Vehicle Longitudinal Following Sliding Mode Control Construction Method

Abstract

The present application relates to a method for constructing vehicle longitudinal following sliding mode control. The method includes: establishing a third-order vehicle dynamics model based on bounded variable parameters; establishing a tracking error model based on a fixed spacing strategy; designing a sliding mode surface and a sliding mode control law based on the third-order vehicle dynamics model; performing motion reachability analysis on the sliding mode surface based on the Lyapunov method and the sliding mode control law to verify whether the current system state is asymptotically reachable along the sliding mode surface, so as to improve the exponential stability of the system; subtracting the sliding mode surfaces of adjacent vehicles based on the Lyapunov method and the tracking error model to obtain the sliding mode motion equations of the inter-vehicle distance error and the speed error, and determining whether the exponent of the sliding mode motion equation of the current system state is stable, so as to improve the exponential chord stability of the system; the above method enables the system to accurately converge to the desired state under complex working conditions and improves the exponential stability of the system.

Description

Technical Field

[0001] This application relates to the field of vehicle longitudinal following control, and particularly to a method for constructing a vehicle longitudinal following sliding mode control. Background Art

[0002] Vehicle longitudinal following control technology is one of the key technologies in modern intelligent transportation systems, aiming to coordinate and optimize the relative positions and speeds between vehicles. Existing vehicle longitudinal following control systems are difficult to ensure the exponential stability of the system under complex working conditions. In other words, the distance and speed between vehicles are difficult to accurately converge to the desired state, which has an adverse impact on the reliability of the vehicle longitudinal following control system. Summary of the Invention

[0003] Based on the above technical problems, this application provides a method for constructing a vehicle longitudinal following sliding mode control, which enables the system to accurately converge to the desired state under complex working conditions and improves the exponential stability of the system.

[0004] This application provides a method for constructing a vehicle longitudinal following sliding mode control, and the method execution includes:

[0005] Establish a vehicle three-order dynamic model based on bounded variable parameters;

[0006] Establish a tracking error model based on a fixed spacing strategy;

[0007] Based on the vehicle three-order dynamic model, design a sliding mode surface and a sliding mode control law;

[0008] Based on the Lyapunov method and the sliding mode control law, perform a motion reachability analysis on the sliding mode surface to verify whether the current system state is asymptotically reachable along the sliding mode surface, so as to improve the exponential stability of the system;

[0009] Based on the Lyapunov method and the tracking error model, subtract the sliding mode surfaces of adjacent vehicles to obtain the sliding mode motion equations of the inter-vehicle distance error and the speed error, and determine whether the exponent of the sliding mode motion equation of the current system state is stable, so as to improve the exponential chord stability of the system.

[0010] The method for constructing a vehicle longitudinal following sliding mode control of this application models the vehicle three-order dynamic model, designs a sliding mode surface and a sliding mode control law to make the system state slide along the sliding mode surface and converge to the desired state, enhancing the robustness of the system; at the same time, the Lyapunov method is used to perform a stability analysis on the system, verify that the system has exponential stability under the action of the sliding mode control law and the exponential chord stability of the sliding mode motion equation, which can improve the exponential stability of the system and ensure that under complex working conditions, the distance and speed between vehicles in the system accurately converge to the desired state.

[0011] Specifically, based on the vehicle three-order dynamic model, designing a sliding mode surface and a sliding mode control law includes:

[0012] Design a sliding mode surface based on the weighted sum of state errors. The formula for the sliding mode surface is:

[0013]

[0014] where \(i = 1,\cdots,N\), \(\Delta\) is the communication time lag. It is assumed that the lag of the \(i\)-th vehicle relative to the \((i - 1)\)-th vehicle and the leading vehicle is equal. \(p_0\) is the longitudinal displacement of the leading vehicle, \(p\) i , \(p\) i-1 are the longitudinal displacements of the \(i\)-th vehicle and the \((i - 1)\)-th vehicle respectively. \(v_0\) is the longitudinal speed of the leading vehicle, \(v\) i , \(v\) i-1 are the longitudinal speeds of the \(i\)-th vehicle and the \((i - 1)\)-th vehicle respectively. \(a_0\) is the longitudinal acceleration of the leading vehicle, \(a\) i , \(a\) i-1 are the longitudinal accelerations of the \(i\)-th vehicle and the \((i - 1)\)-th vehicle respectively. \(L\) i is the length of the \(i\)-th vehicle, and \(q_1\), \(q_2\), \(q_3\), \(q_4\), \(q_5\), \(q_6\) are control parameters.

[0015] Specifically, after designing the sliding mode surface based on the weighted sum of state errors, it includes:

[0016] Based on the vehicle three-order dynamic model, take the derivative of the sliding mode surface to obtain the derivative of the sliding mode surface;

[0017] Based on the derivative of the sliding mode surface, determine the equivalent control law and the switching control law;

[0018] Add the equivalent control law and the switching control law to obtain the sliding mode control law.

