H-infinity and passivity-based control methods for vehicle platoon under semi-markovian switching topology

Abstract

The application discloses a H-infinity and passivity control method for vehicle platoon under semi-Markov switching topology, and first establishes a longitudinal dynamics state space model of a single vehicle.Secondly, the information interaction form between vehicles is described based on graph theory, and the time-varying characteristics between vehicles are described by a semi-Markov process.Then, according to the description result, a distributed controller and a closed-loop system of the vehicle platoon under the semi-Markov switching topology are designed.Finally, for the vehicle platoon under the semi-Markov switching topology, the mean square stability and H-infinity and passivity are analyzed, and the control gain matrix of the distributed controller is solved to obtain the vehicle controller.The application can keep the normal operation state of the platoon system when the disturbance exists, and enables the followers to track the leader, so that the performance index of the platoon control is realized.

Description

Technical Field

[0001] This invention relates to the field of intelligent transportation technology, specifically a method for controlling the H-infinity and passivity of vehicle queues under a semi-Markov switching topology. Background Technology

[0002] In recent years, Intelligent Vehicle Highway Technology (IVHS) has flourished, and vehicle platooning control, as a key technology in IVHS, has attracted increasing research attention. Vehicle platooning involves organizing two or more vehicles into a queue with safe distances and the same speed, which can greatly improve road throughput and reduce fuel consumption.

[0003] In practical communication networks, the communication topology exhibits time-varying characteristics due to limitations in sensor transmission distance, network packet loss, and signal interference. Therefore, studying vehicle queuing control under semi-Markov switched topologies is of great significance.

[0004] Dai et al.'s work, "Exponential consensus of non-linear multi-agent systems with semi-Markov switching topologies," investigated the exponential stability problem of nonlinear multi-agent systems under semi-Markov switching topologies. Huang et al.'s work, "Event-triggered leader-following consensus of multi-agent systems under semi-Markov switching topology with partially unknown rates," further studied the stability problem of multi-agent systems under semi-Markov switching topologies based on event triggering.

[0005] In summary, most research focuses on multi-agent systems under semi-Markov switching topologies, while research on vehicle queuing is relatively limited. Summary of the Invention

[0006] To address the aforementioned problems, this invention proposes a control method for H-infinity and passivity of vehicle queuing under a semi-Markov switching topology. By introducing a semi-Markov stochastic process into the vehicle queuing system, a distributed controller is designed to ensure the mean square stability of the vehicle queuing under the semi-Markov switching topology and to satisfy the given H-infinity and passivity performance in the presence of disturbances.

[0007] To achieve the above objectives, the technical solution provided by this invention is as follows:

[0008] A method for controlling the infinity and passivity of vehicle queues under a semi-Markov switching topology includes the following steps:

[0009] Step 1: Establish a longitudinal dynamic state-space model for a single vehicle.

[0010] Step 2: Describe the information interaction between vehicles based on graph theory, and characterize the time-varying characteristics between vehicles using a semi-Markov process.

[0011] Step 3: Based on the characterization results, design a distributed controller and closed-loop system for the vehicle queue under the semi-Markov switching topology.

[0012] Step 4: For the vehicle queue under the semi-Markov switching topology, firstly, mean square stability and H infinity and passivity analysis are performed. Then, the control gain matrix of the distributed controller is solved using Lyapunov stability theory and the linear matrix inequality method to obtain the vehicle controller and complete the control of the vehicle queue.

[0013] Step 1, establishing the vehicle dynamics state-space model, requires the following specific steps:

[0014] Step 1.1: There are N+1 vehicles in the queue traveling on a smooth road, consisting of a lead vehicle numbered 0 and follower vehicles numbered 1 to N. The longitudinal dynamics of each vehicle involve components such as the engine, braking system, and aerodynamic drag. Its longitudinal dynamics mathematical model is as follows:

[0015]

[0016] Where, p i (t) and v i (t) represent the position and velocity of vehicle i, respectively. For p i The first derivative of (t), η T,i For the mechanical efficiency of the transmission system, r w,i T is the tire radius of the vehicle. i (t) represents the actual driving force of the vehicle, C A,i It is the aerodynamic coefficient, m i Let g be the mass of the vehicle, g be the acceleration due to gravity, and f be the acceleration due to gravity. i τ is the tire rolling resistance coefficient. i Let T be the time delay constant of the vehicle's longitudinal system. des,i (t) represents the desired driving force.

