A performance improvement method for active suspension system tracking control

Abstract

The application designs a performance promotion method of active suspension system tracking control, relates to unknown nonlinear systems, and is applied to the active suspension system of an automobile; an intelligent controller is designed based on the backstepping recursion method and the adaptive control method; firstly, a finite time performance function model is established; secondly, stress analysis is conducted on the quarter active suspension system based on Newton's second law, a dynamics model of the suspension system is established, and the state equation based on the finite time performance function is converted; then, the nonlinear active suspension system is identified by using a neural network; an adaptive law and a controller are designed based on the Lyapunov function and the backstepping recursion method; the system is analyzed in the light of the Lyapunov stability theory, and it is proved that all signals are bounded; finally, simulation experiments are conducted by using MATLAB, and it is proved that the method has good control effect.

Description

Technical Field

[0001] This invention belongs to the field of suspension systems, and specifically relates to a method for improving the performance of tracking control in an active suspension system. Background Technology

[0002] Vehicles play an indispensable role in modern life, and suspension systems are widely used in various high-tech fields such as robotics and aerospace. They are also a crucial component of the physical structure of automobiles. The main functions of vehicle suspension include: isolating the vehicle from road disturbances during driving, maintaining good handling, enhancing the vehicle's adaptability to road conditions, and supporting the vehicle's static mass. Traditional passive suspensions are not particularly effective in terms of ride comfort and driving safety, and perform poorly in complex road conditions. Active suspensions, on the other hand, can adjust in real time to ensure vehicle handling stability, improve passenger safety and ride comfort, and thus achieve multi-objective control.

[0003] With the development of control theory, significant progress has been made in the study of suspension system stability. It is worth noting that most results were obtained under infinite-time conditions. While many existing control methods can guarantee system performance when the time variable approaches infinity, in practical applications, the control objective needs to be achieved within a finite time. Infinite-time control schemes lead to longer transient responses, failing to achieve the aforementioned control objective. Compared to existing infinite-time control, finite-time control can quickly obtain the system's transient performance, and it also boasts advantages such as fast convergence, strong robustness, and high control accuracy.

[0004] During vehicle operation, to ensure good driving performance, the active suspension system outputs external control forces through actuators to suppress vehicle body vibrations, thereby maintaining good shock absorption performance. However, in practical applications, vehicle suspension systems inevitably deviate from their original control performance due to certain uncertainties.

[0005] Because the current theoretical framework is still incomplete and there are not enough application tools available, active suspension systems still encounter many unexpected problems in practical production applications when subjected to external disturbances. How to solve these problems and ensure better system operation has become an important research issue. Summary of the Invention

[0006] In order to effectively improve vehicle performance, this invention proposes a performance improvement method for active suspension system tracking control, addressing the shortcomings of existing technologies.

[0007] A method for improving the performance of tracking control in an active suspension system specifically includes the following steps:

[0008] Step 1: Establish a mathematical model for the finite-time performance function;

[0009]

[0010] in, η T >0 and T>0 are constants; from equation (1), we get F ψ It converges to η at time t = T. T And it remains unchanged thereafter;

[0011] Step 2: Based on the physical structure of the quarter-drive active suspension system and Newton's second law, establish the following kinematic formulas:

[0012]

[0013] Where u is the control input of the active suspension system; D s and D w These represent the displacements of the sprung mass and the unsprung mass, respectively. and D is the corresponding second derivative; r Indicates road surface disturbance; k a and k t These represent the stiffness coefficients of the suspension and tires, respectively; c a and c t These represent the damping coefficients of the suspension and tires, respectively, m b and m s These represent the sprung mass and unsprung mass, respectively; the suspension's elastic force and damping force are F and F, respectively. s =k a (D s -D w )and The elastic force and damping force of the tire are F t =k t (D w -D r )and

[0014] Define the state variable as follows: x1 = D s , x3=D w , The state-space expression for the active suspension system of a quarter-car is then expressed as:

[0015]

[0016] Step 3: Design a suspension controller based on the finite-time performance function established in Step 1, and obtain the adaptive law parameters;

[0017] Step 3.1: First, define the following coordinate transformation:

[0018] e1(t) = x1 - y r (4)

[0019] Where y r It is the expected trajectory, and its first derivative is... and second derivative Both are bounded, and e1(t) is the tracking error;

[0020] The error transformation is defined as:

[0021]

[0022] Where ζ is the error variable, and equation (5) is a strictly increasing and smooth function, we get: -1 < ξ(ζ) < 1.

