzing the critical time delay that the controller can tolerate under different combinations of control parameters, thereby providing a basis for the selection of vehicle controller parameters and reducing the impact of time delay on the stability and performance of the controller.

[0058] 2. This invention utilizes a stability analysis method for time-delay systems to obtain stability criteria for the control system. Before selecting vehicle controller parameters, it can determine whether the control system meets the time-delay stability requirements, reducing the parameter adjustment range of the time-delay controller and allowing the selection of appropriate parameters within this range. This reduces testing costs in practical engineering, effectively ensuring the stability of the vehicle control system and improving vehicle handling and safety.

[0059] 3. This invention primarily calibrates the stability boundaries and time-delay feasible regions of vehicle AFS controller parameters. Its research methods are applicable to other automotive control systems, such as suspension systems, and even other electromechanical systems exhibiting time delays. This method is universally applicable to automotive systems with various time delay magnitudes and parameter combinations, and has promising application prospects. Attached Figure Description

[0060] Figure 1 This is a flowchart of the time-delay feasible region calibration of the AFS feedback controller in the method of the present invention;

[0061] Figure 2 This is a diagram of a two-degree-of-freedom vehicle dynamics model in the method of this invention;

[0062] Figure 3 This refers to the time-delay-independent stability regions of the system under different control parameters in the method of this invention;

[0063] Figure 4a This is a three-dimensional diagram of the time-delay feasible region of the AFS time-delay feedback controller in this invention example;

[0064] Figure 4b This is a planar diagram of the time-delay feasible region of the AFS time-delay feedback controller in an example of the present invention;

[0065] Figure 5 This refers to the LK time-delay-independent stable region corresponding to different vehicle speeds in this invention example. Detailed Implementation

[0066] In this embodiment, a method for calibrating the feasible domain of the time delay of the active front steering (AFS) steering time delay feedback controller parameters includes the following steps:

[0067] The specific calibration flowchart is as follows: Figure 1The diagram illustrates the specific process for calibrating the time-delay feasible region of the AFS feedback controller parameters.

[0068] Step 1: Use equation (1) to establish the motion differential equations of the 2DOF vehicle linear model:

[0069]

[0070] In equation (1): β represents the centroid sideslip angle; The first derivative of the sideslip angle β is represented by γ; γ is the yaw rate. The first derivative of γ is represented by γ; a and b are the lengths of the front and rear axles, respectively; m is the total mass of the vehicle body; V x For longitudinal vehicle speed; I z F is the moment of inertia about the Z-axis; yf F yr Let f be the lateral force of the front wheel and r be the lateral force of the rear wheel, respectively. The vehicle parameters used in this example are shown in Table 1, and its two-degree-of-freedom linear vehicle model is as follows: Figure 2 As shown.

[0071] Table 1 Vehicle Parameters

[0072]

[0073]

[0074] The slip angle α of the front and rear wheels is defined using equation (2). f α r :

[0075]

[0076] In equation (2): δ f Let f be the steering angle of the front wheel;

[0077] Ignoring the influence of nonlinear factors on the vehicle tire lateral force, a simplified model of the linear lateral force of the front and rear wheels of the vehicle is obtained using equation (3):

[0078] F yi =C i α i (3)

[0079] In equation (3): i = f, r represents the front and rear wheels respectively; C i Let i be the lateral stiffness of wheel i;

[0080] Step 2: Use equation (4) to establish a second-order linear vehicle time-delay system model that includes time delay:

[0081]

[0082]

[0083] In equation (4): x(t) represents the state variable at time t, and x(t) = [β, γ] T ; Let represent the first derivative of x(t); τ represents the time delay of the state feedback controller, and τ≥0; Δδ f (t-τ) represents the additional front wheel steering angle considering the time delay effect; A and B are obtained by combining equations (1), (2), (3), and (4), representing the coefficients of the system's state equation; in equation (5): C r C represents the lateral stiffness of the rear wheel r; f Let f be the lateral stiffness of the front wheel; in this example, the vehicle speed is: V x =80km / h.

