Fault-tolerant control system and method for adaptive platooning of heterogeneous vehicles with safety constraints

Abstract

The present invention provides a heterogeneous vehicle adaptive formation fault-tolerant control system and method with safety constraints. The displacement, velocity and acceleration output in real time in the heterogeneous vehicle formation are obtained through a third-order heterogeneous vehicle formation structure module. The displacement trajectory tracking error of two adjacent vehicles in the vehicle formation is kept within a specified area by a specified performance spacing error module. The designed coupling sliding surface module ensures the string stability of the vehicle queue and solves the relationship between the two adjacent vehicles. The fuzzy adaptive law parameters are obtained through a fuzzy adaptive law module. When the fault diagnosis module detects that an electromagnetic actuator has a fault, the influence of the fault on the heterogeneous vehicle formation is solved in the actuator fault fault-tolerant control module. Regardless of whether the electromagnetic actuator fails, the control method ensures that all signals of the vehicle formation are bounded and the spacing error does not exceed its performance boundary. At the same time, based on the coupling sliding surface, the proposed control algorithm can effectively ensure the string stability of the queue.

Description

Technical Field

[0001] The present invention relates to the technical field of vehicle platooning, and in particular to a heterogeneous vehicle adaptive platooning fault-tolerant control system and method with safety constraints. Background Art

[0002] In recent years, with the rapid and healthy development of the automobile industry, vehicle platooning has become one of the hot issues in the current automobile research field. Vehicle platooning can not only achieve shorter vehicle spacing and increase traffic capacity, thereby improving the stability of single vehicles and platoons when driving on the road, but also effectively reduce energy loss and reduce environmental pollution. Therefore, designing control methods for vehicle platooning to improve the overall performance of vehicle platoons has aroused great interest and attention from researchers.

[0003] At present, although many control technologies have been applied to the design of vehicle platoons. However, the existing technologies still have the following problems: in the control of heterogeneous vehicle platoons, although the existing technologies can increase the traffic capacity and improve the safety of vehicle platoons to a certain extent, they do not consider the maximum feasible acceleration and minimum safe distance of vehicles when driving, so that the secondary spacing error between two adjacent vehicles in the vehicle queue cannot be kept within the specified area. At the same time, it does not consider the impact of unexpected actuator failures caused by factors such as overvoltage in the traction transformer, overcurrent in the traction converter, and overheating in the asynchronous motor on the control design of heterogeneous vehicle platoons in actual operation when the vehicle queue error does not exceed its safety constraints. Summary of the invention

[0004] The present invention provides a heterogeneous vehicle adaptive formation fault-tolerant control system and method with safety constraints to overcome the above technical problems.

[0005] In order to achieve the above object, the technical solution of the present invention is:

[0006] A heterogeneous vehicle adaptive platoon fault-tolerant control system with safety constraints, comprising a three-order heterogeneous vehicle platoon structure module, an actuator fault diagnosis module, a fuzzy adaptive law module, a specified performance spacing error module, a coupled sliding surface module, and an actuator fault fault-tolerant control module;

[0007] The three-order heterogeneous vehicle formation structure module is used to obtain the control output force u of the electromagnetic actuator i (t), and according to the control output force u of the electromagnetic actuator i (t) Adjust the driving displacement s of the three heterogeneous vehicle formations i , speed v i and acceleration a i (i=1,2,3);

[0008] The actuator fault diagnosis module is used to obtain the failure rate ρ of the electromagnetic actuator i (t) Bias fault with electromagnetic actuator The failure rate of the electromagnetic actuator ρ i (t) and bias fault Send to the actuator fault tolerance control module;

[0009] The fuzzy adaptive law module is used to obtain the vehicle speed v of the third-order heterogeneous vehicle formation structure module. i and vehicle acceleration a i , and combined with the road slope δ i The rolling resistance and air resistance caused by the calculation of the estimated values ​​of the adaptive parameters of the fuzzy control

[0010] The specified performance spacing error module obtains the driving displacement s of the three heterogeneous vehicle formations. i , and according to the driving displacement s of the three heterogeneous vehicle formations i And the designed performance function h i (t), calculate the tracking error variable ξ with the specified quadratic spacing i (t);

[0011] The designed coupled sliding surface module obtains the specified quadratic spacing tracking error variable ξ i (t), and according to the specified quadratic spacing tracking error variable ξ i (t) Establish the coupled sliding surface ∏ i (t);

[0012] The actuator fault tolerance control module is used to obtain the failure rate ρ of the electromagnetic actuator i (t), bias fault Estimation of Adaptive Parameters in Fuzzy Control And the coupled sliding surface ∏ i (t), and according to the failure rate ρ of the electromagnetic actuator i (t), bias fault of electromagnetic actuator Estimation of Adaptive Parameters in Fuzzy Control And the coupled sliding surface ∏ i (t) Calculate the control output force u of the electromagnetic actuator i (t).

[0013] Furthermore, the three-order heterogeneous vehicle formation structure module includes a vehicle longitudinal dynamics model, a three-order physical model of "displacement-velocity-force", a dynamic model of the leading vehicle in the vehicle formation, and a quadratic spacing error model:

[0014] The vehicle longitudinal dynamics model of the three-order heterogeneous vehicle formation structure module is:

[0015]

[0016] Among them, m i and a i (t) are the mass and acceleration of the i-th vehicle respectively; F i (t) is the actual driving force or braking force of the engine; l i is the known vehicle mechanical efficiency; R i is the known vehicle tire radius; is the air resistance; where ρ is the known air density, C ai is the known drag coefficient, v i (t) is the speed of the ith vehicle, A i is the known cross-sectional area; d i (t) is an unknown external disturbance caused by rough road surface or wind; F fi (t) = m i gf i cos(δ i ) is the rolling resistance, F gi (t) = m i gsin(δ i ) is gravity; where f i is the known rolling resistance coefficient, g is the known acceleration due to gravity, δ i is the random road slope;

[0017] The third-order physical model of "displacement-velocity-force" of the third-order heterogeneous vehicle formation structure module is:

[0018]

[0019] Among them, s i (t) is the driving displacement of the i-th vehicle; Yes i The first derivative of (t); Yes i The first derivative of (t); Yes F i The first derivative of (t); τ i is a known dynamic characteristic time constant used to represent the heterogeneous characteristics of the vehicle formation; u i0 (t) is the throttle or brake control torque input;