[0023] To ensure that the tracking error always remains within the boundary F formed by the proposed finite-time performance function ψ Within (t), the following transformation is given:

[0024] e1 = F ψ (t)ξ(ζ) (6)

[0025] From (4) and (6), the time derivative of e1 is:

[0026]

[0027] Based on (3), Represented as:

[0028]

[0029] in

[0030] Define the following error transformation:

[0031] z2=x2-α1 (9)

[0032] Where α1 is the virtual controller;

[0033] Based on (8) and (9), (3) is rewritten as:

[0034]

[0035] The derivative of the error variable ζ can be further rewritten in the following form:

[0036]

[0037] in It is an unknown function; to estimate this unknown function, the following neural network is used to approximate it:

[0038] Γ1=W1 T R1(χ1)+σ1 (12)

[0039] in, L1 is the ideal weight vector, and L1 is the number of nodes in the neural network. It is the Gaussian function. Let ε1 be the input vector of the neural network and σ1 be the approximation error. Then there exists ε1 > 0 such that |σ1| ≤ ε1.

[0040] Define estimation error This is an estimate of θ1; based on Young's inequality, we get:

[0041]

[0042] Where c1 > 0 is the design parameter, and θ1 = ||W1|| 2 ;

[0043] The virtual controller is designed as follows:

[0044]

[0045] Where k1 > 0 is a constant;

[0046] According to (14), we obtain the following equation:

[0047]

[0048] Step 3.2: The derivative of the error variable z2 can be rewritten in the following form:

[0049]

[0050] in Input vector And |σ²| ≤ ε², ε² > 0. Define the estimation error. This is an estimate of θ²; based on Young's inequality, we get:

[0051]

[0052] Where c2 > 0 are design parameters, and θ2 = ||W2|| 2 .

[0053] The controller is defined as follows:

[0054]

[0055] Where k2 > 0 is a constant;

[0056] According to (18), we obtain the following equation:

[0057]

[0058] Step 4: Adjust the parameters of the controller and adaptive law obtained in Step 3 to achieve the final tracking control:

[0059] By adjusting the controller and the parameter adaptive law, the performance of the quarter active suspension system can reach the predetermined performance index.

[0060] The design and stability verification of the parameter adaptive law for the active suspension system are carried out, specifically as follows:

[0061] Based on the error transformation in step 3, the following Lyapunov function is selected:

[0062]

[0063] Where p1 > 0 are design parameters;

[0064] Based on (10) and (20), the time derivative of V1 is obtained:

[0065]

[0066] Design the adaptive law in V1:

[0067]

[0068] Where λ1>0 are design parameters;

[0069] Substituting (13), (15), and (22) into (21), we get:

[0070]

[0071] Choose a Lyapunov function of the following form:

[0072]

[0073] Based on (10), (16), (23), and (24), the time derivative of V2 is obtained:

[0074]

[0075] Designing the V2 adaptive law:

[0076]

[0077] Where λ2>0 are design parameters;

[0078] Substituting (17), (19), and (26) into (25), we get:

[0079]

[0080] Where a = min{2k1, 2k2, λ1, λ2}, According to the Lyapunov stability criterion, it can be seen from formula (27) that all signals of the closed-loop active suspension system are bounded.

[0081] Beneficial technical effects of the present invention

[0082] This invention combines finite-time control and specified performance control concepts to improve the robustness and control precision of the suspension system, further ensuring its steady-state and transient performance. Simultaneously, it improves vehicle handling stability, enhancing ride comfort and driving safety. The invention first performs a force analysis on a quarter-wave active suspension system, establishing dynamic equations. Then, by selecting appropriate state variables, it establishes a state-space expression and provides a detailed transformation process based on a finite-time performance function. In this invention, a neural network is used to identify the nonlinear active suspension system. The controller and adaptive law designed in this invention offer valuable insights for solving the finite-time specified performance problem of active suspension systems.

[0083] This invention applies the concepts of finite-time control and specified performance control to the controller design of a quarter-wave active suspension system, improving the control performance of the suspension system and demonstrating greater practical application value compared to infinite-time control schemes. The proposed control algorithm ensures vehicle driving safety, ride comfort, and handling stability. Attached Figure Description

[0084] Figure 1 Flowchart of the method according to an embodiment of the present invention;

[0085] Figure 2 A schematic diagram illustrating the changes in vehicle displacement and performance boundaries within 20 seconds according to an embodiment of the present invention;

[0086] Figure 3 A schematic diagram of the vehicle body displacement within 20 seconds according to an embodiment of the present invention;

[0087] Figure 4 A schematic diagram of the dynamic control input u within 20 seconds according to an embodiment of the present invention;

[0088] Figure 5 Within 20 seconds of the embodiment of the present invention and A schematic diagram of the response curve. Detailed Implementation

[0089] The present invention will be further described below with reference to the accompanying drawings and embodiments;

[0090] A method for improving the performance of tracking control in an active suspension system, as shown in the appendix. Figure 1 As shown, the specific steps include:

[0091] Step 1: Establish a mathematical model for the finite-time performance function;

[0092]