[0019] Specifically, based on the Lyapunov method and the sliding mode control law, conduct a motion reachability analysis of the sliding mode surface to verify whether the current system state is asymptotically reachable along the sliding mode surface, including:

[0020] According to the Lyapunov method, take the time derivative of the sliding mode surface to obtain the time derivative of the sliding mode surface; where the formula of the Lyapunov method is: \(S\) i is the sliding mode surface;

[0021] Verify whether the time derivative of the sliding mode surface approaches zero to determine whether the current system state is asymptotically reachable along the sliding mode surface.

[0022] Specifically, based on the Lyapunov method and the tracking error model, subtract the sliding mode surfaces of adjacent vehicles to obtain the sliding mode motion equations of the inter-vehicle distance error and the speed error, and determine whether the exponent of the sliding mode motion equation of the current system state is stable, including:

[0023] Based on the tracking error model, the sliding mode surfaces of adjacent vehicles are subtracted to obtain the sliding mode motion equations of the inter-vehicle distance error and the speed error;

[0024] Verify whether the sliding mode motion equations conform to the preset stability theorem rules to determine whether the exponents of the sliding mode motion equations of the current system state are stable; the preset stability theorem rules are designed based on the Lyapunov method.

[0025] Specifically, the formula of the sliding mode motion equation is:

[0026]

[0027] where \(i = 1,\cdots,N\), \(\Delta\) is the communication time lag, \(e\) 1,i is the position error between the \(i\)-th vehicle and the \((i - 1)\)-th vehicle, \(e\) 2,i is the speed error between the \(i\)-th vehicle and the \((i - 1)\)-th vehicle, \(l_1\), \(l_2\), \(l_3\) and \(l_4\) are the evaluation ratios of the control parameters of the currently selected sliding mode surface.

[0028] Specifically, the formula of the vehicle three-order dynamic model based on the bounded variable parameter is:

[0029]

[0030] where \(i = 1,2,\cdots,N\), \(\Delta_1\) is the fixed system time delay, and it satisfies \([p\) i (s), \(v\) i (s), \(a\) i (s)] T \(=\varphi\) i (s), \(\Delta_1\lt s\leq0\), \(p\) i is the longitudinal driving displacement of the vehicle, \(v\) i is the longitudinal driving speed of the vehicle, \(a\) i is the longitudinal driving acceleration of the vehicle, \(\tau\) i is the time constant of the vehicle, \(u\) i is the throttle / brake input of the vehicle, \(b\) i is the ratio of the air resistance value to the mass value of vehicle \(i\), \(g\) i is the ratio of the vehicle resistance value to the mass value of vehicle \(i\), \(d\) i is the reciprocal of the mass value of vehicle \(i\).

[0031] Specifically, the formula of the tracking error model based on the fixed spacing strategy is:

[0032]

[0033] where \(e\) 1,i is the position error between the \(i\)-th vehicle and the \((i - 1)\)-th vehicle, \(e\)2,i is the speed error between the i-th vehicle and the (i - 1)-th vehicle, e 3,i is the acceleration error between the i-th vehicle and the (i - 1)-th vehicle, p i is the longitudinal driving displacement of the i-th vehicle, p i-1 is the longitudinal driving displacement of the (i - 1)-th vehicle, v i is the longitudinal driving speed of the i-th vehicle, v i-1 is the longitudinal driving speed of the (i - 1)-th vehicle, a i is the longitudinal driving acceleration of the i-th vehicle, a i-1 is the longitudinal driving acceleration of the (i - 1)-th vehicle, d i-1,i is the actual distance between the i-th vehicle and the (i - 1)-th vehicle, d des,i is the ideal distance between the i-th vehicle and the (i - 1)-th vehicle, L i is the length of the i-th vehicle, r i is the ideal distance, adopting a fixed distance strategy, r i is a constant. Description of the Drawings

[0034] Figure 1 is the schematic flow chart of the vehicle longitudinal following sliding mode control construction method according to the embodiment of the present application;

[0035] Figure 2 is the simulation diagram of the leading vehicle action trajectory according to the embodiment of the present application;

[0036] Figure 3 is the simulation diagram of the bounded variable parameter type vehicle position-time characteristic according to the embodiment of the present application;

[0037] Figure 4 is the simulation diagram of the bounded variable parameter type vehicle position error-time characteristic according to the embodiment of the present application;

[0038] Figure 5 is the simulation diagram of the bounded variable parameter type vehicle speed-time characteristic according to the embodiment of the present application;

[0039] Figure 6 is the simulation diagram of the bounded variable parameter type vehicle speed error-time characteristic according to the embodiment of the present application;

[0040] Figure 7 is the simulation diagram of the bounded variable parameter type vehicle acceleration-time characteristic according to the embodiment of the present application;

[0041] Figure 8 is the simulation diagram of the bounded variable parameter type vehicle acceleration error-time characteristic according to the embodiment of the present application;

