CN113485125A - Time-lag-containing vehicle queue stability control method and system suitable for any communication topology - Google Patents

Time-lag-containing vehicle queue stability control method and system suitable for any communication topology Download PDF

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CN113485125A
CN113485125A CN202110921297.5A CN202110921297A CN113485125A CN 113485125 A CN113485125 A CN 113485125A CN 202110921297 A CN202110921297 A CN 202110921297A CN 113485125 A CN113485125 A CN 113485125A
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朱旭
张泽华
梁宇飞
闫茂德
杨盼盼
左磊
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Changan University
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Abstract

The invention discloses a method and a system for controlling the stability of a vehicle queue with time lag, which are suitable for any communication topology.A distributed controller with communication time lag is established based on the communication topology, and a vehicle queue closed-loop control system with time lag is decoupled into a plurality of subsystems by utilizing Jordan transformation of a matrix, so that the method and the system can be used for dealing with the condition that a Laplace matrix of the communication topology simultaneously contains real eigenvalues and conjugate complex eigenvalues and can process the problem of heavy eigenvalues; the method comprises the steps of solving a pure virtual root and a critical time lag of a subsystem characteristic equation, and calculating the root trend of the pure virtual root at the critical time lag, so that an accurate time lag boundary and a stable region of a subsystem can be obtained; the intersection of the stable domains of all the subsystems is taken by comprehensively considering, the stable domain and the accurate time lag boundary of the vehicle queue are finally obtained, the vehicle queue control system can be ensured to stably run in the calculated time lag boundary, the sharing interaction of workshop state information is flexibly controlled, a smaller following distance can be adopted, and the road capacity is increased.

Description

Time-lag-containing vehicle queue stability control method and system suitable for any communication topology
Technical Field
The invention belongs to the field of vehicle queue control, and particularly relates to a method and a system for controlling the stability of a vehicle queue with time lag, which are suitable for any communication topology.
Background
In recent years, with the increase of the holding capacity of motor vehicles, people can travel conveniently, and meanwhile, serious problems of frequent traffic jam, aggravation of energy consumption, air quality reduction and the like are brought. Research shows that the intelligent vehicle realizes the sharing interaction of workshop state information through various wireless communication modes such as vehicle-vehicle communication, vehicle-infrastructure communication and the like, so that the vehicle runs in a queue mode, the driving safety can be improved, the traffic capacity can be increased, and the energy consumption can be reduced.
However, due to objective factors such as communication bandwidth and channel blockage, communication time lag inevitably exists in the communication of the vehicle queue. The existence of the communication time lag causes the rate of change of the state (position, speed, and acceleration state quantity) of the vehicle queue to be related not only to the current time state but also to the past time state. The communication time lag not only has adverse effects on the steady-state and transient-state control performance of the vehicle queue, but also reduces the stability of the system and influences the driving safety. Therefore, how to ensure that the vehicle queue can stably run under the influence of communication time lag is an urgent problem to be solved in the vehicle queue control technology.
Currently, there are still many drawbacks in vehicle queue control system research related to communication time lag. Most of the existing researches are only suitable for several types of special communication topologies, such as a front-car following communication topology, a front-car-navigator following communication topology, a double-front-car-navigator following communication topology, and the like. The special communication topologies have a common characteristic that eigenvalues of Laplace matrixes are real numbers and have no complex numbers, so that stability control and analysis are relatively simple for vehicle queues under the communication topologies. However, vehicle-to-vehicle communication, vehicle-to-infrastructure communication, and the like are flexible, and any form of communication topology can be constructed. For any type of communication topology, the laplacian matrix may contain conjugate complex eigenvalues, or even both real eigenvalues and conjugate complex eigenvalues. More difficult, these real and complex eigenvalues may also be heavy roots. These complications regarding the eigenvalues of the laplace matrix, interleaved with the communication time lag, present serious design and analytical difficulties for vehicle fleet stability control.
Disclosure of Invention
The invention aims to provide a method and a system for controlling the queue stability of a vehicle with time lag, which are suitable for any communication topology and overcome the defects of the prior art.