[0020] The dynamic model of the leading vehicle in the vehicle formation of the three-order heterogeneous vehicle formation structure module is:

[0021]

[0022] Among them, s 0 (t), v 0 (t) and a 0 (t) is the displacement, velocity and acceleration of the lead vehicle in the vehicle formation; Yes 0 The first derivative of (t); Yes 0 The first derivative of (t);

[0023] The quadratic spacing error model of the three-order heterogeneous vehicle formation structure module is:

[0024]

[0025] Among them, z i (t) is the secondary distance error of the vehicles; s i With s i-1 is the displacement of two adjacent vehicles in a platoon of three heterogeneous vehicles; L i is the known heterogeneous vehicle length; κ is the known safety factor; Γ represents the known delay; ρ i0 is the known upper bound of the electromagnetic actuator failure rate; Λ is the known minimum safety distance; A max is the absolute value of the known maximum possible acceleration; and defines the required inter-vehicle distance including the vehicle length for:

[0026]

[0027] Furthermore, the model of the actuator fault tolerance control module is as follows:

[0028]

[0029] Among them, u i (t) is the control output force when the electromagnetic actuator fails; ρ i (t) is the failure rate of the electromagnetic actuator and satisfies 0<ρ 0 (t)≤ρ i (t)≤1;ρ 0 (t) is a known design positive constant; It is a bias fault of the electromagnetic actuator and satisfies is a known bounded positive constant; u i0 (t) is the throttle or brake control torque input;

[0030] Furthermore, the model of the specified performance spacing error module is as follows

[0031]

[0032] in, is a performance function; and satisfies h i0 >0,h i∞ >0 and a i >0 is a known parameter; ξ i (t) is the specified quadratic spacing tracking error variable; z i (t) is the vehicle secondary spacing error.

[0033] A method for adaptive platooning fault-tolerant control of heterogeneous vehicles with safety constraints comprises the following steps:

[0034] Step S1: Obtain the control output force u of the electromagnetic actuator through the three-order heterogeneous vehicle formation structure module i (t), and according to the control output force u of the electromagnetic actuator i (t) Adjust the driving displacement s of the three heterogeneous vehicle formations i , speed v i and acceleration a i (i=1,2,3);

[0035] Step S2: Obtain the vehicle speed v of the third-order heterogeneous vehicle formation structure module through the fuzzy adaptive law module i With vehicle acceleration a i , and combined with the road slope δ i The rolling resistance and air resistance caused by the calculation of the estimated values ​​of the adaptive parameters of the fuzzy control

[0036] The fuzzy adaptive law module estimates the adaptive parameters of the fuzzy control Send to the actuator fault tolerance control module;

[0037] Step S3: Obtain the driving displacement s of the three heterogeneous vehicle formations through the specified performance spacing error module i , and according to the driving displacement s of the three heterogeneous vehicle formations i And the designed performance function h i (t), calculate the tracking error variable ξ with the specified quadratic spacing i (t);

[0038] And the specified performance spacing error module will have a specified quadratic spacing tracking error variable ξ i (t) sent to the actuator fault tolerance control module;

[0039] Step S4: Obtain the specified secondary spacing tracking error variable ξ through the designed coupled sliding surface module i (t), and according to the specified quadratic spacing tracking error variable ξ i (t) Establish the coupled sliding surface ∏ i(t), and the designed coupled sliding surface module couples the sliding surface ∏ i (t) sent to the actuator fault tolerance control module;

[0040] Step S5: Obtain the failure rate ρ of the electromagnetic actuator through the actuator fault diagnosis module i (t) and bias fault of electromagnetic actuator The failure rate of the electromagnetic actuator ρ i (t) and bias fault Send to the actuator fault tolerance control module;

[0041] Step S6: enabling the actuator fault tolerance control module to obtain the failure rate ρ of the electromagnetic actuator i (t), bias fault Estimation of Adaptive Parameters in Fuzzy Control And the coupled sliding surface ∏ i (t);

[0042] And according to the failure rate ρ of the electromagnetic actuator i (t), bias fault of electromagnetic actuator Estimation of Adaptive Parameters in Fuzzy Control And the coupled sliding surface ∏ i (t) Calculate the control output force u of the electromagnetic actuator i (t).

[0043] Furthermore, the model of the fuzzy adaptive law module in step S2 is as follows:

[0044] According to formula (1) and formula (2), we can get:

[0045]

[0046] in, is a i The first derivative of (t); f i (v i ,a i ) is the first intermediate parameter variable; D i is the second intermediate parameter variable; Ξ i =m i gf i cos(δ i )+m i gsin(δ i ), Yes i The first derivative of i is d i (t) is a simplified form; v i vi (t) in simplified form;

[0047] Substituting formula (7) into formula (3), we can get

[0048]

[0049] Among them, u i (t) is the control output force when the electromagnetic actuator fails; ρ i (t) is the failure rate of the electromagnetic actuator; It is a bias fault of the electromagnetic actuator; is a i The first derivative of (t); f i (v i ,a i ) is the first intermediate parameter variable; D i is the second intermediate parameter variable;

[0050] because Ξ i =m i gf i cos(δ i )+m i gsin(δ i ) is an unknown function, which can be approximated by fuzzy logic system.

[0051] in is a known bounded positive constant; ε i (v i ,a i ) is the error term; θ i * is the optimal parameter vector in fuzzy logic control; is the transpose of the optimal parameter vector in fuzzy logic control; is the basis function in the fuzzy logic system;

[0052] According to formula (8), we can know that:

[0053]

[0054] in, is an unknown bounded positive constant; η i is the first intermediate variable; g i is the second intermediate variable;

[0055] The parameter adaptive law model is designed as

[0056]

[0057]

[0058] in, is the basis function in the fuzzy logic system The transpose of 1i and σ 2i is a known positive constant; is a performance function; and satisfies h i0 >0,h i∞ >0 and a i >0 is a known parameter; b i , q and χ i is a known positive constant; is η i Estimates of i For i A simplified form of (t).