[0017] By linearizing the feedback, the desired driving force is obtained: in U is used to represent the acceleration of vehicle i. i(t) is the control input of vehicle i after feedback linearization. The third-order dynamic model of each vehicle is as follows:

[0018]

[0019] Define x i (t)=[p i (t),v i (t),a i (t)] T The longitudinal dynamic state-space model of the i-th vehicle can be expressed as:

[0020]

[0021] Where w i (t) represents the external interference from the vehicle.

[0022] Step 2, which uses a semi-Markov process to describe the communication topology between vehicles, includes the following specific steps:

[0023] Step 2.1: The communication topology of the N following vehicles in the queue is represented by a directed graph. Modeling is performed, among which Represents the vertex set and edge set. This indicates the connectivity between vehicles. (Definition related to E) N Corresponding adjacency matrix When vehicle i can receive the status information of vehicle j, a ij =1, i≠j, otherwise a ij =0. In-degree matrix in diag{·} represents a diagonal block matrix. Directed graph. Laplace matrix The definition is as follows:

[0024]

[0025] Define traction matrix When vehicle i can receive information from the lead vehicle otherwise

[0026] Step 2.2: Introduce a semi-Markov random process to describe the dynamic characteristics of the communication topology. The communication topology of the vehicle queue at time t is described by... To describe, among which and ε θ(t) These represent the adjacency matrix, Laplace matrix, traction matrix, and connectivity between vehicles under a semi-Markov random process, respectively. The mode switching is controlled by a stochastic process {θ(t), t≥0}, and in the state space... Take a value from. Definition Here is the transition matrix of a semi-Markov process, and its transition rates are as follows:

[0027]

[0028] Where lim Δ→∞ o(Δ) / Δ=0, π sl (h) represents the transition rate from mode s at time t to mode l at time t+Δ.

[0029] Step 3, the design of the controller and the closed-loop system, includes the following specific steps:

[0030] Step 3.1: In this invention, the specific objective of vehicle queuing control is to design a controller that ensures all vehicles eventually reach a uniform speed and that adjacent vehicles satisfy a predetermined spacing strategy, i.e.:

[0031]

[0032] Where v0(t) is the speed of the lead vehicle, d i,i-1 Let represent the expected distance between the i-th car and the (i-1)-th car, which is a constant.

[0033] For the following vehicle i, the distributed controller is designed as follows:

[0034]

[0035] Where, x i (t) represents the state of the i-th vehicle. It is the control gain matrix to be designed. And there is d i,j =d i，0 -d j,0 d i，0 This represents the distance between the i-th vehicle and the lead vehicle.

[0036] Step 3.2: For each following vehicle i, define the tracking error as:

[0037]

[0038] in, Let be the position, velocity, and acceleration error of the i-th vehicle, respectively.

[0039] definition Combining equations (2) and (6), the closed-loop system of the vehicle queue can be obtained as follows:

[0040]

[0041] in, I represents the Kronecker product of matrices. N Represents an N×N identity matrix. u0(t) and w0(t) are the input and disturbance of the navigator, respectively. z(t) is the controlled output, which is usually used to measure performance, and C = [1, 0, 0].

[0042] Step 4, solving for the controller's control gain matrix, includes the following specific steps:

[0043] Step 4.1: For vehicles under a semi-Markov switching topology, mean square stability and H infinity and passivity analysis are first performed, and then the control gain matrix of the controller is obtained.

[0044] Construct the following Lyapunov function:

[0045]

[0046] Among them, P θ(t) Let be the positive definite matrix to be found. Then the vehicle queuing closed-loop system (8) is mean square stable and satisfies the conditions of infinite H and passive performance as follows:

[0047] Given a scalar α∈(0,1) and γ>0, if there exists a positive definite symmetric matrix P s P l and K s , so that:

[0048]

[0049] The closed-loop system (8) is mean square stable and satisfies the requirements of infinite H and passive performance.

