A vehicle tire force robust estimation method considering output time delay

Abstract

This invention discloses a robust estimation method for vehicle tire force considering output time delay in the field of key information perception technology for vehicle systems. The method includes the following steps: Step 1: Establishing a continuous-time model of the tire's longitudinal force; Step 2: Considering the output time delay, determining the measurement equation, and compensating for the output time delay in the sensor signal; Step 3: Designing an extended state observer that combines system state estimation and disturbance observation. By estimating the system state and its disturbances in real time and providing feedback, the system state and disturbance terms under the influence of output time delay are obtained. The gain matrix of the extended state observer is dynamically adjusted using an output prediction algorithm. By considering the time delay present in the vehicle system's measured output signal and compensating for the output time delay based on a delay compensation strategy, combined with the ESO disturbance estimation mechanism, the method effectively improves the estimation accuracy of tire force and the robustness of the vehicle system to external disturbances.

Description

Technical Field

[0001] This invention relates to a robust estimation method for vehicle tire force considering output time delay in the field of key information perception technology for vehicle systems. Background Technology

[0002] Tire force is crucial information for active vehicle control. Due to technological and cost limitations, direct measurement of tire force relies on expensive tire force sensors or specialized experimental equipment, which is insufficient for practical applications in ordinary vehicles. Existing feasible alternatives include using tire models or acquiring key vehicle data through sensors and then estimating tire force using a state observer. It's important to note that tire models obtain the mathematical relationship between tire slip angle and tire force by fitting experimental data, but their adaptability to different road conditions is poor. In comparison, estimation methods based on vehicle dynamics models offer a practical and feasible alternative.

[0003] Currently, observer-based tire force estimation methods primarily rely on vehicle dynamics models. These methods typically use measurable vehicle information, such as driving torque, yaw rate, and lateral acceleration, as input, combining tire rotational dynamics and vehicle lateral dynamics models to indirectly obtain tire force. However, in real-world driving conditions, time delays inevitably exist in vehicle systems, which existing dynamics models often ignore, leading to significant modeling errors. Furthermore, the input process of tire driving torque introduces disturbances. Ignoring these time delays and disturbances severely reduces the accuracy and convergence speed of the observer's tire force estimation, and may even cause system oscillations or failures. Extended state observers (ESOs), as observation structures with low model dependence, can effectively estimate the total disturbances introduced by modeling errors or external interferences, demonstrating good robustness in engineering practice. In addition, super-twisting observers (STOs), employing an unknown input observer structure, exhibit strong robustness to model uncertainties and measurement noise, ensuring the accuracy of lateral force estimation. In summary, the performance of tire force estimation based on dynamic model can be improved in two ways: (1) Reasonably compensate for the output time delay in the vehicle system, use ESO to estimate the disturbance term in the input signal and feed it back to eliminate the influence of the disturbance; (2) Combine output time delay prediction compensation and output error, rely on STO for indirect observation when the input is unknown to ensure the robustness of the estimation results. Summary of the Invention

[0004] The purpose of this invention is to provide a robust estimation method for vehicle tire force considering output time delay. It takes into account the time delay in the measurement output signal in the vehicle system, improves the estimation accuracy of vehicle tire force by compensating for the output time delay, and uses an extended state observer (ESO) to estimate disturbances and perform compensation, combined with a super-twisting observer (STO) to estimate lateral forces, which can effectively enhance the robustness of the vehicle system to external disturbances.

[0005] To achieve the above objectives, this invention provides a robust estimation method for vehicle tire force considering output time delay, comprising the following steps:

[0006] Step 1: Establish a continuous-time model of the longitudinal force of the tire;

[0007] Step 2: Consider the output time delay, determine the measurement equation, and compensate for the output time delay in the sensor signal;

[0008] Step 3: Design an extended state observer that combines system state estimation and disturbance observation. By estimating the system state and its disturbances in real time and feeding them back, the system state and disturbance terms under the influence of output time delay are obtained. The gain matrix of the extended state observer is dynamically adjusted in combination with the output prediction algorithm.

[0009] Step 4: Iterate the recursive estimation process based on the extended state observer in Step 3 to obtain the real-time estimation results of the longitudinal force and disturbance of the left front wheel;

[0010] Step 5: Establish a system dynamics model for the front-wheel drive vehicle, considering output time delay. Utilize the longitudinal resultant force of the front wheels obtained in Step 4, and introduce an output time delay interval average prediction term to offset the influence of output time delay. Combined with a super-helical observer, obtain robust estimation results of lateral force in real time.