[0059] Furthermore, the model of the coupled sliding surface module in step S4 is:

[0060] The sliding mode variable of the coupled sliding surface module, i.e., the displacement S of the three heterogeneous vehicle formations i (t) is:

[0061]

[0062] Among them, λ is a known positive constant; ξ i (t) is the specified quadratic spacing tracking error variable; Yes i The first derivative of (t);

[0063] Coupled sliding surface module coupled sliding surface ∏ i for:

[0064]

[0065] Where q>0 is a known constant; and when ∏ i When reaching the sliding surface, S i (i=1,2,3) also reaches the sliding surface at the same time; s i (t) is the driving displacement of the i-th vehicle among the three heterogeneous vehicles; S i+1 (t) is the driving displacement of the i+1th vehicle among the three heterogeneous vehicles;

[0066] According to formula (13), for:

[0067]

[0068] in, Yes i The first derivative of (t); Yes i The second derivative of (t); YesS i The first derivative of

[0069] Based on formula (12), and for:

[0070]

[0071] in, Yes i The first derivative of (t); Yes i The second derivative of (t); Yes i The first derivative of (t); Yes i The second derivative of (t);

[0072] but

[0073]

[0074] in, Yes i The first derivative of YesS i The first derivative of YesS i+1 The first derivative of (t);

[0075] According to (5), we can know

[0076]

[0077] Among them, z i (t) is the vehicle secondary spacing error; Yes i The first derivative of Yes i The second derivative of κ is a known safety factor; Γ represents a known delay; A max is the absolute value of the known maximum feasible acceleration; a i-1 is the acceleration of the i-1th (i=1,2,3)th car;

[0078] so

[0079]

[0080] Among them, M i is a piecewise function;

[0081]

[0082] Furthermore, the model of the actuator fault tolerance control module in step S6 is specifically:

[0083] The Lyapunov function is designed as:

[0084]

[0085] in, is the optimal parameter vector θ in fuzzy logic control i * The squared term of the norm; is an unknown bounded positive constant; η i is the first intermediate variable; and yes and of estimates, and

[0086]

[0087] in,

[0088] From Young's inequality we know that:

[0089]

[0090] in, b i >0 is a known parameter;

[0091]

[0092] Design the control output force u of the electromagnetic actuator i for:

[0093]

[0094] Among them, τ i is a known dynamic characteristic time constant used to represent the heterogeneous characteristics of the vehicle formation; u i for u i (t) abbreviated form; i and χ i is a known positive constant, k i is the known feedback gain; ρ io and K i is a known positive constant and satisfies 1≤1 / ρ io ≤K i ;

[0095] From formula (25), we can know

[0096]

[0097] According to formula (10) and formula (11), we know

[0098]

[0099] Substituting formula (24) and formula (27) into formula (22), we can obtain

[0100]

[0101] Let C = min{2k i ρ i0 ,σ 1i ,σ 2i}, but

[0102]

[0103] in, C is a variable in the first formula representation, and C=min{2k i ρ i0 ,σ 1i ,σ 2i}; V is the designed Lyapunov function V i ; H is a variable in the second formula representation; and V i is the Lyapunov function;

[0104] V(t)=e -Ct V(0)+H / C (30)

[0105]

[0106] Where V is a simplified form of V(t); V(0) is the initial Lyapunov function when time is t=0; t is the time value;

[0107] According to the coupled sliding surface i (t) = qS i (t)-S i+1 (t), when ∏ i By converging the design parameters to the region of the origin, we can obtain:

[0108]

[0109] Then the Laplace transformation is:

[0110] q[(s+λ)E i (s)]=(s+λ)E i+1 (s)(33)

[0111] So the conversion function is

[0112]

[0113] Among them, E i (s) is ξ i Laplace transform of (t); E i+1 Yes i+1 (t) is the Laplace transformation; when q∈(0,1], the string stability of the vehicle queue can be effectively ensured, that is, |z 3 (t)|≤|z 2 (t)|≤|z 1 (t)|;z 1 (t) is the quadratic distance error between the lead vehicle and the first following vehicle; z 2 (t) is the quadratic distance error between the first following vehicle and the second following vehicle; z 3 (t) is the quadratic spacing error between the second and third following vehicles.

[0114] Beneficial effects: The present invention provides a heterogeneous vehicle adaptive formation fault-tolerant control system and method with safety constraints, wherein the displacement, velocity and acceleration output in real time in the heterogeneous vehicle formation are obtained through a third-order heterogeneous vehicle formation structure module; the displacement trajectory tracking error of two adjacent vehicles in the vehicle formation is kept within a specified area by a specified performance spacing error module, thereby ensuring a higher traffic capacity; the designed coupling sliding surface module ensures the string stability of the vehicle queue and solves the relationship between two adjacent vehicles; since there are unmeasurable unknown variables in the system, the fuzzy adaptive law parameters are obtained through the fuzzy adaptive law module, and when the fault diagnosis module detects that the electromagnetic actuator has a fault, the influence of the fault on the heterogeneous vehicle formation is solved in the actuator fault fault-tolerant control module, regardless of whether the electromagnetic actuator fails, the designed control method ensures that all signals of the vehicle formation are bounded and the spacing error does not exceed its performance boundary. BRIEF DESCRIPTION OF THE DRAWINGS

[0115] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0116] Figure 1 It is a schematic diagram of a heterogeneous vehicle adaptive formation fault-tolerant control system with safety constraints disclosed in the present invention;

[0117] Figure 2It is a flow chart of a fault-tolerant control method for adaptive formation of heterogeneous vehicles with safety constraints disclosed in the present invention;

[0118] Figure 3 is the vehicle displacement s in the third-order heterogeneous vehicle formation of the present invention i The curve of

[0119] Figure 4 is the vehicle speed v in the third-order heterogeneous vehicle formation of the present invention i The curve of

[0120] Figure 5 is the vehicle acceleration a in the third-order heterogeneous vehicle formation of the present invention i The curve of

[0121] Figure 6 is the fault-tolerant control output force u in the three-order heterogeneous vehicle formation of the present invention i The curve of

[0122] Figure 7 is the spacing error z in the third-order heterogeneous vehicle formation of the present invention i and performance boundary curves. DETAILED DESCRIPTION

[0123] In order to make the purpose, technical solution and advantages of the embodiments of the present invention clearer, the technical solution in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0124] This embodiment provides a heterogeneous vehicle adaptive platoon fault-tolerant control system with safety constraints, such as Figure 1 As shown, it includes a three-order heterogeneous vehicle formation structure module, an actuator fault diagnosis module, a fuzzy adaptive law module, a specified performance spacing error module, a coupled sliding surface module, and an actuator fault tolerance control module;