[0050] Wherein, the symbol He(M) is defined as M+M T M is an arbitrary matrix. E{·} represents the mathematical expectation symbol, f s (h) represents the probability density function for the dwell time h of mode s, the symbol * denotes the transpose of the elements at the diagonal positions of the matrix, matrix I represents the dimension-matched identity matrix, and we have:

[0051]

[0052]

[0053]

[0054] Step 4.2: Transform the above conditions containing coupling terms into the form of linear matrix inequalities, and derive the gain matrix of the distributed controller equation (6).

[0055] Using the Schul complement lemma, Transform into the following form:

[0056]

[0057] in

[0058] Multiply both sides of equation (11) by the matrix. And its transpose, then using the Schul complement lemma, and letting as well as The following form can be obtained.

[0059]

[0060] in,

[0061]

[0062] Finally, the above inequality (12) is solved using the linear matrix inequality to obtain X. s and Y s At this point, the control gain of the distributed controller...

[0063] The beneficial effects of this invention are as follows: This invention employs feedback linearization technology to construct a longitudinal dynamics model of a single vehicle, and uses a semi-Markov random process to characterize the time-varying characteristics of the inter-vehicle communication topology, presenting a closed-loop system for the vehicle platoon. It also considers a novel H-infinity and passive performance, namely, the ability of the platoon system to maintain normal operation even with disturbances. The control objective of this invention is to design a distributed controller that enables followers to track the leader while simultaneously achieving the desired platoon control performance indicators. Attached Figure Description

[0064] Figure 1 This is a flowchart illustrating the entire method of the present invention.

[0065] Figure 2 These are three communication topology diagrams corresponding to the invention's content.

[0066] Figure 3 This is a diagram illustrating the vehicle-to-vehicle communication topology switching relationship in a specific embodiment of the present invention;

[0067] Figure 4(a) is a simulation diagram of the vehicle position according to a specific embodiment of the present invention;

[0068] Figure 4(b) is a simulation diagram of vehicle position error according to a specific embodiment of the present invention;

[0069] Figure 4(c) is a simulation diagram of vehicle speed error in a specific embodiment of the present invention;

[0070] Figure 4(d) is a simulation diagram of vehicle acceleration error in a specific embodiment of the present invention. Detailed Implementation

[0071] The present invention will now be further described in conjunction with the embodiments and accompanying drawings:

[0072] The method for controlling the H-infinity and passivity of vehicle queuing under a semi-Markov switching topology is as follows: Figure 1 As shown, it includes the following steps:

[0073] Step 1: Establish a longitudinal dynamic state-space model for a single vehicle.

[0074] Step 2: Describe the information interaction between vehicles based on graph theory, and characterize the time-varying characteristics between vehicles using a semi-Markov process.

[0075] Step 3: Based on the characterization results, design a distributed controller and closed-loop system for the vehicle queue under the semi-Markov switching topology.

[0076] Step 4: For the vehicle queue under the semi-Markov switching topology, firstly, mean square stability and H infinity and passivity analysis are performed. Then, the control gain matrix of the distributed controller is solved using Lyapunov stability theory and the linear matrix inequality method to obtain the vehicle controller and complete the control of the vehicle queue.

[0077] Step 1, establishing the vehicle dynamics state-space model, requires the following specific steps:

[0078] Step 1.1: There are N+1 vehicles in the queue traveling on a smooth road, consisting of a lead vehicle numbered 0 and follower vehicles numbered 1 to N. The longitudinal dynamics of each vehicle involve components such as the engine, braking system, and aerodynamic drag. Its longitudinal dynamics mathematical model is as follows:

[0079]

[0080] Where, p i (t) and v i (t) represent the position and velocity of vehicle i, respectively. For p i The first derivative of (t), η T,i For the mechanical efficiency of the transmission system, r w,i T is the tire radius of the vehicle. i (t) represents the actual driving force of the vehicle, CA,i It is the aerodynamic coefficient, m i Let g be the mass of the vehicle, g be the acceleration due to gravity, and f be the acceleration due to gravity. i τ is the tire rolling resistance coefficient. i Let T be the time delay constant of the vehicle's longitudinal system. des,i (t) represents the desired driving force.