In order to achieve the purpose, the invention adopts the following technical scheme:
a vehicle queue stability control method with time lag suitable for any communication topology comprises the following steps:
s1, constructing a vehicle longitudinal dynamic model, a fixed distance strategy and a communication topology of the vehicle queue;
s2, establishing a distributed controller with communication time lag based on communication topology, establishing a vehicle queue closed-loop control system with communication time lag, and decoupling the vehicle queue closed-loop control system with time lag into a plurality of subsystems by using Jordan transformation of a matrix;
s3, establishing an auxiliary characteristic equation of the subsystem through Kronecker sum operation and z substitution, solving the auxiliary characteristic equation to obtain a solution set with a root module value equal to 1, then solving a pure virtual root set of the subsystem characteristic equation, determining asymptotic behavior of the pure virtual root at a critical time lag position through calculating root trend, obtaining a stable region and an accurate time lag boundary of the subsystem, respectively obtaining the stable region and the accurate time lag boundary of each subsystem, obtaining intersection of the stable regions of all the subsystems, obtaining the stable region and the accurate time lag boundary of the vehicle queue closed-loop control system, and obtaining sufficient and necessary conditions for vehicle queue stability.
Further, the vehicle longitudinal dynamics model is as follows:
Figure BDA0003207507580000031
Figure BDA0003207507580000032
wherein i represents the number of the vehicle, the number 0 represents the pilot vehicle, the number of the following vehicles is 1 to N in sequence from front to back, and pi(t)、vi(t)、ai(t) represents the position, speed, and acceleration state quantity of the ith vehicle at time t, xi(T) represents the state quantity of the i-th vehicle at time T, A, B represents the state matrix and input matrix of the i-th vehicle, T represents the inertia constant of the driving mechanism of the vehicle, u represents the inertia constant of the driving mechanism of the vehicle, and u represents the state quantity of the i-th vehicle at time Ti(t) represents a control input of the ith vehicle.
Further, the fixed-distance strategy is used for enabling a control target of the vehicle queue to follow the state of the pilot vehicle and enabling the vehicle queue to run at a desired fixed distance, namely:
Figure BDA0003207507580000033
wherein, | | | | represents taking two norms, v0(t) speed of the pilot vehicle at time t,/i,i-1For the desired spacing from the i-1 st vehicle to the i-th vehicle, there is li,i-1＝-li-1,iSetting l by adopting a fixed-pitch strategyi,i-1Is a fixed constant: li,i-1＝l0，l0Is a constant.
Further, the distributed controller with communication time lag:
Figure BDA0003207507580000034
where, τ is the communication time lag,
Figure BDA0003207507580000035
the position state error, the speed state error and the acceleration state error of the ith vehicle relative to the pilot vehicle at the moment t-tau are respectively; k is a radical ofp、kv、kaRespectively obtaining the position, the speed and the acceleration gain of a controller deployed on the ith vehicle; r isijIndicating whether the ith vehicle can receive the state information of the jth vehicle, if so, rij1, otherwise rij＝0。
Further, a distributed controller with respect to the state error is established based on the distributed controller with communication skew:
Figure BDA0003207507580000041
wherein K ═ Kp,kv,ka]For the control gain vector for each vehicle,
Figure BDA0003207507580000042
for the state error of the ith following vehicle relative to the pilot vehicle at time t,
Figure BDA0003207507580000043
as position state error,/i,0Is the desired distance from the pilot vehicle to the ith vehicle;
Figure BDA0003207507580000044
is a speed state error;
Figure BDA0003207507580000045
is the acceleration state error.
Further, according to the longitudinal dynamics of the vehicle and the distributed controller related to the state error, obtaining a closed-loop dynamics equation of the ith vehicle:
Figure BDA0003207507580000046
further, the state space expression of the closed-loop control system of the vehicle queue with the communication time lag is as follows:
Figure BDA0003207507580000047
wherein, INIs a unit matrix of N-order,
Figure BDA0003207507580000048
representing the Kronecker product, matrix
Figure BDA0003207507580000049
For describing the communication topology of a vehicle fleet,
Figure BDA00032075075800000410
is a Laplace matrix, is used for describing the communication relationship among all the following vehicles,
Figure BDA00032075075800000411
and X (t-tau) represents the lumped state error of the vehicle queue at the moment of t-tau.
Further, the characteristic equation of the vehicle queue closed-loop control system is as follows:
Figure BDA00032075075800000412
where det (-) denotes determinant operation, I3NIs a 3N order unit matrix, and s is a Laplace variable.