[0011] As a further improvement to the present invention, the specific content of step 1 is as follows.

[0012] Step 1.1: By considering the rotational motion of the left front wheel, the rotational dynamics equation of the front-wheel drive vehicle is determined as follows:

[0013] ;

[0014] Among them, I wfl Let ω be the moment of inertia of the left front wheel. fl T represents the rotational speed of the left front wheel. fl F is the torque of the left front wheel. xfl R is the longitudinal force on the left front wheel. fl The radius of the left front wheel;

[0015] Step 1.2: Considering the effects of changes in road conditions and tire relaxation, establish a continuous-time model of the longitudinal force on the left front wheel:

[0016] ;

[0017] Where, the state vector x = [ω fl F xfl ] T T is the matrix transpose, and the control input u = T fl μ is the unmodeled disturbance, A is the system state matrix, and B is the control input matrix;

[0018] , ;

[0019] Step 1.3: Characterize the unmodeled perturbation using the following relationship:

[0020] ;

[0021] Where μ is the unmodeled disturbance, which is a disturbance of known type but unknown magnitude, and A w This is the disturbance type coefficient matrix describing the dynamic behavior of external disturbances, where t is a time parameter.

[0022] As a further improvement to the present invention, the specific content of step 2 is as follows.

[0023] Choosing the rotational speed of the left front wheel as the measurement signal, the corresponding measurement equation can be expressed as:

[0024] ;

[0025] Wherein, the measured output quantity y = ω fl ,C= [1 0].

[0026] Assume that the wheel speed sensor signal has a time-varying and bounded time delay τ during the acquisition and transmission process. k The upper boundary and the lower realm It is known that the formula is... The measurement equation shown can be expressed as:

[0027] ;

[0028] For time-varying and bounded time delays, existing fixed compensation strategies are not applicable. To obtain the measurement value at the current moment, the following delay compensation strategy is used:

[0029] ;

[0030] in, The current compensated measurement value. The compensated state quantity is the output at the current moment, where e refers to the natural constant and θ is the integration variable.

[0031] As a further improvement to the present invention, the specific content of step 3 is as follows:

[0032] Pair The state vector in the original system is extended to obtain an augmented state vector that includes the state variables x of the original system and the perturbation μ. The state equation of the corresponding extended state observer can be expressed as:

[0033] (7);

[0034] Where, the augmented matrix of A augmented matrix of B L is the gain matrix of the observer, and its value should be such that A−LC satisfies the stability criterion of discrete systems.

[0035] As a further improvement of the present invention, in step 3, in order to determine the stability condition of the extended state observer, the observation error is defined as: ;

[0036] in, ;

[0037] Taking the derivative with respect to e1, we get:

[0038] ;

[0039] Based on the system's state equation and observation equation, the derivative of the error can be further expressed as:

[0040] ;

[0041] To analyze the convergence of error e1, and considering the influence of the system's state error, the Lyapunov functional V is defined as:

[0042] ;

[0043] Where, matrix P ∈ R n×n Let R be a positive definite matrix, where R is the real number field and n represents the matrix order.

[0044] Differentiating the Lyapunov functional V, we get:

[0045] ;

[0046] To ensure system stability, the following conditions must be met:

[0047] ;

[0048] Right now:

[0049] ;

[0050] The observer is stable when the observer gain matrix L satisfies the Lyapunov matrix inequality conditions mentioned above.

[0051] As a further improvement to the present invention, the specific content of step 4 is as follows.

[0052] Based on the recursive estimation process of the extended state observer in step 3, the longitudinal force F of the left front wheel can be obtained. xfl The real-time estimation results of the disturbance are then fed back to the corresponding control inputs by the disturbance estimation results output by the extended state observer, so as to construct a disturbance feedback strategy to eliminate the influence of the disturbance.

[0053] Then obtain the longitudinal force F of the right front wheel xfr The real-time estimation results of the longitudinal forces of the left and right front wheels are vectored together to obtain the resultant longitudinal force F of the left and right front wheels. xf .

[0054] As a further improvement to the present invention, the specific content of step 5 is as follows.

