An Improved Sliding Mode Extension Control Method for Active Front Wheel Steering Based on Extended State Observer

Abstract

This invention discloses an improved sliding mode extension control method for active front wheel steering of vehicles based on an extended state observer, relating to the field of active front wheel steering control for electric vehicles. Based on the current characteristic state of the vehicle, the method determines its location and selects different control strategies, outputting a dynamically adjusted additional front wheel steering angle to achieve real-time control of vehicle stability. Step 1: Establish the spatial state equations of the vehicle's two-degree-of-freedom model and obtain the expected values ​​of yaw rate and sideslip angle. Step 2: Design an integral-exponential fast terminal sliding mode controller based on a fast exponential reaching law. Step 3: Use a second-order extended state observer to perform real-time observation and compensation of nonlinear disturbance terms in the vehicle model. Step 4: Form the improved sliding mode extension controller. This method not only improves model accuracy and anti-interference performance, but also further suppresses chattering in sliding mode control, while enhancing the control effect in the global region. It exhibits high accuracy and strong robustness.

Description

Technical Field

[0001] This invention relates to the field of active front wheel steering control for electric vehicles, and in particular to an improved sliding mode extension control method for active front wheel steering based on an extended state observer. Background Technology

[0002] In emergency situations such as rapid evasive maneuvers, vehicle stability must be effectively guaranteed. Therefore, the demand for vehicle stability control in recent years has led to the development of active front-wheel steering technology. However, previous active front-wheel steering control technologies have not been ideal when the vehicle is in an unstable region. Using a single control algorithm for global active front-wheel steering results in poor control performance in certain areas. Commonly used two-degree-of-freedom vehicle models differ significantly from actual models, failing to fully consider the impact of parameter uncertainties and external disturbances on the vehicle. Furthermore, traditional sliding mode control struggles to effectively suppress chattering while maintaining good robustness and approach rate. Summary of the Invention

[0003] To address the above problems, this invention proposes an improved sliding mode extension control method for active front wheel steering of vehicles based on an extended state observer. This method utilizes the vehicle's current characteristic state (including the yaw rate error e1 and the derivative of the yaw rate error). The system determines the region it is in (including stable region, critical stable region, and unstable region), selects different control strategies, and outputs dynamically adjusted additional front wheel steering angle to achieve real-time control of vehicle stability.

[0004] The technical solution of the present invention includes the following steps:

[0005] Step 1: Establish the spatial state equations of the two-degree-of-freedom vehicle model and obtain the expected values ​​of yaw rate and sideslip angle.

[0006] Step 2: Design an integral-exponential fast terminal sliding mode controller based on the fast exponential approach rate;

[0007] Step 3: Use a second-order extended state observer to observe and compensate for the nonlinear disturbance terms in the vehicle model in real time.

[0008] Step 4: Combine extension theory with the sliding mode control designed in Step 3 to form an improved sliding mode extension controller, which controls the vehicle's yaw rate ω and center of mass sideslip angle β.

[0009] Step 1 is as follows:

[0010] Considering potential modeling errors and external disturbances during vehicle operation, the impact of these factors on the system is summarized into a nonlinear disturbance term D, with an additional front wheel steering angle Δδ input.f To adjust the yaw rate and sideslip angle, the spatial state equations of the two-degree-of-freedom vehicle model considering uncertainties are obtained:

[0011]

[0012] In the formula:

[0013]

[0014] In the formula, ω and β are the yaw rate and the sideslip angle, respectively. A is the system state matrix, containing the element: a 11 a 12 a 21 a 22 B is the input matrix, containing the element: b 11 b 21 K1 and K2 are the lateral stiffness of the front and rear tires, respectively. a and b are the distances from the front and rear axles to the vehicle's center of gravity, respectively. z δ is the moment of inertia of the vehicle's center of mass about the z-axis; m is the total mass of the vehicle; f It is the steering angle of the vehicle's front wheels, and δ f =δ sw / i, δ sw is the steering wheel angle, i is the gear ratio; v is the longitudinal velocity at the vehicle's center of gravity; D1 and D2 are nonlinear disturbance terms, both bounded, satisfying:

[0015]

[0016] The expected values ​​of yaw rate and sideslip angle can be obtained using an ideal linear two-degree-of-freedom model:

[0017]

[0018] In the formula, L is the wheelbase between the front and rear axles of the vehicle; It is the stability factor; μ is the road surface adhesion coefficient.