[0125] The three-order heterogeneous vehicle formation structure module is used to obtain the control output force u of the electromagnetic actuator i (t), and according to the control output force u of the electromagnetic actuator i (t) Adjust the driving displacement s of the three heterogeneous vehicle formations i , speed v i and acceleration a i (i=1,2,3); where i=1,2,3 represent following car 1, following car 2 and following car 3 respectively;

[0126] The actuator fault diagnosis module is used to obtain the failure rate ρ of the electromagnetic actuator i (t) Bias fault with electromagnetic actuator The failure rate of the electromagnetic actuator ρ i (t) and bias fault The signal is sent to the actuator fault tolerance control module; the actuator fault diagnosis module is a module that uses the vehicle's own controller (dsPIC30F4011 digital signal controller) as the core chip and uses assembly language for system software programming. When an actuator fault occurs during driving, the failure rate and bias fault signal of the electromagnetic actuator can be obtained in real time;

[0127] The fuzzy adaptive law module is used to obtain the vehicle speed v of the third-order heterogeneous vehicle formation structure module. i and vehicle acceleration a i , and combined with the road slope δ i The rolling resistance and air resistance caused by the calculation of the estimated values ​​of the adaptive parameters of the fuzzy control

[0128] The specified performance spacing error module obtains the driving displacement s of the three heterogeneous vehicle formations. i , and according to the driving displacement s of the three heterogeneous vehicle formations i And the designed performance function h i (t), calculate the tracking error variable ξ with the specified quadratic spacing i (t);

[0129] The designed coupled sliding surface module obtains the specified quadratic spacing tracking error variable ξ i (t), and according to the specified quadratic spacing tracking error variable ξ i (t) Establish the coupled sliding surface ∏ i (t); thereby ensuring the string stability of the vehicle train;

[0130] The actuator fault tolerance control module is used to obtain the failure rate ρ of the electromagnetic actuator i (t), bias fault Estimation of Adaptive Parameters in Fuzzy Control And the coupled sliding surface ∏ i (t), and according to the failure rate ρ of the electromagnetic actuator i (t), bias fault of electromagnetic actuator Estimation of Adaptive Parameters in Fuzzy Control And the coupled sliding surface ∏ i (t) Calculate the control output force u of the electromagnetic actuator i (t).

[0131] The displacement, velocity and acceleration of the heterogeneous vehicle formation are obtained through the third-order heterogeneous vehicle formation structure module; the displacement trajectory tracking error of two adjacent vehicles in the vehicle formation is kept within the specified area by the specified performance spacing error module, thereby ensuring a higher traffic capacity; the designed coupled sliding surface module ensures the string stability of the vehicle queue and solves the relationship between two adjacent vehicles; due to the existence of unmeasurable unknown variables in the system, the fuzzy adaptive law parameters are obtained through the fuzzy adaptive law module. When the fault diagnosis module detects that the electromagnetic actuator has a fault, the actuator fault tolerance control module solves the impact of the fault on the heterogeneous vehicle formation. Regardless of whether the electromagnetic actuator fails, the designed control method ensures that all signals of the vehicle formation are bounded and the spacing error does not exceed its performance boundary. At the same time, based on the coupled sliding surface, the proposed control algorithm can effectively ensure the string stability of the queue.

[0132] In a specific embodiment, the three-order heterogeneous vehicle formation structure module includes a vehicle longitudinal dynamics model, a three-order physical model of "displacement-velocity-force", a dynamic model of the leading vehicle in the vehicle formation, and a quadratic spacing error model:

[0133] The vehicle longitudinal dynamics model of the three-order heterogeneous vehicle formation structure module is:

[0134]

[0135] Among them, m i and a i (t) are the mass and acceleration of the i-th vehicle respectively; F i (t) is the actual driving force or braking force of the engine; l i is the known vehicle mechanical efficiency; R i is the known vehicle tire radius; is the air resistance; where ρ is the known air density, C ai is the known drag coefficient, v i (t) is the speed of the ith vehicle, A i is the known cross-sectional area; d i (t) is an unknown external disturbance caused by rough road surface or wind; F fi (t) = m i gf i cos(δ i ) is the rolling resistance, F gi (t) = m i gsin(δ i ) is gravity; where f i is the known rolling resistance coefficient, g is the known acceleration due to gravity, δ i is the random road slope;

[0136] The third-order physical model of "displacement-velocity-force" of the third-order heterogeneous vehicle formation structure module is:

[0137]

[0138] Among them, s i (t) is the driving displacement of the i-th vehicle; Yes i The first derivative of (t); Yes i The first derivative of (t); Yes F i The first derivative of (t); τ i is a known dynamic characteristic time constant used to represent the heterogeneous characteristics of the vehicle formation; u i0 (t) is the throttle or brake control torque input;

[0139] The dynamic model of the leading vehicle in the vehicle formation of the three-order heterogeneous vehicle formation structure module is:

[0140]

[0141] Among them, s 0 (t), v 0 (t) and a 0 (t) is the displacement, velocity and acceleration of the lead vehicle in the vehicle formation; Yes 0 The first derivative of (t); Yes 0 The first derivative of (t);

[0142] In order to further improve safety and comfort while ensuring the stability of single vehicles and queues, the following secondary spacing error strategy is designed:

[0143]

[0144] Among them, z i (t) is the secondary distance error of the vehicles; s i With s i-1 is the displacement of two adjacent vehicles in a platoon of three heterogeneous vehicles; L i is the known heterogeneous vehicle length; κ is the known safety factor; Γ represents the known delay, which is used to compensate for the delay caused by acceleration and braking in the vehicle queue; ρ i0 is the known upper bound of the electromagnetic actuator failure rate; Λ is the known minimum safety distance; A max is the absolute value of the known maximum possible acceleration; therefore, the required headroom including the vehicle length is defined for:

[0145]

[0146] In a specific embodiment, the model of the actuator fault tolerance control module is as follows:

[0147]

[0148] Among them, u i (t) is the control output force when the electromagnetic actuator fails; ρ i (t) is the failure rate of the electromagnetic actuator and satisfies 0<ρ 0 (t)≤ρ i (t)≤1;ρ 0 (t) is a known design positive constant; It is a bias fault of the electromagnetic actuator and satisfies is a known bounded positive constant; u i0 (t) is the throttle or brake control torque input;

[0149] In a specific embodiment, the model of the specified performance spacing error module is as follows:

[0150] In order to ensure the quadratic spacing error z i (t) can be kept in the specified area, introducing the following error variables:

[0151]

[0152] in, is a performance function; and satisfies h i0 >0,h i∞ >0 and a i >0 is a known parameter; ξ i (t) is the specified quadratic spacing tracking error variable; z i (t) is the quadratic distance error of the vehicles. Therefore, for the continuous function ξ i (t) and the initial condition ξ i (0)∈[0,1), we can see that for When 0≤ξ i When <1, we can get |z i (t)|<h i (t).