[0042] Figure 9 is the simulation diagram of the bounded variable parameter type vehicle inter-vehicle distance-time characteristic according to the embodiment of the present application;

[0043] Figure 10 It is the simulation diagram of the time characteristics of the bounded variable parameter type vehicle sliding mode switching function in the embodiment of the present application;

[0044] Figure 11 It is the simulation result diagram of the position error comparison under different sliding mode boundary layers in the embodiment of the present application;

[0045] Figure 12 It is the simulation result diagram of the speed error comparison under different sliding mode boundary layers in the embodiment of the present application;

[0046] Figure 13 It is the simulation result diagram of the acceleration error comparison under different sliding mode boundary layers in the embodiment of the present application;

[0047] Figure 14 It is the simulation result diagram of the sliding mode switching function comparison under different sliding mode boundary layers in the embodiment of the present application. Specific Embodiments

[0048] The following describes the present invention in detail with reference to specific embodiments.

[0049] As Figure 1 shown, the embodiment of the present application proposes a method for constructing a vehicle longitudinal following sliding mode control. The embodiment of the present application realizes the robustness and exponential stability of the vehicle longitudinal following control system under complex working conditions by designing a sliding mode control algorithm that adapts to unknown parameters and time delay effects.

[0050] The specific technical solution of the embodiment of the present application is as follows:

[0051] Step 1: Establish a vehicle three-order dynamic model based on bounded variable parameters.

[0052] First, establish a vehicle three-order dynamic model with system time delay, as shown in the following formula (1):

[0053]

[0054] where i = 1, 2,... N, p i is the longitudinal displacement of the vehicle, v i is the longitudinal speed of the vehicle, a i is the longitudinal acceleration of the vehicle, are the derivatives of p i , v i , a i respectively, τ i is the time constant of the vehicle, u i is the throttle / brake input of the vehicle, Δ1 is the fixed system time delay, and it satisfies [p i (s), v i (s), a i (s)]T = φ i (s), Δ1 < s ≤ 0, m i , c i , f i are respectively the mass, air resistance system, and vehicle resistance of vehicle i, and satisfy the following formula (2):

[0055] m i,min ≤ m i ≤ m i,max , c i,min ≤ c i ≤ c i,max , f i,min ≤ f i ≤ f i,max (2)

[0056] wherein, m i,min , m i,max is the minimum and maximum values that m i can take, c i,min , c i,max is the maximum and minimum values that c i can take, f i,min , f i,max is the maximum and minimum values that f i can take.

[0057] In the above formula (2), the values are as follows in formulas (3)-(6):

[0058]

[0059] In the above formula (3), let:

[0060]

[0061] c i ' = c i - c i,0 , b i ' = b i - b i,0 , d i ' = d i - d i,0 , g i ' = g i - g i,0 (5)

[0062]

[0063] wherein,

[0064] Therefore, a third-order dynamic model of the vehicle with bounded variable parameters and system time delay is established based on formulas (1)-(6), and the third-order dynamic model of the vehicle is as follows in formula (7):

[0065]

[0066] where \(i = 1, 2,\cdots,N\), \(\Delta_1\) is the fixed system time delay, and it satisfies \([p i (s), v i (s), a i (s)] T =\varphi i (s), \Delta_1 < s\leq0\), \(p i is the longitudinal driving displacement of the vehicle, \(v i is the longitudinal driving speed of the vehicle, \(a i is the longitudinal driving acceleration of the vehicle, \(\tau i is the time constant of the vehicle, \(u i is the throttle / brake input of the vehicle, \(b i is the ratio of the air resistance value to the mass value of vehicle \(i\), \(g i is the ratio of the vehicle resistance value to the mass value of vehicle \(i\), \(d i is the reciprocal of the mass value of vehicle \(i\).

[0067] Step 2: Establish a tracking error model based on the fixed-spacing strategy.

[0068] Establish a tracking error model using the fixed-spacing strategy, as follows in formula (8):

[0069]

[0070] where \(e 1,i is the position error between the \(i\)-th vehicle and the \((i - 1)\)-th vehicle, \(e 2,i is the speed error between the \(i\)-th vehicle and the \((i - 1)\)-th vehicle, \(e 3,i is the acceleration error between the \(i\)-th vehicle and the \((i - 1)\)-th vehicle. \(p i , \(p i-1 are the longitudinal driving displacements of the \(i\)-th vehicle and the \((i - 1)\)-th vehicle respectively, \(v i , \(v i-1 are the longitudinal driving speeds of the \(i\)-th vehicle and the \((i - 1)\)-th vehicle respectively, \(a i , \(a i-1 are the longitudinal driving accelerations of the \(i\)-th vehicle and the \((i - 1)\)-th vehicle respectively, \(d i-1,i is the actual spacing between the \(i\)-th vehicle and the \((i - 1)\)-th vehicle, \(d des,i is the ideal spacing between the \(i\)-th vehicle and the \((i - 1)\)-th vehicle, \(L i is the length of the \(i\)-th vehicle, \(r iThe ideal spacing is adopted with a fixed spacing strategy, which is a constant here.