[0019] Step 2 is as follows:

[0020] Yaw rate is selected as the control variable, and the error term is defined as follows:

[0021] e1=ω-ω d (3)

[0022] The nonlinear IEFTSMC sliding surface s is established as follows:

[0023]

[0024] In the formula, e1(0) represents the initial value of the error variable e1; c3>0, c4>0, 0 <kr <1, q r p r All are odd numbers, and satisfy q r <p r <2q r Ω1 and Ω2 satisfy:

[0025]

[0026] To effectively suppress chattering without affecting its robustness and approach rate, the following fast exponential approach rate is adopted:

[0027]

[0028] In the formula:

[0029]

[0030] In the formula, ε is a positive real number, c5>0 is a positive real number, s0 is the initial value of s, N is a positive integer, α>1 is a positive real number, and p≥2 is a positive integer; it is known that... The value is greater than 0, therefore the stability of the system will not be affected. The influence of the term; at the same time, ε|s0| is introduced to prevent the denominator from being 0 when s approaches the sliding surface;

[0031] By combining equations (1), (4), (6), and (7), we obtain the control law of IEFTSMC:

[0032]

[0033] Where, Δδ f For additional front wheel steering angle.

[0034] Step 3 specifically involves:

[0035] The controller contains a nonlinear disturbance term D2. In practical engineering applications, D2 cannot be directly measured. Therefore, D2 is extended into a new state variable, forming the following second-order system:

[0036]

[0037] In the formula, η(t) is the derivative of the nonlinear disturbance term D2; the following second-order extended state observer can be designed:

[0038]

[0039] In the formula, e2 is the observation error of the ESO on the system state variable ω; z1 is the observation value of the ESO on ω; z2 is the observation value of the ESO on the nonlinear disturbance term D2; β 01 and β 02Here are the error correction coefficients; the expression for the nonlinear function fal(e2,α1,ξ) is shown below:

[0040]

[0041] In the formula, α1 and ξ are the parameters of the nonlinear function, and the observed values ​​of the above formula can be obtained by the extended state observer.

[0042] Combining equations (8) and (10), we obtain the ESO-based IEFTSMC control law:

[0043] Δδ f =(fa 21 β-a 22 ω-z2) / b 21 -δ f

[0044]

[0045] Where, Δδ f For additional front wheel steering angle.

[0046] Step 4 is as follows:

[0047] Step 4.1: Feature extraction;

[0048] The deviation between the ideal yaw rate and the actual yaw rate, and the derivative of the deviation, are selected as characteristic quantities to form the characteristic state.

[0049] Step 4.2: Partitioning of the extension set;

[0050] The horizontal axis represents the selection deviation e1, and the vertical axis represents the derivative of the selection deviation. Choose the vehicle's yaw rate deviation e1 and the allowable range of the deviation derivative as e1, respectively. om and The system's adjustable maximum deviation and the derivative of the maximum deviation are respectively e m and

[0051] Step 4.3: Calculate the degree of correlation;

[0052] Assumption The origin of the characteristic plane is S0(0,0), defined as follows: and Then for any point on the plane Define its correlation function as:

[0053]

[0054] Where \(H(S)\) is the characteristic state correlation function, \(R\) is the classical domain in the graph, \(k_3\) and \(k_4\) are the weighting coefficients of the error and the error change rate respectively;

[0055] Step 4.4, control strategy;

[0056] When \(\{SH(S)\geq0\}\), the corresponding characteristic quantity belongs to the classical domain at this time, and the active front-wheel steering effect is good. Sliding mode control is adopted alone;

[0057] When \(\{S - 1<H(S)\leq0\}\), the corresponding characteristic quantity belongs to the extension domain. In this range, the control performance is usually poor, but it can adjust the characteristic state of the system at this time, change the control output quantity, and thus improve the control performance; The designed controller is:

[0058] \(u_2 = e_1 / k\) c \(-K\) ci \(H(S)\text{sgn}(e_1)\ (14)\)

[0059] Where \(k\) c is the controller gain, \(K\) ci is the control coefficient of the current domain;

[0060] When \(\{SH(S)\leq - 1\}\), the system is working in the unstable region at this time. In order to reduce the time for the vehicle to be in the unstable region, the output amplitude of the controller is taken as the output value of the current controller;

[0061] Denote the output of the above-mentioned sliding mode controller as \(u_1\), then the output of the extended sliding mode controller is:

[0062]

[0063] Where \(\rho\) is the output coefficient.