[0153] A fault-tolerant control method for adaptive platooning of heterogeneous vehicles with safety constraints, such as Figure 2 As shown, the following steps are included:

[0154] Step S1: Obtain the control output force u of the electromagnetic actuator through the three-order heterogeneous vehicle formation structure module i (t), and according to the control output force u of the electromagnetic actuator i(t) Adjust the driving displacement s of the three heterogeneous vehicle formations i , speed v i and acceleration a i (i=1,2,3);

[0155] Step S2: Obtain the vehicle speed v of the third-order heterogeneous vehicle formation structure module through the fuzzy adaptive law module i With vehicle acceleration a i , and combined with the road slope δ i The rolling resistance and air resistance caused by the calculation of the estimated values ​​of the adaptive parameters of the fuzzy control

[0156] The fuzzy adaptive law module estimates the adaptive parameters of the fuzzy control Send to the actuator fault tolerance control module;

[0157] Step S3: Obtain the driving displacement s of the three heterogeneous vehicle formations through the specified performance spacing error module i , and according to the driving displacement s of the three heterogeneous vehicle formations i And the designed performance function h i (t), calculate the tracking error variable ξ with the specified quadratic spacing i (t);

[0158] And the specified performance spacing error module will have a specified quadratic spacing tracking error variable ξ i (t) sent to the actuator fault tolerance control module;

[0159] Step S4: Obtain the specified secondary spacing tracking error variable ξ through the designed coupled sliding surface module i (t), and according to the specified quadratic spacing tracking error variable ξ i (t) Establish the coupled sliding surface ∏ i (t), and the designed coupled sliding surface module couples the sliding surface ∏ i (t) sent to the actuator fault tolerance control module;

[0160] Step S5: Obtain the failure rate ρ of the electromagnetic actuator through the actuator fault diagnosis module i (t) and bias fault of electromagnetic actuator The failure rate of the electromagnetic actuator ρ i (t) and bias fault Send to the actuator fault tolerance control module;

[0161] Step S6: enabling the actuator fault tolerance control module to obtain the failure rate ρ of the electromagnetic actuator i(t), bias fault Estimation of Adaptive Parameters in Fuzzy Control And the coupled sliding surface ∏ i (t);

[0162] And according to the failure rate ρ of the electromagnetic actuator i (t), bias fault of electromagnetic actuator Estimation of Adaptive Parameters in Fuzzy Control And the coupled sliding surface ∏ i (t) Calculate the control output force u of the electromagnetic actuator i (t).

[0163] In a specific embodiment, the model of the fuzzy adaptive law module in step S2 is as follows:

[0164] According to formula (1) and formula (2), we can get:

[0165]

[0166] in, is a i The first derivative of (t); f i (v i ,a i ) is the first intermediate parameter variable; D i is the second intermediate parameter variable; Ξ i =m i gf i cos(δ i )+m i gsin(δ i ), Yes i The first derivative of i is d i (t) is a simplified form; v i v i (t) in simplified form;

[0167] Substituting formula (7) into formula (3), we can get

[0168]

[0169] Among them, u i (t) is the control output force when the electromagnetic actuator fails; ρ i (t) is the failure rate of the electromagnetic actuator; It is a bias fault of the electromagnetic actuator; is a i The first derivative of (t); f i (v i,a i ) is the first intermediate parameter variable; D i is the second intermediate parameter variable;

[0170] because Ξ i =m i gf i cos(δ i )+m i gsin(δ i ) is an unknown function, which can be approximated by fuzzy logic system.

[0171] in is a known bounded positive constant; ε i (v i ,a i ) is the error term; θ i * is the optimal parameter vector in fuzzy logic control; is the transpose of the optimal parameter vector in fuzzy logic control;

[0172] It is the basis function in the fuzzy logic system;

[0173] According to formula (8), we can know that:

[0174]

[0175] in, is an unknown bounded positive constant; η i is the first intermediate variable; g i is the second intermediate variable;

[0176] The parameter adaptive law model is designed as

[0177]

[0178] in, is the basis function in the fuzzy logic system The transpose of 1i and σ 2i is a known positive constant; is a performance function; and satisfies h i0 >0,h i∞ >0 and a i >0 is a known parameter; b i , q and χ i is a known positive constant; is η i Estimates of i For iA simplified form of (t).

[0179] In a specific embodiment, the model of the coupled sliding surface module in step S4 is:

[0180] The sliding mode variable of the coupled sliding surface module, i.e., the displacement S of the three heterogeneous vehicle formations i (t) is:

[0181]

[0182] Among them, λ is a known positive constant; ξ i (t) is the specified quadratic spacing tracking error variable; Yes i The first derivative of (t);

[0183] In addition, for the last following car, when N=3, N is the last following car, that is, when i=3, N=i=3; this paper only studies three following cars, then S N+1 represents the sliding surface of the fourth following vehicle (not existing), z N+1 The quadratic distance error S between the third following car and the fourth following car (not existing) N+1 =0, similarly, This is only explained here; in order to ensure the string stability of the vehicle queue and solve S i (t) and S i+1 (t), the following coupled sliding surface is introduced:

[0184] Coupled sliding surface module coupled sliding surface ∏ i for:

[0185]

[0186] Where q>0 is a known constant; and when ∏ i When reaching the sliding surface, S i (i=1,2,3) also reaches the sliding surface at the same time; s i (t) is the driving displacement of the i-th vehicle among the three heterogeneous vehicles; S i+1 (t) is the driving displacement of the i+1th vehicle among the three heterogeneous vehicles;