[0055] Step 5.1: Considering the lateral and yaw motions of the vehicle, determine the system dynamics equations for the front-wheel-drive vehicle:

[0056] ;

[0057] Where m is the mass of the vehicle, a y F is the lateral acceleration of the vehicle. yfl F yfr F yrl F yrr F represents the resultant lateral force of the four wheels (left front, right front, left rear, and right rear) on the tires. xf Based on the longitudinal resultant force of the left and right front wheels obtained in step 4, δ f For the steering angle of the vehicle's front wheels, I z Let be the vehicle's moment of inertia about the z-axis, r be the vehicle's yaw rate, and a and b be the distances from the vehicle's center of mass to the front and rear axles, respectively.

[0058] Step 5.2, the continuous-time model of the vehicle can be represented as:

[0059] ;

[0060] Wherein, the state vector v y Lateral vehicle speed, unknown input Control input φ = F xf w represents process noise, and the input matrix is ​​unknown. Control input matrix ;

[0061] Step 5.3: Select yaw rate as the measurement signal. The measurement equation considering output time delay is as follows:

[0062] ;

[0063] Where the measurement is g = r, the output matrix is ​​R1 = [0 1], and τ is the bounded output time delay, i.e. τ1 and τ2 are the upper and lower bounds of the output time delay, respectively, and v(t) is the measurement noise;

[0064] Step 5.4: Introduce an output time delay interval average prediction term and perform prediction compensation based on the interval discontinuities of the time-varying output time delay to offset the impact of the output time delay; the output with time delay compensation can be expressed as:

[0065] ;

[0066] in, For outputs with time delay compensation, and These are estimates of the historical states at times t−τ1 and t−τ2, respectively.

[0067] Step 5.5: Define the error at the current time as:

[0068] ;

[0069] Design the following superhelical observer structure:

[0070] ;

[0071] Where K1 and K2 are sliding mode gain matrices, and the sign function is... ;

[0072] The corresponding unknown input estimation law is:

[0073] ;

[0074] When the error approaches 0, the estimated value of the lateral force can be obtained as follows:

[0075] .

[0076] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0077] (1) This invention considers the time delay in the measurement output signal of the vehicle system and compensates for the output time delay based on the delay compensation strategy. Combined with the disturbance estimation mechanism of ESO, it effectively improves the estimation accuracy of tire force and the robustness of the vehicle system to external disturbances.

[0078] (2) The present invention constructs a tire force estimation structure based on a superspiral observer, and combines the output time delay interval average prediction term to predict and compensate for the output time delay, so as to ensure the convergence speed while satisfying the real-time estimation.

[0079] (3) The present invention does not rely on expensive tire force sensors in the estimation process, but only requires conventional vehicle state information, and can achieve low-cost acquisition of vehicle tire force. Attached Figure Description

[0080] Figure 1 This is a block diagram of the system structure of the present invention. Detailed Implementation

[0081] The present invention will be further described below with reference to the accompanying drawings:

[0082] like Figure 1 The method shown is a robust estimation method for vehicle tire force considering output time delay, which includes the following steps:

[0083] Step 1: Establish a continuous-time model of the longitudinal force of the tire;

[0084] Step 1.1: By considering the rotational motion of the left front wheel, the rotational dynamics equation of the front-wheel drive vehicle is determined as follows:

[0085] ;

[0086] Among them, I wfl Let ω be the moment of inertia of the left front wheel. fl T represents the rotational speed of the left front wheel. fl F is the torque of the left front wheel. xfl R is the longitudinal force on the left front wheel. fl The radius of the left front wheel;

[0087] Step 1.2: Considering the effects of changes in road conditions and tire relaxation, establish a continuous-time model of the longitudinal force on the left front wheel:

[0088] ;

[0089] Where, the state vector x = [ω fl F xfl ] T T is the matrix transpose, and the control input u = T fl μ is the unmodeled disturbance, A is the system state matrix, and B is the control input matrix;

[0090] , ;

[0091] Step 1.3: Characterize the unmodeled perturbation using the following relationship:

[0092] ;

[0093] Where μ is the unmodeled disturbance, which is a disturbance of known type but unknown magnitude, and A w This is the disturbance type coefficient matrix describing the dynamic behavior of external disturbances, where t is a time parameter.

[0094] Step 2: Consider the output time delay, determine the measurement equation, and compensate for the output time delay in the sensor signal;

[0095] Choosing the rotational speed of the left front wheel as the measurement signal, the corresponding measurement equation can be expressed as:

[0096] ;

[0097] Wherein, the measured output quantity y = ω fl ,C= [1 0].