[0064] Based on the two-degree-of-freedom vehicle dynamics model considering modeling errors and external disturbances, this invention establishes a stability control dynamics model; conducts hierarchical control based on the dynamics model. The upper layer improves the sliding mode control and designs an integral exponential type fast terminal sliding mode controller (IEFTSMC) based on the fast exponential reaching law (FERL); uses a second-order extended state observer (ESO) to observe and compensate the non-linear disturbance terms in the vehicle model in real time; the lower layer combines the above improved sliding mode control with the extension control to form an improved sliding mode extension controller, judges the area where the vehicle is located according to the current characteristic state of the vehicle, and thus selects different control strategies to output the dynamically adjusted additional front wheel angle of the vehicle. This invention not only improves the model accuracy and anti-interference performance, further suppresses the chattering phenomenon of the sliding mode control, but also improves the control effect in the global area. It has high accuracy and strong robustness. Description of the Drawings

[0065] Figure 1 This invention provides an improved sliding mode extension control strategy framework for vehicle active front wheel steering based on an extended state observer.

[0066] Figure 2 The flowchart of the improved sliding mode extension controller of the present invention is shown.

[0067] Figure 3 This is a comparison chart of the yaw rate ω at 108 km / h, on a wet and slippery road surface (road adhesion coefficient of 0.5), and without crosswinds.

[0068] Figure 4 This is a comparison diagram of the centroid sideslip angle β at 108 km / h, on a wet and slippery road surface (road surface adhesion coefficient of 0.5), and without crosswinds, according to the present invention.

[0069] Figure 5 This is an interference observation diagram for the present invention at 108 km / h, on a wet and slippery road surface (road surface adhesion coefficient of 0.5), and without crosswind.

[0070] Figure 6 This is a comparison chart of the yaw rate ω at 108 km / h, on a wet and slippery road surface (road adhesion coefficient of 0.5), and with crosswinds.

[0071] Figure 7 This is a comparison diagram of the centroid sideslip angle β at 108 km / h, on a wet and slippery road surface (road surface adhesion coefficient of 0.5), and with crosswinds.

[0072] Figure 8 This is an interference observation diagram for the present invention at 108 km / h, on a wet and slippery road surface (road surface adhesion coefficient is 0.5), with crosswinds. Detailed Implementation

[0073] To clearly illustrate the technical features of this patent, the following detailed description is provided through specific embodiments and in conjunction with the accompanying drawings.

[0074] like Figure 1 As shown, this invention provides an improved sliding mode extension control method for active front wheel steering of a vehicle based on an extended state observer, comprising the following steps:

[0075] Step 1: Establish the spatial state equations of the two-degree-of-freedom vehicle model and obtain the expected values ​​of yaw rate and sideslip angle.

[0076] Considering potential modeling errors and external disturbances during vehicle operation, the impact of these factors on the system is summarized into a nonlinear disturbance term D, with an additional front wheel steering angle Δδ input. fTo adjust the yaw rate and sideslip angle, the spatial state equations of the two-degree-of-freedom vehicle model considering uncertainties are obtained:

[0077]

[0078] In the formula:

[0079]

[0080] In the formula, ω and β are the yaw rate and the sideslip angle, respectively. A is the system state matrix, containing the element: a 11 a 12 a 21 a 22 B is the input matrix, containing the element: b 11 b 21 K1 and K2 are the lateral stiffness of the front and rear tires, respectively. a and b are the distances from the front and rear axles to the vehicle's center of gravity, respectively. z δ is the moment of inertia of the vehicle's center of mass about the z-axis; m is the total mass of the vehicle; f It is the steering angle of the vehicle's front wheels, and δ f =δ sw / i, δ sw is the steering wheel angle, i is the gear ratio; v is the longitudinal velocity at the vehicle's center of gravity; D1 and D2 are nonlinear disturbance terms, both bounded, satisfying:

[0081]

[0082] The expected values ​​of yaw rate and sideslip angle can be obtained using an ideal linear two-degree-of-freedom model:

[0083]

[0084] In the formula, L is the wheelbase between the front and rear axles of the vehicle; It is the stability factor; μ is the road surface adhesion coefficient.