[0187] According to formula (13), for:

[0188]

[0189] in, Yes i The first derivative of (t); Yes i The second derivative of (t); YesS i The first derivative of

[0190] Based on formula (12), and for:

[0191]

[0192] in, Yes i The first derivative of (t); Yes i The second derivative of (t); Yes i The first derivative of (t); Yes i The second derivative of (t);

[0193] but

[0194]

[0195] in, Yes i The first derivative of YesS i The first derivative of YesS i+1 The first derivative of (t);

[0196] According to (5), we can know

[0197]

[0198] Among them, z i (t) is the vehicle secondary spacing error; Yes i The first derivative of Yes i The second derivative of; k is a known safety factor; Γ represents a known delay; A max is the absolute value of the known maximum feasible acceleration; a i-1 is the acceleration of the i-1th (i=1,2,3)th car;

[0199] so

[0200]

[0201] Among them, M i is a piecewise function;

[0202]

[0203] In a specific embodiment, the model of the actuator fault tolerance control module in step S6 is specifically:

[0204] The Lyapunov function is designed as:

[0205]

[0206] in, is the optimal parameter vector θ in fuzzy logic control i * The squared term of the norm; is an unknown bounded positive constant; η i is the first intermediate variable; and yes and of estimates, and

[0207]

[0208] in, From Young's inequality we know that:

[0209]

[0210] in, b i >0 is a known parameter;

[0211]

[0212] Design the control output force u of the electromagnetic actuator i for:

[0213]

[0214] Among them, τ i is a known dynamic characteristic time constant used to represent the heterogeneous characteristics of the vehicle formation; u i for u i (t) abbreviated form; i and χ i is a known positive constant, k i is the known feedback gain; ρ io and K i is a known positive constant and satisfies 1≤1 / ρ io ≤K i ;

[0215] And the control output force u of the electromagnetic actuator i According to formula (8), the driving displacement s of the three heterogeneous vehicle formation is adjusted i , speed v iand acceleration a i (i=1,2,3), where is a i The first derivative of (t); a i is a i A simplified form of (t); a i (t) by v i Take the first-order derivative to get; v i By s i Take the first-order derivative to obtain;

[0216] From formula (25), we can know

[0217]

[0218] According to formula (10) and formula (11), we know

[0219]

[0220] Substituting formula (24) and formula (27) into formula (22), we can obtain

[0221]

[0222] Let C = min{2k i ρ i0 ,σ 1i ,σ 2i}, but

[0223]

[0224] in, C is the variable expressed in the first formula, and V is the designed Lyapunov function V i The sum of; H is the variable in the second formula; V i is the Lyapunov function;

[0225] V(t)=e -Ct V(0)+H / C (30)

[0226]

[0227] Where V is a simplified form of V(t); V(0) is the initial Lyapunov function when time is t=0; t is the time value;

[0228] Therefore, all states of the closed-loop system are bounded and the spacing error ξ i (t) is bounded, that is, z i (t) is bounded;

[0229] According to the coupled sliding surfacei (t) = qS i (t)-S i+1 (t), when ∏ i By converging the design parameters to the region of the origin, we can obtain:

[0230]

[0231] Then the Laplace transformation is:

[0232] q[(s+λ)E i (s)]=(s+λ)E i+1 (s) (33)

[0233] So the conversion function is

[0234]

[0235] Among them, E i (s) is ξ i Laplace transform of (t); E i+1 Yes i+1 (t) is the Laplace transformation; when q∈(0,1], the string stability of the vehicle queue can be effectively ensured, that is, |z 3 (t)|≤|z 2 (t)|≤|z 1 (t)|;z 1 (t) is the quadratic distance error between the lead vehicle and the first following vehicle; z 2 (t) is the quadratic distance error between the first following vehicle and the second following vehicle; z 3 (t) is the quadratic spacing error between the second and third following vehicles.

[0236] Through Figure 3 , Figure 4 , Figure 5 , Figure 6 as well as Figure 7 As shown in the figure, when t = 4.5s, the actuator fails and the vehicle displacement s in the third-order heterogeneous vehicle formation is i , the vehicle speed v in the third-order heterogeneous vehicle formation i , the vehicle acceleration a in the third-order heterogeneous vehicle formation i The curve graph shows abnormal changes. By accepting the design of fault-tolerant controller u i After the signal, all the signals of the controlled system (vehicle displacement s in the third-order heterogeneous vehicle formation) i , the speed of vehicles in the vehicle formation v i , vehicle acceleration a in the vehicle formation i ) have achieved convergence and finally stabilized. And the spacing error zi Without exceeding its performance boundary. In addition, based on the coupled sliding surface, the proposed control algorithm can effectively ensure the string stability of the vehicle formation. This means that when there is an electromagnetic actuator failure in the vehicle, the proposed control scheme ensures that the vehicle queue achieves the goal of fast fault tolerance, thereby greatly improving the driving safety of the heterogeneous vehicle formation during operation.

[0237] In the present invention, considering the actuator failure in the operation process of a heterogeneous vehicle formation with high safety constraints is in line with the actual situation, which is more universal and convincing. By transmitting the designed actual control signal back to the heterogeneous vehicle formation with high safety constraints, it can be seen that regardless of whether the electromagnetic actuator failure occurs, the designed adaptive fault-tolerant control method ensures that all signals of the vehicle formation are bounded and the spacing error does not exceed its performance boundary. At the same time, based on the coupled sliding surface, the proposed control algorithm can effectively ensure the stability of the queue string.