[0084] Step 3: Design a vehicle AFS controller with time-delay feedback control using equation (6):

[0085]

[0086] In equation (6), K is the gain of the state feedback controller; x(t-τ) represents the state variable considering time delay, specifically, x(t-τ)=[β(t-τ),γ(t-τ)] T ;k p k d This represents the proportional control coefficient to be determined in the state feedback controller, where k p The value range of k is [-5, 5]; d The value range is [-1, 1].

[0087] Using equation (7), the 2DOF closed-loop control model of the vehicle is obtained:

[0088]

[0089] Step 4: Construct the Lyapunov-Krasovskii functional V(t) at time t using equation (8):

[0090]

[0091] In equation (8), P and S are two n×n dimensional constant matrices; and P = P T >0, S=S T >0;P T S represents the transpose of P. T Let S be the transpose of S; s represent the time variable; x(s) represents the state of the AFS time-delay feedback control system at a past time point s; x T (s) denotes the transpose of x(s); x T (t) denotes the transpose of x(t);

[0092] The first derivative of V(t) is obtained using equation (9). Therefore, the stability condition can be constructed using equation (10):

[0093]

[0094]

[0095] The results of solving the time-delay-independent stability parameter region using the LMI toolbox in MATLAB are as follows: Figure 3 As shown, the dark gray area represents the LK time-delay-independent stability region, where the control system remains stable even with a time delay. Since the time delay τ does not appear in the stability condition, this condition applies to any time delay. Conversely, if the system is in the stability-dependent region, the presence of a time delay may lead to instability in the control system. The next step will be to perform stability analysis on the time-delay-dependent region.

[0096] Step 5: Using the Laplace transform shown in equation (11), obtain the characteristic equation D(λ) of the vehicle active front wheel steering time delay controller with feedback control:

[0097]

[0098] In equation (11), λ represents the eigenvalue of the characteristic equation; P(λ) and Q(λ) are second-order and first-order real coefficient polynomials, respectively;

[0099] Step 6: When τ = 0, the characteristic equation of the vehicle AFS time-delay feedback controller is obtained using equation (12):

[0100]

[0101] According to the Routh–Hurwitz stability condition, we can obtain equation (13):

[0102]

[0103] In different k p k d Under the parameter combination, satisfying the inequality in equation (13) yields one of the conditions for the system to satisfy the time-delay-independent stability parameter region. This condition is the light gray region surrounded by the dashed line, such as... Figure 3 As shown.

[0104] Step 7: When τ>0, let the pure imaginary root λ=iω0, and use equation (13) to separate the real and imaginary parts of equation (11):

[0105] P R (ω0)+iP I (ω0)+[QR (ω0)+iQ I (ω0)][cos(τω0)-isin(τω0)]=0 (14)

[0106] In equation (14), λ represents the root of the complex number; i represents the imaginary unit; ω0 represents the angular frequency; P R (ω0) represents the real part coefficient of P(λ); P I (ω0) represents the imaginary part coefficient of P(λ); Q R (ω0) represents the real part coefficients of Q(λ); Q I (ω0) represents the imaginary part coefficient of Q(λ);

[0107] Setting both the real and imaginary parts of equation (14) to 0, we obtain equation (15), which simplifies to equation (16):

[0108]

[0109]

[0110] In equation (15), R e (ω0) represents the real coefficients of the characteristic equation; I m (ω0) represents the imaginary part coefficient of the characteristic equation;

[0111] Eliminating the trigonometric functions in equation (15), we can obtain the polynomial equation F(ω0) using equation (17):

[0112]

[0113] The specific expressions of each term in equation (17) are shown in equation (18):

[0114]

[0115] Eliminating the trigonometric functions yields the polynomial equation, as shown in equation (19):

[0116]