[0085] Step 2: Design an integral-exponential fast terminal sliding mode controller based on the fast exponential approach rate;

[0086] Yaw rate is selected as the control variable, and the error term is defined as follows:

[0087] e1=ω-ω d (3)

[0088] The nonlinear IEFTSMC sliding surface s is established as follows:

[0089]

[0090] In the formula, e1(0) represents the initial value of the error variable e1; c3>0, c4>0, 0 <k r <1, q r p r All are odd numbers, and satisfy q r <p r <2q r Ω1 and Ω2 satisfy:

[0091]

[0092] To effectively suppress chattering without affecting its robustness and approach rate, the following fast exponential approach rate is adopted:

[0093]

[0094] In the formula:

[0095]

[0096] In the formula, ε is a positive real number, c5>0 is a positive real number, s0 is the initial value of s, N is a positive integer, α>1 is a positive real number, and p≥2 is a positive integer; it is known that... The value is greater than 0, therefore the stability of the system will not be affected. The influence of the term; at the same time, ε|s0| is introduced to prevent the denominator from being 0 when s approaches the sliding surface;

[0097] By combining equations (1), (4), (6), and (7), we obtain the control law of IEFTSMC:

[0098] Δδ f =(fa 21 β-a 22 ω-D2) / b 21 -δ f

[0099]

[0100] Step 3: Use a second-order extended state observer to observe and compensate for the nonlinear disturbance terms in the vehicle model in real time.

[0101] The controller contains a nonlinear disturbance term D2. In practical engineering applications, D2 cannot be directly measured. Therefore, D2 is extended into a new state variable, forming the following second-order system:

[0102]

[0103] In the formula, η(t) is the derivative of the nonlinear disturbance term D2; the following second-order extended state observer can be designed:

[0104]

[0105] In the formula, e2 is the observation error of the ESO on the system state variable ω; z1 is the observation value of the ESO on ω; z2 is the observation value of the ESO on the nonlinear disturbance term D2; β 01 and β 02 Here are the error correction coefficients; the expression for the nonlinear function fal(e2,α1,ξ) is shown below:

[0106]

[0107] In the formula, α1 and ξ are the parameters of the nonlinear function, and the observed values ​​of the above formula can be obtained by the extended state observer.

[0108] Combining equations (8) and (10), we obtain the ESO-based IEFTSMC control law:

[0109] Δδ f =(fa 21 β-a 22 ω-z2) / b 21 -δ f

[0110]

[0111] Step 4: Combine extension theory with the sliding mode control designed in Step 3 to form an improved sliding mode extension controller;

[0112] The establishment of an improved sliding mode extension controller involves the following steps:

[0113] Step 4.1: Feature extraction;

[0114] The deviation between the ideal yaw rate and the actual yaw rate, and the derivative of the deviation, are selected as characteristic quantities to form the characteristic state.

[0115] Step 4.2: Partitioning of the extension set;

[0116] The horizontal axis represents the selection deviation e1, and the vertical axis represents the derivative of the selection deviation. Choose the vehicle's yaw rate deviation e1 and the allowable range of the deviation derivative as e1, respectively. om and The system's adjustable maximum deviation and the derivative of the maximum deviation are respectively e m and

[0117] Step 4.3: Calculate the degree of correlation;

[0118] Assumption The origin of the characteristic plane is S0(0,0), defined as follows: and Then for any point on the plane define its correlation function as:

[0119]

[0120] where \(H(S)\) is the correlation function of the characteristic state, \(R\) is the classical domain in the graph, \(k_3\) and \(k_4\) are the weighting coefficients of the error and the error change rate respectively;

[0121] Step 4.4, control strategy;

[0122] When \(\{SH(S)\geq0\}\), at this time the corresponding characteristic quantity belongs to the classical domain, and the active front-wheel steering effect is good, and the sliding mode control is used alone;

[0123] When \(\{S - 1\lt H(S)\leq0\}\), the corresponding characteristic quantity belongs to the extension domain. In this range, the control performance is usually poor, but it can adjust the characteristic state of the system at this time, change the control output quantity, and thus improve the control performance; the designed controller is:

[0124] \(u_2 = e_1 / k\) c -K ci \(H(S)\text{sgn}(e_1)\ (14)\)

[0125] where \(k\) c is the controller gain, \(K\) ci is the control coefficient of the current domain;

[0126] When \(\{SH(S)\leq - 1\}\), at this time the system works in the unstable region. In order to reduce the time for the vehicle to be in the unstable region, the output amplitude of the controller is taken as the output value of the current controller;

[0127] Denote the output of the above-mentioned sliding mode controller as \(u_1\), then the output of the extended sliding mode controller is:

[0128]

[0129] where \(\rho\) is the output coefficient.