[0071] Step 3: Based on the vehicle's third-order dynamic model, design a sliding mode surface and a sliding mode control law.

[0072] To achieve the exponential stability of the system, the embodiments of the present application design a sliding mode surface and a sliding mode control law so that the system state can slide along the sliding mode surface and finally converge to the desired state.

[0073] The sliding mode surface S i can be designed as the weighted sum of the state errors. The sliding mode surface S i is as shown in the following formula (9):

[0074]

[0075] where Δ in the formula is the communication time lag. Here, it is assumed that the lag of the i-th vehicle relative to the (i - 1)-th vehicle and the leading vehicle is equal. p0 is the longitudinal driving displacement of the leading vehicle, p i , p i-1 are the longitudinal driving displacements of the i-th vehicle and the (i - 1)-th vehicle respectively. v0 is the longitudinal driving speed of the leading vehicle, v i , v i-1 are the longitudinal driving speeds of the i-th vehicle and the (i - 1)-th vehicle respectively. a0 is the longitudinal driving acceleration of the leading vehicle, a i , a i-1 are the longitudinal driving accelerations of the i-th vehicle and the (i - 1)-th vehicle respectively. L i is the length of the i-th vehicle. q1, q2, q3, q4, q5, q6 are undetermined control parameters.

[0076] Combining formulas (1)-(9), take the derivative of the sliding mode surface S i to obtain the derivative S i of the sliding mode surface, as shown in the following formula (10):

[0077]

[0078] where the meanings of the parameters in the formula are as shown in formulas (1)-(9).

[0079] Design the sliding mode control law. First, obtain the equivalent control law and the switching control law of the sliding mode control, and then use the equivalent control law and the switching control law to ensure the sliding mode reachability of the system.

[0080] In the above formula (10), let the equivalent control law be as shown in the following formula (11):

[0081] u i,equ = u i,0 / d i,0 (11)

[0082] Among them,

[0083]

[0084] wherein, the meanings of the parameters in the formula are shown in Formulas (1)-(9).

[0085] Meanwhile, to ensure the reachability of the sliding mode, the switching control law takes the following value in Formula (13):

[0086]

[0087] wherein, λ>0 is the control parameter, and other parameters are shown in Formulas (1)-(6), and in the formula,

[0088]

[0089] Thus, by adding the equivalent control law and the switching control law, the sliding mode control law is obtained as follows in Formula (14):

[0090] u i = u i,equ + u i,N (14)

[0091] For the sliding mode surface and the sliding mode control law obtained in the above steps, the embodiments of the present application further verify whether the sliding mode of the dynamic system (as shown in the above Formula 7) is asymptotically reachable under the sliding mode control law (as shown in the above Formula 14), and verify whether the sliding mode motion equation (as shown in the following Formula 29) obtained through the sliding mode surface conforms to the rules of the preset stability theorem, and prove whether the obtained sliding mode motion equation is exponentially stable. If the sliding mode motion equation conforms to the rules of the preset stability theorem, it is proved that the included spacing error and speed error are also exponentially stable.

[0092] It should be noted that the rules of the preset stability theorem in the embodiments of the present application are to first establish an error state equation of a relatively general time-delay vehicle system, and then establish a stability theorem and a proof process for this state equation. In this way, the stability of the subsequent sliding mode motion equation can be directly determined according to the stability theorem.

[0093] The following is an illustration of the design of the rules of the preset stability theorem.

[0094] Establish an error state equation of the longitudinal following control system of a fixed-time-delay vehicle, which is represented by the following differential equation (15):

[0095]

[0096] wherein, x i ∈R n represents the longitudinal following error vector of the i-th vehicle, A, B, C ∈ R n×n , xT and A T respectively denote the transposes of the vector x and the matrix A. Δ ∈ [0, Δ d represents the time delay of the workshop communication. Assume that the initial condition of the system (Equation 15) is x i (s) = φ i (s), Δ d ≤ s ≤ 0, φ i is bounded and continuous on the interval [-Δ d , 0]. Meanwhile, assume that the system (15) satisfies the existence and uniqueness conditions of the equilibrium point. We adopt the following notations: λ(A) represents the eigenvalue of the matrix A, λ max (A) represents the largest eigenvalue of the matrix A, ||x|| and ||A|| are the 2-norms of the vector x and the matrix A respectively, ||A|| = ({λ: λ is the largest eigenvalue of A T A}) 1 / 2 . The partial derivative of the function v i with respect to x i , where x k,i is the k-th term of x i .