[0081] By linearizing the feedback, the desired driving force is obtained: in U is used to represent the acceleration of vehicle i. i (t) is the control input of vehicle i after feedback linearization. The third-order dynamic model of each vehicle is as follows:

[0082]

[0083] Define x i (t)=[p i (t),v i (t),a i (t)] T The longitudinal dynamic state-space model of the i-th vehicle can be expressed as:

[0084]

[0085] Where w i (t) represents the external interference from the vehicle.

[0086] Step 2, which uses a semi-Markov process to describe the communication topology between vehicles, includes the following specific steps:

[0087] Step 2.1: The communication topology of the N following vehicles in the queue is represented by a directed graph. Modeling is performed, among which Represents the vertex set and edge set. This represents the connectivity between vehicles. Defined with ε. N Corresponding adjacency matrix When vehicle i can receive the status information of vehicle j, a ij =1, i≠j, otherwise a ij =0. In-degree matrix in diag{·} represents a diagonal block matrix. Directed graph. Laplace matrix The definition is as follows:

[0088]

[0089] Define traction matrix When vehicle i can receive information from the lead vehicle otherwise

[0090] Step 2.2: Introduce a semi-Markov random process to describe the dynamic characteristics of the communication topology. The communication topology of the vehicle queue at time t is described by... To describe, among which and E θ(t) These represent the adjacency matrix, Laplace matrix, traction matrix, and connectivity between vehicles under a semi-Markov random process, respectively. The mode switching is controlled by a stochastic process {θ(t), t≥0}, and in the state space... Take a value from. Definition Here is the transition matrix of a semi-Markov process, and its transition rates are as follows:

[0091]

[0092] Where lim Δ→∞ o(Δ) / Δ=0, π sl (h) represents the transition rate from mode s at time t to mode l at time t+Δ.

[0093] Step 3, the design of the controller and the closed-loop system, includes the following specific steps:

[0094] Step 3.1: In this invention, the specific objective of vehicle queuing control is to design a controller that ensures all vehicles eventually reach a uniform speed and that adjacent vehicles satisfy a predetermined spacing strategy, i.e.:

[0095]

[0096] Where v0(t) is the speed of the lead vehicle, d i,i-1 Let d represent the expected distance between the i-th vehicle and the (i-1)-th vehicle, which is a constant. This invention employs a fixed spacing strategy, i.e., d... i,i-1 =d0 is a constant.

[0097] For the following vehicle i, the distributed controller is designed as follows:

[0098]

[0099] Where, x i (t) represents the state of the i-th vehicle. It is the control gain matrix to be designed. And there is d i,j =d i，0 -d j,0 d i，0 This represents the distance between the i-th vehicle and the lead vehicle.

[0100] Step 3.2: For each following vehicle i, define the tracking error as:

[0101]

[0102] in, Let be the position, velocity, and acceleration error of the i-th vehicle, respectively.

[0103] definition Combining equations (2) and (6), the closed-loop system of the vehicle queue can be obtained as follows:

[0104]

[0105] in, I represents the Kronecker product of matrices. N Represents an N×N identity matrix. u0(t) and w0(t) are the input and disturbance of the navigator, respectively. z(t) is the controlled output, which is usually used to measure performance, and C = [1, 0, 0].

[0106] Step 4, solving for the controller's control gain matrix, includes the following specific steps:

[0107] Step 4.1: For vehicles under a semi-Markov switching topology, mean square stability and H infinity and passivity analysis are first performed, and then the control gain matrix of the controller is obtained.

[0108] Construct the following Lyapunov function:

[0109]

[0110] Among them, P θ(t) Let be the positive definite matrix to be found. Then the vehicle queuing closed-loop system (8) is mean square stable and satisfies the conditions of infinite H and passive performance as follows:

[0111] Given a scalar α∈(0,1) and γ>0, if there exists a positive definite symmetric matrix P s and K s , so that:

[0112]

[0113] The closed-loop system (8) is mean square stable and satisfies the requirements of infinite H and passive performance.