Further, Jordan transformation is carried out on the matrix Y, the vehicle queue closed-loop control system with communication time lag is equivalently decomposed into a plurality of low-order subsystems, Kronecker sum operation and z substitution are utilized, and the auxiliary characteristic equation form of the subsystems is established as follows:
Figure BDA0003207507580000051
wherein the matrix Y has q different real eigenvalues and w different complex conjugate eigenvalues,
Figure BDA0003207507580000052
Figure BDA0003207507580000053
z＝e-τsand Z is a complex number, solving a solution set Z of the ancillary feature equationsn＝{z|gn(z) ═ 0, | z | ═ 1 }; solving pure virtual root omega of subsystem characteristic equation, the set of pure virtual root omega uses omeganRepresents; the critical time lag set of the subsystem is:
Figure BDA0003207507580000054
wherein the content of the first and second substances,
Figure BDA0003207507580000055
it is shown that the operation of taking the angle,
Figure BDA0003207507580000056
represents a positive number in the real domain;
determining the asymptotic behavior of the pure virtual root of the subsystem characteristic equation at the critical time lag, and solving the pure virtual root omega of the subsystem, which belongs to omeganWith respect to critical time lag
Figure BDA0003207507580000057
The change trend of (2) is called root trend, and the expression is as follows:
Figure BDA0003207507580000058
where Re represents the real part, sgn represents the sign function,
Figure BDA0003207507580000059
represents the pair s about
Figure BDA00032075075800000510
Calculating partial derivative, wherein i is an imaginary number unit; according to whatAnd the pure virtual root and the critical time lag of the subsystem can be obtained by the auxiliary characteristic equation of the subsystem and the characteristic equation of the subsystem.
A vehicle queue stability control system with time lag suitable for any communication topology comprises a vehicle control model and a vehicle queue stability control model;
the method is characterized by comprising the steps of storing a vehicle longitudinal dynamics model, a fixed spacing strategy and a communication topology, establishing a distributed controller containing communication time lag based on the communication topology, establishing a vehicle queue closed-loop control system containing the communication time lag, and decoupling the vehicle queue closed-loop control system containing the time lag into a plurality of subsystems by using Jordan transformation of a matrix;
the vehicle queue stability control model is used for establishing an auxiliary characteristic equation of a subsystem through Kronecker sum operation and z substitution, solving the auxiliary characteristic equation to obtain a solution set of which the modulus value of a root is equal to 1, then solving a pure virtual root set of the subsystem characteristic equation, determining the asymptotic behavior of the pure virtual root at a critical time lag position through calculating the root trend, obtaining a stable domain and an accurate time lag boundary of the subsystem, respectively solving the stable domain and the accurate time lag boundary of each subsystem, taking an intersection of the stable domains of all the subsystems, and taking and outputting the stable domain and the accurate time lag boundary of the vehicle queue closed-loop control system.
Compared with the prior art, the invention has the following beneficial technical effects:
the invention relates to a control method of vehicle queue stability with time lag, which is suitable for any communication topology.A distributed controller with communication time lag is established based on the communication topology, a vehicle queue closed-loop control system with communication time lag is established, and the vehicle queue closed-loop control system with time lag is decoupled into a plurality of subsystems by using Jordan transformation of a matrix, so that the condition that a Laplace matrix of the communication topology simultaneously contains real eigenvalue and conjugate complex eigenvalue can be dealt with, and the problem of heavy eigenvalue can be dealt with; constructing an auxiliary characteristic equation of the subsystem, solving a pure virtual root and a critical time lag of the characteristic equation of the subsystem, and calculating a root trend of the pure virtual root at the critical time lag to obtain an accurate time lag boundary and a stable region of the subsystem; and comprehensively considering all the subsystems, and taking the intersection of the stable domains of the subsystems to finally obtain the stable domain and the accurate time lag boundary of the vehicle queue. The method can ensure that the vehicle queue control system stably runs in the calculated time lag boundary, flexibly control the sharing interaction of the workshop state information, and can adopt a smaller following distance to increase the road capacity.
The method for controlling the stability of the vehicle queue with the time lag is suitable for any communication topology, and can meet the flexible adjustment requirement of a workshop state information sharing interaction mode. By means of the train-type running of the vehicles, a smaller vehicle following distance can be adopted on the premise of ensuring safety, and therefore the road capacity is increased. Meanwhile, the stability control problem of the vehicle queue closed-loop control system is converted into the stability control problem of all subsystems, so that the calculated amount of the vehicle queue stability control is greatly reduced. By establishing an auxiliary characteristic equation of the subsystem, an infinite characteristic root solution problem is converted into a finite pure virtual root solution problem, and the difficulty of stability control and analysis of the fleet is greatly reduced. The invention can calculate the accurate time lag boundary of the vehicle queue, can ensure that the vehicle queue stably runs in the time lag boundary, and the obtained stability condition is a sufficient necessary condition.