[0097] Therefore, the isolated subsystems of the system are:

[0098]

[0099] Introduce Definition 1, Definition 2 and Lemma 1 as follows:

[0100] Definition 1: The equilibrium point of the system (Equation 15) is chord stable if for any given ε > 0, there exists δ > 0 such that

[0101] Definition 2: The equilibrium point of the system (Equation 15) is exponentially chord stable if the origin is chord stable and there exist constants λ > 0 and M > 0 for all t ≥ 0, satisfying sup i ||x i (t)|| ≤ M||Φ i (s)||e -λt , where

[0102] Lemma 1: Assume and satisfy the differential inequality (17):

[0103] In the above formula, τ1,..., τ i are time delays. For v i (t) ≥ 0, there is g i (·) > 0, γi0 > 0, γ ij ≥ 0.

[0104] If there exists v = (v 1,s , v 2,s ,..., v i,s ) such that:

[0105]

[0106] and inf i {v 10} = α > 0, sup i {v 10} = β > 0, then for any given ε > 0, there exists δ > 0 such that

[0107] Prove as follows using Definition 1, Definition 2, and Lemma 1:

[0108] Take the set:

[0109] γ = {z(l): z i = lv i,s , l > 0, i = 1, 2,..., N}, Ω(z) = {u: 0 ≤ u ≤ z, z ∈ γ}, for any given ε > 0, there exists l0 > 0 satisfying ε = sup i {l0v i,s}, take δ = inf i {l0v i,s}. At the same time, define O = {u: u i = l0v i,s , u j ≤ l0v j,s , j ≠ i, i, j = 1, 2,..., N}, it is easy to obtain that If sup i |v i (s)| ∞ < δ, it is obtained that v i (s) ∈ Ω(z(l0)), and

[0110] Next, prove by contradiction that when t > 0, sup i |v i (t)| ∞ < ε. Assume that there exists a certain moment t0 > 0 such that sup i |v i (t)| ∞ = ε, then there exists a certain moment t1 ∈ [0, t0] and a certain i such that v i (t1) ∈ O, that is, v i (t1) = l0v i,0 , vj (t1) ≤ l0v j,0 For t ∈ [0, t1], j = 1, 2, ..., N, j ≠ i. This means that at t = t1, there is However, from equations (17), (18) and g(·) > 0, γ i0 > 0, γ ij ≥ 0; it is deduced that:

[0111]

[0112] This is in contradiction with So the assumption is not valid, that is, there does not exist a certain moment t0 > 0 such that sup i |v i (t0)| ∞ = ε. Therefore, there is sup i |v i (t)| ∞ < ε. Then for any ε > 0, there exists δ > 0 such that when sup i |v i (s)| ∞ < δ, there is sup i |v i (t)| ∞ < ε.

[0113] According to Theorem 1: If the system (15) satisfies the following conditions (i), (ii) and (iii), then its zero solution is exponentially string stable.

[0114] (i) 0 < ||C|| < 1;

[0115] (ii) For the isolated subsystem (Equation 16), there exists a Lyapunov function v i and constants β l > 0, β h > 0, β p > 0, β q > 0, such that: β l ||x i || 2 ≤ v i ≤ β h ||x i || 2 ,

[0116] Thus, it is proved that: From Equation (15), we have:

[0117]

[0118] where Δ is the communication time lag.

[0119] Let: w i (t) = e ξt v i (t) (20)

[0120] where ξ is an arbitrarily small positive number, and the above formula (20) is an auxiliary function for exponential stability proof.

[0121] Differentiate formula (20) along the system (formula 15), we have:

[0122]

[0123] According to condition (ii) of Theorem 1 above, the formula can be written as:

[0124]

[0125] According to formula (20), we have:

[0126]

[0127] According to the condition it is known that there exist ξ > 0 and e ξ(i-j-1)τ) / 2 w j (t - (i - j - 1)τ) > 0 (j = 1,..., i - 1) such that

[0128]

[0129] Therefore, formula (15) is as follows:

[0130]

[0131] According to Lemma 1 and formula (25) above, for any ε o > 0, there exists δ0 > 0 such that Then the system (15) is chord stable.

[0132] From sup i ||w i (s)|| ∞ < ε0 and α l > 0, we get

[0133]

[0134] Thus, according to Definition 2, it can be obtained that the system (15) is exponentially chord stable.

[0135] Step 4: Based on the Lyapunov method and the sliding mode control law, conduct a motion reachability analysis of the sliding mode surface to verify whether the current system state is asymptotically reachable along the sliding mode surface, so as to improve the exponential stability of the system.

[0136] Using the above preset stability theorem rules, it is proved that under the sliding mode control law (Equation 14), the sliding mode of the dynamic system (Equation 7) is asymptotically reachable.

[0137] According to the Lyapunov method, the time derivative of the sliding mode surface is obtained by taking the time derivative of the sliding mode surface; among them, the formula of the Lyapunov method is: S i is the sliding mode surface; specifically, take the Lyapunov function The derivative along the sliding mode surface (Equation 10) is differentiated with respect to time to obtain the following formula (27):

[0138]

[0139] In the formula, the parameter meanings are shown in (1)-(7), (9).