[0114] Where: the symbol He(M) is defined as = M + M T M is an arbitrary matrix. E{·} represents the mathematical expectation symbol, f s(h) represents the probability density function for the dwell time h of mode s, the symbol * denotes the transpose of the elements at the diagonal positions of the matrix, matrix I represents the dimension-matched identity matrix, and we have:

[0115]

[0116]

[0117]

[0118] prove:

[0119] definition For a if infinitesimal operator of V(ε(t),s), when Then, combining equations (4) and (8) above, we can obtain:

[0120]

[0121] in, From equation (10), it can be seen that, Right now That is, if Γ < 0, then we can obtain According to Lyapunov's stability theory, the closed-loop system (8) is mean square stable at this moment.

[0122] Introduce the following H-infinity and passive performance index functions:

[0123]

[0124] Where α > 0 represents the trade-off between infinite H and passive performance. γ > 0 represents the system's anti-interference capability.

[0125] when When defining a vector Then you can get

[0126]

[0127] Obviously, it can be seen from equation (10) that This ensures that J < 0, therefore we can deduce that:

[0128]

[0129] Integrating both sides of inequality (13), we can obtain the following under zero initial conditions:

[0130]

[0131] It can be seen that the system formula (8) satisfies the given infinite H and passive control performance.

[0132] In summary, it can be derived from Therefore, when Γ < 0. At that time, the closed-loop system is time-mean-square stable; At that time, the system meets the given performance indicators.

[0133] Step 4.2: Transform the above conditions containing coupling terms into the form of linear matrix inequalities, and derive the gain of the distributed controller equation (6).

[0134] Using the Schul complement lemma, Transform into the following form:

[0135]

[0136] in

[0137] Multiply both sides of equation (16) by the matrix. And its transpose, then using the Schul complement lemma, and letting as well as The following form can be obtained:

[0138]

[0139] in,

[0140]

[0141] Finally, the linear matrix inequality toolbox is used to solve the above inequality (17) to obtain X. s and Y s At this point, the control gain of the distributed controller...

[0142] Example:

[0143] (1) Establishment of longitudinal dynamic state-space model

[0144] The dynamic model of this invention is shown in equation (3). Assuming there are 5 vehicles in the vehicle queue, the disturbance of each vehicle is w(t) = 2sin(t), the sampling interval is 0.002s, the time delay constant τ = 0.5, and the simulation duration is 16s.

[0145] (2) Modeling of inter-vehicle communication topology

[0146] Assume that the communication topology between vehicles has three modes, such as Figure 2 As shown. L0 is the lead car, and F1 to F5 represent the 1st to 5th following cars respectively. L0→F1 means that the first following car can receive information from the lead car, that is... F1→F2 means that the second following vehicle can receive the information from the first following vehicle, i.e., a 21 =1. Figure 3 The communication topology randomly switches between three modes (θ(t) = 1, θ(t) = 2, and θ(t) = 3) according to a semi-Markov process. For example, at time 0, it jumps from mode 1 to mode 3 according to probability, and at the next time, it jumps from mode 3 to mode 2. The transition rate matrix of the semi-Markov process is:

[0147]

[0148] Assuming the dwell time of each mode follows a Weibull distribution, the probability density function for mode s is: When s = 1, let b = 2 and c = 2. When s = 2, let b = 1 and c = 3. When s = 3, let b = 1 and c = 2. Through calculation, the expected value of the transfer rate function is obtained as follows:

[0149]

[0150] (3) Design of other relevant parameters

[0151] Given scalars α = 0.01 and γ = 15, initialize each vehicle, where x1 = [30, 2, 0], x2 = [28, 2, 0], x3 = [26, 2, 0], x4 = [24, 2, 0], x5 = [22, 2, 0], and the desired vehicle spacing d = 3m.

[0152] The stability results are shown in Figures 4(a), 4(b), 4(c), and 4(d), which achieve the expected control objective. Figure 4(ad) represents the position, position error, velocity error, and acceleration error between vehicles, respectively. As can be seen from the figures, the method can guarantee the stability of the vehicle platoon.