Furthermore, the designed distributed controller enables the following vehicle to automatically track the state of the pilot vehicle, so that the purpose of automatic longitudinal following of the vehicle is achieved, the traffic capacity is increased, and the driving safety is improved.
Drawings
FIG. 1 is a block flow diagram of a vehicle fleet stability control method in accordance with an embodiment of the present invention.
Detailed Description
The invention is described in further detail below with reference to the accompanying drawings:
as shown in fig. 1, the method for controlling the queue stability of a vehicle with time lag applicable to any communication topology provided by this embodiment mainly includes the following implementation steps:
step 1: the vehicle queue consists of 1 pilot vehicle and N follow-up vehicles, wherein N is a natural number greater than 1, and a mathematical model of the vehicle queue control system is established as follows: the mathematical model comprises a vehicle longitudinal dynamic model, a fixed distance strategy, a communication topology and a distributed controller. The vehicle longitudinal dynamics model describes longitudinal driving characteristics of the intelligent vehicle. The fixed pitch strategy is used to determine the longitudinal distance between vehicles. The communication topology is used for state information interaction between vehicles. And the distributed controller calculates the expected acceleration of the self vehicle through state feedback by utilizing the state information of the other vehicles through the communication topology.
Step 2: establishing a vehicle longitudinal dynamic model in the step 1:
i represents the number of the vehicle, 0 represents the pilot vehicle, and the numbers of the following vehicles are 1 to N, p from front to backi(t)、vi(t)、ai(t) the position, speed and acceleration of the ith vehicle at time t are respectively shown, and the state of the vehicle is xi(t)＝[pi(t),vi(t),ai(t)]TThen, the third order linear model of the vehicle longitudinal dynamics is described as follows:
Figure BDA0003207507580000081
wherein u isi(t) represents the control input of the ith vehicle, and the state matrix A and the input matrix B are respectively as follows:
Figure BDA0003207507580000082
where T is an inertia constant of a drive mechanism of the vehicle.
And step 3: determining the fixed pitch strategy in step 1: the control objective of the vehicle train is to let the following vehicle track the state of the lead vehicle and to let the vehicle train run at a desired fixed distance, i.e.
Figure BDA0003207507580000083
Wherein, | | | | represents taking two norms, v0(t) speed of the pilot vehicle at time t,/i,i-1For the desired spacing from the i-1 st vehicle to the i-th vehicle, there is li,i-1＝-li-1,i. Setting l by adopting a fixed-pitch strategyi,i-1Is a fixed constant: li,i-1＝l0，l0Is a constant.
And 4, step 4: constructing the communication topology in the step 1: treating vehicles as nodes with directed graphs
Figure BDA0003207507580000084
A communication topology representing a vehicle queue, wherein
Figure BDA0003207507580000085
Is a set of nodes representing all following vehicles, viWhere i is 1,2, …, N represents the node represented by the ith vehicle,
Figure BDA0003207507580000086
representing a set of edges in the communication topology, each edge representing a jth vehicle communicating to an ith vehicle. Adjacency matrix is available for information transfer relationships between all following vehicles
Figure BDA0003207507580000087
It is shown that, among others,
Figure BDA0003207507580000088
representing the real number field, rijIndicating whether the ith vehicle can receive the status information of the jth vehicle. If (upsilon)i,υj) E is epsilon, then rij1 is ═ 1; if it is
Figure BDA00032075075800000813
Then r isij0; furthermore, the vehicle cannot communicate by itself, i.e. rij0. The ith node viIn degree of
Figure BDA0003207507580000089
The in-degree moment consisting of the in-degrees of all nodesArray is
Figure BDA00032075075800000810
And the drawings
Figure BDA00032075075800000811
The associated Laplace matrix is
Figure BDA00032075075800000812
In addition, a communication relation available traction matrix between the following vehicle and the pilot vehicle
Figure BDA0003207507580000091
A description is given. If the ith following vehicle can acquire the state information of the pilot vehicle, pi1 is ═ 1; otherwise, pi＝0。
And 5: designing the distributed controller in the step 1. The embodiment considers that the actual communication is influenced by physical conditions such as bandwidth limitation, channel blockage and the like, and time lag exists in information transmission. In order to realize the control target in the step 3, the feedback control is carried out on the vehicle state containing the communication time lag, and a distributed controller containing the communication time lag is established:
Figure BDA0003207507580000092
wherein k isp、kv、kaRespectively controlling the position, the speed and the acceleration gain of each vehicle distributed controller, wherein tau represents communication time lag, and pi(t-τ)、vi(t-τ)、ai(t- τ) represents the position, velocity, and acceleration states of the ith vehicle at time t- τ, respectively.