[0140] When selecting the sliding mode control, it is necessary to ensure that the selected sliding mode surface is reachable. In short, it is proved that the selected sliding mode surface is available, that is, the sliding mode surface can reach 0. From the above formula (27), it can be seen that as long as q3 + q6 > 0 and S i ≠0, That is, when t → ∞, there is S i →0. At this time, the sliding mode of the system is asymptotically reachable. In other words, verifying whether the time derivative of the sliding mode surface approaches zero can determine whether the current system state is asymptotically reachable along the sliding mode surface.

[0141] Step 5: Based on the Lyapunov method and the tracking error model, subtract the sliding mode surfaces of adjacent vehicles to obtain the sliding mode motion equations of the inter-vehicle distance error and the speed error, and determine whether the exponent of the sliding mode motion equation of the current system state is stable to improve the exponential string stability of the system.

[0142] Next, analyze the stability of the inter-vehicle distance error and the speed error on the sliding mode plane. Determine the control parameters q1~q6 set in formula (9). When t → ∞, e 1,i →0, e 2,i →0, e 3,i →0. From S i =0, i = 1,..., N, there is:

[0143]

[0144] Let Define e 1,i =0, e 2,i =0, and the sliding mode motion equations of the inter-vehicle distance error and the speed error are obtained as:

[0145]

[0146] Among them,

[0147]

[0148] and, i = 1, ..., N, Δ is the communication time lag, e 1,i is the position error between the i-th vehicle and the (i - 1)-th vehicle, e 2,i is the speed error between the i-th vehicle and the (i - 1)-th vehicle. Briefly speaking, l1, l2, l3 and l4 are the evaluation ratios of the control parameters of the currently selected sliding mode surface.

[0149] Combined with the error state equation (Formula 15) of the fixed-delay vehicle longitudinal following system above, it can be known that the sliding mode motion equation (Formula 29) is a special case thereof, that is, the time-delay vehicle longitudinal following system applied in the embodiments of the present application is a special case of the fixed-delay vehicle longitudinal following system. The exponential chord stability of the system can be determined according to the above preset stability theorem rules. The sliding mode motion equation (Formula 29) obtained through the sliding mode surface includes the inter-vehicle distance error and the speed error. Then, according to Theorem 1 proposed by the above preset stability theorem rules, if the conditions (i), (ii), and (iii) are satisfied, the obtained sliding mode motion equation (Formula 29) is exponentially stable, and the included distance error and speed error are also exponentially stable.

[0150] Combined with the above steps 1-5, the vehicle longitudinal following sliding mode control construction method of the embodiments of the present application can effectively cope with the changes of bounded unknown parameters through system modeling and sliding mode surface design in terms of enhanced robustness, ensuring that the system can still maintain stable control performance under parameter uncertainty. The design of the sliding mode control law further enhances the adaptability of the system to parameter changes by adjusting the control gain and the equivalent control term. In the aspect of optimizing the time-delay effect in the embodiments of the present application, the vector Lyapunov method is used for stability analysis to ensure that the system can achieve exponential stability in the presence of time delay. This not only solves the problems of control lag and oscillation caused by time delay, but also improves the response speed and accuracy of the system. In the aspect of ensuring exponential stability in the embodiments of the present application, through the sliding mode control law and stability analysis, the exponential stability of the system can be ensured under complex working conditions, which is crucial for the safety and reliability of the vehicle control system. Verifying the exponential stability and exponential chord stability ensures that the distance and speed between vehicles can quickly and accurately reach the desired state.

[0151] Based on the above-established dynamic model, tracking error model, and the designed sliding mode control system, a simulation test of the vehicle longitudinal following control system is carried out. By designing the controller parameters through stability analysis, the time response diagrams of the position, speed, acceleration of each vehicle in the vehicle, and the inter-vehicle distance from the preceding vehicle under different parameters can be obtained.

[0152] Specifically, a simulation experiment was conducted on the constructed sliding mode control system (i.e., Equation 29). The longitudinal vehicle following control system used in the simulation consisted of 5 vehicles, 1 leading vehicle (Vehicle 0) and 4 following vehicles (Vehicles 1 - 4). The vehicle model parameters are shown in Table 1 below, the vehicle model control parameters are shown in Table 2, the vehicle control parameters are shown in Table 3, and the vehicle initial conditions are shown in Table 4. The trajectory of the leading vehicle is as Figure 1 shown, where a) is the acceleration graph and b) is the speed graph. The simulation duration was 30 s and the simulation step size was 0.01 s.