[0117] Step 8: Solve the equation F(ω0)=0 using the built-in functions of MATLAB. If F(ω0) has no positive real roots (except for 0), the vehicle AFS time-delay feedback controller remains stable for any time delay, and the time-delay-independent stability region is obtained. If there is a positive real root ω0, it can be determined that the system is in the time-delay-dependent stability region. Substitute the positive real root ω0 into equation (16) to obtain the current longitudinal vehicle speed V. x The minimum critical time delay τ0 is required to be greater than 0 at different periodic angular frequencies. min :

[0118]

[0119] In equation (20), j represents different periodic angular frequencies, j = 0, 1, 2, 3, ...

[0120] Step 9: By varying the control parameter k p k d The minimum critical time delay feasible region is obtained by solving the following method, and the result is as follows: Figure 4a , Figure 4b As shown, where Figure 4a A three-dimensional plot of the time-delay feasible region of the AFS time-delay feedback controller under different combinations of control parameters; Figure 4b Its two-dimensional planar diagram; from the diagram, we can see that k p k d There exists a minimum critical time delay corresponding to the control parameters. This is the maximum time delay that the system can accommodate to achieve stable control at this time. If the time delay exceeds this value, the control system will diverge and cannot be controlled. Conversely, if the time delay in the system is less than the minimum critical time delay, the control system can converge.

[0121] Step 10: Investigate the LK time-delay indifference region, critical time-delay stability region, and minimum critical time-delay under AFS control parameters at different vehicle speeds. Repeat steps 3-9 by changing the vehicle speed to obtain the results. Figure 5 As shown, the AFS time-delay control system is used at vehicle speed V. x At speeds of 60–120 km / h, the corresponding LK time-delay independent stability region can be obtained by repeating steps 3–9 if the corresponding time-delay feasible region of the control system needs to be solved.

[0122] In this embodiment, an electronic device includes a memory and a processor. The memory stores a program that supports the processor in executing the above-described method, and the processor is configured to execute the program stored in the memory.

[0123] In this embodiment, a computer-readable storage medium stores a computer program, which is executed by a processor to perform the steps of the above method.