[0130] In order to verify the feasibility and effectiveness of this method, vehicle stability control simulations were carried out using this method on the CarSim / Simulink platform, and the simulation results were compared with the traditional sliding mode control (FTSMC) method. The vehicle parameters are shown in Table 1. The double lane change condition is adopted, the vehicle speed is 108 km / h, and the road surface adhesion coefficient is 0.5. Simulation tests are carried out under the conditions of no side wind and with side wind respectively.

[0131] parameter value mass m 1270kg Front wheelbase a 1.015m Rear wheelbase b 1.895m <![CDATA[Moment of inertia I z > <![CDATA[1536.7kg·m 2 ]]> <![CDATA[Cornering stiffness of the front wheel K1]]> -112600N / rad <![CDATA[Lateral stiffness K2 of the rear wheel]]> -89500N / rad Transmission ratio i 17

[0132] Figure 2The flowchart of the improved sliding mode extension controller of the present invention is shown. Figure 3 This is a comparison chart of the yaw rate at 108 km / h, on a wet and slippery road surface (road adhesion coefficient of 0.5), and without crosswinds. Figure 4 This is a comparison diagram of the center of gravity sideslip angle at 108 km / h, on a wet and slippery road surface (road surface adhesion coefficient of 0.5), and without crosswinds. Figure 5 This is an interference observation diagram for the present invention at 108 km / h, on a wet and slippery road surface (road surface adhesion coefficient of 0.5), and without crosswind. Figure 6 This is a comparison chart of the yaw rate at 108 km / h, on a wet and slippery road surface (road adhesion coefficient of 0.5), and with crosswinds. Figure 7 This is a comparison diagram of the centroid side slip angle at 108 km / h, on a wet and slippery road surface (road adhesion coefficient of 0.5), and with crosswinds. Figure 8 This is an interference observation diagram for the present invention at 108 km / h, on a wet and slippery road surface (road surface adhesion coefficient is 0.5), with crosswinds.

[0133] There are many specific ways to implement this invention. The above description is only a preferred embodiment of this invention. It should be noted that for those skilled in the art, several improvements can be made without departing from the principle of this invention, and these improvements should also be considered within the scope of protection of this invention.

Claims

1. An improved sliding mode extension control method for active front wheel steering of a vehicle based on an extended state observer, characterized in that, It includes the following steps: Step 1: Establish the state - space equation of the vehicle two - degree - of - freedom model and obtain the expected values of the yaw rate and the sideslip angle of the center of mass; Step 2: Design an integral exponential - type fast terminal sliding - mode controller based on the fast exponential reaching law; Step 3: Use a second - order extended state observer to observe and compensate for the non - linear interference term in the vehicle model in real time; Step 4: Combine the extension theory with the sliding - mode control designed in Step 3 to form an improved sliding - mode extension controller to control the vehicle yaw rate ω and the sideslip angle β of the center of mass; Step 2 is specifically as follows: Select the yaw rate as the control variable, and define the error term as: e1=ω-ω d (3) Establish the non - linear IEFTSMC sliding - mode surface s as follows: In the formula, e1(0) represents the initial value of the error variable e1; c3>0, c4>0, 0 <k r <1, q r p r All are odd numbers, and satisfy q r <p r <2q r Ω1 and Ω2 satisfy: In order to effectively suppress the chattering phenomenon while not affecting its robustness and reaching rate, adopt the following fast exponential reaching law: In the formula: In the formula, ε is a positive real number, and k s >0 is a positive real number, s0 is the initial value of s, N is a positive integer, α>1 is a positive real number, p≥2 is a positive integer; it is known that... The value is greater than 0, therefore the stability of the system will not be affected. The influence of the term; at the same time, ε|s0| is introduced to prevent the denominator from being 0 when s approaches the sliding surface; By combining equations (1), (4), (6) and (7), the control law of IEFTSMC is obtained: Dd f =(fa 21 β-a 22 ω-D2) / b 21 -d f Where, Δδ f For additional front wheel steering angle; Step 3 is specifically as follows: The controller contains the non - linear interference term D₂. In actual engineering applications, D₂ cannot be directly measured. Therefore, D₂ is expanded into a new state variable to form the following second - order system: In the formula, η(t) is the derivative of the non - linear interference term D₂; the following second - order extended state observer can be designed: In the formula, e2 is the observation error of the ESO on the system state variable ω; z1 is the observation value of the ESO on ω; z2 is the observation value of the ESO on the nonlinear disturbance term D2; β 01 and β 02 Here are the error correction coefficients; the expression for the nonlinear function fal(e2,α1,ξ) is shown below: In the formula, α₁ and ξ are the parameters of the non - linear function, and the observed value of the above formula can be obtained by the extended state observer; By combining equations (8) and (10), the IEFTSMC control law based on ESO is obtained: Dd f =(fa 21 β-a 22 ω-z2) / b 21 -d f Where, Δδ f For additional front wheel steering angle.