[0153] Table 1 Definition of Vehicle Model Parameters

[0154]

[0155] Table 2 Vehicle Model Control Parameters

[0156]

[0157]

[0158] Table 3 Vehicle Control Parameters

[0159]

[0160] Table 4 Vehicle Initial Conditions

[0161]

[0162] According to Table 3, the sliding mode control system (i.e., Equation 29) can be written as the following Equation (31) at this time:

[0163]

[0164] Therefore, there is λ max (A) = -3, ||C|| = 1 / 3. At this time, Condition (i) in Theorem 1 is satisfied. Take the Lyapunov function Let α l = α h = 1, α p = -λ max (A) = 3, α q = 2. At this time, Condition (ii) in Theorem 1 is satisfied. ||B + CA|| = 3Δ, then If Then there is Satisfying Condition (iii) in Theorem 1, the system (31) is exponentially chord - stable.

[0165] The simulation results are as Figures 3 to 10 shown. Among them, the dotted line represents the leading vehicle and the solid line represents the following vehicles.Figure 3 It is a time characteristic diagram of vehicle position with bounded variable parameters, Figure 4 It is a time characteristic diagram of vehicle position error with bounded variable parameters, Figure 5 It is a time characteristic diagram of vehicle speed with bounded variable parameters, Figure 6 It is a time characteristic diagram of vehicle speed error with bounded variable parameters, Figure 7 It is a time characteristic diagram of vehicle acceleration with bounded variable parameters, Figure 8 It is a time characteristic diagram of vehicle acceleration error with bounded variable parameters, Figure 9 It is a time characteristic diagram of vehicle headway with bounded variable parameters, Figure 10 It is a time characteristic diagram of the sliding mode switching function of the vehicle with bounded variable parameters (i.e., the sliding mode diagram). From Figure 5 and Figure 7 it can be seen that the speed and acceleration of the following vehicle can quickly approach the speed of the leading vehicle. At the same time, from Figure 4 , Figure 6 and Figure 8 it can be seen that the position error, speed error and acceleration error of the vehicle in the platoon have a relatively fast convergence speed. From Figure 9 it can be seen that the headway can quickly reach the ideal headway. From Figure 10 it can be seen that the sliding mode can quickly tend to 0. At the same time, according to Figures 3 to 8 it can be seen that this method can stabilize the vehicle when the initial error is not zero.

[0166] Due to the on-demand design characteristics of the sliding mode, the system's sliding mode motion is independent of the parameter changes of the controlled object and external disturbances of the system. The robustness of the sliding mode variable structure control system is stronger than that of general conventional continuous systems. However, in essence, the discontinuous switching characteristics will cause chattering in the system. In practice, chattering is inevitable and cannot be eliminated. Therefore, it can only be weakened to a certain extent. To achieve the purpose of modifying the control term to weaken chattering, the concepts of "quasi-sliding mode" and "boundary layer" are used. The quasi-sliding mode surface refers to the mode in which the system's motion trajectory is restricted within the neighborhood of a certain boundary layer of the ideal sliding mode. In terms of the phase trajectory, the control with the ideal sliding mode is to attract the state points within a certain range to the switching surface, while the quasi-sliding mode control is to attract the state points within a certain range to the neighborhood of a certain boundary layer of the switching surface. Within the boundary layer, the sliding mode does not require to satisfy the existence conditions of the sliding mode. Therefore, the quasi-sliding mode does not require the switching of the control structure on the switching surface. It can be a control system that performs structure transformation on the boundary layer or a continuous state feedback control system that does not perform structure transformation at all. This difference in the implementation of the quasi-sliding mode control fundamentally avoids or weakens chattering, so it has been widely used in practice. There are two common types of quasi-sliding mode control: replacing the sign function in the ideal sliding mode with a saturation function, making the relay characteristic continuous, and replacing the sign function with a continuous function. In this section of the control, the saturation function (sat(S i )) is used to replace the sign function (sgn(S i ))), that is:

[0167]

[0168] where Δ is the boundary layer.

[0169] The simulation results are as shown in Figures 11 to 14 . Among them, Figure 11 a), Figure 12 a) and Figure 13 a) are the simulation results of the position, velocity, and acceleration errors when the boundary layer thickness is 0.2 respectively. Figure 11 b), Figure 12 b) and Figure 13 b) are the simulation results of the position, velocity, and acceleration errors when the boundary layer thickness is 0.4 respectively. By comparing Figures 3 to 8 and Figures 11 to 14 , it can be seen that when the quasi-sliding mode control is adopted, the chattering can be effectively weakened. The larger the value of the boundary layer thickness, the smaller the chattering, the smaller the control gain, and the worse the control effect; the smaller the value of the boundary layer thickness, the larger the chattering, the larger the control gain, and the better the control effect.

[0170] Therefore, the sliding mode control algorithm designed by the vehicle longitudinal following sliding mode control construction method of the embodiments of the present application has good effects in adapting to unknown parameters and optimizing time-delay effects.

[0171] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than limiting the protection scope of the present invention. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that the technical solutions of the present invention can be modified or equivalently replaced without departing from the essence and scope of the technical solutions of the present invention.