[0238] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A fault-tolerant control system for adaptive platooning of heterogeneous vehicles with safety constraints, It is characterized in that It includes a three-order heterogeneous vehicle formation structure module, an actuator fault diagnosis module, a fuzzy adaptive law module, a specified performance spacing error module, a coupled sliding surface module, and an actuator fault tolerance control module; The three-order heterogeneous vehicle formation structure module is used to obtain the control output force u of the electromagnetic actuator i (t), and according to the control output force u of the electromagnetic actuator i (t) Adjust the driving displacement s of the three heterogeneous vehicle formations i , speed v i and acceleration a i ,i=1,2,3; The actuator fault diagnosis module is used to obtain the failure rate ρ of the electromagnetic actuator i (t) Bias fault with electromagnetic actuator The failure rate of the electromagnetic actuator ρ i (t) and bias fault Send to the actuator fault tolerance control module; The fuzzy adaptive law module is used to obtain the vehicle speed v of the third-order heterogeneous vehicle formation structure module. i and vehicle acceleration a i , and combined with the road slope δ i The rolling resistance and air resistance caused by the calculation of the estimated values ​​of the adaptive parameters of the fuzzy control The specified performance spacing error module is used to obtain the driving displacement s of the three heterogeneous vehicle formations. i , and according to the driving displacement s of the three heterogeneous vehicle formations i And the designed performance function h i (t), calculate the tracking error variable ξ with the specified quadratic spacing i (t); The designed coupled sliding surface module is used to obtain the specified quadratic spacing tracking error variable ξ i (t), and according to the specified quadratic spacing tracking error variable ξ i (t) Establish the coupled sliding surface ∏ i (t); The actuator fault tolerance control module is used to obtain the failure rate ρ of the electromagnetic actuator i (t), bias fault Estimation of Adaptive Parameters in Fuzzy Control And the coupled sliding surface ∏ i (t), and according to the failure rate ρ of the electromagnetic actuator i (t), bias fault of electromagnetic actuator Estimation of Adaptive Parameters in Fuzzy Control And the coupled sliding surface ∏ i (t) Calculate the control output force u of the electromagnetic actuator i (t).

2. A heterogeneous vehicle adaptive platoon fault-tolerant control system with safety constraints according to claim 1, It is characterized in that The three-order heterogeneous vehicle formation structure module includes a vehicle longitudinal dynamics model, a three-order physical model of "displacement-velocity-force", a dynamic model of the leading vehicle in the vehicle formation, and a quadratic spacing error model: The vehicle longitudinal dynamics model of the three-order heterogeneous vehicle formation structure module is: Among them, m i and a i (t) are the mass and acceleration of the i-th vehicle respectively; F i (t) is the actual driving force or braking force of the engine; l i is the known vehicle mechanical efficiency; R i is the known vehicle tire radius; is the air resistance; where ρ is the known air density, C ai is the known drag coefficient, v i (t) is the speed of the ith vehicle, A i is the known cross-sectional area; d i (t) is an unknown external disturbance caused by rough road surface or wind; F fi (t) = m i gf i cos(δ i ) is the rolling resistance, F gi (t) = m i gsin(δ i ) is gravity; where f i is the known rolling resistance coefficient, g is the known acceleration due to gravity, δ i is the random road slope; The third-order physical model of "displacement-velocity-force" of the third-order heterogeneous vehicle formation structure module is: Among them, s i (t) is the driving displacement of the i-th vehicle; Yes i The first derivative of (t); Yes i The first derivative of (t); Yes F i The first derivative of (t); τ i is a known dynamic characteristic time constant used to represent the heterogeneous characteristics of the vehicle formation; u i0 (t) is the throttle or brake control torque input; The dynamic model of the leading vehicle in the vehicle formation of the three-order heterogeneous vehicle formation structure module is: Among them, s 0 (t), v 0 (t) and a 0 (t) is the displacement, velocity and acceleration of the lead vehicle in the vehicle formation; Yes 0 The first derivative of (t); Yes 0 The first derivative of (t); The quadratic spacing error model of the three-order heterogeneous vehicle formation structure module is: Among them, z i (t) is the secondary distance error of the vehicles; s i With s i-1 is the displacement of two adjacent vehicles in a platoon of three heterogeneous vehicles; L i is the known heterogeneous vehicle length; κ is the known safety factor; Γ represents the known delay; ρ i0 is the known upper bound of the electromagnetic actuator failure rate; Λ is the known minimum safety distance; A max is the absolute value of the known maximum possible acceleration; and defines the required inter-vehicle distance including the vehicle length for:

3. A heterogeneous vehicle adaptive platoon fault-tolerant control system with safety constraints according to claim 1, It is characterized in that The model of the actuator fault-tolerant control module is as follows: Among them, u i (t) is the control output force when the electromagnetic actuator fails; ρ i (t) is the failure rate of the electromagnetic actuator and satisfies 0<ρ 0 (t)≤ρ i (t)≤1;ρ 0 (t) is a known design positive constant; It is a bias fault of the electromagnetic actuator and satisfies is a known bounded positive constant; u i0 (t) is the throttle or brake control torque input.

4. A heterogeneous vehicle adaptive platoon fault-tolerant control system with safety constraints according to claim 1, It is characterized in that The model of the specified performance spacing error module is as follows in, is a performance function; and satisfies h i0 >0,h i∞ >0 and a i >0 is a known parameter; ξ i (t) is the specified quadratic spacing tracking error variable; z i (t) is the vehicle secondary spacing error.

5. A method for controlling a heterogeneous vehicle adaptive platoon with safety constraints, comprising the heterogeneous vehicle adaptive platoon with safety constraints according to any one of claims 1 to 4, It is characterized in that The following steps are involved: Step S1: Obtain the control output force u of the electromagnetic actuator through the three-order heterogeneous vehicle formation structure module i (t), and according to the control output force u of the electromagnetic actuator i (t) Adjust the driving displacement s of the three heterogeneous vehicle formations i , speed v i and acceleration a i ,i=1,2,3; Step S2: Obtain the vehicle speed v of the third-order heterogeneous vehicle formation structure module through the fuzzy adaptive law module i With vehicle acceleration a i , and combined with the road slope δ i The rolling resistance and air resistance caused by the calculation of the estimated values ​​of the adaptive parameters of the fuzzy control The fuzzy adaptive law module estimates the adaptive parameters of the fuzzy control Send to the actuator fault tolerance control module; Step S3: Obtain the driving displacement s of the three heterogeneous vehicle formations through the specified performance spacing error module i , and according to the driving displacement s of the three heterogeneous vehicle formations i And the designed performance function h i (t), calculate the tracking error variable ξ with the specified quadratic spacing i (t); And the specified performance spacing error module will have a specified quadratic spacing tracking error variable ξ i (t) sent to the actuator fault tolerance control module; Step S4: Obtain the specified secondary spacing tracking error variable ξ through the designed coupled sliding surface module i (t), and according to the specified quadratic spacing tracking error variable ξ i (t) Establish the coupled sliding surface ∏ i (t), and the designed coupled sliding surface module couples the sliding surface ∏ i (t) sent to the actuator fault tolerance control module; Step S5: Obtain the failure rate ρ of the electromagnetic actuator through the actuator fault diagnosis module i (t) and bias fault of electromagnetic actuator The failure rate of the electromagnetic actuator ρ i (t) and bias fault Send to the actuator fault tolerance control module; Step S6: enabling the actuator fault tolerance control module to obtain the failure rate ρ of the electromagnetic actuator i (t), bias fault Estimation of Adaptive Parameters in Fuzzy Control And the coupled sliding surface ∏ i (t); And according to the failure rate ρ of the electromagnetic actuator i (t), bias fault of electromagnetic actuator Estimation of Adaptive Parameters in Fuzzy Control And the coupled sliding surface ∏ i (t) Calculate the control output force u of the electromagnetic actuator i (t).