Claims

1. Vehicle queuing under a semi-Markov switching topology The infinite and passive control method is characterized by, Includes the following steps: Step 1: Establish the longitudinal dynamic state-space model of a single vehicle, as follows: There are a total of The vehicles are traveling on a smooth road, led by vehicle number 0 and vehicle numbered... The vehicle consists of following vehicles; the longitudinal dynamics of each vehicle include the engine, braking system, and aerodynamic drag, and its longitudinal dynamics mathematical model is as follows: (1) in, and Representing vehicles Position and velocity, for The first derivative, For the mechanical efficiency of the transmission system. This is the radius of the vehicle's tires. The actual driving force of the vehicle. It is the aerodynamic coefficient. For the quality of the vehicle, It is the acceleration due to gravity. This is the tire rolling resistance coefficient. Let be the time delay constant of the vehicle's longitudinal system. Driven by expectations; By linearizing the feedback, the desired driving force is obtained: ,in, Indicates vehicle acceleration, It is the vehicle after feedback linearization The control input for each vehicle is as follows: The third-order dynamics model for each vehicle is: (2) definition , No. The longitudinal dynamic state-space model of the vehicle is represented as follows: (3) in For external interference to vehicles, , ; Step 2: Describe the information interaction between vehicles based on graph theory, and characterize the time-varying characteristics between vehicles using a semi-Markov process. The specific process is as follows: Step 2.1: In the queue The communication topology of the following vehicle is represented by a directed graph. Modeling is performed, among which Represents the vertex set and edge set. Indicates the connectivity between vehicles; defines the relationship between vehicles and vehicles. Corresponding adjacency matrix When the vehicle Able to receive vehicle Status information, , ,otherwise in-degree matrix ,in , Represents a diagonal block matrix; a directed graph. Laplace matrix The definition is as follows: Define traction matrix When the vehicle When information from the lead vehicle can be received, ,otherwise ; Step 2.2: Introduce a semi-Markov random process to describe the dynamic characteristics of the communication topology, where the vehicle queue... Communication topology at any time Description, in which , , , and These represent the adjacency matrix, Laplace matrix, traction matrix, and connectivity between vehicles under a semi-Markov random process, respectively. The switching of modes is caused by a random process. Control, and in the state space Take a value from; define Here is the transition matrix of a semi-Markov process, and its transition rates are as follows: (4) in, , , , Indicates in Momental mode arrive Momental mode The transfer rate; Step 3: Based on the characterization results, design the distributed controller and closed-loop system for the vehicle queue under the semi-Markov switching topology. The specific process is as follows: Step 3.1: The objective of vehicle queue control is: (5) in For the speed of the lead car, Indicates the first Car and the The expected distance between vehicles; For following vehicles The designed distributed controller takes the following form: (6) in, For the first The condition of the vehicle It is the control gain matrix to be designed; And there are , Indicates the first The distance between the vehicle and the lead vehicle; Step 3.2: For each following vehicle The tracking error is defined as: (7) in, , , The first The vehicle's position, speed, and acceleration errors; definition , Combining equations (2) and (6), the closed-loop system of the vehicle queue is obtained as follows: (8) in, The Kronecker product of the matrix, represent The identity matrix, , and These are the input and disturbance of the navigator vehicle, respectively; The controlled output is used to measure performance. ; Step 4: For the vehicle queue under the semi-Markov switching topology, solve for the control gain matrix of the distributed controller to obtain the vehicle controller and complete the control of the vehicle queue. The specific process is as follows: Step 4.1: Construct the following Lyapunov function: (9) in, Let be the positive definite matrix to be found; then the vehicle queuing closed-loop system of equation (8) is mean square stable and satisfies The conditions for infinite and passive properties are: Given scalar , If a positive definite symmetric matrix exists , and , so that: (10) Then the closed-loop system (8) is mean-square stable and satisfies Infinite and passive performance; Among them, the definition symbol , For any matrix, , Represents the modality Duration of stay The probability density distribution function, sign This represents the transpose of the elements at the symmetric positions along the diagonal of a matrix. Let represent the identity matrix for matching in any dimension, and we have: ， ， ； Step 4.2: Using the Schul complement lemma, Transform into the following form: (11) in ; Multiply both sides of equation (11) by the matrix. And its transpose, then using the Schul complement lemma, and letting as well as ,have to: (12) in, , ， ； Finally, the above inequality (12) is solved using the linear matrix inequality to obtain the result. and Then the control gain matrix of the distributed controller , .

2. The vehicle queuing under the semi-Markov switching topology according to claim 1 The infinite and passive control method is characterized by, In step 3.1, the desired distance It is a constant value.

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