[0098] Assume that the wheel speed sensor signal has a time-varying and bounded time delay τ during the acquisition and transmission process. k The upper boundary and the lower realm It is known that the formula is... The measurement equation shown can be expressed as:

[0099] ;

[0100] For time-varying and bounded time delays, existing fixed compensation strategies are not applicable. To obtain the measurement value at the current moment, the following delay compensation strategy is used:

[0101] ;

[0102] in, The current compensated measurement value. The compensated state quantity is the output at the current moment, where e refers to the natural constant and θ is the integration variable.

[0103] Step 3: Design an extended state observer that combines system state estimation and disturbance observation. By estimating the system state and its disturbances in real time and feeding them back, the system state and disturbance terms under the influence of output time delay are obtained. The gain matrix of the extended state observer is dynamically adjusted in combination with the output prediction algorithm.

[0104] Pair The state vector in the original system is extended to obtain an augmented state vector that includes the state variables x of the original system and the perturbation μ. The state equation of the corresponding extended state observer can be expressed as:

[0105] (7);

[0106] Where, the augmented matrix of A augmented matrix of B L is the gain matrix of the observer, and its value should be such that A−LC satisfies the stability criterion of discrete systems.

[0107] To determine the stability condition of the extended state observer, the observation error is defined as: ;

[0108] in, ;

[0109] Taking the derivative with respect to e1, we get:

[0110] ;

[0111] Based on the system's state equation and observation equation, the derivative of the error can be further expressed as:

[0112] ;

[0113] To analyze the convergence of error e1, and considering the influence of the system's state error, the Lyapunov functional V is defined as:

[0114] ;

[0115] Where, matrix P ∈ R n×nLet R be a positive definite matrix, where R is the real number field and n represents the matrix order.

[0116] Differentiating the Lyapunov functional V, we get:

[0117] ;

[0118] To ensure system stability, the following conditions must be met:

[0119] ;

[0120] Right now:

[0121] ;

[0122] The observer is stable when the observer gain matrix L satisfies the Lyapunov matrix inequality conditions mentioned above.

[0123] Step 4: Iterate the recursive estimation process based on the extended state observer in Step 3 to obtain the real-time estimation results of the longitudinal force and disturbance of the left front wheel;

[0124] The longitudinal force F of the left front wheel can be obtained. xfl The real-time estimation results of the disturbance are then fed back to the corresponding control inputs by the disturbance estimation results output by the extended state observer, so as to construct a disturbance feedback strategy to eliminate the influence of the disturbance.

[0125] Then obtain the longitudinal force F of the right front wheel xfr The real-time estimation results of the longitudinal forces of the left and right front wheels are vectored together to obtain the resultant longitudinal force F of the left and right front wheels. xf .

[0126] Step 5: Establish a system dynamics model for the front-wheel drive vehicle, considering output time delay. Utilize the longitudinal resultant force of the front wheels obtained in Step 4, and introduce an output time delay interval average prediction term to offset the influence of output time delay. Combined with a super-helical observer, obtain robust estimation results of lateral force in real time.

[0127] Step 5.1: Considering the lateral and yaw motions of the vehicle, determine the system dynamics equations for the front-wheel-drive vehicle:

[0128] ;

[0129] Where m is the mass of the vehicle, a y F is the lateral acceleration of the vehicle. yfl F yfr F yrl F yrrF represents the resultant lateral force of the four wheels (left front, right front, left rear, and right rear) on the tires. xf Based on the longitudinal resultant force of the left and right front wheels obtained in step 4, δ f For the steering angle of the vehicle's front wheels, I z Let be the vehicle's moment of inertia about the z-axis, r be the vehicle's yaw rate, and a and b be the distances from the vehicle's center of mass to the front and rear axles, respectively.

[0130] Step 5.2, the continuous-time model of the vehicle can be represented as:

[0131] ;

[0132] Wherein, the state vector v y Lateral vehicle speed, unknown input Control input φ = F xf w represents process noise, and the input matrix is ​​unknown. Control input matrix ;

[0133] Step 5.3: Select yaw rate as the measurement signal. The measurement equation considering output time delay is as follows:

[0134] ;

[0135] Where the measurement is g = r, the output matrix is ​​R1 = [0 1], and τ is the bounded output time delay, i.e. τ1 and τ2 are the upper and lower bounds of the output time delay, respectively, and v(t) is the measurement noise;

[0136] Step 5.4: Introduce an output time delay interval average prediction term and perform prediction compensation based on the interval discontinuities of the time-varying output time delay to offset the impact of the output time delay; the output with time delay compensation can be expressed as:

[0137] ;

[0138] in, For outputs with time delay compensation, and These are estimates of the historical states at times t−τ1 and t−τ2, respectively.