Claims

1. A method for calibrating the feasible region of time delay of parameters of an active front wheel steering time delay feedback controller, characterized in that, Includes the following steps: Step 1: Use equation (1) to establish the motion differential equations of the 2DOF vehicle linear model: (1) In equation (1): β represents the centroid sideslip angle; The first derivative of the sideslip angle β is represented by γ; γ is the yaw rate. The first derivative of γ is represented by γ; a and b are the lengths of the front and rear axles, respectively; m is the total mass of the vehicle body; V x For longitudinal vehicle speed; I z F is the moment of inertia about the Z-axis; yf F yr These are the lateral forces f for the front wheel and r for the rear wheel, respectively. The slip angle α of the front and rear wheels is defined using equation (2). f α r : (2) In equation (2): δ f Let f be the steering angle of the front wheel; Ignoring the influence of nonlinear factors on the vehicle tire lateral force, a simplified model of the linear lateral force of the front and rear wheels of the vehicle is obtained using equation (3): (3) In equation (3): i = f, r represents the front and rear wheels respectively; C i Let i be the lateral stiffness of wheel i; Step 2: Use equation (4) to establish a second-order linear vehicle time-delay system model that includes time delay: (4) In equation (4): x(t) represents the state variable at time t, and x(t) = [β, γ]. T ; Let represent the first derivative of x(t); τ represents the time delay of the state feedback controller, and τ≥0; Δδ f (t-τ) represents the additional front wheel steering angle considering the time delay effect; by simultaneously solving equations (1), (2), (3), and (4), we obtain the two coefficients A and B of the system's state equation, and we have: (5) In equation (5): C r C represents the lateral stiffness of the rear wheel r; f Let f be the lateral stiffness of the front wheel; Step 3: Design a vehicle AFS controller with time-delay feedback control using equation (6): (6) In equation (6), K is the gain of the state feedback controller; x(t-τ) represents the state variable considering time delay, and x(t-τ) = [β(t-τ), γ(t-τ)]. T ;k p k d This represents the proportional control coefficient to be determined in the state feedback controller; Using equation (7), the 2DOF closed-loop control model of the vehicle is obtained: (7) Step 4: Construct the Lyapunov-Krasovskii functional at time t using equation (8). : (8) In equation (8), P and S are two n×n dimensional constant matrices; and P = P T >0, S=S T >0;P T S represents the transpose of P. T Indicates the transpose of S; Let x(s) represent the time variable; x(s) represents the state of the AFS time-delay control system at a past time point s. Indicates the transpose of x(s); Indicates the transpose of x(t); Using equation (9) to obtain first derivative Thus, the stability condition can be constructed using equation (10): (9) (10) In equation (9), This represents the transpose of x(t-τ); Equation (10) is solved using the LMI toolbox in MATLAB to obtain the results of the time-delay-independent stability parameter region of the control system. Step 5: Using the Laplace transform shown in equation (11), obtain the characteristic equation D(λ) of the vehicle active front wheel steering time delay controller with feedback control: (11) In equation (11), λ represents the eigenvalue of the characteristic equation; P(λ) and Q(λ) are second-order and first-order real coefficient polynomials, respectively; Step 6: When τ=0, the characteristic equation of the vehicle AFS time delay controller is obtained using equation (12): (12) Satisfying the result in equation (12) yields one of the conditions for the time-delay-independent stability parameter region of the system; Step 7: When τ>0, let the pure imaginary root λ = iω0, and use equation (13) to separate the real and imaginary parts of equation (11): (13) In equation (13), i represents the imaginary unit; ω0 represents the angular frequency; P R (ω0) represents the real part coefficient of P(λ); P I (ω0) represents the imaginary part coefficient of P(λ); Q R (ω0) represents the real part coefficients of Q(λ); Q I (ω0) represents the imaginary part coefficient of Q(λ); Setting both the real and imaginary parts of equation (13) to 0, we obtain equation (14), which simplifies to equation (15): (14) (15) In equation (14), R e (ω0) represents the real coefficients of the characteristic equation; I m (ω0) represents the imaginary part coefficient of the characteristic equation; Eliminating the trigonometric functions in equation (14), we can obtain the polynomial equation F(ω0) using equation (16): (16) Step 8: Solve the equation F(ω0)=0 using the built-in functions of MATLAB. If there are no positive real roots other than 0 in the solution, it means that the vehicle AFS time-delay controller remains stable for any time delay, and the time-delay-independent stability region is obtained. If there is a positive real root ω0, it means that the AFS time-delay control system is in the time-delay-dependent stability region. Substitute the positive real root ω0 into equation (15) to obtain the current longitudinal vehicle speed V. x The critical time delay τ0 is given below the critical time delay τ0; and τ0>0, and then the minimum critical time delay τ at different periodic angular frequencies is obtained using equation (17). min : (17) In equation (17), j represents the angular frequency of different periods; Step 9: Change different control parameters k p k d Then, return to step 3 and execute sequentially to obtain the minimum critical time delay feasible region that the controller can encompass in the time delay-independent stability region and the time delay-dependent stability region of the AFS time delay feedback controller under different proportional control coefficients; Step 10: Change the longitudinal speed V x Then, return to steps 3 to 9 in sequence to obtain the minimum critical time delay feasible region that the AFS feedback controller can cover in the time delay-independent stability region and time delay-dependent stability region under different proportional control coefficients under different vehicle speeds.

2. An electronic device, comprising a memory and a processor, characterized in that, The memory is used to store a program that supports the processor in executing the calibration method of claim 1, and the processor is configured to execute the program stored in the memory.

3. A computer-readable storage medium storing a computer program, characterized in that, The computer program is executed by the processor to perform the steps of the calibration method of claim 1.

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