6. A method for adaptive platooning fault-tolerant control of heterogeneous vehicles with safety constraints according to claim 5, It is characterized in that The model of the fuzzy adaptive law module in step S2 is as follows: According to formula (1) and formula (2), it can be obtained: in, is a i The first derivative of (t); f i (v i ,a i ) is the first intermediate parameter variable; D i is the second intermediate parameter variable; Ξ i =m i gf i cos(δ i )+m i gsin(δ i ), Yes i The first derivative of i is d i (t) is a simplified form; v i v i (t) in simplified form; Substituting formula (7) into formula (3), we can get Among them, u i (t) is the control output force when the electromagnetic actuator fails; ρ i (t) is the failure rate of the electromagnetic actuator; It is a bias fault of the electromagnetic actuator; is a i The first derivative of (t); f i (v i ,a i ) is the first intermediate parameter variable; D i is the second intermediate parameter variable; because Ξ i =m i gf i cos(δ i )+m i gsin(δ i ) is an unknown function, which can be approximated by fuzzy logic system. in is a known bounded positive constant; ε i (v i ,a i ) is the error term; θ i * is the optimal parameter vector in fuzzy logic control; is the transpose of the optimal parameter vector in fuzzy logic control; It is the basis function in the fuzzy logic system; According to formula (8), we can know that: in, is an unknown bounded positive constant; η i is the first intermediate variable; g i is the second intermediate variable; The parameter adaptive law model is designed as in, is the basis function in the fuzzy logic system The transpose of 1i and σ 2i is a known positive constant; is a performance function; and satisfies h i0 >0,h i∞ >0 and a i >0 is a known parameter; b i , q and χ i is a known positive constant; is η i Estimates of i For i A simplified form of (t).

7. The method for adaptive platooning fault-tolerant control of heterogeneous vehicles with safety constraints according to claim 5, It is characterized in that The model of the coupled sliding surface module in step S4 is: The sliding mode variable of the coupled sliding surface module, i.e., the displacement S of the three heterogeneous vehicle formations i (t) is: Among them, λ is a known positive constant; ξ i (t) is the specified quadratic spacing tracking error variable; Yes i The first derivative of (t); Coupled sliding surface module coupled sliding surface ∏ i for: Where q>0 is a known constant; and when ∏ i When reaching the sliding surface, S i , i=1,2,3 also reach the sliding surface; s i (t) is the driving displacement of the i-th vehicle among the three heterogeneous vehicles; S i+1 (t) is the driving displacement of the i+1th vehicle among the three heterogeneous vehicles; According to formula (13), for: in, Yes i The first derivative of (t); Yes i The second derivative of (t); YesS i The first derivative of Based on formula (12), and for: in, Yes i The first derivative of (t); Yes i The second derivative of (t); Yes i The first derivative of (t); Yes i The second derivative of (t); but in, Yes i The first derivative of YesS i The first derivative of YesS i+1 The first derivative of (t); According to (5), we can know Among them, z i (t) is the vehicle secondary spacing error; Yes i The first derivative of Yes i The second derivative of κ is a known safety factor; Γ represents a known delay; A max is the absolute value of the known maximum feasible acceleration; a i-1 is the acceleration of the i-1th, i=1, 2, 3rd car; so Among them, M i is a piecewise function; 8. The method for adaptive platooning fault-tolerant control of heterogeneous vehicles with safety constraints according to claim 5, It is characterized in that The model of the actuator fault tolerance control module in step S6 is specifically: The Lyapunov function is designed as: in, is the optimal parameter vector θ in fuzzy logic control i * The squared term of the norm; is an unknown bounded positive constant; η i is the first intermediate variable; and yes and of estimates, and in, From Young's inequality we know that: in, b i >0 is a known parameter; Design the control output force u of the electromagnetic actuator i for: Among them, τ i is a known dynamic characteristic time constant used to represent the heterogeneous characteristics of the vehicle formation; u i for u i (t) abbreviated form; i and χ i is a known positive constant, k i is the known feedback gain; ρ io and K i is a known positive constant and satisfies 1≤1 / ρ io ≤K i ; From formula (25), we can know According to formula (10) and formula (11), we know Substituting formula (24) and formula (27) into formula (22), we can obtain Let \(C = \min\{2k i \rho i0 ,\sigma 1i ,\sigma 2i \}\), then in, C is a variable in the first formula representation, and C=min{2k i ρ i0 ,σ 1i ,σ 2i }; V is the designed Lyapunov function V i ; H is a variable in the second formula representation; and V i is the Lyapunov function; V(t)=e -Ct V(0)+H / C (30) Where V is a simplified form of V(t); V(0) is the initial Lyapunov function when time is t=0; t is the time value; According to the coupled sliding surface i (t) = qS i (t)-S i+1 (t), when ∏ i By converging the design parameters to the region of the origin, we can obtain: Then the Laplace transformation is: q[(s+λ)E i (s)]=(s+λ)E i+1 (s) (33) So the conversion function is Among them, E i (s) is ξ i Laplace transform of (t); E i+1 Yes i+1 (t) is the Laplace transformation; when q∈(0,1], the string stability of the vehicle queue can be effectively ensured, that is, |z 3 (t)|≤|z 2 (t)|≤|z 1 (t)|;z 1 (t) is the quadratic distance error between the lead vehicle and the first following vehicle; z 2 (t) is the quadratic distance error between the first following vehicle and the second following vehicle; z 3 (t) is the quadratic spacing error between the second and third following vehicles.

Images