[0139] Step 5.5: Define the error at the current time as:

[0140] ;

[0141] Design the following superhelical observer structure:

[0142] ;

[0143] Where K1 and K2 are sliding mode gain matrices, and the sign function is... ;

[0144] The corresponding unknown input estimation law is:

[0145] ;

[0146] When the error approaches 0, the estimated value of the lateral force can be obtained as follows:

[0147] .

[0148] This invention first establishes a continuous-time model of tire longitudinal force; then, considering output time delay, it determines the measurement equation and compensates for the output time delay in the sensor signal; next, it designs an extended state observer, combining system state estimation and disturbance observation, and obtains the system state and disturbance terms under the influence of output time delay by estimating the system state and its disturbance in real time and providing feedback, and dynamically adjusts the gain matrix of the extended state observer based on the output prediction algorithm; then, it iterates based on the recursive estimation process of the extended state observer to obtain the real-time estimation results of the longitudinal force and disturbance of the left front wheel; finally, it establishes a system dynamics model of a front-wheel drive vehicle, considering output time delay, and uses the estimated longitudinal resultant force of the front wheel to counteract the influence of output time delay by introducing an output time delay interval average prediction term; combined with a super-spiral observer, it obtains robust estimation results of lateral force in real time.

[0149] This invention is not limited to the above embodiments. Based on the technical solutions disclosed herein, those skilled in the art can make some substitutions and modifications to some of the technical features without creative effort, and all such substitutions and modifications are within the protection scope of this invention.

Claims

1. A robust estimation method for vehicle tire force considering output time delay, characterized in that: Includes the following steps, Step 1: Establish a continuous-time model of the longitudinal force of the tire; Step 2: Consider the output time delay, determine the measurement equation, and compensate for the output time delay in the sensor signal; Step 3: Design an extended state observer that combines system state estimation and disturbance observation. By estimating the system state and its disturbances in real time and feeding them back, the system state and disturbance terms under the influence of output time delay are obtained. The gain matrix of the extended state observer is dynamically adjusted in conjunction with the output prediction algorithm. Step 4: Iterate the recursive estimation process based on the extended state observer in Step 3 to obtain the real-time estimation results of the longitudinal force and disturbance of the left front wheel; Step 5: Establish a system dynamics model for the front-wheel drive vehicle, considering output time delay. Utilize the longitudinal resultant force of the front wheels obtained in Step 4, and introduce an output time delay interval average prediction term to offset the influence of output time delay. Combined with a super-spiral observer, obtain robust estimation results of lateral force in real time.

2. The robust estimation method for vehicle tire force considering output time delay according to claim 1, characterized in that: The specific details of step 1 are as follows: Step 1.1: By considering the rotational motion of the left front wheel, the rotational dynamics equation of the front-wheel drive vehicle is determined as follows: ； Among them, I wfl Let ω be the moment of inertia of the left front wheel. fl T represents the rotational speed of the left front wheel. fl F is the torque of the left front wheel. xfl R is the longitudinal force on the left front wheel. fl The radius of the left front wheel; Step 1.2: Considering the effects of changes in road conditions and tire relaxation, establish a continuous-time model of the longitudinal force on the left front wheel: ； Where, the state vector x = [ω fl F xfl ] T T is the matrix transpose, and the control input u = T fl μ is the unmodeled disturbance, A is the system state matrix, and B is the control input matrix; ， ； Step 1.3: Characterize the unmodeled perturbation using the following relationship: ； Where μ is the unmodeled disturbance, which is a disturbance of known type but unknown magnitude, and A w This is the disturbance type coefficient matrix describing the dynamic behavior of external disturbances, where t is a time parameter.