[0093] in, η T >0 and T>0 are constants; from equation (1), we get F ψ It converges to η at time t = T. T And it remains unchanged thereafter; F ψ The simulation curves are as follows Figure 2 As shown;

[0094] Step 2: Based on the physical structure of the quarter-drive active suspension system and Newton's second law, establish the following kinematic formulas:

[0095]

[0096] Where u is the control input of the active suspension system; D s and D w These represent the displacements of the sprung mass and the unsprung mass, respectively. and D is the corresponding second derivative; r Indicates road surface disturbance; k a and k t These represent the stiffness coefficients of the suspension and tires, respectively; c a and c t These represent the damping coefficients of the suspension and tires, respectively, m b and m s These represent the sprung mass and unsprung mass, respectively; the suspension's elastic force and damping force are F and F, respectively. s =k a (D s -D w )and The elastic force and damping force of the tire are F t =k t (D w -D r )and

[0097] Define the state variable as follows: x1 = D s , x3=D w , The simulation curve of x1 is as follows Figure 3 As shown; the state-space expression of the active suspension system of a quarter-car is then expressed as:

[0098]

[0099] Step 3: Design a suspension controller based on the finite-time performance function established in Step 1, and obtain the adaptive law parameters;

[0100] Step 3.1: First, define the following coordinate transformation:

[0101] e1(t) = x1 - y r (31)

[0102] Where y r It is the expected trajectory, and its first derivative is... and second derivative Both are bounded, and e1(t) is the tracking error;

[0103] The error transformation is defined as:

[0104]

[0105] Where ζ is the error variable, and equation (5) is a strictly increasing and smooth function, we get: -1 < ξ(ζ) < 1.

[0106] To ensure that the tracking error always remains within the boundary F formed by the proposed finite-time performance function ψ Within (t), the following transformation is given:

[0107] e1 = F ψ (t)ξ(ζ) (33)

[0108] From (4) and (6), the time derivative of e1 is:

[0109]

[0110] Based on (3), Represented as:

[0111]

[0112] in

[0113] Define the following error transformation:

[0114] z2=x2-α1 (36)

[0115] Where α1 is the virtual controller;

[0116] Based on (8) and (9), (3) is rewritten as:

[0117]

[0118] The derivative of the error variable ζ can be further rewritten in the following form:

[0119]

[0120] in It is an unknown function; to estimate this unknown function, the following neural network is used to approximate it:

[0121]

[0122] in, L1 is the ideal weight vector, and L1 is the number of nodes in the neural network. It is the Gaussian function. Let ε1 be the input vector of the neural network and σ1 be the approximation error. Then there exists ε1 > 0 such that |σ1| ≤ ε1.

[0123] Define estimation error This is an estimate of θ1; based on Young's inequality, we get:

[0124]

[0125] Where c1 > 0 is the design parameter, and θ1 = ||W1|| 2 ;

[0126] The virtual controller is designed as follows:

[0127]

[0128] Where k1 > 0 is a constant;

[0129] According to (14), we obtain the following equation:

[0130]

[0131] Step 3.2: The derivative of the error variable z2 can be rewritten in the following form:

[0132]

[0133] Where Γ2=W2 T R2(χ2)+σ2, input vector And |σ²| ≤ ε², ε² > 0. Define the estimation error. This is an estimate of θ²; based on Young's inequality, we get:

[0134]

[0135] Where c2 > 0 are design parameters, and θ2 = ||W2|| 2 .

[0136] The controller is defined as follows:

[0137]

[0138] Where k2>0 is a constant, λ2>0 is a design parameter, and the simulation curve of u is as follows: Figure 4 As shown, The simulation curves are as follows Figure 5 As shown;

[0139] According to (18), we obtain the following equation:

[0140]

[0141] Step 4: Adjust the parameters of the controller and adaptive law obtained in Step 3 to achieve the final tracking control:

[0142] By adjusting the controller and the parameter adaptive law, the performance of the quarter active suspension system can reach the predetermined performance index.

[0143] The design and stability verification of the parameter adaptive law for the active suspension system are carried out, specifically as follows:

[0144] Based on the error transformation in step 3, the following Lyapunov function is selected:

[0145]

[0146] Where p1 > 0 are design parameters;

[0147] Based on (10) and (20), the time derivative of V1 is obtained:

[0148]

[0149] Design the adaptive law in V1:

[0150]

[0151] Where λ1>0 are design parameters;

[0152] Substituting (13), (15), and (22) into (21), we get:

[0153]

[0154] Choose a Lyapunov function of the following form:

[0155]

[0156] Based on (10), (16), (23), and (24), the time derivative of V2 is obtained:

[0157]

[0158] Designing the V2 adaptive law:

[0159]

[0160] Where λ2>0 are design parameters;

[0161] Substituting (17), (19), and (26) into (25), we get:

[0162]

[0163] Where a = min{2k1, 2k2, λ1, λ2}, According to the Lyapunov stability criterion, it can be seen from formula (27) that all signals of the closed-loop active suspension system are bounded.