Step 6: and according to the distributed controller with communication time lag in the step 5, giving a distributed controller about state errors. Definition of
Figure BDA0003207507580000093
The state error of the ith following vehicle relative to the pilot vehicle at the moment t, wherein,
Figure BDA0003207507580000094
as position state error,/i,0Is the desired distance from the pilot vehicle to the ith vehicle;
Figure BDA0003207507580000095
is a speed state error;
Figure BDA0003207507580000096
for an acceleration state error, the distributed controller for the state error is:
Figure BDA0003207507580000097
the above equation can be written in a compact form:
Figure BDA0003207507580000098
wherein K ═ Kp,kv,ka]Is the control gain vector for each vehicle.
And 7: obtaining a closed-loop kinetic equation of the ith vehicle according to the third-order linear model of the longitudinal kinetic of the vehicle in the step 2 and the distributed controller in the step 6:
Figure BDA0003207507580000101
and 8: defining a lumped state error vector for the entire vehicle fleet
Figure BDA0003207507580000102
And collective control input
Figure BDA0003207507580000103
Then a state space expression of the closed-loop control system of the vehicle queue with communication time lag can be deduced:
Figure BDA0003207507580000104
wherein, INIs a unit matrix of N-order,
Figure BDA0003207507580000105
representing the Kronecker product, matrix
Figure BDA0003207507580000106
For describing the communication topology of the vehicle queue, X (t-tau) represents the lumped state error of the vehicle queue at the time of t-tau.
The characteristic equation of the vehicle queue closed-loop control system is as follows:
Figure BDA0003207507580000107
where det (-) denotes determinant operation, I3NIs a 3N order unit matrix, and s is a Laplace variable.
For any type of communication topology, the characteristic values of the laplacian matrix may have both real characteristic values and conjugate complex characteristic values, and both the real characteristic values and the complex characteristic values may have multiple roots. The vehicle queue control system constructed in the embodiment adopts the following steps to carry out control design and analysis in order to be applicable to any type of communication topology.
And step 9: according to the communication topology in step 4, it can be known that the matrix Y is an N-order square matrix. Let λ benN is 1,2, …, q is the real number eigenvalue of Y, and λnHas a weight of Δn，n＝1,2,…,q；λn＝αn+iβn，
Figure BDA0003207507580000108
Is a characteristic value of the conjugate complex number of Y, and λnThe same is as deltanWhere n is q +1, q +2, …, and q + w, i is an imaginary unit. The number of real eigenvalues and the number of conjugate complex eigenvalues satisfy the relationship:
Figure BDA0003207507580000109
there is a non-singular matrix
Figure BDA00032075075800001010
So that
Figure BDA0003207507580000111
J is a Jordan matrix having the form:
Figure BDA0003207507580000112
wherein, JnN-1, 2, …, q is a Jordan block corresponding to the real eigenvalues of matrix Y, of the form:
Figure BDA0003207507580000113
Jnn is q +1, q +2, …, q + w being a Jordan block corresponding to the complex eigenvalues of the matrix Y, having the form:
Figure BDA0003207507580000114
step 10: converting the matrix Y into a Jordan matrix according to the step 9, and equivalently decomposing the vehicle queue closed-loop control system in the step 8 into a plurality of low-order subsystems.