Claims

1. A method for constructing a vehicle longitudinal following sliding mode control, characterized in that: The method execution comprises: Establish a third-order vehicle dynamics model based on bounded variable parameters; Establish a tracking error model based on a fixed spacing strategy; Design sliding surface and sliding mode control law based on the third-order dynamics model of the vehicle; Based on the Lyapunov method and sliding mode control law, the kinematic reachability analysis of the sliding surface is performed to verify whether the current system state is asymptotically reachable along the sliding surface, so as to improve the exponential stability of the system. Based on the Lyapunov method and tracking error model, the sliding surfaces of adjacent vehicles are subtracted to obtain the sliding mode motion equation of the inter-vehicle distance error and speed error, and determine whether the exponent of the sliding mode motion equation of the current system state is stable, so as to improve the exponential chord stability of the system; Among them, the error state equation of the system verifying the exponential chord stability is , The following conditions must be met: (i) ； (ii) For isolated subsystems of the current system , , there exists a Lyapunov function and constant ,constant ,constant ,constant , so that: , , ; (iii) ; In the formula, Indicates The longitudinal following error vector of the vehicle, , , , , , , and d is the evaluation ratio of the control parameters of the currently selected sliding surface, is the Lyapunov function about The partial derivative of , Δ represents the communication time lag between vehicles.

2. The method according to claim 1, characterized in that Based on the third-order dynamics model of the vehicle, the sliding surface and sliding mode control law are designed, including: Design a sliding surface based on the weighted sum of state errors. The sliding surface formula is: ； in, , is the communication time lag, assuming that The car is relative to The lag of the leading vehicle is equal to that of the leading vehicle. is the longitudinal displacement of the lead vehicle, , Respectively Car and The longitudinal displacement of the vehicle, is the longitudinal speed of the lead vehicle, , Respectively Car and The longitudinal speed of the vehicle, is the longitudinal acceleration of the lead vehicle, , Respectively Car and The longitudinal acceleration of the vehicle, For the The length of the vehicle, , , , , , is the control parameter.

3. The method according to claim 2, characterized in that After designing the sliding surface based on the weighted sum of state errors, it includes: Based on the third-order dynamics model of the vehicle, the sliding surface is differentiated to obtain the derivative of the sliding surface; Based on the derivative of the sliding surface, the equivalent control law and the switching control law are determined; The sliding mode control law is obtained by adding the equivalent control law and the switching control law.

4. The method according to claim 1, characterized in that: Based on the Lyapunov method and sliding mode control law, the motion reachability analysis of the sliding surface is performed to verify whether the current system state is asymptotically reachable along the sliding surface, including: According to the Lyapunov method, the time derivative of the sliding surface is processed to obtain the time derivative of the sliding surface; the formula of the Lyapunov method is: , is the sliding surface; Verify whether the time derivative of the sliding surface approaches zero to determine whether the current system state is asymptotically reachable along the sliding surface.

5. The method according to claim 1, characterized in that Based on the Lyapunov method and tracking error model, the sliding surfaces of adjacent vehicles are subtracted to obtain the sliding mode motion equations of the inter-vehicle distance error and speed error, and determine whether the exponent of the sliding mode motion equation of the current system state is stable, including: Based on the tracking error model, the sliding mode surfaces of adjacent vehicles are subtracted to obtain the sliding mode motion equations of the inter-vehicle distance error and speed error. Verify whether the sliding mode motion equation complies with the preset stability theorem rules to determine whether the exponent of the sliding mode motion equation of the current system state is stable; the preset stability theorem rules are designed based on the Lyapunov method.

6. The method according to claim 1, characterized in that The equation of motion for sliding mode is: ； in, , is the communication time lag, For the Car and The position error of the vehicle, For the Car and The speed error of the vehicle, , , , , , and It is the evaluation ratio of the control parameters of the currently selected sliding surface.

7. The method according to claim 1, characterized in that The formula of the vehicle third-order dynamics model based on bounded variable parameters is: ； in, , is a fixed system time delay, and satisfies , , is the longitudinal displacement of the vehicle, is the longitudinal speed of the vehicle, is the vehicle longitudinal acceleration, is the time constant of the vehicle, is the vehicle's throttle / brake input, For vehicles i The ratio of air resistance value to mass value, For vehicles i The ratio of vehicle resistance value to mass value, For vehicles i The reciprocal of the quality value.

8. The method according to claim 1, characterized in that The formula for the tracking error model based on the fixed spacing strategy is: ； in, For the Car and The position error of the vehicle, For the Car and The speed error of the vehicle, For the Car and The acceleration error of the vehicle, For the The longitudinal displacement of the vehicle, For the The longitudinal displacement of the vehicle, For the The longitudinal speed of the vehicle, For the The longitudinal speed of the vehicle, For the The longitudinal acceleration of the vehicle, For the The longitudinal acceleration of the vehicle, For the Car and The actual distance between vehicles, For the Car and The ideal distance between vehicles, For the The length of the vehicle, For the ideal spacing, a fixed spacing strategy is adopted, which is a constant.

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