[0164] The sprung mass of the active suspension system selected in this invention is m. b =590kg, unsprung mass m s =59kg, spring constant k of suspension spring a =18000N / m, damping coefficient c a =2200 Ns / m, the elastic coefficient k of the tire t =150000 N / m, damping coefficient c t =1200Ns / m.

[0165] In this invention, the main design parameters are k1 = 150, k2 = 120, c1 = c2 = 0.6, p1 = p2 = 0.4, λ1 = λ2 = 0.5. η T =0.1, T=3, the initial values ​​are selected as x1(0)=0.03, x2(0)=x3(0)=x4(0)=0, the initial values ​​of θ1 and θ2 are both selected as 0.03, and the expected trajectory y r =0. In this invention, the running time t = 20, the amplitude of the given periodic interference signal is 0.02 cm, and the frequency is 50 Hz.

[0166] Simulation experiments show that the adaptive control scheme based on the finite-time performance function exhibits smaller vehicle vibration amplitude, faster convergence speed, and better control performance.

Claims

1. A method for improving the performance of tracking control in an active suspension system, characterized in that, Specifically, the following steps are included: Step 1: Establish the mathematical model of the finite-time performance function, as shown in the following formula: (1)； in, , and It is a constant; by Formula exist Converging at all times And it remains unchanged thereafter; Step 2: Based on the physical structure of the quarter-vehicle active suspension system and Newton's second law, establish the kinematic formulas and obtain the state-space expression of the quarter-vehicle active suspension system, as shown in the following formula: (2)； in, This is the control input for the active suspension system; and These represent the displacements of the sprung mass and the unsprung mass, respectively. and This is the corresponding second derivative; Indicates disturbance to the road surface; and These represent the stiffness coefficients of the suspension and tires, respectively. and These represent the damping coefficients of the suspension and tires, respectively. and These represent the sprung mass and unsprung mass, respectively; the suspension's elastic force and damping force are respectively... and The elasticity and damping force of the tire are respectively and ; Define the following state variables: , , , The state-space expression for the active suspension system of a quarter-car is then expressed as: (3)； Step 3: Design a suspension controller based on the finite-time performance function established in Step 1, and obtain the adaptive law parameters; Step 3.1: First, define the following coordinate transformation: (4)； in It is the expected trajectory, and its first derivative is... and second derivative Both are bounded. ; The error transformation is defined as: (5)； in Let be the error variable, and the equation For a strictly increasing and smooth function, we obtain: , , ; To ensure that the tracking error always remains within the boundary formed by the proposed finite-time performance function Within, the following transformation is given: (6)； From (4) and (6), The time derivative is: (7)； Based on (3), Represented as: (8)； in , ; Define the following error transformation: (9)； in It is a virtual controller; Based on (8) and (9), (3) is rewritten as: (10)； Error variables The derivative can be further rewritten in the following form: (11)； in It is an unknown function; to estimate this unknown function, the following neural network is used to approximate it: (12)； in, It is the ideal weight vector. It is the number of nodes in the neural network. It is the Gaussian function. It is the input vector of the neural network. If it is an approximation error, then it exists. , making ; Define estimation error , yes The estimate; based on Young's inequality, we get: (13)； in These are design parameters. ; The virtual controller is designed as follows: (14)； in It is a constant; According to (14), we obtain the following equation: (15)； Step 3.2: Error Variables The derivative can be rewritten in the following form: (16)； in Input vector ,and , ; Define estimation error , yes The estimate; based on Young's inequality, we get: (17)； in These are design parameters. ; The controller is defined as follows: (18)； in It is a constant; According to (18), we obtain the following equation: (19)； Step 4: Adjust the parameters of the controller and adaptive law obtained in Step 3 to achieve the final tracking control.

2. The method for improving the performance of active suspension system tracking control according to claim 1, characterized in that, Step 4: By adjusting the controller and the parameter adaptive law, the performance of the quarter active suspension system is brought to the predetermined performance target. The design and stability verification of the parameter adaptive law for the active suspension system are carried out, specifically as follows: Based on the error transformation in step 3, the following Lyapunov function is selected: (20)； in These are design parameters; Based on (10) and (20), we get Time derivative: (21)； design Adaptive law in the middle: (22)； in For design parameters; Substituting (13), (15), and (22) into (21), we get: (23)； Choose a Lyapunov function of the following form: (24)； Based on (10), (16), (23), and (24), we obtain Time derivative: (25)； design Adaptive law: (26)； in For design parameters; Substituting (17), (19), and (26) into (25), we get: (27)； in, , According to the Lyapunov stability criterion, it can be seen from formula (27) that all signals of the closed-loop active suspension system are bounded.

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