Making a linear transformation
Figure BDA0003207507580000115
ξ (t) is X (t) the state vector after linear transformation,
Figure BDA0003207507580000116
for transforming the matrix, one obtains
Figure BDA0003207507580000117
Through the process, the vehicle queue closed-loop control system suitable for any communication topology can be equivalently decomposed into q 3-order subsystems and w 6-order subsystems
Figure BDA0003207507580000118
The characteristic equation of the subsystem is as follows:
Figure BDA0003207507580000121
the relationship between the vehicle queue closed-loop control system characteristic equation and the subsystem characteristic equation is as follows:
Figure BDA0003207507580000122
each subsystem must stabilize if the entire vehicle fleet control system stabilizes. Because the subsystem characteristic equation contains a time-lag term e-τsResulting in an infinite number of feature roots for the feature equation. To solve this problem, the present embodiment uses Kronecker sum operation and z substitution (z ═ e)-τsAnd z is a complex number), establishing an auxiliary characteristic equation of the subsystem, and solving the auxiliary characteristic equation to obtain a pure virtual root of the characteristic equation. By the method, an infinite characteristic root solving problem is converted into a finite pure virtual root solving problem, and the difficulty of fleet stability control and analysis is greatly reduced.
Step 11: establishing an auxiliary characteristic equation of the fleet subsystem in the step 10:
Figure BDA0003207507580000123
wherein the content of the first and second substances,
Figure BDA0003207507580000124
Figure BDA0003207507580000125
solution set Z for solving auxiliary characteristic equationn＝{z|gn(z) ═ 0, | z | ═ 1 }. Then, a pure virtual root set of the subsystem characteristic equation can be solved, using ΩnRepresents:
Figure BDA0003207507580000126
wherein
Figure BDA0003207507580000127
Representing a positive number in the real domain, and ω is the pure imaginary root of the subsystem characteristic equation.
The critical skew set of the subsystem is then:
Figure BDA0003207507580000128
wherein the content of the first and second substances,
Figure BDA0003207507580000129
representing an angle-taking operation.
Step 12: determining the asymptotic behavior of the pure virtual root of the subsystem characteristic equation at the critical time lag, namely determining the crossing behavior of the pure virtual root at the virtual axis of the complex plane, wherein the pure virtual root omega of the subsystem needs to be solved to be belonged to omeganWith respect to critical time lag
Figure BDA0003207507580000131
The change trend of (2) is called root trend, and the expression is as follows:
Figure BDA0003207507580000132
where Re represents the real part, sgn represents the sign function,
Figure BDA0003207507580000133
represents to s press
Figure BDA0003207507580000134
And (5) calculating a partial derivative.
Step 13: and obtaining a pure virtual root and a critical time lag of the subsystem according to the auxiliary characteristic equation of the subsystem in the step 11 and the characteristic equation of the subsystem in the step 10. The solving sequence is as follows:
Figure BDA0003207507580000135
and then, judging the asymptotic behavior of the pure virtual root at the critical time lag according to the root trend in the step 12, and counting the number of unstable roots in the region partitioned by the critical time lag, wherein the region with the number of the unstable roots being zero is the stable region of the subsystem, and the time lag at the boundary of the stable region is the accurate time lag boundary of the subsystem.
Step 14: and (3) respectively adopting the stability control process from the step (11) to the step (13) for the state space expressions of all the subsystems in the step (10) to obtain a stable domain and an accurate time lag boundary of each subsystem. And then, taking intersection of the stable domains of all the subsystems, and finally obtaining the stable domain and the accurate time lag boundary of the whole vehicle queue, so that the vehicle queue can be ensured to stably run in the calculated time lag boundary.
And respectively applying the stability control process to each subsystem to a time-lag vehicle queue closed-loop control system suitable for any communication topology to obtain a stable domain and an accurate time-lag boundary of each subsystem. And (3) taking intersection of the stable domains of all the subsystems to finally obtain the stable domain and the accurate time lag boundary of the whole vehicle queue, and ensuring that the vehicle queue stably runs in the calculated time lag boundary.
The invention enables the following vehicle to automatically track the state of the pilot vehicle through the designed distributed controller, thereby achieving the purpose of automatic longitudinal following of the vehicle, increasing the traffic capacity and improving the driving safety. The designed stability control method for the vehicle queue with the time lag is suitable for any communication topology, and can meet the flexible adjustment requirement on the workshop state information sharing interaction mode. By means of the train-type running of the vehicles, a smaller vehicle following distance can be adopted on the premise of ensuring safety, and therefore the road capacity is increased. Meanwhile, the stability control problem of the vehicle queue closed-loop control system is converted into the stability control problem of all subsystems, so that the calculated amount of the vehicle queue stability control is greatly reduced. By establishing an auxiliary characteristic equation of the subsystem, an infinite characteristic root solution problem is converted into a finite pure virtual root solution problem, and the difficulty of stability control and analysis of the fleet is greatly reduced. The invention can calculate the accurate time lag boundary of the vehicle queue, can ensure that the vehicle queue stably runs in the time lag boundary, and the obtained stability condition is a sufficient necessary condition.