2. The improved sliding mode extension control method for active front wheel steering of a vehicle based on an extended state observer according to claim 1, characterized in that, Step 1 is specifically as follows: Considering potential modeling errors and external disturbances during vehicle operation, the impact of these factors on the system is summarized into a nonlinear disturbance term D, with an additional front wheel steering angle Δδ input. f To adjust the yaw rate and sideslip angle, the spatial state equations of the two-degree-of-freedom vehicle model considering uncertainties are obtained: In the formula: In the formula, ω and β are the yaw rate and the sideslip angle of the center of mass respectively; A is the system state matrix, containing the element: a 11 a 12 a 21 a 22 ; B is the input matrix, containing the element: b 11 b 21 ; K₁ and K₂ are the cornering stiffnesses of the front and rear tires respectively; a and b are the distances from the front and rear axles to the position of the vehicle center of mass respectively; I z It is the moment of inertia of the vehicle's center of mass about the z-axis; m is the total mass of the vehicle; δ f It is the steering angle of the vehicle's front wheels, and δ f =δ sw / i, δ sw is the steering wheel angle, i is the gear ratio; v is the longitudinal velocity at the vehicle's center of gravity; D1 and D2 are nonlinear disturbance terms, both bounded, satisfying: The expected values of the yaw rate and the sideslip angle of the center of mass can be obtained through an ideal linear two - degree - of - freedom model: In the formula, L is the wheelbase between the front and rear axles of the vehicle; It is a stability factor; [[ID=ZZZ]]μ is the road adhesion coefficient.

3. The improved sliding mode extension control method for active front wheel steering of a vehicle based on an extended state observer according to claim 1, characterized in that, Step 4 is specifically as follows: Step 4.1: Feature extraction; Select the deviation between the ideal yaw rate and the actual yaw rate, and the derivative of the deviation as the features, Compositional characteristic state Step 4.2: Extension set partitioning; The horizontal axis represents the selection deviation e1, and the vertical axis represents the derivative of the selection deviation. Choose the vehicle's yaw rate deviation e1 and the allowable range of the deviation derivative as e1, respectively. om and The system's adjustable maximum deviation and the derivative of the maximum deviation are respectively e m and Step 4.3: Correlation degree calculation; Assumption The origin of the characteristic plane is S0(0,0), defined as follows: and Then for any point on the plane Define its correlation function as: In the formula, H(S) is the characteristic state correlation function, and R is the classical domain in the graph. k3 and k4 are the weighting coefficients for the error and the rate of change of the error, respectively; Step 4.4: Control strategy; When {S|H(S)≥0}, the corresponding features at this time belong to the classical domain, and the active front wheel steering effect is good. Adopt sliding - mode control alone; When {S|-1 < H(S)≤0}, the corresponding features belong to the extension domain. In this range, the control performance is usually poor, but it can adjust the characteristic state of the system at this time, change the control output, so as to improve the control performance; design the controller as: u2=e1 / k c -K ci H(S)sgn(e1) (14) In the formula, k c For controller gain, K ci These are the control coefficients for the current domain; When {S|H(S)≤ - 1}, the system works in an unstable region. In order to reduce the time for the vehicle to be in the unstable region, take the output amplitude of the controller as the output value of the current controller; Denote the output of the above - mentioned sliding - mode controller as u₁, then the output of the extension sliding - mode controller is: In the formula, ρ is the output coefficient.

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