3. The robust estimation method for vehicle tire force considering output time delay according to claim 2, characterized in that: The specific details of step 2 are as follows: Choosing the rotational speed of the left front wheel as the measurement signal, the corresponding measurement equation can be expressed as: ； Wherein, the measured output quantity y = ω fl C = [1 0]; Assume that the wheel speed sensor signal has a time-varying and bounded time delay τ during the acquisition and transmission process. k The upper boundary and the lower realm It is known that the formula is... The measurement equation shown can be expressed as: ； For time-varying and bounded time delays, existing fixed compensation strategies are not applicable. To obtain the measurement value at the current moment, the following delay compensation strategy is used: ； in, The current compensated measurement value. The compensated state quantity is the output at the current moment, where e refers to the natural constant and θ is the integration variable.

4. A robust estimation method for vehicle tire force considering output time delay according to claim 3, characterized in that: The specific details of step 3 are as follows: Pair The state vector in the original system is extended to obtain an augmented state vector that includes the state variables x of the original system and the perturbation μ. The state equation of the corresponding extended state observer can be expressed as: （7）； Where, the augmented matrix of A augmented matrix of B L is the gain matrix of the observer, and its value should be such that A−LC satisfies the stability criterion of discrete systems.

5. A robust estimation method for vehicle tire force considering output time delay according to claim 4, characterized in that: In step 3, to determine the stability conditions of the extended state observer, the observation error is defined as: ; in, ; Taking the derivative with respect to e1, we get: ； Based on the system's state equation and observation equation, the derivative of the error can be further expressed as: ； To analyze the convergence of error e1, and considering the influence of the system's state error, the Lyapunov functional V is defined as: ； Where, matrix P ∈ R n×n Let R be a positive definite matrix, where R is the real number field and n represents the matrix order. Differentiating the Lyapunov functional V, we get: ； To ensure system stability, the following conditions must be met: ； Right now: ； The observer is stable when the observer gain matrix L satisfies the Lyapunov matrix inequality conditions mentioned above.

6. A robust estimation method for vehicle tire force considering output time delay according to claim 5, characterized in that: The specific details of step 4 are as follows: Based on the recursive estimation process of the extended state observer in step 3, the longitudinal force F of the left front wheel can be obtained. xfl The real-time estimation results of the disturbance are then fed back to the corresponding control inputs by the disturbance estimation results output by the extended state observer, so as to construct a disturbance feedback strategy to eliminate the influence of the disturbance. Then obtain the longitudinal force F of the right front wheel xfr The real-time estimation results of the longitudinal forces of the left and right front wheels are vectored together to obtain the resultant longitudinal force F of the left and right front wheels. xf .

7. A robust estimation method for vehicle tire force considering output time delay according to claim 6, characterized in that: The specific details of step 5 are as follows: Step 5.1: Considering the lateral and yaw motions of the vehicle, determine the system dynamics equations for the front-wheel-drive vehicle: ； Where m is the mass of the vehicle, a y F is the lateral acceleration of the vehicle. yfl F yfr F yrl F yrr F represents the resultant lateral force of the four wheels (left front, right front, left rear, and right rear) on the tires. xf Based on the longitudinal resultant force of the left and right front wheels obtained in step 4, δ f For the steering angle of the vehicle's front wheels, I z Let be the vehicle's moment of inertia about the z-axis, r be the vehicle's yaw rate, and a and b be the distances from the vehicle's center of mass to the front and rear axles, respectively. Step 5.2, the continuous-time model of the vehicle can be represented as: ； Wherein, the state vector v y Lateral vehicle speed, unknown input Control input φ = F xf w represents process noise, and the input matrix is ​​unknown. Control input matrix ; Step 5.3: Select yaw rate as the measurement signal. The measurement equation considering output time delay is as follows: ； Where the measurement is g = r, the output matrix is ​​R1 = [0 1], and τ is the bounded output time delay, i.e. τ1 and τ2 are the upper and lower bounds of the output time delay, respectively, and v(t) is the measurement noise; Step 5.4: Introduce an output time delay interval average prediction term and perform prediction compensation based on the interval discontinuities of the time-varying output time delay to offset the impact of the output time delay; the output with time delay compensation can be expressed as: ； in, For outputs with time delay compensation, and These are estimates of the historical states at times t−τ1 and t−τ2, respectively. Step 5.5: Define the error at the current time as: ； Design the following superhelical observer structure: ； Where K1 and K2 are sliding mode gain matrices, and the sign function is... ; The corresponding unknown input estimation law is: ； When the error approaches 0, the estimated value of the lateral force can be obtained as follows: 。

Images