Claims (10)

1. A vehicle queue stability control method with time lag suitable for any communication topology is characterized by comprising the following steps:
s1, constructing a vehicle longitudinal dynamic model, a fixed distance strategy and a communication topology of the vehicle queue;
s2, establishing a distributed controller containing communication time lag based on a vehicle longitudinal dynamics model, a fixed spacing strategy and a communication topology, establishing a vehicle queue closed-loop control system containing communication time lag, and decoupling the vehicle queue closed-loop control system containing time lag into a plurality of subsystems by using Jordan transformation of a matrix;
s3, establishing an auxiliary characteristic equation of the subsystem through Kronecker sum operation and z substitution, solving the auxiliary characteristic equation to obtain a solution set with a root module value equal to 1, then solving a pure virtual root set of the subsystem characteristic equation, determining asymptotic behavior of the pure virtual root at a critical time lag position through calculating root trend, obtaining a stable region and an accurate time lag boundary of the subsystem, respectively obtaining the stable region and the accurate time lag boundary of each subsystem, obtaining intersection of the stable regions of all the subsystems, obtaining the stable region and the accurate time lag boundary of the vehicle queue closed-loop control system, and obtaining sufficient and necessary conditions for vehicle queue stability.
2. The method for controlling the queue stability of the vehicle with the time lag suitable for any communication topology, according to claim 1, is characterized in that a vehicle longitudinal dynamic model is as follows:
Figure FDA0003207507570000011
Figure FDA0003207507570000012
wherein i represents the number of the vehicle, the number 0 represents the pilot vehicle, the number of the following vehicles is 1 to N in sequence from front to back, and pi(t)、vi(t)、ai(t) represents the position, speed, and acceleration state quantity of the ith vehicle at time t, xi(T) represents the state quantity of the i-th vehicle at time T, A, B represents the state matrix and input matrix of the i-th vehicle, T represents the inertia constant of the driving mechanism of the vehicle, u represents the inertia constant of the driving mechanism of the vehicle, and u represents the state quantity of the i-th vehicle at time Ti(t) represents a control input of the ith vehicle.
3. The method for controlling the stability of the vehicle queue with time lag suitable for any communication topology, according to claim 1, is characterized in that a fixed-distance strategy is used for enabling a control target of the vehicle queue to follow a state of a pilot vehicle and enabling the vehicle queue to run at a desired fixed distance, namely:
Figure FDA0003207507570000021
wherein, | | | | represents taking two norms, v0(t) speed of the pilot vehicle at time t,/i,i-1For the desired spacing from the i-1 st vehicle to the i-th vehicle, there is li,i-1＝-li-1,iSetting l by adopting a fixed-pitch strategyi,i-1Is a fixed constant: li,i-1＝l0，l0Is a constant.
4. The method for controlling the queue stability of the vehicle with the time lag suitable for any communication topology, according to claim 3, is characterized in that a distributed controller with the communication time lag is used for:
Figure FDA0003207507570000022
where, τ is the communication time lag,
Figure FDA0003207507570000023
the position state error, the speed state error and the acceleration state error of the ith vehicle relative to the pilot vehicle at the moment t-tau are respectively; k is a radical ofp、kv、kaRespectively obtaining the position, the speed and the acceleration gain of a controller deployed on the ith vehicle; r isijIndicating whether the ith vehicle can receive the state information of the jth vehicle through the communication topology, if so, rij1, otherwise rij＝0。
5. The method for controlling the queue stability of a vehicle with time lag applied to any communication topology according to claim 4, is characterized in that the distributed controllers with the state error are established according to the distributed controllers with the communication time lag:
Figure FDA0003207507570000024
wherein K ═ Kp,kv,ka]For the control gain vector for each vehicle,
Figure FDA0003207507570000025
for the state error of the ith following vehicle relative to the pilot vehicle at time t,
Figure FDA0003207507570000026
as position state error,/i,0To a desired distance from the pilot vehicle to the i-th vehicle；
Figure FDA0003207507570000031
Is a speed state error;
Figure FDA0003207507570000032
is the acceleration state error.
6. The lag-containing vehicle queue stability control method applicable to any communication topology according to claim 4, wherein a closed-loop kinetic equation of an ith vehicle is obtained according to vehicle longitudinal dynamics and a distributed controller about state errors:
Figure FDA0003207507570000033
7. the method for controlling the stability of the vehicle queue with the time lag, applicable to any communication topology, according to claim 4, is characterized in that a state space expression of the closed-loop control system of the vehicle queue with the time lag is as follows:
Figure FDA0003207507570000034
wherein, INIs a unit matrix of N-order,
Figure FDA0003207507570000035
representing the Kronecker product, matrix
Figure FDA0003207507570000036
For describing the communication topology of a vehicle fleet,
Figure FDA0003207507570000037
is a Laplace matrix, is used for describing the communication relationship among all the following vehicles,
Figure FDA0003207507570000038
and X (t-tau) represents the lumped state error of the vehicle queue at the moment of t-tau.
8. The method for controlling the stability of the vehicle queue with the time lag suitable for any communication topology according to claim 7, wherein a characteristic equation of a vehicle queue closed-loop control system is as follows:
Figure FDA0003207507570000039
where det (-) denotes determinant operation, I3NIs a 3N order unit matrix, and s is a Laplace variable.
9. The method for controlling the stability of the vehicle queue with the time lag suitable for any communication topology, according to the claim 6, is characterized in that a Jordan transformation is carried out on a matrix Y, a vehicle queue closed-loop control system with the communication time lag is equivalently decomposed into a plurality of low-order subsystems, and the auxiliary characteristic equation form of the subsystems is established by using Kronecker sum operation and z substitution, wherein the auxiliary characteristic equation form comprises the following steps:
Figure FDA0003207507570000041
wherein the matrix Y has q different real eigenvalues and w different complex conjugate eigenvalues,
Figure FDA0003207507570000042
Figure FDA0003207507570000043
z＝e-τsand Z is a complex number, solving a solution set Z of the ancillary feature equationsn＝{z|gn(z) ═ 0, | z | ═ 1 }; solving pure virtual root omega of subsystem characteristic equation, the set of pure virtual root omega uses omeganRepresents; the critical time lag set of the subsystem is:
Figure FDA0003207507570000044
wherein the content of the first and second substances,
Figure FDA0003207507570000045
it is shown that the operation of taking the angle,
Figure FDA0003207507570000046
represents a positive number in the real domain;
determining the asymptotic behavior of the pure virtual root of the subsystem characteristic equation at the critical time lag, and solving the pure virtual root omega of the subsystem, which belongs to omeganWith respect to critical time lag
Figure FDA0003207507570000047
The change trend of (2) is called root trend, and the expression is as follows:
Figure FDA0003207507570000048
where Re represents the real part, sgn represents the sign function,
Figure FDA0003207507570000049
represents the pair s about
Figure FDA00032075075700000410
Calculating partial derivative, wherein i is an imaginary number unit; and obtaining the pure virtual root and the critical time lag of the subsystem according to the auxiliary characteristic equation of the subsystem and the characteristic equation of the subsystem.
10. A vehicle queue stability control system with time lag suitable for any communication topology is characterized by comprising a vehicle control model and a vehicle queue stability control model;
the method is characterized by comprising the steps of storing a vehicle longitudinal dynamics model, a fixed spacing strategy and a communication topology, establishing a distributed controller containing communication time lag based on the communication topology, establishing a vehicle queue closed-loop control system containing the communication time lag, and decoupling the vehicle queue closed-loop control system containing the time lag into a plurality of subsystems by using Jordan transformation of a matrix;
the vehicle queue stability control model is used for establishing an auxiliary characteristic equation of a subsystem through Kronecker sum operation and z substitution, solving the auxiliary characteristic equation to obtain a solution set of which the modulus value of a root is equal to 1, then solving a pure virtual root set of the subsystem characteristic equation, determining the asymptotic behavior of the pure virtual root at a critical time lag position through calculating the root trend, obtaining a stable domain and an accurate time lag boundary of the subsystem, respectively solving the stable domain and the accurate time lag boundary of each subsystem, taking an intersection of the stable domains of all the subsystems, and taking and outputting the stable domain and the accurate time lag boundary of the vehicle queue closed-loop control system.
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